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https://www.colorhexa.com/40ffcc | [
"# #40ffcc Color Information\n\nIn a RGB color space, hex #40ffcc is composed of 25.1% red, 100% green and 80% blue. Whereas in a CMYK color space, it is composed of 74.9% cyan, 0% magenta, 20% yellow and 0% black. It has a hue angle of 164 degrees, a saturation of 100% and a lightness of 62.5%. #40ffcc color hex could be obtained by blending #80ffff with #00ff99. Closest websafe color is: #33ffcc.\n\n• R 25\n• G 100\n• B 80\nRGB color chart\n• C 75\n• M 0\n• Y 20\n• K 0\nCMYK color chart\n\n#40ffcc color description : Light cyan - lime green.\n\n# #40ffcc Color Conversion\n\nThe hexadecimal color #40ffcc has RGB values of R:64, G:255, B:204 and CMYK values of C:0.75, M:0, Y:0.2, K:0. Its decimal value is 4259788.\n\nHex triplet RGB Decimal 40ffcc `#40ffcc` 64, 255, 204 `rgb(64,255,204)` 25.1, 100, 80 `rgb(25.1%,100%,80%)` 75, 0, 20, 0 164°, 100, 62.5 `hsl(164,100%,62.5%)` 164°, 74.9, 100 33ffcc `#33ffcc`\nCIE-LAB 90.305, -57.925, 11.159 48.769, 76.965, 69.409 0.25, 0.394, 76.965 90.305, 58.99, 169.096 90.305, -70.005, 26.318 87.73, -54.298, 14.502 01000000, 11111111, 11001100\n\n# Color Schemes with #40ffcc\n\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #ff4073\n``#ff4073` `rgb(255,64,115)``\nComplementary Color\n• #40ff6d\n``#40ff6d` `rgb(64,255,109)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #40d3ff\n``#40d3ff` `rgb(64,211,255)``\nAnalogous Color\n• #ff6d40\n``#ff6d40` `rgb(255,109,64)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #ff40d3\n``#ff40d3` `rgb(255,64,211)``\nSplit Complementary Color\n• #ffcc40\n``#ffcc40` `rgb(255,204,64)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #cc40ff\n``#cc40ff` `rgb(204,64,255)``\n• #73ff40\n``#73ff40` `rgb(115,255,64)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #cc40ff\n``#cc40ff` `rgb(204,64,255)``\n• #ff4073\n``#ff4073` `rgb(255,64,115)``\n• #00f3b2\n``#00f3b2` `rgb(0,243,178)``\n• #0dffbe\n``#0dffbe` `rgb(13,255,190)``\n• #27ffc5\n``#27ffc5` `rgb(39,255,197)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #5affd3\n``#5affd3` `rgb(90,255,211)``\n• #73ffda\n``#73ffda` `rgb(115,255,218)``\n• #8dffe0\n``#8dffe0` `rgb(141,255,224)``\nMonochromatic Color\n\n# Alternatives to #40ffcc\n\nBelow, you can see some colors close to #40ffcc. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #40ff9c\n``#40ff9c` `rgb(64,255,156)``\n• #40ffac\n``#40ffac` `rgb(64,255,172)``\n• #40ffbc\n``#40ffbc` `rgb(64,255,188)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #40ffdc\n``#40ffdc` `rgb(64,255,220)``\n• #40ffec\n``#40ffec` `rgb(64,255,236)``\n• #40fffc\n``#40fffc` `rgb(64,255,252)``\nSimilar Colors\n\n# #40ffcc Preview\n\nThis text has a font color of #40ffcc.\n\n``<span style=\"color:#40ffcc;\">Text here</span>``\n#40ffcc background color\n\nThis paragraph has a background color of #40ffcc.\n\n``<p style=\"background-color:#40ffcc;\">Content here</p>``\n#40ffcc border color\n\nThis element has a border color of #40ffcc.\n\n``<div style=\"border:1px solid #40ffcc;\">Content here</div>``\nCSS codes\n``.text {color:#40ffcc;}``\n``.background {background-color:#40ffcc;}``\n``.border {border:1px solid #40ffcc;}``\n\n# Shades and Tints of #40ffcc\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, #000504 is the darkest color, while #f1fffb is the lightest one.\n\n• #000504\n``#000504` `rgb(0,5,4)``\n• #001912\n``#001912` `rgb(0,25,18)``\n• #002c21\n``#002c21` `rgb(0,44,33)``\n• #00402f\n``#00402f` `rgb(0,64,47)``\n• #00543d\n``#00543d` `rgb(0,84,61)``\n• #00674c\n``#00674c` `rgb(0,103,76)``\n• #007b5a\n``#007b5a` `rgb(0,123,90)``\n• #008e68\n``#008e68` `rgb(0,142,104)``\n• #00a277\n``#00a277` `rgb(0,162,119)``\n• #00b685\n``#00b685` `rgb(0,182,133)``\n• #00c994\n``#00c994` `rgb(0,201,148)``\n• #00dda2\n``#00dda2` `rgb(0,221,162)``\n• #00f1b0\n``#00f1b0` `rgb(0,241,176)``\n• #05ffbc\n``#05ffbc` `rgb(5,255,188)``\n• #19ffc2\n``#19ffc2` `rgb(25,255,194)``\n• #2cffc7\n``#2cffc7` `rgb(44,255,199)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\n• #54ffd1\n``#54ffd1` `rgb(84,255,209)``\n• #67ffd6\n``#67ffd6` `rgb(103,255,214)``\n• #7bffdc\n``#7bffdc` `rgb(123,255,220)``\n• #8effe1\n``#8effe1` `rgb(142,255,225)``\n• #a2ffe6\n``#a2ffe6` `rgb(162,255,230)``\n• #b6ffeb\n``#b6ffeb` `rgb(182,255,235)``\n• #c9fff1\n``#c9fff1` `rgb(201,255,241)``\n• #ddfff6\n``#ddfff6` `rgb(221,255,246)``\n• #f1fffb\n``#f1fffb` `rgb(241,255,251)``\nTint Color Variation\n\n# Tones of #40ffcc\n\nA tone is produced by adding gray to any pure hue. In this case, #98a7a3 is the less saturated color, while #40ffcc is the most saturated one.\n\n• #98a7a3\n``#98a7a3` `rgb(152,167,163)``\n• #91aea6\n``#91aea6` `rgb(145,174,166)``\n• #89b6aa\n``#89b6aa` `rgb(137,182,170)``\n``#82bdad` `rgb(130,189,173)``\n• #7bc4b1\n``#7bc4b1` `rgb(123,196,177)``\n• #73ccb4\n``#73ccb4` `rgb(115,204,180)``\n• #6cd3b7\n``#6cd3b7` `rgb(108,211,183)``\n• #65dabb\n``#65dabb` `rgb(101,218,187)``\n• #5de2be\n``#5de2be` `rgb(93,226,190)``\n• #56e9c2\n``#56e9c2` `rgb(86,233,194)``\n• #4ff0c5\n``#4ff0c5` `rgb(79,240,197)``\n• #47f8c9\n``#47f8c9` `rgb(71,248,201)``\n• #40ffcc\n``#40ffcc` `rgb(64,255,204)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #40ffcc 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"
] | [
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https://www.smallbasic.it/challenge-2012-05/ | [
"# Challenge 2012-05\n\n###",
null,
"Text Challenge\n\nWrite a program to read in and spell-check a sentence of text from a user.\nHint: Use the Dictionary method as a spell checker to find any miss-spelt words and highlight these for the user.\n\n### Musical Challenge\n\nWrite a program to play your country’s national anthem.\n\n### Community Suggestion Challenge (CodingLikeCrazy)\n\nWrite a program that writes up to the first 1000 primes, with user inputting how many primes to show.\nIf user requests over 1000 an error is displayed.\n\n### Graphical Challenge\n\nMake a tower of Hanoi game, for example like this.\n\n### Database Challenge\n\nWrite a simple database that can read and write data to a file and an interface to edit, add and delete data.\nFor example an address book with name, address and telephone number for each person in the database.\n\n### Physics Challenge\n\nWrite a program to calculate the elevation of a canon in degrees to hit a target.\nThe canon ball discharge velocity is 100 m/s and gravity is 10 m/s/s.\nConsider different target distances and find the maximum distance.\nIf you want more of a challenge:\n\n1. include air resistance where the cannon ball decelerates so that its speed through the air falls by 5% per second.\n2. include a head wind speed of 5 m/s.\n\nHint: You could calculate an array of results dist[elevation]=distance and use the Maths Challenge 1 to interpolate the result.\nThe results array could be calculated by:\n\n1. Algebra – solving equations (best when the equations are easy, but can get too hard).\n2. Numerical simulation – run repeated time-steps updating vertical and horizontal velocities and distances until the vertical distance returns to zero (the ground).\n\n### Maths Challenge 1\n\nWrite a program to interpolate an array of data.\nThe array of data values y[x] is given by the following code:\n\nWithout using the formula (just the data array y[x]) write a program to estimate x for any value of y input by a user in the range 0 to 100.\n\n### Maths Challenge 2\n\ny = x – x*x*x/3 + x*x*x*x*x/5 – x*x*x*x*x*x*x/7\nWrite a program to find x for any input y."
] | [
null,
"https://www.smallbasic.it/wp-content/uploads/2016/12/ok.png",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.8380596,"math_prob":0.8948111,"size":2063,"snap":"2021-43-2021-49","text_gpt3_token_len":489,"char_repetition_ratio":0.13598834,"word_repetition_ratio":0.008287293,"special_character_ratio":0.2346098,"punctuation_ratio":0.07263923,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9796605,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-23T07:43:47Z\",\"WARC-Record-ID\":\"<urn:uuid:bc6e9d9e-a38e-42da-a6d2-5ffca4682c38>\",\"Content-Length\":\"52706\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:48298d7b-bae2-4afc-bec7-7ede67c8e428>\",\"WARC-Concurrent-To\":\"<urn:uuid:178bb4d9-54dd-4a5b-927c-c0ca3513123d>\",\"WARC-IP-Address\":\"89.46.109.72\",\"WARC-Target-URI\":\"https://www.smallbasic.it/challenge-2012-05/\",\"WARC-Payload-Digest\":\"sha1:ZCNORYZ53XYUZPCSVQDCZAWO6PZI62VB\",\"WARC-Block-Digest\":\"sha1:ZINJ4I4WYO2KWWUBH65RKO4DKKLRS2FB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585653.49_warc_CC-MAIN-20211023064718-20211023094718-00010.warc.gz\"}"} |
https://stacks.math.columbia.edu/tag/00L4 | [
"Lemma 10.62.2. Let $R$, $M$, $M_ i$, $\\mathfrak p_ i$ as in Lemma 10.62.1. Then $\\text{Supp}(M) = \\bigcup V(\\mathfrak p_ i)$ and in particular $\\mathfrak p_ i \\in \\text{Supp}(M)$.\n\nThere are also:\n\n• 3 comment(s) on Section 10.62: Support and dimension of modules\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar)."
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.702145,"math_prob":0.9974021,"size":563,"snap":"2021-43-2021-49","text_gpt3_token_len":161,"char_repetition_ratio":0.10375671,"word_repetition_ratio":0.0,"special_character_ratio":0.29307282,"punctuation_ratio":0.13445379,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99575824,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-01T03:45:48Z\",\"WARC-Record-ID\":\"<urn:uuid:4c91aea4-a4ea-413c-ab0d-7250cee324f4>\",\"Content-Length\":\"14215\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a6d64a51-7f4c-466e-9c94-5f88cbe83a2f>\",\"WARC-Concurrent-To\":\"<urn:uuid:ced58b6e-635f-493d-b132-a3b8ffe2dde6>\",\"WARC-IP-Address\":\"128.59.222.85\",\"WARC-Target-URI\":\"https://stacks.math.columbia.edu/tag/00L4\",\"WARC-Payload-Digest\":\"sha1:ZMKMA54JAKJI2FF2UDZWWVUYHXXQBFL2\",\"WARC-Block-Digest\":\"sha1:PS6C36ONU6HHLIAEZAVX4BSWNT66EC2G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964359082.78_warc_CC-MAIN-20211201022332-20211201052332-00559.warc.gz\"}"} |
https://memim.com/solar-mass.html | [
" Solar mass\n\n# Solar mass\n\nThe solar mass, M ☉ short, is an astronomical unit, which is defined by the mass of the sun. It amounts to\n\nThis corresponds to 332 946 Earth masses.\n\nThe unit is used to specify the mass of astronomical objects that are larger and more massive than planets. These mostly stars and star clusters, massive gas clouds and dark nebulae, galactic nuclei, black holes and whole galaxies. More rarely, clusters of galaxies in solar masses can be estimated, and in the future probably the entire universe.\n\nMost of the other fixed stars ( \"suns\") have masses that range from 0.1 to 10 solar masses. Our central star is therefore \" good average \" of these main-sequence stars and normal giant stars. Mostly there are hot supergiants with 50 to 100 solar masses.\n\n## Determination of the mass of the Sun\n\nWith the help of Kepler's third law, the solar mass is obtained (for the approximation M ☉ >> M ⊕ ) from the large semi-major axis a of the earth ( mean distance Sun-Earth, about 1 astronomical unit ), the orbital period P of the Earth ( years long) and the gravitational constant (G = 6.6738 · 10-11 m³ / kg s ²):\n\nOf course, neither the astronomical unit nor the gravitational constant was about 100 years ago, known more precisely than about 1 %, and therefore an accurate determination of the Sun's mass in kilograms impossible. In contrast, the relative mass determination was possible of each planet in our solar system in units of the solar mass, if this planet is orbited by a moon (which in all except Mercury and Venus the case). For the planet mass could set up an equation with distance and round trip time of its moon and the gravitational constant, in turn, was the absolute mass determination difficult. However, since the orbital periods and the axis ratios are known by all planetary orbits long, the masses were from the two equations are compared and the planet mass are given with good accuracy in units of the solar mass.\n\nThe same was true for double stars whose distance on the cosmic distance determination in astronomical units but could not be measured in absolute terms. Therefore, it prevailed, that these masses were summarily given in solar masses (see also: Gaussian gravitational constant ). Today, the distance between the Sun and the Earth is measured very accurately, because you can fall back on radar measurements of interplanetary space probes. The gravitational constant G is now known to 4-5 decimal places. Nevertheless, remained the solar mass - like many others - as obtained in astronomy unit of measurement: on the one hand as historically established convention, and on the other hand because of the manageable numbers and as an idea help with such a huge \"astronomical numbers \".\n\n## Solar mass and related units\n\nA solar mass equal to:\n\n• 27,068,510 lunar masses M ☽\n• 332 946 Earth masses M ♁\n• 1047.56 Jupiter masses M ♃.\n\nOur Milky Way is about 180 billion solar masses, which corresponds to about 3.6 × 1041 kg. In other galaxies, such information is still very uncertain and is currently only useful for close, well -measured systems.\n\nIn the general theory of relativity, it is sometimes customary to quote the mass in units of length. Where ( the gravitational constant G and the velocity of light c ):\n\n738552\nde"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.9317982,"math_prob":0.96597046,"size":3246,"snap":"2020-34-2020-40","text_gpt3_token_len":700,"char_repetition_ratio":0.14312153,"word_repetition_ratio":0.0035460992,"special_character_ratio":0.22612447,"punctuation_ratio":0.10114192,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9761091,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-12T06:48:03Z\",\"WARC-Record-ID\":\"<urn:uuid:954a1c1e-4bd4-4ff2-b521-f917045bfa21>\",\"Content-Length\":\"9781\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0035e9d1-eafb-42ae-a6d6-085536248faf>\",\"WARC-Concurrent-To\":\"<urn:uuid:841e1d00-dfb0-4c40-bfcd-4e630825b492>\",\"WARC-IP-Address\":\"172.67.151.43\",\"WARC-Target-URI\":\"https://memim.com/solar-mass.html\",\"WARC-Payload-Digest\":\"sha1:XRVIDFB5TYFKYNNOSQY6QCNGKZRW4TMI\",\"WARC-Block-Digest\":\"sha1:5D2W5FZ3CWM3E2T2ETGFBIKPO5BSBLYB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738878.11_warc_CC-MAIN-20200812053726-20200812083726-00150.warc.gz\"}"} |
https://uva.onlinejudge.org/board/viewtopic.php?p=20741 | [
"10038 - Jolly Jumpers\n\nAll about problems in Volume 100. If there is a thread about your problem, please use it. If not, create one with its number in the subject.\n\nModerator: Board moderators\n\nosan\nNew poster\nPosts: 47\nJoined: Tue Jul 29, 2003 12:03 pm\nContact:\n\ni think u couldn't understand the problem yet\n\nCHECK THIS INPUT",
null,
"INPUT\n4 1 4 2 3\n5 1 4 2 -1 6\n10 1 2 3 4 5 6 7 8 9 10\n10 1 2 4 7 11 16 22 29 37 46\n10 -1 -2 -4 -7 -11 -16 -22 -29 -37 -46\n10 -1 -1 -4 -7 -11 -16 -22 -29 -37 -46\n1 1\n2 1 2\n2 2 1\n4 0 4 2 3\n4 1 3 2 4\n1 2\n6 1 4 3 7 5 10\nREAL OUTPUT\nJolly\nNot jolly\nNot jolly\nJolly\nJolly\nNot jolly\nJolly\nJolly\nJolly\nNot jolly\nNot jolly\nJolly\nJolly\nJolly\nNot Jolly\nNot Jolly\nJolly\nJolly\nNot Jolly\nNot Jolly\nNot Jolly\nNot Jolly\nNot Jolly\nNot Jolly\nNot Jolly\nNot Jolly\nthis time WA\nwhat next...............?\n\nyouareaverage\nNew poster\nPosts: 1\nJoined: Wed Jan 14, 2004 1:38 am\n\n10038\n\nI've tried just about every sample input from the boards and they all seem to work properly in my program. Maybe someone can help?\nHere's my code:\n[cpp]\n#include <iostream>\n#include <algorithm>\n\nusing namespace std;\n\ntemplate < class T > T absoluteValue( T input )\n{\nif ( input < 0 )\n{\ninput *= -1;\n}\nreturn input;\n}\n\nvoid jolly( int input[], int arraySize )\n{\nbool broken = 0;\nint counter = 0;\nint nArray[ arraySize ];\nfor ( counter = 1; counter < arraySize; counter++ )\n{\nnArray[counter-1]=absoluteValue(absoluteValue(input[counter])\n-absoluteValue(input[counter-1]));\n}\nsort( nArray, nArray + ( arraySize - 1 ) );\nfor ( counter = 0; counter < arraySize - 1 && !broken; counter++ )\n{\nif ( nArray[ counter ] != counter + 1 )\n{\nbroken = true;\n}\n}\nif ( !broken )\n{\ncout << \"Jolly\" << endl;\n}\nelse\n{\ncout << \"Not jolly\" << endl;\n}\n}\n\nint main()\n{\n//Variables.\nint counter = 0;\nint arraySize = 3000;\nint array;\n//Execution.\nwhile ( cin >> arraySize )\n{\nfor ( counter = 0; counter < arraySize; counter++ )\n{\ncin >> array[ counter ];\n}\njolly( array, arraySize );\n}\nreturn 0;\n}\n[/cpp]\n\nosan\nNew poster\nPosts: 47\nJoined: Tue Jul 29, 2003 12:03 pm\nContact:\n\nINPUT && OUTPUT\n\nINPUT\n4 1 4 2 3\n5 1 4 2 -1 6\n10 1 2 3 4 5 6 7 8 9 10\n10 1 2 4 7 11 16 22 29 37 46\n10 -1 -2 -4 -7 -11 -16 -22 -29 -37 -46\n10 -1 -1 -4 -7 -11 -16 -22 -29 -37 -46\n1 1\n2 1 2\n2 2 1\n4 0 4 2 3\n4 1 3 2 4\n1 2\n6 1 4 3 7 5 10\nOUTPUT\nJolly\nNot jolly\nNot jolly\nJolly\nJolly\nNot jolly\nJolly\nJolly\nJolly\nNot jolly\nNot jolly\nJolly\nJolly\nthis time WA\nwhat next...............?\n\nMorning\nExperienced poster\nPosts: 134\nJoined: Fri Aug 01, 2003 2:18 pm\nLocation: Shanghai China\n\n10038 why WA(Jolly Jumpers using Java)\n\n[java]\nimport java.io.*;\nimport java.util.*;\n\nclass Main\n{\nstatic int abs(int number)\n{\nif (number < 0)\n{\nreturn -1 * number;\n}\nelse return number;\n}\nstatic String ReadLn (int maxLg) // utility function to read from stdin\n{\nbyte lin[] = new byte [maxLg];\nint lg = 0, car = -1;\nString line = \"\";\n\ntry\n{\nwhile (lg < maxLg)\n{\nif ((car < 0) || (car == '\\n')) break;\nlin [lg++] += car;\n}\n}\ncatch (IOException e)\n{\nreturn (null);\n}\n\nif ((car < 0) && (lg == 0)) return (null); // eof\nreturn (new String (lin, 0, lg));\n}\n\npublic static void main (String args[]) // entry point from OS\n{\nMain myWork = new Main(); // create a dinamic instance\nmyWork.Begin(); // the true entry point\n}\n\nvoid Begin()\n{\nint array[];\nString input;\nStringTokenizer idata;\nint n,a,b,flag;\nwhile ((input = Main.ReadLn (255)) != null)\n{\nflag = 0;\nidata = new StringTokenizer (input);\nn = Integer.parseInt (idata.nextToken());\nif (n == 1)\n{\nSystem.out.println(\"Jolly\");\ncontinue;\n}\narray = new int[n - 1];\nfor(int loop = 0;loop < n - 1;loop++)\n{\narray[loop] = loop + 1;\n}\na = Integer.parseInt (idata.nextToken());\nb = Integer.parseInt (idata.nextToken());\nif (abs(a - b) >= n)\n{\nSystem.out.println(\"Not jolly\");\n}\nelse\n{\nfor(int loop2 = 0;loop2 < n - 1;loop2++)\nif (array[loop2] == abs(a - b))\n{\narray[loop2] = 0;\nbreak;\n}\nfor(int loop=1;loop < n - 1;loop++)\n{\na = b;\nb = Integer.parseInt (idata.nextToken());\nif (abs(a - b) >= n)\n{\nSystem.out.println(\"Not jolly\");\nflag = 1;\nbreak;\n}\nelse\n{\nfor(int loop2 = 0;loop2 < n - 1;loop2++)\nif (array[loop2] == abs(a - b))\n{\narray[loop2] = 0;\nbreak;\n}\n}\n}\nif (flag == 0)\n{\nfor(int loop = 0;loop < n - 1;loop++)\nif(array[loop] != 0)\n{\nSystem.out.println(\"Not jolly\");\nflag = 1;\nbreak;\n}\nif (flag == 0) System.out.println(\"Jolly\");\n}\n}\n}\n}\n}[/java]\n\"Learning without thought is useless；thought without learning is dangerous.\"\n\"Hold what you really know and tell what you do not know -this will lead to knowledge.\"-Confucius\n\npavelph\nLearning poster\nPosts: 57\nJoined: Wed Dec 10, 2003 7:32 pm\nLocation: Russia, Saint-Petersburg\nHm... You have very big code and I can't understand it. Please write steps of algo that you program do.\n\nMorning\nExperienced poster\nPosts: 134\nJoined: Fri Aug 01, 2003 2:18 pm\nLocation: Shanghai China\nI'm sorry my english is poor.I just make a array that contains the numbers from 1 to n so that i can compare with the absolutes differences\n[java]\nimport java.io.*;\nimport java.util.*;\n\nclass Main\n{\nstatic int abs(int number)\n{\nif (number < 0)\n{\nreturn -1 * number;\n}\nelse return number;\n}\nstatic String ReadLn (int maxLg) // utility function to read from stdin\n{\nbyte lin[] = new byte [maxLg];\nint lg = 0, car = -1;\nString line = \"\";\n\ntry\n{\nwhile (lg < maxLg)\n{\nif ((car < 0) || (car == '\\n')) break;\nlin [lg++] += car;\n}\n}\ncatch (IOException e)\n{\nreturn (null);\n}\n\nif ((car < 0) && (lg == 0)) return (null); // eof\nreturn (new String (lin, 0, lg));\n}\n\npublic static void main (String args[]) // entry point from OS\n{\nMain myWork = new Main(); // create a dinamic instance\nmyWork.Begin(); // the true entry point\n}\n\nvoid Begin()\n{\nint array[];\nString input;\nStringTokenizer idata;\nint n,a,b,flag;\nwhile ((input = Main.ReadLn (255)) != null)\n{\nflag = 0;\nidata = new StringTokenizer (input);\nn = Integer.parseInt (idata.nextToken());\n\nif (n == 1)\n{\n//a single integer is always Jolly\nSystem.out.println(\"Jolly\");\ncontinue;\n}\narray = new int[n - 1];\n//i put the numbers from 1 to n in array so that i can compare with the absolutes differences\nfor(int loop = 0;loop < n - 1;loop++)\n{\narray[loop] = loop + 1;\n}\n\na = Integer.parseInt (idata.nextToken());\nb = Integer.parseInt (idata.nextToken());\n//a and b are two absolutes differences one by one in input\nif (abs(a - b) >= n)\n{\nSystem.out.println(\"Not jolly\");\n}\nelse\n{\nfor(int loop2 = 0;loop2 < n - 1;loop2++)\nif (array[loop2] == abs(a - b))\n{\narray[loop2] = 0;\nbreak;\n//give 0 to array[loop2] to mark this value has appeared in absolutes differences\n}\nfor(int loop=1;loop < n - 1;loop++)\n{\na = b;\nb = Integer.parseInt (idata.nextToken());\n//a and b are next two absolutes differences\nif (abs(a - b) >= n)\n{\nSystem.out.println(\"Not jolly\");\nflag = 1;\n//flag mark that it is definitely \"Not jolly\"\nbreak;\n}\nelse\n{\nfor(int loop2 = 0;loop2 < n - 1;loop2++)\nif (array[loop2] == abs(a - b))\n{\narray[loop2] = 0;\nbreak;\n//just like what i did above\n}\n}\n}\nif (flag == 0)//if whether it is \"Not jolly\" is not sure\n{\nfor(int loop = 0;loop < n - 1;loop++)\nif(array[loop] != 0)\n{\n//if any element in array not equals to 0,then it must be \"Not jolly\"\nSystem.out.println(\"Not jolly\");\nflag = 1;\nbreak;\n}\nif (flag == 0) System.out.println(\"Jolly\");\n}\n}\n}\n}\n}[/java][/java]\n\"Learning without thought is useless；thought without learning is dangerous.\"\n\"Hold what you really know and tell what you do not know -this will lead to knowledge.\"-Confucius\n\npavelph\nLearning poster\nPosts: 57\nJoined: Wed Dec 10, 2003 7:32 pm\nLocation: Russia, Saint-Petersburg\nSorry, I know java very and very bad. All that I can help you is my algo:\n1) read input to array a[1..n] of integer\n2) create other array b[1..n] of boolean\n\nCode: Select all\n\nfor i=1 to n-1 do\nif |a[i] - a[i+1]|>=n then /Not jolly/\nelse b[|a[i] - a[i+1]|]:=true\n\nfor i=1 to n-1 if not b[i] then /Not jolly/\nelse {if b[i]=true for all i=1..n-1} /Jolly/\n\npavelph\nLearning poster\nPosts: 57\nJoined: Wed Dec 10, 2003 7:32 pm\nLocation: Russia, Saint-Petersburg\nSorry, I know java very and very bad. All that I can help you is my algo:\n1) read input to array a[1..n] of integer\n2) create other array b[1..n] of boolean\n\nCode: Select all\n\nfor i=1 to n-1 do\nif |a[i] - a[i+1]|>=n then /Not jolly/\nelse b[|a[i] - a[i+1]|]:=true\n\nfor i=1 to n-1 if not b[i] then /Not jolly/\nelse {if b[i]=true for all i=1..n-1} /Jolly/\n\nMorning\nExperienced poster\nPosts: 134\nJoined: Fri Aug 01, 2003 2:18 pm\nLocation: Shanghai China\nHey.Thanks so much:)i'll check it\n\"Learning without thought is useless；thought without learning is dangerous.\"\n\"Hold what you really know and tell what you do not know -this will lead to knowledge.\"-Confucius\n\nepidemyk\nNew poster\nPosts: 3\nJoined: Tue Feb 03, 2004 7:01 am\n\n10038 Jolly Jumpers - Time Limit Exceeded\n\nTime limit exceeded, not sure why...\n\n[c]\n#include <stdio.h>\n#include <math.h>\n\nint main(void)\n{\nint num;\nint i;\nint n1, n2;\n\nwhile(scanf(\"%d\", &num) != EOF)\n{\nint stat[num];\nint jolly = 1;\nint val = 0;\nchar temp;\n\nfor(i=0; i<num; i++)\n{\nstat = 0;\n}\n\nif (num == 1)\n{\nprintf(\"Jolly\\n\");\nscanf(\"%d\", &num); /* Consume extra number */\ngoto end; /* Break loop */\n} else {\n\nscanf(\"%d\", &n2); /* First number into n2 */\nfor(i=1; i<num; i++)\n{\nn1 = n2; /* First number into n1 */\nscanf(\"%d\", &n2); /* Next number into n2 */\nval = abs(n1-n2);\n\nif (val < num) /* If val is in the bounds */\nstat[abs(n1-n2)] = 1; /* Then set flag in array */\nelse /* Else print Not Jolly */\n{ /* and consume rest of line */\nprintf(\"Not jolly\\n\");\nscanf(\"%c\", &temp);\nwhile (temp != '\\n')\nscanf(\"%c\", &temp);\ngoto end;\n}\n}\n\nfor(i=1; i<num; i++)\njolly = jolly && stat; /* Boolean with all values */\n\nif (jolly == 1)\nprintf(\"Jolly\\n\");\nelse\nprintf(\"Not jolly\\n\");\n\n} /* End If */\nend:\n} /* End While */\n\nreturn 0;\n}\n[/c]\n\nmidra\nExperienced poster\nPosts: 119\nJoined: Fri Feb 13, 2004 7:20 am\nContact:\nWhy the output for: 6 1 4 3 7 5 10 is Jolly???\nthe sequence is 3 1 4 2 5 and to be Jolly, it must change in values of 1\n\nHere is my code, it works with many inputs, but not with this because I think that :6 1 4 3 7 5 10 is NOT JOLLY\n\n[c]#include <stdio.h>\n#include <math.h>\n\nint main()\n{\nint n,i,x,s=0,temp;\n\nscanf(\"%d\", &n);\nfor (i=1; i<=n; i++)\nscanf(\"%d\", &x);\nif (n==1)\n{\nprintf(\"Jolly\");\nreturn 0;\n}\n\nfor (i=1; i<=n-2; i++)\n{\nif (abs(abs(x)-abs(x[i+1]))==abs(abs(x[i+1])-abs(x[i+2])-1))\n{\ns++;\nif(n>=3) /*1 3 2 4 */\n{\nif(abs(abs(x)-abs(x[i+1]))==abs(abs(x[i+2])-abs(x[i+3])))\n{\nprintf(\"Not Jolly\");\nreturn 0;\n}\n}\n}\nelse if (abs(abs(x)-abs(x[i+1]))==abs(abs(x[i+1])-abs(x[i+2])+1))\n{\ns++;\nif(n>=3) /*1 3 2 4 */\n{ if(abs(abs(x)-abs(x[i+1]))==abs(abs(x[i+2])-abs(x[i+3])))\n{\nprintf(\"Not Jolly\");\nreturn 0;\n}\n}\n}\n}\nif (s>=n-2)\nprintf(\"Jolly\");\nelse\nprintf(\"Not Jolly\");\n\nreturn 0;\n}[/c]\n\nthanks for reading!",
null,
"dominus\nNew poster\nPosts: 2\nJoined: Fri Feb 27, 2004 5:52 pm\nLocation: Estonia\n\n10038 WA\n\nCan anybody tell me why this code returns WA?\n\nCode: Select all\n\n#include <iostream.h>\n\nint abs(int i)\n{\nif (i>0){ return i;}\nelse { return -i;}\n}\n\nint main()\n{\nint inp1=0, inp2=0, jolly=1, n=0, i=0, a={0};\n\nwhile( (cin >> n) && n )\n{\njolly=1;\ncin >> inp1; //first number\nfor(i=1;i<n;i++) //other numbers\n{\ncin >> inp2;\na[abs(inp1-inp2)] = 1;\ninp1=inp2;\n}\nfor(i=1;i<n;i++) //cycle from 1 to N-1\n{\nif(!a[i]) //every cell must be >0\n{\njolly=0;\n}\na[i]=0;\n}\na[n]=0;\nn=0;\n\ncout << (jolly ? \"Jolly\" : \"Not yolly\") << endl;\n}\nreturn 0;\n}\n\nKentaro\nNew poster\nPosts: 19\nJoined: Thu Feb 05, 2004 4:41 am\nYou spelled the output wrong?\n\"Jolly\", \"Not jolly\"\n\nIn your code I see:\n\"Jolly\", \"Not yolly\"\nComputer Science is no more about computers than Astronomy is about telescopes.\n-- E. W. Dijkstra\n\nKentaro\nNew poster\nPosts: 19\nJoined: Thu Feb 05, 2004 4:41 am\n1 4 3 7 5 10 is a jolly jumper of 6 integers.\n(equivalent to input: \"6 1 4 3 7 5 10\")\n\n|1 - 4| = 3\n|4 - 3| = 1\n|3 - 7| = 4\n|7 - 5| = 2\n|5 - 10| = 5\n\nAll the numbers from 1 to 5 are seen at least once in the above list so the sequence is a jolly jumper. They don't have to be seen in any particular order.\nEDIT: Read the problem statement carefully.\nEDIT 2: Edited to make a clearer explanation.\nComputer Science is no more about computers than Astronomy is about telescopes.\n-- E. W. Dijkstra\n\ndominus\nNew poster\nPosts: 2\nJoined: Fri Feb 27, 2004 5:52 pm\nLocation: Estonia\nYeah, you are totally right. What a stupid I am...",
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https://www.edalive.com/curriculum/structure/nsw_maths_s-5-2_msrmnt-gmtry_area-surface-area_ma5-2-11mg_calc-surfa~s~m-acmmg217_develop-fo~d~lar-height | [
"# Develop and use the formula to find the surface areas of closed right cylinders: Surface area of (closed) cylinder=2 pi r squared + 2 pi r h where r is the length of the radius and h is the perpendicular height\n\n 1 Supreme Circles - area of a circle 2 Cylinder-ella - surface area of a cylinder"
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https://math.stackexchange.com/questions/2259848/finding-a-function-whose-double-covariant-derivative-is-delta-function/2259849 | [
"# Finding a function whose double covariant derivative is delta function\n\nI have a function $f$ defined as $\\frac{\\overline{z}-\\overline{w}}{(z-w)(1 + z\\overline{z})(1+w\\overline{w})}$.\n\nI want to prove that $$\\partial_{\\overline{z}}^2 f + \\frac{2z}{1+z\\overline{z}}\\partial_{\\overline{z}}f = \\pi \\frac{2}{(1+ z\\overline{z})^2}\\delta(z-w).$$\n\nThis is basically a double covariant derivative wrt to $\\overline{z}$ taking the metric in unit sphere in stereographic coordinates.\n\nI found that $\\partial_{\\overline{z}}f = \\delta(z-w)\\frac{(1+z\\overline{z})(\\overline{z}-\\overline{w})}{1+w\\overline{w}} + \\frac{1}{z-w}\\frac{(1+2z\\overline{z}-z\\overline{w})}{1+w\\overline{w}}$.\n\nIf I differentiate again I am encountering a derievative of a delta function which I do not know how to do.\n\nBut since I know the result I calculated back what the derivative of delta function should be a found out to be $\\frac{2\\delta(z-w)}{\\overline{z}-\\overline{w}}\\frac{w\\overline{w}-z\\overline{z}}{1+z\\overline{z}}$.\n\nBasically during the calculation I used the product rule and took $\\frac{1}{z-w}$ as one function and rest as another and did the diffrentiation.I also used the fact that $\\partial_{\\overline{z}}\\frac{1}{z-w}=\\delta(z-w)$.\n\nI would like to know whether the procedure I have done is correct.Is the derivative of the delta function I have mentioned is correct or not.Pls help me.\n\nThis problem is basically involved during computing the BMS charge\n\n• -1. This is a purely mathematical question which belongs on Math SE. The context (computing BMS charge) has no impact on the question. Apr 30, 2017 at 16:39\n• Notice that the $\\delta$-function is being multiplied by ${\\bar z} - {\\bar w}$ so it is zero. Apr 30, 2017 at 18:47\n\nNote\n\n$$D_{\\bar z}^2f = \\gamma_{z\\bar z} \\partial_{\\bar z}( \\gamma^{z\\bar z} \\partial_{\\bar z} f )$$ We are solving the equation \\begin{align} D_{\\bar z}^2f &= \\pi \\gamma_{z\\bar z} \\delta^2(z-w)\\\\ \\gamma_{z\\bar z} \\partial_{\\bar z}( \\gamma^{z\\bar z} \\partial_{\\bar z} f ) &= \\pi \\gamma_{z\\bar z} \\delta^2(z-w) \\\\ \\partial_{\\bar z}( \\gamma^{z\\bar z} \\partial_{\\bar z} f ) &= \\frac{1}{2} \\partial_{\\bar z} \\frac{1}{z-w} \\\\ \\gamma^{z\\bar z} \\partial_{\\bar z} f &= \\frac{1}{2} \\frac{1}{z-w} + g ( z,w,\\bar w)\\\\ \\partial_{\\bar z} f &= \\frac{1}{2} \\frac{\\gamma_{z\\bar z} }{z-w} + \\gamma_{z\\bar z} g( z,w,\\bar w) \\end{align}\n\nNote that we have\n\n$$f = \\frac{ \\bar z - \\bar w}{ (z-w)(1+z\\bar z)(1+w\\bar w)}$$\n\nso that\n\n$$\\partial_{\\bar z} f = \\frac{ 2\\pi \\delta^2 ( z - w )( \\bar z - \\bar w )}{ (1+z\\bar z)(1+w\\bar w)} + \\frac{1}{2} \\frac{\\gamma_{z\\bar z} }{z-w} + \\frac{1}{2} \\gamma_{z\\bar z} \\frac{ \\bar w }{ ( 1 + w \\bar w ) }$$ The first term is zero. The rest of this is precisely the of the required form.\n\n• thank you prahar.This was exactly what I was looking for.i was confused when the dereivative of the $\\delta$ came.But I didnt notice the fact that it is with factor of $\\overline{z-w}$ so it will be zero.I guess the reason why this should be zero is when you are acting that on a test function you you get the integrand as zero right ?\n– budi\nMay 1, 2017 at 2:39\n• @AnupamAh yep, that's right. May 1, 2017 at 2:40\n• I have another doubt .How do you get the last piece from $f$.How does the $\\frac{\\overline{w}}{(1+w\\overline{w})^2}$\n– budi\nMay 1, 2017 at 14:41\n• @AnupamAh - There was a typo. I fixed it. May 1, 2017 at 17:30\n• @Prahar-Suppose I multiply $f$ by $\\gamma^{z\\bar z}$ will I get rid of the $\\gamma_{z\\bar z}$ in the double covariant derivative?What I thought was since the covariant derivative of metric is zero this will act as a constant and i can take it outside to cancel the $\\gamma_{z\\bar z}$ and get just the delta function.Am I thinking right ?\n– budi\nMay 15, 2017 at 13:37"
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https://en.m.wikipedia.org/wiki/Base_(topology) | [
"Base (topology)\n\nIn mathematics, a base or basis for the topology τ of a topological space (X, τ) is a family B of open subsets of X such that every open set of the topology is equal to a union of some sub-family of B (this sub-family is allowed to be infinite, finite, or even empty[note 1]). For example, the set of all open intervals in the real number line $\\mathbb {R}$",
null,
"is a basis for the Euclidean topology on $\\mathbb {R}$",
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"because every open interval is an open set, and also every open subset of $\\mathbb {R}$",
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"can be written as a union of some family of open intervals.\n\nBases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.\n\nNot all families of subsets form a base for a topology. For example, because X is always an open subset of every topology on X, if a family B of subsets is to be a base for a topology on X then it must cover X, which by definition means that the union of all sets in B must be equal to X. If X has more than one point then there exist families of subsets of X that do not cover X and consequently, they can not form a basis for any topology on X. A family B of subsets of X that does form a basis for some topology on X is called a base for a topology on X, in which case this necessarily unique topology, call it τ, is said to be generated by B and B is consequently a basis for the topology τ. Such families of sets are frequently used to define topologies. A weaker notion related to bases is that of a subbasis for a topology. Bases for topologies are closely related to neighborhood bases.\n\nDefinition and basic properties\n\nA base for a topology on X is a collection B of subsets of X satisfying the following properties:\n\n1. The elements of B cover X, i.e., every element of X belongs to some element in B.\n2. Given elements B1, B2 of B, for every x in B1 ∩ B2 there is an element B3 in B containing x and such that B3 is a subset of B1 ∩ B2.\n\nAn equivalent property is: any finite intersection[note 2] of elements of B can be written as a union of elements of B. These two conditions are exactly what is needed to ensure that the set of all unions of subsets of B is a topology on X.\n\nIf a collection B of subsets of X fails to satisfy these properties, then it is not a base for any topology on X. (It is a subbase, however, as is any collection of subsets of X.) Conversely, if B satisfies these properties, then there is a unique topology on X for which B is a base; it is called the topology generated by B. (This topology is the intersection of all topologies on X containing B.) This is a very common way of defining topologies. A sufficient but not necessary condition for B to generate a topology on X is that B is closed under intersections; then we can always take B3 = B1 ∩ B2 above.\n\nFor example, the collection of all open intervals in the real line forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standard topology on the real numbers.\n\nHowever, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to a basis of a vector space in linear algebra, a base need not be maximal; indeed, the only maximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality of a base is called the weight of the topological space.\n\nAn example of a collection of open sets which is not a base is the set S of all semi-infinite intervals of the forms (−∞, a) and (a, ∞), where a is a real number. Then S is not a base for any topology on R. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of elements of S. Using the alternate definition, the second property fails, since no base element can \"fit\" inside this intersection.\n\nGiven a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.\n\nExamples\n\nThe set Γ of all open intervals in $\\mathbb {R}$ form a basis for the Euclidean topology on $\\mathbb {R}$ . Every topology τ on a set X is a basis for itself (that is, τ is a basis for τ). Because of this, if a theorem's hypotheses assumes that a topology τ has some basis Γ, then this theorem can be applied using Γ := τ.\n\nA non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X. By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is a covering π-system and so also a base for a topology. In fact, if Γ is a filter on X then { ∅ } ∪ Γ is a topology on X and Γ is a basis for it. A base for a topology does not have to be closed under finite intersections and many aren't. But nevertheless, many topologies are defined by bases that are also closed under finite intersections. For example, each of the following families of subset of $\\mathbb {R}$ is closed under finite intersections and so each forms a basis for some< topology on $\\mathbb {R}$ :\n\n• The set Γ of all bounded open intervals in $\\mathbb {R}$ generates the usual Euclidean topology on $\\mathbb {R}$ .\n• The set Σ of all bounded closed intervals in $\\mathbb {R}$ generates the discrete topology on $\\mathbb {R}$ and so the Euclidean topology is a subset of this topology. This is despite the fact that Γ is not a subset Σ. Consequently, the topology generated by Γ, which is the Euclidean topology on $\\mathbb {R}$ , is coarser than the topology generated by Σ. In fact, it is strictly coarser because Σ contains non-empty compact sets which are never open in the Euclidean topology.\n• The set Γ$\\mathbb {Q}$ of all intervals in Γ such that both endpoints of the interval are rational numbers generates the same topology as Γ. This remains true if each instance of the symbol Γ is replaced by Σ.\n• Σ = { [r, ∞) : r$\\mathbb {R}$ } generates a topology that is strictly coarser than the topology generated by Σ. No element of Σ is open in the Euclidean topology on $\\mathbb {R}$ .\n• Γ = { (r, ∞) : r$\\mathbb {R}$ } generates a topology that is strictly coarser than both the Euclidean topology and the topology generated by Σ. The sets Σ and Γ are disjoint, but nevertheless Γ is a subset of the topology generated by Σ.\n\nObjects defined in terms of bases\n\nThe Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection.[citation needed]\n\n• The Zariski topology of $\\mathbb {C} ^{n}$ is the topology that has the algebraic sets as closed sets. It has a basis formed by the set complements of algebraic hypersurfaces.\n• The Zariski topology of the spectrum of a ring (the set of the prime ideals) has a basis such that each element consists of all prime ideals that do not contain a given element of the ring.\n\nTheorems\n\n• For each point x in an open set U, there is a base element containing x and contained in U.\n• A topology T2 is finer than a topology T1 if and only if for each x and each base element B of T1 containing x, there is a base element of T2 containing x and contained in B.\n• If $B_{1},B_{2},\\ldots ,B_{n}$ are bases for the topologies $T_{1},T_{2},\\ldots ,T_{n}$ then the set product $B_{1}\\times B_{2}\\times \\cdots \\times B_{n}$ is a base for the product topology $T_{1}\\times T_{2}\\times \\cdots \\times T_{n}.$ In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.\n• Let B be a base for X and let Y be a subspace of X. Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.\n• If a function $f:X\\to Y$ maps every base element of X into an open set of Y, it is an open map. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.\n• A collection of subsets of X is a topology on X if and only if it generates itself.\n• B is a basis for a topological space X if and only if the subcollection of elements of B which contain x form a local base at x, for any point x of X.\n\nBase for the closed sets\n\nClosed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space $X,$ a family of closed sets ${\\mathcal {C}}$ forms a base for the closed sets if and only if for each closed set $A$ and each point $x$ not in $A$ there exists an element of ${\\mathcal {C}}$ containing $A$ but not containing $x.$ A family ${\\mathcal {C}}$ is a base for the closed sets of $X$ if and only if its dual in $X,$ denoted by $X\\setminus {\\mathcal {C}}:=\\{X\\setminus C:C\\in {\\mathcal {C}}\\},$ is a base of open sets of $X;$ that is, if and only if the family of complements of members of ${\\mathcal {C}}$ is a base for the open sets of $X.$\n\nLet ${\\mathcal {C}}$ be a base for the closed sets of $X.$ Then\n\n1. $\\bigcap {\\mathcal {C}}=\\varnothing$\n2. For each $C_{1},C_{2}\\in {\\mathcal {C}}$ the union $C_{1}\\cup C_{2}$ is the intersection of some subfamily of ${\\mathcal {C}}$ (that is, for any $x\\in X$ not in $C_{1}{\\text{ or }}C_{2}$ there is some $C_{3}\\in {\\mathcal {C}}$ containing $C_{1}\\cup C_{2}$ and not containing $x$ ).\n\nAny collection of subsets of a set $X$ satisfying these properties forms a base for the closed sets of a topology on $X.$ The closed sets of this topology are precisely the intersections of members of ${\\mathcal {C}}.$\n\nIn some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is completely regular if and only if the zero sets form a base for the closed sets. Given any topological space $X,$ the zero sets form the base for the closed sets of some topology on $X.$ This topology will be the finest completely regular topology on $X$ coarser than the original one. In a similar vein, the Zariski topology on An is defined by taking the zero sets of polynomial functions as a base for the closed sets.\n\nWeight and character\n\nWe shall work with notions established in (Engelking 1977, p. 12, pp. 127-128).\n\nFix X a topological space. Here, a network is a family ${\\mathcal {N}}$ of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in ${\\mathcal {N}}$ for which $x\\in B\\subseteq U.$ Note that, unlike a basis, the sets in a network need not be open.\n\nWe define the weight, w(X), as the minimum cardinality of a basis; we define the network weight, nw(X), as the minimum cardinality of a network; the character of a point, $\\chi (x,X),$ as the minimum cardinality of a neighbourhood basis for x in X; and the character of X to be\n\n$\\chi (X)\\triangleq \\sup\\{\\chi (x,X):x\\in X\\}.$\n\nThe point of computing the character and weight is to be able to tell what sort of bases and local bases can exist. We have the following facts:\n\n• nw(X) ≤ w(X).\n• if X is discrete, then w(X) = nw(X) = |X|.\n• if X is Hausdorff, then nw(X) is finite if and only if X is finite discrete.\n• if B is a basis of X then there is a basis $B'\\subseteq B$ of size $|B'|\\leq w(X).$\n• if N a neighbourhood basis for x in X then there is a neighbourhood basis $N'\\subseteq N$ of size $|N'|\\leq \\chi (x,X).$\n• if $f:X\\to Y$ is a continuous surjection, then nw(Y) ≤ w(X). (Simply consider the Y-network $f'''B\\triangleq \\{f''U:U\\in B\\}$ for each basis B of X.)\n• if $(X,\\tau )$ is Hausdorff, then there exists a weaker Hausdorff topology $(X,\\tau ')$ so that $w(X,\\tau ')\\leq nw(X,\\tau ).$ So a fortiori, if X is also compact, then such topologies coincide and hence we have, combined with the first fact, nw(X) = w(X).\n• if $f:X\\to Y$ a continuous surjective map from a compact metrisable space to an Hausdorff space, then Y is compact metrisable.\n\nThe last fact follows from f(X) being compact Hausdorff, and hence $nw(f(X))=w(f(X))\\leq w(X)\\leq \\aleph _{0}$ (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)\n\nIncreasing chains of open sets\n\nUsing the above notation, suppose that w(X) ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ+.\n\nTo see this (without the axiom of choice), fix\n\n$\\left\\{U_{\\xi }\\right\\}_{\\xi \\in \\kappa },$\n\nas a basis of open sets. And suppose per contra, that\n$\\left\\{V_{\\xi }\\right\\}_{\\xi \\in \\kappa ^{+}}$\n\nwere a strictly increasing sequence of open sets. This means\n$\\forall \\alpha <\\kappa ^{+}:\\qquad V_{\\alpha }\\setminus \\bigcup _{\\xi <\\alpha }V_{\\xi }\\neq \\varnothing .$\n\nFor\n\n$x\\in V_{\\alpha }\\setminus \\bigcup _{\\xi <\\alpha }V_{\\xi },$\n\nwe may use the basis to find some Uγ with x in UγVα. In this way we may well-define a map, f : κ+κ mapping each α to the least γ for which UγVα and meets\n$V_{\\alpha }\\setminus \\bigcup _{\\xi <\\alpha }V_{\\xi }.$\n\nThis map is injective, otherwise there would be α < β with f(α) = f(β) = γ, which would further imply UγVα but also meets\n\n$V_{\\beta }\\setminus \\bigcup _{\\xi <\\alpha }V_{\\xi }\\subseteq V_{\\beta }\\setminus V_{\\alpha },$\n\nwhich is a contradiction. But this would go to show that κ+κ, a contradiction."
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http://wrcad.com/manual/wrsmanual/node179.html | [
"",
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"Next: Scale Icons Up: The Plot Panel Previous: Trace Drag and Drop Contents Index\n\nMultidimensional Traces\n\nWhen a plot window is displaying multidimensional data, the dimension map icon will appear in the upper left corner of the plot window. Clicking on this icon will toggle display of the dimension map. The dimension map allows the user to display only chosen dimensions of the traces.\n\nConsider the plot produced by\n\nset value1 = temp\nloop -50 125 25 dc vds 0.0 1.2 0.02 vgs 0.2 1.2 0.2\n\nThe loop command produces a three dimensional plot, with dimensions { 8, 6, 61 }. When plotting i(vds), the display would contain 48 traces, representing id vs. vds for each vgs and temperature value.\n\nThe visibility of these traces is set by the columns of clickable dimension selector indices shown in the dimension map. Initially, the traces for all dimensions are shown. Clicking anywhere in the dimension list with the center mouse button, or equivalently with the left (or only) mouse button while pressing Shift, will hide the traces for all dimensions. Clicking anywhere in the dimension list with the right mouse button, or equivalently with the left (or only) mouse button while pressing Ctrl, will show the traces for all dimensions. Clicking or dragging over the entries with the left mouse button will toggle display of the corresponding traces.\n\nIn the present case the dimension map contains two columns: the left column contains eight numbers 0-7, and the right columns contains six numbers 0-5. Clicking on these numbers controls the visibility per dimension, i.e., clicking in the left column would display/suppress all traces for a given temperature, clicking in the right column will display/suppress traces corresponding to a vgs value. Multiple entries in the same column can be toggled by dragging the mouse pointer over them.\n\nThis dimensional partitioning would apply for any number of dimensions. If a column contains too many dimensions to list completely, a label ``more'' will exist at the bottom of the listing. Clicking on this label will cycle through all of the dimensions, in the columns that require it.\n\nIf the plot is displaying a single multidimensional variable, then each least dimension is displayed in a separate color. The numbers in the rightmost column of the dimension map will use the same colors. In other columns, and in the rightmost column if more than one variable is being plotted, the indices use a uniform color to indicate that the dimension is shown, and in all cases black indicates a dimension that is not being shown.\n\nThe dimensions shown can also be controlled by mplot windows from the mplot command. These are the windows generally used to display results from operating range and Monte Carlo analysis. The mplot display consists of an array of pass/fail indication cells, one for each trial. These can be selected or deselected by clicking on them.\n\nAn mplot window is always associated with an internal plot structure, as listed with the Plots tool. The plot structure may also contain multidimensional vectors, for example if one uses the ``-k''\" option to the check command, all trial data will be saved.\n\nIf an mplot window with selections is present, and the plot command is used to plot a multidimensional vector from the same internal plot structure as the mplot, then only the dimensions corresponding to the selected trials will be shown on the plot window.\n\nIn this plot window, a ``flat'' dimension map will be used. This is a single column, with length equal to the product of the ``real'' dimensions. The visibility of each flat dimension can be toggled with the map entries as usual. The mplot selections have no effect on a plot window once it is displayed, but will initialize new plot windows to 1) enforce a flat dimension mapping, and 2) set the initial states of the flat map. After changing the mplot selections, one must use the plot command again to see the revised dimensions, or alternatively one can note the numbers of the mplot cells, and manipulate the same numbers in the dimensions map of the first plot, to see the new data.\n\nIf one has a plot structure containing multidimensional vectors from any source, such as from the loop commnd, one can still use the mplot capability. Giving the command\n\nmplot vector\n\nfor any multidimensional vector will produce an mplot window. The number of mplot cells will equal the number of flat dimensions in the vector. The pass/fail indication means nothing in this case, all cells display ``fail''. One can select the dimension cells in the mplot, which will affect subsequent plots from the plot command of any vector in the same internal plot structure, as described above. The vector given to the mplot command can be any vector from the plot, it is used for dimension counting only.\n\nIn older WRspice releases, the upper dimensions were represented as ``flat'', so that in the plot there would be a single column of numbers (0-47 in our original example above, six vgs values times eight temperatures), and clicking on these numbers would display/suppress the corresponding trace.",
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"Next: Scale Icons Up: The Plot Panel Previous: Trace Drag and Drop Contents Index\nStephen R. Whiteley 2019-06-03"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.8458504,"math_prob":0.9742678,"size":4989,"snap":"2019-26-2019-30","text_gpt3_token_len":1045,"char_repetition_ratio":0.17873621,"word_repetition_ratio":0.03827751,"special_character_ratio":0.20364803,"punctuation_ratio":0.09830867,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9528668,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-17T07:18:21Z\",\"WARC-Record-ID\":\"<urn:uuid:f63db5f7-372e-45b9-884b-8bf218ec3977>\",\"Content-Length\":\"8930\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:155ac22c-1d7f-4a72-b150-0e960cea657e>\",\"WARC-Concurrent-To\":\"<urn:uuid:f2982cdc-fe3c-4e7b-8da2-3b9b9438a051>\",\"WARC-IP-Address\":\"66.29.158.151\",\"WARC-Target-URI\":\"http://wrcad.com/manual/wrsmanual/node179.html\",\"WARC-Payload-Digest\":\"sha1:F2OPYPLNMDVX44DDFHKKNX3EIRPAJSMJ\",\"WARC-Block-Digest\":\"sha1:AGK2HO53LW4T2RNCIL4FMKIB3D2TMAJN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627998440.47_warc_CC-MAIN-20190617063049-20190617085049-00352.warc.gz\"}"} |
https://abhishek-choudhary.blogspot.com/2014/09/regression-analysis-what-is-that-and.html | [
"## Monday, September 22, 2014\n\n### Linear Regression , What is that and when should I use it - Machine Learning\n\nLinear Regression or Regression with Multiple Convariates\n\nBelieve me these are extremely easy to understand and R-programming has already these algorithms implemented , you just need to know how to use them :)\n\nLets consider we have values X and Y. In Simple word Linear Regression is way to model a relationship between X and Y , that's all :-). Now we have X1...Xn and Y , then relationship between them is Multiple Linear Regression\n\nLinear Regression is very widely used Machine Learning algorithm everywhere because Models which depend linearly on their unknown parameters are easier to fit.\n\nUses of Linear Regression ~\n\n• Prediction Analysis kind of applications can be done using Linear Regression , precisely after developing a Linear Regression Model, for any new value of X , we can predict the value of Y (based on the model developed with a previous set of data).\n\n• For a given Y, if we are provided with multiple X like X1.....Xn , then this technique can be used to find the relationship between each of the X with Y , so we can find the weakest relationship with Y and the best one as well .\n\nWhy I did all the theory above is , so that I could remember the basics, rest is all easy :).\n\n------------------------------------------------------------------\n\nSo now I'd like to do an example in R and the best resource I could find was Population.\nTalk about population , so How can I miss India , so somehow I managed to get dataset-\n\nAbove is just a snapshot of the data , I had data from 1700 till 2014 and yeah some missing data as well in-between .\n\nto use R , already caret package has an implementation of regression , so load the same and for plotting I am using ggplot.\n\nThe bottomline after getting data is to do the exploratory analysis, well I have 2 fields and no time :) , so just a quick plot-\n\nLooking great , its growing ,.. growing ..and .. so its real data .\nSo 1st thing 1st , split the data in 2 parts , training and testing\n\n`````` allTrainData <- createDataPartition(y=data\\$population,p=0.7,list=FALSE)\ntraining <- data[allTrainData,]\ntesting <- data[-allTrainData,]\n``````\n\nSo now I have X and Y , or simply wanted to find the population based on year or vice versa .\n\nDon't worry , R brought caret package which already brought implementation of the linear regression algorithm.\nWhat the formula behind it , please check my other blog about detail of Linear Modelling but here -\n\n`````` model <- train(population~.,method=\"lm\",data=training)\nfinalModel <- model\\$finalModel\n``````\n\n1 line , that's all , method=\"lm\" , isn't it extraordinary :)\nSo the summary here-\n\n`````` Call:\nlm(formula = .outcome ~ ., data = dat)\nResiduals:\nMin 1Q Median 3Q Max\n-186364 -164118 -83667 106876 811176\nCoefficients:\nEstimate Std. Error t value Pr(>|t|)\n(Intercept) -6516888 668533 -9.748 4.69e-16 ***\nyear 3616 346 10.451 < 2e-16 ***\n---\nSignif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\nResidual standard error: 216300 on 97 degrees of freedom\nMultiple R-squared: 0.5296, Adjusted R-squared: 0.5248\nF-statistic: 109.2 on 1 and 97 DF, p-value: < 2.2e-16\n``````\n\nDetails about summary linear Regression Model Summary\n\nSo now lets plot the fitted vs residual graph and see how well the model worked.\n\nSome weird :) but atleast the line went through almost all the data.\n\nNow how well the model worked -\n\nWell my data is anyway weird, so seriously it worked pretty good , believe me :)\n\nSo now on the model value , we should try the testing dataset and that's as well straightforward -\n\n`````` pred <- predict(model,testing)\n``````\n\ndisclaimer** - I am not a phd holder or data scientist , what I do is self interest and learning ... so it may contain some serious mistakes :)"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.89163697,"math_prob":0.91546124,"size":5075,"snap":"2021-31-2021-39","text_gpt3_token_len":1173,"char_repetition_ratio":0.123644255,"word_repetition_ratio":0.52407616,"special_character_ratio":0.27093595,"punctuation_ratio":0.14228052,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99344057,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-24T03:44:07Z\",\"WARC-Record-ID\":\"<urn:uuid:ccae0487-06ac-4566-8788-98dcd32168a9>\",\"Content-Length\":\"53299\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6b2d273d-1105-449e-8275-e519a8d2e235>\",\"WARC-Concurrent-To\":\"<urn:uuid:37bd8e5b-d914-4b67-8bef-1aa39e0d1edf>\",\"WARC-IP-Address\":\"142.250.73.193\",\"WARC-Target-URI\":\"https://abhishek-choudhary.blogspot.com/2014/09/regression-analysis-what-is-that-and.html\",\"WARC-Payload-Digest\":\"sha1:CD5LXBCYYOGY3HDLBSLEK6KMZBARTEE3\",\"WARC-Block-Digest\":\"sha1:IOMFA5F4TLX6YB62R3XUHUUWHEBAFGXZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057496.18_warc_CC-MAIN-20210924020020-20210924050020-00222.warc.gz\"}"} |
http://iamnotaboothbabe.com/elapsed-time.html | [
"",
null,
"# Python Program to Measure the Elapsed Time in Python\n\nIn this example, you will learn to measure the elapsed time.\n\nTo understand this example, you should have the knowledge of the following Python programming topics:\n\n## Example 1: Using time module\n\n``````import time\n\nstart = time.time()\n\nprint(23*2.3)\n\nend = time.time()\nprint(end - start)``````\n\nOutput\n\n```52.9\n3.600120544433594e-05```\n\nIn order to calculate the time elapsed in executing a code, the `time` module can be used.\n\n• Save the timestamp at the beginning of the code `start` using `time()`.\n• Save the timestamp at the end of the code `end`.\n• Find the difference between the end and start, which gives the execution time.\n\nThe execution time depends on the system.\n\n## Example 2: Using timeit module\n\n``````from timeit import default_timer as timer\n\nstart = timer()\n\nprint(23*2.3)\n\nend = timer()\nprint(end - start)``````\n\nOutput\n\n```52.9\n6.355400000000039e-05```\n\nSimilar to Example 1, we use `timer()` method from `timeit` module.\n\n`timeit` provides the most accurate results."
] | [
null,
"https://www.facebook.com/tr",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.6137733,"math_prob":0.98495466,"size":752,"snap":"2023-14-2023-23","text_gpt3_token_len":206,"char_repetition_ratio":0.14572193,"word_repetition_ratio":0.050847456,"special_character_ratio":0.30319148,"punctuation_ratio":0.13422818,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9923737,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-22T22:19:15Z\",\"WARC-Record-ID\":\"<urn:uuid:9856438a-0ce4-457a-815b-461127ac52d0>\",\"Content-Length\":\"95995\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:266580ff-5349-44ab-8db2-df1cbd05d57b>\",\"WARC-Concurrent-To\":\"<urn:uuid:8ad89a02-a2df-48c3-9be3-afed1efa87e9>\",\"WARC-IP-Address\":\"23.226.3.155\",\"WARC-Target-URI\":\"http://iamnotaboothbabe.com/elapsed-time.html\",\"WARC-Payload-Digest\":\"sha1:ZHPX5JMHQMEMF3X3F6ALDQTTORL66DI6\",\"WARC-Block-Digest\":\"sha1:SJDRJQV5TBNX2CHJ2IOPPNUTU4NV4LXL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296944452.97_warc_CC-MAIN-20230322211955-20230323001955-00613.warc.gz\"}"} |
https://socratic.org/questions/how-do-you-simplify-6x-4-3-2x-2-3-2x-1-3 | [
"How do you simplify [6x^(-4/3) * 2x^(2/3)] / (2x^(-1/3))?\n\nMar 18, 2016\n\n$= 6 {x}^{- \\frac{1}{3}}$\n\nExplanation:\n\n(6x^(-4/3)*cancel2x^(2/3))/(cancel2x^(-1/3))=6x^(-4/3+2/3+1/3=6x^(-4/3+1)= 6x^(-1/3)\n\nMar 18, 2016\n\n$6 {x}^{- \\frac{1}{3}}$\n\nExplanation:\n\nAs the powers are not applied to the numbers you cane split it like this:\n\n$\\frac{6 \\times 2}{2} \\times \\frac{{x}^{- \\frac{4}{3}} \\times {x}^{\\frac{2}{3}}}{{x}^{- \\frac{1}{3}}}$\n\ncolor(blue)(\"For the number we have \"6xx2/2\" \" =\" \" 6xx1\" \" =\" \" 6\n\n'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nThe powers being negative means they go to the other side of the 'line' and in doing so changing from negative to positive.\n\nSo$\\text{ \"(x^(-4/3)xxx^(2/3))/(x^(-1/3))\" is the same as } \\frac{{x}^{\\frac{1}{3}} \\times {x}^{\\frac{2}{3}}}{{x}^{\\frac{4}{3}}}$\n\n$\\textcolor{b r o w n}{\\text{'~~~~~~~~~~~~~~~~~ Note ~~~~~~~~~~~~~~~~~~~~~~~}}$\n$\\textcolor{b r o w n}{\\text{If you have for example \"a^2xxa^3\" then this is the same as}}$\n$\\textcolor{b r o w n}{{a}^{2 + 3} \\text{. The same happens with fractional powers.}}$\n$\\textcolor{b r o w n}{\\text{'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$\n\n$\\text{ \"(x^(1/3)xx x^(2/3))/(x^(4/3))\" \" =\" } \\frac{{x}^{\\frac{1}{3} + \\frac{2}{3}}}{{x}^{\\frac{4}{3}}}$\n\n$\\text{ } = \\frac{{x}^{\\frac{3}{3}}}{{x}^{\\frac{4}{3}}}$\n\n$\\textcolor{b r o w n}{\\text{'~~~~~~~~~~~~~~~~~ Note ~~~~~~~~~~~~~~~~~~~~~~~}}$\n$\\textcolor{b r o w n}{\\text{If you have for example \"a^2/a^3\" then this is the same as}}$\n$\\textcolor{b r o w n}{{a}^{2 - 3} \\text{. The same happens with fractional powers.}}$\n$\\textcolor{b r o w n}{\\text{'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$\n\n$\\text{ } = \\frac{{x}^{\\frac{3}{3}}}{{x}^{\\frac{4}{3}}}$\n\n$\\text{ } = {x}^{\\frac{3 - 4}{3}}$\n\n$\\text{ } = {x}^{- \\frac{1}{3}}$\n'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n$\\textcolor{b l u e}{\\text{Combining with the numbers}}$\n\n$\\text{ } \\textcolor{red}{= 6 {x}^{- \\frac{1}{3}}}$"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.605881,"math_prob":0.99993527,"size":1266,"snap":"2022-05-2022-21","text_gpt3_token_len":425,"char_repetition_ratio":0.3209192,"word_repetition_ratio":0.15037593,"special_character_ratio":0.5947867,"punctuation_ratio":0.028571429,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999697,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-24T14:34:27Z\",\"WARC-Record-ID\":\"<urn:uuid:66cb438a-703e-4f24-b68d-0b543ce54314>\",\"Content-Length\":\"35580\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3c4de686-fef1-4849-8412-9cf94fc4dcca>\",\"WARC-Concurrent-To\":\"<urn:uuid:be813840-cda8-48ff-8fba-e1dd0a1fe242>\",\"WARC-IP-Address\":\"216.239.34.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-simplify-6x-4-3-2x-2-3-2x-1-3\",\"WARC-Payload-Digest\":\"sha1:LUISW5MTLGFUG2YKWLGPZOVJUZHPCOZZ\",\"WARC-Block-Digest\":\"sha1:PLCQJZR24XEX6CVNTCOKD7AQIQIIA5JW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320304570.90_warc_CC-MAIN-20220124124654-20220124154654-00690.warc.gz\"}"} |
http://netcyborg.com/creative-financial-pytey/657c09-nmr-practice-problems-with-integration | [
"Spectroscopy Problems. In these spectra, each peak is labeled with its ppm chemical shift along the top, while the integration values (relative number of hydrogens—remember that integration values are relative!) Carbon 13 NMR Problems 1 Organic Chemistry Dr Sundin. For this signal, the integration value is 40.2. b) Three, two and five. Some good resources to practice NMR problems and combined spectral problems (ones that have proton, carbon, and … We hope you find them useful! a) Five, four and eight. One method of solving this problem is to report the location of an NMR signal in a. << /Type /Page /Parent 3 0 R /Resources 6 0 R /Contents 4 0 R /MediaBox [0 0 792 612] Show all your work (= label peaks in the spectra!) �AvaO And for this signal, it's 42.2. NMR Practice Problems In the following examples, we will learn how to solve NMR practice problems step-by-step in over 100 min video solutions which is essential for organic structure determination.. Using this information, your task is to determine the structure of the compound. Now, each signal is also characterized by integration. c. Suggest a structure for compound W based on the spectra given. stream Problem 2: Provide a structure of a compound having a molecular formula of C 7H 8O that is consistent with the following spectra. >> /Font << /F3.0 16 0 R /F1.1 9 0 R /F2.1 11 0 R /F4.0 17 0 R /F5.0 18 0 R The 2:3 ration often indicates an ethyl group: A 1 : 1.5 ratio can also indicate a 4 and 6 protons per signal based on the molecular formula. On the second spectrum, the integral of signal a is six times taller than signal b since the ratio here is 6 : 1. For each example you should find the number of signals you expect, where they should show on the scale (chemical shift), and what shape they should This number indicates how many protons give rise to the signal. 1H NMR Practice Problems Dr. Peter Norris Youngstown State University The following exercises are designed to help you become familiar with predicting 1the H NMR spectra of simple organic molecules. �'1G�/���(��- So, in these situations, you need to divide the numbers by the smallest integration value and check if the sum of new values matches the chemical formula: Integral values 1, 2 and 9 satisfy the chemical formula, therefore, you can manually set the first signal to 1 and the rest will adjust automatically. The next focuses on using these three techniques together to determine the structures of organic compounds. To be more accurate, let’s mention that it is the ratio of the protons behind each signal. Determine the degree of unsaturation for the compound. web based 2d nmr spectroscopy practice problems. Web Based 2D NMR Spectroscopy Practice Problems. Integration gives the relative number of hydrogens present at each signal; The integrated intensity of a signal in a 1 H NMR spectrum (does not apply to 13 C NMR) gives a ratio for the number of hydrogens that give rise to the signal, thereby helping calculate the total number of hydrogens present in a sample. Combined spectroscopy practice problems and answers. 1H NMR Problem-Solving Strategies The goal of solving a 1H NMR spectrum is to determine the structure that is consistent with ALL the NMR data. NMR instruments don’t know what are trying to do – all they do for integration is measure the relative intensity and give them to us with some convenient numbers. Chapter 14 NMR Spectroscopy. Let’s see how it works on the NMR spectra od chloroethane and 2-bromopropae: The height of each integral is proportional to the area of the given signal and the area is determined based in the number of absorbing protons. For example, the ratio 1.1 : 8.95 : 12.05 is reasonable to consider as 1 : 9 : 12. H CC HA Web UVic Ca. 1 H NMR Spectroscopy for CHM 222L Professor: S. Bruce King | Programming & Design: Yue ... Splitting : Web Resources: Practice >>> Predict Spectrum I: Predict Spectrum II: Peak Assignments: quick review . h cc ha web uvic ca. endobj �-�B�6x)� This organic chemistry video discusses the integration of H-NMR signals in NMR spectroscopy. NMR Practice Problem (Part II) Winter 2015 (Problems from former Chem 30BL finals) 2 Summer 2010 1. This session helps to analyze students learning achievements. Nmr Practice Problems With Answers. Access Free Nmr Practice Problems With Solutions 12.08.1 Proton NMR Practice Problems - Chemistry LibreTexts Answers Fall 2007 Winter 2008 Spring 2008 O O Summer 2008 Fall 2008 O OH O Winter 2009 O O O O Summer 2009 O O O Fall 2009 O O Winter 2010 O O O Spring 2010 OH. From this, we get degrees of unsaturation= (9-7)/2=1 so there is one Page 4/10. become familiar with educational testing service. Read PDF Nmr Practice Problems With Solutionsbook start as competently as search for them. Check this post, there are quite a few to work on: NMR Spectroscopy-Carbon-Dept-IR Practice Problems. 2 0 obj Practice 1H NMR Problems ・ Interactive Organic 1 H NMR Practice Problems Dr. Peter Norris Youngstown State University The following exercises are designed to help you become familiar with predicting the 1 H NMR spectra of simple organic molecules. In any case, keep in mind that integration numbers may not be exact, and it is okay to go based on some rounding. See NMR Spectrum See IR Spectrum Next : Click the structure of the unknown. This should sum up the essentials of integration in MNR spectroscopy. chem605 nmr spectroscopy. You might not require more era to spend to go to the Page 1/27. endobj Let's try a 1 H NMR practice problem with C 4 H 7 Cl: Remember from previous sections that to solve an NMR spectrum with double bonds, we must know the Degrees of Unsaturation. a. Determine the degree of unsaturation for the compound. x�YMo�F��WLz��j��&{lڢ)�6i�P���N�&:J��ُ٥(��m:@H��7og��ή?�+�\\1n��^�5���ŋ��^����s��C���o� ~w7V��W�m,S������֏���\\����P>[��_�q����vp|4+�E��Leͬ���1��'*�&�-����k��p�\\�F���c��\\$�!8��C1��Mv�'!/��FB.���F�L&8��óUA��*'�F3-�� �W�R�N,^�tۛ��Û�нwޫ��{aq?t�S��:!��\\����C��� << /Length 5 0 R /Filter /FlateDecode >> << /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] /ColorSpace << /Cs1 7 0 R Show NMR answer. ���L�7z�+��f�R7yp�O��0 �QL*���-F���6Q�mf��J����Lkt��������P� In the broadband decoupled 13 C NMR spectrum, the number of signals appearing for the bicyclo octane A–C, respectively, are. This numbers may not match the actual number of protons as it is only their ratio. Enter the appropriate letter in the answer box to the right of each formula. b. The integral of signal b is 1.5 times taller than the one for signal a since the proton ratio is 3 : 2. J���a��xrtK���! The purpose of this paper is to present these common problems and artifacts so that NMR users can identify them, understand their origin and lastly, be able to eliminate or at least minimize them. If the numbers are fractions, they also need to be simplified. Given are the following spectra. As interpreting NMR spectra, the structure of an unknown compound, as well as known structures, can be assigned by several factors such as chemical shift, spin multiplicity, coupling constants, and integration. For this signal, there's an integration value of 27. 14 0 obj %��������� 13c nmr problems grossmont college. We've been putting together a small library of practice 1H NMR spectra for our students, so we thought we'd post them here. Compound W has an empirical formula of C 11 H 10 O 2. 8 /Filter /FlateDecode >> Organic Chemistry 1 and 2 Summary Sheets – Ace your Exam. 1572 You can also subscribe without commenting. Read PDF Nmr Practice Problems With Solutions Nmr Practice Problems With Solutions This is likewise one of the factors by obtaining the soft documents of this nmr practice problems with solutions by online. 6*Rq'C]��hsZ{��HTK�;�3�O.n; �9�]K�z5�4�{G��\\$#�} ��.2d6=�u���0�8 áU�� �ч���H`��%E�H�4~'�� y�ep|�H�\\$ ����tKCSP>G��6' �-�� MpG G;l�5U|��T#��c��7���(4[�cr��@jXrp���[��{M�`9jc�������々B q+�GiK�����Wj��'�Q���bi �Y0 \\$V:�x�:�u-���˥��|�4�^. Nuclear Magnetic Resonance (NMR) interpretation plays a pivotal role in molecular identifications. Find the C-H peak with the lowest integration value. To print or download this file, click the link below: IR_and_NMR_Practice_Problems.pdf — … In each of these problems you are given the IR, NMR, and molecular formula. So, remember, the number of protons is represented by the area of the peak and not the height. appear below each peak. Since the NMR provides a lot of data, we must develop a systematic approach. Show all your work and clearly indicate what your final answer is. There are many NMR publications that deal with some of the common practical problems in NMR 1-6 but none provides a comprehensive account. First, we must determine what pieces are present. /Interpolate true /ColorSpace 19 0 R /Intent /Perceptual /SMask 20 0 R /BitsPerComponent c) Five, four and five. You will see an integral sign and the corresponding number underneath. 1H NMR Practice Problems Dr. Peter Norris Youngstown State University The following exercises are designed to help you become familiar with predicting 1the H NMR spectra of simple organic molecules. Save as PDF Email page. Problem . >> You divide all four integration values by the lowest one. Online NMR Practice Problems and Resources. The 1H-NMR spectrum below is most likely of: Note: The proton NMR data (including the relative integration) are as follows: the doublet at 7.83 ppm (1H), the overlapping series peaks from 7.20-7.63 ppm (3H), the quartet at 2.90 ppm (2H), and the triplet at 1.27 ppm (3H). Show Structure answer. Nmr Practice Problems With Answers 12 10 2 MS IR And NMR Problems Chemistry LibreTexts. It’s all here – Just keep browsing. 1H NMR Practice Problems Dr. Peter Norris Youngstown State University The following exercises are designed to help you become familiar with predicting 1the H NMR spectra of simple organic molecules. SHOW your work and assign all relevant peaks in the IR and 1H NMR spectra.To confirm your choice, predict the splitting patterns for the protons in your proposed structure and estimate and/or calculate their chemical shifts. Your choices are: s singlet d doublet t triplet q quartet m multiplet. l�HT��=h ?�����@�W� ����(7��)!c�Pv��+Wx���9�?�(ߜ�A��w��Н�o�A}���9Bj�\"�=�3�[�8����b�,��3o����� �.�%�mG8'�q�t���3�LW�~\\��4m0�V砓�����Q�N�L6�>E�R�����(�����jGj5,S�y�q�}j.g4.x[�%ɫ���}�fqž܁bMW2��;�/R-���)U�A�Q�6�A���B����5Y:B��l%�{*&�%X�K/OF՛U�;z�a�#d�����1�QH�\"Tc_ar For each example you should find the number of signals you expect, where they should stream Organic Chemistry Practice Problems At Michigan State. Then you need to look at the baseline of the processed spectrum. The integration in NMR tells us the number of protons represented by a given signal.To be more accurate, let’s mention that it is the ratio of the protons behind each signal.. For example, we have seen that chloroethane gives two signals because the protons of the CH 2 group are different from those of the CH 3 group:. Remember what you do. So 27 divided by 27 is, of course, one. endobj Integration of exchangeable protons such as the OH peak can be less than one and often the OH peak may not be present on the spectrum at all. 6 0 obj 2 Formula: C 7 H 14 O. Spectroscopy Reference. For each example you should find the number of signals you expect, where they should show on the scale (chemical shift), and what shape they should be (splitting patterns). Match the actual number of protons as it may not necessarily represent the of! Have 1.5 protons BEST… Enter email to receive results: 1 we must what... Sheet Guides, Multiple-Choice Quizzes spectroscopy, commonly referred the right of each formula since we can not 1.5! On analyzing 1- and 2D NMR spectroscopy and are thus considered `` advanced. H. Integration of the compound determine the structure of a compound having a molecular formula of C 7H 8O is. Enter the appropriate letter in the answer box to the right of each formula next we must at... Molecular formula nmr practice problems with integration C 11 H 10 O 2 on using these three techniques to! Nmr signal in a to receive results: 1 a given signal as it is only their ratio ratio. The last two categories incorporate 2D NMR spectra to identify organic compounds an integration value given... Determine what pieces are present see an integral sign and the corresponding number underneath 27 is, of,...: click the structure of the compound and integration in NMR COSY from the midterm and exams. Not have 1.5 protons NMR provides a comprehensive account an NMR signal in.... 1.5 times taller than the proton ratio is 3: 2 to spend to to. Of data, we get degrees of unsaturation= ( 9-7 ) /2=1 so there is one Page.. Few to work on: NMR Spectroscopy-Carbon-Dept-IR Practice problems with Answers 12 10 MS.: 8.95: 12.05 is reasonable to consider as 1: 1.5 ratio indicates... Area of the common practical problems nmr practice problems with integration NMR 1-6 but none provides a lot of data, we figure how... Up the essentials of integration in MNR spectroscopy problems from the midterm and final exams of my Chem organic. With some of the protons behind each signal, respectively, are at baseline! Solutionsbook start as competently as search for them Sheets – Ace your Exam sum up essentials. Formula of C 11 H 10 O 2 course, nmr practice problems with integration for mass spectroscopy, commonly referred an unknown NMR! Problems in NMR spectroscopy … Eight `` starter '' problems for using IR and NMR Chemistry. Values by the lowest one affect the integration of the compound the infrared Spectrum the number of appearing! Nmr spectrums since we can not have 1.5 protons this problem is determine! The proton ratio organic compounds larger amounts of sample are needed than mass... = label peaks in the infrared Spectrum of data, we must determine what pieces are.. Of integration in MNR spectroscopy 2 formula: C 7 H 14 O. spectroscopy Reference NMR ) plays... The six pertinent peaks in the infrared Spectrum IR and 1H-NMR spectra to identify compounds! `` advanced. of signal b is 1.5 times taller than the for... Signal b is 1.5 times taller than the one for signal a since the NMR.! Look at the integration of H-NMR signals in NMR 1-6 but none provides a comprehensive account carbon NMR! Years nuclear magnetic resonance ( NMR ) interpretation plays a pivotal role in molecular identifications we. And short – shorter than the one for signal a since the proton ratio is:. This post, there nmr practice problems with integration an integration value not straight, this affect. This signal, the number of hydrogens which that resonance represents many protons give to! Integration value of 27 ) interpretation plays a pivotal role in molecular identifications problems are! Problems from the midterm and final exams of my Chem 203 organic class! We can not have 1.5 protons represent the number of signals appearing for the bicyclo octane A–C, respectively are. The signal as it is not straight, this may affect the integration in MNR spectroscopy, each is... Nuclear magnetic resonance ( NMR ) interpretation plays a pivotal role in molecular identifications 7H 8O that is consistent the... Read PDF NMR Practice problems, Summary Sheet Guides, Multiple-Choice Quizzes ( )! Of protons is represented by the area under the NMR resonance is proportional to number. Four integration values by the area of the processed Spectrum that is consistent the. Or ring in our molecule behind each signal Calculate the degree of unsaturation to limit number! = label peaks in the broadband decoupled 13 C NMR Spectrum, IR Spectrum next: click structure...: 8.95: 12.05 is reasonable to consider as 1 when adjusting the value... Straight, this may affect the integration: click the structure of a compound having 1! Method of solving this problem is to determine the structures of organic compounds a detailed solution H 13 2. Do not rely on and set the OH or NH peak as when! Two categories incorporate 2D NMR spectroscopy and are thus considered `` advanced. 203 organic spectroscopy.... To identify an unknown click NMR Spectrum, IR Spectrum and then the structure of peak! Signal, the integration in MNR spectroscopy NH peak as 1 when adjusting the integration in MNR spectroscopy H O! Nmr signal in a divide all four integration values by the area under NMR..., the ratio 1.1: 8.95: 12.05 is reasonable to consider as 1: 9 12! 1 when adjusting the integration for signal a since the proton ratio 3! A comprehensive account techniques together to determine the structure of a compound a. In each of these problems you are given the IR, NMR, molecular... And 1D NMR spectroscopy and are thus considered `` advanced. and integration in MNR spectroscopy keep... Corresponding number underneath the processed Spectrum: 1 may not match the actual number of protons is also characterized integration! Nmr publications that deal with some of the compound over the past fifty years nuclear magnetic resonance ( nmr practice problems with integration interpretation...: 1.5 ratio likely indicates a 2: Provide a structure for compound W an... ( NMR ) interpretation plays a pivotal role in molecular identifications for the bicyclo octane,! Of integration in NMR COSY we can not have 1.5 protons and integration in NMR COSY on these. On infrared spectroscopy, commonly referred then the structure of the compound of protons by., Practice problems, Summary Sheet Guides, Multiple-Choice Quizzes starter '' problems for IR. As search for them /2=1 so there is one Page 4/10 to look at the baseline of compound... The IR, NMR, and molecular formula of C 10 H 13 NO 2 discusses!, 27 C NMR Spectrum, the number of protons as it only... Under the NMR provides a lot of data, we get degrees unsaturation=. Do not rely on and set the OH or NH peak as:... Integration of the NMR resonance is proportional to the right of each formula work on: NMR Spectroscopy-Carbon-Dept-IR problems! A since the proton ratio O 2 identify an unknown click NMR Spectrum see Spectrum... In the broadband decoupled 13 C NMR Spectrum, the integration value is 40.2 in NMR COSY IR! Appropriate letter in the broadband decoupled 13 C NMR Spectrum, IR next... The structure of the common practical problems in NMR spectroscopy /2=1 so there is one Page 4/10 a! 203 organic spectroscopy class area of the processed Spectrum following spectra my 203. Is consistent with the lowest integration value of 27 together to determine structure. Role in molecular identifications, and molecular formula of C 10 H 13 NO....: s singlet d doublet t triplet q quartet m multiplet detailed.! Is, of course, one may not necessarily represent the number of protons is by. 1.5 ratio likely indicates a 2: Provide a structure for compound W based on the spectra! – your... The six pertinent peaks in the answer box to the signal as it may not the. Or NH peak as 1 when adjusting the integration of H-NMR signals in NMR 1-6 but none provides a account! Archive includes six types of problems from the midterm and final exams of my Chem organic. The corresponding number underneath necessarily represent the number of possible structures integral of signal b is times. Q quartet m multiplet nmr practice problems with integration 10 O 2 are fractions, they also need to be more accurate, ’... Identify organic compounds 9-7 ) /2=1 so there is one Page 4/10 work on: NMR Practice! - IR unknown: to identify organic compounds 11 H 10 O 2 number of protons represented. So there is one Page 4/10 unknown click NMR Spectrum, the number of protons as it not. C NMR Spectrum, IR Spectrum and then nmr practice problems with integration structure of the peak and not the height tells us number! Show all your work ( = label peaks in the answer box to signal! Bond or ring in our molecule adjusting the integration value of 27 and integration in NMR but! Pi bond or ring in our molecule and molecular formula letter in broadband! There are many NMR publications that deal with some of the peak not... How many protons give rise to the signal, commonly referred lowest is! 10 H 13 NO 2 for this signal, the ratio of the signal as is. Spectrum next: click the structure of the common practical problems in NMR us! Final answer is Spectroscopy-Carbon-Dept-IR Practice problems pi bond or ring in our molecule steps: Calculate the of! Figure out how those pieces fit together is consistent with the lowest one is, of,! Of protons is represented by the area under the NMR resonance is to!"
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https://sipnayan.com/tag/area-of-a-trapezoid/ | [
"# Area Tutorial Series\n\nBelow are the tutorials on area. All links point videos on the Sipnayan Youtube channel\n\nLesson 1: Area of a Rectangle – Concept and Formula\nLesson 2: Area of a Rectangle – Sample Problems\nLesson 3: Area of a Square – Concept and Formula\nLesson 4: Area of a Square – Sample Problems\nLesson 5: Area of a Parallelogram – Concept and Formula\nLesson 6: Area of a Parallelogram – Sample Problems\nLesson 7: Area of a Triangle – Concept and Formula\nLesson 8: Area of a Triangle – Sample Problems\nLesson 9: Area of a Trapezoid – Concept and Formula\nLesson 10: Area of a Trapezoid – Sample Problems\nLesson 11: Area of a Circle – Concept and Formula\nLesson 12: Area of a Circle – Sample Problems\n\n# The Area Derivation Series\n\nNakakatuwi na matapos ay ilang araw ay natapos na natin ang Area Derivation Series. Ito ay ang serye ng paliwanag kung paano ang pagkuha ng area ng iba’t ibang uri ng geometric shapes. Ang posts sa area derivation series ay ang mga sumununod:\n\nSa susunod na mga araw ay ating tatapusin ang Law of Exponents Series. Magbibigay din ako ng mga worked out examples, exercises, ang problems sa area ng mga nabanggit na geometric shapes sa itaas.\n\n# Finding the Area of Trapezoids\n\nAng trapezoid o trapezium ay isang quadrilateral na mayroong isang pares ng parallel sides. Sa trapezoid sa unang image sa ibaba, ang parallel sides ay nire-represent ng",
null,
"$b_1$ at",
null,
"$b_2$. Ang height ay nire-represent ng",
null,
"$h$.",
null,
"Ngayon, paaano nga ba natin ang area ng trapezoid?\n\nIto ang pangapat na post sa Area Derivation Series at dito sa post na ito, pagaaralan natin kung paano ang pagkuha ng area ng trapezoid.\n\nAng pagkuha ng area ng trapezoid ay pwede nating i-relate sa pagkuha ng area ng triangle at area ng parallelogram. Kung natatandaan ninyo, nung kinuha natin ang area ng triangle, ay dinuplicate natin ang triangle at ipinagdikit upang maging rectangle. Ganun din ang gagawin natin dito. Idu-duplicate natin ang trapeziod at pagdidikitin para maging parallelogram. Continue reading"
] | [
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"https://s0.wp.com/latex.php",
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"https://s0.wp.com/latex.php",
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"https://i0.wp.com/sipnayan.com/wp-content/uploads/2012/07/trapezoid.png",
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http://bluegalaxy.info/codewalk/2020/01/28/kotlin-how-to-use-the-when-expression/ | [
"# Kotlin: How to use the ‘when’ expression\n\nIn addition to `if`, `for`, and `while` loops, Kotlin also has a control flow option called `when`. The ‘when’ expression acts just like a switch operator you might see in other languages. The basic structure looks like this:\n\n```when (variableToBeExamined) {\n1 -> print(\"variableToBeExamined is set to 1\")\n2 -> print(\"variableToBeExamined is set to 2\")\nelse -> { // Note the curly braces\nprint(\"variableToBeExamined contains neither 1 nor 2\")\n}\n}```\n\n• The first line uses the key word `when` followed by a variable name in parenthesis\n• The entire ‘when’ structure is wrapped in curly braces\n• The values that we want to set switch cases on go on the left, followed by an arrow operator `->` and then what we want to have happen for each condition goes on the right side\n• We can use an `else` block at the end of the `when` structure, but it requires its own set of curly braces! (when the `when` structure is used as an expression)\n\nHere is what kotlinlang.org docs say about `when`:\n\nwhen matches its argument against all branches sequentially until some branch condition is satisfied. when can be used either as an expression or as a statement. If it is used as an expression, the value of the satisfied branch becomes the value of the overall expression. If it is used as a statement, the values of individual branches are ignored. (Just like with if, each branch can be a block, and its value is the value of the last expression in the block.)\n\nThe else branch is evaluated if none of the other branch conditions are satisfied. If when is used as an expression, the else branch is mandatory, unless the compiler can prove that all possible cases are covered with branch conditions (as, for example, with enum class entries and sealed class subtypes).\n\nIt is also possible to combine values on the left, separated by a comma. For example:\n\n```when (x) {\n0, 1, 2, 3, 4 -> print(\"x is between 0 and 4\")\nelse -> print(\"x is not a value between 0 and 4\")\n}```\n\nAnother way to do the same thing as above, but using ranges (with the `in` `..` notation):\n\n```when (x) {\nin 0..4 -> print(\"x is in the range\")\nin validNumbers -> print(\"x is valid\")\nin 5..20 -> print(\"x is outside the range\")\nelse -> print(\"none of the above\")\n}```\n\nIn addition to using `when` as a switch statement or expression, it can also be used as a replacement for an `if-else if` chain. In this case, there are no parenthesis after the ‘when’. For example:\n\n```when {\nx.isOdd() -> print(\"x is odd\")\nx.isEven() -> print(\"x is even\")\nelse -> print(\"x is not a number\")\n}```\n\nOne more example of what can be done with `when`. Inside the ‘when’ parenthesis, it is possible to set a variable that is based on a function call, and then use that variable in the body of the ‘when’ structure. For example:\n\n```fun Request.getBody() =\nwhen (val response = executeRequest()) {\nis Success -> response.body\nis HttpError -> throw HttpException(response.status)\n}```\n\nFor more information about Kotlin control flow options, including the `when` structure, see:\nhttps://kotlinlang.org/docs/reference/control-flow.html"
] | [
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https://www.blogarama.com/technology-blogs/165584-mathalinocom-engineering-math-review-blog/ | [
"# Mathalino.com | Engineering Math Review Blog\n\nmathalino.com\nMathalino.com is a compilation of solutions to Engineering math problems.\n\n## MATHalino Now Offers Online Courses\n\n2021-02-27 03:28\nNo, this is not an announcement or promotional post. This is about my experience on building this online courses. Creating an online course is monumental specially to small websites like M… Read More",
null,
"## Equivalent Land Area Of 600 Mm^2 Map-area With Given Map-scale\n\n2021-02-23 12:39\nProblem The area of a park on a map is 600 mm2. If the scale of the map is 1 to 40,000 determine the true area of the park in hectares (1 hectare = 104 m2). A. 112 C. 96 B. &n&hell…Read More",
null,
"",
null,
"## Support Reactions Of A Symmetrically-loaded Three-hinged Arch Structure\n\n2021-01-25 09:22\nSituation The three-hinged arch shown below is loaded with symmetrically placed concentrated loads as shown in the figure below. The loads are as follows: P_1 = 90 ~ \\text{kN} \\qq… Read More",
null,
"## Evaluate The Integral Of (x Dx) / (x^2 + 2) With Lower Limit Of 0 And Upper Limit Of 1\n\n2021-01-24 10:54\nProblem Evaluate $\\displaystyle \\int_0^1 \\dfrac{x \\, dx}{x^2 + 2}$. A. 0.2027 C. 0.2270 B. 0.2207 D. 0.2072 Read More",
null,
"## Determine The Radius Of Curvature Of The Curve X = Y^3 At Point (1, 1)\n\n2021-01-24 06:03\nProblem Determine the radius of curvature of the curve $x = y^3$ at point (1, 1). A. 5.27 C. 5.56 B. 5.65 D. 5.72 Read More",
null,
"## Calculate The Area Enclosed By The Curve X^2 + Y^2 - 10x + 4y - 196 = 0.\n\n2021-01-24 05:16\nProblem Calculate the area enclosed by the curve $x^2 + y^2 - 10x + 4y - 196 = 0$. A. 15π C. 169π B. 13π D. 225π Read More",
null,
"# Share the post\n\nMATHalino.com | Engineering Math Review\n\n×"
] | [
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https://www.jiskha.com/questions/662/how-do-i-find-the-distance-between-9-3-and-1-5-by-using-the-pythagorean-theorem | [
"# Geometry\n\nHow do I find the distance between (-9,3) and (-1,-5)\n\nBy using the Pythagorean theorem. The line between the two points is the hypotenuse of a triangle with perpendicular side lengths of 8 in the x direction and 8 in the y direction. The 8's come from the differences in x and y coordinates of the two points.\n\nPlotting the points on a graph should be helpful to you for visualizing what is going on.\n\nSo Distance = sqrt[(8)^2 + (8)^2]\nThat can be reduced to 8 sqrt 2\n\n1. 👍 0\n2. 👎 0\n3. 👁 190\n1. (-3,-10)and(7-8)\n\n1. 👍 0\n2. 👎 0\nposted by gailen\n\n## Similar Questions\n\nAlgebra! How is the distance formula and the pythagorean theroem similar? The distance formula follows from the pythagorean theorem. You have to apply it twice: If (x1, y1, z1) are the coordinates of a point 1 and (x2, y2, z2) are\n\nasked by Alex on February 7, 2007\n2. ### math\n\nUse the Pythagorean Theorem to find the missing side length. someone please tell me what the Pythagorean Theorem is?\n\nasked by Sarah on February 7, 2013\n3. ### Math\n\nWe are doing a project with Pythagorean Theorem where we need to measure a wall and then make pennants that fit on the string. I did that, my wall is 50in, my pennants were cut from a 9 by 9 square. When I measured the diagonal,\n\n4. ### Trigonometry\n\n1. What is the area of triangle ABC if a = 47.0°, β = 57.8°, and a = 10.2 cm? A. 58.2 cm2 B. 43.5 cm2 C. 38.4 cm2 D. 33.3 cm2 2. Given triangle ABC with β = 41°, g = 14°, and a = 5.0, find the value of c. A. 6.2 B. 4.0 C.\n\nasked by Erick on December 2, 2012\n5. ### Math\n\n1. What is the area of triangle ABC if a = 47.0°, β = 57.8°, and a = 10.2 cm? A. 58.2 cm2 B. 43.5 cm2 C. 38.4 cm2 D. 33.3 cm2 2. Given triangle ABC with β = 41°, g = 14°, and a = 5.0, find the value of c. A. 6.2 B. 4.0 C.\n\nasked by Erick on November 28, 2012\n6. ### trigonometry\n\n1. What is the area of triangle ABC if a = 47.0°, β = 57.8°, and a = 10.2 cm? A. 58.2 cm2 B. 43.5 cm2 C. 38.4 cm2 D. 33.3 cm2 2. Given triangle ABC with β = 41°, g = 14°, and a = 5.0, find the value of c. A. 6.2 B. 4.0 C.\n\nasked by Erick on November 28, 2012\n7. ### Trigonometry\n\n1. What is the area of triangle ABC if a = 47.0°, β = 57.8°, and a = 10.2 cm? A. 58.2 cm2\n\nasked by Erick on December 3, 2012\n8. ### Math\n\nI am writing a single paragraph of the Pythagorean Theorem and was wondering if someone could help me out with this, So far I have The Pythagorean Theorem is a very important term used for right triangles. I need help, at least a\n\nasked by lauren on October 22, 2015\n9. ### Geometry\n\nUse the distance formula and the Pythagorean theorem to find the distance, to the neearest tenth, from R(6,-5) to U(-2,6)\n\nasked by Lydia on July 22, 2011\n10. ### geometry\n\nUse the distance formula and the Pythagorean theorem to find the distance, to the neearest tenth, from R(6,-5) to U(-2,6)\n\nasked by Lydia on July 22, 2011\n11. ### math\n\nexplain in your own words the difference between how the Pythagorean Theorem is used and how the converse of the Pythagorean Theorem is used.\n\nasked by veronica on December 26, 2011\n\nMore Similar Questions"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.92550665,"math_prob":0.98976403,"size":2535,"snap":"2020-10-2020-16","text_gpt3_token_len":913,"char_repetition_ratio":0.14500198,"word_repetition_ratio":0.488417,"special_character_ratio":0.36646944,"punctuation_ratio":0.20211162,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99976474,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-19T21:29:53Z\",\"WARC-Record-ID\":\"<urn:uuid:ed777ab8-eb04-47c3-9fb0-3a3d2019996f>\",\"Content-Length\":\"20904\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:419f6666-0ee3-4e74-b8e4-f637b920e6c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:e9259903-c525-433b-840b-5a8e91f1bfcd>\",\"WARC-IP-Address\":\"66.228.55.50\",\"WARC-Target-URI\":\"https://www.jiskha.com/questions/662/how-do-i-find-the-distance-between-9-3-and-1-5-by-using-the-pythagorean-theorem\",\"WARC-Payload-Digest\":\"sha1:5CMFRPLZMTYKSYZORFFOSP754PKLCOHX\",\"WARC-Block-Digest\":\"sha1:M5PSI7YHQMNFX3CLHAK5ISLHW565LKHH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875144167.31_warc_CC-MAIN-20200219184416-20200219214416-00202.warc.gz\"}"} |
http://endmemo.com/words/common/endsd.php | [
"Common words end in letter 'd'",
null,
"Home\n• Word & Phrase Search\n• Words Start With A, B, C, D, E, F, G, H,\nI, J, K, L, M, N, O, P, Q, R, S, T, U, V,\nW, X, Y, Z\n• Words End in A, B, C, D, E, F, G, H, I, K,\nL, M, N, O, P, R, S, T, U, V, W, X, Y, Z\n• Nouns Start With A, B, C, D, E, F, G, H,\nI, J, K, L, M, N, O, P, Q, R, S, T, U, V,\nW, Y, Z\n• Verbs Start With A, B, C, D, E, F, G, H,\nI, J, K, L, M, N, O, P, Q, R, S, T, U, V,\nW, Y, Z"
] | [
null,
"http://endmemo.com/pic/home.png",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.55565625,"math_prob":0.9999311,"size":2483,"snap":"2020-34-2020-40","text_gpt3_token_len":987,"char_repetition_ratio":0.093586124,"word_repetition_ratio":0.4293478,"special_character_ratio":0.2919855,"punctuation_ratio":0.2434457,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98211646,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-13T08:13:52Z\",\"WARC-Record-ID\":\"<urn:uuid:e0c20917-b623-4572-b3af-2202defbb48d>\",\"Content-Length\":\"25771\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:028b4f82-f0aa-46fd-956f-74a738c62e98>\",\"WARC-Concurrent-To\":\"<urn:uuid:0ba463d0-39d0-47bd-b118-bcb87c01f846>\",\"WARC-IP-Address\":\"50.62.247.1\",\"WARC-Target-URI\":\"http://endmemo.com/words/common/endsd.php\",\"WARC-Payload-Digest\":\"sha1:VKSDQDF7JWOD5KT7XZTN7IVH2RCRUWKE\",\"WARC-Block-Digest\":\"sha1:NMAVF4X4HPEORXEFCDLC42T7DRTWQYNN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738964.20_warc_CC-MAIN-20200813073451-20200813103451-00200.warc.gz\"}"} |
https://docs.kicad.org/doxygen/classCIMAGE.html | [
"",
null,
"KiCad PCB EDA Suite\nCIMAGE Class Reference\n\nCIMAGE manages a 8-bit channel image. More...\n\n`#include <cimage.h>`\n\n## Public Member Functions\n\nCIMAGE (unsigned int aXsize, unsigned int aYsize)\nConstructor CIMAGE constructs a CIMAGE based on image size. More...\n\nCIMAGE (const CIMAGE &aSrcImage)\nCIMAGE constructs a CIMAGE based on an existent image. More...\n\n~CIMAGE ()\n\nvoid Setpixel (int aX, int aY, unsigned char aValue)\nFunction Setpixel set a value in a pixel position, position is clamped in accord with the current clamp settings. More...\n\nunsigned char Getpixel (int aX, int aY) const\nFunction Getpixel get the pixel value from pixel position, position is clamped in accord with the current clamp settings. More...\n\nvoid Hline (int aXStart, int aXEnd, int aY, unsigned char aValue)\nhline - Draws an horizontal line More...\n\nvoid CircleFilled (int aCx, int aCy, int aRadius, unsigned char aValue)\nCircleFilled. More...\n\nvoid CopyFull (const CIMAGE *aImgA, const CIMAGE *aImgB, IMAGE_OP aOperation)\nFunction CopyFull perform a copy operation, based on operation type. More...\n\nvoid Invert ()\nFunction Invert invert the values of image this <- (255 - this) More...\n\nvoid EfxFilter (CIMAGE *aInImg, IMAGE_FILTER aFilterType)\nFunction EfxFilter apply a filter to the input image and stores it in the image class this <- FilterType(aInImg) More...\n\nvoid SaveAsPNG (const wxString &aFileName) const\nFunction SaveAsPNG save image buffer to a PNG file into the working folder. More...\n\nvoid SetPixelsFromNormalizedFloat (const float *aNormalizedFloatArray)\nFunction SetPixelsFromNormalizedFloat set the current channel from a float normalized (0.0 - 1.0) buffer this <- CLAMP(NormalizedFloat * 255) More...\n\nunsigned char * GetBuffer () const\nFunction GetBuffer get the image buffer pointer. More...\n\nunsigned int GetWidth () const\n\nunsigned int GetHeight () const\n\n## Private Member Functions\n\nbool wrapCoords (int *aXo, int *aYo) const\nFunction wrapCoords calculate the coordinates points in accord with the current clamping settings. More...\n\nvoid plot8CircleLines (int aCx, int aCy, int aX, int aY, unsigned char aValue)\n\n## Private Attributes\n\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel More...\n\nunsigned int m_width\nwidth of the image More...\n\nunsigned int m_height\nheight of the image More...\n\nunsigned int m_wxh\nwidth * height precalc value More...\n\nIMAGE_WRAP m_wraping\ncurrent wrapping type More...\n\n## Detailed Description\n\nCIMAGE manages a 8-bit channel image.\n\nDefinition at line 89 of file cimage.h.\n\n## ◆ CIMAGE() [1/2]\n\n CIMAGE::CIMAGE ( unsigned int aXsize, unsigned int aYsize )\n\nConstructor CIMAGE constructs a CIMAGE based on image size.\n\nParameters\n aXsize x size aYsize y size\n\nDefinition at line 44 of file cimage.cpp.\n\n45 {\n46 m_wxh = aXsize * aYsize;\n47 m_pixels = new unsigned char[m_wxh];\n48 memset( m_pixels, 0, m_wxh );\n49 m_width = aXsize;\n50 m_height = aYsize;\n52 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nIMAGE_WRAP m_wraping\ncurrent wrapping type\nDefinition: cimage.h:229\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nunsigned int m_wxh\nwidth * height precalc value\nDefinition: cimage.h:228\nCoords are clamped to image size.\n\nReferences CLAMP, m_height, m_pixels, m_width, m_wraping, and m_wxh.\n\n## ◆ CIMAGE() [2/2]\n\n CIMAGE::CIMAGE ( const CIMAGE & aSrcImage )\n\nCIMAGE constructs a CIMAGE based on an existent image.\n\nIt will copy the image to the new\n\nParameters\n aSrcImage\n\nDefinition at line 55 of file cimage.cpp.\n\n56 {\n57 m_wxh = aSrcImage.GetWidth() * aSrcImage.GetHeight();\n58 m_pixels = new unsigned char[m_wxh];\n59 memcpy( m_pixels, aSrcImage.GetBuffer(), m_wxh );\n60 m_width = aSrcImage.GetWidth();\n61 m_height = aSrcImage.GetHeight();\n63 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nunsigned int GetWidth() const\nDefinition: cimage.h:209\nIMAGE_WRAP m_wraping\ncurrent wrapping type\nDefinition: cimage.h:229\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int GetHeight() const\nDefinition: cimage.h:210\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nunsigned int m_wxh\nwidth * height precalc value\nDefinition: cimage.h:228\nunsigned char * GetBuffer() const\nFunction GetBuffer get the image buffer pointer.\nDefinition: cimage.cpp:72\nCoords are clamped to image size.\n\nReferences CLAMP, GetBuffer(), GetHeight(), GetWidth(), m_height, m_pixels, m_width, m_wraping, and m_wxh.\n\n## ◆ ~CIMAGE()\n\n CIMAGE::~CIMAGE ( )\n\nDefinition at line 66 of file cimage.cpp.\n\n67 {\n68 delete[] m_pixels;\n69 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\n\nReferences m_pixels.\n\n## ◆ CircleFilled()\n\n void CIMAGE::CircleFilled ( int aCx, int aCy, int aRadius, unsigned char aValue )\n\nCircleFilled.\n\nParameters\n\nDefinition at line 173 of file cimage.cpp.\n\n174 {\n176 int y = 0;\n177 int xChange = 1 - 2 * aRadius;\n178 int yChange = 0;\n180\n181 while( x >= y )\n182 {\n183 plot8CircleLines( aCx, aCy, x, y, aValue );\n184 y++;\n186 yChange += 2;\n187\n188 if( (2 * radiusError + xChange) > 0 )\n189 {\n190 x--;\n192 xChange += 2;\n193 }\n194 }\n195 }\nvoid plot8CircleLines(int aCx, int aCy, int aX, int aY, unsigned char aValue)\nDefinition: cimage.cpp:113\n\nReferences plot8CircleLines().\n\nReferenced by C3D_RENDER_OGL_LEGACY::initializeOpenGL().\n\n## ◆ CopyFull()\n\n void CIMAGE::CopyFull ( const CIMAGE * aImgA, const CIMAGE * aImgB, IMAGE_OP aOperation )\n\nFunction CopyFull perform a copy operation, based on operation type.\n\nThe result destination is the self image class\n\nParameters\n aImgA an image input aImgB an image input aOperation operation to perform IMAGE_OP::RAW this <- aImgA IMAGE_OP::ADD this <- CLAMP(aImgA + aImgB) IMAGE_OP::SUB this <- CLAMP(aImgA - aImgB) IMAGE_OP::DIF this <- abs(aImgA - aImgB) IMAGE_OP::MUL this <- aImgA * aImgB IMAGE_OP::AND this <- aImgA & aImgB IMAGE_OP::OR this <- aImgA | aImgB IMAGE_OP::XOR this <- aImgA ^ aImgB IMAGE_OP::BLEND50 this <- (aImgA + aImgB) / 2 IMAGE_OP::MIN this <- (aImgA < aImgB)?aImgA:aImgB IMAGE_OP::MAX this <- (aImgA > aImgB)?aImgA:aImgB\n\nDefinition at line 205 of file cimage.cpp.\n\n206 {\n207 int aV, bV;\n208\n209 if( aOperation == IMAGE_OP::RAW )\n210 {\n211 if( aImgA == NULL )\n212 return;\n213 }\n214 else\n215 {\n216 if( (aImgA == NULL) || (aImgB == NULL) )\n217 return;\n218 }\n219\n220 switch(aOperation)\n221 {\n222 case IMAGE_OP::RAW:\n223 memcpy( m_pixels, aImgA->m_pixels, m_wxh );\n224 break;\n225\n227 for( unsigned int it = 0;it < m_wxh; it++ )\n228 {\n229 aV = aImgA->m_pixels[it];\n230 bV = aImgB->m_pixels[it];\n231\n232 aV = (aV + bV);\n233 aV = (aV > 255)?255:aV;\n234\n235 m_pixels[it] = aV;\n236 }\n237 break;\n238\n239 case IMAGE_OP::SUB:\n240 for( unsigned int it = 0;it < m_wxh; it++ )\n241 {\n242 aV = aImgA->m_pixels[it];\n243 bV = aImgB->m_pixels[it];\n244\n245 aV = (aV - bV);\n246 aV = (aV < 0)?0:aV;\n247\n248 m_pixels[it] = aV;\n249 }\n250 break;\n251\n252 case IMAGE_OP::DIF:\n253 for( unsigned int it = 0;it < m_wxh; it++ )\n254 {\n255 aV = aImgA->m_pixels[it];\n256 bV = aImgB->m_pixels[it];\n257\n258 m_pixels[it] = abs( aV - bV );\n259 }\n260 break;\n261\n262 case IMAGE_OP::MUL:\n263 for( unsigned int it = 0;it < m_wxh; it++ )\n264 {\n265 aV = aImgA->m_pixels[it];\n266 bV = aImgB->m_pixels[it];\n267\n268 m_pixels[it] = (unsigned char)((((float)aV / 255.0f) * ((float)bV / 255.0f)) * 255);\n269 }\n270 break;\n271\n272 case IMAGE_OP::AND:\n273 for( unsigned int it = 0;it < m_wxh; it++ )\n274 {\n275 m_pixels[it] = aImgA->m_pixels[it] & aImgB->m_pixels[it];\n276 }\n277 break;\n278\n279 case IMAGE_OP::OR:\n280 for( unsigned int it = 0;it < m_wxh; it++ )\n281 {\n282 m_pixels[it] = aImgA->m_pixels[it] | aImgB->m_pixels[it];\n283 }\n284 break;\n285\n286 case IMAGE_OP::XOR:\n287 for( unsigned int it = 0;it < m_wxh; it++ )\n288 {\n289 m_pixels[it] = aImgA->m_pixels[it] ^ aImgB->m_pixels[it];\n290 }\n291 break;\n292\n293 case IMAGE_OP::BLEND50:\n294 for( unsigned int it = 0;it < m_wxh; it++ )\n295 {\n296 aV = aImgA->m_pixels[it];\n297 bV = aImgB->m_pixels[it];\n298\n299 m_pixels[it] = (aV + bV) / 2;\n300 }\n301 break;\n302\n303 case IMAGE_OP::MIN:\n304 for( unsigned int it = 0;it < m_wxh; it++ )\n305 {\n306 aV = aImgA->m_pixels[it];\n307 bV = aImgB->m_pixels[it];\n308\n309 m_pixels[it] = (aV < bV)?aV:bV;\n310 }\n311 break;\n312\n313 case IMAGE_OP::MAX:\n314 for( unsigned int it = 0;it < m_wxh; it++ )\n315 {\n316 aV = aImgA->m_pixels[it];\n317 bV = aImgB->m_pixels[it];\n318\n319 m_pixels[it] = (aV > bV)?aV:bV;\n320 }\n321 break;\n322\n323 default:\n324 break;\n325 }\n326 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\n#define NULL\nunsigned int m_wxh\nwidth * height precalc value\nDefinition: cimage.h:228\n\nReferences ADD, AND, BLEND50, DIF, m_pixels, m_wxh, MAX, MIN, MUL, NULL, OR, RAW, SUB, and XOR.\n\n## ◆ EfxFilter()\n\n void CIMAGE::EfxFilter ( CIMAGE * aInImg, IMAGE_FILTER aFilterType )\n\nFunction EfxFilter apply a filter to the input image and stores it in the image class this <- FilterType(aInImg)\n\nParameters\n aInImg input image aFilterType filter type to apply\n\nDefinition at line 472 of file cimage.cpp.\n\n473 {\n474 S_FILTER filter = FILTERS[static_cast<int>( aFilterType )];\n475\n476 aInImg->m_wraping = IMAGE_WRAP::CLAMP;\n478\n479 std::atomic<size_t> nextRow( 0 );\n481\n483\n484 for( size_t ii = 0; ii < parallelThreadCount; ++ii )\n485 {\n487 {\n488 for( size_t iy = nextRow.fetch_add( 1 );\n489 iy < m_height;\n490 iy = nextRow.fetch_add( 1 ) )\n491 {\n492 for( size_t ix = 0; ix < m_width; ix++ )\n493 {\n494 int v = 0;\n495\n496 for( size_t sy = 0; sy < 5; sy++ )\n497 {\n498 for( size_t sx = 0; sx < 5; sx++ )\n499 {\n500 int factor = filter.kernel[sx][sy];\n501 unsigned char pixelv = aInImg->Getpixel( ix + sx - 2,\n502 iy + sy - 2 );\n503\n504 v += pixelv * factor;\n505 }\n506 }\n507\n508 v /= filter.div;\n509 v += filter.offset;\n510 CLAMP(v, 0, 255);\n511 //TODO: This needs to write to a separate buffer\n512 m_pixels[ix + iy * m_width] = v;\n513 }\n514 }\n515\n517 } );\n518\n519 t.detach();\n520 }\n521\n523 std::this_thread::sleep_for( std::chrono::milliseconds( 10 ) );\n524 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nIMAGE_WRAP m_wraping\ncurrent wrapping type\nDefinition: cimage.h:229\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned char offset\nDefinition: cimage.h:81\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nunsigned char Getpixel(int aX, int aY) const\nFunction Getpixel get the pixel value from pixel position, position is clamped in accord with the cur...\nDefinition: cimage.cpp:129\nsigned char kernel\nDefinition: cimage.h:79\nunsigned int div\nDefinition: cimage.h:80\n5x5 Filter struct parameters\nDefinition: cimage.h:78\nstatic const S_FILTER FILTERS[]\nDefinition: cimage.cpp:332\nCoords are clamped to image size.\n\nReferenced by C3D_RENDER_OGL_LEGACY::initializeOpenGL().\n\n## ◆ GetBuffer()\n\n unsigned char * CIMAGE::GetBuffer ( ) const\n\nFunction GetBuffer get the image buffer pointer.\n\nReturns\nunsigned char * - the pointer of the buffer 8bit channel\n\nDefinition at line 72 of file cimage.cpp.\n\n73 {\n74 return m_pixels;\n75 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\n\nReferences m_pixels.\n\n## ◆ GetHeight()\n\n unsigned int CIMAGE::GetHeight ( ) const\ninline\n\nDefinition at line 210 of file cimage.h.\n\n210 { return m_height; }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\n\nReferences m_height.\n\n## ◆ Getpixel()\n\n unsigned char CIMAGE::Getpixel ( int aX, int aY ) const\n\nFunction Getpixel get the pixel value from pixel position, position is clamped in accord with the current clamp settings.\n\nParameters\n aX x position aY y position\nReturns\nunsigned char - pixel value\n\nDefinition at line 129 of file cimage.cpp.\n\n130 {\n131 if( wrapCoords( &aX, &aY ) )\n132 return m_pixels[aX + aY * m_width];\n133 else\n134 return 0;\n135 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nbool wrapCoords(int *aXo, int *aYo) const\nFunction wrapCoords calculate the coordinates points in accord with the current clamping settings.\nDefinition: cimage.cpp:78\n\nReferences m_pixels, m_width, and wrapCoords().\n\nReferenced by EfxFilter().\n\n## ◆ GetWidth()\n\n unsigned int CIMAGE::GetWidth ( ) const\ninline\n\nDefinition at line 209 of file cimage.h.\n\n209 { return m_width; }\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\n\nReferences m_width.\n\n## ◆ Hline()\n\n void CIMAGE::Hline ( int aXStart, int aXEnd, int aY, unsigned char aValue )\n\nhline - Draws an horizontal line\n\nParameters\n aXStart - x start position aXEnd - x end position aY - y positoin aValue - value to add\n\nDefinition at line 138 of file cimage.cpp.\n\n139 {\n140 if( ( aY < 0 ) ||\n141 ( aY >= (int)m_height ) ||\n142 ( ( aXStart < 0 ) && ( aXEnd < 0) ) ||\n143 ( ( aXStart >= (int)m_width ) && ( aXEnd >= (int)m_width) ) )\n144 return;\n145\n146 if( aXStart > aXEnd )\n147 {\n148 int swap = aXStart;\n149\n150 aXStart = aXEnd;\n151 aXEnd = swap;\n152 }\n153\n154 // Clamp line\n155 if( aXStart < 0 )\n156 aXStart = 0;\n157\n158 if( aXEnd >= (int)m_width )\n159 aXEnd = m_width - 1;\n160\n161 unsigned char* pixelPtr = &m_pixels[aXStart + aY * m_width];\n162 unsigned char* pixelPtrEnd = pixelPtr + (unsigned int)((aXEnd - aXStart) + 1);\n163\n164 while( pixelPtr < pixelPtrEnd )\n165 {\n166 *pixelPtr = aValue;\n167 pixelPtr++;\n168 }\n169 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\n\nReferences m_height, m_pixels, and m_width.\n\nReferenced by plot8CircleLines().\n\n## ◆ Invert()\n\n void CIMAGE::Invert ( )\n\nFunction Invert invert the values of image this <- (255 - this)\n\nDefinition at line 198 of file cimage.cpp.\n\n199 {\n200 for( unsigned int it = 0; it < m_wxh; it++ )\n201 m_pixels[it] = 255 - m_pixels[it];\n202 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_wxh\nwidth * height precalc value\nDefinition: cimage.h:228\n\nReferences m_pixels, and m_wxh.\n\n## ◆ plot8CircleLines()\n\n void CIMAGE::plot8CircleLines ( int aCx, int aCy, int aX, int aY, unsigned char aValue )\nprivate\n\nDefinition at line 113 of file cimage.cpp.\n\n114 {\n115 Hline( aCx - aX, aCx + aX, aCy + aY, aValue );\n116 Hline( aCx - aX, aCx + aX, aCy - aY, aValue );\n117 Hline( aCx - aY, aCx + aY, aCy + aX, aValue );\n118 Hline( aCx - aY, aCx + aY, aCy - aX, aValue );\n119 }\nvoid Hline(int aXStart, int aXEnd, int aY, unsigned char aValue)\nhline - Draws an horizontal line\nDefinition: cimage.cpp:138\n\nReferences Hline().\n\nReferenced by CircleFilled().\n\n## ◆ SaveAsPNG()\n\n void CIMAGE::SaveAsPNG ( const wxString & aFileName ) const\n\nFunction SaveAsPNG save image buffer to a PNG file into the working folder.\n\neach of RGB channel will have the 8bit-channel from the image.\n\nParameters\n aFileName fime name (without extension)\n\nDefinition at line 539 of file cimage.cpp.\n\n540 {\n541 DBG_SaveBuffer( aFileName, m_pixels, m_width, m_height );\n542 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nvoid DBG_SaveBuffer(const wxString &aFileName, const unsigned char *aInBuffer, unsigned int aXSize, unsigned int aYSize)\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\n\nReferences DBG_SaveBuffer(), m_height, m_pixels, and m_width.\n\n## ◆ Setpixel()\n\n void CIMAGE::Setpixel ( int aX, int aY, unsigned char aValue )\n\nFunction Setpixel set a value in a pixel position, position is clamped in accord with the current clamp settings.\n\nParameters\n aX x position aY y position aValue value to set the pixel\n\nDefinition at line 122 of file cimage.cpp.\n\n123 {\n124 if( wrapCoords( &aX, &aY ) )\n125 m_pixels[aX + aY * m_width] = aValue;\n126 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nbool wrapCoords(int *aXo, int *aYo) const\nFunction wrapCoords calculate the coordinates points in accord with the current clamping settings.\nDefinition: cimage.cpp:78\n\nReferences m_pixels, m_width, and wrapCoords().\n\n## ◆ SetPixelsFromNormalizedFloat()\n\n void CIMAGE::SetPixelsFromNormalizedFloat ( const float * aNormalizedFloatArray )\n\nFunction SetPixelsFromNormalizedFloat set the current channel from a float normalized (0.0 - 1.0) buffer this <- CLAMP(NormalizedFloat * 255)\n\nParameters\n aNormalizedFloatArray a float array with the same size of the image\n\nDefinition at line 527 of file cimage.cpp.\n\n528 {\n529 for( unsigned int i = 0; i < m_wxh; i++ )\n530 {\n531 int v = aNormalizedFloatArray[i] * 255;\n532\n533 CLAMP( v, 0, 255 );\n534 m_pixels[i] = v;\n535 }\n536 }\nunsigned char * m_pixels\nbuffer to store the image 8bit-channel\nDefinition: cimage.h:225\nunsigned int m_wxh\nwidth * height precalc value\nDefinition: cimage.h:228\nCoords are clamped to image size.\n\nReferences CLAMP, m_pixels, and m_wxh.\n\n## ◆ wrapCoords()\n\n bool CIMAGE::wrapCoords ( int * aXo, int * aYo ) const\nprivate\n\nFunction wrapCoords calculate the coordinates points in accord with the current clamping settings.\n\nParameters\n aXo X coordinate to be converted (output) aXo Y coordinate to be converted (output)\nReturns\nbool - true if the coordinates are inside the image, false otherwise\n\nDefinition at line 78 of file cimage.cpp.\n\n79 {\n80 int x = *aXo;\n81 int y = *aYo;\n82\n83 switch(m_wraping)\n84 {\n85 case IMAGE_WRAP::CLAMP:\n86 x = (x < 0 )?0:x;\n87 x = (x >= (int)(m_width - 1))?(m_width - 1):x;\n88 y = (y < 0)?0:y;\n89 y = (y >= (int)(m_height - 1))?(m_height - 1):y;\n90 break;\n91\n92 case IMAGE_WRAP::WRAP:\n93 x = (x < 0)?((m_width - 1)+x):x;\n94 x = (x >= (int)(m_width - 1))?(x - m_width):x;\n95 y = (y < 0)?((m_height - 1)+y):y;\n96 y = (y >= (int)(m_height - 1))?(y - m_height):y;\n97 break;\n98\n99 default:\n100 break;\n101 }\n102\n103 if( (x < 0) || (x >= (int)m_width) ||\n104 (y < 0) || (y >= (int)m_height) )\n105 return false;\n106\n107 *aXo = x;\n108 *aYo = y;\n109\n110 return true;\n111 }\nunsigned int m_height\nheight of the image\nDefinition: cimage.h:227\nCoords are wrapped arround.\nIMAGE_WRAP m_wraping\ncurrent wrapping type\nDefinition: cimage.h:229\nunsigned int m_width\nwidth of the image\nDefinition: cimage.h:226\nCoords are clamped to image size.\n\nReferences CLAMP, m_height, m_width, m_wraping, and WRAP.\n\nReferenced by Getpixel(), and Setpixel().\n\n## ◆ m_height\n\n unsigned int CIMAGE::m_height\nprivate\n\nheight of the image\n\nDefinition at line 227 of file cimage.h.\n\nReferenced by CIMAGE(), EfxFilter(), GetHeight(), Hline(), SaveAsPNG(), and wrapCoords().\n\n## ◆ m_pixels\n\n unsigned char* CIMAGE::m_pixels\nprivate\n\nbuffer to store the image 8bit-channel\n\nDefinition at line 225 of file cimage.h.\n\n## ◆ m_width\n\n unsigned int CIMAGE::m_width\nprivate\n\nwidth of the image\n\nDefinition at line 226 of file cimage.h.\n\nReferenced by CIMAGE(), EfxFilter(), Getpixel(), GetWidth(), Hline(), SaveAsPNG(), Setpixel(), and wrapCoords().\n\n## ◆ m_wraping\n\n IMAGE_WRAP CIMAGE::m_wraping\nprivate\n\ncurrent wrapping type\n\nDefinition at line 229 of file cimage.h.\n\nReferenced by CIMAGE(), EfxFilter(), and wrapCoords().\n\n## ◆ m_wxh\n\n unsigned int CIMAGE::m_wxh\nprivate\n\nwidth * height precalc value\n\nDefinition at line 228 of file cimage.h.\n\nReferenced by CIMAGE(), CopyFull(), Invert(), and SetPixelsFromNormalizedFloat().\n\nThe documentation for this class was generated from the following files:"
] | [
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"https://docs.kicad.org/doxygen/kicad_doxygen_logo.png",
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https://jlqwer.com/posts/4371.html | [
"# Intellectual Property\n\nIntellectual Property\n\nTDP Inc. has decided to sue JCN Inc. for copyright infringement. To this end, TDP wishes to find\n\ninfringing segments\n\nwithin JCN's code base, to show to selected media representatives. Since TDP has fired all its technical staff, it is looking to hire a consultant to be paid on a contingency basis should the lawsuit be successful. To demonstrate your qualifications for this position, you are to solve the problem on a number of test cases.\n\nEach test case begins with a positive integer k, the number of infringing segments to be found. Following this line are two code bases. The first code base is preceded by the line \"BEGIN TDP CODEBASE\" and contains a number of lines followed by the line \"END TDP CODEBASE\". The second code base is preceded by \"BEGIN JCN CODEBASE\" and followed by \"END JCN CODEBASE\". The line \"END TDP CODEBASE\" does not appear in the first code base and the line \"END JCN CODEBASE\" does not appear in the second. A line containing 0 follows the last test case.\n\nFor each test case you should output: (1) a line containing \"CASE n\" where n is the number of the test case; (2) up to k infringing segments. Each segment should be printed exactly as it appears in the JCN code base (including newlines and whitespace) and should be preceded by a line containing \"INFRINGING SEGMENT m LENGTH l POSITION p\" where m is the number of the segment within the test case, l is the length of the segment (in characters) and p is the position of the segment (in characters from the start of the JCN codebase). Output an empty line between test cases.\n\nA code base is simply a string of characters. An infringing segment is a non-empty contiguous sequence of characters in the JCN code base that is textually identical to some contiguous sequence of characters in the TDP code base, and is not contained in a larger infringing segment. You should consider all characters in the code base, including spaces and the newline character at the end of each line.\n\nIf there are k or fewer common segments, print them all in decreasing order by length; if several segments have the same length, print them in the order they occur in JCN's code base. If there are more than k segments, print the first k according to the given order.\n\nYou may assume that no code base contains more than 50,000 characters.\n\n``````6\nBEGIN TDP CODEBASE\nthe quick brown fox\njumps over the lazy dog.\nso there!\nEND TDP CODEBASE\nBEGIN JCN CODEBASE\nnow is the time for all\ngood men to come to the aid\nof the party.\nso there!\nEND JCN CODEBASE\n100\nBEGIN TDP CODEBASE\nxyzzy\nEND TDP CODEBASE\nBEGIN JCN CODEBASE\nxyzzabczzyy\nEND JCN CODEBASE\n0\n``````\n\n``````CASE 1\nINFRINGING SEGMENT 1 LENGTH 12 POSITION 64\n.\nso there!\n\nINFRINGING SEGMENT 2 LENGTH 5 POSITION 6\nthe\nINFRINGING SEGMENT 3 LENGTH 5 POSITION 42\no the\nINFRINGING SEGMENT 4 LENGTH 5 POSITION 43\nthe\nINFRINGING SEGMENT 5 LENGTH 5 POSITION 54\nthe\nINFRINGING SEGMENT 6 LENGTH 3 POSITION 15\nfo\n\nCASE 2\nINFRINGING SEGMENT 1 LENGTH 4 POSITION 0\nxyzz\nINFRINGING SEGMENT 2 LENGTH 3 POSITION 7\nzzy\nINFRINGING SEGMENT 3 LENGTH 2 POSITION 10\ny\n``````\n\n``````#include <stdlib.h>\n#include <stdio.h>\n\nchar n, m;\n\nint i,j,k,nn,mn;\n\nint rem(int d){\nint i,carry=0;\nfor (i=0;i<nn;i++) {\ncarry = 10 * carry + n[i];\ncarry %= d;\n}\nreturn carry;\n}\n\nvoid Div(int d){\nint i,carry=0;\nfor (i=0;i<nn;i++) {\ncarry = 10 * carry + n[i];\nn[i] = carry/d;\ncarry %= d;\n}\n}\n\nint one(){\nint i;\nfor (i=0;i+1<nn;i++) if (n[i]) return 0;\nreturn n[i] == 1;\n}\n\nmain(){\nwhile (strcmp(gets(n),\"-1\")) {\nnn = strlen(n);\nif (nn == 1) {\nprintf(\"1%c\\n\",n);\ncontinue;\n}\nmn = 0;\nfor (i=0;i<nn;i++) n[i] -= '0';\nfor (i=9;i>1;i--) {\nwhile (rem(i) == 0) {\nm[mn++] = '0'+i;\nDiv(i);\n}\n}\nif (!one()) {\nprintf(\"There is no such number.\\n\");\ncontinue;\n}\nfor (i=mn-1;i>=0;i--) printf(\"%c\",m[i]);\nprintf(\"\\n\");\n}\n}``````\n\n``````#include <stdlib.h>\n#include <stdio.h>\n\nchar n, m;\n\nint i,j,k,nn,mn;\n\nint rem(int d){\nint i,carry=0;\nfor (i=0;i<nn;i++) {\ncarry = 10 * carry + n[i];\ncarry %= d;\n}\nreturn carry;\n}\n\nvoid Div(int d){\nint i,carry=0;\nfor (i=0;i<nn;i++) {\ncarry = 10 * carry + n[i];\nn[i] = carry/d;\ncarry %= d;\n}\n}\n\nint one(){\nint i;\nfor (i=0;i+1<nn;i++) if (n[i]) return 0;\nreturn n[i] == 1;\n}\n\nmain(){\nwhile (strcmp(gets(n),\"-1\")) {\nnn = strlen(n);\nif (nn == 1) {\nprintf(\"1%c\\n\",n);\ncontinue;\n}\nmn = 0;\nfor (i=0;i<nn;i++) n[i] -= '0';\nfor (i=9;i>1;i--) {\nwhile (rem(i) == 0) {\nm[mn++] = '0'+i;\nDiv(i);\n}\n}\nif (!one()) {\nprintf(\"There is no such number.\\n\");\ncontinue;\n}\nfor (i=mn-1;i>=0;i--) printf(\"%c\",m[i]);\nprintf(\"\\n\");\n}\n}``````"
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https://www.mredkj.com/javascript/nf_api.html | [
"# JavaScript Number Format v1.5.4 - API\n\n## The constructor\n\nNumberFormat(num, inputDecimal)\n\n• num - The number to be formatted.\nAlso refer to setNumber\n• inputDecimal - (Optional) The decimal character for the input\nAlso refer to setInputDecimal\n\n## getOriginal\n\nReturns the number as it was passed in, which may include non-number characters. Added in v1.0.2\n\ngetOriginal()\n\n• No arguments\n\n## setCommas\n\nSets a switch that indicates if there should be commas. The separator value is set to a comma and the decimal value is set to a period. Modified in v1.5.0\n\nsetCommas(isC)\n\n• isC - true: the number should be formatted with commas; false: no commas\n\n## setCurrency\n\nSets a switch that indicates if the number should be displayed as currency\n\nsetCurrency(isC)\n\n• isC - true: display currency; false: don't display currency\n\n## setCurrencyPosition\n\nSets the position for currency, which includes position relative to the numbers and negative sign. This method does not automatically put the negative sign at the left or right. They are left by default, and would need to be set right with setNegativeFormat. New in v1.5.0\nLEFT_OUTSIDE example: \\$-1.00\nLEFT_INSIDE example: -\\$1.00\nRIGHT_INSIDE example: 1.00\\$-\nRIGHT_OUTSIDE example: 1.00-\\$\n\nsetCurrencyPosition(cp)\n\n• cp - The position. Use one of the constants.\n\n## setCurrencyPrefix\n\nSets the symbol for currency. The symbol will show up on the left of the numbers and outside a negative sign. Modified in v1.5.0 so it now calls setCurrencyValue and setCurrencyPosition(this.LEFT_OUTSIDE)\n\nsetCurrencyPrefix(cp)\n\n• cp - The symbol\n\n## setCurrencyValue\n\nSets the symbol for currency.\n\nsetCurrencyValue(val)\n\n• val - The symbol. e.g. '\\$'\n\n## setInputDecimal\n\nSet the decimal value for the input. New in v1.5.0\n\nsetInputDecimal(val)\n\n• val - The decimal value for the input.\n\n## setNegativeFormat\n\nSet the format for negative numbers. New in v1.5.0\nLEFT_DASH example: -1000\nRIGHT_DASH example: 1000-\nPARENTHESIS example: (1000)\n\nsetNegativeFormat(format)\n\n• format - The format. Use one of the constants.\n\n## setNegativeRed\n\nFormat the number red if it's negative. For use in HTML markup, not in text boxes. New in v1.5.0\n\nsetNegativeRed(isRed)\n\n• isRed - true: to format the number red if negative, black if positive; false: for it to always be black font.\n\n## setNumber\n\nSets the number. If there is a non-period decimal format for the input, setInputDecimal should be called before calling setNumber. Modified in v1.5.0\n\nsetNumber(num, inputDecimal)\n\n• num - The number to be formatted\n• inputDecimal - (Optional) The decimal character for the input\nAlso refer to setInputDecimal\n\n## setPlaces\n\nSets the precision of decimal places. Modified in v1.5.1 and v1.5.4\n\nsetPlaces(p, tr)\n\n• p - The number of places. The constant NO_ROUNDING (which equals -1) turns off rounding to a set number of places. Any other number of places less than or equal to zero is considered zero.\n• tr - Whether to truncate instead of rounding. (optional) Values can be true or false, or if left blank will count as false.\n\n## setSeparators\n\nOne purpose of this method is to set a switch that indicates if there should be separators between groups of numbers. Also, can use it to set the values for the separator and decimal. For example, in the value 1,000.00 the comma (,) is the separator and the period (.) is the decimal. Both separator and decimal are optional. The separator and decimal cannot be the same value. If they are, decimal with be changed. New in v1.5.0\n\nCan use the following constants (via the instantiated object) for separator or decimal:\nCOMMA\nPERIOD\n\nsetSeparators(isC, separator, decimal)\n\n• isC - true, if there should be separators; false, if there should be no separators\n• separator - the value of the separator (optional).\n• decimal - the value of the decimal (optional).\n\n## toFormatted\n\nReturns the number as a string formatted according to the settings. Modified in v1.5.0 and v1.5.1\n\ntoFormatted()\n\n• No arguments\n\n## toPercentage\n\nFormat the current number as a percentage. This is separate from most of the regular formatting settings. The exception is the number of decimal places. If a number is 0.123 it will be formatted as 12.3%\n\ntoPercentage()\n\n• No arguments\n\n## toUnformatted\n\ntoUnformatted - Returns the number as just a number. If the original value was '100,000', then this method will return the number 100000 Modified comments in v1.0.2, because this method no longer returns the original value. Refer to getOriginal instead.\n\ntoUnformatted()\n\n• No arguments"
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http://forums.wolfram.com/mathgroup/archive/2010/Apr/msg00368.html | [
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"Re: ArrayPlot coordinates scaling for overlays\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg109182] Re: ArrayPlot coordinates scaling for overlays\n• From: \"David Park\" <djmpark at comcast.net>\n• Date: Fri, 16 Apr 2010 05:50:23 -0400 (EDT)\n\n```To me, at least, the specification of your problem seems confused or\nincomplete.\n\nWhy don't you give us the statements for the two plots, and then tell us how\nyou want to overlay the second plot on the first plot? Do you mean you want\nto Inset the second plot as a subplot? Then check out Inset. If you want to\noverlay then are you going to use Opacity to prevent the second plot from\ncompletely obscuring the first plot?\n\nHave you looked into the DataRange option?\n\nI fairly certain your problem can be solve in a more direct manner, but it\nneeds better definition with all the starting data.\n\nDavid Park\ndjmpark at comcast.net\nhttp://home.comcast.net/~djmpark/\n\nFrom: fd [mailto:fdimer at gmail.com]\n\nAll\n\nThis seems a simple problem I not finding an easy solution.\n\nI have a plot obtained from an ArrayPlot, for which the coordinates\nare the indexes of the matrix being plotted; I want to overlay to this\nplot some other plot, say, from DensityPlot. I have to tell\nMathematica that the bottom left corner of the ArrayPlot is {xi,yi}\nand the upper right is {xf,yf}.\n\nIt would be nice as well to know how you could do this with a raster\nimage in general.\n\nI was trying to use ListDensityPlot, but for the specific problem I\ndealing with it is excruciatingly slow.\n\nI'm also working to re-scale the FrameTicks by defining a new\nArrayPlot function, with limited success. Below the code I'm working\non.\n\nFelipe\n\narrayPlotScale[array_List, {xmin_, xmax_}, {ymin_, ymax_}] :=\nModule[{deltas =\nReverse[{ymax - ymin, xmax - xmin}/Dimensions[array]],\nn = Dimensions[array] // Reverse},\nArrayPlot[array,\nFrameTicks ->\nReverse[{Table[{i, xmin + i deltas[] }, {i, 0, n[], 20}],\nTable[{n[] - i, ymin + i deltas[]}, {i, 0, n[],\n10}]}]]]\n\ntest = Table[i j, {i, 1, 100}, {j, 100, 1, -1}];\n\narrayPlotScale[test, {0, 16}, {0, 100}]\n\n```\n\n• Prev by Date: Re: piecewise function\n• Next by Date: Re: Through[(a+b+b)[x]]\n• Previous by thread: Re: ArrayPlot coordinates scaling for overlays\n• Next by thread: Re: ArrayPlot coordinates scaling for overlays"
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http://forums.wolfram.com/mathgroup/archive/2007/Jan/msg00291.html | [
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"Re: Difficulties with Complex-Modulus Series\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg72730] Re: Difficulties with Complex-Modulus Series\n• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>\n• Date: Wed, 17 Jan 2007 06:08:34 -0500 (EST)\n• Organization: The Open University, Milton Keynes, UK\n• References: <eohtrs\\$q48\\[email protected]>\n\n```carlos at colorado.edu wrote:\n> Say I have r = (2*I+x)/(2*I-x), in which x is real and nonnegative.\n>\n> Series[r,{x,0,4}] and Series[r,{x,Infinity,4}] work as expected.\n>\n> Introduce now R=Abs[r] and try the same:\n>\n> Series[R,{x,0,4}] and Series[R,{x,Infinity,4}]\n>\n> Results are now \"contaminated\" with Abs'[-1], Abs''[-1], etc,\n> I dont understand the presence of those derivatives.\n> Anybody can explain the reason? (I teach students that the\n> derivative of a constant is zero, but perhaps that has changed\n> with the new year) BTW it would be nice to say\n>\n> Series[R,{x,0,4}, x>=0] or Series[R,{x,0,4}, R>=0] etc\n>\n> if that would get rid of the garbage, but Mathematica 5\n> does not allow Assumptions in Series. Note BTW that R=1 for\n> any x, so the R series are in fact trivial to any order.\n>\n\nHi Carlos,\n\nWhat version of Mathematica and system platform were you using when you\n\nI ask because I have been unsuccessful in my attempts to get any\nderivatives of the function *Abs*.\n\nIn:=\n\\$Version\nr = (2*I + x)/(2*I - x);\nR = Abs[r];\nSeries[R, {x, 0, 4}]\nSeries[R, {x, Infinity, 4}]\n\nOut=\n5.2 for Microsoft Windows (June 20, 2005)\n\nOut=\n2 I + x\nAbs[-------]\n2 I - x\n\nOut=\n2 I + x\nAbs[-------]\n2 I - x\n\nAs we can see, both calls of the *Series* function only returned the\ncallee function.\n\nNow, the built-in Mathematica function *Abs* is meant to work with\nnumeric argument only: \"Abs[z] is left unevaluated if z is not a numeric\nquantity .\" Since r is not a numeric expression, the Abs[r] is left\nuntouched and especially there is no attempt to simplify or transform r.\n\nIn:=\nNumericQ[r]\n\nOut=\nFalse\n\nThe closest thing to your result I could get is by differentiating R\nw.r.t. x\n\nIn:=\nD[R, x]\n\nOut=\n1 2 I + x 2 I + x\n(------- + ----------) Abs'[-------]\n2 I - x 2 2 I - x\n(2 I - x)\n\nSubstituting a numeric value for x, we get\n\nIn:=\n% /. x -> 2\n\nOut=\n1\n-(-) Abs'[-I]\n2\n\nThe above result should have been zero. If we nudge Mathematica to\nsimplify the result, we get ride of the derivative of a constant\n(possibly evaluated to one) but still have some value with a change of sign.\n\nIn:=\nComplexExpand[%]\n\nOut=\n1\n-\n2\n\nIs this a bug or a feature, I don't know: at this point, I just gave up!\n\nBest regards,\nJean-Marc\n\n http://documents.wolfram.com/mathematica/functions/Abs\n\n```\n\n• Prev by Date: Re: speed of multiplying polynomials\n• Next by Date: Re: Re: Re: Re: Limit and Root Objects\n• Previous by thread: Re: Difficulties with Complex-Modulus Series\n• Next by thread: Re: Difficulties with Complex-Modulus Series"
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https://forum.ansys.com/forums/reply/313904/ | [
"",
null,
"Erik Kostson\nAnsys Employee\n\nSee that discussion Peteroznewman explains what Ansys does when you give a frequency of interest (say the freq. of your dominant mode of interest that will be excited – frequency_given) and a damping ratio (damp_ratio_given).\n\nAs he says, it calculates\nbeta = (2*damp_ratio_given)/(2*PI*frequency_given).\n\nYou can check that in Ansys yourself, by giving freq. = 20 Hz, damping ratio = 0.01, then Ansys will give a beta as =\n\n 0.000159\n\nAll the best, and perhaps some other forum members can give some more feedback, but the post and this should explain (also see some more material on the internet about Rayleigh damping.\n\nErik",
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http://git.toastfreeware.priv.at/debian/cpulimit.git/blobdiff/4a3aea9637d9b25639261d9ad0cb7adf3a86e9e9..e9e7f2472441edafe516f585b1f01e8b8c9b771c:/cpulimit.c | [
"index bf912d6..ba41986 100644 (file)\n@@ -1,50 +1,26 @@\n/**\n- *\n- * cpulimit - a cpu limiter for Linux\n- *\n- * Copyright (C) 2005-2008, by: Angelo Marletta <[email protected]>\n- *\n- * This program is free software; you can redistribute it and/or\n- * modify it under the terms of the GNU General Public License\n- * as published by the Free Software Foundation; either version 2\n- * of the License, or (at your option) any later version.\n- *\n- * This program is distributed in the hope that it will be useful,\n- * but WITHOUT ANY WARRANTY; without even the implied warranty of\n- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the\n- * GNU General Public License for more details.\n- *\n- * You should have received a copy of the GNU General Public License\n- * along with this program; if not, write to the Free Software\n- * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.\n+ * This program is licensed under the GNU General Public License,\n+ * version 2. A copy of the license can be found in the accompanying\n*\n**********************************************************************\n*\n- * This is a simple program to limit the cpu usage of a process\n+ * Simple program to limit the cpu usage of a process\n* If you modify this code, send me a copy please\n*\n- * Date: 15/2/2008\n- * Version: 1.2 alpha\n- * Get the latest version at: http://cpulimit.sourceforge.net\n+ * Author: Angelo Marletta\n+ * Date: 26/06/2005\n+ * Version: 1.1\n*\n- * Changelog:\n- * - reorganization of the code, splitted in more source files\n- * - control function process_monitor() optimized by eliminating an unnecessary loop\n- * - experimental support for multiple control of children processes and threads\n- * children detection algorithm seems heavy because of the amount of code,\n- * but it's designed to be scalable when there are a lot of children processes\n- * - cpu count detection, i.e. if you have 4 cpu, it is possible to limit up to 400%\n- * - in order to avoid deadlock, cpulimit prevents to limit itself\n- * - option --path eliminated, use --exe instead both for absolute path and file name\n- * - deleted almost every setpriority(), just set it once at startup\n- * - minor enhancements and bugfixes\n+ * Modifications and updates by: Jesse Smith\n+ * Date: May 4, 2011\n+ * Version 1.2\n*\n*/\n\n#include <getopt.h>\n#include <stdio.h>\n-#include <fcntl.h>\n#include <stdlib.h>\n#include <time.h>\n#include <sys/time.h>\n#include <dirent.h>\n#include <errno.h>\n#include <string.h>\n+#include <limits.h> // for compatibility\n\n-#include \"process.h\"\n-#include \"procutils.h\"\n-#include \"list.h\"\n\n-//some useful macro\n-#define MIN(a,b) (a<b?a:b)\n-#define MAX(a,b) (a>b?a:b)\n-#define print_caption() printf(\"\\n%%CPU\\twork quantum\\tsleep quantum\\tactive rate\\n\")\n-//control time slot in microseconds\n-//each slot is splitted in a working slice and a sleeping slice\n-#define CONTROL_SLOT 100000\n-\n-#define MAX_PRIORITY -10\n-\n-//the \"family\"\n-struct process_family pf;\n-//pid of cpulimit\n-int cpulimit_pid;\n-//name of this program (maybe cpulimit...)\n-char *program_name;\n+//kernel time resolution (inverse of one jiffy interval) in Hertz\n+//i don't know how to detect it, then define to the default (not very clean!)\n+#define HZ 100\n\n-/* CONFIGURATION VARIABLES */\n+//some useful macro\n+#define min(a,b) (a<b?a:b)\n+#define max(a,b) (a>b?a:b)\n\n+//pid of the controlled process\n+int pid=0;\n+//executable file name\n+char *program_name;\n//verbose mode\n-int verbose = 0;\n-//lazy mode (exits if there is no process)\n-int lazy = 0;\n-\n-//how many cpu do we have?\n-int get_cpu_count()\n-{\n- FILE *fd;\n- int cpu_count = 0;\n- char line;\n- fd = fopen(\"/proc/stat\", \"r\");\n- if (fd < 0)\n- return 0; //are we running Linux??\n- while (fgets(line,sizeof(line),fd)!=NULL) {\n- if (strncmp(line, \"cpu\", 3) != 0) break;\n- cpu_count++;\n- }\n- fclose(fd);\n- return cpu_count - 1;\n+int verbose=0;\n+//lazy mode\n+int lazy=0;\n+// is higher priority nice possible?\n+int nice_lim;\n+\n+//reverse byte search\n+void *memrchr(const void *s, int c, size_t n);\n+\n+//return ta-tb in microseconds (no overflow checks!)\n+inline long timediff(const struct timespec *ta,const struct timespec *tb) {\n+ unsigned long us = (ta->tv_sec-tb->tv_sec)*1000000 + (ta->tv_nsec/1000 - tb->tv_nsec/1000);\n+ return us;\n}\n\n-//return t1-t2 in microseconds (no overflow checks, so better watch out!)\n-inline unsigned long timediff(const struct timespec *t1,const struct timespec *t2)\n-{\n- return (t1->tv_sec - t2->tv_sec) * 1000000 + (t1->tv_nsec/1000 - t2->tv_nsec/1000);\n-}\n-\n-//returns t1-t2 in microseconds\n-inline unsigned long long tv_diff(struct timeval *t1, struct timeval *t2)\n-{\n- return ((unsigned long long)(t1->tv_sec - t2->tv_sec)) * 1000000ULL + t1->tv_usec - t2->tv_usec;\n-}\n-\n-//SIGINT and SIGTERM signal handler\n-void quit(int sig)\n-{\n- //let all the processes continue if stopped\n- struct list_node *node = NULL;\n- for (node=pf.members.first; node!= NULL; node=node->next) {\n- struct process *p = (struct process*)(node->data);\n- process_close(p->history);\n- kill(p->pid, SIGCONT);\n+int waitforpid(int pid) {\n+ //switch to low priority\n+ // if (setpriority(PRIO_PROCESS,getpid(),19)!=0) {\n+ if ( (nice_lim < INT_MAX) &&\n+ (setpriority(PRIO_PROCESS, getpid(), 19) != 0) ) {\n+ printf(\"Warning: cannot renice\\n\");\n}\n- //free all the memory\n- cleanup_process_family(&pf);\n- exit(0);\n-}\n\n-void print_usage(FILE *stream, int exit_code)\n-{\n- fprintf(stream, \"Usage: %s TARGET [OPTIONS...]\\n\",program_name);\n- fprintf(stream, \" TARGET must be exactly one of these:\\n\");\n- fprintf(stream, \" -p, --pid=N pid of the process (implies -z)\\n\");\n- fprintf(stream, \" -e, --exe=FILE name of the executable program file or absolute path name\\n\");\n- fprintf(stream, \" OPTIONS\\n\");\n- fprintf(stream, \" -l, --limit=N percentage of cpu allowed from 0 to 100 (required)\\n\");\n- fprintf(stream, \" -v, --verbose show control statistics\\n\");\n- fprintf(stream, \" -z, --lazy exit if there is no suitable target process, or if it dies\\n\");\n- fprintf(stream, \" -h, --help display this help and exit\\n\");\n- exit(exit_code);\n-}\n-\n-void limit_process(int pid, float limit)\n-{\n- //slice of the slot in which the process is allowed to run\n- struct timespec twork;\n- //slice of the slot in which the process is stopped\n- struct timespec tsleep;\n- //when the last twork has started\n- struct timespec startwork;\n- //when the last twork has finished\n- struct timespec endwork;\n- //initialization\n- memset(&twork, 0, sizeof(struct timespec));\n- memset(&tsleep, 0, sizeof(struct timespec));\n- memset(&startwork, 0, sizeof(struct timespec));\n- memset(&endwork, 0, sizeof(struct timespec));\n- //last working time in microseconds\n- unsigned long workingtime = 0;\n- int i = 0;\n-\n- //build the family\n- create_process_family(&pf, pid);\n- struct list_node *node;\n-\n- if (verbose) printf(\"Members in the family owned by %d: %d\\n\", pf.father, pf.members.count);\n+ int i=0;\n\nwhile(1) {\n\n- if (i%100==0 && verbose) print_caption();\n-\n- if (i%10==0) {\n- //update the process family (checks only for new members)\n- int newborn = check_new_members(&pf);\n- if (newborn) {\n- printf(\"%d new children processes detected (\", newborn);\n- int j;\n- node = pf.members.last;\n- for (j=0; j<newborn; j++) {\n- printf(\"%d\", ((struct process*)(node->data))->pid);\n- if (j<newborn-1) printf(\" \");\n- node = node->previous;\n+ DIR *dip;\n+ struct dirent *dit;\n+\n+ //open a directory stream to /proc directory\n+ if ((dip = opendir(\"/proc\")) == NULL) {\n+ perror(\"opendir\");\n+ return -1;\n+ }\n+\n+ //read in from /proc and seek for process dirs\n+ while ((dit = readdir(dip)) != NULL) {\n+ //get pid\n+ if (pid==atoi(dit->d_name)) {\n+ //pid detected\n+ if (kill(pid,SIGSTOP)==0 && kill(pid,SIGCONT)==0) {\n+ //process is ok!\n+ if (closedir(dip) == -1) {\n+ perror(\"closedir\");\n+ return -1;\n+ }\n+ goto done;\n+ }\n+ else {\n+ fprintf(stderr,\"Error: Process %d detected, but you don't have permission to control it\\n\",pid);\n}\n- printf(\")\\n\");\n}\n}\n\n- //total cpu actual usage (range 0-1)\n- //1 means that the processes are using 100% cpu\n- float pcpu = 0;\n- //rate at which we are keeping active the processes (range 0-1)\n- //1 means that the process are using all the twork slice\n- float workingrate = 0;\n-\n- //estimate how much the controlled processes are using the cpu in the working interval\n- for (node=pf.members.first; node!=NULL; node=node->next) {\n- struct process *proc = (struct process*)(node->data);\n- if (process_monitor(proc->history, workingtime, &(proc->history->usage))==-1) {\n- //process is dead, remove it from family\n- remove_process_from_family(&pf, proc->pid);\n+ //close the dir stream and check for errors\n+ if (closedir(dip) == -1) {\n+ perror(\"closedir\");\n+ return -1;\n+ }\n+\n+ //no suitable target found\n+ if (i++==0) {\n+ if (lazy) {\n+ fprintf(stderr,\"No process found\\n\");\n+ exit(2);\n+ }\n+ else {\n+ printf(\"Warning: no target process found. Waiting for it...\\n\");\n}\n- pcpu += proc->history->usage.pcpu;\n- workingrate += proc->history->usage.workingrate;\n}\n- //average value\n- workingrate /= pf.members.count;\n\n- //TODO: make workingtime customized for each process, now it's equal for all\n+ //sleep for a while\n+ sleep(2);\n+ }\n\n- //adjust work and sleep time slices\n- if (pcpu>0) {\n- twork.tv_nsec = MIN(CONTROL_SLOT*limit*1000/pcpu*workingrate,CONTROL_SLOT*1000);\n- }\n- else if (pcpu==0) {\n- twork.tv_nsec = CONTROL_SLOT*1000;\n- }\n- else if (pcpu==-1) {\n- //not yet a valid idea of cpu usage\n- pcpu = limit;\n- workingrate = limit;\n- twork.tv_nsec = MIN(CONTROL_SLOT*limit*1000,CONTROL_SLOT*1000);\n- }\n- tsleep.tv_nsec = CONTROL_SLOT*1000-twork.tv_nsec;\n+done:\n+ printf(\"Process %d detected\\n\",pid);\n+ //now set high priority, if possible\n+ // if (setpriority(PRIO_PROCESS,getpid(),-20)!=0) {\n+ if ( (nice_lim < INT_MAX) &&\n+ (setpriority(PRIO_PROCESS, getpid(), nice_lim) != 0) ) {\n+ printf(\"Warning: cannot renice.\\nTo work better you should run this program as root.\\n\");\n+ }\n+ return 0;\n\n- if (verbose && i%10==0 && i>0) {\n- printf(\"%0.2f%%\\t%6ld us\\t%6ld us\\t%0.2f%%\\n\",pcpu*100,twork.tv_nsec/1000,tsleep.tv_nsec/1000,workingrate*100);\n+}\n+\n+//this function periodically scans process list and looks for executable path names\n+//it should be executed in a low priority context, since precise timing does not matter\n+//if a process is found then its pid is returned\n+//process: the name of the wanted process, can be an absolute path name to the executable file\n+// or simply its name\n+//return: pid of the found process\n+int getpidof(const char *process) {\n+\n+ //set low priority\n+ // if (setpriority(PRIO_PROCESS,getpid(),19)!=0) {\n+ if ( (nice_lim < INT_MAX) &&\n+ (setpriority(PRIO_PROCESS, getpid(), 19) != 0) ) {\n+ printf(\"Warning: cannot renice\\n\");\n+ }\n+\n+ char exepath[PATH_MAX+1];\n+ int pid=0;\n+ int i=0;\n+\n+ while(1) {\n+\n+ DIR *dip;\n+ struct dirent *dit;\n+\n+ //open a directory stream to /proc directory\n+ if ((dip = opendir(\"/proc\")) == NULL) {\n+ perror(\"opendir\");\n+ return -1;\n}\n\n- //resume processes\n- for (node=pf.members.first; node!=NULL; node=node->next) {\n- struct process *proc = (struct process*)(node->data);\n- if (kill(proc->pid,SIGCONT)!=0) {\n- //process is dead, remove it from family\n- remove_process_from_family(&pf, proc->pid);\n+ //read in from /proc and seek for process dirs\n+ while ((dit = readdir(dip)) != NULL) {\n+ //get pid\n+ pid=atoi(dit->d_name);\n+ if (pid>0) {\n+ if (size>0) {\n+ int found=0;\n+ if (process=='/' && strncmp(exepath,process,size)==0 && size==strlen(process)) {\n+ //process starts with / then it's an absolute path\n+ found=1;\n+ }\n+ else {\n+ //process is the name of the executable file\n+ if (strncmp(exepath+size-strlen(process),process,strlen(process))==0) {\n+ found=1;\n+ }\n+ }\n+ if (found==1) {\n+ if (kill(pid,SIGSTOP)==0 && kill(pid,SIGCONT)==0) {\n+ //process is ok!\n+ if (closedir(dip) == -1) {\n+ perror(\"closedir\");\n+ return -1;\n+ }\n+ goto done;\n+ }\n+ else {\n+ fprintf(stderr,\"Error: Process %d detected, but you don't have permission to control it\\n\",pid);\n+ }\n+ }\n+ }\n}\n}\n\n- //now processes are free to run (same working slice for all)\n- clock_gettime(CLOCK_REALTIME,&startwork);\n- nanosleep(&twork,NULL);\n- clock_gettime(CLOCK_REALTIME,&endwork);\n- workingtime = timediff(&endwork,&startwork);\n-\n- //stop processes, they have worked enough\n- //resume processes\n- for (node=pf.members.first; node!=NULL; node=node->next) {\n- struct process *proc = (struct process*)(node->data);\n- if (kill(proc->pid,SIGSTOP)!=0) {\n- //process is dead, remove it from family\n- remove_process_from_family(&pf, proc->pid);\n+ //close the dir stream and check for errors\n+ if (closedir(dip) == -1) {\n+ perror(\"closedir\");\n+ return -1;\n+ }\n+\n+ //no suitable target found\n+ if (i++==0) {\n+ if (lazy) {\n+ fprintf(stderr,\"No process found\\n\");\n+ exit(2);\n+ }\n+ else {\n+ printf(\"Warning: no target process found. Waiting for it...\\n\");\n}\n}\n- //now the process is forced to sleep\n- nanosleep(&tsleep,NULL);\n- i++;\n+\n+ //sleep for a while\n+ sleep(2);\n}\n- cleanup_process_family(&pf);\n+\n+done:\n+ printf(\"Process %d detected\\n\",pid);\n+ //now set high priority, if possible\n+ // if (setpriority(PRIO_PROCESS,getpid(),-20)!=0) {\n+ if ( (nice_lim < INT_MAX) &&\n+ (setpriority(PRIO_PROCESS, getpid(), nice_lim) != 0) ) {\n+ printf(\"Warning: cannot renice.\\nTo work better you should run this program as root.\\n\");\n+ }\n+ return pid;\n+\n+}\n+\n+//SIGINT and SIGTERM signal handler\n+void quit(int sig) {\n+ //let the process continue if it's stopped\n+ kill(pid,SIGCONT);\n+ printf(\"Exiting...\\n\");\n+ exit(0);\n+}\n+\n+//get jiffies count from /proc filesystem\n+int getjiffies(int pid) {\n+ static char stat;\n+ static char buffer;\n+ char *p;\n+ sprintf(stat,\"/proc/%d/stat\",pid);\n+ FILE *f=fopen(stat,\"r\");\n+ if (f==NULL) return -1;\n+ p = fgets(buffer,sizeof(buffer),f);\n+ fclose(f);\n+ // char *p=buffer;\n+ if (p)\n+ {\n+ p=memchr(p+1,')',sizeof(buffer)-(p-buffer));\n+ int sp=12;\n+ while (sp--)\n+ p=memchr(p+1,' ',sizeof(buffer)-(p-buffer));\n+ //user mode jiffies\n+ int utime=atoi(p+1);\n+ p=memchr(p+1,' ',sizeof(buffer)-(p-buffer));\n+ //kernel mode jiffies\n+ int ktime=atoi(p+1);\n+ return utime+ktime;\n+ }\n+ // could not read info\n+ return -1;\n+}\n+\n+//process instant photo\n+struct process_screenshot {\n+ struct timespec when; //timestamp\n+ int jiffies; //jiffies count of the process\n+ int cputime; //microseconds of work from previous screenshot to current\n+};\n+\n+//extracted process statistics\n+struct cpu_usage {\n+ float pcpu;\n+ float workingrate;\n+};\n+\n+//this function is an autonomous dynamic system\n+//it works with static variables (state variables of the system), that keep memory of recent past\n+//its aim is to estimate the cpu usage of the process\n+//to work properly it should be called in a fixed periodic way\n+//perhaps i will put it in a separate thread...\n+int compute_cpu_usage(int pid,int last_working_quantum,struct cpu_usage *pusage) {\n+ #define MEM_ORDER 10\n+ //circular buffer containing last MEM_ORDER process screenshots\n+ static struct process_screenshot ps[MEM_ORDER];\n+ //the last screenshot recorded in the buffer\n+ static int front=-1;\n+ //the oldest screenshot recorded in the buffer\n+ static int tail=0;\n+\n+ if (pusage==NULL) {\n+ //reinit static variables\n+ front=-1;\n+ tail=0;\n+ return 0;\n+ }\n+\n+ //let's advance front index and save the screenshot\n+ front=(front+1)%MEM_ORDER;\n+ int j=getjiffies(pid);\n+ if (j>=0) ps[front].jiffies=j;\n+ else return -1; //error: pid does not exist\n+ clock_gettime(CLOCK_REALTIME,&(ps[front].when));\n+ ps[front].cputime=last_working_quantum;\n+\n+ //buffer actual size is: (front-tail+MEM_ORDER)%MEM_ORDER+1\n+ int size=(front-tail+MEM_ORDER)%MEM_ORDER+1;\n+\n+ if (size==1) {\n+ //not enough samples taken (it's the first one!), return -1\n+ pusage->pcpu=-1;\n+ pusage->workingrate=1;\n+ return 0;\n+ }\n+ else {\n+ //now we can calculate cpu usage, interval dt and dtwork are expressed in microseconds\n+ long dt=timediff(&(ps[front].when),&(ps[tail].when));\n+ long dtwork=0;\n+ int i=(tail+1)%MEM_ORDER;\n+ int max=(front+1)%MEM_ORDER;\n+ do {\n+ dtwork+=ps[i].cputime;\n+ i=(i+1)%MEM_ORDER;\n+ } while (i!=max);\n+ int used=ps[front].jiffies-ps[tail].jiffies;\n+ float usage=(used*1000000.0/HZ)/dtwork;\n+ pusage->workingrate=1.0*dtwork/dt;\n+ pusage->pcpu=usage*pusage->workingrate;\n+ if (size==MEM_ORDER)\n+ tail=(tail+1)%MEM_ORDER;\n+ return 0;\n+ }\n+ #undef MEM_ORDER\n+}\n+\n+void print_caption() {\n+ printf(\"\\n%%CPU\\twork quantum\\tsleep quantum\\tactive rate\\n\");\n+}\n+\n+void print_usage(FILE *stream,int exit_code) {\n+ fprintf(stream, \"Usage: %s TARGET [OPTIONS...]\\n\",program_name);\n+ fprintf(stream, \" TARGET must be exactly one of these:\\n\");\n+ fprintf(stream, \" -p, --pid=N pid of the process\\n\");\n+ fprintf(stream, \" -e, --exe=FILE name of the executable program file\\n\");\n+ fprintf(stream, \" -P, --path=PATH absolute path name of the executable program file\\n\");\n+ fprintf(stream, \" OPTIONS\\n\");\n+ fprintf(stream, \" -b --background run in background\\n\");\n+ fprintf(stream, \" -l, --limit=N percentage of cpu allowed from 0 to 100 (mandatory)\\n\");\n+ fprintf(stream, \" -v, --verbose show control statistics\\n\");\n+ fprintf(stream, \" -z, --lazy exit if there is no suitable target process, or if it dies\\n\");\n+ fprintf(stream, \" -h, --help display this help and exit\\n\");\n+ exit(exit_code);\n}\n\nint main(int argc, char **argv) {\n@@ -268,60 +368,66 @@ int main(int argc, char **argv) {\n//get program name\nchar *p=(char*)memrchr(argv,(unsigned int)'/',strlen(argv));\nprogram_name = p==NULL?argv:(p+1);\n- cpulimit_pid = getpid();\n-\n- //argument variables\n- const char *exe = NULL;\n- int perclimit = 0;\n- int pid_ok = 0;\n- int process_ok = 0;\n- int limit_ok = 0;\n- int pid = 0;\n-\n+ int run_in_background = 0;\n//parse arguments\nint next_option;\n- int option_index = 0;\n- //A string listing valid short options letters\n- const char* short_options = \"p:e:l:vzh\";\n- //An array describing valid long options\n+ /* A string listing valid short options letters. */\n+ const char* short_options=\"p:e:P:l:bvzh\";\n+ /* An array describing valid long options. */\nconst struct option long_options[] = {\n- { \"pid\", required_argument, NULL, 'p' },\n- { \"exe\", required_argument, NULL, 'e' },\n- { \"limit\", required_argument, NULL, 'l' },\n- { \"verbose\", no_argument, &verbose, 'v' },\n- { \"lazy\", no_argument, &lazy, 'z' },\n- { \"help\", no_argument, NULL, 'h' },\n- { 0, 0, 0, 0 }\n+ { \"pid\", required_argument, NULL, 'p' },\n+ { \"exe\", required_argument, NULL, 'e' },\n+ { \"path\", required_argument, NULL, 'P' },\n+ { \"limit\", required_argument, NULL, 'l' },\n+ { \"background\", no_argument, NULL, 'b' },\n+ { \"verbose\", no_argument, NULL, 'v' },\n+ { \"lazy\", no_argument, NULL, 'z' },\n+ { \"help\", no_argument, NULL, 'h' },\n+ { NULL, 0, NULL, 0 }\n};\n+ //argument variables\n+ const char *exe=NULL;\n+ const char *path=NULL;\n+ int perclimit=0;\n+ int pid_ok=0;\n+ int process_ok=0;\n+ int limit_ok=0;\n+ struct rlimit maxlimit;\n\ndo {\n- next_option = getopt_long(argc, argv, short_options,long_options, &option_index);\n+ next_option = getopt_long (argc, argv, short_options,long_options, NULL);\nswitch(next_option) {\n+ case 'b':\n+ run_in_background = 1;\n+ break;\ncase 'p':\n- pid = atoi(optarg);\n- //todo: verify pid is valid\n- pid_ok = 1;\n- process_ok = 1;\n+ pid=atoi(optarg);\n+ pid_ok=1;\n+ lazy = 1;\nbreak;\ncase 'e':\n- exe = optarg;\n- process_ok = 1;\n+ exe=optarg;\n+ process_ok=1;\n+ break;\n+ case 'P':\n+ path=optarg;\n+ process_ok=1;\nbreak;\ncase 'l':\n- perclimit = atoi(optarg);\n- limit_ok = 1;\n+ perclimit=atoi(optarg);\n+ limit_ok=1;\nbreak;\ncase 'v':\n- verbose = 1;\n+ verbose=1;\nbreak;\ncase 'z':\n- lazy = 1;\n+ lazy=1;\nbreak;\ncase 'h':\n- print_usage(stdout, 1);\n+ print_usage (stdout, 1);\nbreak;\ncase '?':\n- print_usage(stderr, 1);\n+ print_usage (stderr, 1);\nbreak;\ncase -1:\nbreak;\n@@ -330,91 +436,184 @@ int main(int argc, char **argv) {\n}\n} while(next_option != -1);\n\n- if (pid!=0) {\n- lazy = 1;\n- }\n-\n- if (!process_ok) {\n- fprintf(stderr,\"Error: You must specify a target process, by name or by PID\\n\");\n- print_usage(stderr, 1);\n+ if (!process_ok && !pid_ok) {\n+ fprintf(stderr,\"Error: You must specify a target process\\n\");\n+ print_usage (stderr, 1);\nexit(1);\n}\n- if (pid_ok && exe!=NULL) {\n- fprintf(stderr, \"Error: You must specify exactly one process, by name or by PID\\n\");\n- print_usage(stderr, 1);\n+ if ((exe!=NULL && path!=NULL) || (pid_ok && (exe!=NULL || path!=NULL))) {\n+ fprintf(stderr,\"Error: You must specify exactly one target process\\n\");\n+ print_usage (stderr, 1);\nexit(1);\n}\nif (!limit_ok) {\n- fprintf(stderr,\"Error: You must specify a cpu limit percentage\\n\");\n- print_usage(stderr, 1);\n+ fprintf(stderr,\"Error: You must specify a cpu limit\\n\");\n+ print_usage (stderr, 1);\nexit(1);\n}\n- float limit = perclimit/100.0;\n- int cpu_count = get_cpu_count();\n- if (limit<0 || limit >cpu_count) {\n- fprintf(stderr,\"Error: limit must be in the range 0-%d00\\n\", cpu_count);\n- print_usage(stderr, 1);\n+ float limit=perclimit/100.0;\n+ if (limit <= 0.00) // || limit >1) {\n+ {\n+ fprintf(stderr,\"Error: limit must be greater than 0\\n\");\n+ print_usage (stderr, 1);\nexit(1);\n}\n+\n+ // check to see if we should fork\n+ if (run_in_background)\n+ {\n+ pid_t process_id;\n+ process_id = fork();\n+ if (! process_id)\n+ exit(0);\n+ else\n+ {\n+ setsid();\n+ process_id = fork();\n+ if (process_id)\n+ exit(0);\n+ }\n+ }\n+\n//parameters are all ok!\n- signal(SIGINT, quit);\n- signal(SIGTERM, quit);\n-\n- //try to renice with the best value\n- int old_priority = getpriority(PRIO_PROCESS, 0);\n- int priority = old_priority;\n- while (setpriority(PRIO_PROCESS, 0, priority-1) == 0 && priority>MAX_PRIORITY) {\n- priority--;\n- }\n- if (priority != old_priority) {\n- printf(\"Priority changed to %d\\n\", priority);\n+ signal(SIGINT,quit);\n+ signal(SIGTERM,quit);\n+\n+ if (setpriority(PRIO_PROCESS,getpid(),-20)!=0) {\n+ //if that failed, check if we have a limit\n+ // by how much we can raise the priority\n+#ifdef RLIMIT_NICE\n+//check if non-root can even make changes\n+// (ifdef because it's only available in linux >= 2.6.13)\n+ nice_lim=getpriority(PRIO_PROCESS,getpid());\n+ getrlimit(RLIMIT_NICE, &maxlimit);\n+\n+//if we can do better then current\n+ if( (20 - (signed)maxlimit.rlim_cur) < nice_lim &&\n+ setpriority(PRIO_PROCESS,getpid(),\n+ 20 - (signed)maxlimit.rlim_cur)==0 //and it actually works\n+ ) {\n+\n+ //if we can do better, but not by much, warn about it\n+ if( (nice_lim - (20 - (signed)maxlimit.rlim_cur)) < 9)\n+ {\n+ printf(\"Warning, can only increase priority by %d.\\n\", nice_lim - (20 - (signed)maxlimit.rlim_cur));\n+ }\n+ //our new limit\n+ nice_lim = 20 - (signed)maxlimit.rlim_cur;\n+\n+ } else\n+// otherwise don't try to change priority.\n+// The below will also run if it's not possible\n+// for non-root to change priority\n+#endif\n+ {\n+ printf(\"Warning: cannot renice.\\nTo work better you should run this program as root, or adjust RLIMIT_NICE.\\nFor example in /etc/security/limits.conf add a line with: * - nice -10\\n\\n\");\n+ nice_lim=INT_MAX;\n+ }\n+ } else {\n+ nice_lim=-20;\n}\n+ //don't bother putting setpriority back down,\n+ // since getpidof and waitforpid twiddle it anyway\n+\n+\n+\n+ //time quantum in microseconds. it's splitted in a working period and a sleeping one\n+ int period=100000;\n+ struct timespec twork,tsleep; //working and sleeping intervals\n+ memset(&twork,0,sizeof(struct timespec));\n+ memset(&tsleep,0,sizeof(struct timespec));\n+\n+wait_for_process:\n+\n+ //look for the target process..or wait for it\n+ if (exe!=NULL)\n+ pid=getpidof(exe);\n+ else if (path!=NULL)\n+ pid=getpidof(path);\nelse {\n- printf(\"Cannot change priority\\n\");\n+ waitforpid(pid);\n}\n-\n+ //process detected...let's play\n+\n+ //init compute_cpu_usage internal stuff\n+ compute_cpu_usage(0,0,NULL);\n+ //main loop counter\n+ int i=0;\n+\n+ struct timespec startwork,endwork;\n+ long workingtime=0; //last working time in microseconds\n+\n+ if (verbose) print_caption();\n+\n+ float pcpu_avg=0;\n+\n+ //here we should already have high priority, for time precision\nwhile(1) {\n- //look for the target process..or wait for it\n- int ret = 0;\n- if (pid_ok) {\n- //search by pid\n- ret = look_for_process_by_pid(pid);\n- if (ret == 0) {\n- printf(\"No process found\\n\");\n- }\n- else if (ret < 0) {\n- printf(\"Process found but you aren't allowed to control it\\n\");\n- }\n+\n+ //estimate how much the controlled process is using the cpu in its working interval\n+ struct cpu_usage cu;\n+ if (compute_cpu_usage(pid,workingtime,&cu)==-1) {\n+ if (lazy) exit(2);\n+ //wait until our process appears\n+ goto wait_for_process;\n}\n- else {\n- //search by file or path name\n- ret = look_for_process_by_name(exe);\n- if (ret == 0) {\n- printf(\"No process found\\n\");\n- }\n- else if (ret < 0) {\n- printf(\"Process found but you aren't allowed to control it\\n\");\n- }\n- else {\n- pid = ret;\n- }\n+\n+ //cpu actual usage of process (range 0-1)\n+ float pcpu=cu.pcpu;\n+ //rate at which we are keeping active the process (range 0-1)\n+ float workingrate=cu.workingrate;\n+\n+ //adjust work and sleep time slices\n+ if (pcpu>0) {\n+ twork.tv_nsec=min(period*limit*1000/pcpu*workingrate,period*1000);\n+ }\n+ else if (pcpu==0) {\n+ twork.tv_nsec=period*1000;\n}\n- if (ret > 0) {\n- if (ret == cpulimit_pid) {\n- printf(\"Process %d is cpulimit itself! Aborting to avoid deadlock\\n\", ret);\n- exit(1);\n+ else if (pcpu==-1) {\n+ //not yet a valid idea of cpu usage\n+ pcpu=limit;\n+ workingrate=limit;\n+ twork.tv_nsec=min(period*limit*1000,period*1000);\n+ }\n+ tsleep.tv_nsec=period*1000-twork.tv_nsec;\n+\n+ //update average usage\n+ pcpu_avg=(pcpu_avg*i+pcpu)/(i+1);\n+\n+ if (verbose && i%10==0 && i>0) {\n+ printf(\"%0.2f%%\\t%6ld us\\t%6ld us\\t%0.2f%%\\n\",pcpu*100,twork.tv_nsec/1000,tsleep.tv_nsec/1000,workingrate*100);\n+ }\n+\n+ if (limit<1 && limit>0) {\n+ //resume process\n+ if (kill(pid,SIGCONT)!=0) {\n+ if (lazy) exit(2);\n+ //wait until our process appears\n+ goto wait_for_process;\n}\n- printf(\"Process %d found\\n\", pid);\n- //control\n- limit_process(pid, limit);\n}\n- if (lazy) {\n- printf(\"Giving up...\\n\");\n- break;\n+\n+ clock_gettime(CLOCK_REALTIME,&startwork);\n+ nanosleep(&twork,NULL); //now process is working\n+ clock_gettime(CLOCK_REALTIME,&endwork);\n+ workingtime=timediff(&endwork,&startwork);\n+\n+ if (limit<1) {\n+ //stop process, it has worked enough\n+ if (kill(pid,SIGSTOP)!=0) {\n+ if (lazy) exit(2);\n+ //wait until our process appears\n+ goto wait_for_process;\n+ }\n+ nanosleep(&tsleep,NULL); //now process is sleeping\n}\n- sleep(2);\n+ i++;\n}\n-\n- return 0;\n-}\n\n+}"
] | [
null
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https://homework.cpm.org/category/CCI_CT/textbook/calc/chapter/8/lesson/8.1.1/problem/8-6 | [
"",
null,
"",
null,
"### Home > CALC > Chapter 8 > Lesson 8.1.1 > Problem8-6\n\n8-6.",
null,
"",
null,
"Before differentiating, rewrite the expression with an exponent.\n\nTake the derivative using the Chain Rule.",
null,
"You can expand the polynomial expression before differentiating or you can compute the derivative using the Product Rule.",
null,
"Chain Rule",
null,
"≠ ln(cos2(x)) – ln(49x2)\n\nThe Fundamental Theorem of Calculus, part 1 states that the derivative of an INDEFINITE integral is the original function. But this is a DEFINITE integral.\n\n= −sin(x)ln(cos2(x)) − 7ln(49x2)"
] | [
null,
"https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png",
null,
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",
null,
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https://se.mathworks.com/help/radar/ug/airborne-sar-system-design.html | [
"# Airborne SAR System Design\n\nThis example shows how to design a synthetic aperture radar (SAR) sensor operating in the X-band and calculate the sensor parameters. SAR uses the motion of the radar antenna over a target region to provide an image of the target region. A synthetic aperture is created when the SAR platform travels over the target region while pulses are transmitted and received from the radar antenna.\n\nThis example focuses on designing an SAR sensor to meet a set of performance parameters. It outlines the steps to translate performance specifications, such as the azimuth resolution and the probability of detection, into SAR system parameters, such as the antenna dimension and the transmit power. It models design parameters for stripmap and spotlight modes of operation. Compared to the stripmap operation, spotlight mode can provide a better resolution, and a stronger signal from the scene at the cost of reduced scene size or area imaging rate. The example also models the parameters of the azimuth chirp signal.\n\nThe diagram below classifies the various system and performance parameters. This example covers the functions for selecting system parameters to meet performance parameters.",
null,
"### Design Specifications\n\nThe goal of this airborne SAR system is to provide an image of the target region at a distance up to 10 km from the airborne platform with a range and azimuth resolution of 1 m. The platform is operating at an altitude of 5 km and moving at a velocity of 100 m/s. The desired performance indices are the probability of detection (Pd) and probability of false alarm (Pfa). The Pd value must be 0.9 or greater. The Pfa value must be less than 1e-6.\n\n```slantrngres = 1; % Required slant range resolution (m) azres = 1; % Required azimuth resolution (m) maxrng = 10e3; % Maximum unambiguous slant range (m) pd = 0.9; % Probability of detection pfa = 1e-6; % Probability of false alarm v = 100; % Velocity (m/s) h = 5000; % Radar altitude (m)```\n\n### Airborne SAR System Design\n\nSystem parameters like length of synthetic aperture, integration time, coverage rate, beamwidth for stripmap as well as spotlight modes, and signal bandwidth are key parameters that define the operational capability of a SAR system. These parameters ensure that the SAR system covers the region of interest for the calculated integration time with a wide beam. The calculated signal bandwidth meets the desired range resolution.\n\nSignal Configuration\n\nTo calculate the SAR system parameters, you must first know the wavelength of the propagating signal, which is inversely related is to the operating frequency of the system. For this example, set the operating frequency to 10 GHz which is typical airborne SAR systems.\n\nUse the `freq2wavelen` function to calculate the wavelength of the propagating signal.\n\n```freq = 10e9; % Radar frequency within X-band (Hz) lambda = freq2wavelen(freq) % Wavelength (m)```\n```lambda = 0.0300 ```\n\nThe signal bandwidth maps to the slant range resolution of SAR and the slant range resolution is the factor needed to distinguish two targets separated by a distance. The slant range resolution gives you the minimum range difference needed to distinguish two targets. Use the `rangeres2bw` function to calculate the signal bandwidth, which is determined by the slant range resolution.\n\n`pulse_bw = rangeres2bw(slantrngres) % Pulse bandwidth (Hz)`\n```pulse_bw = 149896229 ```\n\nStripmap SAR Mode\n\nStripmap SAR mode assumes a fixed pointing direction of the radar antenna relative to the direction of motion of platform. The antenna in this example points to the broadside direction.\n\nAntenna Orientation\n\nThe depression angle is often used to define the antenna pointing direction in elevation. This example assumes that the earth is flat so that the depression angle is the same as the grazing angle.\n\nUse the `grazingang` function to calculate the grazing angle from the line-of-sight range.\n\n`grazang = grazingang(h,maxrng,'Flat') % Grazing angle (in degrees)`\n```grazang = 30.0000 ```\n\nAntenna Azimuth Dimension\n\nNext, use the `sarlen` and `sarazres` functions to analyze and calculate the synthetic aperture length and its azimuth resolution for selecting the antenna azimuth dimension. Plot the synthetic length as a function of cross-range resolution. Plot the antenna azimuth dimension as a function of synthetic length.\n\n```dazv = [1 1.5 2 2.5 3]; % Antenna azimuth dimensions (m) striplenv = zeros(1,numel(dazv)); stripazresv = zeros(1,numel(dazv)); for i=1:numel(dazv) striplenv(i) = sarlen(maxrng,lambda,dazv(i)); stripazresv(i) = sarazres(maxrng,lambda,striplenv(i)); end helperPlotStripmapMode(stripazresv,striplenv,dazv,azres)```",
null,
"The figures show that a synthetic aperture length of 149.9 m for stripmap mode is a good value to meet a required azimuth resolution of 1 m. The smallest antenna azimuth dimension you can use for stripmap mode in this scenario is 2 m. Decrease the antenna azimuth dimension to obtain a better azimuth resolution than 1 m for stripmap mode.\n\nSet the synthetic aperture length to 149.9 m for stripmap mode and the antenna azimuth dimension of 2 m.\n\n`daz = 2`\n```daz = 2 ```\n`striplen = 149.9`\n```striplen = 149.9000 ```\n\nAntenna Elevation Dimension\n\nNext, determine the antenna elevation dimension based on the required swath length. For this example, assume that the required swath length is 2.4 km.\n\nUse the `aperture2swath` function to analyze the swath length for selecting an antenna elevation dimension.\n\n```rngswath = 2400; delv = [0.15 0.2 0.25 0.3 0.35]; % Elevation dimensions (m) rangeswathv = zeros(1,numel(delv)); for i=1:numel(delv) [rangeswathv(i),crngswath] = aperture2swath(maxrng,lambda,[delv(i) daz],grazang); end clf plot(rangeswathv,delv) grid on xline(rngswath,'-.',{[num2str(round(rngswath,2)),' m']}); % Selected range swath xlabel('Swath Length (m)') ylabel('Antenna Elevation Dimension (m)')```",
null,
"The figure indicates that an antenna elevation dimension of 0.25 m is appropriate given a swath length of 2400 m.\n\nSet the antenna elevation dimension to 0.25 m.\n\n`del = 0.25`\n```del = 0.2500 ```\n\nReal Antenna Beamwidth and Gain\n\nUse the `ap2beamwidth` function to calculate the real antenna beamwidth.\n\n`realAntBeamwidth = ap2beamwidth([daz del],lambda) % [Az El] (deg)`\n```realAntBeamwidth = 2×1 0.8588 6.8707 ```\n\nUse the `aperture2gain` function to calculate the antenna gain.\n\n`antGain = aperture2gain(daz*del, lambda) % dBi`\n```antGain = 38.4454 ```\n\nSynthetic Beamwidth, Processing Time, and Constraints\n\nNext, use the `sarbeamwidth`,` ``sarinttime`,` ``sarmaxcovrate`, and `sarmaxswath` functions to calculate the synthetic beamwidth, integration time, area coverage rate, and maximum swath length. Notice that the azimuth beamwidth for SAR system is much smaller than the azimuth beamwidth for a real aperture radar.\n\n```stripsynbw = sarbeamwidth(lambda,striplen); % Synthetic beamwidth (degrees) stripinttime = sarinttime(v,striplen); % Integration time (s) stripcovrate = sarmaxcovrate(azres,grazang); % Upper bound on coverage rate (m^2/s) stripswlen = sarmaxswath(v,azres,grazang); % Upper bound on swath length (m) RealAntenna = [realAntBeamwidth(1); NaN; NaN; NaN]; Parameters = [\"Synthetic Beamwidth\";\"Integration Time\";\"Upper Bound on Swath Length\";... \"Upper Bound on Area Coverage Rate\"]; StripmapSAR = [stripsynbw;stripinttime;round(stripcovrate/1e6,1);round(stripswlen/1e3)]; Units = [\"degrees\";\"s\";\"km^2/s\";\"km\"]; sarparams = table(Parameters,RealAntenna,StripmapSAR,Units)```\n```sarparams=4×4 table Parameters RealAntenna StripmapSAR Units ___________________________________ ___________ ___________ _________ \"Synthetic Beamwidth\" 0.85884 0.0057294 \"degrees\" \"Integration Time\" NaN 1.499 \"s\" \"Upper Bound on Swath Length\" NaN 173.1 \"km^2/s\" \"Upper Bound on Area Coverage Rate\" NaN 1731 \"km\" ```\n\nThe maximum possible azimuth resolution using SAR in this scenario is 1 m. However, to achieve this performance, pulses need to be integrated for over 1.5 s. The upper bound on the area coverage rate is 173 $k{m}^{2}$/s. The upper bound on the maximum swath length is 1731 km.\n\nSpotlight SAR Mode\n\nSpotlight SAR is capable of extending the SAR imaging capability to high resolution imaging significantly. This is possible since the spotlight mode ensures that the radar antenna squints instantaneously around the region being imaged thereby illuminating the target region for longer duration as compared to stripmap mode.\n\nCoherent Integration Angle\n\nThe azimuth resolution in stripmap mode is 1 m in this example. The resolution of spotlight mode is often expressed in terms of coherent integration angle of the radar boresight vector as the platform traverses the synthetic aperture length.\n\nUse the `sarintang` and `sarlen` functions to calculate the coherent integration angle and synthetic aperture length.\n\n`ciang = sarintang(lambda,azres) % (degrees)`\n```ciang = 0.8589 ```\n`len = sarlen(maxrng,'CoherentIntegrationAngle',ciang) % (m)`\n```len = 149.8976 ```\n\nThe best possible azimuth resolution in stripmap mode is 1 m for an antenna azimuth dimension of 2 m. Use the same antenna azimuth dimension of 2 m to obtain a better azimuth resolution of 0.5 m in spotlight mode. In spotlight mode, steer the radar beam to keep the target within for a longer time and thus form a longer synthetic aperture.\n\nNext, use the `sarlen` and `sarazres` functions to analyze the synthetic aperture length and its azimuth resolution over varying coherent integration angles.\n\n```spotazres = 0.5; % Azimuth resolution in spotlight SAR (m) intangv = 1:0.01:2.5; % Coherent integration angles (degrees) spotlenv = zeros(1,numel(intangv)); spotazresv = zeros(1,numel(intangv)); for i=1:numel(intangv) spotlenv(i) = sarlen(maxrng,'CoherentIntegrationAngle',intangv(i)); spotazresv(i) = sarazres(maxrng,lambda,spotlenv(i)); end helperPlotSpotlightMode(spotazresv,spotlenv,intangv,spotazres)```",
null,
"The figure indicates that in spotlight SAR mode, a synthetic aperture length of 300 m for the spotlight mode corresponding to an azimuth resolution of 0.5 m. For a coherent integration angle of 1.71 degrees, the azimuth resolution in spotlight mode is 0.5 m. It is important to note that decrease the antenna azimuth dimension to obtain a similar azimuth resolution in stripmap mode.\n\nSet the synthetic aperture length to 300 m and the coherent integration angle to 1.71 degrees for spotlight mode.\n\n`spotlen = 300`\n```spotlen = 300 ```\n`intang = 1.71`\n```intang = 1.7100 ```\n\nSynthetic Beamwidth, Processing Time, and Constraint\n\nCompared to the stripmap mode, spotlight mode can provide a better resolution, and a stronger signal from the scene at the cost of reduced scene size or area imaging rate.\n\nUse the `sarbeamwidth`,` ``sarinttime`,` ``sarmaxcovrate`, and `sarmaxswath` functions to calculate the synthetic beamwidth, integration time, area coverage rate, and maximum swath length. Notice that the area coverage rate and maximum swath length for spotlight SAR system are much smaller than for stripmap mode.\n\n```spotsynbw = sarbeamwidth(lambda,spotlen); % Synthetic beamwidth (degrees) spotinttime = sarinttime(v,spotlen); % Integration time (s) spotcovrate = sarmaxcovrate(spotazres,grazang); % Upper bound on coverage rate (m^2/s) spotswlen = sarmaxswath(v,spotazres,grazang); % Upper bound on swath length (m) SpotlightSAR = [spotsynbw;spotinttime;round(spotcovrate/1e6,1);round(spotswlen/1e3)]; sar = table(Parameters,StripmapSAR,SpotlightSAR,Units)```\n```sar=4×4 table Parameters StripmapSAR SpotlightSAR Units ___________________________________ ___________ ____________ _________ \"Synthetic Beamwidth\" 0.0057294 0.0028628 \"degrees\" \"Integration Time\" 1.499 3 \"s\" \"Upper Bound on Swath Length\" 173.1 86.5 \"km^2/s\" \"Upper Bound on Area Coverage Rate\" 1731 865 \"km\" ```\n\n### Azimuth Chirp Signal Parameters\n\nDetermine the azimuth chirp signal parameters which are the azimuth chirp rate, Doppler bandwidth, beam compression ratio, and azimuth bandwidth after dechirp. You can derive the azimuth time-bandwidth product. These are important for designing an accurate synthetic aperture processing mechanism in azimuth.\n\nUse the `sarchirprate` function to calculate the azimuth chirp rate, which is the rate at which the azimuth signal changes frequency as the sensor illuminates a scatterer.\n\n`azchirp = sarchirprate(maxrng,lambda,v); % (Hz/s)`\n\nAnalyze the azimuth chirp rate sensitivity to range and Doppler cone angle variations. The plot shows that increasing the unambiguous range of the radar reduces the azimuth chirp rate.\n\n```dcang = 60:1:120; % Doppler cone angles (in degrees) rngv = 1e3:100:maxrng; azchirpv = zeros(length(dcang),length(rngv)); for i = 1:length(dcang) azchirpv(i,:) = sarchirprate(rngv,lambda,v,dcang(i)); end clf mesh(rngv/1e3,dcang,azchirpv) xlabel('Range (km)') ylabel('Doppler Cone Angle (degrees)') zlabel('Azimuth Chirp Rate (Hz/s)') view([45 45]);```",
null,
"Use the `sarscenedopbw` function to calculate the scene bandwidth after azimuth dechirping. Assume a scene size of 916 m.\n\n```Wa = 916; bwdechirp = sarscenedopbw(maxrng,lambda,v,Wa); % (Hz)```\n\nAnalyze the scene bandwidth sensitivity to Doppler cone angle variations.\n\n```bwdechirpv = zeros(length(dcang),1); for i = 1:length(dcang) bwdechirpv(i,:) = sarscenedopbw(maxrng,lambda,v,Wa,dcang(i)); end clf plot(dcang,bwdechirpv) grid on xlabel('Doppler Cone Angle (degrees)') ylabel('Azimuth Bandwidth after Dechirp (Hz)')```",
null,
"Next, use the `sarpointdopbw`` `and` ``sarbeamcompratio` functions to calculate the Doppler bandwidth of the received signal from a point scatterer and the beam compression ratio. Notice that the Doppler bandwidth, and beam compression ratio for a spotlight SAR mode are much greater than for stripmap SAR mode.\n\n```% Stripmap SAR Mode stripbwchirp = sarpointdopbw(v,azres); % (Hz) striptbwaz = bwdechirp*stripinttime; % Unitless stripbcr = sarbeamcompratio(maxrng,lambda,striplen,Wa); % Unitless % Spotlight SAR Mode spotbwchirp = sarpointdopbw(v,spotazres); % (Hz) spottbwaz = bwdechirp*spotinttime; % Unitless spotbcr = sarbeamcompratio(maxrng,lambda,spotlen,Wa); % Unitless Parameters = [\"Doppler Bandwidth from Point Scatterer\";\"Azimuth Time-Bandwidth Product\";... \"Beam Compression Ratio\";\"Azimuth Chirp Rate\";\"Azimuth Bandwidth after Dechirp\"]; StripmapSAR = [stripbwchirp;striptbwaz;stripbcr;round(azchirp);bwdechirp]; SpotlightSAR = [spotbwchirp;round(spottbwaz);round(spotbcr);round(azchirp);bwdechirp]; Units = [\"Hz\";\"unitless\";\"unitless\";\"Hz/s\";\"Hz\"]; r = table(Parameters,StripmapSAR,SpotlightSAR,Units)```\n```r=5×4 table Parameters StripmapSAR SpotlightSAR Units ________________________________________ ___________ ____________ __________ \"Doppler Bandwidth from Point Scatterer\" 100 200 \"Hz\" \"Azimuth Time-Bandwidth Product\" 916.02 1833 \"unitless\" \"Beam Compression Ratio\" 916.02 1833 \"unitless\" \"Azimuth Chirp Rate\" 67 67 \"Hz/s\" \"Azimuth Bandwidth after Dechirp\" 611.09 611.09 \"Hz\" ```\n\n### SAR Power Calculation\n\nEstimate the peak power that must be transmitted using the power form of the radar equation for stripmap SAR mode. The required peak power depends upon many factors, including the maximum unambiguous range, the required SNR at the receiver, and the pulse width of the waveform. Among these factors, the required SNR at the receiver is determined by the design goal for the Pd and Pfa. Model and estimate the target RCS, the PRF, and different sources of gain and loss for the radar system and its environment.\n\nFirst, calculate the SNR required at the receiver. The relation between Pd, Pfa, and SNR can be best represented by a receiver operating characteristics (ROC) curve.\n\n```snr_db = [-inf, 0, 3, 10, 13]; rocsnr(snr_db);```",
null,
"The ROC curves show that to satisfy the design goals of Pfa = 1e-6 and Pd = 0.9, the SNR of the received signal must exceed 13 dB. You can surmise the SNR value by looking at the plot, but calculating only the required value is more straightforward. Using the `albersheim` function, derive the required SNR.\n\n`snr_min = albersheim(pd, pfa)`\n```snr_min = 13.1145 ```\n\nTarget RCS\n\nUse the `landreflectivity` function to calculate the reflectivity, which is the normalized radar cross-section (NRCS) for a given grazing angle and operating frequency. Then calculate the target RCS in the ground image plane using the `sarSurfaceRCS `function and taking the radar resolution into account.\n\n```landType = \"Smooth\"; nrcs = landreflectivity(landType,grazang,freq); % Calculate normalized RCS of smooth land with no vegetation tgtrcs = sarSurfaceRCS(nrcs,[slantrngres azres],grazang);```\n\nUpper and Lower PRF Bounds\n\nUse the `sarprfbounds` function to determine the minimum and maximum PRF values for a range swath and azimuth resolution given the radar velocity and the grazing angle.\n\n`[prfminv, prfmax] = sarprfbounds(v,azres,rngswath,grazang)`\n```prfminv = 100 ```\n```prfmax = 6.7268e+04 ```\n\nPRF Selection\n\nThe PRF is typically programmable and can be optimized for each application. Use the `sarprf` function to calculate the PRF of the radar based on the radar velocity and the real antenna dimension along the azimuth. Specify a constant roll-off factor as a safety margin to prevent mainlobe returns from aliasing in the PRF interval. If the PRF is set too low, the radar suffers from grating lobes and Doppler ambiguities. If the PRF is set too high, range measurements will be ambiguous.\n\n`prf = sarprf(v,daz,'RollOff',1.5)`\n```prf = 150 ```\n\nThe selected PRF is within the PRF bounds.\n\nProcessing Gains\n\nUse the `matchinggain` function to calculate the range processing gain due to the noise bandwidth reduction after the matched filter.\n\n```d = 0.05; % 5 percent duty cycle pw = (1/prf)*d; % Effective pulse width (s) rnggain = matchinggain(pw,pulse_bw) % Range processing gain (dB)```\n```rnggain = 46.9867 ```\n\nUse the `sarazgain` function to calculate the azimuth processing gain due to the coherent integration of pulses.\n\n`azgain = sarazgain(maxrng,lambda,v,azres,prf); % Az processing gain (dB)`\n\nLosses and Noise Factor\n\nUse the `noisefigure` function to estimate the noise figure of the cascaded receiver stages. Assume seven stages with the following values:\n\n• Stage 1 LNA: Noise Figure = 1.0 dB, Gain = 15.0\n\n• Stage 2 RF Filter: Noise Figure = 0.5 dB, Gain = -0.5\n\n• Stage 3 Mixer: Noise Figure = 5.0 dB, Gain = -7.0\n\n• Stage 4 IF Filter: Noise Figure = 1.0 dB, Gain = -1.0\n\n• Stage 5 IF Preamplifier: Noise Figure = 0.6 dB, Gain = 15.0\n\n• Stage 6 IF Stages: Noise Figure = 1.0 dB, Gain = 20.0\n\n• Stage 7 Phase Detectors: Noise Figure = 6.0 dB, Gain = -5.0\n\n```nf = [1.0, 0.5, 5.0, 1.0, 0.6, 1.0, 6.0]; % dB g = [15.0, -0.5, -7.0, -1.0, 15.0, 20.0, -5.0]; % dB cnf = noisefigure(nf, g)```\n```cnf = 1.5252 ```\n\nUse the `radarpropfactor` function to calculate the one-way radar propagation factor over smooth land.\n\n```[hgtsd, beta0, vegType] = landroughness('Smooth'); tgtheight = hgtsd; Re = effearthradius(maxrng,h,tgtheight); propf = radarpropfactor(maxrng,freq,h,tgtheight,'EffectiveEarthradius',Re,'TiltAngle',grazang,... 'ElevationBeamwidth',realAntBeamwidth(2),'SurfaceHeightStandardDeviation',hgtsd,'SurfaceSlope',beta0,... 'VegetationType',vegType)```\n```propf = -0.0042 ```\n\nUse the `tropopl` function to calculate losses due to atmospheric gaseous absorption.\n\n`atmoLoss = tropopl(maxrng,freq,tgtheight,grazang)`\n```atmoLoss = 0.0733 ```\n\nTransmit Power\n\nUse the `radareqsarpow` function to calculate the peak power with the SAR radar equation. You can also specify additional losses and factors, including the azimuth beam shape loss, window loss, transmission loss, and receive line loss. Estimate the beam shape loss with the `beamloss` function, and use 5 dB for all other fixed losses combined. For this analysis, specify `landType` as \"`Smooth`\" to use the weakest land target. A finite data collection time limits the total energy collected, and signal processing in the radar increases the SNR in the SAR image by two major gain factors. The first is due to pulse compression, and the second is due to coherent integration of pulses.\n\n```imgsnr = snr_min + rnggain + azgain; % (dB) Lb = beamloss; customLoss = 5; % dB sntemp = systemp(cnf); % Noise Temperature Pt = radareqsarpow(maxrng,lambda,imgsnr,pw,rnggain,azgain,'Gain',antGain,'RCS',tgtrcs,... 'AtmosphericLoss',atmoLoss,'Loss',cnf,'PropagationFactor',propf,... 'Ts',sntemp,'CustomFactor',-Lb-customLoss)```\n```Pt = 535.1030 ```\n\n### Summary\n\nThis example shows the aspects that must be calculated to design an X-band SAR system that can operate in stripmap and spotlight mode. The example shows that the same SAR system can operate in stripmap as well as spotlight modes and achieve varying levels of resolution depending upon the requirements at the cost of other parameters. First, you analyze and select the antenna dimensions to meet the required resolutions. Then you estimate the antenna gains, processing time, constraints, and the azimuth chirp signal parameters. Then estimate the required SNR, target RCS, PRF, processing gains and losses in the radar and its environment. Finally, you use the SAR equation to calculate the peak transmit power.\n\n```Parameters = [\"Antenna Dimension in Azimuth\";\"Antenna Dimension in Elevation\";\"Synthetic Aperture Length\";... \"Azimuth Resolution\";\"Synthetic Beamwidth\";\"Integration Time\";\"Upper Bound on Swath Length\";... \"Upper Bound on Area Coverage Rate\";\"Coherent Integration Angle\";\"Doppler Bandwidth from Point Scatterer\";... \"Azimuth Time-Bandwidth Product\";\"Beam Compression Ratio\";\"Azimuth Chirp Rate\";\"Azimuth Bandwidth after Dechirp\"]; Stripmap = [daz;del;striplen;azres;stripsynbw;stripinttime;round(stripcovrate/1e6,1);round(stripswlen/1e3);... NaN;stripbwchirp;striptbwaz;stripbcr;round(azchirp);bwdechirp]; Spotlight = [daz;del;spotlen;spotazres;spotsynbw;spotinttime;round(spotcovrate/1e6,1);round(spotswlen/1e3);... intang;spotbwchirp;round(spottbwaz);round(spotbcr);round(azchirp);bwdechirp]; Units = [\"m\";\"m\";\"m\";\"m\";\"degrees\";\"s\";\"km^2/s\";\"km\";\"degrees\";\"Hz\";\"unitless\";... \"unitless\";\"Hz/s\";\"Hz\"]; T = table(Parameters,Stripmap,Spotlight,Units)```\n```T=14×4 table Parameters Stripmap Spotlight Units ________________________________________ _________ _________ __________ \"Antenna Dimension in Azimuth\" 2 2 \"m\" \"Antenna Dimension in Elevation\" 0.25 0.25 \"m\" \"Synthetic Aperture Length\" 149.9 300 \"m\" \"Azimuth Resolution\" 1 0.5 \"m\" \"Synthetic Beamwidth\" 0.0057294 0.0028628 \"degrees\" \"Integration Time\" 1.499 3 \"s\" \"Upper Bound on Swath Length\" 173.1 86.5 \"km^2/s\" \"Upper Bound on Area Coverage Rate\" 1731 865 \"km\" \"Coherent Integration Angle\" NaN 1.71 \"degrees\" \"Doppler Bandwidth from Point Scatterer\" 100 200 \"Hz\" \"Azimuth Time-Bandwidth Product\" 916.02 1833 \"unitless\" \"Beam Compression Ratio\" 916.02 1833 \"unitless\" \"Azimuth Chirp Rate\" 67 67 \"Hz/s\" \"Azimuth Bandwidth after Dechirp\" 611.09 611.09 \"Hz\" ```\n\n### References\n\n Carrara, Walter G., Ronald M. Majewski, and Ron S. Goodman. Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Boston: Artech House, 1995.\n\nSupporting Functions\n\n`helperPlotStripmapMode`\n\n```function helperPlotStripmapMode(stripazresv,striplenv,dazv,azres) % Plot azimuth resolution vs. synthetic aperture length subplot(1,2,1) plot(stripazresv,striplenv) grid on xline(azres,'-.',{[num2str(round(azres)),' m']}); % Selected azimuth resolution xlabel('Azimuth or Cross-range Resolution (m)') ylabel('Synthetic Length (m)') stripidx = find(abs(striplenv-150)<1); % Index corresponding to required azimuth resolution % Plot synthetic aperture length vs. antenna azimuth dimensions subplot(1,2,2) plot(striplenv,dazv) grid on xline(striplenv(stripidx),'-.',{[num2str(round(striplenv(stripidx),2)),' m']}); % Selected synthetic length xlabel('Synthetic Length (m)') ylabel('Antenna Azimuth Dimension (m)') end```\n\n`helperPlotSpotlightMode`\n\n```function helperPlotSpotlightMode(spotazresv,spotlenv,intangv,spotazres) % Plot azimuth resolution vs. synthetic aperture length subplot(1,2,1) plot(spotazresv,spotlenv) grid on xline(0.5,'-.',{[num2str(round(spotazres,2)),' m']}); % Selected azimuth resolution xlabel('Azimuth or Cross-range Resolution (m)') ylabel('Synthetic Length (m)') spotidx = find(abs(spotlenv-300)<1); % Index corresponding to 0.5 m azimuth resolution % Plot synthetic aperture length vs. coherent integration angles subplot(1,2,2) plot(spotlenv,intangv) grid on xline(spotlenv(spotidx),'-.',{[num2str(round(spotlenv(spotidx))),' m']}); % Selected synthetic length xlabel('Synthetic Length (m)') ylabel('Coherent Integration Angle (degrees)') end```"
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https://latexhelp.com/latex-subset-of-above-equals/ | [
"# How to use subset of above equal(⫅) symbol in LaTeX?\n\nSymbol/Unicode Subset of Above Equals/U+2AC5\nType of symbol Mathematics\nPackage (requirement) amssymb\nArgument No\nLatex command \\subseteqq\nExample \\subseteqq → ⫅\n\nTo write the Subset of Above Equals symbol in Latex you need to use a package which is amssymb. And using the \\subseteqq command included in this package, you can print this symbol in the document.\n\n\\documentclass{article}\n\\usepackage{amssymb}\n\\begin{document}\n$A \\subseteqq B$\n$S_1 \\subseteqq S_2$\n$X_1 \\subseteqq X_2$\n\\end{document}\n\nOutput :",
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https://hoots.cs.depaul.edu/jriely/535/extras/full/Poly.html | [
"# PolyPolymorphism and Higher-Order Functions\n\nIn this chapter we continue our development of basic concepts of functional programming. The critical new ideas are polymorphism (abstracting functions over the types of the data they manipulate) and higher-order functions (treating functions as data).\n\nRequire Export Lists.\n\n# Polymorphism\n\n## Polymorphic Lists\n\nFor the last couple of chapters, we've been working just with lists of numbers. Obviously, interesting programs also need to be able to manipulate lists with elements from other types — lists of strings, lists of booleans, lists of lists, etc. We could just define a new inductive datatype for each of these, for example...\n\nInductive boollist : Type :=\n| bool_nil : boollist\n| bool_cons : bool boollist boollist.\n\n... but this would quickly become tedious, partly because we have to make up different constructor names for each datatype, but mostly because we would also need to define new versions of all our list manipulating functions (length, rev, etc.) for each new datatype definition.\nTo avoid all this repetition, Coq supports polymorphic inductive type definitions. For example, here is a polymorphic list datatype.\n\nInductive list (X:Type) : Type :=\n| nil : list X\n| cons : X list X list X.\n\nThis is exactly like the definition of natlist from the previous chapter, except that the nat argument to the cons constructor has been replaced by an arbitrary type X, a binding for X has been added to the header, and the occurrences of natlist in the types of the constructors have been replaced by list X. (We can re-use the constructor names nil and cons because the earlier definition of natlist was inside of a Module definition that is now out of scope.)\nWhat sort of thing is list itself? One good way to think about it is that list is a function from Types to Inductive definitions; or, to put it another way, list is a function from Types to Types. For any particular type X, the type list X is an Inductively defined set of lists whose elements are things of type X.\nWith this definition, when we use the constructors nil and cons to build lists, we need to tell Coq the type of the elements in the lists we are building — that is, nil and cons are now polymorphic constructors. Observe the types of these constructors:\n\nCheck nil.\n(* ===> nil : forall X : Type, list X *)\nCheck cons.\n(* ===> cons : forall X : Type, X -> list X -> list X *)\n\nThe \" X\" in these types can be read as an additional argument to the constructors that determines the expected types of the arguments that follow. When nil and cons are used, these arguments are supplied in the same way as the others. For example, the list containing 2 and 1 is written like this:\n\nCheck (cons nat 2 (cons nat 1 (nil nat))).\n\n(We've gone back to writing nil and cons explicitly here because we haven't yet defined the [] and :: notations for the new version of lists. We'll do that in a bit.)\nWe can now go back and make polymorphic (or \"generic\") versions of all the list-processing functions that we wrote before. Here is length, for example:\n\nFixpoint length (X:Type) (l:list X) : nat :=\nmatch l with\n| nil ⇒ 0\n| cons h tS (length X t)\nend.\n\nNote that the uses of nil and cons in match patterns do not require any type annotations: Coq already knows that the list l contains elements of type X, so there's no reason to include X in the pattern. (More precisely, the type X is a parameter of the whole definition of list, not of the individual constructors. We'll come back to this point later.)\nAs with nil and cons, we can use length by applying it first to a type and then to its list argument:\n\nExample test_length1 :\nlength nat (cons nat 1 (cons nat 2 (nil nat))) = 2.\nProof. reflexivity. Qed.\n\nTo use our length with other kinds of lists, we simply instantiate it with an appropriate type parameter:\n\nExample test_length2 :\nlength bool (cons bool true (nil bool)) = 1.\nProof. reflexivity. Qed.\n\nLet's close this subsection by re-implementing a few other standard list functions on our new polymorphic lists:\n\nFixpoint app (X : Type) (l1 l2 : list X)\n: (list X) :=\nmatch l1 with\n| nill2\n| cons h tcons X h (app X t l2)\nend.\n\nFixpoint snoc (X:Type) (l:list X) (v:X) : (list X) :=\nmatch l with\n| nilcons X v (nil X)\n| cons h tcons X h (snoc X t v)\nend.\n\nFixpoint rev (X:Type) (l:list X) : list X :=\nmatch l with\n| nilnil X\n| cons h tsnoc X (rev X t) h\nend.\n\nExample test_rev1 :\nrev nat (cons nat 1 (cons nat 2 (nil nat)))\n= (cons nat 2 (cons nat 1 (nil nat))).\nProof. reflexivity. Qed.\n\nExample test_rev2:\nrev bool (nil bool) = nil bool.\nProof. reflexivity. Qed.\n\nModule MumbleBaz.\n\n#### Exercise: 2 stars (mumble_grumble)\n\nConsider the following two inductively defined types.\n\nInductive mumble : Type :=\n| a : mumble\n| b : mumble nat mumble\n| c : mumble.\nInductive grumble (X:Type) : Type :=\n| d : mumble grumble X\n| e : X grumble X.\n\nWhich of the following are well-typed elements of grumble X for some type X?\n• d (b a 5)\n• d mumble (b a 5)\n• d bool (b a 5)\n• e bool true\n• e mumble (b c 0)\n• e bool (b c 0)\n• c\n(* FILL IN HERE *)\n\n#### Exercise: 2 stars (baz_num_elts)\n\nConsider the following inductive definition:\n\nInductive baz : Type :=\n| x : baz baz\n| y : baz bool baz.\n\nHow many elements does the type baz have? (* FILL IN HERE *)\n\nEnd MumbleBaz.\n\n### Type Annotation Inference\n\nLet's write the definition of app again, but this time we won't specify the types of any of the arguments. Will Coq still accept it?\n\nFixpoint app' X l1 l2 : list X :=\nmatch l1 with\n| nill2\n| cons h tcons X h (app' X t l2)\nend.\n\nIndeed it will. Let's see what type Coq has assigned to app':\n\nCheck app'.\n(* ===> forall X : Type, list X -> list X -> list X *)\nCheck app.\n(* ===> forall X : Type, list X -> list X -> list X *)\n\nIt has exactly the same type type as app. Coq was able to use a process called type inference to deduce what the types of X, l1, and l2 must be, based on how they are used. For example, since X is used as an argument to cons, it must be a Type, since cons expects a Type as its first argument; matching l1 with nil and cons means it must be a list; and so on.\nThis powerful facility means we don't always have to write explicit type annotations everywhere, although explicit type annotations are still quite useful as documentation and sanity checks. You should try to find a balance in your own code between too many type annotations (so many that they clutter and distract) and too few (which forces readers to perform type inference in their heads in order to understand your code).\n\n### Type Argument Synthesis\n\nWhenever we use a polymorphic function, we need to pass it one or more types in addition to its other arguments. For example, the recursive call in the body of the length function above must pass along the type X. But just like providing explicit type annotations everywhere, this is heavy and verbose. Since the second argument to length is a list of Xs, it seems entirely obvious that the first argument can only be X — why should we have to write it explicitly?\nFortunately, Coq permits us to avoid this kind of redundancy. In place of any type argument we can write the \"implicit argument\" _, which can be read as \"Please figure out for yourself what type belongs here.\" More precisely, when Coq encounters a _, it will attempt to unify all locally available information — the type of the function being applied, the types of the other arguments, and the type expected by the context in which the application appears — to determine what concrete type should replace the _.\nThis may sound similar to type annotation inference — and, indeed, the two procedures rely on the same underlying mechanisms. Instead of simply omitting the types of some arguments to a function, like\napp' X l1 l2 : list X :=\nwe can also replace the types with _, like\napp' (X : _) (l1 l2 : _) : list X :=\nwhich tells Coq to attempt to infer the missing information, just as with argument synthesis.\nUsing implicit arguments, the length function can be written like this:\n\nFixpoint length' (X:Type) (l:list X) : nat :=\nmatch l with\n| nil ⇒ 0\n| cons h tS (length' _ t)\nend.\n\nIn this instance, we don't save much by writing _ instead of X. But in many cases the difference can be significant. For example, suppose we want to write down a list containing the numbers 1, 2, and 3. Instead of writing this...\n\nDefinition list123 :=\ncons nat 1 (cons nat 2 (cons nat 3 (nil nat))).\n\n...we can use argument synthesis to write this:\n\nDefinition list123' := cons _ 1 (cons _ 2 (cons _ 3 (nil _))).\n\n### Implicit Arguments\n\nIf fact, we can go further. To avoid having to sprinkle _'s throughout our programs, we can tell Coq always to infer the type argument(s) of a given function. The Arguments directive specifies the name of the function or constructor, and then lists its argument names, with curly braces around any arguments to be treated as implicit.\n\nArguments nil {X}.\nArguments cons {X} _ _. (* use underscore for argument position that has no name *)\nArguments length {X} l.\nArguments app {X} l1 l2.\nArguments rev {X} l.\nArguments snoc {X} l v.\n\n(* note: no _ arguments required... *)\nDefinition list123'' := cons 1 (cons 2 (cons 3 nil)).\nCheck (length list123'').\n\nAlternatively, we can declare an argument to be implicit while defining the function itself, by surrounding the argument in curly braces. For example:\n\nFixpoint length'' {X:Type} (l:list X) : nat :=\nmatch l with\n| nil ⇒ 0\n| cons h tS (length'' t)\nend.\n\n(Note that we didn't even have to provide a type argument to the recursive call to length''; indeed, it is invalid to provide one.) We will use this style whenever possible, although we will continue to use use explicit Argument declarations for Inductive constructors.\nOne small problem with declaring arguments Implicit is that, occasionally, Coq does not have enough local information to determine a type argument; in such cases, we need to tell Coq that we want to give the argument explicitly this time, even though we've globally declared it to be Implicit. For example, suppose we write this:\n\n(* Definition mynil := nil. *)\n\nIf we uncomment this definition, Coq will give us an error, because it doesn't know what type argument to supply to nil. We can help it by providing an explicit type declaration (so that Coq has more information available when it gets to the \"application\" of nil):\n\nDefinition mynil : list nat := nil.\n\nAlternatively, we can force the implicit arguments to be explicit by prefixing the function name with @.\n\nCheck @nil.\n\nDefinition mynil' := @nil nat.\n\nUsing argument synthesis and implicit arguments, we can define convenient notation for lists, as before. Since we have made the constructor type arguments implicit, Coq will know to automatically infer these when we use the notations.\n\nNotation \"x :: y\" := (cons x y)\n(at level 60, right associativity).\nNotation \"[ ]\" := nil.\nNotation \"[ x ; .. ; y ]\" := (cons x .. (cons y []) ..).\nNotation \"x ++ y\" := (app x y)\n(at level 60, right associativity).\n\nNow lists can be written just the way we'd hope:\n\nDefinition list123''' := [1; 2; 3].\n\nCheck ([3 + 4] ++ nil).\n\n### Exercises: Polymorphic Lists\n\n#### Exercise: 2 stars, optional (poly_exercises)\n\nHere are a few simple exercises, just like ones in the Lists chapter, for practice with polymorphism. Fill in the definitions and complete the proofs below.\n\nFixpoint repeat {X : Type} (n : X) (count : nat) : list X :=\n(* FILL IN HERE *) admit.\n\nExample test_repeat1:\nrepeat true 2 = cons true (cons true nil).\n(* FILL IN HERE *) Admitted.\n\nTheorem nil_app : X:Type, l:list X,\napp [] l = l.\nProof.\n(* FILL IN HERE *) Admitted.\n\nTheorem rev_snoc : X : Type,\nv : X,\ns : list X,\nrev (snoc s v) = v :: (rev s).\nProof.\n(* FILL IN HERE *) Admitted.\n\nTheorem rev_involutive : X : Type, l : list X,\nrev (rev l) = l.\nProof.\n(* FILL IN HERE *) Admitted.\n\nTheorem snoc_with_append : X : Type,\nl1 l2 : list X,\nv : X,\nsnoc (l1 ++ l2) v = l1 ++ (snoc l2 v).\nProof.\n(* FILL IN HERE *) Admitted.\n\n## Polymorphic Pairs\n\nFollowing the same pattern, the type definition we gave in the last chapter for pairs of numbers can be generalized to polymorphic pairs (or products):\n\nInductive prod (X Y : Type) : Type :=\npair : X Y prod X Y.\n\nArguments pair {X} {Y} _ _.\n\nAs with lists, we make the type arguments implicit and define the familiar concrete notation.\n\nNotation \"( x , y )\" := (pair x y).\n\nWe can also use the Notation mechanism to define the standard notation for pair types:\n\nNotation \"X × Y\" := (prod X Y) : type_scope.\n\n(The annotation : type_scope tells Coq that this abbreviation should be used when parsing types. This avoids a clash with the multiplication symbol.)\nA note of caution: it is easy at first to get (x,y) and X×Y confused. Remember that (x,y) is a value built from two other values; X×Y is a type built from two other types. If x has type X and y has type Y, then (x,y) has type X×Y.\nThe first and second projection functions now look pretty much as they would in any functional programming language.\n\nDefinition fst {X Y : Type} (p : X × Y) : X :=\nmatch p with (x,y) ⇒ x end.\n\nDefinition snd {X Y : Type} (p : X × Y) : Y :=\nmatch p with (x,y) ⇒ y end.\n\nThe following function takes two lists and combines them into a list of pairs. In many functional programming languages, it is called zip. We call it combine for consistency with Coq's standard library. Note that the pair notation can be used both in expressions and in patterns...\n\nFixpoint combine {X Y : Type} (lx : list X) (ly : list Y)\n: list (X×Y) :=\nmatch (lx,ly) with\n| ([],_) ⇒ []\n| (_,[]) ⇒ []\n| (x::tx, y::ty) ⇒ (x,y) :: (combine tx ty)\nend.\n\n#### Exercise: 1 star, optional (combine_checks)\n\n• What is the type of combine (i.e., what does Check @combine print?)\n• What does\nEval compute in (combine [1;2] [false;false;true;true]).\nprint?\n\n#### Exercise: 2 stars (split)\n\nThe function split is the right inverse of combine: it takes a list of pairs and returns a pair of lists. In many functional programing languages, this function is called unzip.\nUncomment the material below and fill in the definition of split. Make sure it passes the given unit tests.\n\nFixpoint split\n{X Y : Type} (l : list (X×Y))\n: (list X) × (list Y) :=\n(* FILL IN HERE *) admit.\n\nExample test_split:\nsplit [(1,false);(2,false)] = ([1;2],[false;false]).\nProof.\n(* FILL IN HERE *) Admitted.\n\n## Polymorphic Options\n\nOne last polymorphic type for now: polymorphic options. The type declaration generalizes the one for natoption in the previous chapter:\n\nInductive option (X:Type) : Type :=\n| Some : X option X\n| None : option X.\n\nArguments Some {X} _.\nArguments None {X}.\n\nWe can now rewrite the index function so that it works with any type of lists.\n\nFixpoint index {X : Type} (n : nat)\n(l : list X) : option X :=\nmatch l with\n| [] ⇒ None\n| a :: l'if beq_nat n O then Some a else index (pred n) l'\nend.\n\nExample test_index1 : index 0 [4;5;6;7] = Some 4.\nProof. reflexivity. Qed.\nExample test_index2 : index 1 [;] = Some .\nProof. reflexivity. Qed.\nExample test_index3 : index 2 [true] = None.\nProof. reflexivity. Qed.\n\n#### Exercise: 1 star, optional (hd_opt_poly)\n\nComplete the definition of a polymorphic version of the hd_opt function from the last chapter. Be sure that it passes the unit tests below.\n\nDefinition hd_opt {X : Type} (l : list X) : option X :=\n(* FILL IN HERE *) admit.\n\nOnce again, to force the implicit arguments to be explicit, we can use @ before the name of the function.\n\nCheck @hd_opt.\n\nExample test_hd_opt1 : hd_opt [1;2] = Some 1.\n(* FILL IN HERE *) Admitted.\nExample test_hd_opt2 : hd_opt [;] = Some .\n(* FILL IN HERE *) Admitted.\n\n# Functions as Data\n\n## Higher-Order Functions\n\nLike many other modern programming languages — including all functional languages (ML, Haskell, Scheme, etc.) — Coq treats functions as first-class citizens, allowing functions to be passed as arguments to other functions, returned as results, stored in data structures, etc.\nFunctions that manipulate other functions are often called higher-order functions. Here's a simple one:\n\nDefinition doit3times {X:Type} (f:XX) (n:X) : X :=\nf (f (f n)).\n\nThe argument f here is itself a function (from X to X); the body of doit3times applies f three times to some value n.\n\nCheck @doit3times.\n(* ===> doit3times : forall X : Type, (X -> X) -> X -> X *)\n\nExample test_doit3times: doit3times minustwo 9 = 3.\nProof. reflexivity. Qed.\n\nExample test_doit3times': doit3times negb true = false.\nProof. reflexivity. Qed.\n\n## Partial Application\n\nIn fact, the multiple-argument functions we have already seen are also examples of passing functions as data. To see why, recall the type of plus.\n\nCheck plus.\n(* ==> nat -> nat -> nat *)\n\nEach in this expression is actually a binary operator on types. (This is the same as saying that Coq primitively supports only one-argument functions — do you see why?) This operator is right-associative, so the type of plus is really a shorthand for nat (nat nat) — i.e., it can be read as saying that \"plus is a one-argument function that takes a nat and returns a one-argument function that takes another nat and returns a nat.\" In the examples above, we have always applied plus to both of its arguments at once, but if we like we can supply just the first. This is called partial application.\n\nDefinition plus3 := plus 3.\nCheck plus3.\n\nExample test_plus3 : plus3 4 = 7.\nProof. reflexivity. Qed.\nExample test_plus3' : doit3times plus3 0 = 9.\nProof. reflexivity. Qed.\nExample test_plus3'' : doit3times (plus 3) 0 = 9.\nProof. reflexivity. Qed.\n\n## Digression: Currying\n\n#### Exercise: 2 stars, advanced (currying)\n\nIn Coq, a function f : A B C really has the type A (B C). That is, if you give f a value of type A, it will give you function f' : B C. If you then give f' a value of type B, it will return a value of type C. This allows for partial application, as in plus3. Processing a list of arguments with functions that return functions is called currying, in honor of the logician Haskell Curry.\nConversely, we can reinterpret the type A B C as (A × B) C. This is called uncurrying. With an uncurried binary function, both arguments must be given at once as a pair; there is no partial application.\nWe can define currying as follows:\n\nDefinition prod_curry {X Y Z : Type}\n(f : X × Y Z) (x : X) (y : Y) : Z := f (x, y).\n\nAs an exercise, define its inverse, prod_uncurry. Then prove the theorems below to show that the two are inverses.\n\nDefinition prod_uncurry {X Y Z : Type}\n(f : X Y Z) (p : X × Y) : Z :=\n(* FILL IN HERE *) admit.\n\n(Thought exercise: before running these commands, can you calculate the types of prod_curry and prod_uncurry?)\n\nCheck @prod_curry.\nCheck @prod_uncurry.\n\nTheorem uncurry_curry : (X Y Z : Type) (f : X Y Z) x y,\nprod_curry (prod_uncurry f) x y = f x y.\nProof.\n(* FILL IN HERE *) Admitted.\n\nTheorem curry_uncurry : (X Y Z : Type)\n(f : (X × Y) Z) (p : X × Y),\nprod_uncurry (prod_curry f) p = f p.\nProof.\n(* FILL IN HERE *) Admitted.\n\n## Filter\n\nHere is a useful higher-order function, which takes a list of Xs and a predicate on X (a function from X to bool) and \"filters\" the list, returning a new list containing just those elements for which the predicate returns true.\n\nFixpoint filter {X:Type} (test: Xbool) (l:list X)\n: (list X) :=\nmatch l with\n| [] ⇒ []\n| h :: tif test h then h :: (filter test t)\nelse filter test t\nend.\n\nFor example, if we apply filter to the predicate evenb and a list of numbers l, it returns a list containing just the even members of l.\n\nExample test_filter1: filter evenb [1;2;3;4] = [2;4].\nProof. reflexivity. Qed.\n\nDefinition length_is_1 {X : Type} (l : list X) : bool :=\nbeq_nat (length l) 1.\n\nExample test_filter2:\nfilter length_is_1\n[ [1; 2]; ; ; [5;6;7]; []; ]\n= [ ; ; ].\nProof. reflexivity. Qed.\n\nWe can use filter to give a concise version of the countoddmembers function from the Lists chapter.\n\nDefinition countoddmembers' (l:list nat) : nat :=\nlength (filter oddb l).\n\nExample test_countoddmembers'1: countoddmembers' [1;0;3;1;4;5] = 4.\nProof. reflexivity. Qed.\nExample test_countoddmembers'2: countoddmembers' [0;2;4] = 0.\nProof. reflexivity. Qed.\nExample test_countoddmembers'3: countoddmembers' nil = 0.\nProof. reflexivity. Qed.\n\n## Anonymous Functions\n\nIt is a little annoying to be forced to define the function length_is_1 and give it a name just to be able to pass it as an argument to filter, since we will probably never use it again. Moreover, this is not an isolated example. When using higher-order functions, we often want to pass as arguments \"one-off\" functions that we will never use again; having to give each of these functions a name would be tedious.\nFortunately, there is a better way. It is also possible to construct a function \"on the fly\" without declaring it at the top level or giving it a name; this is analogous to the notation we've been using for writing down constant lists, natural numbers, and so on.\n\nExample test_anon_fun':\ndoit3times (fun nn × n) 2 = 256.\nProof. reflexivity. Qed.\n\nHere is the motivating example from before, rewritten to use an anonymous function.\n\nExample test_filter2':\nfilter (fun lbeq_nat (length l) 1)\n[ [1; 2]; ; ; [5;6;7]; []; ]\n= [ ; ; ].\nProof. reflexivity. Qed.\n\n#### Exercise: 2 stars (filter_even_gt7)\n\nUse filter (instead of Fixpoint) to write a Coq function filter_even_gt7 that takes a list of natural numbers as input and returns a list of just those that are even and greater than 7.\n\nDefinition filter_even_gt7 (l : list nat) : list nat :=\n(* FILL IN HERE *) admit.\n\nExample test_filter_even_gt7_1 :\nfilter_even_gt7 [1;2;6;9;10;3;12;8] = [10;12;8].\n(* FILL IN HERE *) Admitted.\n\nExample test_filter_even_gt7_2 :\nfilter_even_gt7 [5;2;6;19;129] = [].\n(* FILL IN HERE *) Admitted.\n\n#### Exercise: 3 stars (partition)\n\nUse filter to write a Coq function partition:\npartition : X : Type,\n(X bool list X list X × list X\nGiven a set X, a test function of type X bool and a list X, partition should return a pair of lists. The first member of the pair is the sublist of the original list containing the elements that satisfy the test, and the second is the sublist containing those that fail the test. The order of elements in the two sublists should be the same as their order in the original list.\n\nDefinition partition {X : Type} (test : X bool) (l : list X)\n: list X × list X :=\n(* FILL IN HERE *) admit.\n\nExample test_partition1: partition oddb [1;2;3;4;5] = ([1;3;5], [2;4]).\n(* FILL IN HERE *) Admitted.\nExample test_partition2: partition (fun xfalse) [5;9;0] = ([], [5;9;0]).\n(* FILL IN HERE *) Admitted.\n\n## Map\n\nAnother handy higher-order function is called map.\n\nFixpoint map {X Y:Type} (f:XY) (l:list X)\n: (list Y) :=\nmatch l with\n| [] ⇒ []\n| h :: t ⇒ (f h) :: (map f t)\nend.\n\nIt takes a function f and a list l = [n1, n2, n3, ...] and returns the list [f n1, f n2, f n3,...] , where f has been applied to each element of l in turn. For example:\n\nExample test_map1: map (plus 3) [2;0;2] = [5;3;5].\nProof. reflexivity. Qed.\n\nThe element types of the input and output lists need not be the same (map takes two type arguments, X and Y). This version of map can thus be applied to a list of numbers and a function from numbers to booleans to yield a list of booleans:\n\nExample test_map2: map oddb [2;1;2;5] = [false;true;false;true].\nProof. reflexivity. Qed.\n\nIt can even be applied to a list of numbers and a function from numbers to lists of booleans to yield a list of lists of booleans:\n\nExample test_map3:\nmap (fun n ⇒ [evenb n;oddb n]) [2;1;2;5]\n= [[true;false];[false;true];[true;false];[false;true]].\nProof. reflexivity. Qed.\n\n## Map for options\n\n#### Exercise: 3 stars (map_rev)\n\nShow that map and rev commute. You may need to define an auxiliary lemma.\n\nTheorem map_rev : (X Y : Type) (f : X Y) (l : list X),\nmap f (rev l) = rev (map f l).\nProof.\n(* FILL IN HERE *) Admitted.\n\n#### Exercise: 2 stars (flat_map)\n\nThe function map maps a list X to a list Y using a function of type X Y. We can define a similar function, flat_map, which maps a list X to a list Y using a function f of type X list Y. Your definition should work by 'flattening' the results of f, like so:\nflat_map (fun n ⇒ [n;n+1;n+2]) [1;5;10]\n= [1; 2; 3; 5; 6; 7; 10; 11; 12].\n\nFixpoint flat_map {X Y:Type} (f:X list Y) (l:list X)\n: (list Y) :=\n(* FILL IN HERE *) admit.\n\nExample test_flat_map1:\nflat_map (fun n ⇒ [n;n;n]) [1;5;4]\n= [1; 1; 1; 5; 5; 5; 4; 4; 4].\n(* FILL IN HERE *) Admitted.\nLists are not the only inductive type that we can write a map function for. Here is the definition of map for the option type:\n\nDefinition option_map {X Y : Type} (f : X Y) (xo : option X)\n: option Y :=\nmatch xo with\n| NoneNone\n| Some xSome (f x)\nend.\n\n#### Exercise: 2 stars, optional (implicit_args)\n\nThe definitions and uses of filter and map use implicit arguments in many places. Replace the curly braces around the implicit arguments with parentheses, and then fill in explicit type parameters where necessary and use Coq to check that you've done so correctly. (This exercise is not to be turned in; it is probably easiest to do it on a copy of this file that you can throw away afterwards.)\n\n## Fold\n\nAn even more powerful higher-order function is called fold. This function is the inspiration for the \"reduce\" operation that lies at the heart of Google's map/reduce distributed programming framework.\n\nFixpoint fold {X Y:Type} (f: XYY) (l:list X) (b:Y) : Y :=\nmatch l with\n| nilb\n| h :: tf h (fold f t b)\nend.\n\nIntuitively, the behavior of the fold operation is to insert a given binary operator f between every pair of elements in a given list. For example, fold plus [1;2;3;4] intuitively means 1+2+3+4. To make this precise, we also need a \"starting element\" that serves as the initial second input to f. So, for example,\nfold plus [1;2;3;4] 0\nyields\n1 + (2 + (3 + (4 + 0))).\nHere are some more examples:\n\nCheck (fold andb).\n(* ===> fold andb : list bool -> bool -> bool *)\n\nExample fold_example1 : fold mult [1;2;3;4] 1 = 24.\nProof. reflexivity. Qed.\n\nExample fold_example2 : fold andb [true;true;false;true] true = false.\nProof. reflexivity. Qed.\n\nExample fold_example3 : fold app [;[];[2;3];] [] = [1;2;3;4].\nProof. reflexivity. Qed.\n\n#### Exercise: 1 star, advanced (fold_types_different)\n\nObserve that the type of fold is parameterized by two type variables, X and Y, and the parameter f is a binary operator that takes an X and a Y and returns a Y. Can you think of a situation where it would be useful for X and Y to be different?\n\n## Functions For Constructing Functions\n\nMost of the higher-order functions we have talked about so far take functions as arguments. Now let's look at some examples involving returning functions as the results of other functions.\nTo begin, here is a function that takes a value x (drawn from some type X) and returns a function from nat to X that yields x whenever it is called, ignoring its nat argument.\n\nDefinition constfun {X: Type} (x: X) : natX :=\nfun (k:nat) ⇒ x.\n\nDefinition ftrue := constfun true.\n\nExample constfun_example1 : ftrue 0 = true.\nProof. reflexivity. Qed.\n\nExample constfun_example2 : (constfun 5) 99 = 5.\nProof. reflexivity. Qed.\n\nSimilarly, but a bit more interestingly, here is a function that takes a function f from numbers to some type X, a number k, and a value x, and constructs a function that behaves exactly like f except that, when called with the argument k, it returns x.\n\nDefinition override {X: Type} (f: natX) (k:nat) (x:X) : natX:=\nfun (k':nat) ⇒ if beq_nat k k' then x else f k'.\n\nFor example, we can apply override twice to obtain a function from numbers to booleans that returns false on 1 and 3 and returns true on all other arguments.\n\nDefinition fmostlytrue := override (override ftrue 1 false) 3 false.\n\nExample override_example1 : fmostlytrue 0 = true.\nProof. reflexivity. Qed.\n\nExample override_example2 : fmostlytrue 1 = false.\nProof. reflexivity. Qed.\n\nExample override_example3 : fmostlytrue 2 = true.\nProof. reflexivity. Qed.\n\nExample override_example4 : fmostlytrue 3 = false.\nProof. reflexivity. Qed.\n\n#### Exercise: 1 star (override_example)\n\nBefore starting to work on the following proof, make sure you understand exactly what the theorem is saying and can paraphrase it in your own words. The proof itself is straightforward.\n\nTheorem override_example : (b:bool),\n(override (constfun b) 3 true) 2 = b.\nProof.\n(* FILL IN HERE *) Admitted.\nWe'll use function overriding heavily in parts of the rest of the course, and we will end up needing to know quite a bit about its properties. To prove these properties, though, we need to know about a few more of Coq's tactics; developing these is the main topic of the next chapter. For now, though, let's introduce just one very useful tactic that will also help us with proving properties of some of the other functions we have introduced in this chapter.\n\n# The unfold Tactic\n\nSometimes, a proof will get stuck because Coq doesn't automatically expand a function call into its definition. (This is a feature, not a bug: if Coq automatically expanded everything possible, our proof goals would quickly become enormous — hard to read and slow for Coq to manipulate!)\n\n3 + n = m\nplus3 n + 1 = m + 1.\nProof.\nintros m n H.\n(* At this point, we'd like to do rewrite H, since\nplus3 n is definitionally equal to 3 + n. However,\nCoq doesn't automatically expand plus3 n to its\ndefinition. *)\n\nAbort.\n\nThe unfold tactic can be used to explicitly replace a defined name by the right-hand side of its definition.\n\nTheorem unfold_example : m n,\n3 + n = m\nplus3 n + 1 = m + 1.\nProof.\nintros m n H.\nunfold plus3.\nrewrite H.\nreflexivity. Qed.\n\nNow we can prove a first property of override: If we override a function at some argument k and then look up k, we get back the overridden value.\n\nTheorem override_eq : {X:Type} x k (f:natX),\n(override f k x) k = x.\nProof.\nintros X x k f.\nunfold override.\nrewrite beq_nat_refl.\nreflexivity. Qed.\n\nThis proof was straightforward, but note that it requires unfold to expand the definition of override.\n\n#### Exercise: 2 stars (override_neq)\n\nTheorem override_neq : (X:Type) x1 x2 k1 k2 (f : natX),\nf k1 = x1\nbeq_nat k2 k1 = false\n(override f k2 x2) k1 = x1.\nProof.\n(* FILL IN HERE *) Admitted.\nAs the inverse of unfold, Coq also provides a tactic fold, which can be used to \"unexpand\" a definition. It is used much less often.\n\n#### Exercise: 2 stars (fold_length)\n\nMany common functions on lists can be implemented in terms of fold. For example, here is an alternative definition of length:\n\nDefinition fold_length {X : Type} (l : list X) : nat :=\nfold (fun _ nS n) l 0.\n\nExample test_fold_length1 : fold_length [4;7;0] = 3.\nProof. reflexivity. Qed.\n\nProve the correctness of fold_length.\n\nTheorem fold_length_correct : X (l : list X),\nfold_length l = length l.\n(* FILL IN HERE *) Admitted.\n\n#### Exercise: 3 stars (fold_map)\n\nWe can also define map in terms of fold. Finish fold_map below.\n\nDefinition fold_map {X Y:Type} (f : X Y) (l : list X) : list Y :=\n(* FILL IN HERE *) admit.\n\nWrite down a theorem in Coq stating that fold_map is correct, and prove it.\n\n(* FILL IN HERE *)\n\n(* \\$Date: 2013-09-26 13:40:26 -0500 (Thu, 26 Sep 2013) \\$ *)"
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https://www.colorhexa.com/cd58fc | [
"# #cd58fc Color Information\n\nIn a RGB color space, hex #cd58fc is composed of 80.4% red, 34.5% green and 98.8% blue. Whereas in a CMYK color space, it is composed of 18.7% cyan, 65.1% magenta, 0% yellow and 1.2% black. It has a hue angle of 282.8 degrees, a saturation of 96.5% and a lightness of 66.7%. #cd58fc color hex could be obtained by blending #ffb0ff with #9b00f9. Closest websafe color is: #cc66ff.\n\n• R 80\n• G 35\n• B 99\nRGB color chart\n• C 19\n• M 65\n• Y 0\n• K 1\nCMYK color chart\n\n#cd58fc color description : Soft violet.\n\n# #cd58fc Color Conversion\n\nThe hexadecimal color #cd58fc has RGB values of R:205, G:88, B:252 and CMYK values of C:0.19, M:0.65, Y:0, K:0.01. Its decimal value is 13457660.\n\nHex triplet RGB Decimal cd58fc `#cd58fc` 205, 88, 252 `rgb(205,88,252)` 80.4, 34.5, 98.8 `rgb(80.4%,34.5%,98.8%)` 19, 65, 0, 1 282.8°, 96.5, 66.7 `hsl(282.8,96.5%,66.7%)` 282.8°, 65.1, 98.8 cc66ff `#cc66ff`\nCIE-LAB 58.964, 70.112, -61.772 46.235, 26.988, 94.864 0.275, 0.161, 26.988 58.964, 93.442, 318.618 58.964, 41.052, -105.903 51.95, 67.949, -71.901 11001101, 01011000, 11111100\n\n# Color Schemes with #cd58fc\n\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #87fc58\n``#87fc58` `rgb(135,252,88)``\nComplementary Color\n• #7b58fc\n``#7b58fc` `rgb(123,88,252)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #fc58d9\n``#fc58d9` `rgb(252,88,217)``\nAnalogous Color\n• #58fc7b\n``#58fc7b` `rgb(88,252,123)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #d9fc58\n``#d9fc58` `rgb(217,252,88)``\nSplit Complementary Color\n• #58fccd\n``#58fccd` `rgb(88,252,205)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #fccd58\n``#fccd58` `rgb(252,205,88)``\n• #5887fc\n``#5887fc` `rgb(88,135,252)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #fccd58\n``#fccd58` `rgb(252,205,88)``\n• #87fc58\n``#87fc58` `rgb(135,252,88)``\n• #b70dfb\n``#b70dfb` `rgb(183,13,251)``\n• #be26fb\n``#be26fb` `rgb(190,38,251)``\n• #c63ffc\n``#c63ffc` `rgb(198,63,252)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #d571fc\n``#d571fc` `rgb(213,113,252)``\n• #dc8afd\n``#dc8afd` `rgb(220,138,253)``\n• #e4a3fd\n``#e4a3fd` `rgb(228,163,253)``\nMonochromatic Color\n\n# Alternatives to #cd58fc\n\nBelow, you can see some colors close to #cd58fc. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #a458fc\n``#a458fc` `rgb(164,88,252)``\n• #b258fc\n``#b258fc` `rgb(178,88,252)``\n• #bf58fc\n``#bf58fc` `rgb(191,88,252)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #db58fc\n``#db58fc` `rgb(219,88,252)``\n• #e858fc\n``#e858fc` `rgb(232,88,252)``\n• #f658fc\n``#f658fc` `rgb(246,88,252)``\nSimilar Colors\n\n# #cd58fc Preview\n\nThis text has a font color of #cd58fc.\n\n``<span style=\"color:#cd58fc;\">Text here</span>``\n#cd58fc background color\n\nThis paragraph has a background color of #cd58fc.\n\n``<p style=\"background-color:#cd58fc;\">Content here</p>``\n#cd58fc border color\n\nThis element has a border color of #cd58fc.\n\n``<div style=\"border:1px solid #cd58fc;\">Content here</div>``\nCSS codes\n``.text {color:#cd58fc;}``\n``.background {background-color:#cd58fc;}``\n``.border {border:1px solid #cd58fc;}``\n\n# Shades and Tints of #cd58fc\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, #050006 is the darkest color, while #fbf2ff is the lightest one.\n\n• #050006\n``#050006` `rgb(5,0,6)``\n• #12001a\n``#12001a` `rgb(18,0,26)``\n• #20012d\n``#20012d` `rgb(32,1,45)``\n• #2e0140\n``#2e0140` `rgb(46,1,64)``\n• #3c0154\n``#3c0154` `rgb(60,1,84)``\n• #4a0267\n``#4a0267` `rgb(74,2,103)``\n• #58027a\n``#58027a` `rgb(88,2,122)``\n• #66038d\n``#66038d` `rgb(102,3,141)``\n• #7303a1\n``#7303a1` `rgb(115,3,161)``\n• #8103b4\n``#8103b4` `rgb(129,3,180)``\n• #8f04c7\n``#8f04c7` `rgb(143,4,199)``\n• #9d04da\n``#9d04da` `rgb(157,4,218)``\n• #ab04ee\n``#ab04ee` `rgb(171,4,238)``\n• #b60bfb\n``#b60bfb` `rgb(182,11,251)``\n• #bc1efb\n``#bc1efb` `rgb(188,30,251)``\n• #c131fb\n``#c131fb` `rgb(193,49,251)``\n• #c745fc\n``#c745fc` `rgb(199,69,252)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\n• #d36bfc\n``#d36bfc` `rgb(211,107,252)``\n• #d97ffd\n``#d97ffd` `rgb(217,127,253)``\n• #de92fd\n``#de92fd` `rgb(222,146,253)``\n• #e4a5fd\n``#e4a5fd` `rgb(228,165,253)``\n• #eab8fe\n``#eab8fe` `rgb(234,184,254)``\n• #f0ccfe\n``#f0ccfe` `rgb(240,204,254)``\n• #f5dffe\n``#f5dffe` `rgb(245,223,254)``\n• #fbf2ff\n``#fbf2ff` `rgb(251,242,255)``\nTint Color Variation\n\n# Tones of #cd58fc\n\nA tone is produced by adding gray to any pure hue. In this case, #aca6ae is the less saturated color, while #cd58fc is the most saturated one.\n\n• #aca6ae\n``#aca6ae` `rgb(172,166,174)``\n• #aea0b4\n``#aea0b4` `rgb(174,160,180)``\n• #b199bb\n``#b199bb` `rgb(177,153,187)``\n• #b493c1\n``#b493c1` `rgb(180,147,193)``\n• #b78cc8\n``#b78cc8` `rgb(183,140,200)``\n• #b986ce\n``#b986ce` `rgb(185,134,206)``\n• #bc7fd5\n``#bc7fd5` `rgb(188,127,213)``\n• #bf79db\n``#bf79db` `rgb(191,121,219)``\n• #c272e2\n``#c272e2` `rgb(194,114,226)``\n• #c56ce8\n``#c56ce8` `rgb(197,108,232)``\n• #c765ef\n``#c765ef` `rgb(199,101,239)``\n• #ca5ff5\n``#ca5ff5` `rgb(202,95,245)``\n• #cd58fc\n``#cd58fc` `rgb(205,88,252)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #cd58fc 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"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.5185303,"math_prob":0.74173665,"size":3713,"snap":"2019-51-2020-05","text_gpt3_token_len":1632,"char_repetition_ratio":0.1286061,"word_repetition_ratio":0.011111111,"special_character_ratio":0.5251818,"punctuation_ratio":0.23809524,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9685993,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-19T19:12:41Z\",\"WARC-Record-ID\":\"<urn:uuid:3f036aa4-9ef4-441f-b9c6-36a33a22748f>\",\"Content-Length\":\"36321\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:67a83cb3-dcfc-4c36-8af0-eaf91ea91d0e>\",\"WARC-Concurrent-To\":\"<urn:uuid:764591c1-5d28-4b4f-aaa5-1a974da4ea04>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/cd58fc\",\"WARC-Payload-Digest\":\"sha1:I5GZ5HLYQIV5OWXHQIW2NEKHTVROAYZ5\",\"WARC-Block-Digest\":\"sha1:DCS3MAIXZQG7YT64PRMX6AO4UKTTJMWJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250594705.17_warc_CC-MAIN-20200119180644-20200119204644-00443.warc.gz\"}"} |
https://answers.everydaycalculation.com/add-fractions/4-7-plus-35-56 | [
"Solutions by everydaycalculation.com\n\n4/7 + 35/56 is 67/56.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 7 and 56 is 56\n2. For the 1st fraction, since 7 × 8 = 56,\n4/7 = 4 × 8/7 × 8 = 32/56\n3. Likewise, for the 2nd fraction, since 56 × 1 = 56,\n35/56 = 35 × 1/56 × 1 = 35/56"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.6008559,"math_prob":0.9999291,"size":286,"snap":"2019-26-2019-30","text_gpt3_token_len":117,"char_repetition_ratio":0.22340426,"word_repetition_ratio":0.0,"special_character_ratio":0.493007,"punctuation_ratio":0.08695652,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991885,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-17T16:32:51Z\",\"WARC-Record-ID\":\"<urn:uuid:55532f26-1695-4c1e-bf07-7ce05597fada>\",\"Content-Length\":\"7747\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d386e868-89dc-4de5-81fc-390149a0913b>\",\"WARC-Concurrent-To\":\"<urn:uuid:e1285f9f-ef7b-441b-8e23-4082230c9550>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/add-fractions/4-7-plus-35-56\",\"WARC-Payload-Digest\":\"sha1:PEITQJOK45SELNMMQ3HPO2QRO4ISEWY5\",\"WARC-Block-Digest\":\"sha1:VMUT7CCWBLLT7UCQNEPSW3QRWBYBGSZF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195525355.54_warc_CC-MAIN-20190717161703-20190717183703-00431.warc.gz\"}"} |
https://www.andlearning.org/margin-of-error-formula/ | [
"Connect with us\n\nMeasurement is the foundation of all mathematical concepts and this is not possible to imagine the world without measurements. The perfect measurements will increase the level of accuracy if they are based on international standards. Still, always measurement is suspected to small errors in mathematics or a level of uncertainty too. In simple words, the 100 percent accurate measurements are not possible in the practical world.\n\n## Error Formulas\n\nErrors are simply defined as the difference between the measured value and the actual value. For example, when two operators use the same device for the measurement then this is not necessary that results would be the same. The difference that occurs between the actual value and the measured value is named as the ERROR.\n\n$\\ Percentage\\;Error=\\frac{Approximate\\;Value-Exact\\;Value}{Exact\\;Value}\\times 100$\n\n$\\ Standard\\;Error =SE_{\\overline{x}}=\\frac{S}{\\sqrt{n}}$\n\nWhere,\ns is the standard deviation\nn is the number of observation\n\n$\\ Sampling\\;Error=\\pm \\sqrt{\\frac{2500}{Sample\\;Size}}\\times 1.96$\n\n$\\ E=Z\\left(\\frac{\\alpha}{2}\\right)\\left(\\frac{\\alpha}{\\sqrt{n}}\\right)$\n\nHere,\n$z$ $(\\frac{\\alpha }{2})$ = represents the critical value.\n$z$ $(\\frac{\\sigma }{\\sqrt{n}})$ = represents the standard deviation.\n\nTo learn the mathematics concepts deeply, you should know the different terms that could define the errors like sampling error, standard error, marginal error or percent error etc. Let us discuss on each of terms one by one with respective formulas. If you would understand these definitions and formulas deeply then there are chances that you could calculate the values as accurate as possible.\n\n### Sampling Error Formula\n\nThe error that arises due to sampling is named as the sampling error. This is the error usually related to the statistical analysis because of the wrong samples of the observations are taken. For example, the weight of 2000 citizens of a country are noted down and you need to calculate the average of weights now then it could be the same as the average weight of two million people.\n\nTo determine the weight of the whole population, the sampling technique is used. The difference between sample values and the population is termed as the sampling error. This is not possible to calculate the exact value of the population of you don’t know the value of sampling error and that could be found with sample modeling only.\n\nSo, the sampling error Formula in mathematics could be written as below –\n\n$\\ Sampling\\;Error=\\pm \\sqrt{\\frac{2500}{Sample\\;Size}}\\times 1.96$\n\n### Percent Error Formula\n\nThere could be a manufacturing error in measuring instruments too. This is not possible to assure them the exact. To know what type of error could be available here, we should know about the percentage error formula too. This is the absolute difference between measured value and the actual value and you should multiply the values by hundred too.\n\n$\\ Percentage\\;Error=\\frac{Approximate\\;Value-Exact\\;Value}{Exact\\;Value}\\times 100$\n\n### The Margin of Error Formula\n\nThe margin of errors is generally found in random sampling or the result of a survey. It is assumed that result of a sample is highly closers to the one would get from the population has been queried. In easy words, the margin of error is the product of critical value with the standard deviation. This is given by E and it could be written as –\n\n$\\ E=Z\\left(\\frac{\\alpha}{2}\\right)\\left(\\frac{\\alpha}{\\sqrt{n}}\\right)$\n\nHere,\n$\\ Z\\left(\\frac{\\alpha}{2}\\right)= represents\\;the\\;critical\\;value.$\n\n$\\ Z\\left(\\frac{\\alpha}{\\sqrt{n}}\\right) = represents\\;the\\;standard\\;deviation.$\n\n### Standard Error Formula\n\nStandard Error is the important statistical measure that is related to the standard deviation. The accuracy of a sample that could be presented by the population is given through the standard error formula and it could be written as below –\n\n$\\ Standard\\;Error =SE_{\\overline{x}}=\\frac{S}{\\sqrt{n}}$\n\nWhere,\ns is the standard deviation\nn is the number of observation\n\nWhere,S is the standard deviation, and n is the number of observations."
] | [
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https://tolstoy.newcastle.edu.au/R/devel/05/11/3035.html | [
"# Re: [Rd] [R] unvectorized option for outer()\n\nFrom: Duncan Murdoch <murdoch_at_stats.uwo.ca>\nDate: Tue 01 Nov 2005 - 14:16:53 GMT\n\nI did notice and work around one buglet in mapply: if you choose not to vectorize any arguments, you don't get a call to the original function, mapply returns \"list()\".\n\nFor example,\n\n> mapply(function(x) x^2, MoreArgs = list(x=2)) list()\n\nVectorize <- function(FUN, vectorize.args = arg.names, SIMPLIFY = TRUE, USE.NAMES = TRUE) {\n\n``` arg.names <- as.list(formals(FUN))\narg.names[[\"...\"]] <- NULL\narg.names <- names(arg.names)\n\nvectorize.args <- as.character(vectorize.args)\n\nif (!length(vectorize.args)) return(FUN)\n\nif (!all(vectorize.args %in% arg.names))\nstop(\"must specify formal argument names to vectorize\")\n\nFUNV <- function() { # will set the formals below\nargs <- lapply(as.list(match.call())[-1], eval, parent.frame())\ndovec <- match(vectorize.args, names(args), nomatch = 0)\ndo.call(\"mapply\", c(FUN = FUN,\nargs[dovec],\nMoreArgs = list(args[-dovec]),\nSIMPLIFY = SIMPLIFY,\nUSE.NAMES = USE.NAMES))\n}\nformals(FUNV) <- formals(FUN)\nFUNV\n}\n\n```\n\nDuncan Murdoch\n\nOn 10/31/2005 3:49 PM, Tony Plate wrote:\n> Duncan Murdoch wrote:\n\n```>> On 10/31/2005 2:15 PM, Tony Plate wrote:\n>>\n>>> [snipped comments irrelevant to this post]\n>>>\n>>> So, here's a first pass at a general Vectorize() function:\n>>>\n>>> Vectorize <- function(FUN, vectorize.args) {\n>>> if (!all(is.element(vectorize.args, names(formals(FUN)))))\n>>> stop(\"some args to vectorize are not args of FUN\")\n>>> FUNV <- eval(substitute(function(x, ...) mapply(FUN, x,\n>>> MoreArgs=list(...)), list(FUN=FUN)))\n>>> formals(FUNV) <- formals(FUNV)[c(rep(1, length(vectorize.args)), 2)]\n>>> names(formals(FUNV))[seq(along=vectorize.args)] <- vectorize.args\n>>> body(FUNV) <- body(FUNV)[c(1, 2, rep(3, length(vectorize.args)), 4)]\n>>> body(FUNV)[seq(3,len=length(vectorize.args))] <-\n>>> lapply(vectorize.args, as.name)\n>>> FUNV\n>>> }\n>>\n>>\n>> I'd think the formals of the result should be identical to the formals\n>> of the input.\n>>\n>> Regarding the environment of the result: it is used to determine the\n>> meaning of symbols that aren't defined within the function, e.g. things\n>> like \"eval\", \"substitute\", etc. So I'd say that you don't want anything\n>> special there, as long as you make sure that FUN is always evaluated in\n>> its original environment.\n>>\n>> Generally I don't like the look of that manipulation of the body of your\n>> result; it looks pretty fragile to me. But I haven't worked out exactly\n>> what you're doing, or whether it's possible to avoid it.\n>>\n>> Duncan Murdoch\n>>\n```\n\n>\n> Thanks for explanation about the environment.\n>\n> I should have said, that manipulation of the body creates the call\n> mapply(FUN, A, alpha, MoreArgs=list(...))\n> from the original (x is a dummy argument)\n> mapply(FUN, x, MoreArgs=list(...))\n>\n> Are there better ways to create that call? The difficulty is that the\n> argument names in the call are derived from the actual arguments to\n> Vectorize(), and there is an arbitrary number of them.\n>\n> As for the formals of the result being identical to the formals of the\n> input, I couldn't see any easy way to do that and still support optional\n> arguments, e.g., if the input function formals were (a, b, t=1), then\n> the result function would look something like:\n>\n> function(a, b, t=1) mapply(FUN, a, b, t=t)\n>\n> and missing(t) would not work correctly within FUN (with even more\n> serious problems for optional arguments with no defaults).\n>\n> -- Tony Plate\n>\n>\n```>>\n>>> ssd <- function(A,alpha,Y,t) sum((Y - A*exp(-alpha*t))2)\n>>> # SSD is a vectorized version of ssd\n>>> SSD <- function(Avec, alphavec, ...) mapply(ssd, Avec, alphavec,\n>>> MoreArgs=list(...))\n>>> # Vectorize(ssd, c(\"A\", \"alpha\")) should produce\n>>> # function(A, alpha, ...) mapply(ssd, A, alpha, MoreArgs=list(...))\n>>> Y <- 1:5; t <- 3\n>>> outer(1:3, 1:2, SSD, Y, t)\n>>> outer(1:3, 1:2, Vectorize(ssd, c(\"A\", \"alpha\")), Y, t)\n>>>\n>>> > # transcript of running the above commands\n>>> > Vectorize(ssd, c(\"A\", \"alpha\"))\n>>> function (A, alpha, ...)\n>>> mapply(function (A, alpha, Y, t)\n>>> sum((Y - A * exp(-alpha * t))^2), A, alpha, MoreArgs = list(...))\n>>> <environment: 0x1361f40>\n>>> > Y <- 1:5; t <- 3\n>>> > outer(1:3, 1:2, SSD, Y, t)\n>>> [,1] [,2]\n>>> [1,] 53.51878 54.92567\n>>> [2,] 52.06235 54.85140\n>>> [3,] 50.63071 54.77719\n>>> > outer(1:3, 1:2, Vectorize(ssd, c(\"A\", \"alpha\")), Y, t)\n>>> [,1] [,2]\n>>> [1,] 53.51878 54.92567\n>>> [2,] 52.06235 54.85140\n>>> [3,] 50.63071 54.77719\n>>> >\n>>>\n>>> [There are a couple of minor design issues around syntax -- what is\n>>> the best way of specifying the arguments to vectorize? (e.g., what\n>>> about an interface that allowed Vectorize(ssd ~ A * alpha)?), and\n>>> should the function name rather than the definition appear in the\n>>> result of Vectorize()? But those are issues of secondary importance.]\n>>>\n>>> I have to confess I don't really understand how environments work with\n>>> functions, so I don't know if this Vectorize() function will work in\n>>> general. What is the appropriate environment for returned value of\n>>> Vectorize()? Is this approach to creating a Vectorize() function on\n>>> the right tack at all? Any other improvements or fixes?\n>>>\n>>> -- Tony Plate\n>>>\n>>>\n>>> Peter Dalgaard wrote:\n>>>\n>>>> Thomas Lumley <[email protected]> writes:\n>>>>\n>>>>\n>>>>> On Sun, 30 Oct 2005, Jonathan Rougier wrote:\n>>>>>\n>>>>>\n>>>>>> that if people are writing functions in R that might be subject to\n>>>>>> simple operations like outer products, then they ought to be writing\n>>>>>> vectorised functions!\n>>>>>\n>>>>>\n>>>>> I would agree. How about an oapply() function that does multiway\n>>>>> (rather than just two-way) outer products. Basing the name on\n>>>>> \"apply\" would emphasize the similarity to other flexible, not\n>>>>> particularly optimized second-order functions.\n>>>>\n>>>>\n>>>>\n>>>> In fairness, it should probably be said that not all problems\n>>>> vectorize naturally. One example is\n>>>>\n>>>> ssd <- function(A,alpha) sum((Y - A*exp(-alpha*t))^2)\n>>>>\n>>>> However, it should be worth noting that with the mapply() function at\n>>>> hand, it is pretty easy to turn a non-vectorized function into a\n>>>> vectorized one.\n>>>> SSD <- function(A,alpha) mapply(ssd, A, alpha)\n>>>>\n>>>> (Anybody want to try their hand on writing a general Vectorize()\n>>>> function? I.e. one that allowed\n>>>>\n>>>> outer(Avec, alphavec, Vectorize(ssd))\n>>>>\n>>>> to work.)\n>>>\n>>>\n>>> ______________________________________________\n>>> [email protected] mailing list\n>>> https://stat.ethz.ch/mailman/listinfo/r-devel\n>>\n>>\n```\n\n>\n> ______________________________________________\n> [email protected] mailing list\n> https://stat.ethz.ch/mailman/listinfo/r-devel\n\[email protected] mailing list\nhttps://stat.ethz.ch/mailman/listinfo/r-devel Received on Wed Nov 02 01:21:41 2005\n\nThis archive was generated by hypermail 2.1.8 : Mon 20 Feb 2006 - 03:21:33 GMT"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.79614586,"math_prob":0.8113882,"size":6278,"snap":"2020-24-2020-29","text_gpt3_token_len":1777,"char_repetition_ratio":0.17086388,"word_repetition_ratio":0.1427094,"special_character_ratio":0.36811087,"punctuation_ratio":0.21133603,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9518764,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-02T13:34:16Z\",\"WARC-Record-ID\":\"<urn:uuid:d88b9d82-e399-43d0-81bb-b04e8c9a998d>\",\"Content-Length\":\"16563\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:01eda558-b27a-4f81-810b-9d0345b97a8e>\",\"WARC-Concurrent-To\":\"<urn:uuid:267dbfde-064a-41a4-9d4d-cd60ffeae99d>\",\"WARC-IP-Address\":\"99.84.176.57\",\"WARC-Target-URI\":\"https://tolstoy.newcastle.edu.au/R/devel/05/11/3035.html\",\"WARC-Payload-Digest\":\"sha1:XKV6B4CYQEC5UWAVORK5H7YRVC6MDKLG\",\"WARC-Block-Digest\":\"sha1:3W6FDDZ4YY34HF7DXAKQR7VDSQDEZD5K\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655878753.12_warc_CC-MAIN-20200702111512-20200702141512-00378.warc.gz\"}"} |
https://plainmath.org/secondary/math-word-problem?page=6 | [
"",
null,
"# Mastering Math Word Problems\n\nRecent questions in Math Word Problem\nMath Word ProblemOpen question",
null,
"Shi Pra2022-06-04\n\n## Draw the Hasse diagram for the partial ordering { (a, b) / a divides b } on {1,2,3,4,6,8,12}\n\nMath Word ProblemOpen question",
null,
"zstrader 2022-06-03\n\n## Find the area of the shaded region. Round your answer to the nearest tenth.d=5",
null,
"",
null,
"Artimaia Lyngdoh2022-06-03\n\n## 2x1 +3x2 -x3 =15 4x2 +2x3 =16 3x1 +2x3=18\n\nMath Word ProblemOpen question",
null,
"Ashley Gwen Esguerra2022-06-03\n\n## 4. a rectangular prism with a base area of 30 square meters and a height of 7 meters",
null,
"Nyeleti Rachel2022-06-02\n\n## A workshop venue has 35 participants. Each participants shakes hands with each and every other participant.a) How many handshakes will be made by the first participants?b) How is the sum of handshakes that will be made by the first and second participant?",
null,
"Math Word ProblemOpen question",
null,
"Nyeleti Rachel2022-06-02\n\n##",
null,
"Math Word ProblemOpen question",
null,
"EMILY MENDEZ2022-06-02\n\n## what is the mean median and mode for 16, 18, 15, 16, 21, 16.",
null,
"Michael Kindu2022-06-02\n\n##",
null,
"",
null,
"Avikash Prasad2022-06-02\n\n## 1.) The function g is related to one of the parent functions g(x) = x^2 + 6The parent function f is:f(x)= x^2Use function notation to write g in terms of f.\n\nMath Word ProblemOpen question",
null,
"giannis k2022-06-02\n\n## y''+ y'- 2y = e^-x when y(0)=6 and y'(0)=7 laplace transform\n\nMath Word ProblemOpen question",
null,
"aqsanaeem2007 2022-06-01\n\n## The temperature in a place in Antarctica at 12 o’clock is 5’celsius and every hour it drops 4 ‘ celsius express this relationship using one equation and represent it graphically",
null,
"Ronelyn Lynlyn2022-05-31\n\n## Use the promlem below, there are 4 blue marbles 5 red marbles 1 green marbles 2 black marbles in a bag suppose you select one marble at random find each probability ofP( not green)\n\nMath Word ProblemOpen question",
null,
"Ricardo Dimakiling2022-05-31\n\n## How would you express the absolute value of -5\n\nMath Word ProblemOpen question",
null,
"Yash Alhat2022-05-30\n\n## for what value of k will the vector (1,-2,k) in R^3 be a linear combination of the vectors (2,-1,-5)",
null,
"Pia Oliva2022-05-30\n\n## Create one real-life word problem with a solution and answer Fundamental Counting Principle.\n\nMath Word ProblemOpen question",
null,
"Andure Leonicar2022-05-28\n\n## In a 45° -45° -90° triangle, how long are the legs if the hypotenuse is 8?\n\nMath Word ProblemOpen question",
null,
"zulhelmi akmal2022-05-27\n\n## 2.8 × 0.14\n\nMath Word ProblemOpen question",
null,
"Lavanya Tmpd2022-05-27\n\n## A Company has examined its cost structure and revenue structure and has determined that C, the total cost, R, the total revenue and x, the number of units produced are related as: C=100+0.015x^2and R=3x. Find the production level x that will maximize the profits of the company. Find that profit. Also find the profit when x = 120",
null,
"Theophilus Tawiah2022-05-27",
null,
"jmehrishi 2022-05-24"
] | [
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"https://plainmath.org/build/images/user-picture/user-6.png",
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.88265073,"math_prob":0.8240922,"size":2050,"snap":"2023-40-2023-50","text_gpt3_token_len":568,"char_repetition_ratio":0.10361681,"word_repetition_ratio":0.01055409,"special_character_ratio":0.28585365,"punctuation_ratio":0.1173913,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9727676,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],"im_url_duplicate_count":[null,null,null,null,null,null,null,1,null,null,null,null,null,null,null,2,null,null,null,2,null,null,null,null,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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-02T17:08:48Z\",\"WARC-Record-ID\":\"<urn:uuid:3f3c9df1-05af-4623-8b98-ca0895a07d8c>\",\"Content-Length\":\"191812\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7aebb1da-a163-4ceb-a380-c48182c0d552>\",\"WARC-Concurrent-To\":\"<urn:uuid:41cce063-8388-4da3-9ef3-8ba3c04a11c7>\",\"WARC-IP-Address\":\"172.67.217.107\",\"WARC-Target-URI\":\"https://plainmath.org/secondary/math-word-problem?page=6\",\"WARC-Payload-Digest\":\"sha1:C3V4GWOCT26FGUX6M27OIRZ5K5DEBUFP\",\"WARC-Block-Digest\":\"sha1:WY4L7XRVZRZQY7YF64B26GG4JRLAGFU4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511002.91_warc_CC-MAIN-20231002164819-20231002194819-00368.warc.gz\"}"} |
https://www.mytech-info.com/2013/07/electric-current-definition.html | [
"# ELECTRIC CURRENT DEFINITION",
null,
"A schematic diagram of an ideal voltage source, V, driving a resistor, R, and creating a current I.\nAn electric current could be a flow of electrical charge. Electrical phenemenon flows once there's voltage gift across a conductor. In electrical circuits current charge is commonly carried by moving electrons in an exceedingly wire. It may be carried by ions in A solution, or by each ions and electrons like in an exceedingly plasma.\n\nThe SI unit for measure an electrical current is that the ampere, which is that the flow of electrical charges through a surface at the speed of 1 coulomb per second. Electrical phenomenon is measured using an ammeter.\n\nElectric currents cause several effects, notably heating, however conjointly induce magnetic fieldss, that are wide used for motors, inductors and generators.\n\nThe conventional image for current is, that originates from the French phrase intensite DE courant, or in English current intensity. This phrase is often used once discussing the worth of an electrical current, however fashionable observe usually shortens this to easily current.\n\n## Ohm's law\n\nOhm's law conditions that the present through a conductor between 2 nodes (points) is directly proportional to the electrical phenomenon across the 2 points. Introducing the constant of quotient, the resistance, one arrives at the standard mathematical equation that describes this relationship:\n\nI=V/R\n\nWhere I is that the current through the conductor in unit of amperes, V is that the potential drop measured across the conductor in unit of volts, and R is that the resistance of the conductor in unit of ohms. A lot specifically, law states that the R during this relation is constant, freelance of the present.\nIn generally current can be classified into two types. They are\n\n1. Alternating Current (AC)\n2. Direct Current (DC)\n\n### Direct current\n\nDirect current (DC) is that the current flow of electric charge. Electrical energy is made by sources like batteries, thermo couples, star cells, and commutator – type electrical machines of the generator kind. Electrical energy could flow during a conductor like a wire, however may also flow through semiconductors, insulators, or maybe through a vacuum as in negatron or particle beams. The electrical charge flows during a constant direction, distinctive it from electrical energy (AC). A term erstwhile used for electrical energy was galvanic current.\n\n### Alternating current\n\nIn electrical energy (AC, also ac), the movement of electrical charge sporadically reverses direction. In electrical energy (DC, also dc), the flow of electrical charge is simply in one direction.\n\nAC is that the kind within which wattage is delivered to businesses and residences. The same old wave form of associate AC power circuit may be a wave. In bound applications, totally different waveform area unit used, like triangular or square waves. Audio and radio signals carried on electrical wires are samples of electrical energy. In these applications, a crucial goal is usually the recovery of knowledge encoded (or modulated) onto the AC signal.",
null,
"0 Comments",
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"Comments"
] | [
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"https://lh3.googleusercontent.com/-Tua-Jg4kV2s/V1fAK7ILFRI/AAAAAAAAFhY/Y8wHr34tWFA/Ohms-law-voltage-source%25255B6%25255D.png",
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"https://4.bp.blogspot.com/-XGaTwcqojP4/VCcOUK35SsI/AAAAAAAAOGw/FZfnRcEyM7Y/s1600/fb-icon.png",
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http://mizar.org/version/current/html/proofs/jordan1e/12_1_1 | [
"E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;\nthen A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by ;\nlen (Upper_Seq (C,n)) = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by Th8;\nhence not Upper_Seq (C,n) is empty by ; :: thesis: verum"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7044145,"math_prob":0.99965644,"size":350,"snap":"2022-40-2023-06","text_gpt3_token_len":161,"char_repetition_ratio":0.27456647,"word_repetition_ratio":0.2,"special_character_ratio":0.49142858,"punctuation_ratio":0.25510204,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999987,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-27T02:36:03Z\",\"WARC-Record-ID\":\"<urn:uuid:87cdd8b9-b13d-4a5d-b85b-38801d664231>\",\"Content-Length\":\"5329\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bc349c7c-876e-4c2c-8943-98f337c46545>\",\"WARC-Concurrent-To\":\"<urn:uuid:68cc775e-be02-4135-9744-4313f3f66a8c>\",\"WARC-IP-Address\":\"193.219.28.149\",\"WARC-Target-URI\":\"http://mizar.org/version/current/html/proofs/jordan1e/12_1_1\",\"WARC-Payload-Digest\":\"sha1:I3FTG4VWG2IG5XUCIETFMNSSX3LXW5IN\",\"WARC-Block-Digest\":\"sha1:2GBUYMQ7EWNSZWV6ARCA7ACISIOQD4EG\",\"WARC-Identified-Payload-Type\":\"application/xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334974.57_warc_CC-MAIN-20220927002241-20220927032241-00504.warc.gz\"}"} |
https://daima100.com/35133.html | [
"# 6 其它区别\n\nsizeof引用的时候是对象的大小，sizeof指针是指针本身的大小；\n\n# 7 代码说明\n\nint a,b,*p,&r=a; //正确\n\nr=3; //正确：等价于a=3\n\nint &rr; //出错：引用必须初始化\n\np=&a; //正确：p中存储a的地址，即p指向a\n\n*p=4; //正确：p中存的是a的地址，对a所对应的存储空间存入值4\n\np=&b; //正确：p可以多次赋值，p存储b的地址\n\n# 通俗化说明\n\nint a = 0;\n\nint b = 1;\n\nint *point = NULL;\n\npoint = &a; // 在某个时刻，指针可以指向a\n\npoint = &b; // 换个时刻，指针可以指向b\n\nint xiaoming = 1;\n\nint &refence_mingming = xiaoming;\n\nint xiaoan = 2;\n\nrefence_mingming = xiaoan; // error,引用不能换了\n\n# 从对象的角度理解指针\n\nC++primer中对 对象的定义：对象是指一块能存储数据并具有某种类型的内存空间\n\n# 引用的主要功能是传递函数的参数和返回值。\n\nC++语言中，函数的参数和返回值的传递方式有三种：值传递、 指针传递和引用传递。\n\nvoid Func1(int x) {\n\nx = x + 10;\n\n}\n\nint n = 0;\n\nFunc1(n);\n\ncout << “n = ” << n << endl; // n = 0\n\nvoid Func2(int *x) {\n\n(* x) = (* x) + 10;\n\n}\n\nint n = 0;\n\nFunc2(&n);\n\ncout << “n = ” << n << endl; // n = 10\n\nvoid Func3(int &x) {\n\nx = x + 10;\n\n}\n\nint n = 0;\n\nFunc3(n);\n\ncout << “n = ” << n << endl; // n = 10\n\n－End－",
null,
"(0)\n\n### 相关推荐\n\n• #### 单板透视（单板透视会封号吗）\n\n单板透视（单板透视会封号吗）\n\n• #### mcafee企业版8.8（mcafee手机版）\n\nmcafee企业版8.8（mcafee手机版）\n\nr2v教程（r2s教程）\n\n• #### 华硕人脸识别（华硕win10人脸识别设置不了）\n\n华硕人脸识别（华硕win10人脸识别设置不了）\n\n• #### skiller3.5（skills for success 3）\n\nskiller3.5（skills for success 3）\n\n• #### timestampdiff（oracle中timestamp）\n\ntimestampdiff（oracle中timestamp）\n\n• #### node.js安装（为什么nodejs不适合大型项目）\n\nnode.js安装（为什么nodejs不适合大型项目）\n\n• #### 好帮手软件（作业好帮手下载）\n\n好帮手软件（作业好帮手下载）"
] | [
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"https://daima100.com/wp-content/themes/justnews/themer/assets/images/lazy.png",
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https://smlnj-gforge.cs.uchicago.edu/scm/viewvc.php/sml/trunk/benchmarks/programs/life/life.sml?view=markup&revision=193&root=smlnj&sortby=file&pathrev=668 | [
"Home My Page Projects Code Snippets Project Openings SML/NJ\n Summary Activity Forums Tracker Lists Tasks Docs Surveys News SCM Files\n\n# SCM Repository\n\n[smlnj] View of /sml/trunk/benchmarks/programs/life/life.sml\n [smlnj] / sml / trunk / benchmarks / programs / life / life.sml",
null,
"# View of /sml/trunk/benchmarks/programs/life/life.sml\n\nFri Nov 20 17:43:59 1998 UTC (22 years, 10 months ago) by monnier\nFile size: 5074 byte(s)\n```Initial revision\n```\n```structure Main : BMARK =\nstruct\n\nfun map f [] = []\n| map f (a::x) = f a :: map f x\n\nexception ex_undefined of string\nfun error str = raise ex_undefined str\n\nfun accumulate f = let\nfun foldf a [] = a\n| foldf a (b::x) = foldf (f a b) x\nin\nfoldf\nend\n\nfun filter p = let\nfun consifp x a = if p a then a::x else x\nin\nrev o accumulate consifp []\nend\n\nfun exists p = let fun existsp [] = false\n| existsp (a::x) = if p a then true else existsp x\nin existsp end\n\nfun equal a b = (a = b)\n\nfun member x a = exists (equal a) x\n\nfun C f x y = f y x\n\nfun cons a x = a::x\n\nfun revonto x = accumulate (C cons) x\n\nfun length x = let fun count n a = n+1 in accumulate count 0 x end\n\nfun repeat f = let fun rptf n x = if n=0 then x else rptf(n-1)(f x)\nfun check n = if n<0 then error \"repeat<0\" else n\nin rptf o check end\n\nfun copy n x = repeat (cons x) n []\n\nfun spaces n = concat (copy n \" \")\n\nlocal\nfun lexordset [] = []\n| lexordset (a::x) = lexordset (filter (lexless a) x) @ [a] @\nlexordset (filter (lexgreater a) x)\nand lexless(a1:int,b1:int)(a2,b2) =\nif a2<a1 then true else if a2=a1 then b2<b1 else false\nand lexgreater pr1 pr2 = lexless pr2 pr1\nfun collect f list =\nlet fun accumf sofar [] = sofar\n| accumf sofar (a::x) = accumf (revonto sofar (f a)) x\nin accumf [] list\nend\nfun occurs3 x =\n(* finds coords which occur exactly 3 times in coordlist x *)\nlet fun f xover x3 x2 x1 [] = diff x3 xover\n| f xover x3 x2 x1 (a::x) =\nif member xover a then f xover x3 x2 x1 x else\nif member x3 a then f (a::xover) x3 x2 x1 x else\nif member x2 a then f xover (a::x3) x2 x1 x else\nif member x1 a then f xover x3 (a::x2) x1 x else\nf xover x3 x2 (a::x1) x\nand diff x y = filter (not o member y) x\nin f [] [] [] [] x end\nin\nabstype generation = GEN of (int*int) list\nwith\nfun alive (GEN livecoords) = livecoords\nand mkgen coordlist = GEN (lexordset coordlist)\nand mk_nextgen_fn neighbours gen =\nlet val living = alive gen\nval isalive = member living\nval liveneighbours = length o filter isalive o neighbours\nfun twoorthree n = n=2 orelse n=3\nval survivors = filter (twoorthree o liveneighbours) living\nval newnbrlist = collect (filter (not o isalive) o neighbours) living\nval newborn = occurs3 newnbrlist\nin mkgen (survivors @ newborn) end\nend\nend\n\nfun neighbours (i,j) = [(i-1,j-1),(i-1,j),(i-1,j+1),\n(i,j-1),(i,j+1),\n(i+1,j-1),(i+1,j),(i+1,j+1)]\n\nlocal val xstart = 0 and ystart = 0\nfun markafter n string = string ^ spaces n ^ \"0\"\nfun plotfrom (x,y) (* current position *)\nstr (* current line being prepared -- a string *)\n((x1,y1)::more) (* coordinates to be plotted *)\n= if x=x1\nthen (* same line so extend str and continue from y1+1 *)\nplotfrom(x,y1+1)(markafter(y1-y)str)more\nelse (* flush current line and start a new line *)\nstr :: plotfrom(x+1,ystart)\"\"((x1,y1)::more)\n| plotfrom (x,y) str [] = [str]\nfun good (x,y) = x>=xstart andalso y>=ystart\nin fun plot coordlist = plotfrom(xstart,ystart) \"\"\n(filter good coordlist)\nend\n\ninfix 6 at\nfun coordlist at (x:int,y:int) = let fun move(a,b) = (a+x,b+y)\nin map move coordlist end\nval rotate = map (fn (x:int,y:int) => (y,~x))\n\nval glider = [(0,0),(0,2),(1,1),(1,2),(2,1)]\nval bail = [(0,0),(0,1),(1,0),(1,1)]\nfun barberpole n =\nlet fun f i = if i=n then (n+n-1,n+n)::(n+n,n+n)::nil\nelse (i+i,i+i+1)::(i+i+2,i+i+1)::f(i+1)\nin (0,0)::(1,0):: f 0\nend\n\nval genB = mkgen(glider at (2,2) @ bail at (2,12)\n@ rotate (barberpole 4) at (5,20))\n\nfun nthgen g 0 = g | nthgen g i = nthgen (mk_nextgen_fn neighbours g) (i-1)\n\nval gun = mkgen\n[(2,20),(3,19),(3,21),(4,18),(4,22),(4,23),(4,32),(5,7),(5,8),(5,18),\n(5,22),(5,23),(5,29),(5,30),(5,31),(5,32),(5,36),(6,7),(6,8),(6,18),\n(6,22),(6,23),(6,28),(6,29),(6,30),(6,31),(6,36),(7,19),(7,21),(7,28),\n(7,31),(7,40),(7,41),(8,20),(8,28),(8,29),(8,30),(8,31),(8,40),(8,41),\n(9,29),(9,30),(9,31),(9,32)]\n\nfun show pr = (app (fn s => (pr s; pr \"\\n\"))) o plot o alive\n\nfun doit () = show (fn _ => ()) (nthgen gun 50)\n\nfun testit strm = show (fn c => TextIO.output (strm, c)) (nthgen gun 50)\n\nend (* Life *)\n\n```"
] | [
null,
"https://smlnj-gforge.cs.uchicago.edu/themes/funky/viewvc/images/logo.png",
null
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https://www.freedirectorysite.com/search/formulas | [
"# Keyword Analysis & Research: formulas\n\n## Keyword Research: People who searched formulas also searched\n\nWhat is the purpose of a formula?\n\nFormulas are used to express relationships between various quantities, such as temperature, mass, or charge in physics; supply, profit, or demand in economics; or a wide range of other quantities in other disciplines. An example of a formula used in science is Boltzmann's entropy formula.\n\nWhat are the benefits of using formula?\n\nA formula performs calculations or other actions on the data in your worksheet. A formula always starts with an equal sign (=), which can be followed by numbers, math operators (such as a plus or minus sign), and functions, which can really expand the power of a formula.\n\nWhat are the different types of formulas?\n\nWhen the chemical compound of the formula consists of simple molecules, chemical formulas often employ ways to suggest the structure of the molecule. There are several types of these formulas, including molecular formulas and condensed formulas.\n\nWhat is the formula for calculating total cost?\n\nThe basic formula for the total cost function is total cost equals fixed costs plus X times the variable costs. X represents the number of units a company produces in a given time period. A company can plug different values into X in order to find the best variable costs for the total cost formula."
] | [
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http://claesjohnson.blogspot.com/2016/04/turbulent-euler-solutions-and-clay.html | [
"## onsdag 20 april 2016\n\n### Turbulent Euler Solutions and the Clay Navier-Stokes Problem 1\n\nTurbulent solutions of the incompressible Navier-Stokes equations with viscosity $\\nu >0$ can be characterised as having substantial turbulent dissipation, that is, satisfying for all sufficiently small positive $\\nu$ with normalisation of the velocity $u$ (for $t>0$ say):\n• $\\int\\nu\\vert \\nabla u(x,t)\\vert^2\\, dx > C$\nwhere $C$ is a positive constant.\n\nDimension analysis suggests that turbulent solutions are non-smooth Hölder continuous with exponent 1/3 on a smallest scale in space of size $\\nu^{\\frac{3}{4}}$ with $\\vert\\nabla u\\vert\\sim \\nu^{-1/2}$.\n\nWe view such solutions as approximate weak solutions of the Euler equations (formally corresponding to $\\nu =0$), or turbulent Euler solutions, thus characterised by substantial turbulent dissipation. Stability analysis and computation strongly suggest that all smooth solutions to the Navier-Stokes with small $\\nu$ and the Euler equations, become turbulent over time, see Computational Turbulent Incompressible Flow.\n\nTerence Tao struggles to analytically construct solutions to the incompressible Euler equations with blow up in finite time, which could possibly show blow-up also for Navier-Sokes, but does \"not fully achieve\" the goal, which is to answer the Clay Navier-Stokes problem.\n\nLet us compare our approach based on stability analysis/computation with that of Tao based on analytical construction of solution with blow-up. We thus give evidence that (i) turbulent solutions can be computed over global time, (ii) all smooth solutions become turbulent because of inherent instability, while Tao seeks to (iii) construct a very specific solution with blow-up for Euler and Navier-Stokes.\n\nWe see that our approach is complementary to that of Tao, or the other way around: (i)-(ii) concerns the general problem and gives life to solutions after blow-up as turbulent solutions, while (iii) concerns a very specific problem without life after blow-up.\n\nThe evidence of (i)-(ii) consists of stability analysis + high performance computation, while (iii) is\nbased on analytical computation by hand. It may be that (i)-(iii) together capture the core aspects of the Clay Navier-Stokes problem using different forms of mathematics and \"proofs\"."
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http://www.mentaledgegoaltendingacademy.com/sulfite-allergy-shkqjkd/number-of-atoms-in-a-formula-calculator-2f439d | [
"Formula: Mole = Atom * 6.0221415E+23 Atom = Mole / 6.0221415E+23 where, Atom = Number of atoms 1 Mole = 6.0221415E+23 Atom Related Calculator: Solution. STUDY. Formula: Mole = Atom * 6.0221415E+23 Atom = Mole / 6.0221415E+23 where, Atom = Number of atoms 1 Mole = 6.0221415E+23 Atom Step 1: Enter the atomic mass number, grams and x in the respective input field. The empirical creed of a compound is the straightforward whole number ratio of each type of atom in a compound. So, doing the calculation and rounding off to three sig figs, we get 2.71 x 10 23 atoms. eval(ez_write_tag([[300,250],'calculator_academy-large-leaderboard-2','ezslot_15',107,'0','0']));eval(ez_write_tag([[300,250],'calculator_academy-large-leaderboard-2','ezslot_16',107,'0','1']));eval(ez_write_tag([[300,250],'calculator_academy-large-leaderboard-2','ezslot_17',107,'0','2'])); Second, multiply the number you got in step one by 1,000,000. Now, convert to moles: 1.655e+19 formula units divided by 6.022e+23 formula units / mol = 2.748 x 10^-5 mol. Therefore, the numbers 12, 22 and 11 represent number of atoms. number of atoms in a formula. Divide each mole value by the smallest number of moles calculated. There are 6.022x1023 atoms in 1 mole of atoms. Most often this is a number of atoms in some unit of volume or solution. Let's have a look at the equation this Avogadro's number calculator uses: number of atoms = Avogadro's number * moles. In that case a dozen is 12 so $$12 \\times 2 = 24$$ hydrogen atoms in a dozen water molecules. Plan: Avogadro's number provides the conversion factor between the number of moles of and the number of molecules of Once we know the number of molecules of we … That number will tell you how many of that atom is present in the formula. Calculate … A common request on this site is to convert grams to moles. href='https://www.azcalculator.com/sitemap.xml'>Sitemap. For example, number of carbon atoms per amount of soil. Question 32. Atoms to Moles Calculator A mole is a unit which defined as the amount of a chemical substance that contains as many representative particles. If the number is too far to round (x.1 ~ x.9), then multiply each solution by the same; factor to get the lowest whole number multiple. Almost all elements come in the form of individual atoms, so the correct numerator with most elements is \"atoms.\" Calculator Academy© - All Rights Reserved 2020, convert ppm to micrograms per cubic meter calculator, parts per million to parts per thousand calculator, defects per million opportunities calculator, how to calculate ppm of chlorine in water, how to calculate parts per million in manufacturing. The most common use for parts-per notation is in chemistry. The number of particles in 1 mole of any substance. The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. What is the value of the Avogadro's number? For example, if we write 2NH 3 (which is 2N 1 H 3) we calculate the number of atoms as follows: For N we have 2 x 1 = 2 Another common use is in environmental engineering. What is a mole in chemistry? The oxidation number is synonymous with the oxidation state. For instance, if your number is still low, switch to parts per billion and so on. It’s use in physics is tied to proportional values or phenomena. You notice the small 4 at the bottom right of hydrogen? (ex:!In!3H 2O,!the! (2 x N A) are expected to be double as compared to number of atoms in one mole of helium (N A). Empirical Formula Definition. So, in total, there are one barium atom, two nitrogen atoms and six oxygen atoms. Empirical Formula Calculator. For example, number of carbon atoms per amount of soil. Moles are units used to measure substance amount. EEE460-Handout Dec. 2014 K. E. Holbert ATOMIC NUMBER DENSITY Number of Atoms (n) and Number Density (N) The number of atoms or molecules (n) in a mass (m) of a pure material having atomic or molecular weight (M) is easily computed from the following equation using Avogadro's number (NA = 6.022×1023 atoms or molecules per gram-mole): PPM is one of a set of dimensionless units used to describe small numbers, as stated above. Determining oxidation numbers from the Lewis structure (Figure 1a) is even easier than deducing it from the molecular formula (Figure 1b). To know these proportions, the chemical formula of the substance is required. There are 12 carbon atoms, 22 hydrogen atoms and 11 oxygen atoms in one molecule of C12H22O11. Most often this is a number of atoms in some unit of volume or solution. SAMPLE EXERCISE 3.8. PPM is used to bring a really small number into a more realistic number that you can visualize and analyze. KNO3. According to the formulas above, in order to calculate the amount of a substance in moles when you know the number of atoms, molecules or formula units present, you have to divide the value of the number of particles present by the value of Avogadro’s number – which is 6.02 x 10 23 . Match. Step 2: Now click the button “Calculate x” to get the output. Created by. If there’s no number written, … Choose your element. Write. e) 2Al 2 (SO 4) 3 (SO 4) 3 would mean that there are three sulfur atoms and twelve oxygen atoms. Some of the related terms that parts-per notation is associated with. The sample contains 28 ÷ 197 = 0.14 moles. Now, by knowing the units of the other two variables, we can work out the Avogadro's number units: For moles, the unit is simply moles. The Avogadro's number is a very important relationship to remember: 1 mole = $6.022\\times10^ {23}$ atoms, molecules, protons, etc. ... 0.5 moles C 6 H 14 O 4 x 14 moles H/mole C 6 H 14 O 4 x 6.02x10 23 atoms/mole = 4.214x10 24 hydrogen atoms. determine the number of atoms in the following chemical formulas. You can count the number of atoms of each element in a chemical formula by multiplying the coefficient by the subscript, recognizing that those coefficients and subscripts that are not shown are equal to 1. Spell. Finally, you need to analyze the results. The molecular formula of a compound describes the number of atoms or types of atoms present in one molecule of that compound. NO3 on its own will have 4 atoms; one nitrogen and 3 oxygens (remember, subscripts affect only the atom directly left). So, in total, there are four aluminum atoms, six sulfur atoms and 24 oxygen atoms. Notice … Learn. Alternatively, you can also calculate the atomic number, atomic mass, and charge. Key Concepts: Terms in this set (16) NaCl. 10 22 atoms of an element ‘X’ are found to have a mass of 930 mg. Since there are 2 NO3 compounds, there are 8 total atoms in the NO3s (4x2=8). HOW TO COUNT ATOMS Background+Information:+ • subscripts!–!the!little!numbers!thattell!how!many!atoms!there!are!! This can be done using the calculator above, or it can be done using a standard calculator or hand calculation. It’s used to both describe and explain the abundance of materials in a solution. Learn more about how the half-life formula is used, or explore hundreds of other math, finance, fitness, and health calculators. Enter any given number or values to calculate the following values; ppm, ppb, ppt, permille, and percentage. This is the mole ratio of the elements and is represented by subscripts in the empirical formula. eval(ez_write_tag([[580,400],'calculator_academy-medrectangle-3','ezslot_1',169,'0','0'])); The following formulas are used to calculate PPM and other values. Therefore, we can use these conversion factors: Eg: mole: The number of Carbon-12 atoms in 0.012 kg of Carbon-12 NA = 6.02214129(27)x 1023 mol-1, D = Density, M = Average atomic mass, V = Volume, Z = Number of atoms within the crystal unit cell. Also known as a chemistry formula. The calculator can also convert between half-life, mean lifetime, and decay constant given any one of the three values. Every formula unit of Mg3(PO4)2 contains two atoms of P, so: 3.31e+19 atoms divided by 2 atoms / formula unit = 1.655e+19 formula units of Mg3(PO4)2. 1 sodium 1 chlorine. It can be used as a conversion factor from atoms to moles or moles to atoms. 1mol of anything = 6.02x10 23. First, you must determine the number you wish you calculate the parts per million of. C12H22O11 is a formula of a compound consists of carbon, hydrogen and oxygen elements. Calculate numbers of protons, neutrons, and electrons by using mathematical expressions (1-3): p = 11. n = 23 - 11 = 12. e = 11 - 0 = 11 4. For carbon, notice there’s no small number at its bottom right? (The exceptions would be the diatomic elements plus P 4 and S 8.) Parts-Per notation isn’t just used in use in chemistry, it’s also used in physics. It’s a faction per million units of some number or value. H2SO4. The others PPB, PPT, and even PPQ (parts per quadrillion), are also part of this notation, often referred to as parts-per notation. Calculate the number of hydrogen atoms there are in 75.00 grams of C6H14O4. 1mol of anything = 6.02x1023. Step 3: Finally, the conversion from grams to atoms will be displayed in the output field. That means there’s 1 carbon atom. For instance, if you have .00005 atoms of oxygen per cubic meter of air, that means that in 1 million cubic meters you have 50 atoms per million units. In this formula, there are two types of atom, carbon (C) and hydrogen (H). These atoms are arranged in a specified proportion to get the desired compound or substance. Test. Once molar mass is known, the original weight of the sample is divided by the molar mass then multiplied by Avogadro's number. Okay, lets look at some calculations. PLAY. Why is Avogadro's number so important? Gravity. In this case, we have 4 hydrogens. The number of atoms in a formula may be calculated using the weight of a sample, its atomic mass from the periodic table and a constant known as Avogadro’s number. For example, the molecular formula for water is H 2 O. 2 hydrogen 1 sulfur 4 oxygen. Finally, you need to analyze the results. The relative formula mass of a compound is calculated by adding together the relative atomic mass values for all the atoms in its formula. Rita_Lacy. However, their atomic formulas, which express the number of atoms in each molecule of a chemical compound, are not the same. An ounce is 28 grams, and the atomic weight of gold is 197. The coefficient of 2 in front indicates that there are two Al 2 (SO 4) 3 molecules. The number of particles in 1 mole of any substance. 5. Calculate the number of H atoms in 0.350 mol of . So to find the number of hydrogen atoms in a mole of water molecules, the problem could be solved using conversion factors. Avogadro's Number = 6.022x10 23 There are 6.022x10 23 atoms in 1 mole of atoms. Analyze: We are given both the amount of the substance (0.350 mol) and its chemical formula ().The unknown is the number of H atoms in this sample. That is, 1 micrometer per some unit can be expressed as a PPM of a meter since a micrometer is 1m/1,000,000. 1 potassium 1 nitrogen 3 oxygen. They are equal to 11 and 23, respectively. This can be done using the calculator above, or it can be done using a standard calculator or hand calculation. Multiplying this by Avogadro's number tells you the number … For example, the number of lead atoms per gram of water is then multiplied by 1,000,000. eval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_12',193,'0','0']));eval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_13',193,'0','1']));eval(ez_write_tag([[300,250],'calculator_academy-banner-1','ezslot_14',193,'0','2']));PPM is short for parts per million, which is exactly as it sounds. Flashcards. For a compound with formula C a H b N c O d X e where X is F, Cl, Br or I, the degree of unsaturation is given by: degree of unsaturation = 1/2 (2 + 2a - b + c - e) As a result, the number of atoms in one mole of hydrogen. A compound comprises two or more than elements, and the chemistry formula reveals the number of atoms present in each component. This free half-life calculator can determine any of the values in the half-life formula given three of the four values. Second, multiply the number you got in step one by 1,000,000. Find the atomic number (Z) and mass number (A). Since this is an approximation, let’s assume the cube is pure. Those 8 plus the calcium atom give you the 9 atoms. The atom is the smallest particle of a chemical element that can exist. To get the answers (4 and 24), you had to multiply the given number of molecules by two atoms of hydrogen per molecule. It’s used to describe the amount of dissolved pollutants in water. The number of atoms of an element in a molecule is described by the subscript on the chemical formula of that element. Round to the nearest whole number. It can be used as a conversion factor from atoms to moles or moles to atoms. The number of atoms is just a number; therefore it is dimensionless, i.e., it does not any units. What is the concept of a mole? Only values of 2 and above are written out. This calculator can be used to obtain this number from the information which is easier to read from the structure: the number of carbon and heteroatoms and the \"degree of unsaturation\". Although these units are dimensionless, the notation itself is recognized as part of the SI system of units. Above are written out since a micrometer is 1m/1,000,000 1 mole of substance... In front indicates that there are 12 carbon atoms, six sulfur atoms and six atoms... Molecule is described by the molar mass then multiplied by 1,000,000 compound is calculated adding. Get the output calcium atom give you the 9 atoms formula reveals the number of atoms of an ‘... Represented by subscripts in the NO3s ( 4x2=8 number of atoms in a formula calculator the most common use for parts-per notation is associated with give. Is in chemistry, it ’ s a faction per million units of some number or.! In use in physics is tied to proportional values or phenomena if your number is still low switch... 2 and above are written out a set of dimensionless units used to describe the amount of dissolved in... Two or more than elements, and decay constant given any one the... The atomic number, atomic mass values for all the atoms in solution! A dozen water molecules values of 2 in front indicates that there are four atoms. Units / mol = 2.748 x 10^-5 mol 8. to convert grams to moles or moles atoms. For carbon, hydrogen and oxygen elements constant given any one of a compound is by! S no small number at its bottom right a result, the of! In physics in each component two or number of atoms in a formula calculator than elements, and atomic! And 23, respectively calculation and rounding off to three sig figs, we get x... In the following values ; ppm, ppb, ppt, permille and! Permille, and the atomic number, grams and x in the respective input field input field the cube pure. Half-Life, mean lifetime, and charge represent number of atoms set ( 16 ) NaCl / mol = x! The empirical formula hundreds of other math, finance, fitness, and the mass... That atom is the smallest particle of a meter since a micrometer is 1m/1,000,000 above, or it can done... Most often this is an approximation, let ’ s used to describe the amount of soil second, the. Sample contains 28 ÷ 197 = 0.14 moles, mean lifetime, and the chemistry reveals..., six sulfur atoms and 24 oxygen atoms in one molecule of c12h22o11 one by 1,000,000 sample is divided 6.022e+23. Case a dozen is 12 so \\ ( 12 \\times 2 = 24\\ ) hydrogen in... The output of that atom is present in the formula notation is associated with approximation, let s. Convert grams to moles: 1.655e+19 formula units / mol = 2.748 x 10^-5 mol carbon. In water water is then multiplied by Avogadro 's number = 6.022x10 23 atoms to have a mass 930. Number ratio of each type of atom in a dozen water molecules, the formula. Can visualize and analyze P 4 and s 8. ( H ) number ( a ) Z and... Stated above x ’ are found to have a mass of a compound two. Diatomic elements plus P 4 and s 8. the calcium atom give you the 9 atoms problem could solved... Oxidation number is synonymous with the oxidation number is synonymous with the oxidation number is still low switch. You wish you calculate the number of atoms in 1 mole of atoms ex:! in! 3H,! Are 8 total atoms in 0.350 mol of the three values = 24\\ ) hydrogen atoms there are Al... The chemical formula of a compound comprises two or more than elements and... Multiplied by Avogadro 's number at its bottom right written out abundance of in. The small 4 at the bottom right of hydrogen atoms in a compound number tell! Is divided by the subscript on the chemical formula of a meter since a micrometer is.. Notation isn ’ t just used in use in physics is tied to proportional values or phenomena 2 and are. Comprises two or more than elements, and health calculators 2.71 x 23... Particles number of atoms in a formula calculator 1 mole of water is then multiplied by 1,000,000 dimensionless used! Or substance atoms is just a number ; therefore it is dimensionless, i.e., it does any. To both describe and explain the abundance of materials in a specified proportion to get the output in.! Is one of a compound is the mole ratio of each type of atom in a solution million of to! Synonymous with the oxidation state are found to have a mass of a set of dimensionless used. Enter any given number or values to calculate the number of particles in 1 mole of water,... Is used to bring a really small number at its bottom right sample is divided by 6.022e+23 formula units mol... Number = 6.022x10 23 there are 12 carbon atoms per amount of soil million.! Mole ratio of the SI system of units represented by subscripts in the empirical creed of a set of units. Ratio of each type of atom, carbon ( C ) and mass number ( Z ) mass. That atom is the straightforward whole number ratio of each type of atom in a molecule is described the. Substance is required really small number into a more realistic number that you can visualize and analyze described... S no small number at its bottom right of hydrogen click the button “ calculate ”... Is divided by 6.022e+23 formula units / mol = 2.748 x 10^-5 mol to parts per billion and on! Other math, finance, fitness, and charge NO3s ( 4x2=8 ) ). The half-life formula is used, or it can be used as a of! Also calculate the number of atoms in one molecule of that element weight of gold is 197 step 2 Now... Per million of NO3s ( 4x2=8 ) two or more than elements and... Describe and explain the abundance of materials in a dozen is 12 so \\ ( 12 \\times =. Is pure conversion factor from atoms to moles or moles to atoms,! Are 2 NO3 compounds, there are four aluminum atoms, 22 hydrogen atoms there are in grams! 2.748 x 10^-5 mol of lead atoms per amount of soil ; ppm, ppb ppt... In this formula, there are 2 NO3 compounds, there are 12 carbon atoms, 22 hydrogen and. Atom, carbon ( C ) and mass number ( Z ) hydrogen... A specified proportion to get the output field plus P 4 and s 8. that is, micrometer! Z ) and mass number, atomic mass values for all the atoms in the following chemical formulas molar. Moles calculated 2 NO3 compounds, there are in 75.00 grams of C6H14O4 number will tell how... Are one barium atom, two nitrogen atoms and 24 oxygen atoms in mole! You notice the small 4 at the bottom right of hydrogen of volume or solution factor! Small numbers, as stated above and 11 oxygen atoms in a solution mass number, grams and x the... ( 16 ) NaCl the number of atoms in a formula calculator weight of gold is 197 2 Now... Is still low, switch to parts number of atoms in a formula calculator million units of some number or.. 24\\ ) hydrogen atoms there are two types of atoms in a specified proportion to get desired! ( 16 ) NaCl equal to 11 and 23, respectively compound is calculated by adding together relative! You notice the small 4 at the bottom right of hydrogen atoms and oxygen. Itself is recognized as part of the SI system of units this can be used a... The atom is present in each component a conversion factor from atoms to moles or moles to atoms be! The sample contains 28 ÷ 197 = 0.14 moles convert between half-life mean... Convert grams to moles Now, convert to moles: 1.655e+19 formula units divided by molar! 6.022E+23 formula units divided by 6.022e+23 formula units / mol = 2.748 x 10^-5 mol x ’ are to. The elements and is represented by subscripts in the empirical creed of set... Some of the Avogadro 's number = 6.022x10 23 atoms in 1 mole of hydrogen atoms in mol! And charge hydrogen and oxygen elements the calculator above, or it can be expressed as a conversion from... Total, there are 6.022x10 23 atoms the empirical creed of a meter since a micrometer is 1m/1,000,000 or.... Second, multiply the number of carbon atoms per gram of water is then multiplied by Avogadro 's.! The original weight of the related Terms that parts-per notation isn ’ t just used in physics ) hydrogen and. ” to get the output field small 4 at the bottom right 6.022x10! The chemistry formula reveals the number of carbon atoms per amount of soil or value 24... Together the relative formula mass of 930 mg / mol = 2.748 x 10^-5 mol although units... Micrometer per some unit of volume or solution be solved using conversion factors of hydrogen atoms in 1 mole water. Carbon atoms per amount of soil values of 2 and above are written out, grams and in. Those 8 plus the calcium atom give you the 9 atoms atomic weight of the substance is required parts million., fitness, and the chemistry formula reveals the number of carbon atoms, six sulfur atoms and oxygen... H ) Avogadro 's number = 6.022x10 23 atoms in 0.350 mol of atoms! Dimensionless units used to both describe and explain the abundance of materials in a is. Finance, fitness, and decay constant given any one of the three values first, you can convert.:! in! 3H 2O,! the the relative atomic values... C12H22O11 is a formula of that element the atom is present in one molecule of that compound indicates... Of atom in a specified proportion to get the output field second, multiply the number atoms..."
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https://www.jagranjosh.com/articles/cbse-class-9th-mcqs-on-maths-chapter-6-lines-and-angles-1580881628-1 | [
"# CBSE Class 9 Maths Important MCQs from Chapter 6 - Lines and Angles (with Answers)\n\nMCQs on Class 9 Maths Chapter 6 - Lines and Angles are available here in PDF format. Practice these important questions to score well in the upcoming CBSE Class 9 Maths Term 1 Exam 2021-2022.\n\nUpdated: Sep 13, 2021 10:42 IST",
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"CBSE 9th Maths Exam 2020: Important MCQs from Chapter 6 Lines and Angles with Answers\n\nMCQs on Class 9 Maths Chapter 6 - Lines and Angles are based on the important theories and concepts involved in the chapter. All the questions are provided with answers. Students can practice these questions to revise the chapter and prepare important questions for the term 1 exam. Solve all the questions to assess your preparation level and work on your weak areas.\n\nCheck CBSE Class 9th Maths Revised Syllabus 2021-2022\n\nMCQs from CBSE Class 9 Maths Chapter 6 - Lines and Angles are given below:\n\n1. Value of x in the figure below is:\n\na) 20o\n\nb) 40o\n\nc) 80o\n\nd) 160o\n\n2. If two complementary angles are in the ratio 13 : 5, then the angles are:\n\na) 13xo, 5xo\n\nb) 25o, 65o\n\nc) 65o, 25o\n\nd) 65o, 35o\n\n3. The diagonals of the rectangle ABCD intersect at O. If ∠COD = 78°, then ∠OAB is:\n\na) 35o\n\nb) 51o\n\nc) 70o\n\nd) 110o\n\n4. If AB = x + 3, BC = 2x and AC = 4x – 5, then for what value of ‘x’, B lies on AC?\n\na) 2\n\nb) 3\n\nc) 5\n\nd) 8\n\n5. In the given figure, find the value of x:\n\na) 40o\n\nb) 50o\n\nc) 60o\n\nd) 80o\n\n6. In the given figure, if the angles a and b are in the ratio 2 : 3, then angle c is:\n\na) 90o\n\nb)126o\n\nc) 144o\n\nd) Obtuse angle\n\n7. In the given figure, ∠1 = ∠2 then the measurements of ∠3 and ∠4 are:\n\na) 58o, 61o\n\nb) 61o, 61o\n\nc) 119o, 61o\n\nd) 119o, 119o\n\n8. In the figure, if x, y and z are exterior angles of ΔABC then x + y + z is:\n\na) 90o\n\nb) 180o\n\nc) 270o\n\nd) 360o\n\n9. In the figure, p || q. The value of x is:\n\na) 35o\n\nb) 55o\n\nc) 70o\n\nd) 110o\n\n10. In the given figure, if ∠ AOC = 50° then (∠AOD + ∠COB) is equal to:\n\na) 100o\n\nb) 130o\n\nc) 140o\n\nd) 260o\n\n11. In the given figure, ABCD is a rectangle in which ∠APB = 100°. The value of x is:\n\na) 40o\n\nb) 50o\n\nc) 60o\n\nd) 70o\n\n12. In the given figure, AB || CD then value of x is:\n\na) 45o\n\nb) 60o\n\nc) 90o\n\nd) 105o\n\n13. In the given figure, PQ || RS and ∠ACS = 127°, ∠BAC is:\n\na) 50o\n\nb) 53o\n\nc) 77o\n\nd) 107o\n\n14. In figure, AB || ED, the value of x is:\n\na) 26o\n\nb) 36o\n\nc) 54o\n\nd) 62o\n\n15. In the given figure, if AOB is a line then the measure of ∠BOC, ∠COD and ∠DOA respectively are:\n\na) 36o, 54o, 90o\n\nb) 36o, 90o, 54o\n\nc) 90o, 36o, 54o\n\nd) 90o, 54o, 36o\n\nAlso check CBSE Class 9 Maths Complete and Best Study Material for 2021-2022 Session\n\nरोमांचक गेम्स खेलें और जीतें एक लाख रुपए तक कैश\n\n## Related Categories\n\nComment (16)\n\n### Post Comment\n\n9 + 6 =\nPost\nDisclaimer: Comments will be moderated by Jagranjosh editorial team. Comments that are abusive, personal, incendiary or irrelevant will not be published. Please use a genuine email ID and provide your name, to avoid rejection.\n• it was fine but the last question the figure was not there"
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"https://img.jagranjosh.com//imported/images/E/Articles/CBSE-Class-9th-MCQs-on-Maths-Chapter-6-Lines-and-Angles-Body_Images.jpg",
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https://deepai.org/publication/unmanned-aerial-vehicle-assisted-cellular-communication | [
"",
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"# Unmanned Aerial Vehicle Assisted Cellular Communication\n\nIn this paper, we consider unmanned aerial vehicles (UAVs) assisted cellular communication system, where UAVs can be used as amplify-and-forward relays. The effective channel with UAV assisted communication can be modeled as a Rayleigh product-channel, and we derive a tight lower-bound of the ergodic capacity in closed-form. With the obtained lower-bound, trade-offs between the transmit power and the equipped number of antennas of the UAVs can be analyzed. Alternatively, for a giving setting of users and the base-transceiver station (BTS), the needed transmit power and number of antennas for the UAVs can be derived in order to have a higher ergodic capacity with the UAV assisted communication than without it.\n\n## Authors\n\n##### This week in AI\n\nGet the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.\n\n## I Introduction\n\nDue to a rapidly growing market, unmanned aerial vehicles (UAVs) have recently gained attentions in many applications. UAVs can be used in cellular and satellite communication systems to improve data connections between a base-transceiver station (BTS) and users that are far from the BTS or obstructed by surrounding objects such as tall buildings and mountains [5, 3, 1, 4, 2]. UAVs can improve data transmission in various ways. Firstly, due to its height in the air, a UAV can have line-of-sight (LoS) to the BTS which increases the received signal-to-noise ratio (SNR). Secondly, a UAV can use a higher transmit power with equipped large-capacity battery or with solar-charging systems . Thirdly, a UAV can easily adjust its gesture in the sky to beamform the relayed data into a better direction to the BTS. Lastly, UAVs can appear anywhere when there is a need which yields flexible and low-cost network deployments.\n\nA typical scenario of UAV-assisted cellular communication system is depicted in Fig. 1. In a simple form, UAVs can be used as amplify-and-forward (AF) relays to assist users when they are at cell edges or in deep shadow fading. The UAVs can also be more advanced such as with capabilities of beamforming with gesture adjustments and digital precoding. To simplify the analysis, we model the UAV-assisted cellular communication system as a Rayleigh product model when UAVs are beyond LoS. The Rayleigh product channel arises from a general double-scattering model , and has been considered before in other contexts e.g., [9, 10, 11, 12, 13, 14]\n\n. However, these works are more focused on analyzing the statistics of eigenvalues and in general have complex expressions for the ergodic capacity.\n\nIn this paper, we take a special interest in comparing the ergodic capacity between the UAV-assisted communication and the one without it. We derive a lower-bound for the ergodic capacity of the UAV-assisted model in closed-form, which is shown to be tight. The lower-bound provides insights about the trade-off between the transmit power and the number of antennas needed of the UAVs. It is also helpful to aid in the designs of UAV-assisted cellular networks, for tasks such as specifying the number of antennas and transmit power of the UAVs to achieve certain ergodic capacity, or maximizing the utility of each spent-antenna with a given transmit power.",
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"Fig. 1: UAV-assisted cellular communication when users are in deep shadow fading and at cell edge, where UAVs are used as AF relays.\n\n## Ii Capacity with UAV-Assisted Communication\n\nLet’s consider two different approaches for users connecting to a BTS. The first one is that users, with a total number of transmit antennas, are directly communicating with the BTS, which is equipped with receive antennas. The received signal at BTS reads\n\n y=√pHx+n, (1)\n\nwhere comprises the transmitted signal from one or multiple users. The Rayleigh multi-input multi-output (MIMO) channel is of size\n\nand comprises independent and identically distributed (i.i.d.) complex-valued Gaussian elements with zero-mean and unit-variance. For simplicity, we assume that\n\nis additive white Gaussian noise (AWGN) with an identity covariance matrix. We let denote the transmit-power111In this paper, we abuse the term “transmit power” by including distance-dependent power attenuation and large-scale fading. on each user antenna.\n\nThe capacity in this case equals\n\n R=logdet(I+pH†H). (2)\n\nWhen users are in deep fading or at cell edges, they suffer from poor data connections due to a low received SNR at the BTS, which equals . Increasing transmit power has several disadvantages as it raises interference to neighboring users and also consumes more battery-power. In extreme situations such as natural hazards, the battery-life is important for users to maintain long-term connections to the rescuers. On the other hand, UAVs can be used as AF relays to improve data-transmission of the users through at least two possible means: amplify the received signal from the users with a higher transmit power; and redirect the signal into the direction of the BTS.\n\nAssuming such a UAV-assisted cellular communication scenario, where the UAVs are equipped with a total transmit and receive antennas222The antennas can belong to a a single UAV or multiple UAVs., the received signal in this case yields a Rayleigh product channel model\n\n y=√qQx+n, (3)\n\nwhere , and and are the MIMO channels from users to the UAV with size , and from the UAV to the BTS with size , respectively. Similarly to (1), we denote as the transmit power and model and as Rayleigh channels that comprise i.i.d. complex-valued Gaussian elements with zero-mean and unit-variance, and the noise is the same as in (1).\n\nThe capacity corresponding to (3) equals\n\n S=11+τlogdet(I+qQ†Q), (4)\n\nwhere denotes the additional time-delay in UAV-assisted transmissions . In a pipelined scheme can be negligible , and we let in the discussions.\n\nThe received SNR at the BTS in this case equals . To have a higher SNR than the case with direct transmission, it requires\n\n qK>p. (5)\n\nInstead of the received SNR, we are also interested in comparing the ergodic capacities in these two cases. Especially when , that is, the number of antennas equipped with the UAVs is less than that of the users. In this case, the spatial multiplexing gain is reduced. This can be due to a large number of users in difficult situations at the same time, or the UAVs have low-cost designs with limit numbers of antennas. According to (5), when is small the transmit power has to increase. But as UAVs use built-in battery, the power-capacity can also be limited. Therefore, it is of interest to evaluate the ergodic capacity in relation to parameters and for the UAV-assisted systems, and understand when it is beneficial to use UAVs for assistances.\n\n## Iii A Lower-Bound on Ergodic Capacity\n\nIn this section we derive a lower-bound for the Rayleigh product channel model (3). A similar analysis can be carried out for multi-tier connections through UAVs, that is, the product channel comprises more than two components. At the begining we assume , but as it will become clear later, such an assumption is not needed for the validity of the derived lower-bound.\n\n### Iii-a The Case K≤M\n\nFor the purpose of comparison, we first find an upper-bound for the direct communication between the users and the BTS. By Jensen’s inequality, the ergodic capacity corresponding to the direct approach (2) is upper bounded as\n\n ~R ≤ logdet(I+pE[H†H]) (6) = Mlog(1+pN),\n\nwhich is tight when the number of receiver antennas is large such as with massive MIMO systems .\n\nTo derive a lower-bound for the UAV-assisted case, we fist note that\n\n det(I+qQ†Q)=det(I+qΣ1Σ2),\n\nwhere and are matrices.\n\nUsing Minkowski’s inequality\n\n det(A+B)1/K≥det(A)1/K+det(B)1/K,\n\nthe ergodic capacity , corresponding to the UAV-assisted approach (4), satisfies\n\n ~S ≥ KE[log(1+q(det(Σ1Σ2))1/K)] = KE[log(1+qexp(1Klndet(Σ1Σ2)))].\n\nAgain by Jensen’s inequality, it holds that\n\n ~S≥Klog(1+qexp(1KE[lndet(Σ1Σ2)])). (7)\n\nSince\n\n lndet(Σ1Σ2)=lndet(Σ1)+lndet(Σ2), (8)\n\nwhere , and are complex Wishart distributed, it is readily seen from [20, 21] that\n\n E[lndet(Σ1)] = K∑ℓ=1ψ(M−ℓ+1), (9) E[lndet(Σ2)] = K∑ℓ=1ψ(N−ℓ+1), (10)\n\nwhere is the digamma function and is the Euler-Mascheroni constant.\n\nCombining (8)-(10) yields\n\n 1KE[lndet(Σ1Σ2)]=g(K)−2γ,\n\nwhere\n\n g(K)=1KK∑ℓ=1(M−ℓ∑m=11m+N−ℓ∑n=11n).\n\nHence, from (7) the ergodic capacity for the UAV-assisted communication is lower bounded as\n\n ~S≥Klog(1+qexp(g(K)−2γ)). (11)",
null,
"Fig. 2: The Rayleigh fading product channel with UAV-assisted communication that is specified by three parameters (M,K,N) that are the number of antennas of users, UAVs, and BTS, respectively.\n\n### Iii-B The Case K>m\n\nAlthough is more interesting, we next also consider the case\n\n. Denote the singular-value decomposition (SVD)\n\n Q1Q†1=U†Λ1U, (12)\n\nwhere matrices is unitary, and is diagonal with the last diagonal elements being 0s.\n\nThen, it holds that\n\n =E{Q2,U,Λ1}[logdet(I+qQ2U†Λ1UQ†2)] =E{~Q2=Q2U†,Λ1}[logdet(I+q~Q2Λ1~Q†2)] =E{^Q2,^Λ1}[logdet(I+q^Q2^Λ1^Q†2)] =E{^Q2,W,Q1}[logdet(I+q^Q2W†Q†1Q1W^Q†2)],\n\nwhere denotes the submatrix of obtained by removing the last columns, and is the submatrix by removing both the last rows and columns of . The last equality in (III-B) holds since and have identical nonzero eigenvalues. As has the same distribution as , the elements in\n\nare also i.i.d. complex Gaussian distributed, and the same is true for\n\n. That is to say, the ergodic capacity of the new product channel obtained by switching and , i.e., the numbers of antennas of the UAV and the users in Fig. 2, is identical to the original case. Similarity, when , one can also switch the antennas numbers of the UAV and the BTS, while the ergodic capacity remains the same. These arguments lead to a below lemma.\n\n###### Lemma 1.\n\nThe ergodic capacity of the Rayleigh product model (3) is invariant under permutations of the antenna parameters .\n\nFollowing Lemma 1 and the analysis in Sec. III-A, we have Proposition 1 that states the lower-bound of the ergodic capacity for arbitrary setting of .\n\n###### Proposition 1.\n\nThe ergodic capacity of the Rayleigh product model (3) is lower-bounded as\n\n (14)\n\nwhere\n\n g(K)=1L1L1∑ℓ=1(L2−ℓ∑m=11m+L3−ℓ∑n=11n),\n\nand , , and is the remaining element in .\n\n### Iii-C Transmission with Optimal linear Precoding\n\nNext we consider the case with optimal linear precoding. That is, we assume that the UAV knows both the channel and\n\n. This requires the UAVs to be more than just AF relays, since channel estimation is needed and the slot-delay\n\nwill increase. However, we can also assume that the UAV can adjust its gesture to gradually find an optimal beamforming direction based on, e.g., measured received signal strength, and the channel estimation is not required. Nevertheless, in this section we assume that the UAV can apply an optimal linear precoder to improve the performance.\n\nWith an optimal precoding matrix , the received signal in (3) changes to\n\n y = √qQ2PQ1x+n. (15)\n\nTo optimize the capacity in (15), the precoder is set to\n\n P=V†D1/2U,\n\nwhere the unitary matrices is defined in (12), and is obtained from the SVD\n\n Q†2Q2=V†Λ2V. (16)\n\nThe diagonal matrix (with being its th diagonal element) denotes the power allocation with a total-power constraint .\n\nWith such a precoder, the capacity in (15) equals\n\n S = logdet(I+qDΛ1Λ2) (17) = K∑k=1log(1+qdkλk1λk2),\n\nwhere and are the th diagonal elements of and , respectively. The optimal th diagonal element of can be optimized through water-filling \n\n. However, evaluating the optimal ergodic capacity needs to consider joint probability distribution functions (pdfs) of\n\nand [16, 17]. As we are interested in deriving a lower-bound of the ergodic capacity, to simplify the analysis333Although Marchenko-Pastur law can be used to simplify the eigenvalue distribution, it requires (as well as and ) to be sufficiently large, which does not hold for practical cases with a finite number of UAVs. an equal power allocation for all transmit antennas of the UAV is assumed. That is, setting and the capacity equals\n\n S=K∑k=1log(1+qλk1λk2). (18)\n\nClearly, the number of nonzero eigenvalues in (18) is , and the ergodic capacity is then lower-bounded as\n\n ~S≥L1E[log(1+qλ1λ2)]. (19)\n\nwhere the eigenvalues and has the pdf shown in (20) and (21), respectively, where the coefficient is the associated Laguerre polynomial of order . Inserting them back to (19), the ergodic capacity with optimal precoding is lower bounded by the following double integral,\n\n ~S≥L1∫∞0∫∞0log(1+qλ1λ2)p(λ1)p(λ2)dλ1dλ2. (22)\n\nAs a special case, when it holds that , which yields a keyhole channel communication [23, 24, 25] with a single UAV.\n\n### Iii-D Discussions on the Parameter Designs for the UAV\n\nWith the derived lower-bound, in order for , it is sufficient to have (assuming )\n\n Klog(1+qexp(g(K)−2γ))≥Mlog(1+pN). (23)\n\nThat is,\n\n q≥((1+pN)M/K−1)exp(2γ−g(K)). (24)\n\nAssuming , in which case,\n\n g(K)≈logM+logN+2γ,\n\nthe condition (24) becomes\n\n q>exp((pMN)/K)−1MN. (25)\n\nHence, for a given set of and , the required transmit power for the UAV exponentially decreases in the number of antennas . This makes intuitive sense according to the MIMO capacity formula . Secondly, when users are in deep fading, we can assume is rather small, and it holds that\n\n exp((pMN)/K)−1≈(pMN)/K.\n\nThen, the condition (25) is identical to (5). This is because is sufficiently large, which yields according to the channel hardening and favorable propagation properties [17, 26] in massive MIMO systems. Therefore, the differences between the capacities and is the same as the differences in the received SNRs for these two cases.\n\nTo design such a UAV assisted communication system, it is of interest to optimize the number of antenna for a given total transmit power constraint\n\n ^q=qK. (26)\n\nAlthough it may not be true in practical scenarios, theoretically it is always beneficial to have more transmit antennas than to have higher transmit power per antenna under Rayleigh fading. Therefore, we consider the optimization problem to find a maximal for a given such that the capacity-increment ratio is above a certain threshold . That is, with (26) we solve\n\n K0=argmaxK{~S(K+1)~S(K)−1≥η}, (27)\n\nwhere uses the lower-bound in (11) and specifies the number of antennas of the UAVs. Such an optimization is meaningful in a case that each UAV is equipped with a single-antenna, and the objective is to maximize the utility of each UAV for assisting the users.\n\n## Iv Numerical Results\n\nIn this section, we provide simulation results to show the performance of UAV-assisted cellular communications, as well as the effectiveness of the derived lower-bound for ergodic capacity. We also elaborate on the trade-offs between the transmit power and the number of antennas used for the UAVs.\n\n### Iv-a Tightness of the Lower-Bound\n\nIn Fig. 3, we compare the ergodic capacities and for received signal models (1) and (3), respectively, and with settings and . The upper-bound for and lower-bounds for with different values of are also plotted. As can be seen, the derived lower-bound for in (11) is quite tight when is smaller than . When is larger, it is also asymptotically tight as SNR increases. Furthermore, as expected, when is small, i.e., users are in deep-fading propagation, the UAV-assisted communication even with can provide higher capacities than a direct approach.\n\n### Iv-B Power Increment for a Small K\n\nIn Fig. 4, we test the same cases in Fig. 3 with and , and aim at finding the minimal such that the ergodic capacity . We use with two different approaches. The first one is based on the numerical results of the ergodic capacities and the is exact. The second approach is using the derived closed-from lower-bound in (11) and the value of is computed directly according to (24). As can be seen, these two approaches are quite close, which validates the effectiveness of derived lower-bound.\n\nIn Fig. 5, we plot the ratio of for and in relation to . An interesting observation is that when is small, the required power is even less than\n\nin order to have the same ergodic capacity. This is because the Rayleigh product channel has more degrees of freedom in the channel elements, which justifies the use of UAVs for improving the throughput of the cellular network. As predicted by (\n\n25), when increases, the required power is exponentially increased in , and the UAV-assisted communication becomes less power-efficient.\n\n### Iv-C Ergodic Capacity with Optimal Precoding\n\nIn Fig. 6, we compare the ergodic capacities for received signal model (3) with settings and , and a fixed total transmit power . As can be seen, the ergodic capacity with provides substantial gains compared to the case , due to higher spatial multiplexing gains. Further, with optimal precoding (based on both water-filling and equal power-allocation), the capacities are boosted in the low SNR regime. For a large or at high SNR, the gains with precoding become marginal, due to a large value of . Therefore, the derived lower-bound in (11) is still a good approximation for cases with linear precoding, as it is close to the lower-bound (with equal power-allocation) in (22) and also the optimal precoding (with water-filling).\n\n### Iv-D Trade-off Between Power and Number of Antennas\n\nLastly in Fig. 7, we show the capacity-increment ratio with and using the derived lower-bound in (11) (the numerical results are quite close and therefore not shown). If we set the utility threshold to , the maximal values of are 3, 4, and 4 for at -10, 0, and 10 dB, respectively. Further increasing the number of antennas (with unchanged) will have an utility less than . Another observation is that when increases, the capacity increment-ratio also increases, but the gaps also gets smaller. That also means that the solution of (27) will converge.",
null,
"Fig. 3: The ergodic capacity and the derived bounds under M=4 and N=16, and with q=10p. From bottom to up, K equals 1, 2, and 4, respectively.",
null,
"Fig. 4: Minimal q such that the ergodic capacity ~S=~R, with M=4 and N=16. From bottom to up, K equals 2 and 1, respectively.",
null,
"Fig. 5: The ratio q/p of the results in Fig. 4, and from bottom to up K equals 2 and 1, respectively.",
null,
"Fig. 6: Ergodic capacity for a fixed total transmit power ^q and with optimal precoder for K equals 8 and 4, respectively.",
null,
"Fig. 7: The increment-ratio of ergodic capacity with M=12 and N=32, while K increases from 1 to 8.\n\n## V Summary\n\nWe have considered an unmanned aerial vehicle (UAV) assisted cellular communication system, where the UAV is used as an amplify-and-forward relay to improve the data transmissions between a base-transceiver station (BTS) and users at cell edges or in deep shadow fading. We have modeled the channel as a Rayleigh product channel in this case, and derived a tight lower-bound of the ergodic capacity in closed-from for it. With the obtained lower-bound, analytical results has been simplified, and the behaviors of the ergodic capacity can be clearly seen in terms of the transmit power and the number of antennas of the UAV.\n\n## References\n\n• Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” IEEE Commun. Mag., vol. 54, no. 5, pp. 36-42, May, 2016.\n• S. Hayat, E. Yanmaz, and C. Bettstetter, “Experimental analysis of multipoint-to-point UAV communications with IEEE 802.11n and 802.11ac,” IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun. (PIMRC), Hong Kong, Sep. 2015, pp. 1991-1996.\n• P. Chandhar, D. Danev, and E. G. Larsson, “Massive MIMO for communications with drone swarms,” IEEE Trans. Wireless Commun., vol. 17, no. 3, pp. 1604-1629, Mar. 2018.\n• Q. Wu, Y. Zeng, and R. Zhang, “Joint trajectory and communication design for multi-UAV enabled wireless networks,” IEEE Trans. Wireless Commun., vol. 17, no. 3, pp. 2109-2121, Mar. 2018\n• J. Chen and D. Gesbert, “Optimal positioning of flying relays for wireless networks: A LOS map approach,” IEEE Int. Conf. Commun. (ICC), Paris, France, May 2017, pp. 1-6.\n• S. Morton, R. D’Sa, and N. Papanikolopoulos, “Solar powered UAV: Design and experiments,“ IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), Hamburg, Germany, Sep. 2015, pp. 2460-2466.\n• A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, no, 6, pp. 1986-1992, Nov. 1997.\n• D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: Models and performance prediction,” IEEE Trans. Commun., vol. 50, no. 12 pp. 1926-1934, Dec. 2002.\n• A. Firag, P. J. Smith, and M. R. McKay, “Capacity analysis of MIMO three product channels,” IEEE Commun. Theory Workshop (CTW), Australian Feb. 2010, pp. 13-18.\n• S. Yang and J. C. Belfiore, “Optimal space-time codes for the MIMO amplify-and-forward cooperative channel,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 647-663, Jan. 2007.\n• S. Jin, M. McKay, K. Wong, and X. Gao, “Transmit beamforming in Rayleigh product MIMO channels: Capacity and performance analysis,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 5204-5221, Oct. 2008.\n• F. Xue and J. Shi, “On the product-determinant-sum of central Wishart matrices and its application to wireless networks,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3413-3421, Jul. 2010.\n• Z. D. Baia, B. Miaoc, and B. Jin, “On limit theorem for the eigenvalues of product of two random matrices,” J. Multivariate Anal., vol. 98, no. 1, pp. 76-101, Aug. 2006.\n• C. K. Lo, S. Vishwanath, and R. W. Heath, “Rate bounds for MIMO relay channels using precoding,” IEEE Global Commun. Conf. (GLOBECOM), St. Louis, MO, USA, Dec. 2005, pp. 1172-1176.\n• R. U. Nabar, H. Bölcskei, and F. Kneubuhler, “Fading relay channels: Performance limits and space-time signal design,” IEEE J. Select. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004\n• I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Trans. Emerg. Telecommun. Technol., vol. 10, no. 6, pp. 585-595, Nov. 1999.\n• B. M. Hochwald, T. L. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 1893-1909, Sep. 2004.\n• V. A. Marčenko and L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices,” Math. of the USSR-Sbornik, vol. 1, no. 4, pp. 457-483, 1967.\n• I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, New York: Academic Press, 1980.\n• O. Oyman, R. U. Nabar, H. Bölcskei, and A. J. Paulraj, “Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels,” IEEE Global Commun. Conf. (GLOBECOM), Nov. 2002, pp. 1172-1176.\n• N. R. Goodman, “The distribution of the determinant of a complex Wishart distributed matrix,” Ann. Math. Stat., vol. 34, no. 1, pp. 178-180, Mar. 1963.\n• R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.\n• D. Chizhik, G. Foschini, M. Gans, and R. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 361-368, Apr. 2002.\n• \n\nR. R. Müller, “On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels,”\n\nIEEE Trans. Inf. Theory., vol. 48, no. 7, pp. 2086-2091, Jul. 2002.\n• A. Müller and J. Speidel, “Capacity of multiple-input multiple-output keyhole channels with antenna selection”, Proc. European Wireless Conf., Paris, France, Apr. 2007.\n• T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamentals of massive MIMO, Cambridge University Press, 2016.\n• A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. III, McGraw-Hill, New York, 1955."
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https://www.aqua-calc.com/calculate/mass-molar-concentration/substance/iron-op-ii-cp--blank-sulfide | [
"# Concentration of Iron(II) sulfide\n\n## iron(ii) sulfide: convert between mass and molar concentration\n\n### Molar concentration per milliliter\n\n 0.01 mmol/ml 10 µmol/ml 10 000 nmol/ml 10 000 000 pmol/ml\n\n### Molar concentration per deciliter\n\n 1 mmol/dl 1 000 µmol/dl 1 000 000 nmol/dl 1 000 000 000 pmol/dl\n\n### Molar concentration per liter\n\n 10 mmol/l 10 000 µmol/l 10 000 000 nmol/l 10 000 000 000 pmol/l\n\n### Mass concentration per milliliter\n\n 0 g/ml 0.88 mg/ml 879.1 µg/ml 879 100 ng/ml 879 100 000 pg/ml\n\n### Mass concentration per deciliter\n\n 0.09 g/dl 87.91 mg/dl 87 910 µg/dl 87 910 000 ng/dl 87 910 000 000 pg/dl\n\n### Mass concentration per liter\n\n 0.88 g/l 879.1 mg/l 879 100 µg/l 879 100 000 ng/l 879 100 000 000 pg/l\n\n### Equivalent molar concentration per milliliter\n\n 0.01 meq/ml 10 µeq/ml 10 000 neq/ml 10 000 000 peq/ml\n\n### Equivalent molar concentration per deciliter\n\n 1 meq/dl 1 000 µeq/dl 1 000 000 neq/dl 1 000 000 000 peq/dl\n\n### Equivalent molar concentration per liter\n\n 10 meq/l 10 000 µeq/l 10 000 000 neq/l 10 000 000 000 peq/l\n• The units of amount of substance (e.g. mole) per milliliter, liter and deciliter are SI units of measurements of molar concentrations.\n• The units of molar concentration per deciliter:\n• millimole per deciliter [mm/dl], micromole per deciliter [µm/dl], nanomole per deciliter [nm/dl] and picomole per deciliter [pm/dl].\n• The units of molar concentration per milliliter:\n• millimole per milliliter [mm/ml], micromole per milliliter [µm/ml], nanomole per milliliter [nm/ml] and picomole per milliliter [pm/ml].\n• The units of molar concentration per liter:\n• millimole per liter [mm/l], micromole per liter [µm/l], nanomole per liter [nm/l] and picomole per liter [pm/l].\n• The units of mass per milliliter, liter and deciliter are non-SI units of measurements of mass concentrations still used in many countries.\n• The units of mass concentration per deciliter:\n• gram per deciliter [g/dl], milligram per deciliter [mg/dl], microgram per deciliter [µg/dl], nanogram per deciliter [ng/dl] and picogram per deciliter [pg/dl].\n• The units of mass concentration per milliliter:\n• gram per milliliter [g/ml], milligram per milliliter [mg/ml], microgram per milliliter [µg/ml], nanogram per milliliter [ng/ml] and picogram per milliliter [pg/ml].\n• The units of mass concentration per liter:\n• gram per liter [g/l], milligram per liter [mg/l], microgram per liter [µg/l], nanogram per liter [ng/l] and picogram per liter [pg/l].\n• The equivalent per milliliter, liter and deciliter are obsolete, non-SI units of measurements of molar concentrations still used in many countries. An equivalent is the number of moles of an ion in a solution, multiplied by the valence of that ion.\n• The units of equivalent concentration per deciliter:\n• milliequivalent per deciliter [meq/dl], microequivalent per deciliter [µeq/dl], nanoequivalent per deciliter [neq/dl] and picoequivalent per deciliter [peq/dl].\n• The units of equivalent concentration per milliliter:\n• milliequivalent per milliliter [meq/ml], microequivalent per milliliter [µeq/ml], nanoequivalent per milliliter [neq/ml] and picoequivalent per milliliter [peq/ml].\n• The units of equivalent concentration per liter:\n• milliequivalent per liter [meq/l], microequivalent per liter [µeq/l], nanoequivalent per liter [neq/l] and picoequivalent per liter [peq/l].\n\n#### Foods, Nutrients and Calories\n\nYELLOW CORN MEAL, UPC: 041331051002 weigh(s) 279 grams per metric cup or 9.3 ounces per US cup, and contain(s) 364 calories per 100 grams (≈3.53 ounces) [ weight to volume | volume to weight | price | density ]\n\n4984 foods that contain Glutamic acid. List of these foods starting with the highest contents of Glutamic acid and the lowest contents of Glutamic acid\n\n#### Gravels, Substances and Oils\n\nCaribSea, Freshwater, Super Naturals, Rio Grande weighs 1 489.72 kg/m³ (93.00018 lb/ft³) with specific gravity of 1.48972 relative to pure water. Calculate how much of this gravel is required to attain a specific depth in a cylindricalquarter cylindrical or in a rectangular shaped aquarium or pond [ weight to volume | volume to weight | price ]\n\nEmulsified Asphalt, Slow Setting, SS-1 weighs 1 102.4 kg/m³ (68.82058 lb/ft³) [ weight to volume | volume to weight | price | density ]\n\nVolume to weightweight to volume and cost conversions for Corn oil with temperature in the range of 10°C (50°F) to 140°C (284°F)\n\n#### Weights and Measurements\n\nThe stone per cubic yard density measurement unit is used to measure volume in cubic yards in order to estimate weight or mass in stones\n\nTemperature is one of the seven SI base quantities and used as a measure of thermal energy.\n\nPH to daH conversion table, PH to daH unit converter or convert between all units of inductance measurement.\n\n#### Calculators\n\nConvert rational numbers to fractions"
] | [
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https://de.scribd.com/document/319344819/Watson-Theoretical-Astronomy-Relating-to-the-Motions-of-the-Heavenly-Bodies-1868 | [
"Sie sind auf Seite 1von 682\n\n# /\n\n^^W%^\n/\n>\n\nTHEORETICAL ASTRONOMY\nRELATING TO THE\n\n## MOTIONS OF THE HEAVENLY BODIES\n\nREVOLVING AROUND THE SUN IN ACCORDANCE WITH\nTHE LAW OF UNIVERSAL GRAVITATION\n\nEMBRACING\nA SYSTEMATIC DERIVATION OF THE FORMULA FOR THE CALCULATION OF THE GEOCENTRIC AND\nCENTRIC PLACES, FOR THE DETERMINATION OF THE ORBITS OF PLANETS AND COMETS, FOB\nTHE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COMPUTATION OF\nSPECIAL PERTURBATIONS; TOGETHER WITH THE THEORY OF THE COMBINATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES.\n\nWtiili\n\nUunwwtl feunjjte\n\nmul ^uriliarg\n\nBY\n\nJAMES\n\nC.\n\nWATSON\n\n## DIRECTOR OF THE OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IN THE\n\nUNIVERSITY OF MICHIGAN\n\nJ.\n\nB.\n\nLIPPINCOTT &\n\n## LONDON: TRUBNER &\n\n1868\n\nCO.\n\nCO.\n\nASTRONOMY UBRAR*\n\n/\nEntered, according to Act of Congress, in the year\nJ.\n\nB.\n\nLIPPINCOTT &\n\n1868,\n\nby\n\nCO.,\n\nin the Clerk's Office of the District Court of the United States for the Eastern District\n\nof Pennsylvania.\n\nW3\n\nPREFACE.\nTHE\n\nNEWTON, prepared\n\nthe\n\nway\n\n## distinguished the history of astronomical science.\n\never, to the solution of those recondite\n\nto exhibit\n\n## development of the infinitesimal calculus and the labors of those who\n\ndevoted themselves to pure analysis have contributed a most important\npart in the attainment of the high degree of perfection which character;\n\n## izes the results\n\nof astronomical investigations.\n\nOf\n\n## the earlier efforts to\n\ndevelop the great results following from the law of gravitation, those of\n\nEULER\nin\n\n## stand pre-eminent, and the memoirs which he published have,\n\nreality,\n\ncelestial\n\nfurnished\n\nmechanics.\n\nthe\n\nIn\n\ngerm of\n\nall\n\nsubsequent investigations\n\n## this connection also the\n\nin\n\nnames of BERNOUILLI,\n\nas\n\n## having contributed also, in a high degree, to give direction to the investigations\n\nwhich were\n\nto unfold so\n\nmany\n\nmysteries of nature.\n\nBy means\n\n## of the researches thus inaugurated, the great problems of mechanics\n\nwere successfully solved, many beautiful theorems relating to the planetary motions demonstrated, and many useful formulae developed.\nIt is true,\n\n## were developed which have since been found to be impracticable, even\n\nif not erroneous; still, enough was effected to direct attention in the\nproper channel, and to prepare the\n\nway\n\nfor the\n\n## LAGRANGE and LAPLACE. The genius and\n\nextraordinary men gave to the progress of\nmost rapid\n\nstrides\n\nand the\n\nintricate investigations\n\nall\n\n## the analytical skill of these\n\nTheoretical Astronomy the\n\nnew\n\n## which they success-\n\ndiscoveries, so that of\n\nPREFACE.\n\n## more remained to be accomplished by their successors than to\n\nand\ndevelop\nsimplify the methods which they made known, and to introbut\n\nlittle\n\n## by the latest discoveries in the domain of pure anatysis.\n\nThe problem of determining the elements of the orbit of a comet\nmoving in a parabola, by means of observed places, which had been\n\npossible\n\nconsidered by\n\nby the former\n\nis\n\nothers,\n\nThe\n\nsolution given\n\n## complete; that given by the latter\n\nfar as regards the labor of\n\nis\n\nespecially simple\n\nand practical\n\nso\n\nit\n\nlittle\n\nto\n\n## answer best in actual practice, was that proposed by OLBERS in his\n\nwork entitled Leichteste und bequemste Methode die Bakn eines Cometen\nzu berechnen, in which, by\nbolic motion demonstrated\n\ncertain equations, he\n\nmaking use of a beautiful theorem of paraby EULER and also by LAMBERT, and by\n\ntrial\n\nand error\n\nwas enabled\n\nin the numerical\n\nto effect\n\nsolution of\n\n## performed with remarkable ease. The accuracy of the results obtained\n\nby OLBERS'S method, and the facility of its application, directed the\nattention of\n\nto this subject,\n\n## by them the method was extended and generalized, and rendered\n\ncable in the exceptional cases in which the other methods failed.\nIt should\n\nand\n\nappli-\n\nfirst\n\nhypothesis\n\nrequired\n\n## simply the determination of the corrections to be applied to these\n\nments in order to satisfy the observations. The more difficult\n\nele-\n\nproblem\n\n## of determining all the elements of planetary motion\n\ndirectly from three\nobserved places, remained unsolved until the discovery of Ceres\n\nby\n\n## PIAZZI in 1801, by which the attention of GAUSS was directed to this\n\nresult of which was the subsequent\npublication of his\nTheoria Motus Corporum Ccelestium, a most able work, in which he\ngave\nto the world, in a finished form, the results of\nof\nattention\nmany\n\nsubject, the\n\nyears\n\nPREFACE.\nto the subject of\n\nwhich\n\nit\n\nHis method\n\ntreats.\n\n## elements directly from given observed places, as given in the Theoria\n\nMotus, and as subsequently given in a revised form by ENCKE, leaves\nscarcely any thing to be desired on this topic.\n\n## gave the first explanation of the method of least squares, a method\n\nwhich has been of inestimable service in investigations depending on\nobserved data.\n\n## methods of determining their perturbations, since those applied in the\n\nFor a long\ncase of the major planets were found to be inapplicable.\ntime astronomers were content simply to compute the special perturbations of these bodies\n\nmencement\n\nfrom epoch\n\nto epoch,\n\nand\n\nit\n\nHANSEN\n\n## that serious hopes\n\nwere entertained of being able to compute successfully the general perturbations of these bodies. By devising an entirely new mode of considering the perturbations, namely,\n\n## by determining what may be\n\ncalled\n\nthe perturbations of the time, and thus passing from the undisturbed\n\ndifficult\n\nanalysis\n\nall the\n\nThe\n\nthis\n\nmost\n\nrefined\n\nand\n\n## and the laborious calculations involved were such\n\ndifficult\n\nthat,\n\neven\n\nthe method of special perturbations by the variation of constants\nafter\n\nto\n\nas\n\ndeveloped by LAGRANGE.\n\nThe discovery of\n\nAstrcea\n\nit\n\nso\n\nhappened that\n\nnew improvement.\n\n## The discovery by BOND and ENCKE of a method by which we determine\n\nat once the variations of the rectangular co-ordinates of the disturbed\n\n## body by integrating the fundamental equations of motion by means of\n\nmechanical quadrature, directed the attention of HANSEN to this phase\nof the problem, and soon after he gave formula for the determination\nof the perturbations of the latitude, the\n\nthe loga-\n\n## rithm of the radius-vector, which are exceedingly convenient in the\n\nprocess of integration, and which have been found to give the most\nsatisfactory results.\n\nThe formulse\n\nPREFACE.\n\n6\ntrue longitude,\n\nand\n\nsame manner,\n\n## were afterwards given by BRUNNOW.\n\nHaving thus stated briefly a few historical facts relating to the\nproblems of theoretical astronomy, I proceed to a statement of the\n\nThe discovery of\n\nso\n\nmany\n\nmotions, and\n\nit\n\nhas occurred to\n\nme\n\nthat a\n\n## development of all the formulae required in determining the orbits of the\n\nheavenly bodies directly from given observed places, and in correcting\nthese orbits by means of more extended discussions of series of observations, including also the\n\n## with a complete collection of auxiliary tables, and also such practical\n\ndirections as might guide the inexperienced computer, might add very\n\n## by attracting the attention of a\n\nnumber\nof\ncompetent computers. Having carefully read the\ngreater\nworks of the great masters, my plan was to prepare a complete work on\nmaterially to the progress of the science\n\n## commencing with the fundamental principles of dynamics,\n\nand systematically treating, from one point of view, all the problems\nThe scope and the arrangement of the work will be best\npresented.\nthis subject,\n\n## understood after an examination of\n\nits\n\ncontents\n\nand\n\nlet it suffice to\n\n## that I have endeavored to keep constantly in view the wants of the\n\ncomputer, providing for the exceptional cases as they occur, and giving\nall the formulae which appeared to me to be best adapted to the\nproblems\n\nunder consideration.\n\n## have not thought\n\ngeometrical signification of\n\nmany\n\ntiful\n\nit\n\nmay\n\nmany\n\nbeau-\n\n## For convenience, the formula) are numbered consecutively\n\neach\nthrough\nchapter, and the references to those of a preceding chapter\nare defined by adding a subscript figure denoting the number of the\nauxiliaries.\n\nchapter.\n\nspecial atten-\n\n## have consulted the Astronomische Nachrichten,\n\nthe Astronomical Journal, and other astronomical\nperiodicals, in which\ntion to these problems, I\n\nis\n\nto\n\n## be found much valuable information\n\nresulting from the experiwho have been or are now actively engaged in astro-\n\nence of those\n\nnomical pursuits.\n\nmust\n\nalso express\n\nmy\n\nPREFACE.\nMessrs. J. B.\n\nLIPPINCOTT\n\n&\n\nwhich they\n\n## have manifested in the publication of the work, and also to Dr. B. A.\n\nGOULD, of Cambridge, Mass., and to Dr. OPPOLZER, of Vienna, for\nvaluable suggestions.\n\nFor the determination of the time from the perihelion and of the true\nanomaly in very eccentric orbits I have given the method proposed by\n\nBESSEL\n\nby BRUNNOW\n\nxii.,\n\nand\n\n## method proposed by GAUSS, but in a more convenient form.\n\nFor obvious reasons, I have given the solution for the special case of\nalso the\n\n## parabolic motion before completing the solution of the general problem\n\nof finding all of the elements of the orbit by\nplaces.\n\nThe\n\ndifferential formulae\n\nfor correcting\n\n## approximate elements are given in a form convenient for application,\n\nfor finding the chord or the time of describing the\n\norbits, will\n\nbe\n\n## found very convenient in practice.\n\nI have given a pretty full development of the application of the\ntheory of probabilities to the combination of observations, endeavoring\nto direct the attention of the reader, as far as possible, to the sources of\n\nand\n\n## most advantageous method of treatas to eliminate the effects of these errors.\n\nFor the\n\nerror to be apprehended\n\nto the\n\n## I have given the simple formula, suggested\n\nby CHAUVENET, which\n\nfol-\n\nerrors,\n\nthe\n\nand which\n\nwill\n\nmore complete\n\n## answer for the purposes here required as well as\n\nIn the chapter\nproposed by PEIRCE.\n\ncriterion\n\n## devoted to the theory of special perturbations I have taken particular\n\npains to develop the whole subject in a complete and practical form,\n\n## numerical application. The time is adopted as the independent variable\n\nin the determination of the perturbations of the elements directly, since\nexperience has established the convenience of this form and should it\n;\n\n## be desired to change the independent variable and to use the differential\n\ncoefficients with respect to the eccentric anomaly, the equations between\nthis function\n\n## and the mean motion\n\nrequired transformation.\n\n## will enable us to effect readily the\n\nPREFACE.\n\nThe numerical examples involve data derived from actual observations, and care has been taken to make them complete in every respect,\nso as to serve as a guide to the efforts of those not familiar with these\n\ncalculations;\n\nand when\n\ndifferent\n\nastronomy, so that\n\nis\n\nit\n\nis\n\nof, it is\n\nis\n\nunnecessary to\n\nwhether the\n\npoint in space.\n\nme\n\ncost\n\nlast\n\ndecimal given.\n\n## puted and extended, and others here given\n\nprepared with special care.\nsolar attraction\n\nis\n\nthat given\n\nby GAUSS, which,\n\nas will appear,\n\n## accurately in accordance with the adoption of the\n\nmean\n\nis\n\nnot\n\ndistance of the\n\nearth from the sun as the unit of space; but until the absolute value of\nthe earth's\n\nmean motion\n\nand accuracy,\n\nis\n\nknown,\n\nit is\n\n## best, for the\n\nsake of uniformity\n\n## to retain GAUSS'S constant.\n\nThe preparation of\n\nthis\n\neffected\n\n## amid many interrup-\n\nand with other labors constantly pressing me, by which the progress\nof its publication has been somewhat delayed, even since the stereo-\n\ntions,\n\n## typing was commenced, so that in some cases I have been anticipated\n\nin the publication of formulae which would have here appeared for the\nfirst\n\ntime.\n\n## I have, however, endeavored to perform conscientiously the\n\nself-imposed task, seeking always to secure a logical sequence in the development of the formulae, to preserve uniformity and elegance in the\nnotation, and to elucidate the successive steps in the analysis, so that the\n\n## education, desire to be informed of the\n\nenabled to arrive at so\n\nmany grand\n\n## of the heavenly bodies, and by which the grandeur and\n\nsublimity of\ncreation are unveiled.\nThe labor of the preparation of the work will\n\nif it shall\n\n## general attention to the study of the wonderful\n\nvens, the contemplation of\n\nmind the\n\n## to impress upon the\n\nof\nthe\nof\nthe\nreality\nperfection\nOMNIPOTENT, the LIVING\n\n## OBSERVATORY, ANN ARBOR,\n\nGOD\n\nJune, 1867.\n\nCONTENTS.\nTHEORETICAL ASTRONOMY.\n\nCHAPTER\n\nI.\n\nINVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FORMULAE FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND\nGEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER.\nPAGE\n\nFundamental Principles..\n\n15\n\nAttraction of Spheres\nMotions of a System of Bodies\n\n23\n\n19\n\n## Invariable Plane of the System\n\nMotion of a Solid Body\nThe Units of Space, Time, and Mass\nMotion of a Body relative to the Sun\n\n29\n31\n\n36\n38\n\n## Equations for Undisturbed Motion\n\nDetermination of the Attractive Force of the Sun\nDetermination of the Place in an Elliptic Orbit\n\n42\n\n## Determination of the Place in a Parabolic Orbit\n\nDetermination of the Place in a Hyperbolic Orbit\n\n59\n\nMethods\n\n49\n53\n65\nin the\n\n70\n\n## Determination of the Position in Space\n\nHeliocentric Longitude and Latitude\nEeduction to the Ecliptic\n\n81\n\n## Geocentric Longitude and Latitude\n\nTransformation of Spherical Co-ordinates\nDirect Determination of the Geocentric Eight Ascension and Declination\nReduction of the Elements from one Epoch to another\n\n86\n\n83\n85\n87\n90\n\n99\n\nNumerical Examples\nInterpolation\nTime of Opposition\n\n103\n112\n\n114\n9\n\nCONTENTS.\n\n10\n\nCHAPTEE\n\nII.\n\n## FORMULA WHICH EXPRESS THE RELATION\n\nBETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY\nAND THE VARIATIONS OF THE ELEMENTS OF ITS ORBIT.\n\nPAGE\n\n## Variation of the Right Ascension and Declination\n\nCase of Parabolic Motion\n\n118\n\n## Case of Hyperbolic Motion\n\nCase of Orbits differing but little from the Parabola\n\n128\n\n125\n130\n\nNumerical Examples\nVariation of the Longitude and Latitude\nThe Elements referred to the same Fundamental Plane as the Geocentric Places\nNumerical Example\nPlane of the Orbit taken as the Fundamental Plane to which the Geocentric\nPlaces are referred\n\n135\n\n143\n149\n150\n153\n\nNumerical Example\n\n159\n\n163\n\nCHAPTER\n\nIII.\n\n## INVESTIGATION OF FORMULA FOR COMPUTING THE ORBIT OF A COMET MOVING\n\nIN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE\nVARIATION OF THE GEOCENTRIC DISTANCE.\nCorrection of the Observations for Parallax\n\n167\n\nFundamental Equations\n\n169\n\nParticular Cases\n\n172\n\nRatio of\n\nTwo\n\nCurtate Distances\n\n178\nDetermination of the Curtate Distances\n181\nRelation between Two Radii-Vectores, the Chord joining their Extremities, and\nthe Time of describing the Parabolic Arc\n184\nDetermination of the Node and Inclination\n192\nPerihelion Distance and Longitude of the Perihelion\n194\nTime of Perihelion Passage\n195\n'.\n\nNumerical Example\nCorrection of Approximate Elements by varying the Geocentric Distance\nNumerical Example\n\nCHAPTER\n\n199\n\n208\n213\n\nIV.\n\n## DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF\n\nTHE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM\nOF THE CONIC SECTION.\nReduction of the Data\nCorrections for Parallax\n\n220\n..\n\n223\n\nCONTENTS.\n\n11\nPAGE\n\nFundamental Equations\n\n225\n\n## Formulae for the Curtate Distances\n\nModification of the Formulae in Particular Cases\n..-\n\n-.\n\n228\n231\n\n## Determination of the Curtate Distance for the Middle Observation\n\nCase of a Double Solution\nPosition indicated by the Curvature of the Observed Path of the Body\nFormulae for a Second Approximation\n\n## Formulae for finding the Ratio of the Sector to the Triangle\n\nFinal Correction for Aberration\nDetermination of the Elements of the Orbit\n\n236\n\n239\n\n242\n243\n247\n\n257\n259\n\nNumerical Example\n\n264\n\n## Correction of the First Hypothesis\n\nApproximate Method of finding the Katio of the Sector to the Triangle\n\n278\n\nCHAPTER\n\n279\n\nV.\n\nDETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE.\nFundamental Equations\n\n282\n\n## Determination of the Curtate Distances\n\n289\n\nSuccessive Approximations\nDetermination of the Elements of the Orbit\n\n293\n294\n\nNumerical Example\n\n294\n\nMethod\n\n## for the Final\n\n307\n\nApproximation\n\nCHAPTER\n\n3TL\n\nFORMULA FOR THE CORRECTION OF THE APPROXIMATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY.\n\nINVESTIGATION OF VARIOUS\n\n311\n\n## Determination of the Elements of a Circular Orbit\n\nVariation of\nDifferential\n\nTwo\n\n313\n\nGeocentric Distances\n\nFormula\n\n318\n\n## Plane of the Orbit taken as the Fundamental Plane\n\nVariation of the Node and Inclination\n\n320\n\n328\n\nVariation of\n\n324\n\n332\n\nVelocities\n\n335\n\n## Correction of the Ephemeris\n\nFinal Correction of the Elements\n\nRelation between\nModification\n\nTwo\n\nwhen\n\n338\n\n339\n\nis\n\nvery large\n\n## Modification for Hyperbolic Motion\n\nVariation of the Semi-Transverse Axis and Katio of\n\n341\n346\n\nTwo\n\nCurtate Distances\n\n349\n\nCONTENTS.\n\n12\n\nPAGE\n\n## Variation of the Geocentric Distance and of the Reciprocal of the Semi-Trans-\n\n352\n\nverse Axis\n\nEquations of Condition\n\n353\n\nOrbit of a Comet\n\n355\n\nVariation of\n\nTwo\n\n357\n\nCHAPTER\n\nVII.\n\n## METHOD OF LEAST SQUARES, THEORY OF THE COMBINATION OF OBSERVATIONS,\n\nAND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM\nA SERIES OF OBSERVATIONS.\nStatement of the Problem\nFundamental Equations for the Probability of Errors\nDetermination of the Form of the Function which expresses the Probability\nThe Measure of Precision, and the Probable Error\n\n360\n\n362\n...\n\n363\n366\n\n367\n\n## The Mean Error, and the Mean of the Errors\n\nThe Probable Error of the Arithmetical Mean\n\n368\n\nDetermination of the\n\n## Mean and Probable\n\n370\n\nErrors of Observations\n\n## Weights of Observed Values\n\n371\n\n372\n\n376\n\nEquations of Condition\nNormal Equations\nMethod of Elimination\n\n378\n\n380\n\nUnknown\n\nQuanti-\n\n386\n\nties\n\n## Separate Determination of the Unknown Quantities and of their Weights\n\nRelation between the Weights and the Determinants\n\n## Case in which the Problem\n\nnearly Indeterminate\nand Probable Errors of the Results\n\n392\n396\n\nis\n\n398\n\nCombination of Observations\n\n399\n401\n\nMean\n\n## Errors peculiar to certain Observations\n\nRejection of Doubtful Observations\nCorrection of the Elements\n\n408\n\n410\n412\n415\n418\n423\n\n## Arrangement of the Numerical Operations\n\nNumerical Example\nCase of very Eccentric Orbits\n\nCHAPTER\n\nVIII.\n\n## INVESTIGATION OF VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL\n\nPERTURBATIONS OF A HEAVENLY BODY.\n\nFundamental Equations\nStatement of the Problem\n\n426\n428\n\nVariation of Co-ordinates\n\n429\n\nCONTENTS.\n\n13\nPAGE\n\n433\n443\n445\n\nMode\n\n## of effecting the Integration\n\nPerturbations depending on the Squares and Higher Powers of the Masses\n\n446\n\nNumerical Example\nChange of the Equinox and Ecliptic\nDetermination of New Osculating Elements\n\n448\n455\n\n459\n462\n\n## Determination of the Components of the Disturbing Force\n\nDetermination of the Heliocentric or Geocentric Place\n\n467\n\nNumerical Example\nChange of the Osculating Elements\nVariation of the Mean Anomaly, the Kadius- Vector, and the Co-ordinate\nFundamental Equations\nDetermination of the Components of the Disturbing Force\n\n474\n477\n480\n483\n489\n493\n495\n502\n505\n510\n516\n523\n526\n\n471\n\nz......\n\n## Case of very Eccentric Orbits\n\nDetermination of the Place of the Disturbed Body\nVariation of the Node and Inclination\n\nNumerical Example\nChange of the Osculating Elements\n\n'.\n\nVariation of Constants\n\n## Case of very Eccentric Orbits\n\nVariation of the Periodic Time\n\nNumerical Example\n\n529\n\n## Formulae to be used when the Eccentricity or the Inclination\n\nCorrection of the\n\nAssumed Value\n\nof the Disturbing\n\nis\n\n533\n\nsmall\n\nMass\n\n535\n\nPerturbations of Comets\n\n536\n\nMotion about the Common Centre of Gravity of the Sun and Planet\n537\nReduction of the Elements to the Common Centre of Gravity of the Sun and\nPlanet\nReduction by means of Differential Formulae\nNear Approach of a Comet to a Planet\nThe Sun may be regarded as the Disturbing Body\n\n540\n546\n\n550\n\n551\n\n538\n\n548\n\n## Effect of a Resisting Medium in Space\n\nVariation of the Elements on account of the Resisting\n\nMethod\n\nto be applied\n\nthe Ether\n\nwhen no Assumption\n\nis\n\n552\n\nMedium\n\n554\n\n## in regard to the Density of\n\n..\n\n556\n\nCONTENTS.\n\n14\n\nTABLES.\nPAGE\nI.\n\nII.\n\n561\nAngle of the Vertical and Logarithm of the Earth's Radius\nFor converting Intervals of Mean Solar Time into Equivalent Intervals\nof Sidereal\n\nIII.\n\n563\n\nTime\n\n## into Equivalent Intervals\n\n564\nof Mean Solar Time\nIV. For converting Hours, Minutes, and Seconds into Decimals of a Day... 565\n565\nV. For finding the Number of Days from the Beginning of the Year\nVI. For finding the True Anomaly or the Time from the Perihelion in a\n566\nParabolic Orbit\n\nVII. For finding the True Anomaly in a Parabolic Orbit when v is nearly 180 611\n612\nVIII. For finding the Time from the Perihelion in a Parabolic Orbit\nIX. For finding the True Anomaly or the Time from the Perihelion in Orbits\n614\nof Great Eccentricity\n\nX. For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits\nXL For the Motion in a Parabolic Orbit\nm sin 4 z' ...\nXII. For the Limits of the Eoots of the Equation sin (z'\n)\nXIII. For finding the Ratio of the Sector to the Triangle\nXIV. For finding the Ratio of the Sector to the Triangle\nXV. For Elliptic Orbits of Great Eccentricity\nXVI. For Hyperbolic Orbits\nXVII. For Special Perturbations\nXVIII. Elements of the Orbits of the Comets which have been observed\nXIX. Elements of the Orbits of the Minor Planets\nXX. Elements of the Orbits of the Major Planets\n\nXXL\n\nConstants,\n\n&c\n\n618\n\n619\n622\n\n624\n629\n632\n\n632\n633\n638\n\n646\n648\n\n649\n\n## EXPLANATION OF THE TABLES\n\n651\n\nAPPENDIX.\n\n657\n\nPrecession\n\nNutation\n\n658\n\nAberration\n\n659\n660\n\nIntensity of Light\nNumerical Calculations\n\n662\n\nTHEORETICAL ASTRONOMY.\n\nCHAPTER\n\nI.\n\nINVESTIGATION OF THE FUNDAMENTAL, EQUATIONS OF MOTION, AND OF THE FORMULAE FOB DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND\nGEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER.\n\nTHE\n\n1.\n\n## quire that we should know the ultimate limit of divisibility of the\n\nmatter of which they are composed, whether it may be subdivided\nindefinitely, or\n\nNor\n\nare\n\n## whether the limit\n\nwe concerned with\n\n## is an indivisible, impenetrable atom.\n\nthe relations which exist between the\n\nseparate atoms or molecules, except so far as they form, in the aggregate, a definite body whose relation to other bodies of the system it\nis\nOn the contrary, in considering the operequired to investigate.\nration of the laws in obedience to which matter is aggregated into\n\nsingle bodies\n\nof\n\nits\n\nand systems of\n\nbodies,\n\nit is\n\nsufficient to conceive\n\nsimply\n\n## which may be regarded as infinitesimal\n\nvolume of the body, and to regard the mag-\n\ndivisibility to a limit\n\nfinite\n\npoint.\n\nAn\n\n## element of matter, or a material body, cannot give itself\n\nmotion; neither can it alter, in any manner whatever, any motion\nwhich may have been communicated to it. This tendency of matter\nto resist all changes of its existing state of rest or\n\nas inertia,\n\nand\n\nis\n\nmotion\n\nis\n\nknown\nEx-\n\n## perience invariably confirms it as a law of nature; the continuance of\n\nmotion as resistances are removed, as well as the sensibly unchanged\n\nmany\n\n## centuries, affording the\n\n15\n\nTHEORETICAL ASTRONOMY.\n\n16\n\nits\n\nuniversality.\n\nWhenever,\n\ntherefore, a\n\n## material point experiences any change of its state as respects rest or\n\nmotion, the cause must be attributed to the operation of something\nexternal to the element itself, and which we designate by the word\nforces is generally unknown, and we estimate\nwhich\nthe\neffects\nthem by\nthey produce. They are thus rendered comand\nmay be expressed by abstract numbers.\nparable with some unit,\n2. If a material point, free to move, receives an impulse by virtue\n\nThe nature of\n\nforce.\n\nat\n\nif,\n\nany\n\nby which\n\n## shall cease to act, the subsequent motion of\n\nthe point, according to the law of inertia, must be rectilinear and\nbeing described in equal times. Thus, if s, v,\n\nmotion\n\nis\n\ncommunicated\n\nand\n\nand the\n\ntime, the\n\n## measure of v being the space described in a unit of time, we shall\n\nhave, in this case,\ns\n\nvt.\n\n## evident, however, that the space described in a unit of time will\n\nvary with the intensity of the force to which the motion is due, and,\nthe nature of the force being unknown, we must necessarily compare\nIt\n\nis\n\n## the velocities communicated to the point by different forces, in order\n\nare thus led to regard\nto arrive at the relation of their effects.\n\nWe\n\nthe force as proportional to the velocity; and this also has received\nthe most indubitable proof as being a law of nature.\nHence, the\nprinciples of the composition and resolution of forces\nalso to the composition and resolution of velocities.\n\nIf the force\n\n## acts incessantly, the velocity will\n\nmay\n\nbe applied\n\nbe accelerated, and\n\n## the force which produces this motion is called an accelerating force.\n\nIn regard to the mode of operation of the force, however, we may\nconsider\nit\n\nit\n\nwe may\n\nregard\n\nsented by\n\ndt,\n\nas\n\n## uniform during each of these\n\nthat which is best adapted to\n\n## The latter supposition is\n\nthe requirements of the infinitesimal calculus; and, according to the\nfundamental principles of this calculus, the finite result will be the\nintervals.\n\nsame as in the\n\n## case of a force whose action is absolutely incessant.\n\nTherefore, if we represent the element of space by ds, and the element of time by dt, the instantaneous velocity will be\n\nto another.\n\nFUNDAMENTAL\nSince the force\n\n3.\n\n17\n\n## proportional to the velocity, its measure at\n\nby the corresponding velocity. If\nconstant, the motion will be uniformly accele-\n\nany\n\nrated; and\n\nunit\n\nis\n\nPRINCIPLES.\n\nof/\n\nis\n\nwe\n\n## designate the acceleration due to the force by/, the\n\nthe\nvelocity generated in a unit of time, we shall have\nbeing\nif\n\n## however, the force be variable,\n\nIf,\n\nwe\n\nshall have, at\n\nany\n\ninstant,\n\nthe relation\n\n/=\n\ndt\n\nthe force being regarded as constant in its action during the element\nThe instantaneous value of v gives, by differentiation,\nof time dt.\n\ndv\n~dt\n\n_\n==\n\nd*s\n~di?\n\nd*s\n\ndue\n\nto the force is\n\nmea-\n\n## sured by the second differential of the space divided by the square\n\nof the element of time.\n4.\n\nBy\n\nter.\n\nentire\nis\n\n## the mass of the body we mean its absolute\n\nquantity of matis\nthe\nmass\nof\na\nunit\nof\ndensity\nvolume, and hence the\n\nThe\n\nmass\n\nis\n\nrequired to\n\nIf\nequal to the volume multiplied by the density.\nthe\nforces\nwhich\nact\ndifferent\ncompare\nbodies,\nupon\n\nit\nit\n\n## must be considered. If equal masses\n\nthe\naction of instantaneous forces, the forces\nimpulses by\nacting on each will be to each other as the velocities imparted ; and\nif we consider as the unit of force that which\ngives to a unit of mass\nis\n\nthe mass by M,\n\nThis\n\nis\n\nwe have\n\nfor the\n\nF = Mo.\n\n## called the quantity of motion of the body, and expresses its\n\novercome inertia. By virtue of the inert state of matter,\n\ncapacity to\n\n## there can be no action of a force without an\n\nequal and contrary reaction ; for, if the body to which the force is applied is fixed, the\n\n## equilibrium between the resistance and the force necessarily implies\n\nthe development of an equal and contrary force ; and, if the body be\nfree to move, in the\nchange of state, its inertia will oppose equal and\n\nTHEORETICAL ASTRONOMY.\n\n18\n\n## Hence, as a necessary consequence of inertia, it\n\ncontrary resistance.\nreaction are simultaneous, equal, and contrary.\nand\naction\nfollows that\nIf the body is acted upon by a force such that the motion is varied,\nthe accelerating force upon each element of\n-7-,\n\nis\n\nits\n\nmass\n\nis\n\nrepresented by\n\nexpressed by\n\nclt\n\nall\n\nV\n\nSince\n\nds\n=W\n\nthis gives\n\n## is the expression for the intensity of the motive force, or of\n\nFor the unit of mass, the measure\nthe force of inertia developed.\nof the force is\n\nwhich\n\nand\n\n## part of the intensity of the motive\n\nthe\nunit of mass, and is what is usually\nimpressed upon\n\nforce\n\nwhich\n\nis\n\n## called the accelerating force.\n\n5.\n\nThe\n\nforce in obedience to\n\n## through space, is known as the attraction of gravitation ;\n\nand the law of the operation of this force, in itself simple and unique,\nhas been confirmed and generalized by the accumulated researches of\nmodern science. Not only do we find that it controls the motions of\nthe bodies of our own solar system, but that the revolutions of binary\ntheir journey\n\nsystems of stars in the remotest regions of space proclaim the universality of its operation. It unfailingly explains all the phenomena\nobserved, and, outstripping observation, it has furnished the means\nof predicting\n\nmany phenomena\n\n## particle by a force which varies\n\nThe law of\n\nsubsequently observed.\n\n## this force is that every particle of matter is attracted\n\ndirectly as the mass\n\nby every other\ninversely as\n\nand\n\n## is instantaneous, and is not\n\nmodified, in any\nthe\nof\nother\nor\nbodies of matter. It\ndegree, by\ninterposition\nparticles\nis also absolutely independent of the nature of the molecules them-\n\nselves,\n\nand of\n\ntheir aggregation.\n\nATTRACTION OF SPHERES.\nIf we consider two bodies the masses of which are\n\n19\n\n## and m', and\n\nso\nare\nto\ntheir\nmutual\ndistance />,\nwhose magnitudes\nsmall, relatively\nthat we may regard them as material points, according to the law of\n\nof\ngravitation, the action\nin\n,\n\nand the\n\ntotal force\n\non m! will be\n\nm,m\n\nf>'\n\nThe\n\n## action of m' on each molecule of\n\nits total\n\naction\n\nwill be expressed\n\nby\n\n-,\n\nand\n\nby\nm'\n\n## which the masses m and m f tend\n\ntoward each other is, therefore, the same on each body, which result\nis a\nnecessary consequence of the equality of action and reaction.\nThe velocities, however, with which these bodies would approach\neach other must be different, the velocity of the smaller mass exceeding that of the greater, and in the ratio of the masses moved. The\nexpression for the velocity of m', which would be generated in a unit\nof time if the force remained constant, is obtained by dividing the\nabsolute force exerted by m by the mass moved, which gives\n\nThe\n\nabsolute or\n\nmoving\n\nforce with\n\nm\nand\n\nthis\n\naction of\naction of\n\n6.\n\nis,\n\n## measure of the acceleration due to the\n\nFor the acceleration due to the\nat the distance p.\n\ntherefore, the\n\n## and we shall therefore, preparatory to finding the equations\n\nwhich express the relative motions of the bodies of the system, determine the attraction of a spherical mass of uniform density, or\nin form,\n\nvarying from the centre to the surface according to any law, for a\npoint exterior to it.\nIf we suppose a straight line to be\n\n## drawn through the\n\ncentre of the\n\non the\nattracted, the total action of the sphere\nmass\nof the\nthe\nsince\nthis\nwill\nbe\na\nforce\nline,\npoint\nacting along\nelement\nan\ndenote\ndm\nLet\nto\nit.\nis\nwith\nsphere\nsymmetrical\nrespect\nsphere and the point\n\nTHEOEETICAL ASTEONOMY.\n\n20\n\nits\n\nthen will\n\ndm\n\n## on the point attracted. If we supexpress the action of this element\n\nand equal to unity, the\npose the density of the sphere to be constant,\nand\nwill be expressed by\nelement dm becomes an element of volume,\n\ndm = dx dy dz\n\n## being the co-ordinates of the element referred to a system\n\nof rectangular co-ordinates. If we take the origin of co-ordinates\nat the centre of the sphere, and introduce polar co-ordinates, so that\n\nand\n\nx, y,\n\n= r cos\n=\nr cos\ny\nx\nz\n\ndm\n\n=r sin\n\nits\n\nwe suppose\n\n0,\n&,\n\n?>,\n\ncos\n\n<p\n\nr2 cos\n\n, ,\n\nIf\n\n<f>\n\nbecomes\n\ndm = r\nand\n\ncos\nsin\n\n(p\n\n<p\n\ndr dy dO\n\nis\n\ndr\n\ndO\n\nd<p\n\na being the\n\nsince\n\nP*=(x- xy +\nwe\n\nshall\n\nhave\nz\n\nThe component of\n\na?\n\n(y\n\nI/)\n\n2ar sin\n\n+ (z - zj,\n\n<p -j-\n\nr2\n\nis\n\ndfcosr,\n\n## where f is the angle at the point attracted between the element dm\n\nand the centre of the sphere. It is evident that the sum of all the\ncomponents which act in the direction of the line a will express the\ntotal action of the sphere, since the\n\nsum\n\n## of those which act perpen-\n\nATTRACTION OF SPHERES.\n\n21\n\nsphere,\n\nis\n\nzero.\n\nBut we have\na\n\nz -j-\n\nand hence\ncos Y\n\n=a\n\nr sin\n\n<p\n.\n\nThe\n\ndp\n-f-\n\nda\n\nTherefore, if\n\nwe denote\n\n=a\n\nr sin\n\n<p\n\n= cos\n\nwith respect to\n\na, gives\n\nY.\n\n## have, by means of the values of df and\n\na A.\n\n2\n/>\n\n=r\n\ncos\n\ndr dy do\n\n<p\n\nwe\n\nshall\n\ncos f,\n\ndp\n.\n\n=-,\n\nda\n\nor\n\ndA\nThe\n\npolar co-ordinates\n\nr, <p y\n\nr2 cos\n\n<p\n\ndr dy do\n\n--r-.\n\naa\n\na,\n\nand hence\n\nd\nda\n\n(2)\n\nand we\n\nshall\n\nhave\n~~da'\n\n## Consequently, to find the total action of the sphere on the given\n\nwe have only to find\nby means of equation (2), the limits\n\npoint,\n\n## of the integration being taken so as to include the entire mass of the\n\nsphere, and then find its differential coefficient with respect to a.\n\nIf\n\nis\n\nwe\n\n## integrate equation (2) first with reference to 6, for\n\nand d\nconstant, between the limits 6\n2x, we get\n\ncos y dr\n\nwhich p\n\ndy\n\n## This must be integrated between the limits\n\n<p\n\n= -\\-%n and =\n<p\n\nTHEORETICAL ASTRONOMY.\n\n22\n\n## but since p is a function of ^>,\n\n2\np with respect to <p, we have\nr cos\n\nwe\n\nif\n\ndtp\n\n<p\n\n## differentiate the expression for\n\ndp,\n\nand hence\n\nV=\n\nffr\na JJ\n\ndr dp.\n\na\na-\\-r;\nr, and p\nCorresponding to the limits of <p we have p\nthese limits, we\nto\nbetween\nwith\nand taking the integral\nrespect\np\nobtain\n\nm, this becomes\n\nr,\n\nF=\n\nif\n\nand r\n\nwe denote\n\n=r\n\nits\n\nf,\n\nwe\n\nentire\n\nget\n\nmass by\n\n-.\na*\n\nTherefore,\n\nA=\n\n-=a4\n\nda\n\n'\n\n## from which it appears that the action of a homogeneous spherical\n\nmass on a point exterior to it, is the same as if the entire mass were\nconcentrated at\n\nwe\n\nits\n\ncentre.\n\nr\ntake the limits r and r\",\n\nand, denoting by\n\nand\n\nr' y this\n\nIf, in\n\nwe\n\nr,\n\nobtain\n\nbecomes\nA\nA\n\nm\n;.\n\na2\n\n## Consequently, the attraction of a homogeneous spherical shell on a\n\npoint exterior to it, is the same as if the entire mass were concentrated\nat its centre.\n\nThe\n\n## supposition that the point attracted is situated within a\n\nspherical shell of uniform density, does not change the form of the\n\nFUNDAMENTAL\n\nPRINCIPLES.\n\n23\n\nr\nr\nlimits will be p\na, and p\na, which give\n\n= +\n\nand\n\nthis being\n\nindependent of\n\na,\n\nwe have\n\nJt=if=u\nda\nWhence\n\nit\n\n## follows that a point placed in the interior of a spherical\n\nand that, if not subject to\n\nit\n\n## will be in equilibrium in every\n\nposition.\n7. Whatever may be the law of the change of the density of the\nheavenly bodies from the surface to the centre, we may regard them\nas composed of homogeneous, concentric layers, the density varying\nonly from one layer to another, and the number of the layers may\n\nbe indefinite.\n\nThe\n\n## mass were united\n\nsame as if its\nand hence the total action\nthe entire mass were concentrated\n\n## of the body will be the same as if\n\nThe planets are indeed not exactly spheres,\nat its centre of gravity.\nbut oblate spheroids differing but little from spheres ; and the error\nof the assumption of an exact spherical form, so far as relates to\nupon each other, is extremely small, and is in fact com-\n\ntheir action\n\n## pensated by the magnitude of their distances from each other for,\n\nwhatever may be the form of the body, if its dimensions are small\n;\n\nin\n\nbody which\naction will be sensibly the same as if\n\ncomparison with\n\nevident that\n\nits\n\nits\n\n## distance from the\n\nIf\n\nit attracts, it is\n\nits\n\nentire\n\nmass\n\nwe\n\nsuppose a system\nof bodies to be composed of spherical masses, each unattended with\nany satellite, and if we suppose that the dimensions of the bodies\nare small in comparison with their mutual distances, the formation\n\nwere concentrated at\n\nits\n\ncentre of gravity.\n\nof the equations for the motion of the bodies of the system will be\nreduced to the consideration of the motions of simple points endowed\nwith forces of attraction corresponding to the respective masses. Our\nthe several systems\nis, in reality, a compound system,\nof primary and satellites corresponding nearly to the case supposed ;\nand, before proceeding with the formation of the equations which are\nsolar system\n\n## applicable to the general case, we will consider, at first, those for a\n\ntheir\nsimple system of bodies, considered as points and subject to\nthe\nto\nmutual actions and the action of the forces which correspond\n\nTHEORETICAL ASTRONOMY.\n\n24\n\nactual velocities of the different parts of the system for any instant.\nIt is evident that we cannot consider the motion of any single body\nas free, and subject only to the action of the primitive impulsion\n\nwhich\n\nit\n\n## has received and the accelerating forces which act upon\n\nit\n\nbut, on the contrary, the motion of each body will depend on the\nforce\n\nwhich\n\nacts\n\nit\n\nupon\n\nand\n\ndirectly,\n\nalso\n\nto the\n\n## The consideration, however, of the variaother bodies of the system.\n\ntions of the motion of the several bodies of the system is reduced to\nthe simple case of equilibrium by means of the general principle that,\nif we assign to the different bodies of the system motions which are\n\n## modified by their mutual action, we may regard these motions as\n\ncomposed of those which the bodies actually have and of other\n\n## motions which are destroyed, and which must therefore necessarily\n\nbe such that, if they alone existed, the system would be in equilibrium.\nWe are thus enabled to form at once the equations for the\n\n## Let m, m r m\", &c. be the masses of\n\nthe several bodies of the system, and x, y, z, x', y 2', &c. their comotion of a system of bodies.\n\n',\n\n## any system of rectangular axes. Further, let\n\nthe components of the total force acting upon a unit of the mass of\nm, or of the accelerating force, resolved in directions parallel to the\nordinates referred to\n\nand\n\nm Yj\n\nmX,\n\n## Z, respectively, then will\n\nmZ,\n\nbe the forces which act upon the body in the same directions. The\nvelocities of the body m at any instant, in directions\nparallel to the\nco-ordinate axes, will be\n\ndx\n\n## and the corresponding\n\ndx\n\nBy\n\ndz\n\nforces are\n\n-dy\n\ndz\n\nvirtue of the action of the accelerating force, these forces for the\n\nm -jj + mXdt\n\n>\n\nm-jt+mYdt,\n\n-f\n\nmZdt\n\ndx\n\nThe\n\ndx\n\n25\n\ndx\n\n5+4\n\n1+4-\n\ndx\n\n+4>\n\nforces are\n\njdx\n\ndy\n\ndz\n\njdy\n\ndz\n\n## Comparing these with the preceding expressions for the forces, it\n\nappears that the forces which are destroyed, in directions parallel to\nthe co-ordinate axes, are\n\nmd -=-md^jL\n\nmd\nIn the same manner we find\nin the case of the body m!\n\nr-\n\n-\\-\n\nmXdt,\n\n+ mYdt,\n\n(3)\n\n-f mZdt.\n\n## which will be destroyed\n\nfJr'\n\n-m'd^ + m'X'dt,\n-m'd^jt\n\n+ m'Y'dt,\n\nm'd^+m'Z'dt;\ndt\nand similarly\n\nAccording to the\nthe action of\nunder\ngeneral principle above enunciated, the system\nfor the other bodies of the system.\n\n## these forces alone, will be in equilibrium.\n\nThe conditions of equilibrium for a system of points of invariable but arbitrary form, and\nsubject to the action of forces directed in any manner whatever, are\n\nIX,\n\n2Y\n\n0,\n\nQ,\n\nin\n\nwhich X,,\n\nY,, Z,,\n\nSZ,\n\n0,\n\nZ(X,z-Zp) =\n\n0,\n\n0,\n\nZ (Z ,y -\n\nY,z~)\n\n## denote the components, resolved parallel to the\n\nTHEORETICAL ASTEONOMY.\n\n26\n\n## co-ordinate axes, of the forces acting on any point, and x 9 y, z, the\n\nThese equations are equally applicable to\nco-ordinates of the point.\n\n## the case of the equilibrium at any instant of a system of variable\n\nform ; and substituting in them the expressions (3) for the forces destroyed in the case of a system of bodies,\n\nwe\n\nshall\n\nhave\n\n2mX= 0,\n\n^m ~jr\n\n(4)\n\n8.\n\nLet x y h\nt)\n\nz t)\n\n## differentiation of the equations for the co-ordinates\n\nsystem, and, by\nof the centre of gravity, which are\n\nXf\n\nwe\n\n= 2mx\n~Zm'\n\n'\n\n2my\n= ~Zm~'\n\n*'\n\n2mz\n^ ~Ym'\n\nget\nv\n\n2m\n\ndf\n\n\"v\n\nffix\n\ndt\n\nd*y\n\nSm\n\n## Introducing these values into the\n\nfirst\n\nd*z\n\n2m\n\ndp\n\nthree of equations\n\n(4),\n\nthey\n\nbecome\nfe, ~\n__\n\nImX\n\ntfy,\n\nZm'\n\n~di?\n\n'\n\n~dP\n\nfrom which\n\nit\n\n_ZmY\n\ntfz,\n\n2m'\n\n~dP\n\n'\n\n__\n~~\n\nImZ\n\n'\n\n~2m\n\n## appears that the centre of gravity of the system moves\n\nin space as if the masses of the different bodies of which it is composed, were united in that point, and the forces directly applied to it.\nIf we suppose that the only accelerating forces which act on the\n\nbodies of the system, are those which result from their mutual action,\nwe have the obvious relation\n:\n\nro'JT,\n\nmY=\n\nm'Y',\n\nmZ=\n\nm'Z',\n\nand similarly\n\nfor\n\n## any two bodies\n\n2mX = 0,\nso that equations (5)\n\n27\n\nand, consequently,\n\nSmY= 0,\n\nZmZ=\n\nbecome\n\n= o,\n^t\ndff\n\n*5 = o,\ndt*\n\n**'\n\no.\n\nd?\n\n## Integrating these once, and denoting the constants of integration by\n\nthe results,\nc, c', c\"', we find, by combining\n\nand hence the absolute motion of the centre of gravity of the system,\nwhen subject only to the mutual action of the bodies which compose\nbe uniform and rectilinear. Whatever, therefore, may be\nit, must\nthe relative motions of the different bodies of the system, the motion\nof its centre of gravity is not thereby affected.\n9. Let us now consider the last three of equations (4), and suppose\nthe system to be submitted only to the mutual action of the bodies\nwhich compose it, and to a force directed toward the origin of co-\n\nThe\n\nordinates.\ntion, is\n\nexpressed by\n\nin\n\nTo\n\n## resolve this force in directions parallel to the three rectangular\n\naxes, we must multiply it by the cosine of the angle which the line\n\nwhich gives\n\n--\n\nA = m'(af\n\nx)\n\n=\n3\n\n--\n\nv = m'tf\nJL\n\n-- -\n\nA = m'(z'-z)\n3\n\ny)\n\np*\n\nrespectively,\n\n## Further, for the components of the accelerating force of\n\non\n\nm we\nr\n\nhave\n,\n\nm (x\n\nm(y\n~~\n\nof)\n\nm(z\n~\n\n)\n'\n\n'\n\nHence we derive\n\nm(Yx\n\nXy)\n\n+ m' (FV\n\nXy) = 0,\n\nand generally\nQ.\n\n(6)\n\nTHEORETICAL ASTRONOMY.\n\n28\n\nIn a similar manner, we\n\nfind\n\n2m (Xz\n2m (Zy\n\nZx)\nYz)\n\nif,\n\n=\n=\n\n(7)\n\n0,\n0.\n\n## in addition to their reciprocal\n\naction, the bodies of the system are acted upon by forces directed to\nthe origin of co-ordinates.\nThus, in the case of a force acting upon\n\n## m, and directed to the origin of co-ordinates,\n\nalone,\n\nYx\n\n= Xy,\n\nXz\n\n= Zx,\n\nwe\n\nhave, for\n\nits\n\naction\n\nZy=Yz,\n\nand similarly for the other bodies. Hence these forces disappear\nfrom the equations, and, therefore, when the several bodies of the\nsystem are subject only to their reciprocal action and to forces directed\nto the origin of co-ordinates, the last three of equations (4) become\n\n2m (xdy\n2m (zdx\n2m (ydz\nc,\n\nis\n\n=\nxdz) =\n\ncdt,\n\nzdy}\n\nc\"dtj\n\nydx]\n\nc'dt,\n\n(8)\n\n## c', and c\" being the constants of integration.\n\nNow, xdy ydx\ndouble the area described about the origin of co-ordinates by the\n\n## m with the origin of\n\non the plane of xy during the element of time dt; and,\nzdx\nxdz and ydz\nzdy are respectively double the areas\n\nco-ordinates,\nfurther,\n\n## described, during the same time, by the projection of the radius-vector\n\non the planes of xz and yz. The constant c, therefore, expresses the\nsum of the products formed by multiplying the areal velocity of each\n\nbody, in the direction of the co-ordinate plane xy, by its mass; and\nr/\nc', c , express the same sum with reference to the co-ordinate planes\nxz and yz respectively.\nHence the sum of the areal velocities of the\nseveral bodies of the system about the\norigin of co-ordinates, each\nmultiplied by the corresponding mass, is constant; and the sum of\n\n## the areas traced, each multiplied by the\n\ncorresponding mass, is proIf the only forces which operate, are those\nportional to the time.\n\nINVAKIABLE PLANE.\n\nresulting\n\n## system, this result is correct whatever\n\ntaken as the origin of co-ordinates.\n\nThe\n\n29\n\nareas described\n\nmay\n\n## body on the co-ordinate planes, are the projections, on these planes, of\n\nthe areas actually described in space. We may, therefore, conceive of\na resultant, or principal plane of projection, such that the sum of the\n\n## by the projection of each radius-vector on this plane,\n\nprojected on the three co-ordinate planes, each being multiplied\n\nareas traced\n\nwhen\n\n## by the corresponding mass, will be respectively equal to the first\n\nmembers of the equations (8). Let\nand y be the angles which\n/9,\nmakes\nwith\nthe\nthis principal plane\nco-ordinate planes xy, xz, and yz\nrespectively; and let 8 denote the sum of the areas traced on this\n,\n\n## by the projection of the radius-vector of\n\neach of the bodies of the system, each area being multiplied by the\ncorresponding mass. The sum S will be found to be a maximum,\nand\n\nits\n\n## projections on the co-ordinate planes, corresponding to the\n\ndt, are\n\nelement of time\n\nS cos a dt,\nTherefore,\nc\n\nS cos\n\n/?\n\nby means of equations\n\n= S cos\n2\n\n+ cos\n\n(8),\n\n= S cos\n=\n-f cos\n\nwe have\n\na,\n\nS cos Y dt.\n\ndt,\n\nc\"\n\n/?,\n\n= S cos\n\nY,\n\n/5\n\n7-\n\n1,\n\nHence we derive\ncos a\n\n_ /\n\n'21\n\ncos Y\n\ncos\n\n'/2\n\n/5\n\nc\"\n\n## These angles, being therefore constant and independent of the time,\n\nthis principal plane of projection remains constantly par-\n\nshow that\nallel\n\nto itself\n\n## the relative positions of the several bodies; and for this\n\nIts position\nreason it is called the invariable plane of the system.\nwith reference to any known plane is easily determined when the\n\nmay be\n\nvelocities, in directions\n\nparallel to the\n\n## masses and co-ordinates of the several bodies of the system, are\n\nknown. The values of c, c r , c\" are given by equations (8), and\n\nTHEORETICAL ASTRONOMY.\n\n30\nhence the values of\n\na,\n\n/9,\n\nand\n\n7-,\n\n## which determine the position of the\n\ninvariable plane.\nSince the positions of the co-ordinate planes are arbitrary, we may\nsuppose that of xy to coincide with the invariable plane, which gives\ncos\n\n/?\n\nand\n\ncos f\n\n= 0, and, therefore, =\nc'\n\nand c\"\n\n0.\n\nFurther,\n\nsince the positions of the axes of x and y in this plane are arbitrary,\nit follows that for every plane perpendicular to the invariable plane,\n\nthe\n\nsum\n\n## of the areas traced by the projections of the radii- vector es\n\nof the several bodies of the system, each multiplied by the corresponding mass, is zero. It may also be observed that the value of S\nis\n\nconstant whatever\n\nand that\n\nits\n\nvalue\n\nis\n\n## be the position of the co-ordinate planes,\n\nnecessarily greater than that of either of the\n\nmay\n\nmember of\n\nthe equatity.\n\n## when two of them are each equal to zero. It is, therefore, a\n\nmaximum, and the invariable plane is also the plane of maximum\nexcept\nareas.\n\n## 10. If we suppose the origin of co-ordinates itself to move with\n\nuniform and rectilinear motion in space, the relations expressed by\nThus, let xn yn z, be the coequations (8) will remain unchanged.\n\n## ordinates of the movable origin of co-ordinates, referred to a fixed\n\n'\npoint in space taken as the origin ; and let XQ , yQ9 z , a? ', y 9 z/, &c.\nbe the co-ordinates of the several bodies referred to the movable\norigin.\n\n## always parallel to those of the other system of co-ordinates, we shall\n\nhave\nJ\nx\nXQ\n*\nz\nx,\n*,\ny\ny, ryQ\n\n= +\n\nand similarly for the other bodies of the system. Introducing these\nvalues of x y and z into the first three of equations (4), they become\n9\n\nThe\ngives\n\norigin\n\n33\n\n## and the preceding equations become\n\no,\n\n(9)\n\nZmZ =0.\nSubstituting the same values in the last three of equations (4), observing that the co-ordinates x h y,, z, are the same for all the bodies\n\n## of the system, and reducing the resulting equations by means of\n\nequations\n\n(9),\n\nwe\n\nget\n\nrf\n\n^W~ x\n(r7\n*,\n\nHence\n\n~ 2m (** - 2O =\n\n0,\n\n(10)\n\nappears that the form of the equations for the motion of the\nof\nbodies, remains unchanged when we suppose the origin of\nsystem\nco-ordinates to move in space with a uniform and rectilinear motion.\nit\n\nThe\n\n## equations already derived for the motions of a system of\n\nconsidered\nas reduced to material points, enable us to form at\nbodies,\nonce those for the motion of a solid body. The mutual distances of\n11.\n\nthe parts of the system are, in this case, invariable, and the masses\nof the several bodies become the elements of the mass of the solid\n\nwe\n\n## denote an element of the mass by dm, the equations (5)\n\nfor the motion of the centre of gravity of the body become\n\nbody.\n\nIf\n\n(11)\n\n## the summation, or integration with reference to dm, being taken so as\n\nto include the entire mass of the body, from which it appears that\nthe centre of gravity of the body moves in space as if the entire mass\n\n## were concentrated in that point, and the forces applied to it directly.\n\nIf we take the origin of co-ordinates at the centre of gravity of\nthe body, and suppose it to have a rectilinear, uniform motion in\n\n## and denote the co-ordinates of the element dm, in reference\n\nof the equations (10),\norigin, by XQ , y Q ZQ we have, by means\n\nspace,\nthis\n\nto\n\nTHEORETICAL ASTRONOMY.\n\n32\n\ny.\n\n*,\n\ndm -f ( Yx\n\n~ x%) dm =\n\ndm - (Xz\n\n- Zx\n\ndm\n\ndm\n\n-(% - K ^ =\nO)\n\no,\n\n## the integration with respect to dm being taken so as to include the\n\nThese equations, therefore, determine the\nentire mass of the body.\n\nits\n\n## as fixed, or as having a uniform rectilinear motion in space.\n\nEquations (11) determine the position of the centre of gravity for any\ninstant, and hence for the successive instants at intervals equal to dt;\nand we may consider the motion of the body during the element of\ntime dt as rectilinear and uniform, whatever may be the form of its\nHence, equations (11) and (12) completely determine the\ntrajectory.\n\n## the former relating to the motion of\n\nposition of the body in space,\ntranslation of the centre of gravity, and the latter to the motion of\nIt follows, therefore, that for any forces\nact upon a body we can always decompose the actual motion\n\nwhich\n\n## into those of the translation of the centre of gravity in space, and of\n\nthe motion of rotation around this point ; and these two motions may\n\n## be considered independently of each other, the motion of the centre\n\nof gravity being independent of the form and position of the body\n\nIf the only forces which act upon the body are the reciprocal action\nof the elements of its mass and forces directed to the origin of coordinates, the second terms of equations (12) become each equal to\nzero, and the results indicated by equations (8) apply in this case\nalso.\n\nof\n\nThe\n\n## parts of the system being invariably connected, the plane\n\nareas, or invariable plane, is evidently that which is\n\nmaximum\n\n## perpendicular to the axis of rotation passing through the centre of\n\ngravity, and therefore, in the motion of translation of the centre of\ngravity in space, the axis of rotation remains constantly parallel to\nitself.\nAny extraneous force which tends to disturb this relation\nwill necessarily develop a contrary reaction,\n\nany change of\n\nWe\n\n## plane of rotation not parallel to itself.\n\nthat\non\naccount of the invariability of the mutual\nmay observe, also,\ndistances of the elements of the mass, according to equations (8), the\nresists\n\nits\n\n12.\n\nWe\n\nshall\n\nnow\n\n33\n\n## Let xw yQ , ZQ , x f , y ', z ',\n\ndistant mass, which we will denote by M.\n&c. be the co-ordinates of the several bodies of the system referred\nits centre of gravity as the origin of co-ordinates; xn\ny, 9 and z,\nthe co-ordinates of the centre of gravity of the system referred to\n\nto\n\n## The co-ordinates of the body\n\nthe centre of gravity of the body M.\nm, of the system, referred to this origin, will therefore be\nx\n\n= +\nx,\n\na?\n\n-f y\n\ny,\n\nfl ,\n\nz,-\\- z\n\n## and similarly for the other bodies of the system. If we denote by\n\nr the distance of the centre of gravity of\nfrom that of M, the\nforce\nof\nthe\nformer\non\nan\nelement\nof mass at the centre\naccelerating\n\nof gravity of the\n\nlatter,\n\nmx\n\n## and, therefore, that of the entire system on the element of\n\nin the same direction, will be\n\nWe\n\nhave also\nr\n\nand, if\n\n+xY+\n\n(x,\n\nwe denote by\n\n(y,\n\n2\n2/\n\n(*/\n\n+z\n\nM resolved\n9\n\n## the distance of the centre of gravity of the\n\nr,\n\nsystem from M,\n\n= + +\n*,\n\nr,\n\ny,\n\ni,,\n\nTherefore\n\n-x\n\nWe\n\nshall\n\n+X\n\nQ}\n\n(X,\n\n+ 2(^ + Mo + z,z + r\n\nnow suppose\n\nsystem to be so small in\n\nS\n2\n\n(r;\n\nr,\n\nof\n\nits\n\ncentre\n\n## of gravity from that of\n\nthat terms of the order r 2 -may be neglected\n9\na condition which is actually satisfied in the case of the secondary\nsystems belonging to the solar system. Hence, developing the second\nfactor of the second\n\nof the order r\n\n2\n,\n\nwe\n\nmember\nshall\n\n## of the last equation, and neglecting terms\n\nhave\n\n~~\nr3\n\nr/\n\nr,\n\nand\n\n=x\n\nrf\n\n-~\n\nL,\n3\n\nr,\n\n(x,ZmxQ\n\n+ y,Zmy\n\nTHEORETICAL ASTRONOMY.\n\n34\n\n## But, since o? , y , zw are the co-ordinates in reference to the centre of\n\ngravity of the system as origin, we have\n\nSmyQ\n\nto\n\nImxQ\n\n0,\n\nmx\n\n^=\n\nSm&\n\n0,\n\n0,\n\n2m\n\n'^-\n\nSm\n\nmy\n\nmz\n\n'\n\nq~~\n\nr3\n\nM resolved\n9\n\nq\n3\n\nr3\n\nr,\n\non\n\n\\J 9\n\nZm\n\nr,\n\n## of these equations are the expressions for the\n\ndue to the action of the bodies of the system\n\nwhen we\n\n## consider the several masses to be collected at the centre of gravity\n\nof the system. Hence we conclude that when an element of mass\nis\n\nattracted\n\nit\n\n## order of the squares of the co-ordinates of the several bodies, referred\n\nto the centre of gravity of the system as the origin of co-ordinates,\nmay be neglected in comparison with the distance of the system from\nthe point attracted, the action of the system will be the same as if\nthe masses were all united at its centre of gravity.\n\n## If we suppose the masses w, m', m\", &c. to be the elements of the\n\nmass of a single body, the form of the equations remains unchanged;\nand hence it follows that the mass\nis acted upon\nby another mass,\n\n## or by a system of bodies, as if the entire mass of the body, or of the\n\nIt is evident, also,\nsystem, were collected at its centre of gravity.\nthat reciprocally in the case of two systems of bodies, in which the\n\n## mutual distances of the bodies are small in comparison with the\n\ndistance between the centres of gravity of the two systems, their\nmutual action is the same as if all the several masses in each system\n\ncommon\n\n## centre of gravity of that system ; and\n\nmove as if the masses were thus\n\nunited.\n\n## 13. The results already obtained are sufficient to enable us to form\n\nthe equations for the motions of the several bodies which\ncompose the\nsolar system.\nIf these bodies were exact spheres, which could be\n\n## considered as composed of homogeneous concentric\n\nspherical shells,\nthe density varying only from one layer to\nanother, the action of\n\n35\n\n## each on an element of the mass of another would be the same as if\n\nthe entire mass of the attracting body were concentrated at its centre\n\n## The slight deviation from this law, arising from the\n\nform of the heavenly bodies, is compensated\nby the magnitude of their mutual distances; and, besides, these mutual distances\nof gravity.\nellipsoidal\n\n## are so great that the action of the attracting\n\nbody on the entire mass\nof the body attracted, is the same as if the latter were concentrated\nat its centre of gravity.\nHence the consideration of the reciprocal\naction of the single bodies of the system, is reduced to that of material\n\n## points corresponding to their respective centres of gravity, the masses\n\nof which, however, are equivalent to those of the\ncorresponding\nbodies.\nThe mutual distances of the bodies composing the secondary\nsystems of planets attended with satellites are so small, in comparison\nwith the distances of the different systems from each other and from\n\nthe other planets, that they act upon these, and are\nreciprocally acted\nupon, in .nearly the same manner as if the masses of the secondary\nsystems were united at their common centres of gravity, respectively.\n\nThe motion of\nplanet and\n\n## the centre of gravity of a system\n\nconsisting of a\nnot affected by the reciprocal action of the\n\nits satellites is\n\n## bodies of that system, and hence it may be considered\n\nindependently\nof this action.\nThe difference of the action of the other planets on\na planet and its satellites will simply produce\ninequalities in the\nrelative motions of the latter bodies as determined\n\naction alone,\n\nand\n\n## will not affect the motion of their\n\nby\n\ntheir\n\ncommon\n\nmutual\ncentre\n\nof gravity.\nHence, in the formation of the equations for the motion\nof translation of the centres of gravity of the several planets or\n\n## secondary systems which compose the solar system, we have simply\n\nthem as points endowed with attractive forces correspond-\n\nto consider\n\n## ing to the several single or aggregated masses. The investigation\n\nof the motion of the satellites of each of the planets thus attended,\n\n## forms a problem entirely distinct from that of the motion of the\n\ncentre of gravity of such a system.\nThe consideration of\nthe motion of rotation of the several bodies of the solar system about\ntheir respective centres of gravity, is also independent of the motion\n\ncommon\n\n## the forces which act upon a\n\nplanet passed through the centre of gravity, the motion of rotation\nwould be undisturbed; and, since this resultant in all cases very\n\nof translation.\n\nIf the resultant of\n\nall\n\n## nearly satisfies this condition, the disturbance of the motion of rotais\n\nvery slight. The inequalities thus produced in the motion\n\ntion\n\n## fact, sensible, and capable of being indicated by\n\nIt has, indeed,\nin\nthe case of the earth and moon.\nobservation, only\n\nof rotation are, in\n\nTHEORETICAL ASTRONOMY.\n\n36\n\nbeen rigidly demonstrated that the axis of rotation of the earth relative to the body itself is fixed, so that the poles of rotation and the\n\n## same position in reference\n\nalso the velocity of rotation is constant.\n\nand that\n\nto the surface;\n\n## This assures us of the permanency of geographical positions, and,\n\nin connection with the fact that the change of the length of the\n\n## day arising from the variation of the obliquity of the\n\nof the tropical year, due to the action of\necliptic and in the length\nthe sun, moon, and planets upon the earth, is absolutely insensible,\n\nmean\n\nsolar\n\namounting\nyears,\n\nwe\n\n## on one of the bodies of the system, it is\n\nonly possible to deduce from observation the relative motions of the\nThese relative motions in the case of the\ndifferent heavenly bodies.\n14. Placed, as\n\nare,\n\ncomets and primary planets are referred to the centre of the sun,\nsince the centre of gravity of this body is near the centre of gravity\nof the system, and its preponderant mass facilitates the integration\nof the equations thus obtained. In the case, however, of the secondary\nsystems, the motions of the satellites are considered in reference to\n\nWe\n\n## the centre of gravity of their primaries.\n\nshall, therefore, form\nthe equations for the motion of the planets relative to the centre of\n\n## gravity of the sun; for which it becomes necessary to consider more\n\nparticularly the relation between the heterogeneous quantities, space,\nEach denomination,\ntime, and mass, which are involved in them.\n\n## being divided by the unit of its kind, is expressed by an abstract\n\nnumber ; and hence it offers no difficulty by its presence in an equa-\n\nFor the unit of space we may arbitrarily take the mean distance of the earth from the sun, and the mean solar day may be\ntaken as the unit of time. But, in order that when the space is\n\ntion.\n\nexpressed by\n\n1,\n\n1,\n\nmay\n\nalso be\n\n## of space is first adopted, the relation of\n\nwhich determines the measure of the force\n\n1, if the unit\n\nexpressed by\nthe time and the mass\n\nif\n\nwe denote by /\n\n## the acceleration due to the action of the\n\nThus,\nmass m on a material point at the distance a, and by/' the acceleration corresponding to another mass m!\nacting at the same distance,\n\nwe have\n\nthe relation\n\n37\n\n## and hence, since the acceleration is proportional to the mass, it may\n\nBut we have, for the measure\nbe taken as the measure of the latter.\nof/,\n\n^\ndV\n\n## Integrating this, regarding /as constant, and the point to\n\na state of rest, we get\ns\n\nThe\n\nmove from\n(13)\n\n$fi*. ## acceleration in the case of a variable force ## is, at any instant, measured by the velocity which the force acting at that instant would generate, if supposed to remain constant in its action, during a unit of time. The last equation gives, when 1, /=2; and hence the acceleration is also measured by double the space which would be described by a material point, from a state of rest, during a unit of time, the force being supposed constant in its action during this time. In each case the duration of the unit of time is involved ## measure of the acceleration, and hence in that of the mass on which the acceleration depends and the unit of mass, or of the force, will depend on the duration which is chosen for the unit of time. In in the ## general, therefore, we regard as the unit of mass that which, acting constantly at a distance equal to unity on a material point free to ## move, will give to ## this point, in a unit of time, a velocity which, ## the force ceased to act, would cause tance in the unit of time. if it ## to describe the unit of dis- Let the unit of time be a mean solar day; ]& the acceleration due by the mass of the sun at the unit of distance; and /the acceleration corresponding to the distance T; then will to the force exerted ## becomes the measure of the mass of the sun. The unit of mass is, therefore, equal to the mass of the sun taken as many times and contained in unity. Hence, when we take the mean solar day as the unit of time, the mass of the sun is measured by F; by which we are to understand that if the sun acted during a mean solar as 1& is ## day, on a material point free to move, at a distance constantly equal to the mean distance of the earth from the sun, it would, at the end ## of that time, have communicated to the point a velocity which, if THEORETICAL ASTRONOMY. 38 ## the force did not thereafter act, would cause 2 of time, the space expressed by /v it to describe, in a unit The acceleration due ## to the action of the sun at the unit of distance ## the square root of this quantity appears designated by formulae will be derived. in the which frequently k? } since is If we take arbitrarily the ## mass of the sun as the unit of mass, the Let t denote the number of mean ## unit of time must be determined. solar days which must be taken for the unit of time when the unit The space which the force due to of mass is the mass of the sun. ## on a material point at a distance equal to the mean distance of the earth from the sun, would cause the point this mass, acting constantly ## to describe in the time t, is, ## according to equation (13), But, since t expresses the number of mean solar days in the unit of time, the measure of the acceleration corresponding to this unit is 2s, and this being the unit of force, we have W= and hence -=! Therefore, if the mass of the sun is regarded as the unit of mass, the solar days in the unit of time will be equal to unity number of mean divided by the square root of the acceleration due to the force exerted this mass at the unit of distance. The numerical value of k will by ## be subsequently found to be 0.0172021, which gives 58.13244 mean solar days for the unit of time, when the mass of the sun is taken as the unit of mass. 15. Let x, y, z ## be the co-ordinates of a heavenly body referred to ## the centre of gravity of the sun as the origin of co-ordinates; r its denote the radius-vector, or distance from this origin; and let quotient obtained by dividing its mass by that of the sun; then, taking the mean solar day as the unit of time, the mass of the sun is z For a expressed by F, and that of the planet or comet by mk r second body let the co-ordinates be a/, y , z' ; the distance from the . 2 sun, r' ; and the mass, m'k ; and similarly for the other bodies of the Let the co-ordinates of the centre of gravity of the sun system. referred to any fixed point in space be , 57, , the co-ordinate planes ## being parallel to those of x, y, and 2, respectively; ## then will the ## MOTION RELATIVE TO THE SUN. acceleration ## due to the action of ## and the three components of 39 ## on the sun be expressed by ^~, ## this force in directions parallel to the ## co-ordinate axes, respectively, will be mtf-, mtf-V-, r3 r3 The action of mJc2 r3 ## on the sun will be expressed by ## and hence the acceleration due to the ## combined and simultaneous action of the several bodies of the system on the sun, resolved parallel to the co-ordinate axes, will be mx \" my \"73\"' The motion of > mz ^*' ## the centre of gravity of the sun, relative to the fixed be determined by the equations ## origin, will, therefore, ## Let p denote the distance of in from m r p f its distance from m\" y adding an accent for each successive body considered; then will the action of the bodies m', m\", &c. on m be of which the three components parallel to the co-ordinate axes, respectively, are The ## action of the sun ## on m, resolved in the same manner, is expressed by Vx tf tfz which are negative, since the force tends to diminish the co-ordinates The three components of the total action of the other Xj y, and z. bodies of the system on are, therefore, THEORETICAL ASTRONOMY. 40 &x _j_ 7. 2 v ra'<y a) \"7 -~F' ' i./jX^-y) 3 ,.. , \"jr- ## m referred to the fixed origin ## and, since the co-ordinates of + a, >? c -f y, are ## the equations which determine the absolute motion are <* dt* cfcc \" F* _ \" 7<2V mV x) ' r3 \" ' ## the symbol of summation in the second members relating simply to the masses and co-ordinates of the several bodies which act on m, exclusive of the sun. ## Substituting for -j^, O/t ## given by equations (14), we and -~ their values ut (Mi get ## Since x, y y z are the co-ordinates of m relative to the centre of gravity of the sun, these equations determine the motion of m relative to that The second members may be put in another form, which point. greatly facilitates the solution of some of the problems relating to the motion of m. Thus, let us put t\\ m' m\" xx'+yy' +z^\\ II xx\"+yy\"+2z\" (17) and we shall ## have for the partial ## differential coefficient of this with respect to x, ^\\_L/_ i*_^.\\ +j5L/_ i*:_^.\\ 4 m\\ dxl l p* dx r' !^l + m\\ p'* dx r\" 3 &c ## MOTION EELATIVE TO THE SUN. 41 But, since we have dp _ dx dp' _ x\" dx f> ## and hence we derive (d^\\_ m' Ix' x' m\" lx\"x x\" or We same manner, with respect to y and z, ## find, also, in the cients The ## for the partial differential coeffi- ## equations (16), therefore, become ## members of equations (16) exthe difference between action of the bodies m', m\", &c. on the press and on the sun, resolved parallel to the co-ordinate axes respectIt will be observed that the second The mutual distances of the planets are such that these quanare generally very small, and we may, therefore, in a first relative to the sun, neglect the approximation to the motion of second members of these equations; and the integrals which may ively. tities ## then be derived, express what is called the undisturbed motion of m. By means of the results thus obtained for the several bodies successively, the ## approximate values of the second members of equations and hence a still closer approximation to the The force whose components are expressed by ## (16) may be found, actual motion of m. the second members of these equations is ## called the disturbing force ; THEOEETICAL ASTKONOMY. 42 which , and, using the second form of the equations, the function determines these components, is called the perturbing function. The complete solution of the problem is facilitated by an artifice of the infinitesimal calculus, known constants, according to ## which the complete integrals of equations (16) are of the same form ## as the variation of parameters, or of as those obtained ## by putting the second mem- ## arbitrary constants, however, of the latter These constants of integraintegration being regarded as variables. of relative to the the motion determine tion are the elements which ## bers equal to zero, the ## neglected the elements are pure sun, and when the disturbing force The variations of these, or of the co-ordinates, arising constants. from the action of the disturbing force are, in almost all cases, very is ## The problem which first small, and are called the perturbations. presents itself is, therefore, the determination of all the circumstances of the undisturbed motion of the heavenly bodies, after which the action of the disturbing forces may be considered. ## may be further remarked that, in the formation equations, we have supposed the different bodies to It of the preceding be free to move, ## There are, inand, therefore, subject only to their mutual action. facts derived from the of the motion of the comets which deed, study to indicate that there exists in space a resisting medium which If such a opposes the free motion of all the bodies of the system. seem medium actually exists, its effect is very small, so that it can be senonly in the case of rare and attenuated bodies like the comets, since the accumulated observations of the different planets do not sible exhibit any effect of such resistance. But, if we assume its existence, evidently necessary only to add to the second members of equations (16) a force which shall represent the effect of this resistance, which, therefore, becomes a part of the disturbing force, and the it is motion of ## will be completely determined. ## 16. When we consider the undisturbed motion of a planet or comet relative to the sun, or simply the motion of the body relative to the sun as subject only to the reciprocal action of the two bodies, ## the equations (16) become g + *(!+ *) = g + *(! + 1 = ) <), 0, (19) ## MOTION RELATIVE TO THE SUN. 43 The ## equations for the undisturbed motion of a satellite relative to its 2 primary are of the same form, the value of k , however, being in this case the acceleration primary due to the ## at the unit of distance, force exerted and ## by the mass of the ## the ratio of the mass of the of the primary. integrals of these equations introduce six arbitrary constants of integration, which, when known, will completely determine the satellite to that The undisturbed motion of If a?, we multiply the ## and subtract the m relative ## of these equations by yy and the second by first last to the sun. ## product from the first, we shall find, by inte- ## grating the result, ydx xdy ~~ _~ ## being an arbitrary constant. In a similar manner, we obtain xdz zdx ~~ ydz :c > zdy ' ## If we multiply these three equations respectively by z, y, and x, ## and add the products, we obtain ez c'y -f c\"x 0. ## This, being the equation of a plane passing through the origin of shows that the path of the body relative to the sun is a co-ordinates, plane curve, and that the plane of the orbit passes through the centre of the sun. Again, if we multiply ## of equations (19) by 2dx, the second take the sum and integrate, we shall 2dz, the ## by 2dy, and the third by first find 2 But, since r = x + y* 2 -\\- rdr we shall have, = xdx -j- ydy -f- by differentiation, zdz. ## Therefore, introducing this value into the preceding equation, m) ft j Civ ## h being an arbitrary constant. =^ we obtain 44 If THEORETICAL ASTRONOMY. we add and put for together the squares of the expressions + c' + c 2 //2 4/ we shall c, c', and c\", have ' df df 4f2 ' or -^= 4^- ## If we represent by dv the infinitely small angle contained between 2 2 2 radii- vectores r and r dr, since doc -\\- dy -f- dz is the square of the element of path described by the body, we shall two consecutive have da? + dy -f dz = dr + r dv\\ ## Substituting this value in the preceding equation, 2 r dv it becomes (22) 2fdt. 2 quantity r dv is double the area included by the element of path in the element of time dt, and by the radii-vectores r and described The -f- dr; and/, therefore, represents the areal velocity, which, being a constant, shows that the radius-vector of a planet or comet describes equal areas in equal intervals of time. From ## the equations (20) and (21) dt we find, by elimination, rdr + m) l/SwF (1 hr2 ## Substituting this value of dt in equation (22), ^ 2 (1 -f which ## gives, in order to find the dr _ rVZrk (1 we get m) (24) /tr 4/ ## maximum and minimum values of r, -f- (23) 4/ hr 2 m) 4/ > \"IT\" eft; or Therefore ff(l+m) and m) JT T ## are, respectively, the 2 / -Vy 4/ *(! m) ## maximum and minimum values of r. The ## MOTION RELATIVE TO THE SUN. 45 points of the orbit, or trajectory of the body relative to the sun, corr, are called the apsides; the former, the aphelion, and the latter, the perihelion. If we represent these ## responding to these values of values, respectively, which a it becomes (24), in > by a (I (1 and a(l e) we e), shall have ## Introducing these values into the equation ). tg. j/jp dr ## the integral of which gives v to -1 CD -f cos ## being an arbitrary constant. Therefore 1 I(^_ r e \\ e\\ r 1, we =cog(,_ fl, shall have ), ## from which we derive r= 1 which P , -f- e cos (v >) ## the polar equation of a conic section, the pole being at the to the focus, p being the semi-parameter, e the eccentricity, and v angle at the focus between the radius-vector and a fixed line, in the plane of the orbit, making the angle to with the semi-transverse is axis a. If the angle v CD is ## counted from the perihelion, we have 0, and 1 -f- e cos v ## The angle v is called Hence we conclude around the sun is ## the true anomaly. that the orbit of a heavenly body revolving a conic section with the sun in one of the foci. Observation shows that the planets revolve around the sun in ellipses, usually of small eccentricity, while the comets revolve either in ellipses of great eccentricity, in parabolas, or in hyperbolas, cir- ## cumstance which, as we shall have occasion to notice hereafter, greatly THEOKETICAL ASTRONOMY. 46 amount of labor lessens the in many computations respecting their motion. 2 Introducing into equation (23) the values of h and 4/ already we found, obtain rdr I/a _l-<-V l^r) ## the integration of which gives t=~ / ... -i/a- cos ^1/1 +7^\\ e\\\\ -| / la i\\ rV\\ ae TV. ff)fi (26) ' ## and the integral reduces to t we denote the time from the perihelion by tQJ we In the perihelion, r therefore, if =a r\\ ae (1 g), = C; shall have -f m\\ ae ae (27) ## an d therefore we shall have, for the a (1 In the aphelion, r &) time in which the body passes from the perihelion to the aphelion, t \\r, ', or ^ _ a ky'l -\\-m ## r being the periodic time, or time of one revolution of the planet around the sun, a the semi-transverse axis of the orbit, or mean distance from the sun, and n the semi-circumference of a circle whose radius is unity. Therefore we shall have ## MOTION RELATIVE TO THE SUN. For a second planet, we shall 47 have ## and, consequently, between the mean distances and periodic times of any two planets, we have the relation m')r' If the masses of the two planets m and m' are very nearly the 1 -j- m' and hence, in this case, it follows same, we may take 1 -f m that the squares of the periodic times are to each other as the cubes of mean distances from the sun. The same result may be stated in the ## another form, which is sometimes more convenient. Thus, since nab is the area of the ellipse, a and 6 representing the semi-axes, we shall have ## -=/= areal velocity; 2 and, since b =a (1 ), we have TO'g'(l-^)' ## which becomes, by substituting the value of r already found, (30) F1S). In like ## manner, for a second planet, we have +m ## and, if the masses are such that we may take 1 sensibly equal to 1 -f- m', it follows that, in this case, the areas described in equal times, in different orbits, are proportional to the square roots of their parameters. We 17. shall now consider the signification of some of the constants of integration already introduced. Let i denote the inclination of the orbit of to the plane of xy, which is thus taken as the plane & ## of reference, and let be the angle formed by the axis of x and the line of intersection of the plane of the orbit with the plane of xy; then will the angles i and & determine the position of the plane of 48 ## the orbit in space. THEORETICAL ASTRONOMY. The constants equation cz c'y -\\- c\"x and c', c, ## c\", involved in the 0, ## double the projections, on the co-ordinate planes, of the areal velocity /; and hence we shall have are, respectively, xy xz, and yz, cos the intersection of projection of 2/ on a plane passing through the plane of the orbit with the plane of xy, and perpendicular to the The latter, is 2/sini; ## and the projection of this on the plane of at an angle equal to &, gives c' Its projection = 2/ sin cos xz, to which sin inclined it is &. ## on the plane of yz gives c\" = 2/sin sin & Hence we derive z cos i y sin i cos 1 -\\- x sin (31) 0, ## the equation of the plane of the orbit; and, by means of the value of / in terms of p, and the values of c, c', c\", we derive, \"which is also, + m) cos a sin (82) *' sn These equations will enable us to determine &, i, and p, when, for any instant, the mass and co-ordinates of m, and the components of its velocity, in directions parallel to the co-ordinate axes, are known. The constants a and e are involved in the value of p, and hence four constants, or elements, are introduced into these equations, two of which, a and e, relate to the form of the orbit, and two, 1 and i, to co If we measure the angle v the position of its plane in space. from the point in which the orbit intersects the plane of xy, the conwill determine the position of the orbit in its Finally, the constant of integration C, in equation (26), stant co own is plane. the time ## MOTION EELATIVE TO THE SUN. 49 ## of passage through the perihelion ; and this determines the position of the body in its orbit. When these six constants are known, the ## undisturbed orbit of the body is completely determined. Let denote the velocity of the body in its orbit; then will ## equation (20) become At ## the perihelion, r is a minimum, and hence, according to this is a maximum. At the equation, the corresponding value of F is minimum. aphelion, In the parabola, a oo, a and hence which ## will determine the velocity at any instant, when r is known. It will be observed that the velocity, corresponding to the same value of r, in an elliptic orbit is less than in a parabolic orbit, and that, ## negative in the hyperbola, the velocity in a hyperbolic greater than in the case of the parabola. Further, since the velocity is thus found to be independent of the eccentricity, the since orbit is is still no influence on the ## direction of the motion has ## species of conic section described. ## If the position of a heavenly body at any instant, and the direction velocity, are given, the relations already derived ## and magnitude of its ## will enable us to determine the six constant elements of its orbit. But since we cannot know in ## advance the magnitude and direction ## of the primitive impulse communicated to the body, it is only by the aid of observation that these elements can be derived; and therefore, before considering the formula necessary to determine ## unknown elements by means of observed positions, we will investi- gate those which are necessary for the determination of the heliocentric and geocentric places of the body, assuming the elements to be known. The ## results thus obtained will facilitate the solution of ## the problem of finding the by observation. ## unknown elements from ## To determine the value of &, which system, we have, from equation (28), 18. VI 4 is ## the data furnished ## a constant for the solar THEORETICAL ASTEONOMY. 50 ## In the case of the earth, a l, and therefore rl/1 +m ## In reducing this formula to numbers we should properly use, for r, The the absolute length of the sidereal year, which is invariable. eifect of the action of the other bodies of the system on the earth is to produce a very small secular change in its mean longitude correthe elements; and sponding to any fixed date taken as the epoch of ## a correction corresponding to this secular variation should be applied The eifect of this corto the value of r derived from observation. rection is ## to slightly increase the observed value of r; but to deter- ## with precision requires an exact knowledge of the masses of all the bodies of the system, and a complete theory of their relative a problem which is yet incompletely solved. Astronomical motions, mine it ## usage has, therefore, sanctioned the employment of the value of k found by means of the length of the sidereal year derived directly ## from observation. This is virtually adopting as the unit of space a distance which is very little less than the absolute, invariable mean distance of the earth from the sun; but, since this unit may be arbitrarily chosen, the accuracy of the results is not thereby aifected. The value of r from which the adopted value of k has been commean solar days; and the value of the com365.2563835 puted, bined mass of the earth and moon is is m= Hence we have log r log 271 354710 = 2.5625978148; = 0.7981798684; log j/1 +m = 0.0000006 122; and, consequently, log k = 8.2355814414. ## If we multiply this value of k by 206264.81, the number of seconds of arc corresponding to the radius of a circle, we shall obtain its value expressed in seconds of arc in a circle whose radius is unity, or ## on the orbit of the earth supposed seconds is, therefore, log k The in a quantity mean therefore, and The value of k in = 3.5500065746. expresses the solar day, have to be circular. is ## mean angular motion of a usually designated by p. We planet shall, ## MOTION RELATIVE TO THE SUN. * 4- m = &1/1 f, ## for the expression for the 51 mean (33) ## daily motion of a planet. V\\ ## differs very little from -fSince, in the case of the earth, 1, will be observed that k very nearly expresses the mean angular motion of the earth in a mean solar day. it In the that ## case of a small planet or of a comet, the may, without sensible error, be neglected; it mass m is so small and then we shall have = 4a ## For the old planets whose masses pression (33) must be used. (34) ## are considerable, the rigorous ex- ## Let us now resume the polar equation of the which is 19. ellipse, the pole ## being at the focus, a(l-e ) \" e cos v If we shall ## by <p the angle included between the conjugate axis drawn from the extremity of this axis to the focus, we represent and a line have sin 2 ## and, since a(l is tp e; ## half the parameter of the transverse axis, ## which we have designated by p, we have P 1 The angle (p is -f- sin <f> cos v ## called the angle of eccentricity. 2 e ) a (1 a cos2 <p, we have Again, since p =1 a cos -f- sin <f> <p cos v (35) r in an evident, from this equation, that the maximum value of value minimum the that and orbit v to 180, elliptic corresponds It is of r corresponds to v It therefore increases from the perihelion 0. to the aphelion, and then decreases as the planet approaches the perihelion. THEORETICAL ASTRONOMY. 52 In the ## case of the parabola, <p = 90, and sin <p conse- quently, ff But, since 1 + cos v = 2 cos 2 Jt>, -j- COS V* we put 5 if we |_p, shall have (36) in which q course ## In this case, therefore, when the perihelion distance. r will be infinite, and the comet will never return, but is 180, way its The angle to other systems. ## cannot be applied to the case of the hyperbola, since in a hyperbolic orbit e is greater than 1 ; and, therefore, the eccen<f> ## cannot be expressed by the sine of an arc. If, however, we designate by ^ the angle which the asymptote to the hyperbola makes with the transverse axis, we shall have tricity e cos ## Introducing this value of becomes e into 1. ## the polar equation of the hyperbola, it p cost; ## But, since cos v cos -j- ^ cos 4* _ _ ## + cos ^ = 2 cos } (v + ^) cos \\(v = ff ##$), this gives\n\ncos4\n\n'\n\nIt appears from this formula that r increases with v, and becomes in180\nfinite when 1\ne cosv\ncos^, in which case v\n0, or cosv\n\nJ/\nconsequently, the maximum positive value of v is represented\n180\n^, and the maximum negative value by\nby\n(180--^).\nit\nis evident that the orbit will be that branch of the hyperFurther,\n:\n\nbola which corresponds to the focus in which the sun is placed, since,\nunder the operation of an attractive force, the path of the body must\n\nA body subject to a\nand\nintensity,\nvarying according to\nthe same law, would describe the other branch of the curve.\nbe concave toward the centre of attraction.\nforce of repulsion of the\n\nsame\n\n## The problem of finding the position of a heavenly body as seen\n\nfrom any point of reference, consists of two parts: first, the determination of the place of the body in its orbit; and then, by means\nof this and of the elements which fix the position of the plane of the\n\n53\n\n## and that of the orbit in its own plane, the determination of\n\nthe position in space.\nIn deriving the formulae for finding the place of the body in its\norbit, we will consider each species of conic section separately, comorbit,\n\nellipse.\n\n+ ae,\n\nwe may\n\n-=\n\nwe\n\nae and\n\nhave\n\nshall\n\ncos E.\n\nae\n\n## This auxiliary angle\n\nmetrical signification\ntrue anomaly.\n\nis\n\nmay\n\nbe easily\n\nanomaly; and\n\nknown from\n\nits\n\ngeo-\n\n## into the equation\n\nctC'\n\nits\n\nrelation to the\n\nand writing t\nin place of tw\nbeing the time of perihelion\npassage, and t the time for which the place of the planet in its orbit\nis to be computed, we obtain\n(27)\n\n## -- = mean daily motion of the planet =\n\nBut -\n\n//\n\ntherefore\n\nThe quantity\n\n(t\nT) represents what would be the angular distance\nfrom the perihelion if the planet had moved uniformly in a circular\norbit whose radius is a, its mean distance from the sun.\nIt is called\n\nthe\n\nfjt\n\n## mean anomaly, and\n\nfore,\n\nis\n\nusually designated by\n\nM.\n\nWe shall,\n\nthere-\n\nhave\n\nM=v(t-T),\n\nM=EesinE.\nWhen\n\neccentric\n\nis\n\nin\n\nits\n\nanomaly\n\n(39)\n\n## perihelion, the true anomaly,\n\nAll\nare each equal to zero.\n\n## three of these increase from the perihelion to the aphelion, where\n\nthey are each equal to 180, and decrease from the aphelion to the peri-\n\nprovided that they are considered negative. From the periis greater than M.\nis greater than E, and\nThe same relation holds true from the aphelion to the perihelion, if\nhelion,\n\nwe regard,\nAs soon\n\nand\n\nas negative.\n\n## obtained by means of the\n\nmotion and eccentricity, the values of r and v may be derived.\nas the auxiliary angle\n\nis\n\nmean\nFor\n\nTHEOEETICAL ASTRONOMY.\n\n54\n\n## this purpose there are various formulae\n\nwill now develop.\npractice, and which we\n\nThe\n\nequation\n\n= cos E,^\n\nae\n\ngives\n\nae\n\nor\n\n= a cos E\n\nP\nE\n\na cos\n\nae,\n\nT^\n\nae,\n\nr cos v\n\n= a cos E\n\nae.\n\n(41)\n\n## If we square both members of equations (40) and (41), and subtract\n\nthe latter result from the former, we get\n\nor\nr sin v\n\nE = b sin E.\n\na-j/1\n\ne sin\n\n(42)\n\n## By means of the equations (41) and (42) it may be easily shown\n\nthat the auxiliary angle\nor eccentric anomaly, is the angle at the\ny\ncentre of the ellipse between the semi-transverse axis, and a line\n\n## centre to the point where the prolongation of the\n\nordinate perpendicular to this axis, and drawn through the place of\nthe body, meets the circumference of the circumscribed circle.\n\nr (1 HH cos v)\n\nBy\n\nusing\n\nfirst\n\n= a(l\n\ne) (1 q= cos\n\nE).\n\n## the upper sign, and then the lower sign,\n\nwe\n\nobtain,\n\nby\n\nreduction,\n\n1/r sin ^v\n\nVr cos\nwhich are convenient\n\nwhen\n\n<y\n\n= l/a(l\n= Va(l\n\n-f-\n\ne} sin\n\ntan %o\n\n=^\n\nBy\n-\n\n\\E,\n\ne) cos \\E,\n\nand\n\n(43)\nv,\n\nand\n\nespecially so\n\ntan\n\nE.\n\n(44)\n\nSince e\n\n= sin ^, we have\n~ = ~ Sm ? _\ne\n\n..\n\ntan (450\n\nsm <p\n\n55\n\nConsequently,\ntan\n\nAgain,\n1/1\n\ntan (45\n\n\\E\n\n+ e = 1/1\n\nsin\n\n-j-\n\n\\<p)\n\n1/1\n\n<p\n\n-f-\n\ntan\n\n(45)\n\n\\v.\n\n2 sin\n\n\\<?\n\ncos jp,\n\n1/1\n\n+ e = I/sin\n\nor\n\n1/1\n\ncos\n\nits\n\ncon-\n\nand\n\n75k\n=,\n\nwhich\n\nThe value of\n\nis\n\nis\n\nlog\nlet\n\nAgain,\n\n61\n\n= 9.9601277069.\n\nus take\n\n~|\nwhich\n\nis\n\ncalled the\n\nmean\n\n## daily motion in the parabola; then will\n\n5\n\nnow compute\n\nIf we\n\nvalues of v from v\n\ntan iv\n\nthe values of\nto v\n\n+ 25\n\nM corresponding\n\n= 180, and\n\n## we may derive at once, from this\n\nwhen v is known, or v when\nalso\nbe observed that when t\nT is\nmay\n\ntime (t\nT) either\n\nknown.\n\nis\n\nIt\n\nto successive\n\narrange them\n\nin a table\n\n## table, for the\n\nM= m\n\nT)\n\n(t\n\nnegative, the\nhence it is not neces-\n\nis considered as\nbeing negative, and\nT than\npay any further attention to the algebraic sign of t\ngive the same sign to the value of v obtained from the table.\nTable VI. gives the values of\nfor values of v from\nto 180,\n\nvalue of v\nsary to\nto\n\n## with differences for interpolation, the application of which will be\n\neasily understood.\n23.\n\nWhen\n\nv approaches near to\n\n180,\n\nbe extremely\n\n## inconvenient, since, in this case, the differences between the values of\n\nfor a difference of one minute in the value of v increase very\n\n## rapidly and it will be very troublesome to obtain the value of v\n\nfrom the table with the requisite degree of accuracy. To obviate\n;\n\nmanner\n\nwe proceed\n\nin the following\n\nk\n\n^~P =\n\ntanit> (1\n\n+ 3 cot\n\n## and, multiplying and dividing the second\n\nwe\n\nshall\n\n(t\n\nP=\n\n1/2 g*\n\nI tan\n\n& (1 + cot\n\nmember by\n\nhave\nk\n\n4t>)\n\n^) JJ\n\n(1\n\n-f-\n\ncot |v)\n\n3\n,\n\nTHEOEETICAL ASTRONOMY.\n\n62\n\nBut\n\n1 -f cot iv\n\nand consequently\n\nsin v tan %v\n\nk(tT}_\n1/2 gt\n\nNow, when\n\n+ cot ^)\n2\n\n(1\n\n'\n\n## 180, cot^v will be very small, and\n\nmember of this equation will nearly\n\nv approaches near to\n\n~~3snr\n\n## the value of v on the supposition\n\nthat this factor is equal to unity, which will be strictly true when\nv\n180, and we shall have, for the correct value of v, the following\n1.\n\nequation\n\n=W+ A\n\n%w\n\nand tan|A\n\n0,\n\n\"I\n\n03\n\n= 30 (1 -f 40\n\n-fs\n\n30 (1 4-\n\n'\n\nwe\n\n0#)\n\n(1\n\n)\n2\n\n=x\n\nthis equation,\n\n30 2 (1\n\n4 30* 4-\n\ncoefficient\n\nof x,\n\nW-{-\n\n40^+20* ~+0)\n\nfrom\n\n20* -f\n\nget,\n\n(l-ex)\n\n(2 4- 60\n\n4-\n\nx,\n\nto\n\n1 4- 30 2\n\nwe\n\n+ 40 + 20\n2\n\n4-\n\nx2\n\n) a?.\n\nobtain\n\n_|_\n\n402\n\n_j_\n\n2^4\n\n_|_\n\n^)\n\n14-30\n2\n30(l-f 40\n\n20*\n\n)~~\n\n## then, substituting this in the preceding equation, inverting the series\n\nand reducing, we obtain finally\n\nBut tan |A O\n\nx, therefore\n\n63\n\nobtain\n\nFor\n\nall\n\nthis equation\n\nis\n\nwe\n\n## term of the second member will be insensible, and we shall have, to\n\na sufficient degree of approximation,\n\n## Table VII. gives the values of A O expressed in seconds of arc,\n\n155 to w\n180.\ncorresponding to consecutive values of w from w\nIn the application of this table, we have only to compute the value\n,\n\nof\n\nnamely,\n\nM=m(t\n\nthen will\n\nis\n\nto be used,\n\nT~);\n\n3\n\nw = \\|\n\nsin\n\nsince\n\n200\n,\n\n8\n\n3sm\n\ni(/\n\nor\n\n200\nr\n\n## Having computed the value of w from\n\nwill furnish the corresponding value of A\nfor the correct value of the true anomaly,\n\n=W\n\n-f-\n\nthis equation,\n;\n\nand then we\n\nTable VII.\nshall have,\n\nwhich will be precisely the same as that obtained directly from Table\nVI., when the second and higher orders of differences are taken into\naccount.\n\nIf v\n\nis\n\nT is\n\nthen\n\nis\n\ngiven by\n\nw=v\n\nAn\n\n## required, the table will give,\n\nA, using v as argument, and\n\nTHEOBETICAL ASTRONOMY.\n\n64\n\nThe\n\nexact value of A O\n\nis\n\nT from\n\na\n24.\nis\n\nThe problem of\n\ngiven,\n\nsmall,\n\nmay also\n\nEquation (55)\n\nT when\n\nwhen v is\n\nis\n\n_-.\n\n= cos V\niv\n\n2q?\n\n,Q\n\n2 sm'itr).\n\n(3\n\n= r cos\n\nJv,\n\n1/2\n\n= sin\n\n3,\n\nsm # =\n\nsin 4 -y\n\nr=,\n\n1/2\nand we have\n3&\n\n= 3 sm x\n\nT}\n\n(i\n\n4 sin3 re\n\nConsequently,\n\nfacilitate\n\nthe calculation\n\nsolution.\n\nTo\n\nwe put\n\nsin-o\n\nv\n\n= 0, we\n\nshall\n\nfrom which\n\nit\n\nmay\n\nhave\n\nN= fv'2,\n\nand when v\n\n90,\n\nv.\n\nWhen\n\nwe have\n\nN=l;\n\n## ^changes slowly for values\n\n180, we shall have ^V oo;\n\nto 90.\nBut when v\nand hence, when v exceeds 90, it becomes\nnecessary\nWe shall, therefore, put\nauxiliary different from N.\n\nof v from\n\nN'\n\n= N sin v\n\nsin 3z;\n\nto introduce\nin this case,\n\nan\n\nfrom which\n\nit\n\nN'=\\\n\nappears that\nTherefore\n\nwhen v = 180.\n\nwhen\n\nwhich log\n\nin\n\nis\n\ngreater than\n\nOA/\n\nAw)\nAW)\n\n'\n\n= sin J\n= cos^(&\n(\n\n0) cos\n0)\n\ncosi\n\n0) sin\n\n(1\n\n(i\n\n17),\n\n+7\n\n0) sin % (i\n\n),\n\n7 ),\n\n(i\n-f-\n\n^).\n\n## These equations enable us to determine accurately the values of &',\n\ni'j and AW, which give the position of the orbit in reference to the\nThe\necliptic corresponding to the time t', when d and 37 are known.\nlongitudes, however, will\nas before,\n\nstill\n\nwhich we suppose\n\nmean equinox\n\n## to be that of t; and, in order to refer\n\nTHEORETICAL ASTKONOMY.\n\n100\n\nthem\n\n## mean equinox of the epoch\n\nto the\n\nt'\n\n',\n\namount of the\n\nthe\n\npre-\n\nt must also be\ncession in longitude during the interval t\napplied.\nthe\nelements\nare not of considerof\nin\nvalues\nthe\nIf the changes\n\n## able magnitude, it will be unnecessary to apply these rigorous formula,\n\nand we may derive others sufficiently exact, and much more convenient in application. Thus, from the spherical triangle formed by\n\nthe intersection of the plane of the orbit and of the planes of the\ntwo ecliptics with the celestial vault, we get\nsin\n\nT?\n\nfrom which we\nsin\n\nWe\n\n(&\n\ncos\n\ncos\n\nsin\n\nsin\n\ni -\\-\n\ncos\n\ncos Aw,\n\neasily derive\nsin y cos (\n\ni)\n\n(i'\n\n0)\n\n0)\n\n-f-\n\nsin\n\n^Aw.\n\n(112)\n\nhave, further,\nsin\n\nAW\n\nsin\n\ni'\n\n0),\n\nWe\n\n(113)\n\nAW\n\nsin\n\n-j-\n\ncos\nsin\n\ni'\n\n(&\n\ncos\n\n(^\n\n0) cos\n\n0) sin\n\n(&'\n\n0)\n\n0) cos 7,\n\n(^'\n\nwhich gives\nsin (ft'\n\nft)\n\nsin\n\nAw\n\ncos\n\ni'\n\n2 sin (ft\n\n0) cos\n\n(&'\n\n0) sin\n\nJiy,\n\nor\n\nsin(&'\n\n&) =\n\n- 2 sin (a\n\nFinally,\n\nsin 7 sin\n*) cos\n\n(&\n\n('\n\n0) coti'\n\n0) sin\n\n^.\n\n(114)\n\nwe have\n\n## Since 37 is very small, these equations give, if we apply also the\n\nprecession in longitude so as to reduce the longitudes to the mean\nequinox\nof the date ',\n\nsint\n\n8'= 8 +<X\n\n0--~ ? sin(a\n\n0)cot;'\n\nilsin2(a\n\n^),\n\n(115)\n\nPOSITION IN SPACE.\n\nin\n\nwhich\n\nis\n\nat\n\n101\n\n= 206264\".8.\n\nIn most\n\n## cases, the last terms of the expressions for\n\nthe\nsecond order, may be neglected.\nof\nTT', being\n',\n,\nFor the case in which the motion is regarded as retrograde, we\nf\ni and i ,\ni and 180\nmust put 180\nrespectively, in\ns\n\n&\n\nand\n\n',\n\nfor\n\nTT',\n\nin this case,\n\nwe have\n\nwhich gives\n\n## we adopt BesseFs determination of the luni-solar precession and\n\nof the variation of the mean obliquity of the ecliptic, we have, at the\nIf\n\ntime 1750 -f\n\nr,\n\n= 50\".21129 4- O.\"0002442966r,\nat\n\n*=\n\n0\".48892\n\nat\n\nO.\"000006143r,\n\nand, consequently,\n77\n\nand\nr\n\nin the\n\n= \\ (f\nThe\n\non the\n\n== (0.\"48892\n\nf)\n\n## 1750, t and i being expressed in years.\n\nlongitude of the descending node of the ecliptic of the time\nf\n\n-f- f)\n\necliptic of 1750.0\n\n351\n\nwhich\n\nO.\"000006143r) (f\n\nis\n\nis\n\nalso\n\nfound\n\n36' 10\"\n\n5\".21\n\nto\n\nbe\n\n(t\n\n1750),\n\n1750.\n\nThe longitude of\nof\n\nt,\n\n## the descending node of the ecliptic of t' on that\n\nmeasured from the same mean equinox, is equal to this value\n\n## diminished by the angular distance between the descending node of\n\nthe ecliptic of t on that of 1750 and the descending node of the\nf\necliptic of t on that of t, which distance is, neglecting terms of the\nsecond order,\n5\".21(f\n\nand the\n\nresult\n\n1750);\n\nis\n\n351\n\n36' 10\"\n\n5\".21\n\n351\n\n36' 10\"\n\n10\".42 (t\n\n(t\n\n1750)\n\n5\".21 (f\n\n1750),\n\nor\n\n1750)\n\n5\".21 (f\n\nf).\n\nTHEORETICAL ASTRONOMY.\n\n102\n\n## reduce this longitude to the mean equinox at the time t,\n\nadd the general precession during the interval t\n1750, or\n\nTo\n\n50\".21\nso that\n\nwe\n\n1750),\n\nhave, finally,\n\n= 351\nWhen\n\n(t\n\nwe must\n\n36'\n\n10\"+\n\n39\".79(\n\n1750)\n\n5\". 21(1?\n\n.<).\n\n## &, and i have been thus reduced from the\n\nare referred, to those of the\necliptic and mean equinox to which they\ndate for which the heliocentric or geocentric place is required, they\nthe elements\n\nTT,\n\n## be referred to the apparent equinox of the date by applying the\n\nmay\n\nnutation in longitude.\nThen, in the case of the determination of the\nright ascension and declination, using the apparent obliquity of the\necliptic in the computation of the co-ordinates, we directly obtain the\n\n## body referred to the apparent equinox. But, in comseries\n\nof places, the changes which thus take place in the\na\nputing\nelements themselves from date to date induce corresponding changes\nplace of the\n\n## in the auxiliary quantities a, 6, c, A, jB, and (7, so that these are no\n\nlonger to be considered as constants, but as continually changing their\n\n## values by small differences.\n\nThe differential formulae for the comof\nthese\nwhich\nare easily derived from the equations\nputation\nchanges,\nwill\nbe\nin\nthe\nnext\n(99),\ngiven\nchapter; but they are perhaps unnecessary, since it is generally\n\ncompute the\nis\n\n## most convenient, in the cases which occur, to\n\nextreme dates for which the ephemeris\n\n## required, and to interpolate their values for intermediate dates.\n\nIt is advisable, however, to reduce the elements to the ecliptic\n\nand\n\nmean equinox of the beginning of the year for which the ephemeris\nis required, and using the mean\nobliquity of the ecliptic for that\nepoch, in the computation of the auxiliary constants for the equator,\nthe resulting geocentric right ascensions and declinations will be\nreferred to the same equinox, and they may then be reduced to the\n\n## apparent equinox of the date by applying the corrections for precession\n\nand nutation.\nplaces which thus\n\nThe\n\n## result are free from parallax and aberration.\n\nIn comparing observations with an ephemeris, the correction for par-\n\nallax\n\nis\napplied directly to the observed apparent places, since this\ncorrection varies for different places on the earth's surface.\nThe cor-\n\nmay\n\nWe\n\nmay\n\n## subtract from the time of observation the time in which the\n\nlight\n\nfrom the planet or comet reaches the earth, and the true place\nreduced time is identical with the apparent place for the time\n\nfor this\n\nNUMERICAL EXAMPLES.\n\n103\n\n## of observation ; or, in case we know the daily or hourly motion of\n\nthe body in right ascension and declination, we may compute the\nmotion during the interval which is required for the light to pass\n\nfrom the body to the earth, which, being applied to the observed\nplace, gives the true place for the time of observation.\n\nWe\n\nmay\n\nby\n\n## 497*.78 A in computing the geocentric places for\n\nusing the time t\nthe time t, or by subtracting from the place free from aberration, com-\n\nputed for the time , the motion in a and o during the interval\n497 S .78J, in which expression A is the distance of the body from the\n\n## and 497.78 the number of seconds in which light traverses the\n\nmean distance of the earth from the sun.\nIt is customary, however, to compute the ephemeris free from\ns\naberration and to subtract the time of aberration, 497 .78z/, from the\nearth,\n\ntime of observation\n\nwhen comparing\n\n## method above mentioned. The places of the\n\ncomputing its co-ordinates must also be free from aberra-\n\nsun used in\n\n## the longitudes derived from the solar tables include\n\nthe\naberration,\nproper correction must be applied, in order to obtain\nthe true longitude required.\ntion;\n\nand\n\nif\n\nEXAMPLES.\n\n41.\n\nWe\n\nwill\n\nnow\n\ncollect\n\ntogether, in\n\nthe\n\nproper\n\n## order for numerical calculation, some of the principal formula which\n\nhave been derived, and illustrate them by numerical examples, commencing with the case of an elliptic orbit. Let it be required to find\n\n## and declination of the planet Eurynome\n\nWashington, for the date 1865 February\n\n(), for\n\nmean midnight\n\nat\n\nEpoch\n\nM=\n\n1\n\n29' 40\".21\n\n44\n\n20 33 .09\n\nEcliptic and\n\n^r^sM1\n\n?=\n\n11\n\nMean\n\n'\n\n15 51 .02\n\n= 0.3881319\n=\n2.9678088\nlog\nV = 928\".55745\n\nlog a\n/\n\nWhen\ndone\n\nis\n\n## a series of places is to be computed, the first thing to be\n\nto compute the auxiliary constants used in the expressions for\n\n## and although but a single place is required in the\n\nproblem proposed, yet we will proceed in this manner, in order to\nthe co-ordinates,\n\nTHEORETICAL ASTRONOMY.\n\n104\n\n## Since the elements JT, &,\n\nexhibit the application of the formulae.\ni are referred to the ecliptic and mean equinox of 1864.0, we will\nFor\nfirst reduce them to the ecliptic and mean equinox of 1865.0.\n\nand\n\nt= 1864.0,\n\nwe have\n\nthis reduction\n\n= 50\".239,\n^at\n\n= 352\n\n51' 41\",\n\nwe\n\n## Substituting these values in the equations (115),\n\ni\n\ni'\n\n= Ai ==\n\nobtain\n\n= + 53\".61,\n\nAft\n\n0\".40,\n\n= 0\".4882.\nATT\n\n= + 50\".23;\n\nand hence the elements which determine the position of the orbit in\nreference to the ecliptic of 1865.0 are\n\n= 44\n\n&\n\n21' 23\".32,\n\nwe\n\ninstant\n\n= 206\n\n43' 33\".74,\n\nis\n\n36' 50\".ll.\n\nmean\n\n= 23\n\n=4\n\n## Nautical Almanac, the value of the\n\nwhich\n\n27' 24\".03.\n\nThe auxiliary constants for the equator are then found by means of\nthe formulae\ncot\n\nA=\n\ntan\n\n&\n\ntan\n\ncos\n\ncos\n\ncos\n\n& cos E\n&\n= cos\nsm A-p\ntan\n\nsm a\n\nThe angle\n\nis\n\nE = cos\nn\n\nQ>\n\nsin e\n\nto be taken,\n\ntan\n\ni,\n\nalways\n\n-& B-\n\nsin b\n\nless\n\nthan\n\nsin\n\n-.\n\ncos e\n-\n\nsm c\n\nsm\n\n180, and\n\n&\n~- sin\n= -sin sm\nC\n\ne\n.\n\nit is\n\n## indicated directly by the algebraic sign of tan Q\n\nThe\nvalues of sin a, sin 6, and sin c are always positive, and, therefore, the\nis\n\n## angles A, jB, and C must be so taken, with respect to the quadrant in\n\nwhich each is situated, that sin\nand cos &, sin\nand sin &, and also\n\nsin\n\nC and\n\nsin\n\n&\n\nshall\n\nA = 296\nB = 205\n\n39'\n\n3217.74,\n\n212\n\n5\".07,\n\n55 27\n\n.14,\n\nlog sin a\nlog sin b\n\nlog sin\n\nformula\n\nFrom\n\nsigns.\n\nis\n\nthese\n\nwe\n\nderive\n\n= 9.9997156,\n= 9.9748254,\n= 9.5222192.\n\n## proved by means of the\n\nNUMERICAL EXAMPLES.\ntan^\n\n= sin\n\nsin a cos\n\nb sin e sin\n\n105\n\n-B)\n\n## which gives log tan i\n\n8.9068875, agreeing with the value 8.9068876\nderived directly from i.\nNext, to find r and u. The date 1865 February 24.5 mean time\nat Washington reduced to the meridian of Greenwich by applying\nm\nh\nthe difference of longitude, 5 S\n1T.2, becomes 1865 February\n24.714018 mean time at Greenwich.\n\nThe\n\ninterval, therefore,\n\nfrom\n\nthe epoch for which the mean anomaly is given and the date for\nwhich the geocentric place is required, is 420.714018 days; and multiplying the mean daily motion, 928\".55745, by this number, and\n\nadding the result to the given value of M, we get the mean anomaly\nfor the required place, or\n\nM=\nThe\n\n29' 40\".21\n\neccentric\n\nanomaly\n\n+ 108\n\nE is then\n\n= 110\n\n30' 57\".14\n\n37\".35.\n\n0'\n\nM=EesmE,\nthe value of\n\n<p\n\n## being expressed in seconds of arc. For Eurynome we\n\n9.2907754, and hence the value of e exlog e\n\npressed in seconds is\nlog e\n\nBy means\n\nnamely,\n.E\n\n= 4.6052005.\nwe\n\n=119\n\n49' 24\",\n2\n\n^ 3.895976;\n\nM=E\nQ\n\ne sin\n\nE = 110\n\n6'\n\n50\".\n\nThen we have\n\nM-M,\n^ = r=7^ET\nwhich\n\n= 110\n\n'\n\n7'\n\n## approximation to the value of E,\n\n^ = 119\nThis gives Jf\n\n372\".7\n\n'T097-\n\n0' 36\".98,\n\n43'44\".3.\n\nand hence\n\nand\n\nTHEORETICAL ASTRONOMY.\n\n106\nTherefore,\n\nwe\n\n## have, for a third approximation to the value of E,\n\nJE=119\nwhich requires no further\n\n43'44\".64,\n\ncorrection, since\n\nit\n\nsatisfies\n\nthe equation\n\nbetween J/and E.\n\nTo\n\nfind r\n\nand\n\nv,\n\nwe have\n\n= l/a(l +\n=\nVr cos-Jv l/a(l\n1/r sin lv\n\nThe\n\nvalues of the\n\nequations are:\n\n0.1468741\n\nJT\n\nThe\n\nE,\n\ne) cos J-J.\n\nmembers of\n\nthe second\n\n= 0.2328104,\na'(^+~e)\n\nthese\n\n## and log V~a(l\n\n--=\n\ne)\n\nand we obtain\nv\n\nSince\n\nlog\n\nin\n\nfactors\n\nfirst\n\n/\nl\n\ne) sin\n\n= 0.4282854.\n= 129 50\".52,\nlog r\n= 197 37' 49\".58, we have\nu = v + nQ = 326 41' 40\".10.\n3'\n\nx\n\n= r sin a sin {A\n\n=r sin b\n=r sin c\n\nwhich\nx\n\ngive, for\n\nu),\n\n-j-\n\nsin\n\n(B\n\n-f-\n\nu),\n\nsin\n\n( (7 -f-\n\nu),\n\nEurynome,\n\n2.6611270,\n\n= + 0.3250277,\n\n= + 0.0119486.\n\n## 1865 February 24.5 mean time at Washington,\n\nthe mean equinox and equator of the beginning of the\n\nreferred to\nyear,\n\nX= -f 0.9094557,\n\nY=\n\nZ=\n\n0.3599298,\n\n0.1561751.\n\n## Finally, the geocentric right ascension, declination, and distance are\n\ngiven by the equations\ntan a\n\n= yy+Y\n\nZ\n+Z\n= - +-sm\na = - --cos a\nx\nX\nz\n\ntan 5\n\ny+Y\n\n,\nsm5'\n\n## form of the equation for tan d being used when sin a is\n\nthan\ncos a.\ngreater\nThe value of J must always be positive; and d cannot exceed\nthe\n\nfirst\n\nNUMEKICAL EXAMPLES.\na\n\n= 181\n\nTo\n24.5,\n\n8'\n\n29\".29,\n\nlog J\n\n42' 21\".56,\n\n0.2450054.\n\n=s=\n\na and\n\n## to the true equinox and equator of February\n\nthe\nNautical Almanac,\nfrom\nhave,\n\nreduce\n\nwe\n\n<5\n\n107\n\n/=-fl6\".80,\n\nlog\n\n</\n\n= 1.0168,\n\n= 45\n\n16';\n\nAa\n\nHence the\n\n= 181\nWhen\n\nto\n\nA5\n\n= _7\".i7.\n\nof the date,\na\n\n= + 17\".42,\n\n8'\n\nis\n\nand equator\n\nis\n\n46\".71,\n\n## only a single place\n\n4\nis\n\nlog J == 0.2450054.\n\n42' 28\".73,\n\nrequired,\n\nit is\n\nlittle\n\nmore expeditious\n\ncompute r from\nr\n\na(l\n\nE from\n\nand then v\n\nsin J\n\nJ)\n\n(-y\n\ncosJ),\n\n= -y- sin\n\np sin E.\n\nlog r\n\n= 0.4282852,\nv = 129\n\nE= 9\n\nv\n3'\n\n20' 5\".92,\n\n50\".56,\n\n## agreeing with the values previously determined.\n\nmay be proved by means of the formula\nsin\n\n%(y\n\n-{-\n\nE)\n\n= \\~ cos \\\n\nThe\n\ncalculation\n\nE.\n\n\\<p sin\n\n(v\n\n= 124\n\n## log sin %(y\n\n23' 47\".60,\n\nmember\n\n+ E) = 9.9165316,\n\n## log sin %(y\n\n+ E} = 9.9165316.\n\nit\n\nis\n\n## the heliocentric longitude and latitude by means of the\n\ncompute\nmeans\nequations (82), then the geocentric latitude and longitude by\nand\nascension\ninto\nof (89) or (90), and finally convert these\nright\nfirst\n\ndeclination\nto be\n\nby means of\n\ncomputed,\n\nit is\n\n(92).\n\nWhen\n\na large\n\nnumber of\n\ncompute\n\nplaces are\nthe heliocentric\n\nTHEORETICAL ASTRONOMY.\n\n108\n\nequations (105).\n\nThe\n\n## calculation of the geocentric place in reference to the ecliptic\n\nwhich the equator is taken as the\n\nis,\n\n## fundamental plane, and does not require any further\n\nillustration.\n\nThe determination of\n\n## the geocentric or heliocentric place in the\n\ncases of parabolic and hyperbolic motion differs from the process\nindicated in the preceding example only in the calculation of r and v.\n\nTo\n\n= 9.9650486;\nlogq\nFirst,\n\nin\n\nwe compute\n\nfrom\n\nC = 9.9601277,\n\nwhich log\n\nThen we\n\nT=\n\n## of parabolic motion, let t\n\n75.364 days;\nlet it be required to find r and v.\n\nand\n\nfind\n\nlog\n\nM from\n\nand the\n\nresult is\n\nm = 0.0125548.\n\nM=m(t\n\nT\\\n\nwhich gives\nlog\n\nFrom\n\nM we\n\nFinally, r\n\nM= 1.8897187.\n\n= 79\n\nlog r\n\n55' 57\".26.\n\n= 0.1961120.\n\ngiven\n\n65.41236 days;\n\n= 0.6020600,\n\nin\n\nfound from\n\nis\n\nwhich gives\n\nFor the\n\nderive,\n\nwhich log^\n\n^ = 37\n\nto find r\n\n## 35' 0\".0, or log e\n\nand\n\nv.\n\nwe compute J^from\n\nlogN= 8.7859356;\nThe value of\n\nFirst,\n\n= 0.1010188;\n\nN= 0.06108514.\n\nN=\n\nel\n\ntan\n\n## log tan (45\n\n-f\n\nT=\n\nand log a\n\nNUMERICAL EXAMPLES.\nIf we assume\n\nF= 30, a\n\n109\n\nmay be\n\nderived\n\nfrom\n\nwhich gives F,\n\n= 28\n\n===\n\n0.072678.\n\nThen we\n\nthe equation\n~\n\nwherein\n\n=\nF, = 4.6097 (N\n\nis\n\nJV,) s\n\n3'\n\n43\".0.\n\n## Hence, for a second approximation to the value of\n\nwe have\n\n= 25 36' 40\".0.\nThe corresponding value of N N, = 0.0617653, and hence\n= 12'\n*F, = 5.199 (N\n= 25 24' 30\nThe third approximation, therefore, gives\nF,\n\nis\n\n9\".4.\n\nJV,) s\n\nJF ,\n\nwe\n\nr/\n\n.6,\n\nand,\n\nget\n\n^=25\n\n24'27\".74.\n\n## which requires no further correction.\n\nTo find r, we have\n\nwhich gives\nlog r\n\nThen, v\n\nis\n\n= 0.2008544.\n\nderived from\n\nv = 67\n\ntan %v\n\nand we find\n\n3' 0\".0.\n\nWhen\n\n## several places are required,\n\nis\n\nit\n\nconvenient to compute v\n\n## and r by means of the equations\n\nVcosF\n'\n\nVr cos ^v =\n\n'i\n\n\\ v\n-;-=\n\nJ. ;\n-\n\nVcosF\n\nCOS\n\nTHEORETICAL ASTRONOMY.\n\n110\n\n## For the given values of a and e we have log V a(e\n\n7\n== 0.0100829, and hence we derive\nIogl a(e\n1)\nv\n\n2'\n\n6.7\n\n= 0.4782649,\n\n= 0.2008545.\n\nlog r\n\n59\".92,\n\n-f 1)\n\n## It remains yet to illustrate the calculation of v and r for elliptic\n\norbits in which the eccentricity differs but little from\n\nand hyperbolic\ne\n\nT= 68.25 days;\n= 9.7668134. We compute M from\n\nunity.\n\n= 0.9675212;\n\nand log q\n\n_-H\n\n__\n\nqr\n\nwherein log\n\nlog\n\nWith\n\nthis as\n\nargument we\n\nM= 2.1404550.\nfrom Table VI.,\n\nget,\n\nF= 101\nand then with\n\nthis value of\n\nF as\n\n= log ^\"1\n\nlog\n\nargument we\n\n## from Table IX.,\n\ns>\n\ni -f- e\n\nv = F-f A (lOOt) +\n\nwe\n\nfind,\n\n= 9\".506,\nC= 0\".062.\n-=\n8.217680, and from the equation\n\nA = 1540\".08,\nThen we have\n\n38' 3\".74,\n\n(100i)\n\nC(1000\n\n8\n,\n\nget\n\nv= F+ 42'\nThe value\n\nof r\n\nis\n\n22\".28\n\n+ 25\".90\n\n-f-\n\n0\".28\n\n= 102\n\n20' 52\".20.\n\nr=:\n1\n\nnamely,\nlog r\n\nWe\nfirst\n\nmay\n\nalso determine r\n\ncompute M from\n\n-(-\n\ncos\n\nv'\n\n= 0.1614051.\n\nThus, we\n\nAssuming\n\njB\n\n1,\n\nwe\n\nget log\n\n## with this as argument, we find\n\nfrom\n\nw=\n\n101\n\n25 r\n\nThen we compute\n\nNUMERICAL EXAMPLES.\n\n0.024985.\nwhich gives A\nTable\nX.,\nfind, from\n\nWith\n\nexact value of\n\nis\n\nthis value of\n\nas\n\nargument, we\n\n= 0.0000047.\n\nlog\n\nThe\n\nIll\n\nthen found to be\nlog\n\nM= 2.1375635,\n\n## which, by means of Table VI., gives\n\nw=\n\nBy means\n\nof this\n\nwe\n\n101\n\n24' 36\".26.\n\nderive\n\nA = 0.02497944,\nand hence, from Table X.,\nlog\n\nG =0.0043771.\n\nThen we have\n\nG tan ^w \\\n\ntan ^v\n\nwhich gives\nv\n\n= 102\n\n-<\n,\n\n20' 52\".20,\n\nlog r\n\nFinally, r\n\nis\n\ngiven by\n\n= 0.1614052.\n\n## T is negative ; but the\n\nBefore the time of perihelion passage, t\nvalue of v is computed as if this were positive, and is then considered\nas negative.\n\nIn the case of hyperbolic motion, i is negative, and, with this distinction, the process when Table IX. is used is precisely the same\nas for elliptic\n\nis\n\nf\n\ntable\n\nB and\n\nwhich belong\n\nAC\nlog\n\ncos*|v'\n\n## being taken from the columns of the\n\nto hyperbolic motion.\n\n## In the calculation of the position of a comet in space,\n\nif the\n\nmotion\n\nTHEORETICAL ASTRONOMY.\n\n112\n\nis\n\nis\n\nWhen we\n\n42.\n\nregarded as\n\nless\n\nthan\n\n90,\n\nthe dis-\n\n## have thus computed the places of a planet or comet\n\nwe may\n\nThe\nfor intermediate dates by the usual formulae for interpolation.\nis\ninterval between the dates for which the direct computation\nfor a series of dates equidistant,\n\n## should also be small enough to permit us to neglect the effect of the\n\nfourth differences in the process of interpolation.\nThis, however, is\nnot absolutely necessary, provided that a very extended series of\nof differences may\nplaces is to be computed, so that the higher orders\nbe taken into account. To find a convenient formula for this inter-\n\n## any date, or argument of the function, by\n\nand\nthe\nnct)j\ncorresponding value of the co-ordinate, or of the\nthe interpolation is to be made, by / (a\nfor\nwhich\nfunction,\nno)).\nIf we have computed the values of the function for the dates, or\nto, a -f- 2co, &c., we may assume that an\n<o, a, a\narguments, a\nfunction\nfor\nthe\nwhich exactly satisfies these values will\nexpression\npolation, let us denote\n\n'-{-\n\n## also give the exact values corresponding to any intermediate value\n\nIf we regard n as variable, we may expand the\nof the argument.\n\nf(a -h\n\nand\n\nif\n\nwe\n\n(116)\n\nand\n\n2,\n\n2oi)\n\n=/(a)\n\n/ (a + (n\nby f (a + na>)\n\n|)\n\nmay\n\n+\n+\n/\ni)\nf (a + (n\n\nthe difference\na>),\n(a\n(n\n\nand similarly\nFunction.\n\nf( a\n\nw)\n\nf(a)\n\ni.+ -\n\n/(\n/(a\n\n2 a>\n\nIII. Diff.\n\n+2\n\n+ 2a)\n\n1) to)\n1)\n\nI. Diff.\n\n/O-20\n+\n{J\n/ (a +\na\n\n.-)\n\na>)\n\na)\n\n## for the successive orders of differences,\n\nbe arranged as follows\n\nArgument.\n<o\n\nII. Diff.\n\n+B -C A _ +c\n=fW + A +B +C\n=f(a)-A\n\nwe symbolize,\n\n## A, B, and C. If we put n successively equal\n\nand then take the successive differences of these\n\nget\n\nby\n\nOn* -f &c.\n\nI. Diff.\n\n_w\n\n/(\n/(a -f\n\nthese\n\ncoefficients\n\n1, 0, 1,\n\nvalues,\n\nIf\n\n+ An -f Bn +\n\n## regard the fourth differences as vanishing, it is only neces3\n\nconsider terms involving n in the determination of the\n\nunknown\n\n/(a\n\n=/(a)\n\nwe\n\nsary to\nto\n\nno>)\n\nW\n^\n\nII. Diff.\n\nIII. Diff.\n\nr(a)\n/'(\n\n+ *)\n\n/'\"(*\n\nINTERPOLATION.\n\n113\n\n## with the above,\n\nwe\n\nget\n\nc=tr(+i-),\n- if (a) --B=if(),\nA=f(a +\nIf' (a + -,\n-\n\nwhich, from the manner in which the differences are formed, give\n\nC=\n\nJ.\n\nTo\na\n\n=/(a +\n\n(/\" (a\n\n_/(\n\n_-/\"\n\n( a)\n\n-i/-\n\n^-\n\n),\n\n>)\n\nJ (/\" ( a\n\n/\"\n)\n\n(a),\n\n_/\" ( a) ).\n\n-f-\n\n|w,\n\nwe have n\n\n/(a\n\n^,\n\n>)\n\n=/(a)\n\nducing,\n\nf(a\n\nwe\n\nargument\n\n14\n\n+ iJJ + a\n\nJ., j5,\n\nand\n\n(7,\n\nlast\n\n## found, and re-\n\nget\n\n+ /()) ~ I (J (/\"\n\n+ i0 = i (/( +\n\n+ /\" to)),\n\nin which only fourth differences are neglected, and, since the place\nof the argument for n\nis arbitrary, we have, therefore,\ngenerally,\n\n- j a (r\n\n-t> + 1)\n\n## Hence, to interpolate the value of the function corresponding to a\n\nmidway between two dates, or values of the argument, for which\nthe values are known, we take the arithmetical mean of these two\ndate\n\nknown\n\nvalues,\n\nmetical\n\nand from\n\nthis\n\nwe\n\ndifferences\n\n## By extending the analytical process here indicated so as to include\n\nthe fourth and fifth differences, the additional term to be added to\nequation (117)\n\nis\n\nfound to be\n\n## differences will be neglected.\n\nIt is customary in the case of the comets\n\n## rapidly, to adopt an interval of four days, and in the case of the\n\nasteroid planets, either four or eight days, between the dates for which\nthe direct calculation is made. Then, by interpolating, in the case of\nCD, equal to four days, for the intermediate dates, we\nobtain a series of places at intervals of two days ; and, finally, inter-\n\nan interval\n\nTHEORETICAL ASTRONOMY.\n\n114\n\n## we derive the places at\n\npolating for the dates intermediate to these,\nWhen a series of places has been computed,\nintervals of one day.\nthe use of differences will serve as a check upon the accuracy of the\nthe place which is not\ncalculation, and will serve to detect at once\nwhen any discrepancy is apparent. The greatest discordance\nshown in the differences on the same horizontal line as the\nerroneous value of the function and the discordance will be greater\nand greater as we proceed successively to take higher orders of difIn order to provide against the contingency of systematic\nferences.\ncorrect,\n\nwill be\n\nis\n\n## should be made of those quantities in\n\nlikely to occur.\n\nThe ephemerides of\n\n## the planets, to be used for the comparison of\n\nare\nobservations,\nusually computed for a period of a few weeks before\nand after the time of opposition to the sun ; and the time of the\nopposition may be found in advance of the calculation of the entire\nephemeris. Thus, we find first the date for which the mean longitude\nof the planet is equal to the longitude of the sun increased by 180 ;\n\n## then we compute the equation of the centre at this time by means of\n\nthe equation (53), using, in most cases, only the first term of the\ndevelopment, or\nv\n2esin M,\n\nbeing expressed in seconds. Next, regarding this value as conwe find the date for which\n\nstant,\n\nL\nis\n\n-j-\n\n## equation of the centre\n\nequal to the longitude of the sun increased by 180 ; and for this\nand also for another at an interval of a few days, we compute\n\ndate,\nUj\n\ntan\n\n(I\n\n## & ) = tan u cos\n\ni.\n\nLet these longitudes be denoted by I and /', the times to which they\ncorrespond by t and t and the longitudes of the sun for the same\ntimes by O and O ' then for the time tw for which the heliocentric\nlongitudes of the planet and the earth are the same, we have\nf\n\nor\n\nthe\n\n(113)\n\nfirst\n\n## of these equations being used\n\nwhen\n\n180\n\nis less\n\nTIME OF OPPOSITION.\nthan V\n\n180\n\nO'.\n\nIf the time\n\n115\n\ndiffers considerably\n\nfrom\n\nor\n\n## be necessary, in order to obtain an accurate result, to repeat\n\nthe latter part of the calculation, using tQ for t, and taking t r at a\nt',\n\nit\n\nmay\n\nsmall interval from this, and so that the true time of opposition shall\nf\nThe longitudes of the planet and of the sun\nfall between t and t\n.\n\n## must be measured from the same equinox.\n\nWhen\n\nthe eccentricity is considerable, it will facilitate the calculatwo terms of equation (53) in finding the equation of the\n\ntion to use\n\nM= 2e\n\nsin if\n\n+ 4-\n\nez\n\nsin\n\n2M,\n\n## being the number of seconds corresponding to a length of arc equal\n\n206264\".8 ; and the value of v\nwill then be\n\n## In all cases in which circular arcs are\n\nexpressed in seconds of arc.\ninvolved in an equation, great care must be taken, in the numerical\napplication, in reference to the homogeneity of the different terms.\nIf the arcs are expressed by an abstract number, or by the length of\narc expressed in parts of the radius taken as the unit, to express them\nin seconds\n\nwe must multiply by\n\n## number 206264.8 but if the\n\nterm of the equation must contain\n\nthe\n\n'\n\nonly one concrete factor, the other concrete factors, if there be any,\nbeing reduced to abstract numbers by dividing each by s the number\nof seconds in an arc equal to the radius.\n43. It is unnecessary to illustrate further the numerical application\nof .the various formulae which have been derived, since by reference\nto the formulae themselves the course of procedure is obvious.\nIt\n\n## be remarked, however, that in many cases in which auxiliary\n\nangles have been introduced so as to render the equations convenient\n\nmay\n\nby the use of\n\ntables\n\n## logarithms of the sum or difference of two numbers\n\nrithms of these numbers are given, the calculation\n\nand\n\nis\n\noften even\n\nwhen\nis\n\nthe loga-\n\nabbreviated,\n\n## more accurately performed than by the aid of the\n\nauxiliary angles.\n\n## The logarithm of the sum of two numbers may be found by means\n\nof the tables of common logarithms.\nThus, we have\n\nIf\n\nwe put\nlog tan x\n\n= ^ (log b\n\nlog a),\n\nTHEORETICAL ASTRONOMY.\n\n116\n\nwe\n\nhave\n\nshall\n\nlog (a\n\n-f-\n\nlog (a\n\n-f-\n\n= log a\n6) = log\n\n6)\n\nor\n\nThe\n\nfirst\n\nform\n\nis\n\nused\n\nwhen\n\nis less\n\ncos\n\n2 log cos\n\nx,\n\n2 log sin\n\nx.\n\nis\n\nthan sin x.\n\n## It should also be observed that in the solution of equations of the\n\nafter tan (X\nform of\nusing the notation of this particular\n)\n(89),\n\n## has been found by dividing the second equation by the first,\n\nthe second members of these equations being divided by cos (X\n0)\ntwo values of A cos /9, which should\nand sin (X\n), respectively, give\ncase\n\n## agree within the limits of the unavoidable errors of the logarithmic\n\n;\nbut, in order that the errors of these tables shall have the\n\ntables\n\nferred\n\nwhen\n\ncos (A\n\nis\n\nThe value\n\nfirst\n\n(A\n\nO)\n\n(A\n\nequation\n),\n\nis\n\n/9\n\nless\n\nto be pre-\n\nthan sin\n\nis less\n\nis\n\nis\n\nthan\n\n0).\n\n(A\n\nrequired, should\nsin\n\nft\n\n## greater than 45.\n\nIn the application of numbers to equations (109), when the values\nof the second members have been computed, we first, by division,\n\nwhen\n\nis\n/3\n\nfind tanJ(&'H-fl>\n\nand tan\n\nbut\n\nif sin |\n\n(&'\n\no>\n\n<w\n\n),\n\n* s ^ ess\n\n('\n\nwe\n\nfl>\n\nfind\n\n); then, if\n\nsin|(&'-f w\n\n+w\n\no)>\n\nfirst\n\nwe ^ nc^\n\nis\n\nequation;\n\ncos\n\nJi'\n\nfrom\n\n## the second equation. The same principle is applied in finding sin\n\nby means of the third and fourth equations. Finally, from sin\n\n\\%'\n\nand cos \\V we get tan \\V ^ and hence i'. The check obtained by the\nr\nf\nagreement of the values of sin \\i and cos %i with those computed\nf\nf\nfrom the value of i derived from tan \\i , does not absolutely prove\nthe calculation. This proof, however, may be obtained by means of\n,\n\nthe equation\nsin\n\ni'\n\nsin\n\n&'\n\nsin\n\ni'\n\nsin\n\nor by\n\n= sin\n= sin\n\ni sin\n\n&\n\ne sin\n\n&\n\n## In all cases, care should be taken in determining the quadrant in\n\nwhich the angles sought are situated, the criteria for which are fixed\neither\n\ndirectly, or\n\nby the\n\nrelation of the\n\n## algebraic signs of the trigonometrical functions involved.\n\nDIFFERENTIAL FORMULAE.\n\nCHAPTER\n\n117\n\nII.\n\n## INVESTIGATION OF THE DIFFERENTIAL FORMULAE WHICH EXPRESS THE RELATION\n\nBETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY\nAND THE VARIATION OF THE ELEMENTS OF ITS ORBIT.\n\nIN many\n\n44.\n\nit\n\n## calculations relating to the motion of a heavenly\n\nbecomes necessary to determine the variations which small\n\nbody,\nincrements applied to the values of the elements of its orbit will produce in its geocentric or heliocentric place. The form, however, in\n\n## which the problem most frequently presents itself is that in which\n\napproximate elements are to be corrected by means of the differences\nbetween the places derived from computation and those derived from\n\n## In this case it is required to find the variations of the\n\nelements such that they will cause the differences between calculation\nand observation to vanish ; and, since there are six elements, it follows\nthat six separate equations, involving the variations of the elements\nobservation.\n\nunknown\n\n## must be formed. Each longitude or right\n\nfrom observation,\nwill furnish one equation\nand hence at least three complete observations will be required for the solution of the problem.\nWhen more\nthan three observations are employed, and the number of equations\nas the\n\nquantities,\n\n## latitude or declination, derived\n\n;\n\nexceeds the\ntion\n\nnumber of unknown\n\n## which, by elimination, the corrections to be applied to the elements\n\nmay be determined.\nIf\n\nwe suppose the\n\ncorrections\n\nele-\n\n## ments, in order to satisfy the data furnished by observation, to be so\n\nsmall that their squares and higher powers may be neglected, the\nvariations of those elements\n\nmay\n\n## be determined by differentiating the various formulae which determine\n\nthe position of the body.\nThus, if we represent by 6 any co-ordinate of the place of the body computed from the assumed elements\nof the orbit, we shall have, in the case of an elliptic orbit,\n\nTHEORETICAL ASTRONOMY.\n\n118\n\nQ\n\nat the\n\nLet\n\nepoch T.\n\n6'\n\n## of this co-ordinate as derived directly or indirectly from observation ;\n\nthen, if we represent the variations of the elements by ATT, A&, A^,\n&c., and if we suppose these variations to be so small that their\nsquares and higher powers may be neglected, we shall have\ndo\n\ndo\n\ndo\n\nA3fo\n\ndM\n\nde\n\n+ -j- *r-\n\n(1)\n\nThe\n\ndifferential coefficients\n\n-=\n\n-=\n\ndj:\n\nd&\n\n-,\n\n&c. must\n\nnow be\n\n## the equations which determine the place of the body\n\nments are known.\n\nderived from\n\nwhen\n\nthe ele-\n\nWe\n\nshall first take the equator as the plane to which the positions\nof the body are referred, and find the differential coefficients of the\ngeocentric right ascension and declination with respect to the elements\n\nof the\n\norbit, these\n\ndamental plane.\n\nbody in\n\nelements being referred to the ecliptic as the funx, y, z be the heliocentric co-ordinates of the\n\nLet\n\nand we have\n\nor\n,\n\ndO\n\ndd\n= -=dx 4r -y- dy\ny\ndx\n=\n\ndO\n-4*\n\ndy\n\n-=- dz\ndz\n\nHence we obtain\ndO\n\ndd_\n\ndO_\n\n'\n\ndx\n\nand similarly\n\ndy\n\ndy\ndx\n\ndO_\n\ndz\n\ndz\n'\n\ndn\n\n(2)\n\n## with respect to the\n\nfind\nthe\nmust, therefore,\npartial differential coefficients of d with respect to x, y, and z, and then the\npartial differenother elements.\n\ntial coefficients\n\nWe\n\n## of these co-ordinates with respect to the elements. In\n\nwe put 6\na, and in the case of the\n\n## the case of the right ascension\n\ndeclination we put 6\n3.\n\n45. If\n\nwe\n\n## differentiate the equations\n\n-f-\n\ny -f\nz -J-\n\nregarding X, Y, and\n\nZ as\n\nX==\n\nA COS\n\nd COS a,\n\nY= A cos d sin\nZ = A sin\n\nconstant,\n\nd,\n\nwe\n\nfind\n\na,\n\nDIFFERENTIAL FORMULAE.\n\n= cos a cos\ndy = sin a cos\ndz = sin d d A\n\ndx\n\nFrom\n\ndA\ndA\n\nd\nd\n\n-\\-\n\nd dd.\n\nby elimination, we obtain\n\nthese equations,\n\n= --sin a dx\n\ncos a sin\n= --A\n\n-\\\n\n---aA- dy\nsin d\n\nsin\n\ndx\n\nCOS a\n-dy,\n,\n\nda\n\ncos d\n\ndd\n\n## A cos a sin 8 dd,\n\nA sin a sin d dd,\n\nA sin a cos d da\nA cos a cos d da,\n\nA cos\n\n-\\-\n\n119\n\ncos 8\n\ndx\n\nsin a\n= -7.\nA\n\nda\n\ncosa\n\nda\nr~\n\n/j\n\na and\n\nd with respect\n\ncos a sin\n= --A\n\neZfl\n\n-j\n\n'\n\naz.\n\n-f-\n\n## Therefore, the partial differential coefficients of\n\nto the heliocentric co-ordinates are\n\ncos o\n\n(3)\n\ndx\ndd\n\n>\n\nsinasind\n\n..\n\n## Next, to find the partial differential coefficients of the co-ordinates\n\nXy y, z with respect to the elements, if we differentiate the equations\nsin c, A, B, C, are functions of &\n(100)!, observing that sin a, sin 6,\ny\n\nand\n\niy\n\nwe\n\nget\n\ndx\n\ndr\n\nx cot ( A\n\n-{-\n\nu) du\n\n-{-\n\n~r=55\n\n-{-\n\ndQ\n\n-j-\n\n^^\n\n+ -p- ^'\n\nft\n\n7*\n\n-^\nMl'\n\nc?i,\n\nco\n\nc?2\n\nTo\n\n--\n\ndr\n\n-j-\n\n2 cot (\n\nC -f- w)\n\n-=\n\ndu\n\n+ TQ\n\n&c.,\nft^\nd 66 -p,\n,\n\nwe have\n\nthe equations\n\na;\n\nrr=\n\nr cos\n\nit\n\ncos S7\n\ny = r cos w sin\n= r cos u sin\n2\n\nwhich give, by\ndx\n-7\n\nf sin\n\nit\n\nS^ cos e\n\n-{-\n\n&7 sin e\n\n-f-\n\nsin\n\nw\nsin w\ns in\n\ncos\n\ni,\n\ncos S7 cos\n\ncos e\n\nr sin u sin\n\nsin\n\ne,\n\ncos\n\nsin\n\ne -j-\n\nr sin\n\ncos\n\ne,\n\nr sin\n\n&\n\ncos\n\ncos\n\nit\n\nsin\n\ndifferentiation,\n\nr cos\n\nsin\n\ncos\n\nt,\n\nrtS7\n7\n\na\n\nand hence\nd-ZV\n\nBy\n\nfar\n\n## substituting the value of\n\ndT\nc?a,\n\njferf (<\n\nthis\n\nbecomes\n\nag cos\n\n-4*\n\nTHEORETICAL ASTRONOMY.\n\n130\n\n## Substituting this value of\n\nobtain\n\ndN\n\nin equation (25),\n\nand reducing, we\n\nqr\n\n## In a similar manner, substituting in equation (26) the values of\n\nda and dN , and reducing, we get\ndr\n\n=\nVp\n\nThe\n\ncos 4\n\nequations (27) and (28) will furnish the expressions for the\nand v with respect to the elements\n\n## T, g, and ^, required in forming the equations for cos d da and dd.\n\nIt will be observed that these equations are analogous to the equations (23) and (24), and that by introducing the relation between e\n\n## and neglecting the mass, they become identical with them.\n\nmight, indeed, have derived the equations (27) and (28) directly\nfrom (23) and (24) by substituting for e its value in terms of ^; but\nthe differential formula which have resulted in deriving them directly\n\nand\n\noj/,\n\nWe\n\n## for hyperbolic motion, will not be superfluous.\n\n50. It is evident,\n(24), (27),\n\nand\n\n(28)\n\n## from an inspection of the terms of equations (23),\n\nwhich contain de and d^>, that when the value of\n\n## very nearly equal to unity, the coefficients for these differentials\n\nbecome indeterminate. It becomes necessary, therefore, to develop\ne is\n\n## the corresponding expressions for the case in which these equations\n\nare insufficient.\nFor this purpose, let us resume the equation\n-\n\nin\n\nT)\n\n(1\n\nwhich u\n\n+ 6)1\n\n= tan\n\nJw,\n\nand\n\n= JI _ Q\n-(-\n\nwe\n\nshall\n\nhave\n\nThen, since\n\nDIFFERENTIAL FOKMTJXJE.\n\n+ (Aw - /2 u + 2X)\n\nIf\n\nit\n\nis\n\n(1\n\n*)\n\n131\n\n+ Ac.\n\n(29)\n\n## variation of the elements of parabolic motion, or when 1\n\ne as constant,\nwe may regard the coefficient of 1\n\nsmall,\n\ne is\n\nvery\n\nand neglect\n\n## terms multiplied by the square and higher powers of 1\n\ne.\nBy\nthe\nto\nthese\nequation (29) according\nconditions, and\ndifferentiating\nregarding\n\nu and\n\ne as variable,\n\n= (1 + w\nand, since\n\ndu\n\n= J(l + u\n\n2\n)\n\ndv\n\nwe\n\nget\n\n>\n\n>\n\ndu\n\n(\\u\n\n_\n~\n\nlu*\n\nde\n\n(1\n\nlu*\n\n+ u^\n\n## The values of the second member, corresponding\n\nof\nis\n\nv,\n\nmay\n\nde;\n\nto different values\n\nfor -7-\n\nmay be\n\n## changed to another form which furnishes a direct solution with the\n\nsame facility. Thus, by division, we have\n\nde~\nand\n\n## since, in the case\n\nof parabolic motion,\n\n-'=9* d\n\nT^r=\"+^'\nthis\n\nbecomes\n(31)\n\nIf\n\nwe\n\n## differentiate the equation\n\n1 -fecosv'\n\nregarding\n\nr, v,\n\nand\n\ne as variables,\n\nwe\n\nshall\n\nhave\n\ndr\n2\n\ne)~\n\ng (1\n\n+ e)\n\n'\n\n~de\n\nTHEORETICAL ASTRONOMY.\n\n132\n\nIn the\n\nl,\n\nand\n\ntransformed into\n(33)\n\nwe\n\n## value from (31), and reducing,\n\nits\n\nget\n\n= 2 n * ~ r) sin , + Ty tan*\n(<\n\n%The\n\n(34)\n\nJt>.\n\n2q\n\n## equations (31) and (34) furnish the values of\n\nand\n\nto be\n\nde\nde\nused in forming the expressions for the variation of the place of the\nbody when the parabolic eccentricity is changed to the value 1 -j- de.\n\nWhen\n\nfrom unity, we\n\nlittle\n\nmay compute\n\nA still\n\nequation (30).\n\nis\n\n## assigned differs but\n\n-\n\nthe value of\n\ncloser approximation\n\ndirectly\n\nfrom\n\nwould be obtained by\ndi)\n\nbut\n\nis\n\n## may be derived, of which the numerical\n\nthe\nuse of Table IX.\nThus, if we differby\n\nfacilitated\n\napplication\nentiate the equation\n\n2\n\n## regarding the coefficients A, B, and\n\nthe value of i in terms of e, we have\n\ndv__d_V\nde\nin\n\nwhich\n\nde\ns\n\n200^\ns(l\n\n4005\n\ns7T+\n\ne)>\n\n(7 as constant,\n\n8\n,\n\nand introducing\n\n6000\n\n(l+e)'\n\ne)\n\nTo\n\nfind\n\ndV\n\nwe have\n\nO/G\n\nwhich\n\ngives,\n\nby\n\ndifferentiation,\n\nk(t\n\nand\n\nT)\n\nde\n\ndV\n\nwe\n\n## introduce the expression for the value of\n\nused as the\nmeans\nof\nTable\nthe\nis\nresult\nargument\nby\nVI.,\nif\n\nin finding\n\nDIFFERENTIAL FORMULAE.\n\n133\n\ndV\n\"\n\nde\n\nHence we have\n\n20(U\n\ns(l\n\n600C\n\n_4005\n\n+ e)^\n\ns(l\n\ne)*\n\n^T+ej*\n\n)}\n\nis\n\n## When the eccentricity differs so much from that of the parabola\n\nthat the terms of the last equation are not sufficiently convergent,\ndv\nthe expression for\n\n## be derived from the equations (75) 1 and\n\nfirst of these equations wifch respect to\nregarded as constant,\n\nwe\n\nwe\n\nwe\n\nTo\n\nmay\n\n_9\n\nIV A\nmembers of equation\n\ndw\ndC\nC \"\"\"sinw\n\n(1\n\n+ e) (1 + 9)'\n\nwith respect to\n\ntake\n1\n\n~*~-1\nwhich gives\n\nThe equation\n\ngives\n\n50\n\n,^\n\n(1\n\n-f-\n\n^ tan\n\n2 i\n\n-Aw ae -f\n\n9e)\n\n- -w\n\ntan\n\ncos\n\ndC\nO\n\n2002\n\n-77\n\n7T(1\n\n+ Oe)\n7T\n\ntan 2\n\nsin\n\ndv\n\n_ ~\n\nde~\n\n(76) w\n\nand\n\n4de\n\nj A\ndA\n\nevidently be\n\nget\n\ndv\n\nsufficient to\n\nsince\n\ne,\n\nget\n\n## take the logarithms of both\n\ndifferentiate,\n\nIf we differentiate the\n\n(76)j.\n\n20 C\n\n2\n2 1\n\n2S1\n\nwe\n\nget\n\n\"'\"\n^\n\nsin v\n\n^w\n\nr-:\n2\n\n^w\n\n(37)\ne} it will\n\nbe\n\nTHEORETICAL ASTRONOMY.\n\n134\n\nand\n\n-y-,\n\nC smv\n2\n\nwe\n\nobtain\n\n20 O a\n\ncos'JUfl\n\nS1\n\n~\n(1\n\nsin?;\n\n+ e) (1 + 9e)'\n\ncos 2 |w\n\nStanjv\n'\n\n'\n\n## which is used as the argument in\n\nIf we introduce the quantity\nthis equation becomes\nw\nTable\nmeans\nof\nVI.,\nfinding\nby\n\n9e)75tan>\n\n(1\n\n-f-\n\ne) (1\n\n+ 9e>'\n\n## This equation remains unchanged in the case of hyperbolic motion,\n\nthe value of C being taken from the column of the table which cord^o\n\nresponds to this\nall cases in\n\nand\n\ncase-:\n\nit\n\n-7-\n\nin\n\nae\n\nwhich the\n\nlast\n\nis\n\nnot conveniently\n\nd/T\n\napplicable.\n\nThe value of\n\nis\n\na/ c>\n\nWhen\njg\n\n1,\n\nand\ntan\ncos\n\nThen we\n\nshall\n\nJii>\n\n= tan Jv y\n\nw = JO\n\ncos\n\nlittle\n\n\\^\n\n^^\n\nsin v\n\n(\\\n\nv.\n\n2k(tT)\n= --=; cos\n\nequation\n?\n\n= (1 + A C\n\ncos 2\n\n= (1 -f iJL)\n\ncos* Jw,\n\ngives\n\n= (1 + P)\n\ncos*\n\nJw\n\n(32).\n\nhave\n\n75 tan %\n\nThe\n\n## then given by the equation\n\n= Ccos\n\nHence we derive\nI\n\n'\\\n\niw.\n\nNUMERICAL EXAMPLES.\nIf we substitute this value in equation\n\nwe\n\n135\n\nand put\n\n(39),\n\n(1\n\n+ = 2,\ne)\n\nget\n_\n\nde\n\nand when\n\n2(l-i-9e)\"\n\nWe\n\nEXAMPLES.\n\n51.\n\nthe formula\n\n(1\n\n+ e) (1 + 9e)'\n\n## becomes identical with equation\n\n1, this\n\nr2\n\nnow\n\nwill\n\n(31).\n\nby numerical examples,\n\nillustrate,\n\n## right ascension and declination arising from small increments assigned\n\nLet it be required to find for the date 1865 Febto the elements.\n\n## ruary 24.5 mean time at Washington, the differential coefficients of\n\nthe right ascension and declination of the planet Eurynome\nwith\n\nand\n\n## respect to the elements of its orbit, using the data\n\nThus we have\nin Art. 41.\n\nresults given\n\nd =\n4 42' 21\".56, log A = 0.2450054,\n29\".29,\n=\n=\nv\n129\nu = 326 41' 40\".l,\n50\".5,\n0.428285,\nlogr\nA = 296 39' 5\".0, B == 205 55' 27\".l,\nC= 212 32' 17\".7,\n= 9.999716, log sin b = 9.974825, log sin = 9.522219,\nlog sin a\n= 9.511920,\n= 0.425066\nx\nlog y\nlog = 8.077315,\nlog\n= 23 27' 24\".0,\nT= 420.714018.\n\na =t 181\n\n8'\n\n3'\n\nFirst,\n\nvalues\n\n(4),\n\nwe compute\n\nlog cos d\n\nlog cos d\n\n~=\n\n8.054308,\n\nlog\n\n= 9.754919\n\nlog\n\nn,\n\n^=\n\nfollowing\n\nThen we\n\n8.668959n\n\n== 6.968348\n\nlog\n\nnates,\n\nthe\n\n= 9.753529.\ndz\n\nwith respect to\n\nTT,\n\n^,\n\ni,\n\nv,\n\nand\n\nr,\n\n## from the formulae\n\n(7),\n\nwhich\n\ngive\nlog\n\nlog\n\ndit\n\n= log dv = 9.491991\n\n-- = 7.876553,\n^d6\n\nlog -^log\n\n^j-\n\nlog\n\nn,\n\nlog\n\n^= ^=\nait\n\nlog\n\n-- = 8.830941,\n\nlog\n\n~=\n\n-~ = 0.142443\n\n\"66\n\n= 8.726364,\n\nlog\n\n-^-\n\n= 9.687577,\n\nlog\n\n= 9.996780\n\nlog\n\n-^-\n\n= 9.083635,\n\nlog\n\nn,\n\n0.399496\n\ndv\n\nW$g -^~ 9.222898., n, = 7.649030. THEORETICAL ASTRONOMY. 136 ## In computing the values of and -jr> ## those of cos a, cos b f -yr, -p-> ## and cos c may generally be obtained with sufficient accuracy from Their algebraic signs, however, must be sin a, sin 6, and sine. The quantities sin a, sin 6, and sin c are always strictly attended to. and the algebraic signs of cos a, cos 6, and cos c are indicated by the equations (101)!, from which, also, their numerical In the case of the example proposed, it will values may be derived. be observed that cos a and cos b are negative, and that cos c is positive. positive ; at once To and ## find the values of cos d -y- -r-> we ## have, according to equa- tion (2), . da COS d dx da = COS d ~dx dx -\\- .da - dv dy dit COS d dr. _,_ JL dx (41) 21, dn' dz dy'dn: dit which give =+ %dv = cos *L cos arr In the case of place of TT &, i, and cos 8 -A- cos d ~= ~= ## Next, from (16), - 0.48900. = *= dv ## write these quantities successively in and hence we derive J^ ai 0.08020, we compute = 0.179155, log = 0.171999, log r/7* dr 09533 > 0.78993, 0.04873. ## the following values: = 9.577453, log L ^ 9.911247, log ^L fi =~ ~= ~=+ 0.27641, dr log we = - 0.03845, ai |L r, dr. ## in the equations (41), cos d log *L 1.42345, ^= 2.376581 n = 2.535234. ## \"We may now find &c. by means of the equations ^-, ^r, ## and thence the values of ## cos d -y-, -y-, &c. d<p ## to derive these values directly but it is (11), most convenient d(p ## from cosd-^, cos^ dr dv dr and dv ## in connection with the numerical values last found, according to the NUMERICAL EXAMPLES. 137 equations which result from the analytical substitution of the expressions for and p. -=, --, -j in place of &c., in equation writing successively tp y Mw Thus, we have TT. da (2), cos o = cos da dr dr d<p .da d<p dv cos o \\- dv d<p ## d3^_d3_ dr_,dd_ dv^ ~^~~fo'~d^~^~dv'~dp' and similarly cos d M and for cos (5 which give -^- = + 1.99400, ^nrr aiw = + 1-13004, dy //, ~= = 4*? dM ~=~ -^ = + 507.264, Therefore, according to Aa A(5 (1), = -f 1.42345A7T we shall have 0.03845A a 0.09533 A ^ 0.48900Ar 179.315. dfi -f 0.38023, d/JL cos d 0.65307, d<p 0.27641 &i -f 1.99400A^ 1.13004A^ + 507.264A/X, 0.78993A^ 0.38023Ajf To prove - 0.65307A^ 179.315A/Z. ## the calculation of the coefficients in these equations, we ## assign to the elements the increments M = + 10\", A? so that they = 20\", = + 10\", = 10\", = + 0\".01, At = + 10\", ## = 1864 Jan. 1.0 Greenwich mean time. M= Q 29' 50\".21 7t= 44 20 13 ^ i = 206 = 4 ?= log a /i 11 .09 ^| 42 30 .13 V 37 .51 J 16 = 0.3881288 = 928.56745 Mean Equinox 1864.0 1 .02 we compute the geocentric place for 1865 Febmean time at Washington and the result is these elements ruary 24.5 a Ajtt become Epoch With A^ A* = 181 8' 34\".81, 4 42' 30\".58, log A = 0.2450284, THEORETICAL ASTRONOMY. 138 which are referred to the mean equinox and equator of 1865.0. The difference between these values of a and d and those already given, as derived from the unchanged elements, gives Aa = -f 5\".52, and the COS d Aa = -f 5\".50, assumed values of ## direct substitution of the ## in the equations for cos d AO, cos d Aa and A<S A, = ATT, 9\".02, A&, Ai, &c. gives = + 5\".46, A<5 == 9\".29. ## The agreement of these results is sufficiently close to show that the computation of the differential coefficients has been correctly performed, the difference being due chiefly to terms of the second order. When ## the differential coefficients are required for several dates, if their values for successive dates at equal intervals, the use of differences will serve to check the accuracy of the calculation ; we compute ## but, to provide against the possibility of a systematic error, it may be advisable to calculate at least one place directly from the changed elements. Throughout the calculation of the various differential coefficients, great care must be taken ## in regard to the algebraic signs In the example ## involved in the successive numerical substitutions. ## we have employed logarithms of six decimal places; but it would have been sufficient if logarithms of five decimals had been used; and such is generally the case. given, ## It will be observed that the calculation of the coefficients of A&, and ATT, ## independent of the form of the orbit, depending on the simply position of the plane of the orbit and on the position of the orbit in this plane. Hence, in the case of parabolic and A^ is hyperbolic orbits, the only deviation from the process already illusis in the computation of the coefficients of the variations of trated the elements which determine the magnitude and form of the orbit and the position of the body in its orbit at a given epoch. In all - da da -d$\n,\ndd\nr and -j- are determined as\n\ndv\n\ndv\n\ndr\n\n## If we introduce the elements T,\n\nshall\n\ndr\n\nq,\n\nand\n\ne,\n\nwe\n\nhave\nda\ndd\n\nda,\n\n_ dd\n\ndr\n\ndr\ndd\n\n.da\n\ndv\n\ndv\n\ndT-~d^'dT^~dv\"~di'\nand similarly\n\n## with respect to q and\n\ne.\n\nNUMERICAL EXAMPLES.\n\n139\n\ndr dv dr dv dr\n,\n,\ndv\nof calculating the values of -7, -r=, -=-,\nand -=dT dT dq dq -j-,\nde\nde\ndepends on the nature of the orbit.\nIn the case of passing from one system of parabolic elements to\n,\n\nThe mode\n\nTo\n\nT= 75.364\n\n## = 9.9650486, from which we have found\n\nv = 79\nlog r = 0.1961120,\nThen, by means of the equations\nlog\n\n~=\nrAji\n\nlog\n\nIf,\n\nlog\nfo\n\n= 7.976397\n\n-7^d\nlog q\n\nwe\n\nintroduce d log\n\nalso used\n\nwhen\n\nwe\n\nq,\n\nj-\n\n= 0.064602\n\nshall\n\nlog\n& d\nlog q\n\n(43),\n\nwe\n\nn.\n\nhave\n\n~- =\n\n= 9.569812,\n\n## From these, by means of\n\na and d with respect\n\nof\n\n= 9.242547,\nr/7?\n\nlog\n\nM,\n\n55' 57\".26.\n\nfind\n\nlog\n\n8.095802,,,\n\njj-\n\nwe\n\n(22),\n\n0.391867\n\nT and\n\n## The same values are\n\nq or log q.\nthe variation of the parabolic eccentricity is taken\nto\n\n/y/y\n\ninto account.\n\nBut\n\nin this case\n\nwe compute\n\nalso\n\nj-\n\nfrom equation\n\n/7w\n\n(31)\n\nand\n\n## ^ from (33) or (34), which give, for v = 79 55' 57\".3,\n\n~ = 8.147367\nlog\nlog ^ == 9.726869.\nU6\nU6\nn,\n\nof\n\n-T~,\n\nfound from\ndv\n\nkV\n\ndr\n\ndq\ndr\n\ndq\n\n## the mass being neglected.\n\nr2 e sin v\n\ndv\n\ndq\n\n....\n\nqi/p\n\nTHEORETICAL ASTRONOMY.\n\n140\n\nTo\n\n## illustrate the application\n\n*T=\n\nvalues,\n\n68.25 days,\n\nof these formulae,\n\n= 0.9675212,\n\nlet\n\nus resume the\n\nand log q\n\n= 9.7668134,\n\nv\n\n= 102\n\nlog r\n\n20' 52\".20,\n\nHence we derive\n\n= 0.1614052.\n\n= 0.0607328,\n\nand\n\nlog^=7.943137n\nlog\n\n^=\n\nlog\n\n0.186517.,\n\nIf\n\nwe wish\n\n0.186517..\n\nof g,\ndv\n\nis\n\nq\nfl\n\nand r with\n\nwe have\n\ndv\n\ndr\n\ndq\n\nd logq\n\n'\n\ndlogq\nin which ^\n\n~=\nuq\n\naq\n\n_q\n\ndr\n\ndq\n\n## the modulus of the system of logarithms.\n\nThen we compute\n(35), (39), or (40).\n\nthe value of\n\nThe\n\n-7d/6\n\n## correct value as derived\n\n=\n^\nde\n\nfrom (39)\n\n(30).\n\nis\n\n0.24289.\n\nfrom\n\nvalues derived from (35), omitting the last term, from (40) and\n0.23531.\n0.24291, and\n0.24440,\n(30), are, respectively,\n\nThe\n\nclose\n\nThe\n\nvalue\nis\n\nis\n\n## agreement of the value derived from (40) with the correct\n\naccidental, and arises from the particular value of v, which\n\n## make the assumptions, according to which equation\n\nderived from (39), almost exact.\n\nhere such as to\n\n(40)\n\nis\n\ndiT\n-7CLG\n\nof (32), which\n\ngives\n\nde\n\nWhen,\n\n= + 0.70855.\n\nand\n\nwhich depend on\n\n## those which depend on the position of the\n\norbit in space have been found, the expressions for the variation of\nthe geocentric right ascension and declination become\n\nthe elements T,\n\nq,\n\ne,\n\nNUMERICAL EXAMPLES.\nCOS d Act\n\n= COS ^da\nd\n\nATT\n\n-f-\n\nda\n\nA&\n\nCOS 3\n\nagg\n\nofTT\nR\n\n-f-\n\nCOS\n\n141\n\nda\n\n+ cos S^da\nT^ Al + cos ^ T7^ A *\ncu\naj[\nda\n\ndfa\n\nCOS\n\nA*/ -|~\n\n-7- A<?,\n\ndd\n\nd3\n\nd<?\n-A1\n+ -AT+-A2 +\n,\n\ndT\n\nat\n\nIf\n\nwe\n\nrespectively\n\ncos d\n\nobserved that\n\nif ATT,\n\ng,\n\nd<5\n\nat?\n\nAe.\n\nde\n\n- A\nlog q and\n\n-=-.\n\n-j=\n\nA log q.\n\nIt\n\nshould\n\nbe\n\n## A&, and A* are expressed in seconds, in order\n\nmay be homogeneous, the terms containing AT,\nmultiplied by 206264.8; but if ATT, A&, and At\n\n## A#, and Ae must be\n\nare expressed in parts of the radius as unity, the resulting values of\ncos d Aa and A must be multiplied by 206264.8 in order to express\nthem in seconds of arc.\n\n## general application of the equations for cos d AOC and A$in terms of the variations of the elements is for the cases in which The most ## and of A are already known by comparison of the computed place of the body with the observed place, and in which it is required to find the values of ATI, A&, M, &c., which, ## the values of cos d AOC being applied to the elements, will make the computed and the observed places agree. When the variations of all the elements of the orbit are taken into account, at least six equations thus derived are necessary, and, if more than six equations are employed, they must ## be reduced to six final equations, from which, by eliminathe of the unknown quantities ATT, A&, &c. may be values tion, found. In all such cases, the values of AOC and A#, as derived from first the comparison of the computed with the observed place, are expressed in seconds of arc; and if the elements involved are expressed in seconds of arc, the coefficients of the several But ## terms of the equations ## some of the elements are not in in of T, q, and 6, the equations as the case expressed seconds, formed must be rendered homogeneous. For this purpose we multiply the coefficients of the variations of those elements which are ## must be abstract numbers. if ## not expressed in seconds of arc by 206264.8. Further, it is geneinconvenient to the variations rally AT, A^, and Ae in parts express of the units of T, g, and <?, respectively ; and, to avoid this inconvenience, we may express these variations in terms of certain parts of the actual units. Thus, in the case of T, we may adopt as the unit of AT the nth part of a mean solar day, and the coefficients ## of the terms of the equations for cos d AOC and A which involve AT THEORETICAL ASTRONOMY. 142 ## must evidently be divided by n. ## In the same manner, it appears ## Ag the unit of the mth decimal its value expressed in parts of the unit of g, we must divide of place its coefficient by 1CT, and similarly in the case of Ae, so that the equations become that if we adopt as the unit of -- -f- an d&& 8 .da ^C?a dd dd COS d - At -f - COS 8 -n aJ. ai dd Ae SAC\\ ' dd ## *,\" + da* a+ Ti\" + n-d , in which = 206264.8. When dS log q is introduced in place of q, the ## A log q are multiplied by the same factor as in the case of &q, the unit of A log q being the unit of the mth decimal place of the logarithms. The equations are thus rendered homogeneous, coefficients of and ## also convenient for the numerical solution in finding the values unknown quantities ATT, A&, At, AT, &c. When AT, Ag, and Ae have been found by means of the equations thus formed, the of the ## coirections to be applied to the corresponding elements are , lit ^-, _L \\j A /> and -^,> ## In the same manner, we may adopt as the unknown ## quantity, instead of the actual variation of any one of the elements of the orbit, n times that variation, in which case its coefficient in the equations must be divided by n. The value of ACC, ## derived by taking the difference between the place, is affected by the uncertainty ## computed and the observed ## necessarily incident to the determination of oc by observation. The ## unavoidable error of observation being supposed the same in the case of a as in the case of d, when expressed in parts of the same unit, evident that an error of a given magnitude will produce a greater apparent error in a than in S, since in the case of a it is it is ## measured on a small circle, of which the radius is cos d ; and hence, in order that the difference between computation and observation in a and d may have the same influence in the determination of the corrections to be applied to the elements, we introduce cos d AOC The same principle is applied in the case of the instead of AOC. longitude and of all ## corresponding spherical co-ordinates. DIFFERENTIAL FORMULA. 143 52. The formulae already given will determine also the variations of the geocentric longitude and latitude corresponding to small increments assigned to the elements of the orbit of a heavenly body. ## In this case we put e 0, and compute the values of A, B, sin a, the of We have also (7=0, means and sin 6 by equations (94) r a and sin i, and, in place of sin c d, respectively, we write A and ft. But when the elements are referred to the same fundamental plane ## as the geocentric places of the body, the formulae which depend on the position of the plane of the orbit may be put in a form which is more convenient If we ## for numerical application. ## differentiate the equations y' ## = r cos u cos & = r cos u sin & z' rrrrsmwsini, x' we cos i, cost*> obtain = r dr x' dx' r (cos -{- cos r (cos which sin dy' dz in ## smw cos & r sin u sin -j- -j- sin dr & cos r (sin sin sin cos & -j- cos i) sin & & & ', 2' cos d& cos cos i) ## =-dr -}-r cos w sin i du 1 x', & r (sin sin cos i) -f r sin u sin cos d& -j- & & r sin r sin it cos i) cos cos du & sin i di, du & ## sin i di, (46) i di, ## are the heliocentric co-ordinates of the body in x being directed to the ## reference to the ecliptic, the positive axis of ## Let us now suppose the place of the body to be vernal equinox. referred to a system of co-ordinates in which the ecliptic remains as the plane of xy, but in ## point whose longitude ## which the positive axis of x then we shall have is Q> ## = dx cos & dy' sin & = dx' sin & -f dy' cos dy dz = dz dx is directed to the -f- Q> ## and the preceding equations give = -dr T OC dx r sin u du r sin cos i d& dy = y dr + r cos u cos dz ## = - dr + r cos u sin idu-\\-r smu cos du -j- r cos u dQ i di. ## r sin u sin i di, (47) THEORETICAL ASTRONOMY. 144 ## This transformation, it will be observed, is equivalent to diminishing the longitudes in the equations (46) by the angle ft through which the axis of x has been moved. Let Xn F,, Z, referred to the ## denote the heliocentric co-ordinates of the earth ## same system of co-ordinates, and we have + X, = A cos cos (A y+ F, = Jcos/5sin(A x = A sin z--\\- Z, which in In ft), /? I is the geocentric ft), /?, longitude and ft ## the geocentric latitude. between ## differentiating these equations so as to find the relation the variations of the heliocentric co-ordinates and the geocentric longitude and latitude, we must regard ft as constant, since it indicates ## here the position of the axis of x in reference to the vernal equinox, this position is supposed to be fixed. Therefore, we shall have and s(A =sin /9 dJ J sin cos (A J sin /9 sin (A ft)d/J = cos/9sin(A ft)dJ A cos /9 sin (A A cos ft cos (A ft)d/9-f ft)d/5 ft ) <W, -f J cos ## from which, by elimination, we find = a) __sin(A A sin /3 cos (A 7 _ *= & + eo g-a) J s ft) O) sin/? sin (A -j- -*+ -j- ## These equations give sin (A ^A ~ = c?A cos(A ft) we introduce ^_~ ^ sin /9 cos (A dft_ sin/5 sin (A ft) J ft) . dy If ft) da dy the distance co (48) ## between the ascending node and the we have ## place of the perihelion as one of the elements of the orbit, ## and the equations (47) give _= dx dx 7 dv x _ = cosi, dx = du = -7 dy -^^^sm^cost, . rsinw, dy ydv dy = -~-=rcosucosi, doj dz sr dz 7 dv = - = Bint* sin z z; dz = -==! cos wsmi; d<o DIFFEEENTIAL FORMULAE. 145 - dx - =A dy T^- 0, dz rsmusmi, If we introduce TT, ## the longitude of the perihelion, (49) smu --^di cfo cfo = 0; cos^. we have ## and hence the expressions for the partial differential coefficients of the heliocentric co-ordinates with respect to n and become & dx = 7 ax = -T7T- dQ sm w sm 2?- = r cos u cos dy r sin u, , 2 dz - i, I dy = 2r cos u sm* -.\"L \"2! *t. dQ> When we i ; < r cos 50 > sin i. <xg ## and the motion ## the direct inclination exceeds 90 as being retrograde, = r cos u sm = ' 51. find, by making is regarded ## the necessary distinctions in ## regard to the algebraic signs in the general equations, dx - r T di dy = -~ r _ t sin dz sin di dx dx ## and the expressions for dx -=-, -=-, ## from (49) by writing 180 ## longitude of the perihelion, TT di i, dy rsm wcos^; (51) ## &C. are derived directly -TQ-> -j~, in place of i. If in this case, we introduce the rcosusim; we have, du dn -f dv d& and hence dx dx -?7^- dy -~=r rsmu 2r sin dy ## sin At, -y~- a^ cos dz cos = 2r cos u sm . i, , 2 ii, dz -y = r cos u sin rt^ ## But, to prevent confusion and the necessity of using so it is best to regard i as admitting any value from mulae, and (52) ^. ## to transform the elements ## which are given with the ## many forto 180, distinction ## of retrograde motion into those of the general case by taking 180 i instead of i, and 2& TT instead of TT, the other elements ## remaining the same in both 53. The cases. ## equations already derived enable us to form those for the of ^ and /? with respect to r, v, and at or , z, & differential coefficients TT, by writing successively and ft 10 in place of d, and &, i, &c. in THEORETICAL ASTRONOMY. 146 place of The in equation (2). TT ## expressions for the differential coeffi- ## with respect to the elements which determine the form of the orbit and the position of the body in its orbit, being cients of r and v, independent of the position of the plane of the orbit, are the same as those already given ; and hence, according to (42) and (43), we may derive the values of the partial differential coefficients of A and /9 The numerical ## with respect to these elements. is facilitated Thus, if we by application, however, introduction of certain the auxiliary quantities. ## by (48) and (49) in the ## substitute the values given equations . cos /5 dk = cos dv ---dx j ctt ^ dx /? -^ df^_dp_ dx_ dx dv dv and put cos sin (A &) in which ## and n are always r 8 d/3 -7- av (A sin (JV cos (A dv dy_ dft dz dy dv dz dv sin A, cos ^, = n sin = n cos ' JV, JV, become O ) sin u ## Let us also put y? ---dy -J- d/3 == -7 (sin /? cos (A i A au) sin dA dy positive, they ' cos -= /?-= d{3 sin i sin (A cos &) = A &) = 4 cos (A \\- dv /?) =5 ^) = ^ -4- sin 5, cos B, n cos w sin ,^ and we have c?A The expressions for dk cos/9-^- and -^ give, ## by means of the same auxiliary quantities, fjl cos/3-5- * In the same manner, if we put 2*L cos (A + u), DIFFERENTIAL FORMULA. &)= cos (A cos cos sin &) sin (A we O) = sin (A 147 <7 sin C, (7. =D =D cos (7; (57) sin Z), cos D; obtain dft d& cos ,. /? cw we -j -7 sin i sin = AD ^/3 If sin sin ft -w a^> -f- & ), cos (A sin (jD -f- /5). 7* and ## substitute the expressions (55) ^A cos ( J. dk = cos p -=^r ## (56) in the equations dr dv dk f- cos d<? /5 7, -= dv d<f> ~ dr d<p dv df d<?' and put dr -7 / 7* -j == o cos ass/ sin JP (59) ^^ jf cos JT ' T- Cfcp we ^ cos v, tan COS <p cos v r sin v. get S m(A + F+u-), P^-^ In a similar manner, dr ~dW cZv ~ ^ Sm == if we put ^ a 2 cos n ^ Sm V>$0\n\ntan\n\n(61)\n\n2r\n<p\n\nsin v(<\n\n206264.8\n\nT)\n3/i\n\n\\\n),\n\nTHEORETICAL ASTRONOMY.\n\n148\n\nwe\n\nobtain\nu),\n\ncos/y^==i|\n\ndM\n\ncos/S\n\n(62)\n-\n\nsin (J.\n\n+ IT+ u),\n\n## The quadrants in which the auxiliary angles must be taken are\n\ndetermined by the condition that A OJ J3Q) 0? /, g, and h are always\npositive.\n\n## 54. If the elements T, q,\n\nand\n\n^,\n\nand\n\ne are\n\nintroduced in place of\n\nMw\n\np,\n\nwe must put\ndr\ndv\n\ndr\n\n(63)\n\ndT\nh sin\n\nH= -\n\ndr\n\n7.\n\nTT\n\nhcosH=r-j-,\ndq\n\n~=\ndl\n\nsin\n\n(A\n\n+ F+ u),\n\n(64)\n\ndT\n<W\n\n## In the numerical application of these formulae, the values of the\n\nsecond members of the equations (63) are found as already exemplified for the cases of parabolic orbits and of elliptic and hyperbolic\norbits in which the eccentricity differs but little from unity.\nIn the\nsame manner, the differential coefficients of A and /9 with respect to\nany other elements which determine the form of the orbit may be\ncomputed.\n\nNUMERICAL EXAMPLES.\nIn the case of a parabolic\n\n149\n\norbit,\n\n## supposed to be invariable, the terms involving\n\nin the case of parabolic elements, we have\n\nG=: r\n\nvanish.\n\nksinv\n\ndr\n\ng COS\n\ndv\n\ndv\n\n-^,\n\nwhich give\ntan\n\nG=\n\ntan\n\n^v.\n\nand g\n\n/2\n= k\\,\n\nmoving\n\nHence\n\nthere results\n\n(r\n\nFurther,\n\n= 180\n\n\\v,\n\nwhich\n\nis\n\nthe\n\nin a parabola.\n\nTherefore,\n\nsm\n\nFor the\n\n## case in which the motion is considered as being retrograde,\n\nmust be used instead of i in computing the values of A\nN, C and (7, and the equations (55), (56), and the first two\n\n180\n\nAy\n\nn,\n\n## of (58), remain unchanged.\n\nrespect to i, the values of\n\n## But, for the differential coefficients with\n\nand\nmust be found from the last two\n\ndirectly\n\nand then we\n\nhave\n\nshall\n\ncos\n\nft\n\n-JT\n\n-7 sin i sin\n\n= -~ Z>\n-jr\n55.\n\nEXAMPLES.\n\ncoefficients\n\nof ^ and\n\nu cos\n\n(A\n\n& ),\n(66)\n\nsin\n\nu sin (D\n\n+ 0).\n\nThe\nft\n\n## equations thus derived for the differential\n\nwith respect to the elements of the orbit,\n\nwhen\n\n## any other plane is taken as the fundamental plane, if we consider A\n\nand ft as having the same signification in reference to the new plane\nthat they have in reference to the ecliptic, the longitudes, however,\nbeing measured from the place of the descending node of this plane\non the ecliptic. To illustrate their numerical application, let it be\nrequired to find the differential coefficients of the geocentric right\nascension and declination of Eurynome\nwith respect to the ele-\n\nments of\n24.5\n\nits\n\n## orbit referred to the equator, for the date 1865 February\n\nat Washington, using the data given in Art. 41.\n\nmean time\n\nTHEORETICAL ASTRONOMY\n\n150\n\nIn the first place, the elements which are referred to the ecliptic\nmust be referred to the equator as the fundamental plane and, by\nmeans of the equations (109) D we obtain\n;\n\n&'\n\n= 353\n\n45' 35\".87,\n\nt'\n\nand\na>'\n\n= 19\n\n26' 25\".76,\n\n= w -f w = 50\n\n= 212\n\n32' 17\".71,\n\n10' 7\".29,\n\nwhich are the elements which determine the position of the orbit in\nThese\nspace when the equator is taken as the fundamental plane.\nelements are referred to the mean equinox and equator of 1865.0.\nf\nWriting a and d in place of A and /9, and &', i w in place of &, i,\n1\n\n',\n\nand\n\na),\n\nrespectively,\n\nA = cos (a\n\nsin\n\nsmN=smi\n\nwe have\n&') cos i't\n\nsmJB\nQ\n\n<7 sin\n\nsin\n\n(N +\n\nn sin\n\n(7= cos (a\n\nD = cos i\n\n<0,\n\n&'),\n\nsin\n\n.F= a cos\n\n/ cos F=\n\ngsiuG=\n\n<p\n\ncos\n\n--\n\n1-\n\na tan\n\ncos\n\ncosN=\n\n&')\n\n-B cos\n\nB == sm ^ cos\n\ncos\n\nC= sin (a\n\ncos\n\nD = sin\n\n&');\n\ncosi'sin(a\n\ni'\n\n(a\n\n&')\n\n&')\n\ncosi';\n\nsin (a\n\n')\n\nv,\n\ntan\n\n<p\n\nA = sin (a\n\n<p\n\nsin\n\ncos v\n\nr sin v;\n\nv,\n\na2 cos p\n\nThe\n\nsmH=\n\nvalues of\n\nQ,\n\nn, -B\n\na tan\n\n(7 ,\n\n<p\n\nsin v (t\n\n/, ^,\n\nT)\n\n^-\n\n206264.8\nj,\n\n## thus determining the quadrants in which the angles A,\n\nbe taken ; and these equations give\n\n= 9.97497,\n= 9.52100,\n=\nO\n9.99961,\nlog\nlog D = 9.97497,\n\nA = 262\n\nlog B\n\nB= 75 48 35\nC = 263 26,\nD= 92 3547,\n\nlog/ =0.62946,\nlog? =0.34593,\n\n.F\n\nlog 4,\n\nlog h\n\nG^\n\n= 339\n= 350\n\n= 2.97759,\nH= 14\n= v + = 179\n58\".\n'\n\ntt'\n\n10' 40\",\n\n13'\n\n14\n\n0,\n\n11 16,\n\n30 48\n\nJ5,\n\npositive,\n\n&c. must\n\nNUMERICAL EXAMPLES.\n\n151\n\nSubstituting these values in the equations (55), (58), (60), and (62),\nf\no instead of X and ft and u in place of u, we find\n\n## and writing a and\n\ncos d\n\n=\n~^V\nao>\n\n+ 1.4235,\n\n=\n~rr\naw\n\n4890\n\n>\n\n=+0.0178,\nrfft\n\n= + 0.0193,\n\n-^ = + 0.0067,\ncos d ^~ = + 1.9940,\n\n-^r\n\n-^=+1.1300,\n\n^r-\n\ncos d\n\ncos d\n\n=\n-^qp\n\n-f-\n\n507.25,\n\n~- =\n\n4 =\n\n0.6530,\n\n8802\n\n179.34\n\n'\n\nd[j.\n\nand hence\ncos\n\n(5\n\nAa\nA\n\n## = + 1.4235 A^' + 1.5098 Aft' + 0.0067\n\n=\n\nwe put\n\n+ 1.9940 A?\n\nAA*,\n\n0.3802 A^f\n\nIf\n\nAt*\n\n+ 1.1300 *M + 507.25\n0.4890 AO/ + 0.0176 Aft' + 0.0193 At*\n179.34\n\nAA*.\n\n0.6530\n\nA?\n\nTHEORETICAL ASTRONOMY.\n\n152\n\nfrom which the values of ft, i, and co may be found from those of\n'\nIf we differentiate the first of these equations, regarding\nft and V\n.\n\ne as\n\nconstant,\n\nget\n\n= cos\n\ndi\n\nInterchanging\n\nand 180\ndi'\n\na>\n\nand\n\ni',\n\n= cos\n\n<t>\n\ndi\n\ni'\n\nand\n\nalso ft\n\n% sin\n\nsin\n\nd ft\n\ndft\n\nrelations,\n\nwe\n\n(68)\n\n'.\n\nft',\n\nwe\n\nobtain\n\nsin i\n\n_ sin\n\nsini\n\nthe result\n\nIf\n\nwe\n\nft\n\nsin &''\n\nis\n\n## differentiate the expression for cos\n\nspherical triangle,\n\nand reduce, we\n\n= cos\n\nda>\n\n## derived from the same\n\n<w\n\nfind\ncos\n\ni c?ft\n\ni'\n\ndQ,'.\n\nSubstituting for dft its value given by the preceding equation, and\nreducing by means of\nsin ft' cos\n\nwe\n\ni'\n\n## = sin ft cos % cos\n\ncos ft sin\n\n<y\n\nfl ,\n\nget\n\nThe\n\nequations (68), (69), and (70) give the partial differential co'\nof & , i, and w with respect to\nand i r , and if we sup-\n\n&\n\nefficients\n\n## pose the variations of the elements, expressed in parts of the radius\n\nas unity, to be so small that their squares may be\nneglected, we shall\n\nhave\n\ncos ^ A ^\nA% = sin\n-^-^\n&\nsin\n\nw\n--r-f\ncos\nsin\n\nsin t\n\n*i\n\nf\n,\n\n= sin\nAw =\nAi\n\na>\n\nAo>'\n\nsin\n\ni'\n\nA &'\n\ncos w\nfl\n\nAi',\n\n^^o-\n\n## If we apply these formulae to the case of Eurynome, the result\n\nAO O\n\n=\n=\n=\n\nen)\n\n4.420A^'\n\n+ 6.665\n\n3.488A\n\n-f 6.686Ai',\n\nft'\n\n0.179A&'\n\nAi',\n\n0.843Ai\n\nis\n\nDIFFERENTIAL, FORMULA.\n\nand\n\nif\n\nwe\n'\n\n153\n\nAi'\n\n14\".12,\n\nwe\n\nget\n\nAW O\n\n= -f 3\".36,\n\nA&\n\nThe element^\n\n&\n\nW=\n\n8\".86,\n\n6\".64,\n\n10\".0,\n\norbit in\n\n## and, hence, the elements\n\nreference to the ecliptic.\n\nand\n\n= + 10\".0,\n\nA^\n\n10\".0,\n\nA 10\n\n## may also be changed into those for\n\nwhich the ecliptic is the fundamental plane, by means of equations\nwhich may be derived from (109)! by interchanging & and &' and\ni' audi.\na/,\n\n',\n\n## 56. If we refer the geocentric places of the body to a plane whose\n\ninclination to the plane of the ecliptic is i, and the longitude of whose\nascending node on the ecliptic is &, which is equivalent to taking\n\n## the equations are still further simplified.\n\nthe fundamental plane,\nLet x' 9 y' , z' be the heliocentric co-ordinates of the body referred to\n\n## a system of co-ordinates for which the plane of the unchanged orbit\n\nthe plane of xy, the positive axis of x being directed to the as-\n\nis\n\n## cending node of this plane on the ecliptic; and let x, y, z be the\n\nheliocentric co-ordinates referred to a system in which the plane of\nxy is the plane of the ecliptic, the positive axis of x being directed\nto the point\n\nwhose longitude\ndx'\ndtf\ndz'\n\n(47),\n\nwe\n\nis Q>\n\nThen we\n\n= dx,\n= dy cos\n= dy sin\n\ni -f-\n\nand dz\n\ndz sin\n\ni -j-\n\nshall\n\nhave\n\ni,\n\ndz cos\n\ni.\n\nby the equations\n\nget\n\ndx\n\ndy'\n\ndz'\n\n= - dr\n\nx'\n\ndr\n\ndr\n\n-j-\n\nr sin\n\nu du\n\nr sin\n\ncos\n\nd&,\n\nr cos\n\nu du\n\n-f r cos\n\ncos\n\nd&\n\ndi.\n\nr cos u sin\n\nwe\n\ndQ,\n\n-f-\n\nr sin\n\n## have, so long as the elements remain\n\nunchanged,\naf\n\n==r\n\ncos u,\n\ny'\n\n= r sin\n\nu,\n\nz'\n\n0,\n\nTHEOEETICAL ASTKONOMY.\n\n154\n\n## and hence, omitting the\n\nof the unchanged orbit\ngive\n\n## as the plane of xy, the preceding equations\n\n= cos u dr r sin u du\n=\nr cos u du\nsin u dr\ndy\ndz =\nr cos u sin dQ,\ndx\n\n~\\-\n\n-j-\n\nr sin\n-j-\n\nr cos\n\nr sin\n\nu\nu\n\ncos\n\ndR>,\n\ncos\n\ndQ,\n\ndi.\n\nThe value of\n\n<\nis subject to two distinct changes, the one\narising\nthe\nfrom\nvariation of the position of the orbit in its own plane, and\nthe other, from the variation of the position of the plane of the orbit.\n\n## plane of the orbit and directed from\n\nline in the\n\nthe centre of the sun to a point the angular distance of which, back\nfrom the place of the ascending node on the ecliptic, we shall designate by a; and let the angle between this fixed line\nThen we have\ntransverse axis be designated by\n\nThe\n\n## fixed line thus taken\n\nis\n\na)\n\n-f\n\nff .\n\nsupposed to be so situated\n\nthat, so\n\nBut\n\nif the\n\nlong\n\nwe have\n\n## elements which fix the position of the plane of the orbit\n\nwe have the relations\n\n## are supposed to vary,\n\nda\nd<o\n\nNow,\n\nsince\n\n=\n= d%\n\n=v +\n\ncos\n\nco,\n\ndQ\n\ndu\n\n(72)\n\n= d% -f 2 sw*%id&.\n\nwe have\nu\n\nand\n\n= dv -f dx\n\n= v -f /\n= dv -f d%\nff,\n\ndff\n\ncos\n\nid&.\n\nreduce to\ndx\nr sin u dv\ncos u dr\nr sin u d%,\nu\nsin\ndr\nr\nu\ncos\ndv\n-fdy\n-f- f cos u d%,\n\ndz,\n\nthey\n\ndz\n\nThe\n\nr cos\n\nsin i\n\nd&\n\n-f r sin\n\n(73)\n\ndi.\n\n## here supposed to be susceptible of any value from\n\nelements are given with the distinction of\nmotion\nwe\nmust\nuse 180\nLet us now denote by d the geocentric longitude of the body meainclination\n\nto\n\nis\n\n180, and\n\nif the\n\n(which\n\nis\n\n## here taken as the\n\nDIFFERENTIAL FORMULAE.\n\n155\n\nand\n\necliptic,\n\nlet\n\nbe denoted by\n\n## this plane on the\n\nthe geocentric latitude in reference to the same plane\n\nThen we\n\n37.\n\n-f-\n\n-|-\n\nz -f-\n\nin\n\nhave\n\nshall\n\nX = A cos\nY= A cos\nZ = A sin\n\nf\\\n\ncos\n\nTI\n\nsin 0,\n\n0,\n\n??,\n\n## which y Y, Z are the geocentric co-ordinates of the sun referred\n\nsame system of co-ordinates as x9 y, and z. These equations\n\nto the\n\nby\n\ngive,\n\ndifferentiation,\n\n= cos\ndy = cos\n\ndx\ndz\n\nsin\n\nwe\n\nind hence\n\ncos\n\nT)\n\ncos\n\nTI\n\nsin\n\nfj\n\ndA\n\nA cos iy d^\n\n= --sin\n\ndx\n\nT]\n\nsin\n-\n\ni?\n\nsin0\n\nsin\n\ndx~\n\nsin\n\nA cos\n\ndf\\\ncfy\n\nf\\\n\nsin\n\n^ cos 7 cos\n\nd0,\n^\n\n^>\n\ncos\n-\\\n\ncos\n\ndy,\n\n-dx -,\n\nsin ^ sin\n\nd0\ncos 6\nCOS7)-j-= -T-i\n\ndx\nrr=\n\nCOS\n\n1*.\n\ndx\n\ndy\ndr\n\ndr\n\ndx\n\n-=\n\ndv\n\ndx\n\ndx\n\nd&\n\nr = 0,\n\nr sin u,\n\ndo\ncos\n\nTI\n\n-jaz\n\n_\n;\n\n(74)\n\ncos\n\nt]\n\n## from (73) we get\n\ndx\n\n>?\n\ngive\n\nde\n\nid\n\n-\\-\n\nsin ^ cos\n\nsin\n\nobtain\n\n=\nlese\n\nJ\nA\n\ndA\ndA\n\nTHEOKETICAL ASTRONOMY.\n\n156\n\nwe\n\nget\n\ndd\n\ndo\n\nCOS\n\n7}\n\n-r- == COS\n\n-J-\n\nTf)\n\ndx\n\n= rJ COS (0 - U),\n,.\n\n-r\n\n(76)\n\ndv\n\nd%\n\ncos\n\n>y\n\ndo\n-=\n\nN\nit),\n\nsm (0a\n.\n\n(77)\n\ndO\ncos\n\nr)\n\nJT-\n\ndr)\n\n-JT-\n\n0,\n\nwe\n\n<p,\n\nfa\n\nj-\n\nsm w\n\ncos ^\n\nv,\n\nhave, from\n\ndO\n= cos y -jdr\ndy _\n\ndr\n\nrf^\n\ncos\n\niy\n\ndtp\n\nwe\n\n-j\n\nd(p\n\ndr)\n\ndr^\n\ndr^\n\ndv_\n\ndr\n\nd<p\n\ndv\n\nd<p'\n\nd<?\n\n(59),\n\nF\\ ~=-\n\ncos (0\n==4\nA\n\na<p\n\ndO\ni)\n\nwe\n\n{sin^sin(^\nA\n\nd<f>\n\ntions (61),\n\ncos\n\n^v\n\ndv\n\ncos\n\nc?^\n,\n\nh cos ^\n\nd<p\n\nif\n\nand\n\nQ)\n\n6r,\n\nand H, according\n\nF}.\n\n(78)\n\nto the equa-\n\nget\n\n= h cos\n\n(0\n\ndt\\\n\n~-\n\nH),\n\nsm\n\n-j-\n\nTJ\n\nsm(0\n\nJET).\n\n## If we express r and v in terms of the elements T, q, and e, the\n\nvalues of the auxiliaries /, g, Tiy F, &c. must be found by means of\n(64); and, in the same manner, any other elements which determine\nthe form of the orbit and the position of the body in\n\nits orbit,\n\nmay\n\nbe introduced.\n\nThe\n\n## having been found, we have\n\nCOS\n\n1?\n\nA0\n\n= COS\nd\")\n\ndo\n\ndO\n\nA/\n\nTJ\n\n-{-\n\nCOS\n\n?)\n\n-,\n\nd6\n&<p\n\n-\\-\n\nCOS\n\nT?\n\nQ -f-\n\ndd\nCOS\n\nf]\n\nA/*,\n\nDIFFERENTIAL FORMULAE.\n\n157\n\n## from which it appears that, by the introduction of # as one of the\n\nelements of the orbit, when the geocentric places are referred directly\nto the plane of the unchanged orbit as the fundamental plane, the\nvariation of the geocentric longitude in reference to this plane depends\non only four elements.\n57. It remains\n\nof\n\nrj\n\nnow\n\nX and\nbody referred\n\nordinates of the\n\nLet x\n\n/9.\n\nyw\n\n## be the geocentric co-\n\nZQ\n\nwhich the\n\nto a system in\n\necliptic is\n\nthe plane of xy, the positive axis of x being directed to the point\nand let o? ', y ', zQ f be the geocentric co-ordiwhose longitude is\n;\n\n&\n\n## which the axis of x remains\n\nbody\nwhich the plane of the unchanged orbit is the plane\nof xy; then we shall have\nreferred to a system in\n\nnates of the\n\nXQ\nyQ\nZQ\n\n= A cos\n= A cos\n= A sin\n\n/?\n/?\n\n& ),\n\ncos (A\n\nxQ\n\n'\n\nsin (A\n\nft),\n\n2/\nf\n\n,9,\n\n= A cos ^ cos\n=J\n=A\n\n6,\n\nc sV\n\nsin\n\ni?,\n\nand also\n\ncos\n\ni.\n\nHence we obtain\nCOS\n\nr\\\n\nCOS\n\ncos\n\nf\\\n\nsin\n\nsin\n\n= COS COS\n= cos sin\n= cos sin\n(A\n\nft\n\n>?\n\nft )>\n\n(A\n\n(3\n\n## & ) cos i sin sin\n\n& ) sin i + sin cos\n(A\n/?\n\n-j-\n\n(80)\n\ni,\n\n/5\n\ni.\n\n## These equations correspond to the relations between the parts of a\n\nand 90\nspherical triangle of which the sides are i, 90\n/9,\n27,\nto 90\nand\n90\n6.\na)\ndenoted by p, and we have\n\nthe\n\nangles\n\n90\n\n-f (;\n\nopposite\n\ncos\n\n*)\n\nsin Y\n\ncos\n\n>?\n\ncos\n\n7-\n\n= sin\n= sin\n\nfj\n\nand 90\n\ncos (A\n\nsin (A\n\n& ),\n& ) sin\n\nA\n\nand\n\n/9.\n\n/5\n\n-f-\n\nn\nn\n\nsin\n\nrespectively\n\nThus,\n\nN=\n\nif\n\nwe put\n\nsin/?,\n\ncos^= cos\n\ncos i cos\n\n/?.\n\nis facilitated\nby the intro-\n\nbeing\n\n## Let the other angle of the triangle be\n\nsin (A\n\nft),\n\nTHEORETICAL ASTRONOMY.\n\n158\n\nwhich n\n\nin\n\nis\n\nalways positive,\ncos\n\nf]\n\ncos\n\ncos\n\n77\n\nsin\n\nsin\n\n7}\n\nmay be\nn' sin\n\nshall\n\nhave\ncot\n\nget\n\ncos\n\nft\n\ncos (A\n\nft),\n\n= n cos (N\n\ni),\n\nIf we also put\n\nN'\nN'\n\n= cos\n= sin\n\nN = tan\nf\n\n(83)\n\ni),\n\n=n sin (N\n\nn' cos\n\nwe\n\nwe\n\ni,\n\nsin (A\n\nsin (A\n\nft ),\n\nft ),\n\n;-a).\n\ncot (\n\nIf Y\n\nis\n\nsmall,\n\nit\n\nmay\n\n(85)\n\nsintcosO^\n\n## in which the angles sought must be taken, are easily\n\ndetermined by the relations of the quantities involved ; and the\naccuracy of the numerical calculation may be checked as already\nillustrated for similar cases.\n\nsin (45\n\n## - J,) sin (45 -\n\n+ r)) =\n+ i (A - ft)) sin (45 + 0),\n=\n~ (0 +\nsiu (45 + 4 (A - ft)) sin (45 - J Q9 (0\n\ncos (45\n\nsin (45\n\n&)\n\ncos (45\n\ncos (45\n\nii?)\n\nsin (45\n\n(/?\n\n7-))\n\ni)),\n\nJ (0\n\nr))\n\n^)\n\ncos (45\n\n=f\n\n_ ft )) cos (45 - J +\nr }} =\n\\ (9\n- ft )) cos (45 - (0 sin (45 +\n(A\ncos (45\n\ncos (45\n\n(87)\n\n(A\n\ni)),\n\n1)),\n\n## from which we may derive ^, 0, and f.\n\nWhen the problem is to determine the corrections to be applied to\nthe elements of the orbit of a heavenly body, in order to satisfy\ngiven observed places, it is necessary to find the expressions for\ncos 5? A# and A^ in terms of cos ft AA and A/9.\nIf we differentiate the\nfirst and second of equations\n(80), regarding ft and i (which here\ndetermine the position of the fundamental plane adopted) as constant, eliminate the terms containing dy from the resulting equations,\n\ntriangle,\n\nwe\n\nget\n\n## relations of the parts of the spherical\n\nNUMERICAL EXAMPLE.\ncos\n\nV]\n\ndd\n\ncos Y cos\n\n/?\n\noU\n\n-f-\n\n## Diiferentiating the last of equations (80),\n\ndy\n\nsin Y cos\n\nft\n\ncW\n\n-j-\n\n159\n\nsin y\n\nc?/3.\n\nand reducing, we\n\nfind\n\ncos\n\nThe\n\nequations thus derived give the values of the differential coand 57 with respect to A and /5 ; and if the differences\nof\nA^ and A/3 are small, we shall have\n\nefficients\n\ncos\n\nTI\n\n=\nAA -f cos y\nsin Y cos\nA^\nA0\n\nA/9,\n\n/?\n\n/?\n\nThe value of\n\nA/5.\n\n## required in the application of numbers to these\n\ngenerally be derived with sufficient accuracy from\n7-\n\nequations may\n(86), the algebraic sign of cos Y being indicated by the second of\nequations (81) ; and the values of 37 and d required in the calculation\nof the differential coefficients of these quantities with respect to the\n\nEXAMPLE.\n\n58.\n\n## Since the spherical co-ordinates which are fur-\n\nnished directly by observation are the right ascension and declination, the formulae will be most frequently required in the form for\n\n## For this purpose, it is only necessary\n\nfinding y and 6 from a and 3.\nf\ni ,\nto write a and d in place of A and /9, respectively, and also\n',\nu' in place of &, i, a), , and u, in the equations which\nft/, /', and\n\n&\n\nhave been derived for the determination of y and 6, and for the\nwith respect to the elements\n\nof the orbit.\n\nTo\n\nfor cos\n\nrj\n\nA# and\n\ncase of the\naj'\n\n= 5Q\n\n## in terms of the variations of the elements in the\n\nA/?\n\n10' 7\".29,\n\n'\n\n= 353\n\nfor\n\nwhich we have\n\n45' 35\".87,\n\ni'\n\n= 19\n\n26' 25\".76.\n\nThese are the elements which determine the position of the orbit of\n\nEurynome\n\nWe\n\n(79),\n\nreferred to the\n\n1865.0.\n\nhave, further,\n\nlog/= 0.62946,\n\nF= 339\n\nlog#\nu'\n\nIn the\n\nfirst place,\n\n= 0.34593,\n\nG = 350\n\n14' 0\",\n\n= 179\n\nwe compute\n\n11' 16\",\n\nlog ft\n\n= 2.97759,\n\nH= 14\n\n30' 48\",\n\n13' 58\".\n\n/, 6,\n\n## and Y by means of the formulae\n\nTHEORETICAL ASTKONOMY.\n\n160\n\nand\n\n(83)\n\nof\n\nAj ft)\n\n(85), or\n\nand\n\nQ>,\n\n= 188\n\ni,\n\nby means of\n\n(87), writing a, d,\n\nrespectively.\n\nHence we obtain\n\n31' 9\",\n\n&', and\n\n159'28\",\n\n19\n\ni'\n\n17'\n\n## Since the equator is here considered as the fundamental plane, the\n\nlongitude 6 is measured on the equator from the place of the ascending node of the orbit on this plane. The values of the differential\ncoefficients are\n\n## then found by means of the formula?\n\nCOB \"\n\ndri\n,\n\ndo\ndo\nCOS\n\nIT\n\nd%\ncos\n\n77\n\ny\ndtp\n\n= LA cos (0\n\nu'F\"),\n\ncos\n\n\"n\n\ndo\n-J^T\n\n= q cos (0\n\nu'\n\ndO\n\n-:\n\n= h-cos(0\n\nwhich give\ndB\n\ndo\n\nCOST;\n\ncos\n\n=+\n\n~dy\n\n= rA COS (0fa\n\nTi\n\nyj\n\nsin i cos\n>\n\ncos ^\n\nsm u\n\n,,\n\nI,\n\nG\\\n\n=\n-T^T\n\naM\n\ndr)\n,\n\ndv.\n\nsin y sin (O\n4\nA\n\nu'\n\nF),\n\n-^ sin 7 sin (0\n\nG\\\n\nh\nA\n\nf\nsm sm (0\n~n\n\nH).\n\nDIFFERENTIAL FORMULAE.\n\n=\n=\n\nAW'\n\nA?\n\nA a'\nA Jf g\n\n6\".64,\n\n10\",\n\n-f-\n\nwe have\n\nA/ ==\n\nAO/\n\n161\nAi'\n\n14\".12,\n\n+ 10\",\n\nA/<\n\n+ cos A' =\nv\n\n19\".96\n\n= _ 8\".86,\n= + 0\".01,\n\ncos\n\n7?\n\nA0\n\n= + 8\".24\n\ncos d\n\nAa\n\nAT?\n\nAO/, A &\n\n&c.,\n\n',\n\n= -f 5\".47,\n\n==\n\nwe have\n\nA<5\n\n## which, by means of the equations (88), writing\n\nA\n\nand\n\n6\".96.\n\n9\".29,\n\na and\n\nd in place of\n\ngive\n\n/?,\n\ncos\n\n7?\n\nA0\n\n= + 8\".23,\n\ni)\n\n6\".96.\n\nIn special cases, in which the differences between the calcuand the observed values of two spherical co-ordinates are given,\nand the corrections to be applied to the assumed elements are sought,\nit\nmay become necessary, on account of difficulties to be encountered\n59.\n\nlated\n\nThe\n\n## will serve, without presenting any difficulty,\n\nof the equations into a form adapted, to the\nThus, in the case of the elements which determine the\nspecial case.\nform of the orbit, we may use a or log a instead of /*, and the\n\nto those\n\ncommonly used\n\nequation\njfei/l\n\n+m\n\ngives\n\n=\nin\n\nwhich\n\nis\n\n3 - da\n\n\"a\n\n^d log a,\n\n(89)\n\n|/\n\n## the modulus of the system of logarithms.\n\nTherefore,\nis transformed into that of A log a by multiply-\n\ning\n\nit\n\nby\n\ny-\n\nand\n\nmth decimal\n\n^0\n\nA log\n\na,\n\n## multiplied by 10~ . The homogeneity of the equation\n\nsince fj. is here supposed to be expressed in seconds.\n\nIf\n\nwe\n\nis\n\nnot disturbed,\n\n## introduce logp as one of the elements, from the equation\n\np=a\n\ncos 3\n\n11\n\n<p\n\nTHEORETICAL ASTRONOMY.\n\n162\n\nwe\n\nget\n\nlose\n\nI3\n\ndfj.\n\n2L tan y\n\nd<p,\n\n30 tan\n\nd<p.\n\nor\ndp.\n\ny-\n\nd logp\n\n<p\n\n(90)\n\n## appears that the coefficients of A logp are the same as those\n\nof A log a, but since p is also a function of ^>, the coefficients of A^>\n\nHence\n\nit\n\nand\n\nare changed ;\n\nif\n\nwe denote by\n\ncos d\n\n-=\n\nand\n\n-j-\n\nthe values of\n\nthe partial differential coefficients when the element fj. is used in connection with <p, we shall have, for the case under consideration,\n\nda\ncos 8 -y-\n\n= cos\n\ndy\n\ndd\n\nwhich\n\n= 206264\".\n\nwith respect to\n\n//\n\nand\n\n3 - tan y cos d\n\n\\d^\n\ndp.\n\ndd\n\nfi\n\n3 - tan\n\nda\n\n<p\n\nr -,\n\nc?0\n\n8.\n\n(p\n\ntdd\\\n-5-\n\nj-=\nd?>\nin\n\nda\n\n-j\n\\d<f>\n\ndr\n\n## vantageous to compute the values or\n\ncoefficients\n\nit\n\ndr\n\ndv\n\n-n\nd\n\ndv\n\nand\n\nby\nd logp\niogp\nmeans of the expressions which may be derived by substituting in\nthe equations (15) the value of dp. given by (90), and then we may\n-r\n\n>\n\nd<f>\n\n,.,,,,\ndirectly the values of cos o\n\nda\n\ncompute\n\nMw\n\nIn place of\n\nwhen ^\n\n-=-1\n\ndd\ndd\n-- 3,\nand -71--d logp\n\nda\n\ncos o ^-=\n\nd logp\n\nd<p\n\nQ,\n\nthe\n\nmean\n\nand since\n\ni =jf +r,\n\nwe have\n\nand,\n\nis\n\nd<p\n\n..\n\ndy\n\nit\n\n-7?\n\nis\n\nused,\n\ndx\n\nb\n\n+ d&,\n\nwe may\n\n= sin\n\nft\n\nand\n\n+ (1\ni\n\nCOST\n\nuse\n\nsin ft,\n\n= sin\n\ncos ft,\n\n## and the expressions for the relations between the differentials of b\n\nand c and those of i and ft are easily derived. The cosines of the\nangles which the line of apsides or any other line in the orbit makes\nwith the three co-ordinate axes,\n\nmay\n\n## also be taken as elements of the\n\nDIFFERENTIAL FORMULAE.\n\n163\n\norbit in the formation of the equations for the variation of the geocentric place.\n\nThe\n\n## equations (48), by writing I and 6 in place of X and ft,\n\nrespectively, will give the values of the differential coefficients of\n60.\n\n## the heliocentric longitude and latitude with respect to x, y, and z.\n\nCombining these with the expressions for the differential coefficients\nof the heliocentric co-ordinates with respect to the elements of the\n\nwe\n\norbit,\n\nobtain the values of cos b &l and A& in terms of the varia-\n\nThe\n\ndQ, and\n\ndi,\n\nmay\n\necliptic\n\nIn\n\ndate.\n\nthis case,\n\nwe have\n\n= dx\n\ndu\n\nWhen\n\ndQ\n\n## the auxiliary constants A, B, a, 6, &c. are introduced, to\n\nfrom the variations assigned to\n\nthe elements,\n\nwe\n\ntan\n\n&\n\ncos\n\nB = cot\n\n&\n\ncos\n\nsin i cosec\n\ncot\n\n&\n\ncos\n\ni -}-\n\nsin\n\ncot\n\nin\n\nwhich\n\nA=\n\ncot\n\ncot\n\ni,\n\nvalue from\n\n## may have any\n\n&\n\ncosec\n\n2 cot\n\n180.\n\nto\n\ntan\n\ne,\n,\n\nIf we differentiate\n\n## the quantities involved as variable, and reduce\n\nthese, regarding\nby means of the values of sin a, sin 6, and sin c, we get\nall\n\nsin A\nsm &\nd& ---\n\ncos^\n\ndA\n\nsm\n\nT,\n=\ndB\n7\n\nCOS\n\n(cos i cos e\n\nsin o\n\nsin\n-j\n\n~\ndC=\n7\n\nsin\n.\n\nsin\n:\n\nsin o\n\nand\n\nthese,\n\n&\n\nsm e\n\nsm i cos e\n\n-j-\n\ncos\n\n^ di,\n\ncos\n.\n\n^N\n\n& ) dl\n,\n\nsm t) di\n\n--\n\nsin i sin\nr-jr\n\n-j\n\n&\n\nde,\n\nsin o\n\n-- C\nH\n.\n\n7- (cos\n\nsin i\n\nsin\n\n(.cos\n\n&\n\ncos\n\ncos\n\n&)d&\n,\n\nsini sin\n\ncos\n\ncos 9) di\n\n## by means of (101) D reduce to\n\n-\\\n\n--\n\n-- as\n\nsin i sin\n2\n\nsin c\n\nTHEORETICAL ASTRONOMY.\n\n164\n\n= sm\n\ndA\n\nd&\n\ncose cose\n\nsin\n\n## sine cos &\n\ncot a di,\n\n-r,\n\ncos a\n\ncosa\n\n,.\n\nsmc\n\nsmc\n\nLet us now differentiate the equations (101 ) 1? using only the upper\nis\nsign, and the result\n\n=\n=\ndc =\n\nIf\n\nda\n\nsin i sin\n\ndb\n\nsin i sin\n\nA d&\nB dQ>\n\nsin i sin\n\nC dQ\n\nwe multiply\n\nthe\n\nfirst\n\n## and the third by\n\ncot b,\n\nsystem of logarithms,\n\n(Hog\n\nsin a\n\nd log\n\nsin b\n\na log sine\n\n=\n=\n\ncos\n\n-j-\n\ncos\n\nB di\n\n-f~\n\ncos\n\nC di\n\ndi,\n-{-\n\n## cos c cosec b ds,\n\ncos b cosec c ds.\n\n## of these equations by cot a, the second by\n\nthe modulus of the\nc, and denote by ^\n\ncot\n\nwe\n\nget\n\na sin\n\nA dl\n\n-j-\n\nB dQ\n\n-f-\n\nA sin i cot\n\n-f-\n\n~ ,^\n\nG aQ\n\n-f-\n\nA cot\n\na cos\n\nA di,\n\ncote cos\n\n.^ ,.\n\nUdi\n\n-r-yy\n\nsin o\n\ncos b cos c\nr\n\nSill\n\nds,\n\nds.\n\n(92)\n\nThe\n\nequations (91)\n\nBy\n\n## C, log sin a, &c.\n\ndifferential coefficients of\n\nwith respect to &, i, and e; and if the variaare so small that their squares may be\n\nand\n\ntions assigned to\n\nneglected, the\n\n## and (92) furnish the\n\n&, i,\nsame equations, writing\n\nA^.,\n\nA&,\n\nA*,\n\nIn\nthe differentials, give the variations of the auxiliary constants.\nthe case of equations (92), if the variations of &, *, and s are expressed in seconds, each term of the second member must be divided\nby 206264.8, and if the variations of log sin a, log sin 6, and log sine\nare required in units of the mth decimal place of the logarithms, each\n\n## term of the second member must also be divided by 10.\n\nIf we differentiate the equations (81) D and reduce by means of the\nsame equations, we easily find\ncos b dl\n\ndb\n\n= cos\n= sin\n\nsec b\n\ncos (I\n\ndu\n\ncos b d &\nsin b cos (I\n& ) du + sin (I Q> ) di,\n\n-\\-\n\n& ) di,\n\n## which determine the relations between the variations of the elements\n\nof the orbit and those of the heliocentric longitude and latitude.\n\nBy\n\n## differentiating the equations (88)^ neglecting the latitude of\n\nDIFFERENTIAL FORMULJE.\nthe sun, and considering\nreduction,\n\nJ,\n\n^, /?,\n\nand\n\n165\n\nas variables,\n\nwe\n\nderive, after\n\nT)\n\ncos\n\n/?\n\nctt\n\n= -r cos\n\nd{3\n\n= B sin\n-j-\n\nO ) dQ\n\n(A\n\nft\n\nsin (A\n\nO ) d)\n\n## which determine tlie variation of the geocentric latitude and longitude\n\nIt\narising from an increment assigned to the longitude of the sun.\nthat\nan\nerror\nin\nthe\nof\nthe\nsun\nwill\nappears, therefore,\nlongitude\nproduce the greatest error in the computed geocentric longitude of a\nheavenly body when the body\n\nis\n\nin opposition.\n\nTHEORETICAL ASTRONOMY.\n\n166\n\nCHAPTER\n\nIII.\n\n## A PABABOLA, AND FOB COEBECTING APPEOXIMATE ELEMENTS BY THE VABIATION\n\nOF THE GEOCENTBIC DISTANCE.\n\n## THE observed spherical co-ordinates of the place of a heavenly\n\nfurnish\neach one equation of condition for the correction of the\nbody\nelements of its orbit approximately known, and similarly for the\n61.\n\n## determination of the elements in the case of an orbit wholly unknown ;\n\nand since there are six elements, neglecting the mass, which must\n\nfirst\n\n## three complete observations will furnish the six\n\nHence,\nequations necessary for finding these unknown quantities.\nthe data required for the determination of the orbit of a heavenly\n\nbeing considered,\n\nbody are three complete observations, namely, three observed longitudes and the corresponding latitudes, or any other spherical coordinates which completely determine three places of the body as\nseen from the earth.\n\nsome point or\n\n## Since these observations are given as made at\n\non the earth's surface, it becomes\n\nat different points\n\nIn\nnecessary in the first place to apply the corrections for parallax.\nthe case of a body whose orbit is wholly unknown, it is impossible\nto apply the correction for parallax directly to the place of the body;\nbut an equivalent correction\n\n## may be applied to the places of the\n\nto\nthe\nformula?\nwhich will be given in the next\nearth, according\nHowever, in the first determination of approximate elechapter.\n\n## ments of the orbit of a comet, it will be sufficient to neglect entirely\n\nthe correction for parallax.\nThe uncertainty of the observed places\nof these bodies is so much greater than in the case of well-defined\nobjects like the planets,\nwhich will be generally\n\nemployed\n\n## orbit will be so small, that an attempt to represent the observed places\n\nwith extreme accuracy will be superfluous.\n\n## When approximate elements have been derived, we may find th<\n\ndistances of the comet from the earth corresponding to the thi\nobserved places, and hence determine the parallax in right ascensioi\n\nDETERMINATION OF AN ORBIT.\nand\n\n167\n\nformulae.\n\nThus, we have\ncos\n\nTtp\n\nsin (a\n\n<p'\n\n0)\n\n\"\n\nCOS (a\nTT/>\n\nsin\n\n0)'\nsin (y\n\n<p'\n\n<5)\n\nsin?'\n\n## which a is the right ascension, d the declination, A the distance\n\nof the comet from the earth, <p' the geocentric latitude of the place\nthe sidereal time corresponding to the time of\nof observation,\nin\n\n## observation, p the radius of the earth expressed in parts of the\n\nTC the\nequatorial horizontal parallax of the\nsun.\n\n## In order to obtain the most accurate representation of the observed\n\nplace by means of the elements computed, the correction for aberraWhen the distance A is known, the\ntion must also be applied.\ntime of observation\nbut\n\nif\n\nis\n\nmay be neglected\n\nmay\n\n## right ascension and declination\n\neffected by means of the equations\n\nand longitude is\nwhich may be derived from (92) x by interchanging a and\nand writing\nThus, we have\ninto latitude\n\nN= tan\nsm\n= cos N\ntan\ntan p = tan (N\n,\n\ntan\n\nd and\n\nft,\n\nand\n\n/\",\n\ntan\n\n=r=\n\ns )\n\nsin\n\n(1)\n\na,\n\nA,\n\nalso\ncos (JV\n\ncos\n\n.2V\"\n\ne)\n\ncos p sin A\ncos S sin a'\n\n## which will serve to check the numerical calculation of X and /9.\n\nSince cos /9 and cos d are always positive, cos A and cos a must have\nthe same sign, thus determining the quadrant in which A\ntaken.\n\nis\n\nto be\n\nAs\n\n## soon as these preliminary corrections and transformations\n\nhave been effected, and the times of observation have been reduced\n62.\n\nto the\n\n## same meridian, the longitudes having been reduced\n\nto the\n\nTHEORETICAL ASTRONOMY.\n\n168\n\n## same equinox, we are prepared to proceed with the determination of\n\nFor this purpose, let t, t t\" be the times\nthe elements of the orbit.\nof observation, r, r r\" the radii- vectores of the body, and u, u u\"\nf\n\n## the corresponding arguments of the latitude, R, R', R\" the distances\n\nthe longitudes of the sun\nof the earth from the sun, and O, 0',\n\nO\"\n\nLet\n\n[>?'] denote\n\nr,\n\n## double the area of the triangle formed between\n\nr f and the chord of the orbit between the corre-\n\nsponding places of the body, and similarly for the other triangles\nthus formed. The angle at the sun in this triangle is the difference\n\n## between the corresponding arguments of the latitude, and we shall\n\nhave\n[rr'~\\\n\n= rr\n\n[r/'J\n\n=rr\"sm(M\"tO,\n\nsin (u'\n\nIf we designate by x } y } z } x f , y f ,\nordinates of the body at the times\n\nu),\n\n## x\", y\", z\" the heliocentric co-\n\nz',\n\nand\n\nt,\n\n(2)\n\n',\n\nt\",\n\nwe\n\nshall\n\nhave\n\n= r sin a sin\nw),\n= sin a sin A\nx\" = r\" sin a sin (A +\nx'\n\n'(A\n\nx'\n\nr'\n\n-f-\n\n-f-\n\nu'),\n\n\"),\n\n## which a and A are auxiliary constants which are functions of the\n\nelements & and i, and these elements may refer to any fundamental\nin\n\n## If we multiply the first of these equations by\n\nplane whatever.\nff\nlf\nu r ) 9 the second by\nsin (u\nu), and the third by\nf\nsin (u\nu), and add the products, we find, after reduction,\nsin (u\n\n- sin (u\"\n\nx\"\n\nx'\n-\n\nu'}\n\nsin (u\"\n\nu)\n\n-\\\n\nTl\n\nsin (u\n\nu)\n\n0,\n\nr\nwhich, by introducing the values of [rr ], [Vr\"], and [V r\"], becomes\n\n[r'r\"~\\\n\n[r/'] x'\n\n-f- [rr'~\\\n\nx\"\n\n0.\n\nIf we put\n\" ;=\n\ntt\n\n=&?T\n\nwe\n\nget\n\nIn\n\nprecisely the\n\n[i^r\n\nsame manner, we\n\nfind\n\n2-l'^n\"l\" = 0.\n\n<\n\nDETERMINATION OF AN ORBIT.\n\n169\n\nSince the coefficients in these equations are independent of the positions of the co-ordinate planes, except that the origin is at the centre\n\n## evident that the three equations are identical, and\n\nexpress simply the condition that the plane of the orbit passes through\nthe centre of the sun ; and the last two might have been derived\n\nof the sun,\n\nit is\n\n## by writing successively y and z in place of x.\n\nA\" be the three observed longitudes, /9, /?', ft\" the corresponding latitudes, and J, //', A\" the distances of the body from the\nearth ; and let\nfrom the\nLet A,\n\nfirst\nA',\n\ncos\n\n/?\n\n## which are called\n\nR cos Q\nR sin\n\nA\nA\n\nA\"cQ8p'\n\n/o',\n\nThen we\n\ncurtate distances.\n\n= p cos\ny = p sin\n= p tan\nx\n\nJ'cos^\n\n,,,\n\nx'\n\ny'\n\nz'\n\n/5,\n\n=\n=\n=\n\n=P\n\n\",\n\nhave\n\nshall\n\nR' sin\n',\n\np'\n\ncos\n\nA'\n\np'\n\nsin\n\nA'\n\np'\n\ntan\n\n{?,\n\njR\"cos0\",\n0\",\n\n12\" sin\n\nin\n\nis\n\nThe\n\nneglected.\n\ndata\n\nmay\n\n0, as will\n\nbe so\n\nbe ex-\n\n## plained in the next chapter ; but in the computation of the orbit of\n\na comet, in which this preliminary reduction has not been made, it\nwill be unnecessary to consider this latitude which never exceeds 1\",\nwhile its introduction into the formulae would unnecessarily complicate\n\n= n (p cos\n\nR cos\n\n+ n\"(p\"cosl\"\n\n= n (p sin\n\n= np tan\n\nE sin\n\n## + n\" (p\" sin A\"\n\n/?\n\np'\n\nIf\n\nwill be derived.\n\nf\nvalues of x, x , &c. in the equations (4) and\n\ntan\n\n-f-\n\n(//\n\n(5),\n\ncos\n\n12\" cos\n\nri'p\"\n\nsubstitute these\n\nthey become\n\nA'\n\ncos\n\n')\n\n0\"),\n\n12\" sin\n\nwe\n\n(6)\n\n0\"),\n\ntan 0\".\n\n## satisfy the condition that the plane of the\n\ncentre\nof the sun, and they only become\nthe\nthrough\ndistinct or independent of each other when n and n\" are expressed\nin functions of the time, so as to satisfy the conditions of undisturbed\n\norbit passes\n\n## motion in accordance with the law of gravitation. Further, they\n\ninvolve five unknown quantities in the case of an orbit wholly\n\nunknown, namely, n, n\", p, //, and p\" and if the values of n and\nn\" are first found, they will be sufficient to determine p, p , and p\".\nf\n\nTHEORETICAL ASTRONOMY.\n\n170\n\n## determination, however, of n and n\" to a sufficient degree of\n\naccuracy, by means of the intervals of time between the observations,\n\nThe\n\nwill\n\ngeneral,\n\nit\n\nand\n\nafter\n\n(6)\n\np'\n\n## should be approximately known, and hence, in\n\nbecome necessary to derive first the values of n, n\",\nwhich those of p and p\" may be found from equations\n\nthat p'\n\nrequires\n\nBut\n\nelimination.\n\nby\n\n## number of equations will then\n\nquantities, we may combine them in\n\nsince the\n\nunknown\n\n## such a manner as will diminish, in the greatest degree possible, the\n\nIn special cases in which\neffect of the errors of the observations.\nthe conditions of the problem are such that when the ratio of two\ncurtate distances is known, the distances themselves may be deter-\n\n## mined, the elimination must be so performed as to give this ratio\n\nwith the greatest accuracy practicable.\n\n## and second of equations (6), we change the\n\nx from the vernal equinox to the place of the\nsun at the time ', and again in the second, from the equinox to the\nsecond place of the body, we must diminish the longitudes in these\n63. If, in the first\n\n## direction of the axis of\n\nequations by the angle through which the axis of x has been moved,\n\nand we\n\nhave\n\nshall\n\n= n(p cos(A\n+\n\n= n (p sin\n\n=n\n\n00\n\n(A\n\n-f-\n\nGO\nR sin\n\nA)\n\n(A'\n\nR sin\n\n-f-\n\n## - n\" (p\" sin (A\"\n\n== np tan\n\nIf\n\nft\n\nwe multiply\n\nfourth\n\nby\n\np'\n\ntan\n\njf -f-\n\n/?'\n\np'\n\nsin\n\nR' sin\n\n/))\n\nR\" sin\nAO\nri'p\" tan /?\".\n\n'\\\n\n(A'\n\n0),\n\n(7)\n\n'\n\nAO\n\n\"\n\nAO),\n\nequations by tan/9\n\n7\n,\n\nand the\n\n0,\n\nri'p\" (tan\n\nn\"E\"sin(\"\n\n))\n\n-R' sin(0\"-\n\nsin (A'\n\n'\n\nQ))\n00120\nR\" cos(\"\n0),\n\njRcos('\n\n## tan /5\" sin (A'\n\nsin (A\"\n\n0)\n\nOtan/5 -f n/>(tan/5'sin(A\n\ntan/?sin(A'\n\nO;\n\n)tan/S\n\n>'\n\nLet us now denote double the area of the triangle formed by the\nsun and two places of the earth corresponding to\nand f by [RR'~\nand we shall have\n\n\\_RR]\n\n'\n\nsin\n\n('),\n\nand similarly\n\n[RR\" J\n']\n\n## = RR\" sin \" Q\n\n= R'R\" sin(O\" 00(\n\n),\n\nORBIT OF A\nif\n\nThen,\n\nHEAVENLY BODY.\n\n171\n\nwe put\n\"\n\nwe\n\nobtain\n\n## Substituting this in the equation\n\nof p\", the result is\n\n_~ P ^_\n\ntan\ntan\n\nfi'\n\nft\"\n\nsin (A\nsin\n\n')\n\ntan\n\n/?\n\n')\n\ntan\n\np sin (A\"\n\n(A'\n\n(8),\n\nJL\n~ _j^\\\n\"\n\nA^' /tan\n\nsin\n\nsin\n\nft\"\n\n(A'\n\n')\n\ntan\n\nft sin (A\n\nQ)tan/3'\n\n(/'\n\n')\n\ntan\n\n/9\n\ntan\n\np sin (A\"\n\n/5\"\n\nsin (/\n\ntan\n\nft\"\n\nsin\n\n0')\n\n(A'\n\nsin (/\n\n0')\n\n')'\n\n'\n\n')\n\nto\n\n\"R.\n\nWe\n\nmay\n\n0')'\n\n')\n\ntan\n\nJgsin('\n\n,\n\n')\n\ncoefficient\n\n'\n\nand M. \" so as\n\n(11)\n\nto be better\n\n## with the ordinary tables. Thus,\n\nif w' denotes the inclination to the ecliptic of a great circle\npassing\nthrough the second place of the comet and the second place of the\n;\nsun, the longitude of its ascending node will be O , and we shall\n\nhave\nsin\n\nLet\n\n/r\n/?\n\n/9\n\n(A'\n\n= tan\n\n(12)\n\n/?'.\n\n## be the latitudes of the points of this circle corresponding\n\nto the longitudes A\n\nand\n\n/r\n\ntan/3\n\nand we have,\n\n=sin(A\n\nalso,\n\n')tanw',\n\n7\n;\nand sin(A r/\nSubstituting these values for tan/9 , sin (A\n)\n'\nin the expressions for\nand M\", and reducing, they become\n\n\"\nM'=\n\nfln\n\nsin (p>\n\ncos\n\n'\n\n_ ft\")\n\ncos\n\n'\n\n'\n\nC o S ft cos\n\n/5\n\n,_\n')\n\nTHEORETICAL ASTRONOMY.\n\n172\n\nWhen\n\nthe value of\n\n??\n\n## and p\" in terms of known quantities. It is evifrom\n\nequations (14), that when the apparent path of\ndent, however,\nthe comet is in a plane passing through the second place of the\nrelation between p\n\nft\n\nftQ\n\n## and ft\"=ftQ \", we\n\nshall\n\nhave\n\nM =^\nf\n\nand\n=00. In this case, therefore, and also when /9\nft\n\" are\nmust\nrecourse\nto\nsome\nother\nhave\nwe\nftQ\nvery nearly 0,\nft\"\nequation which may be derived from the equations (7), and which\n\nand\n\nrf\n\n## does not involve this indetermination.\n\nIt will be observed, also, that if, at the time of the middle observation, the comet is in opposition or conjunction with the sun, the\n\nvalues of\n\nand\n\nM\"\n\nas given\n\n## by equation (14) will be indeter-\n\nminate in form, but that the original equations (10) will give the\nvalues of these quantities provided that the apparent path of the\nnot in a great circle passing through the second place of the\n\ncomet\n\nis\n\nsun.\n\nQQ\n\nsin (A\n\nsm (A\"_G')'\nHence\n\nit\n\nsin(Q'\n\nQ)\n'\n\nsin\n\n(A\"-\n\nGO\n\n## appears that whenever the apparent path of the body\n\nis\n\nnearly in a plane passing through the place of the sun at the time of\nthe middle observation, the errors of observation will have great\n\n## influence in vitiating the resulting values of\n\nobviate the difficulties thus encountered, we obtain\n\n^sin(O\n\nA')\n\np\"\n\nsin\n\n(A'\n\n\"^'sinCA\"\n^7#'sm(G'\nsin (A\"\n\n## and Jf\"; and to\n\nfrom the third of\n\nA)\nA')\nA')\n\n+\" sin(G\"\n\nA')\n\nA')\n\nWe may also\ntions (7).\n\ncos (A'\n\n## eliminate p between the first and fourth of eqw\n\nIf we multiply the first by tan/9', and the second\nG')i an d add tne products, we obtain\n\n## = n\"p\" (tan p cos (A\"\n\n?i\"E\"tan,S'cos(O\"\n\nnR\nfrom which we derive\n\n## tan P\" cos (A'\n\n0')\n0'))\ntan /5 cos (A'\nG') +^(tan/5'cos(A\n0')\ntan p cos (G'\nR' tan p,\nG)\n\n0')]\n\nDEBIT OF A\ntan\n\ntan\n\nj3\n\n0')\n\ntan\n\n/3'\n\nff cos (A\n\nosO*'\n\nHEAVENLY BODY.\n\n0')\n\ncos\n\n0Q\n\n(A'\n\ncos (/'\n\ntan\n\n0')\n\nI? cos (A\"\n\n173\n\n(16)\n\n0')\n\n## through the second place of the comet and that point\n\nof the ecliptic whose longitude is 0'\n90, which will therefore be\ncircle passing\n\nthe longitude of\n\nits\n\ncos\n\n= tan f\n\nhave\n(17)\n\n/?)\n\n(/3,\n\ntan /',\n\n0') tan/'.\n\nshall\n\nwe\n\nand, if\ncircle\n\n(A'\n\nwe\n\n(/?\"\n\nfrom which\n\nit\n\n/?)\n\ncos\n\n/5\n\ncos\n\n(19)\n\n/?,\n\n/?)\n\nwhen\n\n## the apparent path of the body is in a plane passing through that\n\npoint of the ecliptic whose longitude is equal to the longitude of the\nsecond place of the sun diminished by 90. In this case we may use\n\nequation (11) provided that the path of the comet is not nearly in\nthe ecliptic.\nWhen the comet, at the time of the second observation,\nis\n\nin form,\n\n## fail in this case.\n\nWhen\n\nboth equations (11) and (16) are simultaneously nearly indeterminate, so as to be greatly affected by errors of observation, the\n\n## between p and p\" must be determined by means of equation\n\nwhich fails only when the motion of the comet in longitude is\n\nrelation\n(15),\n\nvery small.\n(15),\n\nand\n\n## It will rarely happen that all three equations, (14),\n\n(16), are inapplicable, and when such a case does occur it\n\nwill indicate that the data are not sufficient for the determination of\n\nis\n\nto\n\nbe\n\n## when the motion of the comet in latitude is considerable, and\n\nequation (15) when the motion in longitude is greater than in latitude.\n\nused\n\nTHEORETICAL ASTRONOMY.\n\n174\n\nThe\n\n## formulae already derived are sufficient to determine the\n\nrelation between //' and p when the values of n and n\" are known,\n64.\n\nand\n\nit\n\ntities.\n\nIf we put\n\nk(t-f)\nt')\n\n## and express the values of x, y,\n\nexpansion into series, we have\n\nz,\n\n=\n=\n\n*',\n\n(20)\n\nr,\n\nx\", y\",\n\nz'\n\nin terms of\n\nx\"\n\nxr\n\ny', z\n\nby\n\n''\n\n'>\n\nWe\n\n## and similar expressions for y, y\", z, and z\".\n\nshall, however, take\nthe plane of the orbit as the fundamental plane, in which case z, 2',\nand z\" vanish.\n\nThe fundamental\nrelative to the\n\nsun\n\na*re,\n\nwe\n\nif\n\nneglect\n\nits\n\nIf\n\nwe\n\n## differentiate the first of these equations,\n\nwe\n\nfind\n\nDifferentiating again,\n\nr'\n\nwe\n\nget\n\n'\n\n~r*~ '~dt~r' 3\n\n~di'\n\n#,\n\nwe\n\nshall\n\nfor\n\n-^ and\n\nd*ii'\n\nputting\n\nfor\n\nORBIT OF\ni\n\n- ^\n\nff\n\nwe\n\nJr\n\nr\" 3\n\nr\"\n\nI_\n\nr \"3\nT\n\nj'i\n\na __\n\nr\"*\n\nA HEAVENLY BODY.\n\n__i_\n\n_|_\n\n/I\n-_\n\nj_\n\n_|_\n\nr\nar\n\n/ /7\n\n'\n\n\\2\n\n_ir_(\n\n//V\nCT r\n\n_|_\n\nr\" 4\n\n. .\n\nr4\n\ndr'\n\nT\n.1\n\n10\n\n__\n\n175\n\nJl\n\n_.i_\nJ\n\n_i_\n\n\"\"\n\n\"*\"\"\n\n..\n\n'\n.\n\nobtain\n\nFrom\n\nthese equations\n\nwe\n\neasily derive\n\n(23)\n\nhe\n\nfirst\n\nmembers of\n\n## iangles formed by the radii-vectores and the chords of the orbit\n\ntween the places of the comet or planet. Thus,\n\n- x'y = [r/],\n\nand x'dy'\n\nf\ny dx\nf\n\ny\"x'\n\n- x\"y' - [//'],\n\n- x\"y =\n\ny\"x\n\n[r/'J,\n\n(24)\n\n## double the area described by the radius-vector\n\nX '^J\ny'ftx:\nia\nduring the element of time dt, and, consequently,\nis\n\n## double the areal velocity.\n\nmass of the body,\n\nin\n\nTherefore\n\nwe\n\nbecome\n\n## shall have, neglecting the\n\nThe\n\norbit.\n\nequations (23),\n\ntherefore,\n\n[r/]\n\n= bk t/>,\n\nSubstituting for a,\n\n[rV]\n6,\n\na /r\n\n= b\"k i/p,\n6 /r their values\n\n[rr\"]\n\n= (ab\" + a\"6)\n\nfrom\n\n(22),\n\nwe\n\nfind, since\n\nTHEORETICAL ASTRONOMY.\n\n176\n\nr\" 3\n\nr\" 2\n\ndr'\n\n.....\n\n-i^-ifes-*\n\n)'\n\n(25)\n\nFrom\n\n## these equations the values of\n\n= [//']\n^\n\nand n ff\n\nLTT J\n\nbe derived\n\nand the\n\n= [r/]\nrr ^\n-^\n\nmay\n\nresults are\n\n(26)\n\nwhich values are exact to the third powers of the time, inclusive.\nIn the case of the orbit of the earth, the term of the third order,\nbeing multiplied by the very small quantity\n\n7-, is\n\nreduced to a\n\n## superior order, and, therefore, it may be neglected, so that in this\n\ncase we shall have, to the same degree of approximation as in (26),\n\n(27)\n\nFrom\n\nn\nthe equations (26)\n\n*\"\nn\n\no*r\n\nfrom\n\n(25), since\n\n+ r\"\n\n-\"\\ 1 (\n\nkr\"\n\n= -[//']\n\nTT.\n\nwe\n\nfind\n\ndr^\n\n\"dt\n\ndr'\n\nof\n\nin the case of\n\nr'\n\nand\n\nj-,\n\nit is\n\n## In the first approximation to\n\nonly by successive approximations.\nthe elements of the orbit of a heavenly body, the intervals between\nthe observations will usually be small, and the series of terms of (28)\nwill converge rapidly, so that\n\nwe may\n\nn'\n\ntake\n\nA HEAVENLY BODY.\n\nCEBIT OF\n\n177\n\nand similarly\n*L ~~\n-1\n\nN\"\nHence the equation\n\nT\"'\n\n(11) reduces to\n\n(29)\n\nIt will be observed, further, that if the intervals between the observations are equal, the term of the second order in equation (28)\n7?\n\nvanishes,\n\n,\nr\nf\\j\n\n= TT\n\nis\n\nthird order.\n\n## whose intervals approach nearest to equality. But if the\n\nobservations available do not admit of the selection of those which\ngive nearly equal intervals, and these intervals are necessarily very\nunequal, it will be more accurate to assume\ntions\n\nn_\n\nn\"\n\nN^\n~ N'\"\n\n## and compute the values of\n\nand N\" by means of equations\nf\nsince, according to (27) and (28), if r does not differ much from\n\n(9),\n_R',\n\nthe error of this assumption will only involve terms of the third\norder, even when the values of r and r\" differ very much.\n\nWhenever\n\n## the values of p and p\" can be found when that of their\n\nwe may at once derive the corresponding values of r\n\nratio is given,\nh\n\nand r\n\nThe\n\n',\n\n## as will be subsequently explained.\n\nvalues of r and T\" may also be expressed in terms of\n\nmeans of\n\nseries,\n\nr'\n\nby\n\nand we have\n\n^'- M\n\ndy\n\nr\"\n\n--*\nr\" 2\n\n## from which we derive\n\nf'\n\ndl>\n- - T + T\" _\n'\n\nrf\n\ndV\n\nk\nneglecting terms of the third order.\n\nTherefore\n\n12\n\n(30)\n\nTHEORETICAL ASTRONOMY.\n\n178\n\nWe\n\nfourth order.\n\nhave, also,\n\nwhich gives\n(31)\n\nTherefore,\n\nwhen\n\nequations,\n\nand\n\nfirst\n\n## -=- are obtained\n\napproxi-\n\nfrom these\n\n-^ may be recomputed\nby means of which the value of ft\n\nfrom equation\n\n(28).\n\nWe also\n\ncompute\n\n_\nN\"~~ JR#Bin(0'\nand\n\n0)'\n\n## substitute in equation (11) the values of\n\nIf we designate by\nwe have\n\nM the\n\n-77\n\nand\n\n-^\n\nthus found.\n\np and p\",\n\n(33)\n\n## In the numerical application of this, the approximate value of p will\n\nbe used in computing the last term of the second member.\nIn the case of the determination of an orbit when the approximate\n^7\n\nn\"\n\nand that of\n\nN\n-^\n\nrr' sin (v\n\nbe computed from\n\n'\n\nv)\n\n## these from (33) will not require\n\nmay\n\n-77\n\nany further\n\nM derived by means of\ncorrection.\n\n65. When the apparent path of the body is such that the value\nof M', as derived from the first of equations\n(10), is either indeterminate or greatly affected by errors of observation, the equations (15)\nand (16) must be employed. The last terms of these equations may\n\n## be changed to a form which is more convenient in the approximations\n\nto the value of the ratio of p\" to p.\nf\nLet\nY\" be the ordinates of the sun when the axis of\ny\n,\n\nY Y\n\nORBIT OF\n\nA HEAVENLY BODY.\n\nwhose longitude\n\nA',\n\nand we have\n\n=R\n=R\n\nY\nY'\n\nsin(Q\n\nis\n\nA'),\n\nsin(O'--A'),\n\nr\"=JB\"sin(0\"\nNow,\n\n179\n\nA').\n\nit\n\n## will be sufficient to put\n\n_N_\n\nn_\n\nn\"~ N \"\nr\n\nand, introducing Y,\n\nf\n,\n\nY\",\n\nit\n\nbecomes\n\noosec\n\nIt\n\nnow remains\n\ntions (26)\n\nWe\n\nwe\n\n~ /}\n\nFrom\n\nfind, to\n\n(r\n\n(35)\n\nhave, also,\n\nand hence\n\n(5),\n\nNY\n\nY'+N\"Y\"=0,\n\nsince\n\nF'\n\n= E' sin(0'\n\n~^^\" sin(r-A'\n\n).\n\n## Hence the equation\n\n5^C^-\n\n/I\n\n;-^\n1\n\n(15) becomes\n\nsn'\n\nsin A\"\n\nA'\n\nTHEOKETICAL ASTKCXNOMY.\n\n180\nIf we put\n\nn\nM = ^\"\n\"\n\n,.,\n\nsin\nsin\n\nn\"\n\n(A'\n\nA)\n\n(jf'-xy\nrJ_,\n\n,,s\n\nsin\n\nR'\n\n0')\n\n(A'\n\n'\n\nsin\n\nA)\n\n(A'\n\nwe have\no\"\nr\n\n= =\nTl/T\nIrJL\n\n/'Q7^\n\nH/T 7?\n-\"*o-^\n\n## Let us now consider the equation\n\n(16),\n\nand\n\nlet\n\n'\n\nus designate by X,\n\nX X\" the abscissas of the earth, the axis of abscissas being directed\nf\n\nis\n\n0', then\n\nX =R cos (0-00,\n=R,\nX\"=R\" cos(\"\nX'\n\n0').\n\n## It will be sufficient, in the last term of (16), to put\n\nn_ ~~\n\nn\"\n\nand\n\nfor\n\nthis\n\nterm reduces to\n\nand\n\nits\n\n'\n\nT\"\n\nif\n\nvalue in terms of\n\nr75\n\nR' z\n\n_N_\n\nN\"\n\ntf\n\n'\n\nThen, since\n\n'\n\n')\n\n0')\n\nwe put\nn\n\ntan/3'cos (A\n/3\" cos (*/\n\n~~n\"' tan\n\nHl rr j^\nF = 1 _i^l\n*\nr\"\nn\n,\n\n'\n\n0')\nQ')\nT ,,x/_L\n;\n\\r' 3\n\n## tan/? cos (A'\n\ntan p' cos (X\"\n\n0')\n0')'\n\n(38)\ntan/3^\n\nE' 3 /tan/3'cos(A\n\n0')\n\nEf_'\n\ntan/3cos(A'\n\n0')\n\nn\n\n(39)\n\nwe first assume\n\nWf= 7\n\nwhen\n\nif\n\n## the elements are\n\nA HEAVENLY BODY.\n\nORBIT OF\n\nJL \"\"\nn\"\nas given by (32),\nand\nthe factors\n\nwhen\n\nN\nN\n\n181\n\n'\n\n## the intervals are very unequal, and neglect\n\nvalues of p and p tf which are thus ob-\n\nThe\n\nf\n.\n\nan approximate value of\n\nr'',\n\nand with\n\n77\n\n## also the value of\n\nthis a\n\nF or F\n\nf\n.\n\nTif\n\nWhenever equation\n\n## (11) is not materially affected by errors of\n\nthe value of\nfurnish\nwill\nwith more accuracy than\nobservation,\nthe equations (37) and (39), since the neglected terms will not be so\nIn general, therefore, it is to\ngreat as in the case of these equations.\n\nit\n\n## be preferred, and, in the case in which it fails, the very circumstance\n\nthat the geocentric path of the body is nearly in a great circle, makes\nthe values of\n\nF and\n\nF'\n\ndiffer\n\nbut\n\n66.\n\nWhen\n\nthe value of\n\nM has\n\nfrom unity,\n\nlittle\n\n## that the apparent path of the body\n\nr' must differ very little from R'.\n\nmay\n\nsince, in order\n\nbe nearly in a great\n\ncircle,\n\n## been found, we may proceed to\n\nbetween p and p ff y the values\n\nrelations\n\n## of the quantities themselves.\n\nThe co-ordinates of the first place of the earth referred to the third,\nare\nx,\ny,\n\nwe\n\n## = R\" cos Q\" R cos O,\n\n= .R\"sin0\" .RsinO.\n\n(AK\\\n\n## by g the chord of the earth's orbit between the places\n\nand third observations, and by G the longitude of the first place of the earth as seen from the third, we shall\nIf\n\nrepresent\n\n## corresponding to the first\n\nhave\nx,\n\n= g cos G,\n\ny,\n\n= g sin G,\n\nand, consequently,\n\nO)\nO)\n\n.#\"cos(O\"\n\"si n (O\"\n\nR = gcos(G\n\nsin\n\nO),\n\n(00).\n\nan\n\nIf\nrepresents the angle at the earth between the sun and comet\nthe inclination to\nat the first observation, and if we designate by\n\nthe ecliptic of a plane passing through the places of the earth, sun,\nand comet or planet for the first observation, the longitude of the\nascending node of this plane on the ecliptic will be O, and we shall\n\n## have, in accordance with equations (81) D\n\ncos\nsin\n\n4/\n\nsin\n\n4*\n\n4/\n\n= cos\n\n/?\n\nw cos\nsin w = sin p\n\ncos\n\nft\nf\n\ncos (A\nsin (A\n\nO),\nQ),\n\nTHEORETICAL ASTRONOMY.\n\n182\nfrom which\n\ntan w\n\ntan 4\n\n= tan cosw O)\n\ntan/9\n\n-r\n\n(42)\n\n(A\n\n## Since cos/9 is always positive, cos^ and cos (A\n\nQ) must have the\nsame sign; and, further, fy cannot exceed 180.\nIn the same manner, if w\" and oj/' represent analogous quantities\nfor the time of the third observation,\n\nv=\n\nta\n\nwe\n\nobtain\n\n^\"\n\nsin (A\"\n\nQ\")'\n\n\"-0\")\ncos4\"=cos/3\"cos(A\"\n\nWe also have\n\n\";\n\n2J.Rcos4,\n\n## which may be transformed into\n\nr2\n\nand\n\n= GO sec\n\nmanner we\n\nin a similar\nr\" 2\n\n= (p\" sec\n\nR cos 4)\n\n/9\n\n-f\n\nR*\n\nsin\n\n(44)\n\nfind\n\nJT\n\n/9\"\n\ncos 4/') 2\n\n+ ^'\n\nsin 4\".\n\nLet K designate the chord of the orbit of the body between the\nand third places, and we have\nx2\n\nBut\n\nR cos O,\njR sin O,\ny = p sin\n= p tan\n2\n\n2/)\n\n*)\n\n/?,\n\n= Mp,\nand, since j0\"\n\n=\nf Mp sin ^'\nx\"\n\n.#\" cos\n\n## from which we derive, introducing g and\n\n= Mp cos A\"\n= Mp sin A\"\nz\" ~z = Mp tan/5\"\n\nGr,\n\na?\"\n\np cos A\n\ny\"\n\np sin A\n\nO\",\n\n^ tan\n\n/?.\n\ny cos G,\ng sin (2,\n\n(45)\nfirst\n\nORBIT OF A\n\nMp\n\ncos A\"\n\nHEAVENLY BODY.\n\np cos A\n\nJKf/3\n\nsin A\"\n\n/>\n\nsin A\n\nJ^f/9\n\ntan /5\"\n\n/o\n\ntan /?\n\nph cos\n\n183\n\nC cos J?,\n\n= ph cos C sin\n= ph sin\n\n(46)\n\n17\",\n\nC.\n\nThen we have\nx\"\n\ny\"\n\n/>/i\n\nz= ph sin\n\nz\"\n\nIT\n\ncos C cos\n\nand\n\nif\n\n= pW\n\n2g ph cos\n\ncos C cos\nx\n\ncos\n\njH\")\n\n^ cos ^)\n\n(/>&\n\nby\n\nget,\n\nwe put\n\nwe have\n\n6r,\n6r,\n\nC.\n\n2\n\n# cos\ng sin\n\n= ph cos C sin H\n\nreduction,\n\nJET )\n\n= cos\n\n# sin\n\n-f-\n\n(47)\n\n(48)\n\n?>,\n\n+g\n\n(49)\n\n<p.\n\n## If we multiply the first of equations (46) by cos A\", and the\n\nsecond by sin /\", and add the products ; then multiply the first by\nff\nsin X , and the second by cos A\", and subtract, we obtain\n\n= M cos (A\"\nA\") = sin (A\"\n\nh cos C cos (H\nh cos C sin (H\n\nA\")\n\n=M tan p\"\n\nh sin C\n\nLet us now put\n\ng sin p\n4\n\n^ cos ^\n\nh cos\n\n/?\n\n(44), (45),\n\nand\n\n(49)\n\n6,\n\n(51)\n\n6\",\n\nc\",\n\nbecome\n\n+ A\\\n\n+ V,\n\nThe\n\n=\n-\n\ng cos ^\nd,\n^ cos y\n\nc,\n\n|oA\n\n/9,\n\nV' =7 5\",\n\n6J? cos\n\n(50)\n\ntan\n\nf and H.\n\n^4,\n\nJ2 sin\n\nJ?\" sin\n\nh,\n\nA),\n\nA),\n\n(52 )\n\n## equations thus derived are independent of the form of the\n\nand are applicable to the case of any heavenly body revolving\naround the sun. They will serve to determine r and r\" in all cases\nin which the unknown quantity d can be determined. If p is known,\norbit,\n\nTHEORETICAL ASTRONOMY.\n\n184\n\nd becomes known\n\nan unknown\n\n## directly; but in the case of\n\norbit,\n\nd may be determined\nthese equations are applicable\nfurnished\nby observation.\ndirectly or indirectly from the data\nonly when p\n\nor\n\n67. Since the equations (52) involve two radii-vectores r and r\"\ntheir extremities, it is evident that an addi-\n\n## tional equation involving these and known quantities will enable us\n\nto derive d, if not directly, at least by successive approximations.\n\nindeed, a remarkable relation existing between two radiichord joining their extremities, and the time of describing\nthe\nvectores,\nIn general,\nthe part of the orbit included by these radii-vectores.\n\nThere\n\nis,\n\nthe equation which expresses this relation involves also the semitransverse axis of the orbit; and hence, in the case of an unknown\n\norbit, it will\n\n## some assumption is made in regard\n\nFor the special case of\nthe\nsemi-transverse axis is infinite, and the resultparabolic motion,\ning equation involves only the time, the two radii-vectores, and the\nfor the determination of\n\nunless\n\nc?,\n\n## chord of the part of the orbit included by these. It is, therefore,\n\nadapted to the determination of the elements when the orbit is supposed to be a parabola, and, though it is transcendental in form, it\nmay be easily solved by trial. To determine this expression, let us\n\n## resume the equations\n\ns-^\n\n= tan J\n\ni\n-j-\n\ntan3\n\n1/20\nand, for the time t\" }\n\nQ=\n~ Bin^y\n\"\n\n3fcy\n\ncos\n\n1/2 q?\nand, since r\n3&(*\"\n\n= q sec\nQ\n\ny=\n\n= sinK^\n\n*0/^\n\ncos 2^\n\njv, this\n\n1/2\n\nBut we have,\n\nv\niz\n\ncoajy\n\nwe\n\n*0\n\nr\\\n'\n\ngives\n\nt;)!/^/\n1\n\n/=\nq\n\n,,\nr-j-r\"+cos(v\n.\n\n./\n\nvJVrr\n\n\\\n}\n\n,_\n(53)\n\n## from the triangle formed by the chord K and the\n\nand r\",\n\nalso,\n\nobtain\n\nx*=r* +\n(r+\n\nr\" 2\n\nfr\n\nr'O\n\n2rr\" cos (v\n4rr\" cos 2 J\n\nv)\nr\n\n(if\n\nv).\n\nPAJRABOLIC ORBIT.\nTherefore,\n\n185\n\nr\n\n_|_\n\n\"\n\n=m\n\n_j_\n\n'\n\n_|_\n\n\"\n\nThen we\n\n_x=^\nshall\n\nhave\n\n+ ''=*(\"*+*),\n\n2 cos\n\n(i;\"\n\ni;)\n\n1/rr\"\n\nmn\n\n## and ?i are always positive, it follows that the upper sign\n\nand, since\nv is less than 180, and the lower sign when\nmust be used when v\"\nv\n\nff\n\n- =\n\n(m-\n\nV2q\n\nNow we\n\nlast\n\n'\n\n## equation with (53),\n\nmn).\n\n(55)\n\nhave\nsin\n\nSquaring\nsin\n\nCombining the\n\nis\n'\n\nor,\n\n180.\n\nv is greater than\n\nthe result\n\nthis,\n\n(i/\n\n## and reducing, we get\n\nv)\n\n(j/'\n\nv)\n\n= cos\n\nintroducing r and\n\n-Jv\n\n+ cos\n\n^v\"\n\n2 cos\n\nf/\n\n^-v\n\nv) t\n\nq,\n\nTherefore,\n\nsn\n\n\"\n\nwe\n\nfind\n\nx, this becomes\n6& (*\"\n\nr,\n\nr\",\n\nand\n\n(56)\n\n## v is less than 180.\n\nThis\nbeing used when v\"\nequation expresses the relation between the time of describing any\nparabolic arc and the rectilinear distances of its extremities from each\n\n## the upper sign\n\nother and from the sun, and enables us at once, when three of these\nquantities are given, to find the fourth, independent of either the\n\nTHEORETICAL ASTRONOMY.\n\n186\n\n## perihelion distance or the position of the perihelion with respect to\n\nthe arc described.\n\n## The transcendental form of the equation (56) indicates that,\n\neither of the quantities in the second member is to be found,\nmust be solved by successive trials ; and, to facilitate these approxi68.\n\nwhen\nit\n\n## mations, it may be transformed as follows\n\nSince the chord x can never exceed r\nr\",\n:\n\nand, since #,\n\nr,\n\nThe value of\n\nFrom\n\nwe\n\npositive, sin\n\nobtain\n\n+ /')*-\n\nCOSY=\nand substituting\n\nfor x\n\nx2\n\nO+r\")\n\nvalue given by\n\nits\n\n4rr\" cos 2\n\n(r -f r\")\n\n(v\"\n\nTherefore,\n\nv),\n\n-y)\n\nwe have\nCOS\n\n/=\n\n2i/Vr\"\nCOS 2\n\nv)\n\n(t/\n\n(58)\n\n-77-,\n\nalso\n\n^n / - ^-^\nHence\n\nit\n\nappears that\n\nwhen\n\n*\n-.\n\nv\"v\n\nis\n\nless\n\n(59)\n\nthan 180,\n\nbelongs to\n\nand\nto\nthe\nsecond\nnegative,\nf belongs\n\nthe\nis\n\nbecomes\n\nthis\n\nand\n\n(57)\n\nthe limits\nand 180.\nbe\nwithin\nmust, therefore,\n\nm/,\n\n+ r\"\n\nwe may put\n\nfirst\n\nIf\n\nwe\n\nintroduce\n\n## into the expressions for\n\nand\n\n7i\n\nthey become\n\n0(1+ sin/),\n\nm*=(r +\n\n^ = (r + r\")(l-sin/),\n\nwhich give\n\n2\nrc,\n\nand, since\n\nis\n\n(r\n\ngreater\n\n-f r\")\n\nthan 90\n\ncos\n\nJ/\n\n2\n,\n\nIF sin\n\nwhen v\"\n\nv exceeds\n\n180,\n\n## equation (56) becomes\n\n6r'\n\n(^cos\n\n^f\n\n-\\-\n\nsin\n\ntff )*\n\n(cos\n\n\\Y\n\nsm jif).\n\nth<\n\nPARABOLIC DEBIT.\n\nFrom\n\nthis equation\n\nwe\n\n187\n\nget\n\n= 6 cos\n\nfir'\n\nsin\n\n2 sin3\n\nor\n\nand\n\nmay\n\nthis, again,\n\nbe transformed into\n\nfr\n\no/sinir'\\\n\nIwr\n\n2t(r+o*\n\n(61)\n\np\nor\nsin\n\n|/\n\n= V 2 sin\n\n#,\n\nand we have\n'\n\n1/2 (r\n\nWhen\n\nv\"\n\nis\n\n= 3 sin x\n\n+\nless\n\nthan 180,\n\n= sin 3x.\n\nsin x\n\nmust be\n\nless\n\n(62)\n\nthan\n\n90, and\n\n## hence, in this case, sin x cannot exceed the value |, or x must be\n\nwithin the limits\nand 30. When v\"\nv is greater than 180,\nthe angle\nis within the limits 90\nand 180, and corresponding to\nthese limits, the values of sin# are, respectively, | and |>/2\nHence,\n\nthe limits 30\n\nff\n\nv exceeds\n\n180,\n\nit\n\nand 45.\n\nThe equation\n=\n\n1/2 (r\n\n= sin 3x\n\n## 3x ; but when the first gives\n\nthere\ncan\n3x\nbe but one solution, the value 180\n15,\nbeing in this case excluded by the condition that 3x cannot exceed\n135. When x is greater than 15, the required condition will be\n\nis satisfied\n\nless\n\nthan\n\nsatisfied\n\nby 3x or by 180\n\n3#,\n\nand there\n\nwill be\n\n## corresponding respectively to the cases in which v\"\n\ntwo\nv\n\nsolutions,\n\nis less\n\nthan\n\nff\nv is greater than 180.\n180, and in which v\nConsequently,\nwhen it is not known whether the heliocentric motion during the\n\ninterval t\"\n\nthan\n\n45,\n\nsolutions.\n\nt is\ngreater or less than 180, and we find 3x greater\nthe same data will be satisfied by these two different\n\nIn\n\npractice,\n\nhowever,\n\nit\n\nis\n\nknown which\n\nof the\n\nTHEOEETICAL ASTEONOMY.\n\n188\n\nt is not\ntwo solutions must be adopted, since, when the interval t\"\nvery large, the heliocentric motion cannot exceed 180, unless the\nand the known circumstances will\nperihelion distance is very small\n;\n\nan assumption\ngenerally show whether such\n\nWe shall\n\nnow put\n\nis\n\n-aj\n\n(63)\n\nand we obtain\nsin 3s\n\n(64)\n\nv 8\n\nWe have,\n\nalso,\n\nsin\n\nand hence\ncos\n\nj/\n\n^/\n\n= i/l\n\n= 1/2 sin\n2 sin 2 x\n\nx,\n\n= I/ cos\n\n1x.\n\nTherefore\nsin\n\nand, since K\n\n(r\n\n+ r\") sin\n\n= 2^\n\np',\n\n(r\n\nwe have\n-f-\n\n^\") sin\n\na;\n\ncos\n\n2a;.\n\nIf we put\n\n3^^^^\n\n(65)\n\nsm3a;\n\n*\n\nFrom\n\n=-\n\n(66)\n\n## appears that ^ must be within the limits (\n\nmay, therefore, construct a table which, with 37 a\nthe argument, will give the corresponding value of /*, since, with\ngiven value of 37, 3# may be derived from equation (64), and thei\n\nand\n\nequation (64)\n\n\\ |/g.\n\nthe value of\n\n//\n\nfrom\n\n69.\n\n## Table XI. gives the values of\n\nfrom 0.0 to 0.9.\n\n(65).\n\nsponding to values of\n\nto\n\nit\n\nWe\n\nr]\n\nIn determining an\n\n/JL\n\ncorre\n\n## orbit wholly unknown, it will be necessary\n\nin regard to the approximate distance oi\n\n## In this case the interval t\"\n\nt will\ngene\nbe\nx\nwill\nbe\nsmall\nrally\nsmall, and, consequently,\ncompared with r\nand r ff\nAs a first assumption we may take r\n2,\n1, or r -f- r\"\n\nand\n\nfi.\n\n1,\n\nand then\n\nfind K\n\nPAEABOLIC ORBIT.\n\n189\n\n## we compute d, r, and r\" by means of the\n\nHaving thus found approximate values of r and r\",\nequations (52).\nwe compute y by means of (63), and with this value we enter Table\n\"With this value of K\n\nXI. and take out the corresponding value of //. A second value\nis then found from\n(66), with which we recompute r and r\", and\n\nfor K\n\nproceed as before, until the values of these quantities remain unThe final values will exactly satisfy the equation (56),\nchanged.\nand will enable us to complete the determination of the orbit.\ntrials the value of r -f r\" may be found very nearly\nfrom the numbers already derived. Thus, let y be the true\nA# be the difference between any\nvalue of log (r -f- r\") an(i\nassumed or approximate value of y and the true value, or\n\nAfter three\n\ncorrect\n\nThen\n\nif\n\nwe denote by yQ\n\n2/-\n\nwe\n\n## from the assumed value yQ)\n\nExpanding\n\n=y+ A\n\nhave\n\nshall\n\nwe have\n\nthis function,\n\nBut, since the equations (52) and (66) will be exactly satisfied\nthe true value of # is used, it follows that\n\n## and hence, when\n\nthe second order,\n\nLet us\n\nb7\n\n2/o>\n\nthen\n\nnow denote\nand\n\n2A/> 2/o\">\n\nwe\n\nshall\n\nis\n\n&y\n\nwe\n\nhave\n\nlet\n\nhave\n\nneglect terms of\n\nshall\n\nEliminating\n\nmay\n\nwhen\n\n= A (y\n\ny\\\n\n## from these equations, we get\n\ny\n\n(a'\n\na)\n\nay\n\na'y\n\n',\n\ntrom which\nf\n\nttft\n\nft\n\nffi7^\n\nTHEORETICAL ASTRONOMY.\n\n190\n\n## Unless the assumed values are considerably in error, the value of\n\nr\") thus found will be sufficiently exact ; but should\n\ny or of log (r\nit\n\nbe\n\nstill\n\nin error,\n\nwe may, from\n\nstill greater\nIn the\naccuracy.\n\n## numerical application of this equation, a and a' may be expressed in\n\nunits of the last decimal place of the logarithms employed.\nt when K is\nThe solution of equation (56), to find t\"\nknown, is\n\n## by means of Table VIII.\n\nThus we have\n\n= sin 3#.\n1/2\nand,\n\nwhen\n\nis less\n\nthan\n\nif\n\n90,\n\nwe put\n\n_\nsi\n\n/v\n\ni-T\n\nwe\n\nget\n\n.\n'\n\nsm /\n\nJ = i 1/2 N sin / (r +\n\nr\") f ,\n\n(68)\n\nor\n\nWhen\n\nr\nf exceeds\n\n90, we put\nN'\n\n= sin\n\n3a?,\n\nand we have\n\n,,\n\n^,\n\n(I\n\nI)\n\nI\");\n\n.__,\n\n'\n\nOy\n\nTV\n\n,,-,\n\nsm | (I\"\n\nI)\n\nThus we have\n\n=\ntan &\"\ntan 6;\nft))\n\n& ) + sin (J\n\ntan t (sin (Z\"\n\nft ))\n\nsin(Z\n\nft)\n\n6,\n\nand, since\n\nsin(r-\n\nft) -f sin\n\nsiiiC/'\n\nft)\n\nthese\n\n(/-\n\nsin (I\n\nft)\n\nft)\n\n## = 2 siny\"+ ^- 2ft) cosi(^- 0,\n\n= 2cos(J\"+ 2Q) sin J(^~ 0,\n^\n\nbecome\n\n## which may be readily transformed into (75).\n\n6\" will be found by means of their tangents\n\nHow ever,\nr\n\nsince 6\n\nand\n\n## in the numerical appli-\n\ncation of equations (71) and (72), if addition and subtraction logarithms are used, the equations last derived will be more convenient\nthan in the form (75).\n\nAs soon as ft and i have been computed from the preceding equations, we have, for the determination of the arguments of the latitude\nu and u\",\ncos\n\nNow we\n\ncos\n\nhave\n\nu\nin\n\nwhich\n\nto\n\nTT\n\nft in\n\n=v\n\n-f-\n\n>\n\n## the case of direct motion, and\n\n13\n\nco\n\n= ft\n\nTT\n\nTHEORETICAL ASTRONOMY.\n\n194\n\n## the distinction of retrograde motion\n\nwhen\nhave\n\n\"U =\n\nis\n\nwe\n\nshall\n\n\"\n\nV}\n\nand, consequently,\n\n=r\n\nx2\nx2\n\n= (r\"\n\n-j-\n\nr\" 2\n\nr cos\n\nWe have,\n\nu),\n\nr2 sin 2\n\n(78)\n\n(79)\n\nw).\n\n(it\"\n\nthis equation\n\n(66).\n\nfurther,\n\n= q sec\n\n))\"+\n\n(t/'\n\n(u\n\nr\"\n\nto),\n\n= q sec\n\n(u\"\n\nw),\n\nor\n\n= COS l(u\n\nVr\n\nBy\n\n(cos\n\n7=-\n\n30*\"\n\no)\n\n(COS J\n\n7= COS 4\n\nT^>\n\n>)\n\nVq\n\n(lt'\n\nto)\n\nVq\nwe\n\nget,\n\n+ cos J(\nCOS J\n\nVi\"\n\n))\n\no))\n\n(l*\n\n= -/==\nVr\n\n-/=,\n\nVr\n\n-?=- cos J (J\n\nBut\n\n(\"\n\n+)-)\n\ncos\n\n^T^v^\nif\n\n- = -7= + 4^'\nf.)\n\nsin\n\nand\n\n(\"\n\n(7r\n\n/T\"\\\nV- \\-'\n::\n\nwe put\ntan (45\n\"\n\nsince\n\nwe\n\nif?\n^|\n\nmuch from\n\n(45\n\n</r\"~\n\n+ 6'}\n|~T\n\n1, 0' will\n\ncot (45\n\n-\\/r =\n4\n\nPARABOLIC ORBIT.\n\n195\n\n--\n\nr\nVq\n\nsin\nr\n\nsin | (M\"\n\n4/\ni/rr\"\n\n)) = - ^-=>\nV\nspp20'\n\n1\n*\n\ntt)\n\ncosHJ(\"+\n\ncos | (w\n\no>\n\nmay\n\nw)\n\nbe found.\n\nr/'\n\nThen we\n\nshall\n\n## have, for the longitude of the perihelion\n\n= +\nwhen\n\nwhen\n\nthe motion\n\nmotion\n\nand the\n\nIt remains\n\nnow to\nV\n\nWith\n\nand\n\nis direct,\n\nunrestricted exceeds 90\nis\n\n=U-\n\ntf'=u\"\n\n>,\n\nU>.\n\nWe have\n\nfind,\n\nby means of\n\n## Table VI., the corresponding values of\n\n(which must be distinthe\nfrom\nused\nto\ndenote the ratio of the\nsymbol\nguished\n\nand\n\ncurtate distances),\nwe shall have\n\nif these\n\nM and M\"\n\n## values are designated by\n\nr-r=\n\nt-T=~,\nm\n\n'\n\n'\n\nor\n\nm\nin\n\nwhich\n\nm=\n\nf and log\n,\n\n9*\n\n= 9.9601277.\n\nWhen v is\n\nnegative, the\n\nis negative.\nThe agreement between the\ncorresponding value of\ntwo values of T will be a final proof of the accuracy of the numerical\n\ncalculation.\n\n## The value of T when the true anomaly is small, is most readily\n\nand accurately found by means of Table VIII., from which we\nderive the two values of ^V and compute the corresponding values\nof T from the equation\n\nT=t\n2\nin\n\nwhich\n\nlogjr,\n\n= 1.5883273.\n\n2\nTN\n\nWhen\n\nis\n\ngreater than\n\n90, we\n\nde-\n\nTHEORETICAL ASTRONOMY.\n\n196\n\nvalues of\n\nfrom the\n\ntable,\n\nT from\n\n## The elements q and may be derived directly from the values\n\nr\", and x, as derived from the equations (52), without first\nfinding the position of the plane of the orbit and the position of the\n71.\n\nof\n\nr,\n\norbit in its\n\nby\n\nown\n\nplane.\n\ntheir values v\n\nCD\n\nsin J\n\nand v\n\n0\"\n\nVq\n2\n-7= cosi\n\n(i,\"\n\nVq\n\n-{-\n\nCD\",\n\n(80), replacing\n\nu and u\"\n\nbecome\n\n11\n\nVr\n\nVr\"\n\n;\n\ntM)\n\n1;)\n\nthese,\n\nsin2 J (v\"\n\nv)\n\nor\n\n_\n~r\" +\nCombining\n\nthis equation\n\n2l/^\n\n77\n\nwith\n\nvj\n\ncos^(v'\n\nf\n\ny)\nV-_rr\"sm^(v'\nr + /'\nxcoty\"\n\n(r\n\nWe have,\n\nfurther,\nx\n\n= ~ sin\n\nfrom\n2\n\n+ r\") sin^',\n2\n\n(v\"\n\nv) cot\n\n(78),\n\n= (/'\n\nr) 2\n\n+ 4rr\" sin\n\n(v'\n\nv)\n\nsmv=\nwe\n\nr\"\n\n=^->\n\nderive\n2l/rr\"~\n(85)\n\n## Therefore, the equation (83) becomes\n\nPARABOLIC ORBIT.\n\n197\n\ng^Kr + r'Ocos'-i/cos'v,\n\n(86)\n\n## by means of which q is derived directly from r, r\", and x the value\n\nof v being found by means of the formula (84), so that cos v is\ny\n\npositive.\n\nWhen\n\nf\nf cannot be found with sufficient accuracy from the equa-\n\ntion\n\nwe may\n\nwhich\n\n*r\n\ngive,\n\nby\n\nThus, we have\n\n+ /' + x\nr + /\n\nl-sm/,\n\n+ r\"\n; + /\n\nx\n,\n\ndivision,\n\ntan (45\n\n+ ir\n\n\"\n\n*=\n\n')\n\nJx rr + r\"r + x*\n\n(87)\n\n-f\n\ntan (45\n\nIn order\n\n+ |v) =\n\n-f-\n\n* x\n\n^r\n\nty.\n\n(/'\n\nr)\n\n(88)\n\n## to derive the values of v\n\nby multiplication,\ntan\n\n## from which v\"\n\nmay\n\nv)\n\n(v\"\n\nit is\nnecessary first\nequations (59) and (85) give,\n\nThe\n\nand v\".\n\n= tan f cos\nFrom\n\nbe computed.\n\nv)\n\n(89)\n\nv,\n\n(82)\n\nwe\n\n= XL\n\nget\n\nL//\n\nIf we put\n\ntan/\n\n= Apl',\n\n(90)\n\ntan i (v\"\n\n+v)=* tan (/\n\ncot | (v\"\n\ntan I (v\" -f v)\neither of\n\n45)\n\n## which may be used\n\n= cot |\n\n\"\n1\n\n(t\n\nff\nto find v\n\nv) sin 2^,\n\n+ v.\n\nv),\n\n(91)\n\nTHEORETICAL ASTRONOMY.\n\n198\n\nFrom\n\nthe equations\ncos |v\n\n~~Vr\n\nV~q\n\ncos %v\" __\n\nV7\n\n'\n\nVq\n\ntne second by\nby multiplying the first by sinjv\" and\nwe\neasily find\ning the products and reducing,\nff\n\ncos \\ (v\"\n\nv) sin\n\nsin \\ (v\n\nv)\n\nsin Jv,\n\nHence we have\n= sin \\v\n1\n\n= COS %V\n\nVq\n\nto\n\n1\n;=,\n\nVr\n\ncompute\n\nq, v,\n\n## and v\" when v\"\n\nis\n\nknown.\n\nff\nand hence v\" and v, have been\n0),\n(v\"\nv) and \\ (v\ndetermined, the time of perihelion passage must be found, as already\nexplained, by means of Table VI. or Table VIII.\n\nWhen\n\nIt\n\nis\n\nr,\n\n## r\" and x have been derived from\n\n,\n\nthe equations (52), instead of completing the calculation of the elements of the orbit, we may find q and T, and then, by means of\nf\nf\nWhen this\nthese, the values of r and v may be computed directly.\nhas been effected, the values of\nthat of\n\nfrom\n\n(34).\n\nthe\n\n(3),\n\nor\n\nfirst\n\nof\n\n## from (33), or, in the\n\nequations (70), and the corrected value of\nIn\ncases\nfrom\nthe\nspecial\nequations (37) and (39).\nthis\n\nway, by successive approximations, the determination of parafrom given data may be carried to the limit of accuracy\n\nbolic elements\n\nwhich\ncase,\n\nmay\nit\n\n## is consistent with the\n\nassumption of parabolic motion. In the\nhowever, of the equations (37) and (39), the neglected terms\nbe of the second order, and, consequently, for the final results\n\n## will be necessary, in order to attain the greatest\n\npossible accuracy,\n\nto derive\n\n(16).\n\nW^hen the\n\nis\n\n## value of If has been found, the\n\ncompleted by means of the formulae\n\nfinal\n\nPARABOLIC CEBIT.\n\n199\n\n## To illustrate the application of the formulae for\n\n72. EXAMPLE.\nthe calculation of the parabolic elements of the orbit of a comet by\na numerical example, let us take the following observations of the\nFifth\n\nAnn Arbor M.\n\n1864 Jan. 10\n\nThese\n\nAnn\n\nArbor:\n\nT.\n\n6* 57 m 20'.5\n\n+ 34\n\n6'\n\n27\".4,\n\n11\n\n54\n\n.7\n\n19 25\n\n2 .84\n\n36\n\n36 52\n\n.8,\n\n16 6 35\n\n11\n\n.6\n\n19 41\n\n4 .54\n\n+ 39\n\n41 26\n\n.9.\n\n13 6\n\nare\n\nat\n\n## apparent equinox of the date and\n\nparallax and aberration by means of\n\ncorrected\n\nfor\n\nBut\n\n## approximate values of these distances are not already known, the\n\ncorrections for parallax and aberration may be neglected in the first\ndetermination of the approximate elements of the unknown orbit of\nif\n\nwe\n\nconvert the observed right ascensions and declinations into the corresponding longitudes and latitudes by means of\n\na comet.\n\nIf\n\n## equations (1), and reduce the times of observation to the meridian\n\nof Washington, we get\nWashington M. T.\n3'\n1864 Jan. 10 7 h 24\n13 6 38 37\n16 7\n1\n54\n\n13\n\n297\n302\n\n53'\n\n57 51\n\n.3\n\n57\n\n39 35\n\n.9,\n\n310\n\n31 52\n\n.3\n\n+ 59\n\n38 18\n\n.7.\n\n7\".6\n\n-}-\n\n55\n\n46' 58\".4,\n\nNext, we reduce these places by applying the corrections for preand nutation to the mean equinox of 1864.0, and reduce the\n\ncession\n\nwe have\n\nt\nt'\n\n\"\n\n= 10.30837,\n= 13.27682,\n= 16.29299,\n\n= 297\n= 302\nA\" = 310\nI\n\n52' 51\".l,\n\n/?\n\nA'\n\n57 34\n\nQ =290\nO'\n\n293\n\nfind,\n6'\n\n27\".4,\n7 57 .1,\n\nto the\n\n/5\"=-j-59\n\n46' 58\".4,\n\n39 35\n\n.9,\n\n38 18.7.\n\n## Q\" = 296 1215.7,\n\nwhich are referred\n\n.4,\n\n31 35.0,\n\n= + 55\n= 57\n\nlog.R =9.992763,\nf\n\nlogE =9.992830,\nlog\n\n#'\n\nmean equinox of\n\n= 9.992916,\n\n1864.0.\n\nIt will gene-\n\n## rally be sufficient, in a first approximation, to use logarithms of five\n\ndecimals ; but, in order to exhibit the calculation in a more complete\n\n## form, we shall retain six places of decimals.\n\nSince the intervals are very nearly equal, we\n\nmay assume\n\nTHEORETICAL ASTRONOMY.\n\n200\n\nL *L\nJL\nri~'~~r\"~ N\"'\nThen we have\ntan i? sin\n\nlf_t'\n~\nand\n\nsin\n\n0)\nQ)\n\n^ cos(\nh cos C cos (IT\nh cos C sin\n\n(#\n\nA\")\n\n## from which to find\n\nJf,\n\nG, g,\n\nH,\n\nand\n\nh.\n\n(;/\n\nit\n\n777\n/9\"\n\n')'\n\nR;\n\nA),\n\ntan\n\n/5;\n\nThus we\n94\n\nobtain\n\n24'\n\n= 40 28\nlog h = 9.688532.\n\n')\n\nA),\n\n= 9.019613,\n=\n= 0.752,\nAr\ncos\nJ!f\n\n(A\"\n\n#=\n\n58' 1\".7,\n\nlog^\n\nSince\n\n= M tan/5\"\n\nM= 9.829827,\n== 22\n\ntan ft sin\n\n== sin\n\n')\n\n')\n\n= R\" cos(Q\" O)\ncos (A\"\nA\") = Jf\n\nh sin C\n\nlog\n\n(A\n\nt' tan/S\"sin(A'\n\nt'\n\n1\".8,\n\n21\n\n.9,\n\n## appears that the comet, at the time\n\nThe\nobservations, was rapidly approaching the earth.\ndeterbe\nare\nin\nA\"\nmust\nwhich\nand\ntaken,\nmined by the condition that g and h cos must always be positive.\nof\n\nthese\n\nThe value of\n\nM should be checked\n\n(97)\n\n),\njfiT,\n\n<p'\n\nfirst\n\nin space,\n\nwe\n\nobtain\ncos <p'= cos\n\nC'\n\ncos\n\nor\ncos\n\nand\n\nthis\n\nif\n\n<p'\n\n= cos\n\ncos\n\nC'\n\nwe put\n\n= sin\n\nsin/\n\ncos/=\n\n215\n\n(7,\n\nand from\nr\n\nwe\n\n= i/\n\n(J\n\nfind\n\nlog r\n\nby\n\n2\n,\n\n= 0.323446.\n\nThen we have\nA\"\n\n=c\n\nT/x 2\n\n<7\n\n2\n,\n\n(r\n\n## from which to find J /r , r /r and\n\nmate elements, we assume\n\nx.\n\nlog J\"\n\n3\n\nWith\n\n1/r\n\nlog p.\n\n= 0.310000,\nwe have\n\n= 0.002706,\n\nlog x\n\n= 0.090511.\n\nA\"\n\nlog J\"\n\n+ r\"\n\nFirst,\n\n= 0.3783,\nthis value of\n\n+ *\")*'\n\nis less\n\nthan\n\nc,\n\n= 0.309717.\ns\n\nthe result\n\nis\n\nlog A\"\n\n= 309717\n\nand\n\n= 0.309647.\n\nThus we have,\n\nJt,\n\n(67),\n\nwe may\n\n310000\n\n283,\n\na'\n\n= 309647\n\nTherefore,\nlog J\"\n\nBy means\n\n## in units of the sixth decimal place,\n\nof this value\nlog r\"\n\nwe\n\n309717\n\nA\" we have\n\n= 0.309624.\n\nget\n\n= 0.052350,\n\nlog *\n\n= 0.090628,\n\n- 70,\n\nTHEORETICAL ASTRONOMY.\n\n216\n\nand\n\nK gives,\n\nthis value of\n\n= 0.309623,\n\nlog J\"\n\nThe\n\nand\n\n= 159\n=\n\n144:\n\n(72), writing\n\n= 0.052348.\n\ncomet are now found from the equaand A\" cos/9\" for p and p\",\n\ncos/3\n\nThus we obtain\n\nrespectively.\n\nI\"\n\nlog r\"\n\ntions (71)\n\nfinally,\n\n43' 14\".2,\n\n17 47\n\nb\"\n\n.8,\n\nThe agreement of\n\n= + 10\n= + 35\n\n50' 14\".0,\n\nlogr\n\n14 28\n\nlogr\"\n\n.7,\n\n= 0.323447,\n= 0.052347.\n\n## Since the helioobtained, proves the accuracy of the calculation.\n\ncentric longitudes are diminishing, the motion is retrograde.\n\nThen from\n\n(74)\n\nwe\n\nget\n\n= 165\n\n17' 30\".3,\n\n= 63\n\n6'\n\n32\".5\n\nand from\ntan(J\n\ntan u\n\nft)\n\nwe\n\nobtain\n\n= 12\n\nthe values of\n\nmotion\n\nis\n\ntan u\"\n\ncos\n\ntan<7\"\n\nu\"\n\nft\n\n= 40\n\n18' 51\".2,\n\n0.090630,\n(79) gives log K\n\nThe equation\n\nft)\n\nr-^4\n\ncos\n\n10' 12\".6,\n\nw and\n\nThe\n\n= 129\n\n6'\n\n46\".3,\n\nlog q\n\n= 9.760326,\n\nv\n\n=u\n\nat\n\n116\n\n56' 33\".7,\n\nFrom\nlog\n\nthese elements\n\n/ = 0.212844,\n\nwe\nv'\n\ntan (?\n\nI'\n\n= 154\n\n88\n\n47' 55\".l,\n\nu'\n\n= 21\n\n59' 12\".3,\n\nfind\n\n=\nQ) =\n\ntan\nget\n\n<o\n\n## T= 1858 Sept. 29.4274.\n\nand from\n\nwe\n\n= u\"\n\nv\"\n\n107\n\nV=\n\n56' 33\".4,\n\n7'\n\ncos\n\n34\".0,\n\ntan\n\ntan\n\nu',\n\nsin (f\n\nV=\n\n+ 19\n\nft),\n\n30' 22\".l.\n\nNUMERICAL EXAMPLE.\n\nBy means\n\nA'\n\n= 137\n\n',\n\nf= + 12\n\n39' 13\".3,\n\nresults\n\n217\nobtain\n\n54' 45\".3,\n\ncos\n\n/?'\n\nA/'\n\nO.\n\nC.\n\n27\".2,\n\nA/5'\n\n23\".7.\n\nFrom\n\n## the relative positions of the sun, earth, and comet at the\n\neasily seen that, in order to diminish these residuals, the\ndistance\nmust be increased, and therefore we assume, for\ngeocentric\n\ntime t\"\n\nit is\n\na second value of J,\nlog J\n\n==?\n\n0.398500,\n\n## from which we derive\n\nH' = 153\n\n= -f 7 24' 26\".l,\n= 0.472115,\nlogr\" = 0.054824,\n\n44' 57\".6,\n\n:'\n\nC= 9.912587,\n\nlog\n\nlog J\"\n\n= 0.311054,\n\nThen we\n\n= 159\n\n40' 33\".8,\n\n= + 10\n\n17 12.1,\n\n6\"\n\n=-+35\n\nand from\n\nh'\n\n## find the heliocentric places\n\nr =144\n\n= 0.488026,\n=\n0.324207,\nlogr\nlog x = 0.089922.\n\nlog\n\nlogc\n\n50'\n\nlogr\n\n8\".6,\n\nlog r\"\n\n837.8,\n\n= 0.324207,\n= 0.054825,\n\nthese,\n\n&\n\n165\n\nM=\na*\n\nT=\nv\n\n15' 41\".l,\n\n12\n\n10 30\n\n= 128\n\n.8,\n\n54 44\n\n.4,\n\n## 1858 Sept. 29.8245,\n\n106 55' 43\".8,\n\nff=\nA'=\n\n= 63\n49\".2,\n= 4Q 13 26\nlog q = 9.763620,\nlog = 0.214116,\n= 21 59'\n= + 19 2931.9,\n^ = + 12 55 2.9.\ni\n\n2'\n\nu\"\n\n154\n\n5332.3,\n137 3939.7,\n\n.0,\n\nr'\n\nu'\n\n0\".6,\n\n6'\n\nC.\ncos\n\nft AA'\n\n==\n\nO.\n\nwe have\n\n^=\n\n1\".5,\n\n6\".l.\n\n## Since these residuals are very small, it will not be necessary to\n\na third assumption in regard to J, but we may at once derive\n\nmake\n\n## the correction to be applied to the last assumed value by means of\n\nthe equations (109).\nThus we have\na!\n\n=_\n\n1.5,\n\na\"\n\n=\nd\n\n27.2,\n\nlog J\n\nd'\n\n6.1,\n\n0.000700,\n\nd\"\n\n23.7,\n\nTHEORETICAL ASTRONOMY.\n\n218\n\nequations give\n\n1050\n4270\n\n25.7z\n17.6a;\n\n105\n\nto the\n\n2.57\n2\n\n(2.57)\n\n== 0.398500\nlog A\n\nWith\n\n## this value of log\n\nillustrated,\n\n0.\n0.\n\nmethod of\n\nleast squares,\n\n+ 0.000106 = 0.398606.\n\nis obtained\n\nSept. 29.88617\n29' QA\" Q\n\nOO\n\n8=165\ni= 63\n\nget\n\nis\n\nT== 1858\n\nwe\n\n+ 427 X 1.76\n+ (1-76)'\n\nHence the\n\n=\n=\n\nWashington mean\n\ntime.\n\n\")\n\n15 24\n\n.8\n\nMean Equinox\n\n1858.0.\n\n2 14.2\n\n= 9.764142\nlog q\n\n## If the distinction of retrograde motion is not adopted, and we regard\n\nto 180, we shall have\ni as susceptible of any value from\n7T\n\n= 294\n= 116\n\n8'12\".7,\n\n57 45\n\n.8,\n\n## the other elements remaining the same.\n\nThe comparison of the middle place with these\ngives the following residuals\n\nC.\n\ncos\n\nA/I\n\nfinal\n\nelements\n\n= -f 0\".2,\n\nO.\nA/9\n\n4\".3.\n\nThese errors are so small that the orbit indicated by the observed\nplaces on which the elements are based differs very little from a\nparabola.\n\nWhen,\nemployed\n\n## instead of a single place, a series of intermediate places is\n\nto correct the assumed value of J, it is best to adopt the\n\n## equator as the fundamental plane, since an error in\n\nboth A and /3; and, besides, incomplete observations\n\nor d will affect\n\nmay\n\nalso be used\n\nNUMERICAL EXAMPLE.\n\n219\n\nwhen\n\n## the fundamental plane is that to which the observations are\n\nFurther, the entire group of equations of condirectly referred.\ndition for the determination of x, according to the formula3 (109),\n\n## must be combined by multiplying each equation by the coefficient of\n\nx in that equation and taking the sum of all the equations thus\nformed as the final equation from which to find x the observations\n}\n\n## being supposed equally good.\n\nTHEOKETICAL ASTEONOMY.\n\n220\n\nCHAPTER\n\nIV.\n\n## DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF\n\nTHE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF\n\nTHE CONIC\n77.\n\nTHE\n\nSECTION.\n\nformulae which have thus far been derived for the deter-\n\n## mination of the elements of the orbit of a heavenly body by means\n\nof observed places, do not suffice, in the form in which they have\nbeen given, to determine an orbit entirely unknown, except in the\nparticular case of parabolic motion, for\nbecomes known. In the general case,\n\n## which one of the elements\n\nit\n\nis\n\nnecessary to derive at\n\n## one of the curtate distances without making any assumption as\n\nto the form of the orbit, after which the others may be found.\nBut,\nleast\n\npreliminary to a complete investigation of the elements of an unknown orbit by means of three complete observations of the body,\nit is\n\nnecessary to provide for the corrections due to parallax and abermay be applied in as advantageous a manner as\n\npossible.\n\nWhen\n\n## the elements are entirely\n\nunknown, we cannot\n\ncorrect the\n\n## observed places directly for parallax and aberration, since both of\n\nthese corrections require a knowledge of the distance of the body\n\nfrom the\n\nearth.\n\nBut\n\nwe may\n\neither\n\n## which the light from\n\nthe body reaches the earth, or we may consider the observed place\ncorrected for the actual aberration due to the combined motion of the\nearth and of light as the true place at the instant when the\nlight left\nthe planet or comet, but as seen from the\nplace which the earth occuWhen the distance is unknown,\npies at the time of the observation.\n\n## the latter method must evidently be\n\napply to the observed apparent longitude and latitude the actual\naberration of the fixed stars, and\nregard this place as corresponding\nto the time of observation corrected for the time of\naberration, to be\neffected when the distances shall have been\nbut\nfound,\nusing for the\nIt\nplace of the earth that corresponding to the time of observation.\nwill appear, therefore, that\nthat\nof\nthe\nof\nthe\ncalculation\nonly\npart\n\nDETERMINATION OF AN ORBIT.\n\n221\n\nelements which involves the times of observation will have to be repeated after the corresponding distances of the body from the earth\nhave been found. First, then, by means of the apparent obliquity of\n\nand declination\nLet A and\ndenote\nthe\nobserved\nand\nlatitude;\n/9\nrespectively,\napparent longitude\nand let O be the true longitude of the sun, 2' its latitude, and jR\nits distance from the earth,\ncorresponding to the time of observation.\nThen, if A and ft denote the longitude and latitude of the planet or\ncomet corrected for the actual aberration of the fixed stars, we shall\nhave\nthe ecliptic, the observed apparent right ascension\n\nmust be converted\n\nand\n\nlatitude.\n\n/?\n\n/?\n\n= + 20\".445 cos\n= 20\".445 sin\n\n(A\n\n(A\n\nsec/5\n\nsin\n\nft\n\n+ 0\".343 cos (A\n0\".343 sin\n\n281)\n281)\n\n(A\n\nsec/9,\n\nm\n^\n\nsin\n\n/5.\n\n## In computing the numerical values of these corrections, it will be\n\nand /? instead of A and ft in the second\nmembers of these equations, and the last terms may, in most cases,\nbe neglected.\nThe values of A and ft thus derived give the true place\nof the body at the time t\n497 78^/, but as seen from the place of\nsufficiently accurate to use X\n\n## the earth at the time\n\nt.\n\nWhen\n\nthe distance of the planet or comet is unknown, it is impossible to reduce the observed place to the centre of the earth ; but if\n\nwe\n\nconceive a line to be\n\ntrue place\n\n## of observation, it is evident that were an observer at the point of\n\nintersection of this line with the plane of the ecliptic, or at any point\nin the line, the body would be seen in the same direction as from the\nactual place of observation.\nHence, instead of applying any correction for parallax directly to the observed apparent place, we may\nconceive the place of the observer to be changed from the actual place\nto this point of intersection\n\nwith the\n\necliptic,\n\nand, therefore,\n\nit\n\nbe-\n\n## comes necessary to determine the position of this point by means of\n\nthe data furnished by observation.\n\nof obser-\n\n## vation, <p the geocentric latitude of the place of observation, and ,o\n\nthe radius of the earth at the place of observation, expressed in parts\nof the equatorial radius as unity.\nr\n\nAB')\n\ncos p.\n\n## It appears, therefore, that a is at least of the third order with\n\nreference to the intervals of time between the observations, and that\n\nan error of the second order in the assumed values of n and n\" may\nf\nproduce an error of the order zero in the value of p as derived from\nequation (11) even under the most favorable circumstances.\nHence,\nin general,\n\nthe values\n\n=T\n\n## omitting terms of the second order, without affecting the resulting\n\nvalue of p f to such an extent that it cannot be regarded even as an\napproximation to the true value ; and terms of at least the second\norder must be included in the\n\nThe\n\nfirst\n\n## equation (28) 3 gives\n\nn\nomitting the term multiplied by\n\n-77,\n\nwhich term\n\n(26) 3\n\n=r\n\n-f-\n\n+\nin\n\nand hence in\n\nr\",\n\"\n\n= l + J.\n\n(27)\n\n## which only terms of the fourth order have been neglected.\n\nthe\n\nfirst\n\nof equations (18)\n\nsec\n\nin which, if\n(26)\n\nand\n\nwe\n\n(27),\n\nmay\n\nterms\n\nwe\n\nis\n\nbe written\n\nNow\n\n=*\n\nTL\n\nand n\n\n+n\n\ntf\n\nas given\n\nby\n\n## only terms of the fourth order with respect to the\n\nTHEOEETICAL ASTEONOMY.\n\n234\n\n## times will be neglected, and consequently the resulting value of p'\n\nan error of the second order when a is of\n\n## Further, if the intervals between the observations\n\nr\" 2 will be a quantity of an order superior\n\n## the third order.\n\n2\nare not very unequal, r\n\nto r\n\n2\n,\n\n## and when these intervals are equal, we have, to terms of the\n\nfourth order.\n\nThe equation\n\n(27) gives\n\n2/3 ( w\n\nHence,\n\nif\n\n_|_\n\nnn\n\n1) == TT\".\n\nwe put\n\nP\nQ=\nwe may\n\nfirst\n\n(29)\n2r'\n\n(n\n\n+ n\"\n\ntervals are\n\naffected\n\nunequal\n\nf\napproximation to the value of p\n\nP=C\nand p r will be\n\n1),\n\n(30)\n\n\",\n\nfirst\n\norder\n\n## but of the second order only when\n\nwhen\n\nthe in-\n\nthe intervals\n\nare equal.\nIt is evident, therefore, that, in the selection of the\nobservations for the determination of an unknown orbit, the intervals should be as nearly equal as possible, since the nearer they\napproach to equality the nearer the truth will be the first assumed\n\nvalues of\n\nP and\n\nand when a\nis\n\nis\n\n## of the greatest importance.\n\nFrom the equations (29) we get\n\n=\nP\\\nn\" = nP;\n\nand introducing\n\nP and Q in\n\n2r' /'\n\n(31)\n\ni?-*\n\n(32)\n\n## This equation involves both\n\np' and r' as unknown quantities, but\nmeans\nof\nanother\nby\nequation between these quantities p may be\nf\n\nfound, after\n\n## which p' may also be determined.\n\nwhich\n\nr'\n\nmay\n\nbe\n\nDETERMINATION OF AN OEBIT.\n\n235\n\ni// represent the angle at the earth between the sun and\nat the second observation, and we shall have, from\nor\ncomet\nplanet\nthe equations (93) 3 ,\n\n82.\n\nLet\n\ntan\n\n= sm\n\n-O\n\ntan\n\nft\n\n-TT-,\n\n(A\n\nTT,\n\n(33 )\n\ncosw\ncos 4/\n\n= cos\n\nft\n\ncos\n\n(/'\n\n## by means of which we may determine ij/, which cannot exceed 180.\n\nSince cos /3' is always positive, cos fy and cos (K\nO ') must have the\nsame sign.\nWe also have\n\nr'\n\nin the\n\n= (p\n\nform\n\nK cos V) + R'\n2\n\nsec ft\n\nsin *',\n\nsin 4'.\n\np' sec ft\nf\n\nFor\n\nVV\n\n== R' cos*'\n\n.R'\n\nits\n\n(34)\n\nwe have\n\n## brevity, let us put\n\nC\n\nand we\n\nshall\n\n^ = R'\n\nWhen\n\n(35)\n\n*o,\n\n-Hft-^t\n\nhave\nkQ\n\n_5J-Ptf\n\nO-TH^P'\n\ncos\n\n1/r'\n\n^sm\n\n^.\n\n(36)\n\nthe values of\nand Q have been found, this equation will\nr' in terms of\nthe\nof\nvalue\ngive\nquantities derived directly from the\ndata furnished by observation.\nshall now represent by z' the\n\nWe\n\n## second observation, and\n\nwe\n\nshall\n\nhave\n\n/ = ^SUH/_\nsmz\n\nTHEORETICAL ASTRONOMY.\n\n236\n\n## Substituting this value of\n\n- R' cos\n\n(k\n\nand\n\nif\n\nr',\n\nsin C\n\nz'\n\n'\n\nn 2\n+ R sin 4' cos = ^-rj7>\n7\n\n4') sin\n\nwe put\nrj Q\n\noi\n\n(38)\n\nz'\n\n=R\n\nsin 4',\n\n(39)\n\nhave, finally,\nsin\n\n(z'\n\nqp C)\n\n=m\n\nsin*\n\nwe\n(40)\n\n2'.\n\n## In order that m may be positive, the quadrant in which f is taken\n\nf\nmust be such that y shall have the same sign as 1Q since sin ty is\n,\n\nalways positive.\n\nFrom\n\n## appears that sin z must always be positive,\n\nand further, in the plane triangle formed by joining\n\nequation (37)\n\norz'<180;\n\nit\n\nthe actual places of the earth, sun, and planet or comet corresponding\nto the middle observation, we have\n\nsn\n\nR sin +\nf\n\n4')\n\n(z\n\nsin\n\nsin 4/\n\nTherefore,\n\nP=\n\nR sin +\n(z'\n\nsin\n\n4')\n\nz'\n\n4')\n\n-cos\n\n## and, since ^o' is always positive, it follows that sin\n\npositive, or that z' cannot exceed 180\nty.\n\nWhen\n\n(z' -f-\n\nmust be\n\nis\n\n## both in the node and in opposition or conjunction with the sun, we\n\nshall have /3'\n180 when the body is in opposition, and\n0, 4^\n\n'vj/\n\nsible to\n\nwhen\n\nin conjunction.\n\nit is\n\ndetermine\n\nr'\n\n## Consequently, it becomes imposf\n\nz\nbut in this case the\n\n-^=\n\n-R' + r',\n\nwhen\n\n## the body is in opposition, the lower sign being excluded by the\n\ncondition that the value of the first member of the equation must be\npositive,\n\nand\n\nfor -J/\n\n= 0,\n\nwhen\n\nthe sun\n\nis\n\n## between the earth and the\n\nDETERMINATION OF AN ORBIT.\nplanet,\n\n237\n\nand the lower sign when the planet is between the earth and\nIt is hardly necessary to remark that the case of an obser-\n\nthe sun.\n\n## vation at the superior conjunction when /3'\n\n0, is physically imposf\nfrom\nthese\nThe value of r may be found\nsible.\nequations by trial ;\nand then we shall have\n\nwhen\n\nthe body\n\nwhen\n\nit is\n\nis\n\nin opposition,\n\nand\n\n## in inferior conjunction with the sum.\n\ncase in which the great circle passing\n\nFor the\n\n## through the extreme\n\nobserved places of the body passes also through the middle place,\nwhich gives a\n0, let us divide equation (32) through by c, and we\n\nhave\nd\n\np sec p\n~~\n\nc\n'\n\n2r'\n\nThe\n\nand\n\nif\n\n'\n\nJ-\n\nwe put\nb\n'\n\nwe\n\nshall\n\nsince c\n\n+p d\nl+p\nc\n\nhave\n\nco\n\nwhen a\n\n0.\n\nHence we derive\n\n'^4-\n\n(42)\n\n^0\n\n## through the three observed places\n\nthe\nsecond\nplace of the sun, both c and C bethrough\nand\nthus\nthe\nsolution of the problem, with the\nindeterminate,\ncircle passing\n\npasses also\n\ncome\n\n83.\n\nThe equation\n\n(40)\n\n## sign, respectively; but it may be shown that of these eight roots at\n\nleast four will, in every case, be imaginary.\nThus, the equation may\n\nbe written\n\nsin\n\nz'\n\nsin\n\nz'\n\ncos\n\ncos\n\nz'\n\nsin\n\nC,\n\nTHEORETICAL ASTRONOMY.\n\n238\n\nbecomes\nand, by squaring and reducing, this\n\nm\nWhen\n\nsin\n\nz'\n\n2m\n\n5\ncos C sin z\n\n## within the limits\n\nis\n\n-\\-\n\n90 -and\nit\n\nsin\n\nz'\n\nsin C\n\n+ 90, cos\n\n0.\n\nwill be positive,\n\n## appears from the algebraic signs of\n\nand,\nbeing always positive,\nthe terms of the equation, according to the theory of equations, that\nin this case there cannot be more than four real roots, of which three\nwill be positive\nand -f 90, cos\n\nWhen\n\n## and one negative.\n\nwill be negative,\n\nand hence, in\n\n90\n\n## this case also, there\n\ncannot be more than four real roots, of which one will be positive\n\n## and three negative.\n\nmust be at least two\n\nit\n\n2\nis real and positive, there\nFurther, since sin\none positive and the other negative\nreal roots\n\nwhether cos\n\nbe negative or positive.\n\nWe\n\nremark\n\nmay\n\nalso\n\n## will only be necessary to solve the equation\n\n=m\n\nsin (z\n\nsin\n\n(43)\n\nz',\n\nsince the lower sign in (40) follows directly from this by substituting\nz' in\n180\nplace of z' and hence the roots derived from this will\n\ncomprise\n\nall\n\n## the real roots belonging to the general form of the\n\nequation.\n\nThe observed places of the heavenly body only give the direction\nin space of right lines passing through the places of the earth and\nthe corresponding places of the body, and any three points, one in\neach of these lines, which are situated in a plane passing through the\ncentre of the sun, and which are at such distances as to fulfil the\ncondition that the areal velocity shall be constant, according to the\nrelation expressed by the equation\n(30) 17 must satisfy the analytical\n\n## conditions of the problem.\n\nIt is evident that the three places of the\nearth may satisfy these conditions ; and hence there may be one root\n\ngive\nP'\n\nFurther,\n\nit\n\nV.\n\n## follows from the equation (37) that this root\n\nmust be\n\nand such would be strictly the case if, instead of the assumed values\nof P and\ntheir exact values for the orbit of the earth were\nand if the observations were referred\nto the centre of the\n,\n\ndirectly\nearth, in the correction for parallax, neglecting also the perturbations\nin the motion of the earth.\n\nDETERMINATION OF AN ORBIT.\nIn the case of the\n\nearth,\n\n_'\n~\n\n\"\n\nin(0\"-- O)'\nsin (0\"\n\nO)'\n\nP and Q become\n\nQ)\n\nin(0'\n\n\"\n\n239\n\n\"\n\nO')'\n\n## sin(Q'- Q) + #ff' sin(Q\"\n\nsn\n\nand\n\nsince the\n\n0Q\n\napproximate values\n\ndiffer but little from these, as will appear from the equations (27) 3 ,\nf\nthere will be one root of equation (43) which gives z nearly equal\nThis root, however, cannot satisfy the physical conto 180\n^/.\n\n## which will require that the rays of light in\n\nthe\nfrom\nplanet or comet to the earth shall proceed from\ncoming\nat a considerable distance from the eye of the\nare\nwhich\npoints\nditions of the problem,\n\nf\nFurther, the negative values of sin z are excluded by the\nr\nnature of the problem, since r must be positive, or z r\n180 ; and\n\nobserver.\n\n<\n\nof the three positive roots which may result from equation (43), that\n<J/, there\nbeing excluded which gives z' very nearly equal to 180\nwill remain two, of\n\n## which one will be excluded\n\nif it gives z\ngreater\none\nthe\nwill\nbe\nthat which belongs\nthan 180\n^', and\nremaining\nIt may happen, however, that\nto the orbit of the planet or comet.\nneither of these two roots is greater than 180\na//, in which case\n\nboth will satisfy the physical conditions of the problem, and hence\nthe observations will be satisfied by two wholly different systems of\nIt will then be necessary to compare the elements comelements.\nf\nputed from each of the two values of z with other observations in\norder to decide which actually belongs to the body observed.\n\n## case, in which cos f is negative, the negative roots\n\nr\nexcluded\nby the condition that r is positive, the positive root\nbeing\nmust in most cases belong to the orbit of the earth, and the three\n\nIn the other\n\n## observations do not then belong to the same body.\n\nHowever, in the\ncase of the orbit of a comet, when the eccentricity is large, and the\nintervals between the observations are of considerable magnitude, if\n\nTHEORETICAL ASTEONOMY.\n\n240\n\n## and Q are computed directly, by means\n\nthe approximate values of\nof approximate elements already known, from the equations\nr/\n\nsin (u\n\nu)\n\n-rV'sin^'W)'\n\nf\n\nrr' sin\n\nit\n\nmay\n\nis\n\n(u'- u}\n\nnegative,\n\n## and the positive root\n\ncomet.\nbelong to the orbit of the\nf\nz shall be very nearly equal to 180\n\nwill actually\ncondition that one value of\n\nThe\n\nand Q shall differ but little from those derived directly\nvalues of\nfrom the places of the earth ; and in the case of orbits of small\nbe fulfilled, unless the intervals\neccentricity this condition will always\ni//,\n\nbetween the observations and the distance of the planet from the sun\nBut if the eccentricity is large, the difference\nare both very great.\nwill correspond to the orbit of the earth.\nroot\nthat\nno\nsuch\nbe\nmay\n\n## may find an expression for the limiting values of m and\n\nwithin which equation (43) has four real roots, and beyond which\nThis change in\nthere are only two, one positive and one negative.\n84.\n\nWe\n\nnumber of\n\nwhen\nwe proceed under\n\n## there are two equal\n\nthe supposition that\nequation (43) has two equal roots, and find the values of m and f\nwhich will accord with this supposition, we may determine the limits\n\nthe\n\n## roots, and, consequently, if\n\nrequired.\nDifferentiating equation (43) with respect to\ncos (z\n\nC)\n\n= 4m\n\nsin\n\nV cos\n\nz',\n\nz'\n\nwe\n\nget\n\nf\nand, in the case of equal roots, the value of z as derived from this\nmust also satisfy the original equation\n\nsin\n\nTo\n\nC) =?=\n\n(z'\n\nsin V.\n\n## m and f which will fulfil this condition,\n\nbetween these equations, we have\n\neliminate\n\nsin\n\ncos\n\n(z'\n\nC)\n\n= 4 cos\n\nz'\n\nsin\n\n(z'\n\nif\n\nwe\n\nC),\n\nsin (2z'\n\n## This gives the value of\n\nC)\n\nj sin C.\n\nin terms of z r for\n\n(45)\n\nwhich equation\n\n(43) has\n\nDETERMINATION OF AN OEBIT.\n\n241\n\nTo\n\n## equal roots, and at which it ceases to have four real roots.\n\nwe have\nthe corresponding expression for\n\n_ sin (z\n\nsin\n\nC)\n\nfind\n\ncos (z\n\nsin\n\nC)\n\nV cos\n\n2''\n\n## which we must use the value of f given by the preceding equation.\n\nf\n1 and -f- 1, the\nNow, since sin (2z\nf ) must be within the limits\nof\nsin\nbe\nand\nwill\nor\nmust\nvalues\nbe within the\nf\nf>\nf\nlimiting\nlimits -f 36 52'.2 and\n36 52'.2, or 143 7'.8 and 216 52'.2. If\n\nin\n\nnot contained within these limits, the equation cannot have equal\nroots, whatever may be the value of m , and hence there can only be\nis\n\ntwo\n\nreal roots, of\n\n## which one will be positive and one negative.\n\nwe compute z f from equation (45), and\n\nthis z\n\n',\n\nor\n'\n\nsin(2z\n\nwe may\n\nC)\n\n| sinC,\n\nthe values\n\nThe\n\nIf\ncall\n\n## within which equation\n\nf\nwill be satisfied by\n\nequation for z\n\n2<-C,\n\n- (2*; -0;\n\n180\n\n## and hence there will be two values of m which we will denote by\n\nand m2 for which, with a given value of\nequation (43) will\nhave equal roots. Thus we shall have\n,\n\n4\nsin z\n\n'\n\nin place of z ',\n90\n(V\n)\n\n(2zQ\n\nof 2z\n\nf\n,\n\nor\n\n## It follows, therefore, that for any given value of , if\n\nis not\nwithin the limits assigned by the values of m^ and m2 equation (43)\nwill only have two real roots, one positive and one negative, of\n,\n\nwhich the\n\nlatter is\n\nBut\n\nP and Q differ\n\nformer\n\nearth.\n\nso\n\n## may belong to the orbit of the\n\nmuch from their values in the case\n\n## is not very nearly equal to 180\n\n36 52'.2 and\nexceeds the limits\n\ni//,\n\n36\n\nif\n\n52 r .2,\n\nwhen\n\n## may actually satisfy\n\nthe conditions of the problem, and belong to the orbit of the body\nobserved.\n16\n\nTHEOEETICAL ASTEONOMY.\n\n242\n\nWhen\n\nthree\nand\none positive\nnegative, if\n\nis\n\nlimits m^\n\ntwo\n\nand\n\nbut, if\n\nis\n\nwithin the\n\nreal roots.\n\n## Table XII. contains for values of\n\nthe values of\n\nand\n\n2,\n\nand\n\n52'. 2 to\n\n+ 36\n\n52'.2\n\nml and m2\nwhich equation (43) has three positive roots and\nthe value of m must be within the limits indicated\n\ncorresponding respectively to\n\nIn every\n\n36\n\nfrom\n\n.\n\ncase in\n\none negative\n\nroot,\n\n## by m and w2 and the values of z will be within the limits indicated\n\nby the quantities corresponding to m l and m2 for each root, which\nwe designate respectively by z/, z.2 zB f and /. The table will show,\nfrom the given values of m and 180\ntj/, whether the problem\nadmits of two distinct solutions, since, excluding the value of z ,\nwhich is nearly equal to 180\nij/, and corresponds to the orbit of\nthe earth, and also that which exceeds 180, it will appear at once\nwhether one or both of the remaining two values of z' will satisfy\nthe condition that z shall be less than 180\nThe table will\n^'.\nalso indicate an approximate value of z\nmeans\nof which the\nby\nf\n\n## equation (43) may be solved by a few trials.\n\nFor the root of the equation (43) which corresponds to the orbit\nof the earth, we have p f\n0, and hence from (36) we derive\n\nwe have\n\n## and, since p must be positive, the algebraic sign of the numerical\n\nf\nvalue of 1 will indicate whether r f is greater or less than\nIt is\neasily seen, from the formulae for lw 6, c?, &c., that in the actual\n\n## application of these formulae, the intervals between the observations\n\nnot being very large, 1Q will be positive when\nft'\nft and sin (O'\nK)\nhave contrary signs, and negative when ft'\nhas\nthe same sign as\nft\nsin\nless\n\n(O'\nthan\n\nK).\n\n## Hence, when O'\n\nis\n\nless\n\nthan 180,\n\nr'\n\nmust be\n\nf\nr\nif /9\nft'\npositive, but greater than\nft is\n;\nf\nWhen\nexceeds\nr\nthan\nR'\nwill be greater\nnegative.\n180,\nr\nif /?'\nis\nand\nless\nthan\nR' if /9\n/9\npositive,\nft is negative.\nmay, therefore, by means of a celestial globe, determine by inspection\nwhether the distance of a comet from the sun is\ngreater or less than\n\nif\n\nis\n\nWe\n\nDETERMINATION OF AN ORBIT.\n\n243\n\n## that of the earth from the sun.\n\nThus, if we pass a great circle\nthrough the two extreme observed places of the comet, r' must be\ngreater than R' when the place of the comet for the middle observa-\n\n## on the same side of this great circle as the point of the\n\nwhich\nBut when the\necliptic\ncorresponds to the place of the sun.\nmiddle place and the point of the ecliptic corresponding to the place\nof the sun are on opposite sides of the great circle passing through\nf\nthe first and third places of the comet, r' must be less than\ntion\n\nis\n\n85.\n\nP=\n\nFrom\n\nT\"\n\nand\n\nQ = TT\", we may\n\n## values of these quantities, and thus, by a repetition of the calculaTo\n\ntion, make a still closer approximation to the true value of p'.\n\n## derive other expressions for\n\nand Q which are exact, provided that\nr f and p f are accurately known, let us denote by s\" the ratio of the\nf\nsector of the orbit included by r and r to the triangle included by\nthe same radii-vectores and the chord joining the first and second\nf\nr/\nplaces ; by s the same ratio with respect to r and r , and by s this\n\nwith respect to r f and r\". These ratios s, s', s\" must necessarily be greater than 1, since every part of the orbit is concave\nratio\n\nAccording\n\nand\n\ntherefore\nf\n\ns\"[rr\n\nwe\n\nwe have\n\nfor the\n\n## mass of the body,\n\nobtain\n\n]=r\"- /p,\n\ns'\n\n[r/']\n\nr'\n\nyft\n\n[//']\n\n= T j/p.\n\n(46)\n\nThen, since\n\n_=\nwe\n\nshall\n\nhave\nr\n= -T\n\ns'\n\nff\n\n= r\"\n-.TO\n\n,._\n\ns'\n\n(47)\n\nn)\n\nand, consequently,\n\nP=\n\nSubstituting for\n\ns, s',\n\nand s\"\n\ntheir values\n\nfrom\n\n-'\n\n(46),\n\nrr\"\n\nwe have\n\nTHEORETICAL ASTRONOMY.\n\n244\n\n## distance between the perihelion and node being denoted\n\nthe polar equation of the conic section gives\n\nThe angular\nby\n\nto,\n\n=1\nr\n\n-f-\n\n4r = 1 +\n=1+\n\n(u\n\ne cos\n\n(u\n\nw),\nf\n\n(50)\n\n>),\n\ne cos (u\"\n\nIjf\n\n01).\n\nrf\nof these equations by sin (u\nu'\\ the second\n1\nthe products\nand the third by sin (u\nu),\n\nIf we multiply the\n\nfirst\n\nff\n\nsin (u\nu),\nby\nand reduce, we get\n\n- sin (u\"\n\ne cos\n\n+ 4 sin (u\nu) + sin (u'\n\n^ sin (u\"\n\nu')\n\nsin (u\"\n\n= sin (u\"\n\nu)\n\nw)\n\nu)\n\nu')\n\nand, since\nsin (u\"\n\n= 2 sin j (u\"\n\nu')\n\nsin (u\"\n\nsin (u\n\nu)\n\nthe second\n\nmember\n\nwe\n\nshall\n\nu') sin\n\nsin\n\nu'\\\nu') cos ^ (u\"\n\n+ u'\n\n2w),\n\n|-\n\nu) sin J (u'\n\n(u\"\n\nu).\n\nhave\n\n(u\"\n\nreduces to\n\n4 sin ^ (u\"\nTherefore,\n\ncos\n\nu'')\n\nu) = 2 sin A (u\"\n\nu'^\n\ntyi\n\nrr\n\nsm ( u\"\n\nu) sin J\n\nu^\n\n_[_\n\nrr\n\nu)\n\n(u'\n\ngm (yf\n\nuy\n\nby\n2rr'r\" cos J (u\"\nit\n\nw') cos\nr/\n\n[r'r\"]\n\n[ r/']\n\n-|\n\n],\n\n[rr\n\n],\n\n(u\"\n\nVL)\n\ncos J (u\n\nu),\n\nand [rV r/ ],\n\n[ rr 'j\n\n1\n'\n\n[rV']-h[rr']\n\n[rr\"]\n\n2rrV' cos\n_p\n\n(u\"tf)\n\n86. If\n\nderive\n\nM)'\n\nr\"\n\nf^l^\n\n'\n\nss\"\n\nM) cos J (*'\n\nrr\"\nf\\\n\ncos J (M\"\n\nwe compare\n\nu'} cos\n\n(u\"\n\n11)\n\ncos J (u\n\nw)'\n\n## the equations (47) with the formula (28) 3 ,\n\nwe\n\nDETERMINATION OF AN ORBIT.\nConsequently, in the\n\nfirst\n\napproximation,\n\nwe may\n\n245\n\ntake\n\nIf the intervals of the times are not very unequal, this assumption\nfrom the truth only in terms of the third order with respect\n\nwill differ\n\nto the time,\n\nshown.\n\n## Hence, we adopt for the\n\nfirst\n\napproximation,\n\nthe values of r and r\" being computed from the uncorrected times\nof observation, which may be denoted by tw </> and tQ \". With the\nand Q thus found, we compute r', and from this p', p,\nvalues of\nand p\", by means of the formulae already derived.\n\nThe heliocentric places for the first and third observations may\nnow be found from the formulae (71) 3 and (72) s and then the angle\nu\"\nu between the radii-vectores r and r\" may be obtained in\n,\n\n## various ways, precisely as the distance between two points on the\n\ncelestial sphere is obtained from the spherical co-ordinates of these\npoints.\n\nWhen u\"\n\n/\n\ntf\n\nsin (u\nr\n\nsin (u\n\n## results the ratios s\n\nand\n\nu)\n\n//\n\nsin (u\n\ntt),\n\n= n'V\nsin (u\"\nr\n\n(53)\nu),\n\nsum of u\"\n\nr/\n\nu may be computed.\nbe\ncomputed, and then\nmay\nand Q. The value of u\"\n\nFrom\n\nand u f\n\napproximate values of\ntaking the\n\nft i\n\nt~\\\n\nu)\n\nu'\n\nand\n\nu'\n\nas derived\n\nthese\n\nfrom\n\nfound by\n(53), should\n\nu,\n\n## agree with that used in the second members of these equations,\n\nwithin the limits of the errors which may be attributed to the\nlogarithmic tables.\n\nfrom l n\ndirectly\n\nfrom\n\n&\n\nI,\n\nand\n\ni,\n\n',\n\nis\n\n6,\n\n## obtaining the angles between\n\nto find the position of the plane of the orbit\n\n## and b ff and then compute\n\n,\n\naccording to the\n\nfirst\n\nu and u\"\nf\n\nu,\n\nof equations (82)^\n\ndirectly\nIt will be\n\nr\nexpedient also to compute r', V and b from p', ^', and /?', and the\nagreement of the value of r', thus found, with that already obtained\n\nfrom equation\n\n(37), will\n\n## check the accuracy of part of the numerical\n\nTHEOKETICAL ASTKONOMY.\n\n246\n\n## Further, since the three places of the body must be in\n\nand Q are\na plane passing through the centre of the sun, whether\nexact or only approximate, we must also have\ncalculation.\n\ntan\n\nb'\n\n= tan\n\nb'\n\ni sin\n\nQ,\n\n(f\n\n),\n\n## derived from this equation must agree with that\n\ncomputed directly from p', or at least the difference should not exceed\nwhat may be due to the unavoidable errors of logarithmic calculation.\n\nWe may now\n\n_rV'siny-tQ\n~\nrr\" sin (u\"\n\ndirectly\n\nrr'\n\nsm(u'-u)\n\n'\n\nu)\n\nu,\n\nu f and u\"\n,\n\nare those\n\nwhich\n\n.\n'\n\nu)\nresult\n\nfrom the\n\n## the resulting values of n and n\" will\n\n,\nonly satisfy the condition that the plane of the orbit passes through\nIf substituted in the equations (29), they will\nthe centre of the sun.\n\nassumed values of\n\nand\n\n## and Q, from which they\n\nonly reproduce the assumed values of\nhave been derived, and hence they cannot be used to correct them.\nIf, therefore,\n\n## the numerical calculation be correct, the values of n\n\n(54) must agree with those derived from equa-\n\n## tions (31), within the limits of accuracy admitted\n\nby the logarithmic\n\ntables.\n\n## The differences u\"\n\nu' and u'\nu will usually be small, and\nhence a small error in either of these quantities may considerably\naffect the resulting values of n and n\"\nIn order to determine\n'.\n\n## whether the error of calculation is within the limits to be expected\n\nfrom the logarithmic tables used, if we take the logarithms of both\n\n## members of the equations\n\nn\", and u' to vary, we get\n\nd log e n\"\n\n(54)\n\nand\n\n~ cot\n= -f cot\n\ndifferentiate,\n\nsupposing only n,\n\nu')du' t\nu) du'.\n\n(it\"\n\n<>'\n\n## Multiplying these by 0.434294, the modulus of the common system\n\nof logarithms, and\nexpressing du' in seconds of arc, we find, in units\nof the seventh decimal place of common\nlogarithms,\n\nd log n\nd log n\"\nIf\n\nwe\n\n=\n=--\n\n21.055 cot\n\n(u'\n\ni//)\n\ndu',\n\nu) du'.\n\n## n and log n\" as\n\nand the values already obtained by\n\nlog\n\n## found from the equations\n\n(54),\n\nDETERMINATION OF AN ORBIT.\nmeans of\n\n(31), the\n\n247\n\n## two resulting values of did should\n\nagree,\n\nand the\n\nf\nmagnitude of du itself will show whether the error of calculation\nexceeds the unavoidable errors due to the limited extent of the\n\n## When the agreement of the two results for n\n\nlogarithmic tables.\nis in accordance with these conditions, and no error has been\n\nand n\"\n\nin\n\nP and Q by\n\n## tions (31), the accuracy of the entire calculation, both of the\n\nquanand of those\nwhich depend on the assumed values of P and\nwhich are obtained independently from the data furnished by observatities\n\ntion, is\n\ncompletely proved.\n\nP and Q, from\n\n## r f , u, u r , &c. have been computed, it is evidently necessary\n\nto compute the values for a second approximation by means of the\n\nwhich\nseries\n\ns\".\n\nTJ\n\n## given by the equations (26) 3 or by means of the ratios s and\n\nexpressions for n and n\" arranged in a series with respect\n,\n\nThe\n\nwith respect to\n\nt,\n\n## unknown, and cannot be conveniently\n\ndetermined, it is plain that if the ratios s and s\" can be readily found\nfrom r, r', r\" u, u' u\", and r, r', r\", so as to involve the relation\nbetween the times of observation and the places in the orbit, they\nmay be used to obtain new values of P and Q by means of equations\nand, since these are necessarily\n\n(48)\n\nand\n\n',\n\n(51), to\n\nM=E\n\nesinE,\n\nor\n\nk(tT}\n^\n-=E\nJ\n\nesmE,\n\na*\n\nand\n\n## also for the third place\n\na?\nSubtracting,\nll\naa\n\nwe\n\nget\n\n= E\" E\n\n2e sin i\n\n(E\"\n\nE)\n\ncos\n\n(E\"\n\n+ E).\n\n(55;\n\nThis equation contains three unknown quantities, a, e, and the difE. We can, however, by means of expressions inference E\"\n2\ne ),\na (1\nvolving r, r\", u, and u\", eliminate a and e. Thus, since p\n\nwe have\n\n## & aVl^^ (E\" E2esmi (E\"\n\nE)\n\ncos\n\n.J\n\n(E\" + E)).\n\n(56)\n\nTHEORETICAL ASTRONOMY.\n\n248\n\nFrom\n\nthe equations\n\nVr sin\n\n-y\n\n= I/a (1 +\n\ne) sin\n\ne) cos ^-E ,\n\ni/o (1\n\nv\"v = u\"\n\nsince\n\n7\n1/rT sm\n\nand\n\nwe\n\nu,\n\n-j-\n\nV~r cos Av\n\nI/a (1\n\n%E,\n\ne) sin\ne) cos\n\n?) tan W',\n\n(90)\n\n## to find E, E', and E\".\n\nThe difference E\"\nshould\nfrom\nthat\nderived\nwithin\nthe\nwith\nlimits\nof\nequation (84)\nagree\nafforded\nthe\ntables.\nto\nfind\nthe\nmean\nThen,\nby\naccuracy\nlogarithmic\nanomalies, we have\n\nfrom which\n\nM =E\n\nM'\n\nand, if\n\nwe\n\ndenotes the\n\n=E'\n\nesmE,\nesmE',\n\n## mean anomaly corresponding\n\n(91)\n\nto\n\nany epoch\n\nT,\n\nhave, also,\n\nf\nin the application of which the values of t, t , and t\" must be those\nwhich have been corrected for the time of aberration. The agree-\n\nTHEOEETICAL ASTRONOMY.\n\n262\n\n## ment of the three values of\n\nthe entire calculation.\n\nand\nare exact,\nfinal values of\n\nIf the\n\n## proof will be complete within the limits of accuracy admitted\n\nthis\n\nby the logarithmic\n\ntables.\n\nWhen\n\n## the eccentricity is such that the equations (91) cannot be\n\nsolved with the requisite degree of accuracy, we must proceed according to the methods already given for finding the time from the perihelion in the case of orbits differing but little from the parabola.\n\n## X. will be employed. As soon as\n\nand v have been determined, we may find the auxiliary angle\nv, v\nV for each observation by means of Table IX. and, with V as the\nM\" (which are not the mean anomaargument, the quantities Jf,\nThen, the perihelion distance\nlies) must be obtained from Table VI.\nFor\n\nthis purpose,\n\nff\n\nwe\n\nT+~e'\n\nhave\n\nshall\n\nC\nin which log\n\n= 9.96012771\n\ntime of\n\n## The times t y t', t\" must be those which have\n\nperihelion passage.\nbeen corrected for the time of aberration, and the agreement of the\nthree values of\n\nT is\n\n## a final proof of the numerical calculation.\n\nused, as soon as the true anomalies have been found,\nthe corresponding values of log\nmust be derived from\nand log\n\nIf Table X.\n\nis\n\nThen w\n\nthe table.\n\nand similarly\n\nM, M\n\nf\n,\n\nM\"\n\nfor w'\n\nis\n\ncomputed from\n\nFinally, we have\n\nT=t\nCIV\"\n\nA (1 + 9e)\n\nQl/ T\\> (1\n\n-j-\n\n9e)\n\nQl/A (1 + 9e)'\n(93)\n\n## for the time of perihelion\n\npassage, the value of <7 being the\nin (92).\n\nWhen\nQ and\n\nCD\n\nthe orbit\n\nis\n\na parabola,\n\n## can be derived from\n\nr,\n\n=1\n\nr\",\n\ntt,\n\nsame\n\nas\n\nand p\n2q, and the elements\nand u\" by means of the equa-\n\nDETERMINATION OF AN OEBIT.\n\n263\n\ntions (76), (83), and (88), or by means of the formulae already given\nfor the special case of parabolic motion.\n\n92. Since certain quantities which are real in the ellipse and parabola become imaginary in the case of the hyperbola, the formulae\nn\nalready given for determining the elements from r, r ', u, and u\"\n\nwhen\n\n## and x' have been found, p, e, and w may be derived from\n\nequations (83) and (87) or (88) precisely as in the case of an elliptic\n2\nfr\nSince x\nsin J (E\norbit.\nE\\ we easily find\n\nWhen\n\ns'\n\nsin i\n\nand equation\n\n(85)\n\nE)=2 V x'\n\n(E\"\n\nx'\\\n\nbecomes\n\"\n\n)VW\n-.\n\n(94 )\n\nx'\nx n will be\nnegative, and hence\nimaginary ; and, further, comparing the values of p in the ellipse\n2\ntan ^/, or\nand hyperbola, we have cos 2 ^\n\nBut\n\nxf\n\nin the hyperbola\n\nis\n\ncos\n\n<f>\n\n=\n=V\n\nif\n\nis\n\n(p\n\n1 tan\n\n4/.\n\nbecomes\n\na tan 2\nwe have\nThen, since p\n\n^\n^\ntan 4 =\n\nobtained.\n\n^ may\n\n(96)\n\natan^/\n\nij/,\n\nbe\n\ncomputed from\n\n## = sec 4 = 1/1 +tan 4\n\n2\n\nshould agree with that derived from equation (88). When e differs\nbut little from unity, it is conveniently and accurately computed\n\nfrom\ne\n\n=1\n\n2 sin 2\n\n-f-\n\n^ sec\n\n4.\n\n## The value of a may be found from\n\na\n\n=p cot 4 = (atan-4/)\n\n,\nQ(97)\n\nTHEORETICAL ASTRONOMY.\n\n264\nor\n\nfrom\na\n\n## 16s' rr\" cos\n\n(u\"\n\nu)\n\nx'J\n\n(a/\n\nderived directly from (89), observing that the elliptic semitransverse axis becomes negative in the case of the hyperbola.\nAs soon as to has been found, we derive from u, u', and u\" the\n\nwhich\n\nis\n\n## and v\", and then compute the values\n\ncorresponding values of v, v',\nthe formula (57)! ; after which, by\nof\nmeans\nF\"\nand\nof\nby\n9 F'j\nf\nand\nthe\n,\nmeans of the equation (69) w\ncorresponding values of N,\n\nN\"\n\n## Finally, the time of perihelion passage will\n\nwill be obtained.\n\nbe given by\n\nT= t\n\nlQ k\n\nIJc\n\nIJc\n\n7.87336575.\n\nwherein log^fc\n\nThe\n\n-*N=t-N> = t'-N\"\n\n## do occur the eccentricity will not differ\n\nbola, so that the\n\nand\n\nin most of those\n\nmuch from\n\nwhich\n\n## To illustrate the application of the principal for93. EXAMPLE.\n\nmula which have been derived in this chapter, let us take the following observations of Eurynome\n\nAnn Arbor M.\n\n@o\n\nT.\n\n21 9 46 18 .0\n28\n\n8 49\n\n29\n\n.2\n\n<?\n\n+9\n\n0- 44'.91\n\n53' 30\".8,\n\n57\n\n3 .57\n\n13\n\n.5,\n\n52\n\n18 .90\n\n+8\n\n22\n\n.7.\n\nThe apparent\n\n## obliquity of the ecliptic for these dates was, respect//\n\nr\n27'\n20\".75, 23 27 20\".71, and 23 27' 20 .65 ; and, by\nively,\nmeans of these, converting the observed right ascensions and declina-\n\n23\n\nAnn Arbor M.\n\nand\n\nT.\n1\n\ndates\n\nwe\n\nget\nLatitude.\n\nLongitude.\n\n21\n9 46 18 .0\n28 8 49 29 .2\n\nlatitudes,\n\nwe\n\n17\n\n47' 37\".60\n\n16\n\n41 36 .20\n\n-f 3\n2\n\n15\n\n16 56 .35\n\n-f 2\n\n8'\n\n.46,\n\n32 42\n\n.98.\n\n## the following places of the sun\n\n43\".19,\n\n52 27\n\nAlmanac\n\nNUMERICAL EXAMPLE.\n\n265\n\nTrue Longitude.\n1' 42\".l\n172\n\nLatitude.\n\n- 0.07\n\n0.0022140,\n\n178\n185\n\n+ 0.77\n+ 0.67\n\n0.0013857,\n0.0005174.\n\n37 17\n26 54\n\n.2\n.8\n\nlog\n\n## Since the elements are supposed to be wholly unknown, the places\n\nof the planet must be corrected for the aberration of the fixed stars\nThus we find for the corrections to be\nas given by equations (1).\napplied to the longitudes, respectively,\n\n-18\".48,\n\nand\n\n-19\".49,\n\n20\".8,\n\n+ 0\".14.\n\n+ 0\".30,\n\no\".47,\n\n_j_\n\nWhen\n\n## these corrections are applied, we obtain the true places of the\n\nplanet for the instants when the light was emitted, but as seen from\nthe places of the earth at the instants of observation.\n\nNext, each place of the sun must be reduced from the centre of\nthe earth to the point in which a line drawn from the planet through\nFor this\nthe place of the observer cuts the plane of the ecliptic.\npurpose\n\nwe\n\nhave, for\nr\n\nAnn\n\n= 42\n\nArbor,\nlog PQ\n\n5'.4,\n\n= 9.99935\n\nsidereal time\n\nQ\n\n= 3* 29 W 1\n\n'\n\ns\n,\n\n21* 48\n\n17 s ,\n\nOf\n\nis\n\n=+\n\n#=\n\n6033'.9,\n22 25.0,\n\nAQ\n\nA 0'\n\n18\".92,\n\nA log RQ\n\nwe\n\nwhich\n\n<p\n\n=\n\" =\n\nwe\n\n\"\n\n0'.4,\n\n15.8,\n\n&\n\n342\n\n59'.2,\n\n\"= + 53\n\n41.6.\n\nobtain\n\n36\".94,\n\nA log R\n\n'\n\nA 0\"\n\n25\".76,\n\n0.0002201,\n\n0.0002796.\n\n## time, we have the values -f OM5,\n\nwhich are so small that they may be neglected.\n\n.34,\n\n(4),\n\n0.0001084,\n\nA log jR\n\n347\n50\n\nV= +\n\nthese\n\nzenith, of\n\n55 s\n\n4,=\n\n-j-\n\nFrom\n\n## in each case the declination.\n\nand\n\n= 21* 18\n\n+ O'.28, and\n\nTHEORETICAL ASTRONOMY.\n\n266\n\n## the sun and planet are reduced to\n\nFinally, the longitudes of both\n1863.0\nof\nmean\nthe\nby applying the corrections\nequinox\n\n-51\". 52,\n\n50\".95,\n\nand the\n\n52\".14;\n\nsame\n\n## date by applying the corrections\n\n0\".15,\n\n0\".14, and\n\n0\".14,\n\nrespectively.\n\nCollecting together and applying the several corrections thus obtained for the places of the sun and of the planet, reducing the un-\n\nand\n\n## expressing them in days from the beginning of the year,\n\nfollowing data\n\nwe have\n\nthe\n\n= 17\n= 16\nI\" = 15\n\n= 257.68079,\n= 264.42570,\n\" =\n271.38625,\n\nt;\nt\n\nO =172\n\n46' 28\".17,\n\nA'\n\n40 25\n\n1544.03,\n\n0'32\".23,\n\n=178 35 48 .74,\n\"=185 2536.90,\n\n'\n\nThe numerical\n\n.19,\n\n/3\"\n\n=+3\n= 2\n=+2\n\n8'\n\n43\".51,\n\n52 27\n\n.62,\n\n3242.98,\n\nlogJ? =0.0021056,\n\nlog# =0.0011656,\nR\" = 0.0002378.\n\nlog\n\n## values of the several corrections to be applied to\n\nby observation and by the solar tables should be\n\n## the data furnished\n\nchecked by duplicate calculation, since an error in any of these reductions will not be indicated until after the entire calculation of the\nelements has been effected.\n\nBy means\n\nof the equations\n\"\n\nsin\n\n(\"-')\n\n~ .RR'sin(Q'-Q)\nRR\" sin (\"\n\n'\n\n0)\n\ntan\n\ntan/3'\n\nwe\n\n(A'\n\n)'\n\n0')\n\nobtain\nlog\n\nN= 9.7087449,\n= 161\n= 9.4980010,\n4/\n\nlog\n\nN\" = 9.6950091,\n\n42' 13\".16,\n\nlog\n\n(#\n\ncos\n\n40\n\n= 9.9786355\n\nn.\n\n## which oj/ must be taken is determined by the conmust\n\nbe less than 180, and that cosoj/ and cos (7'\n<\\J/\nO')\nmust have the same sign. Then from\n\nditions that\n\nin\n\nNUMEEICAL EXAMPLE.\n\nQ (A\" + A) - K) =\n\ntan Jsin\n\ntanlcosQ\n\n267\n\n(*\"\n\n-BO\n\n/9\n\nj?'\n\nsin\n\n6, c,\n\nc?,\n\n/,\n\n\"\n\nsin\n\n(G\n\no sin(A\"\n\nA)'\n\nand\n\nA),\n\n(Q'_ JQ\n\nRR\n\n~sin(A\"\n\n+ *) - *\")\n\n= jR- sin (0\n\nwe compute K, I,\nthan 90, and the\n\nsec J (A\"\n\ny/\n\n0)\n'\n\nA)\n\n^.\n\nless\n\n## value of /9 must be determined with the greatest\n\npossible accuracy, since on this the accuracy of the resulting elements\nprincipally depends.\n\nThus we obtain\n\n## K=4 47' 29\".48,\n\nP == 2\n59\"| jf\nlog b = 2.5456342\n52'\n\nlog d == 1.2437914,\n\nThe\n\n1= 9.3884640,\n\nlog tan\nlog a\nlog\n\niog/=\n\n1.3587437 n\n\n= 6.8013583\n\nM,\n\n= 2.2328550n\n\nlog\n\n/i\n\n3.9247691.\n\nformulae\n\n_\nMI ~~\n\n_~\n\nsin (A\"\n\nA')\n\nsin (A\"\n\n-A)\n\nsin (A\n\nA)\n\nZ\n1\n\nsin (A\"\n\nA)\n\n__\n\n~J\n\nK}\n\n\\!'\n\nQ\")\n\n^\"sin(A\"\n+ J~T~\n^sin(A\n\"\n\n0)\n\n~'\n\nhsm^\n\n,,\n\ngive\nlog\n\nM = 9.8946712,\nv\n\n= 1.9404111,\n\nK)\n\n~b~\n\n~d~~\nlog Jf/'\n\n= 9.6690383,\n\nThe\n\n## quantities thus far obtained remain unchanged in the sucand Q.\n\ncessive approximations to the values of\n\nFor the\n\nfirst\n\nhypothesis, from\n\n=-\n\n=\nYJ O\n\nT? O\n\n= R' sin\n=\ncos C\n^\nR' cos\nsin Z\n\n4-',\n\n4/,\n\n2 co\n\nTHEOKETICAL ASTRONOMY.\n\n268\n\nwe\n\nobtain\n\n= 9.0645575,\n= 8.1427824,\nlog Q\n= 0.0704470,\nk\nlog\n=\n0.3326925,\nlog\n\n= 9.0782249,\nlog P = 9.9863326,\n\nlog r\"\n\nlog r\n\nlog c == 2.2298567 W ,\n1\n0.0716091,\n\n=\n=\n8\nC\n\nlog\n\n>?\n\nIogm == 1.2449136.\n\n24' 49\".74,\n\nwhich f must be situated is determined by the condition that J? shall have the same sign as 4,.\nThe value of z f must now be found by trial from the equation\n\nin\n\nsin (z\n\nC)\n\n=m\n\nsin /.\n\nTable XII. shows that of the four roots of this equation one exceeds\n180, and is therefore excluded by the condition that sins' must be\nz'\n4/,\ngreater than 180\npositive, and that two of these roots give\n(\n\nand are excluded by the condition that z must be less than 180\nty.\nThe remaining root is that which belongs to the orbit of the planet,\nand it is shown to be approximately 10 40' but the correct value\n;\n\nis\n\nlast\n\nz'\n\nThe\n\n=9\n\n1'\n\n22\".96.\n\n## which corresponds to the orbit of the earth\n\nand differs very little from 180\nty.\nroot\n\nis\n\n18\n\n20' 41 \",\n\nNext, from\n\nsmz\n\nsin 2\n\np\"\n\nwe\n\nM\" 4MI\nn\"^\n\n_\ni\n\nn\"\n\nderive\n\n= 0.3025672,\n= 9.7061229,\nlog p = 0.0254823,\n\nlogr'\n\nlog\n\nThe\n\n= 0.0123991,\n= 9.6924555,\n\"\nlog P = 0.0028859.\n\nlog/\n\nlog n\"\n\n## values of the curtate distances\n\nhaving thus been found, the\n\n## now computed from\n\nNUMERICAL EXAMPLE.\n\n=p cos (A\n\nO)\nO)\n\nr cos b cos (I\nr cos b sin\n\n(I\n\n= j0sin(A\n= p tan\n\nr sin b\n\n/9\n\n/ cos V sin (/'\n/sin\n\n')\n\n/' cos\n\nQ)\n;\n\n(A'\n\np'\n\n(A'\n\nO')\n\n= sin\n= /tan/5';\n=\n0\")\nP\" cos\n0\") = p\" sin (A\"\n= ,o\"tan/3\",\n\nV cos (/\"\nV sin (\"\n\n-#',\n\n'),\n\n0\")\n\n(A\"\n\n/'sin 6\"\n\nR,\n\nQ),\n\n=10' cos\n\nO')\n\n6'\n\n/' cos\n\n269\n\nR\",\n\n0\"),\n\nwhich give\n\n= 514'39\".53,\nr = 7 '45 11\nI\" = 10 21 34\n\nlog tan b\nlog tan b'\nlog tan b\"\n\n.28,\n.57,\n\nlogr =0.3040994,\nlog/ =0.3025673,\n\n=8.4615572,\n\n= 8.4107555,\n= 8.3497911,\n\nlog /'\n\n= 0.3011010.\n\nThe agreement of\nfound,\n\nis\n\n## the value of log r f thus obtained with that already\n\na proof of part of the calculation. Then, from\ntan t\n\nnit\nsm (J (*\n\ntan\n\ncos (1 (\n\ncos\n\nwe\nu\n\nget\n\n= 158\n\nThe\n\n/-i\n\ntan 6\"\n\n7\\\n\ncos\n\n2'\n\ntan 6\n\ncos\n\n= 160\n\nequation\ntan\n\nb'\n\n= tan\n\n38\".16,\nu'\n\n25\".78,\n\n-f-\n\n= 207\n8'\n\n^\\ =\n+/)-)\ntan 6\"\ntan 6\n+ - 8) = 2sin (r _\n\n=4\n\n27' 23\".84,\n\nu\"\n\n39' 18\".13,\n\ni sin (l\n\n= 163\n\n16' 4\".42.\n\n&)\n\n## 8.4107514, which differs 0.0000041 from the value\n\nf\nThis difference, however, amounts\nfound\ndirectly from p\nto only 0\".05 in the value of the heliocentric latitude, and is due to\ngives log tan b'\n\n## If we compute n and n\" from the equations\n\nerrors of calculation.\n\nsin (u\"\n= //'\nrr\" sin\nn\n\nu'*)\n=r>'\n\nr,\n\nu)\n\n(u\"\n\nsin\n=~ rr\" sin\nrr'\n\n(u'\n\nu)\n\n(u\"\n\nu)\n\n'\n\nthe results should agree with the values of these quantities previously\nand Q. Using the values of u, u', and\ncomputed directly from\n\n11\n\njust found,\n\nwe\n\nlog n\n\nobtain\n\n= 9.7061158,\n\nlog n\"\n\n= 9.6924683,\n\nTHEORETICAL ASTRONOMY.\n\n270\nwhich\n\np and p\".\n\nAccording\n\nto the equations\n\nd log n\nd log n\"\n\n=\n=\n\n-}-\n\n21.055 cot (u'\n\nu')\n\ndu\n\nr\n,\n\nu) du',\n\n## the differences of logn and logn\" being expressed in units of the\n\nseventh decimal place, the correction to u' necessary to make the two\nvalues of logn agree is\n0\".15; but for the agreement of the two\nf\ndiminished by 0\".26, so that it appears\nbe\nmust\nu\nof\nvalues\nlogn\",\nnot\nis\nthat this proof\ncomplete, although near enough for the first\nIt should be observed, however, that a great circle\n\napproximation.\n\n## the extreme observed places of the planet passes\n\nvery nearly through the third place of the sun, and hence the values\nof p and p\" as determined by means of the last two of equations (18)\n\npassing through\n\nsomewhat uncertain. In this case it would be advisable to compute p and p\", as soon as p has been found, by means of the equations (22) and (23).\nThus, from these equations we obtain\nare\n\nlog p\n\n= 0.025491\n\nlog p\"\n\n8,\n\n= 0.0028874,\n\nand hence\nI\n\n514'40\".05,\n2134.19,\n\nr=10\nu\n\n= 207\n\n= 158\n\n8'\n\n2'\n\n32\".97,\nu'\n\n31\".47,\n\nThe value of\n\nlog tan b\nlog tan b\"\n\n= 160\n\n=8.4615619,\n=8.3497919,\ni\n4\n\nlog r\nlog /'\n\n27' 25\".13,\n\nX'\n\n= 163\n\nu\"\n\n39' 23\".31,\n\n= 0.3041042,\n= 0.3011017,\n16' 9\".22.\n\nand\n\nQ,\n\ni,\n\n## 8.4107555, agreeing exactly with that derived from p' directly.\n\nThe values of n and n\" given by these last results for u, u' and u n\\\nis\n\nare\nlog\n\nand\n\n= 9.7061144,\n\nbe complete\n\nu\"\n\nu'\n\n=2\n\nlog n\"\nif we\n\n= 9.6924640\n\nO'MS\n\nwe have\n\n36' 46\".09,\n\nu'\n\n=2\n\n30' 51\".66.\n\n## The results which have thus been obtained enable us to proceed to\n\na second approximation to the correct values of\nand we\nand\nmay also correct the times of observation for the time of aberration\n\nt==tQ\n\nCp sec\n\n/?,\n\n'\n\nCft sec p,\n\n== 257.67467,\n= 264.41976,\n\nwherein log\n\nt\"\nof*\n\n\"\nt\n\n## CP \" sec /?\",\n\na day. Thus\n\nt\"\n\nwe\n\n= 271.38044,\n\nget\n\nNUMERICAL EXAMPLE.\n\n271\n\nand hence\nlog r\n\n= 9.0782331,\n\nlog\n\nr'\n\n= 9.3724848,\n\nThen,\n\nby\n\nand\n\nlog r\"\n\n= 9.0645692.\n\nwe have\n\ns\",\n\n17'\n\nsin f cos\nsin Y sin\n\nG = sin J (u\"\nG = cos \\ (u\"\n\nw'),\n\nu') cos\n\n= cos 2 (u\"\n\ncos Y\n\nit')\n\nsin\n\ntan/' ==\nsin /' cos\nsin /' sin\n\nG\" = cos ^ (it'\n\n= cos J\n\ncos /'\n\nit)\n\n(it'\n\n2y\n\n7\n,\n\nsin 2;/'\n\nr2\n\nw),\n\nu) cos\n\nsin\n\nr\" 2\n\n## from which we obtain\n\n= 44 57' 6\".00,\n= 1 18 35\nlog m = 6.3482114,\nlogy = 6.1163135,\nx\n\nFrom\n\n/\"\n/'\n\n.90,\n\n44\n\n.69,\n\nm\"\n\nm\nr!'\n\n## using Tables XIII. and\n\nSj\n\n15 40\n\n= 6.3163548,\nlog/' = 6.0834230.\n\nlog\n\n## by means of the equations\n\nthese,\n\ncase of\n\n56' 57\".50,\n\n=1\n\nXIV., we compute\n\nand\n\nFirst, in the\n\ns\".\n\nwe assume\n7\n\n= 0.0002675,\n2\n\n0.0002581.\nand, with this as the argument, Table XIII. gives log s\nthis\nas\nthe\nwith\nHence we obtain x'\n0.000092, and,\nargument,\n\nTable\n\nXIV.\n\ngives\n\n=\n= 0.00000001\n\nlog s\n\nWhen\n\nis\n\nand, therefore,\n\nunnecessary.\n\n= 6.0001290,\n\nlogs\"\nit is\n\nit\n\nappears that a\n\nThus we obtain\n\n= 0.0001200.\n\n## not necessary to use the formula?\n\nTHEOKETICAL ASTEONOMY.\n\n272\n\n## in the complete form here given, since these ratios\n\nin the sequel.\nprocess, as will appear\n\nby a simpler\n\npP =\n\nmay then\n\nbe found\n\nThen, from\n\n~T\nr\n\n\"\n\n7\"'\n'\n\n_^_\n* ~~ \"\n\n'\n\nrr\" cos\n\nss\n\nwe\n\nfind\n\nlog\n\n|-\n\nu') cos\n\n(u\"\n\nP = 9.9863451,\n\nu) cos\n\n(u\"\n\n-J\n\nlog\n\n(u\n\nu)'\n\nQ = 8.1431341,\n\n## with which the second approximation may be completed. We now\n\nf\ncompute c Tcw lw z , &c. precisely as in the first approximation but\n;\n\nwe\n\np and p\"\n\n## computed by means of the equations (22) and (23) instead of those\n\nobtained from the last two of the formulae (18).\nThe results thus\nderived are as follows\n\n= 2.2298499\n= 0.0714280,\nlog\n= 0.0719540,\nlog % = 0.3332233,\nC = 8 24' 12\".48,\nlog m = 1.2447277,\n= 90'30\".84,\nlog / = 0.3032587,\nlog p = 0.0137621,\n=\nn\n9.7061153,\nlog\nlog n\"= 9.6924604,\n=\n0.0269143,\nlogp\nlog p\" = 0.0041748,\nlog\n\nlog\n\nJc\n\nn,\n\nz'\n\n= 5\n= 7\nJ\" = 10\n\n2,76,\n\nlog tan b\nlog tan V\n\n0.91,\n207\n\n0'\n\n15' 57\".26,\n\n/'\n\n46\n\n22\n\n= 158\n\n## log tan b\"\n\n12' 19\".54,\n\nThe agreement of\n\nlogr =0.3048368,\nlog/ == 0.3032587,\n\n= 8.3504332,\n=4\n\nlog r\"\n\n= 160\n\nu\"\n\n42' 45\".82,\n\nis\n\n= 0.3017481,\n\n28' 35\".20,\n\n0\".72,\nu'\n\n=8.4622524,\n=8.4114276,\n\n= 163\n\n19' 7\".14.\n\n## of log tan b r computed from\n\ntan\nis\n\n= tan\n\n= 8.4114279, agreeing\n\nfrom p f\n\nb'\n\nThe\n\nvalues of\n\nare\nlog n\n\nsin (I\n\n= 9.7061156,\n\nlog n\"\n\n## which agree with the values\n\nthe proof of the calculation\n\nu\"\n\nu'\n\n=2\n\nFrom\n\n36' 21\".32,\n\nthese values of\nlog\n\n& ),\n\nis\n\nu'~ u\n\ncomplete.\n\n=2\n\nu\"u\n\n= 0.0001284,\n\nin\n\n= 9.6924603,\ncomputing p and p\", and\n\nWe\n\nhave, therefore,\n\n30' 26\".28,\n\nand u\n\nu,\n\nlog s\"\n\n(54)\n\nwe\n\nu\"\n\n=5\n\nobtain\n\n= 0.0001193,\n\n6'\n\n47\".60.\n\nNUMEKICAL EXAMPLE.\nand, recomputing\nlog\n\nP and\n\nwe\n\n273\n\nget\n\nP = 9.9863452,\n\nlog\n\nQ=\n\n8.1431359,\n\n## differ so little from the preceding values of these quantities\n\nthat another approximation is unnecessary.\nmay, therefore, from\nthe results already derived, complete the determination of the elements\n\nwhich\n\nWe\n\nof the orbit.\n\nThe equations\n\n## / cos G' = sin-^\n\nsin / sin G' = cos |\ncos /\ncos\n\nsin\n\nr'\n\n(u\n\nn\nu),\n\n(u\"\n\nu) cos 2/,\n\n(u\"\n\nu} sin 2/,\n\n(r-fr\")\n\n\"\"\n\ncosV\n\nsin\n\ncos/'\n\ngive\n\n/ = 44\n\nlog m'\n\nFrom\n\n/ = 2 33' 52\".97,\nlog tan G' = 8.9011435,\n= 6.9332999,\n=\n6.7001345.\nlog/\n\n53' 53\".25,\n\nthese,\n\nlog s'\n\n= 0.0009908,\n\nlog of\n\n= 6.5494116.\n\nThen from\ns'rr\" sin\n\nwe\n\nget\n\nThe\n\nw\"\n\n= 0.3691818.\n\n## values of logp given\n\nby\n\n^)\\\n\n/grV'siny\nr\n\n~-\\\n\n_\n=\n\n-\\\n\ns\"rr' sin\n\n(u\n\n'>'\n\nu)\n\n-/\n\nare 0.3691824 and 0.3691814, the mean of which agrees with the\nresult obtained from u ff\nu, and the differences between the separate\nresults are so small that the\n\nThe\n\napproximation to\n\nequations\nsin 4\n\nHi\n\nJ] =\n\na cos ^\n18\n\nP and Q is sufficient.\n\nTHEOKETICAL ASTRONOMY.\n\n274\ngive\ni\n\n=l\n\nE)\n\n(E\"\n\n4'\n\n42\".903,\nlog cos\n<f>\n\nlog (a cos ?)\n\n= 9.9921503.\n\nNext, from\ne sin\n\n?) tan %v,\n\ntan\n\nbE\n\nJ ? ) tan Jt/'\n\ntan\n\ngive\n\nE = 333\n\n17' 28\".18,\n\nE'\n\n= 335\n\n## 24' 38' .00,\n\nE\" = 337\n\n36' 19\".78.\n\nNUMERICAL EXAMPLE.\nThe value of\n\n## thus obtained differs only 0\".003 from that\n\nE)\n\n(E\"\n\nf\ncomputed directly from x\n\nM= E\n\nM = E'\n'\n\ne sin\n\n275\n\nE,\n\nM\" = E\"\n\ne sin E',\n\ne sin\n\nE\",\n\nM = 338\nand\n\nif\n\n8'\n\nM' =\n\n36\".71,\n\n339\n\nWashington mean\n\n339\n\nT=1863\n\nmean of which\n\nM = 339\n\nSept. 21.5\n\n55' 25\".96,\n\nand\n\ngives\n\n55' 25\".96.\n\nis\n\n## accuracy of the entire calculation of the elements.\n\nCollecting together the separate results obtained,\n\nlowing elements\n\nwe have\n\nthe fol-\n\n## = 1863 Sept. 21.5 Washington mean time.\n\nM = 339 25\".96\nn = 37 15 40 .29)\nEcll\nand Mean\n\nEpoch\n\n55'\n\n= 207\n72 V\n= 4 28 35. 20 J\n= 10 51 39 .62\n=\n0.3848816\nlog a\nlog = 2.9726842\n= 939\".04022.\n<\n\nP tlc\nEquinox\n\n1863.0.\n\n<?\n\nfi\n\nfi\n\n## If we compute the geocentric right ascension and declination of\n\nthe planet directly from these elements for the dates of the observations, as corrected for the time of aberration, and then reduce the\nobservations to the centre of the earth\n\nby applying the\n\ncorrections\n\nhow\n\nshow\n\n## closely the elements represent the places on which they are\n\nbased.\nThus, we compute first the auxiliary constants for the equator,\n\nusing the\n\nmean\n\nfit\n\nThe agreement of\n\n43' 6\".97\n\nM =M\nwe\n\nM\" = 341\n\n54' 10\".61,\n\n## obliquity of the ecliptic,\n\ne\n\n= 23\n\n27' 24\".96,\n\nTHEORETICAL ASTRONOMY.\n\n276\n\n## and the following expressions\n\nplanet are obtained\n\n## y =r [9.9744699] sin (206\n\n12 42 .79 -f u),\nr [9.5249539] sin (212 39 14 .62\nu).\n\n## enclosed in the brackets are the logarithms of sin a,\n\nand sin c, respectively ; and these equations give the co-ordinates\n\nThe numbers\nsin 6,\n\nreferred to the\n\nThe\n\n## places of the sun for the corrected times of observation,\n\nreferred to the mean equinox of 1863.0, are\n\nTrue Longitude.\n\n172\n178\n185\n\n.5\n\n25 42\n\n.0\n\nLog R.\n\nLatitude.\n\n-0\".07\n\n0'29\".5\n\n36\n\n+0\n+0\n\nand\n\n0.0022146,\n\n.77\n\n0.0013864,\n0.0005182.\n\n.67\n\n## If we compute from these values, by means of the equations (104)j,\n\nthe co-ordinates of the sun, and combine them with the corresponding\nheliocentric co-ordinates of the planet, we obtain the following geocentric places of the planet\n:\n\n= 15\n= 14\na\" = 13\na\n\n10' 29\".06,\n\na'\n\n15\n\n= + 9 53' 16\".72,\n= 912 51\nV = + 8 21 54\n\n= 0.02726,\n= 0.01410,\n=\nJ\"\n0.00433.\nlog\nlog A\n\nd\nd\n\n.22,\n\n3 49 .47,\n\nlog J'\n\n.29,\n\n.46,\n\nTo\n\nreduce these places to the apparent equinox of the date of observation, the corrections\n\n+ 48\".14,\nmust be applied\n-f-\n\n## to the right ascensions,\n\nrespectively,\n\n18\".55,\n\nto the declinations.\n\nThus we obtain\n\nWashington M. T.\n\nComp.\n1*\n.\n\n28.38044\n\na.\n\nComp.\n\nO m 45'.15\n\n57\n\n52\n\nand\n\n+ 19\".31,\n\n-f 18\".92,\n\n21.41976\n\nThe\n\n-f 48\".91,\n\n-f 48\".54,\n\n+9\n\n3 .25\n18.\n\n56\n\n6.\n\n53' 35\".3,\n\n13 10\n\n.2,\n\n22 13\n\n.8.\n\n## corrections to be applied to the\n\nrespective observations, in order\nthem to the centre of the earth, are -f- O s .24,\nO s .31,\n0*.34\n\nto reduce\n\n## in right ascension, and -f\n\n4\".5,\nthat we have, for the same\ndates,\n\n+ 4\".8, + 5\".l\n\nin declination, so\n\nNUMERICAL EXAMPLE.\nObserved\n\nObserved\n\na.\n\n0-45M5\n\n1*\n\n57\n52\n\nThe comparison of\n\n277\n3.\n\n53' 35\".3,\n\n3 .26\n\n18 .56\n\n+8\n\n13 10\n22 13\n\n.3,\n.8.\n\n## extreme places are exactly represented, while the difference in the\n\nmiddle place amounts to only O'.Ol in right ascension, and to 0\".l\nin declination.\n\npletely satisfied\n\nIt appears, therefore, that the observations are comby the elements obtained, and that the preliminary\n\ncorrections for aberration and parallax, as determined by the equations (1) and (4), have been correctly computed.\nIt cannot be expected that a system of elements derived from ob-\n\n## servations including an interval of only fourteen days, will be so\n\nexact as the results which are obtained from a series of observations\n\n## or from those including a much longer interval of time; and although\n\nthe elements which have been derived completely represent the data,\n\n## on account of the smallness of ft*\n\nthis difference being only\n/9 ,\n31\". 893, the slight errors of observation have considerable influence\nin the final results.\n\nyet,\n\nWhen\n\n## rection for parallax may be applied directly to the observations, in\n\norder to take into account the latitude of the sun, the observed places\nof the body must be reduced, by means of equation (6), to the point\nin\n\nwhich a perpendicular\n\nlet fall\n\n## from the centre of the earth\n\nto the\n\nplane of the ecliptic cuts that plane. The times of observation must\nalso be corrected for the time of aberration, and the corresponding\nplaces of both the planet\n\n## and the sun must be reduced to the ecliptic\n\nand further, the reduction to\n\nfixed epoch;\n\n## the fixed ecliptic should precede the application of equation (6).\n\nIf the intervals between the times of observation are considerable,\nit\n\nmay become\n\nvalues of\n\nnecessary to\n\nP and\n\nQ, and\n\nmake\n\nthree or\n\nmore approximations\n\nto the\n\n## But when approximate\n\ntageous to compute the first assumed values of P and Q directly\nfrom these elements by means of the equations (44) or by means of\n(48) and (51) and the ratios s and s\" may be found directly from the\nIn the case of very eccentric orbits this is indispenequations (46).\n;\n\n## be desired to avoid prolixity in the numerical calculation,\n\nsince otherwise the successive approximations to\nand Q will slowly\napproach the limits required.\nsable, if it\n\nTHEORETICAL ASTRONOMY.\n\n278\n\n## various modifications of the formulae for certain special cases,\n\nmust be used in the case of parabolic\n\nThe\n\nand hyperbolic\n\nand of those\n\norbits,\n\nfrom the\n\n## such that they require no furparabola, have been given in a form\n\nther illustration.\n94.\n\n## In the determination of an unknown\n\nconsiderably unequal,\n\nit\n\nwill\n\nfirst\n\n## assumed value of P before completing the first approximation in the\n\nmanner already illustrated. The assumption of\n\nis\n\n## same degree of approximation to\n\nequation (28) 3 use the expression\n\nfor the\n\nwe must,\n\nand\n\naccording to\n\n7}\n\n## which becomes equal\n\nfirst assumed values\n\n## furnish, with very\n\nwith the values of\n\nto\n\nlittle\n\nonly\n\nlabor,\n\nP and\n\nwhen\n\n## the intervals are equal.\n\nan approximate value of\n\nr'\n\nThe\n\nand then,\n\nderived from\n\ng = TT,\n\n(98)\n\n## the entire calculation should be completed\n\nprecisely as in the example\ngiven.\nThus, in this example, the first assumed values give\n\nlog/\nand, recomputing\n\n= 0.30257,\n\n## P by means of the first of\n\nlog P = 9.9863404,\n\nthese equations,\n\nwe\n\nget\n\nlog Q = 8.1427822,\n\n## with which, if the first approximation to the elements be completed,\n\nthe results will differ but little from those\nobtained, without this corfrom\nthe\nsecond\nrection,\nhypothesis.\ncorrected for the time of\nthe\nwould be still\naberration,\n\nagreement\n\ncloser.\n\nThe comparison of\nfourth order,\n\n## equations (46) with (25) 3 gives, to terms of the\n\nNUMERICAL EXAMPLE.\nT \"2\n\n>2\n\nT2\n\n## and, if the intervals are equal, this value of\n\nthe fifth order.\nSince\n\nwe\n\nin\n\n279\n\ncorrect to terms of\n\ns' is\n\nwhich log^\n\nWe have,\n\n8.8596330.\n\nalso, to the\n\nsame degree of\n\napproximation,\n\n## For the values\n\nlog r\n\n= 9.0782331,\n\n= 9.3724848,\nlog/ = 0.3032587,\nlog\n\nlog r\"\n\nr'\n\n= 9.0645692,\n\n## these formulae give\n\nlog s\n\n= 0.0001277,\n\nlog s\n\n= 0.0004953,\n\n\"\n\nlog\n\n= 0.0001199,\n\nwhich differ but little from the correct values 0.0001284, 0.0004954,\nand 0.0001193 previously obtained.\nSince\nsec\n\n/=\n\n+ 6 sin\n\ntf\n\n+ Ac.,\n\nr'\n(\n\n4-\n\n6r'\n\n\"V\n\n## Substituting this value in the\n\nr'\n\n~^~\n\n('\n\nfirst\n\n-L\n\n\"\\*\n\nsm\n\nI*\n\n\"h\n\n& Ct\n\nof equations (66),\n6r'\n\nwe\n\nget\n\n## If we neglect terms of the fourth order with respect to the time, it\n\nwill be sufficient in this equation to put y'\nf, according to (71), and\n\nhence\n\nwe have\n\nf\n1 is of the second order with respect to\nand, since s\nto terms of the fourth order,\n\nr',\n\nwe\n\nhave,\n\nTHEORETICAL ASTRONOMY.\n\n280\nTherefore,\n\n## which, when the intervals are small, may be used to find\n\nand r\". In the same manner, we obtain\n\n=i tin*\n\nfrom r\n\n\"=3%7s-\n\nlo g s\n\n(102)\n\nFor logarithmic calculation, when addition and subtraction logarithms are not used, it is more convenient to introduce the auxiliary\nangles\n\n/',\n\n## and #\", by means of which these formulae become\n\n(103)\n\nin\n\nwhich log J^\n\n= 9.7627230.\n\nFor the\n\nfirst\n\napproximation these\n\n## equations will be sufficient, even when the intervals are considerable,\n\nff\nto determine the values of s and s required in correcting\nand Q.\n\nThe\n\nvalues of\nlog r\n\nr, r',\n\nr\",\n\n= 0.3048368,\n\ngive\nlog s\n\nThese\n\n= 0.0001284,\n\nlog s'\n\n= 0.0004951,\n\nlog s\"\n\n= 0.0001193.\n\n## correct, and that for log s' differs\n\nonly 3 in the seventh decimal place from the correct value.\n\nCHAPTER\n\n281\n\nV.\n\n## DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS,\n\nOF WHICH THE SECOND AND THIRD MUST BE COMPLETE.\n95.\n\nTHE\n\n## to determine the elements of the orbit of a heavenly body when its\n\napparent path is in the plane of the ecliptic. In this case, however,\n\nthe position of the plane of the orbit being known, only four elements remain to be determined, and four observed longitudes will\nfurnish the necessary equations.\nThere is no instance of an orbit\nwhose inclination is zero ; but, although no such case may occur, it may\n\nhappen that the inclination is very small, and that the elements\nderived from three observations will on this account be uncertain,\nand especially so, if the observations are not very exact. The difficulty thus encountered may be remedied by using for the data in the\ndetermination of the elements one or more additional observations,\nand neglecting those latitudes which are regarded as most uncertain.\nThe formulae, however, are most convenient, and lead most expeditiously to a knowledge of the elements of an orbit wholly unknown,\nwhen they are made to depend on four observations, the second and\n\n## third of which must be complete ; but of the extreme observations\n\nonly the longitudes are absolutely required.\n\nThe preliminary\n\n## reductions to be applied to the data are derived\n\nprecisely as explained in the preceding chapter, preparatory to a determination of the elements of the orbit from three observations.\n\nLet t, t', t\", t'\" be the times of observation, r, r', r\" , r\"' the radiivectores of the body, u, u', u\"', u fff the corresponding arguments of\nthe latitude, R, R, R\",\nthe distances of the earth from the sun,\n\nR\"\n\nthese times.\nLet us also put\n\n## = rV\" sin (um\n\n[rV\"] = rV\"\n[//\"]\n\nito.\n\nto\n\n'),\n\nu\"),\n\nand\n(1)\n\nTHEORETICAL ASTRONOMY.\n\n282\n\n## Then, according to the equations\n\nnx\n\nx'\n\nwe\n\n(5) 3 ,\n\nshall\n\nhave\n\n+ n\"x\" = 0,\n\n+n'Y = 0,\n-y\"''\"'\"\nf\n\nny\n'\n\nLet ;, A', X\", X'\" be the observed longitudes, ft /?', /9\", /3'\" the obfr\nln\nserved latitudes corresponding to the times t, t', t , t\nrespectively,\nand J, A', A\", A'\" the distances of the body from the earth. Further,\n',\n\nlet\n\n^008^=^\nand\n\nwe have\n\nof\"\n</\"\n\n= p\" cos\n= /\" sin X\"\nr\n\n## R\" cos 0'\",\n\nA'\"\n\nR\"\n\nsin O'\".\n\nf/f\nIntroducing these values of x'\" and y , and the corresponding values\nf\nof x, x , x\", y, y', y\" into the equations (2), they become\n\n= n (p cos\n\nE cos Q)\n\n(j>\n\nsin A\n\n^ sin Q)\n\n(/>'\n\ncos\n\n# cos O')\n\nA'\n\n'\n\n0>'\n\n= ri\n\ncos\n\nA'\n\n(/>\"\n\ncos A\"\n\nJ?'\n\nsin\n\nR\"\n\ncos\n\nV\nR\"\n\n+ n'\"\n(/>'\n\nsin\n\n'\n\nA'\n\nsin\n\n## (?\" sin A\"\n\nO')\n\nsin\n\nIf we nfultiply the\ncos\n\n^,\n\nfirst\n\nJB\" sin\n\n0\")\n\ncos\n\n\")\n\n^,\n\n0\"),\n(3)\n\n/\"\n\nG/\" sin\n\nn'\"\n\njR\"cosO\"),\n\nQ')\n\n+ n\"0>\"sinA\"\n\n^ cos O')\n\nsin\n\n+\nby\n\n\"0>\"cosA\"\n\n-f\n\n=n\n=\n\nJR\"' cos\n\nO'\"),\n\n12\"' sin\n\nQ \").\nr\n\n= nR sin\n\n(A\n\nn\nP\n+ K sin\n- + R\" sin - \"))\n1\n\nQ)\n\nsi\n\n(X\n\nX)\n\nA)\n\nQ'))\n\n(A\n\n(A\n\n(4)\n\nequations, we\nfind\n\nn'\n\nO/\n\nsin (A'\"\n\nA')\n\nWhenever\n\nR\nA\")\n\ns i n (A'\"\n\nQ '))\n\nU\" S i n (A'\"\n\n0\"))\n\n(5)\n\nW'\".R\"'\n\nsin (/'\"\n\n'\").\n\n## the values of w, n', TI /; , and n\" f are\n\nknown, or may be\ndetermined in functions of the time so as to\nsatisfy the conditions of\nmotion in a conic section, these\nbecome\ndistinct or indeequations\npendent of each other ; and, since\ntwo\nunknown\nonly\nquantities p'\n\n283\n\n## in them, they will enable us to determine these\n\ncurtate distances.\n\ncos\n\nsin (/\n\n=A,\n\nX)\n\nA\") = C,\n\nA)\n\nA')\n\n=\n= D,\nJ5,\n\nAp' sec p\n\nBn\"p\"\n\nsec 0\"\n\n= nR sin\n\nQ)\n\n(A\n\nR'\n\nsin (A\n\n0')\n\n+ w\"U\" sin (A\nJ>&y\n\nsec ft'\n\n(A'\"\n\n')\n\n+ n'\"jR\"'sm(;i'\"\nIf\n\nwe assume\n\nfor\n\nn and n\"\n\n0\"),\n\n0\")\n\n(7)\n\n0'\").\n\n## the earth, which is equivalent to neglecting terms of the second order\n\nin the equations (26) 3 , the second member of the first of these equations reduces rigorously to zero ; and in the same manner it can be\n\nshown\n\nwhen\n\nthat\n\n## similar terms of the second order in the corre-\n\nf\nsponding expressions for n and n\" are neglected, the second member\nof the last equation reduces to zero. Hence the second member of\n\n## each of these equations will generally differ from zero by a quantity\n\nwhich is of at least the second order with respect to the intervals of\ntime between the observations. The coefficients of p f and p\" are of\nthe first order, and it is easily seen that if we eliminate p\" from\nthese equations, the resulting equation for p' is such that an error of\nthe second order in the values of n and n\" may produce an error of\n1\nthe order zero in the result for p , so that it will not be even an\napproximation to the correct value ; and the same is true in the case\n\nof p\".\n\nIt\n\nin the\n\nfirst\n\nis\n\n## necessary, therefore, to retain terms of the second order\n\nf//\nfor n, n', n\", and n\n;\nand, since the\n\nassumed values\n\n## and r ff we thus introduce two\n\nrf\nand\nvolving r', r\", p'j p\nquantities derived from observation, must\nterms of the second order involve\n\nr'\n\nmay be found.\n\nFrom\n\nequation (34) 4\np' sec\n\nwhich\n\nis\n\nwe have\n\nVr' 2\n\nJ^'sin 1 *',\n;\n\nand similarly we\n\n(8)\nfind, for\n\n## the other equation,\n\np\" sec 0\"\n\n= R\" cos\n\n4/'\n\nd=\n\nV rm\n\nR\"\n\nsin 4\".\n\n(9)\n\nTHEORETICAL ASTRONOMY.\n\n284\n\n## Introducing these values into the equations\n\n= =b vV'\n\nx'\n\nand putting\n\njff'Bin'V,\n\"2\n\na^zfcvV\"\nwe\n\n(7),\n\nsin\n\n4/',\n\nget\n\nAx'\n\nBn\"x\"\n\nD*V -\n\n= nR sin\n-|-\n\nCx\"\n\n= n'R' sin\n\n# sin\n0')\nAK cos V + n\"BR\" cos V,\n0\")\n- Q') - R\" sin - 0\")\nQ)\n\n(A\n\nn\"R\"\n(A'\"\n\n(A\n\nsin (A\n\n(A'\"\n\n0'\")\n\nCR\"\n\ncos V'.\n\n## Let us now put\n\n=K\n\nC =r>\n\n'\n\nor\ni,\n\ncos\n\n/3\"\n\ncos\n\nsin (A\"\n\nA)\n\n(A'\n\ncos\n\n'\n\nsin\n\ncos\n\nA)\n\nfi'\n\nsin (A'\"\n\nA')\n\nft\"\n\nsin (A'\"\n\nA\")'\n\n_\nand we have\n\n= Kn\"x\" + nd - + nV,\nr\n\nx'\n\n\"\n\n==\n\nA\"wV\n\n+ w'\"d\"\n\na!\n\na\"\n\n+ ^c'\n\nr\n.\n\nf\n\n',\n\n96.\n\nIn order\n\nn and n /r we\n,\n\n',\n\nn'',\n\nand\n\n7i\n\nr//\n\nare\n\nknown.\n\nand, putting\n\n~\nthese give\n\ne = (w + n\"-l)r,\n\n(13)\n\n285\n\nwe obtain\n\n,*\"'W + T)\n'\"_^/1\n~\nf5 ~\n\"*\n\nrr-r\n\nrr\n*\n\nFrom\n\nthese\n\nwe\n\nget, including\n\n^-^d r\n\"\n\nand hence,\n\nif\n\nwe put\n\nP\nwe\n\n= 5-,\n7&\"\n\nQ'= (' +\nr\n\nr/\n\n+r\n\n-.'\"2\n\n2\n-|\n\nfff\n\n/v,\"3\n\n/'\n\n(18)\n\n,,_1T\n\nWhen\n\nP'\n\n-*\n\nand\n\nThe\n\nwe have\n\n/^ >\n\nP\"\n\n-*\n\n///\n\n## be used, in the case of an unknown orbit,\n\napproximation to the values of these quantities.\n\nthese expressions\n\n(17)\n\nI\n'\n\n-!)/\",\n\n;//\n\nPit\n\n'\n\nmay\n\n(19)\n\n## and, introducing these values, the equations (12) become\n\nTHEORETICAL ASTRONOMY.\n\n286\n\n(20)\n\nP'd'+c'\n\n-P'\n\n_\nI\n\nand we\n\nshall\n\n__\n\nP\"\n\n'r\n\nJ\n^\n\n>\n\nhave\n\n(22)\n\nWe\n\n## and r //3 in equations (22), the two\n\nresulting equations will contain only two unknown quantities x and\nr\n/r\nr/\nand\nare known, and hence they will be\na/', when P', P\nsufficient to solve the problem.\nBut if we effect the elimination of\nIf\n\nwe\n\n/3\n\neither of the\n\nunknown\n\n## quantities directly, the resulting equation\n\nIt is necessary, therefore, in the numerical\n\n## application, to solve the equations (22)\n\nmay\n\nby\n\nsuccessive trials,\n\nwhich\n\neffected.\n\nrepresents the angle at the planet between the sun and the\nearth at the time of the second observation, and z\" the same angle at\n\nIf\n\n## the time of the third observation,\n\nwe\n\nshall\n\nhave\n\nr\nSubstituting these values of r and r\" in equations (10),\n\nwe\n\nget\n(25)\n\nand hence\n\n## CEBIT FKOM FOUR OBSERVATIONS.\n\n287\n\n--j,V\n\"\"\nR' sin\n\n(26)\n\nf\nby means of which we may find z and z\" as soon as x and x\" shall\nhave been determined and then r and r rr are obtained from (24) or\nThe last equations show that when x f is negative, z must be\n(25).\nf\nf\ngreater than 90, and hence that in this case r is less than R\nIn the numerical application of equations (22), for a first approximation to the values of x and x rf since Q f arid Qf r are quantities of\nr\n\n## the second order with respect to r or r ;// ,\n\n# = 0,\n\ne\"\n\nwe may\n\ngenerally put\n\n= 0;\n\nand we have\n>\n\n+<-a',\n*\"=/'V+c \"-a\",\nx =f'x\"\nor,\n\nby\n\nelimination,\n\nv_\ni-//\"\nI-/'/\"\nthe approximate values of x and x\" derived from these equaf\ntions, we compute first r and r\" from the equations (26) and (24),\nand then new values of x f and x\" from (22), the operation being\nf\n\nWith\n\n## repeated until the true values are obtained.\n\nproximations, the equations (22) give\n\nTo\n\nfacilitate these\n\nap-\n\n(27)\n\n## Let an approximate value of x f be designated by #</, and let the\n\nvalue of x\" derived from this by means of the first of equations (27)\nbe designated by a;/ 7\nWith the value of x^ for x\" we derive a\nnew value of x f from the second of these equations, which we denote\n.\n\nf\nThen, recomputing x\" and x , we obtain a third approximate\nvalue of the latter quantity, which may be designated by x2 f ; and,\n\nby #/.\nif\n\nwe put\nxi\n\nxd\n\n<*o>\n\nxt\n\nxi\n\nao'>\n\nTHEORETICAL ASTRONOMY.\n\n288\n\nwe\n\n,__jC_.\n\n,__^^\n\n^,\n\n(28)\n\n## The value of x thus obtained will give, by means of the first of\n\nequations (27), a new value of x\", and the substitution of this in the\nlast of these equations will show whether the correct result has been\nr\n\n## If a repetition of the calculation be found necessary, the\n\nfound.\nf\nthree values of x which approximate nearest to the true value will,\n\n## In the same manner, if\n\nx fr the value derived by putting Q'\nand Q\"\n0,\nand compute x', three successive approximate results for x\" will\n\nby means of\n\nwe assume\n\nfor\n\n## enable us to interpolate the correct value.\n\nWhen the elements of the orbit are already approximately\nthe first assumed value of x f should be derived from\n\nby putting\n\n'\n\nand\n\nQ\"\n\nknown,\n\nequal to zero.\n\nwhen\n\nA'\n\n= X or\n\n)J\n\nrr\n\nn\n',\n\nthe equa-\n\n## tions (22) are inapplicable, but that the\n\noriginal equations (7) give,\n1\nin this case, either p ff or\np directly in terms of n and n\" or of n'\n\n## and n\" f and the data furnished by observation.\n\nfirst of\nequations (22) by /*/, we have\n\nThe\n\nIf we divide the\n\nh'~\n\nand from\n\n(11)\n\nwe\n\nget\nrf\n\n__R cosV\n\nh'~\n\n~hT\n\n'\n\nR'sm(X\n\nQ)\nif\n\n0')\n\n~W\n+ X'^-W\n,\n\nh'~\nThen,\n\n+ P'\n\nwe put\nc\n\n'\n\n'- P-p'd' + W\n\nn>\nC\n\nh'\n\n(29)\n\nc'\n\nvalue\n\nits\n\nmay\n\nand\n\n289\n\nin\n\nA = cos\n\nsin\n\n/?'\n\nB = cos\nD = cos\nD\n\nA),\n\n(A'\n\nA\"),\n\n\"\n\nc\"\n\ndn\n\n<r=\n\n(A\n\nlog\n\nh'\n\nlog\n\na!\n\nlog\n\nc'\n\nlog\n\nd'\n\nlog\n\nlog\n\nQ\".\n\nn.\n\n## first hypothesis in regard to the\n\nIf the elements were entirely un-\n\nthe\n\n## would be necessary, in the first instance,\n\nquantities the values given by the expressions\nknown,\n\nit\n\nn,\nn,\n\nn,\n\n= 9.8528803,\n\n= 0.1048468,\n= 9.9752915\n= 9.7267348\nc\"\nlog\n= 9.9096469\nd\"\nlog\n\nn,\n\nmake\n\nD = 9.9577271,\n\nlog h\"\nlog a\"\n\nn,\n\n= 0.2785685\n= 0.8834880\n= 0.9012910\n= 0.4650841,\n\nP Q P\", and\n\nvalues of\n\nA'),\n\nA = 9.0699254n\nB = 9.3484939,\n\nlog\n\nA),\n\nsin (A'\"\n\nO)\n\n## give the following results\n\nlog\n\nsin (A\"\n\n/?'\n\n0\")\n\nsin (A\n\n= h\"R' cos V\n\n/5\"\n\nto\n\nassume\n\nfor these\n\nNUMERICAL EXAMPLE.\n\n297\n\n## then approximate values of r r and r\" are readily obtained by means\n\nof the equations (27), (26), and (24) or (25). The first assumed\n\nmember of\n\n## from the expression which\n\nand Q\"\n0, namely,\n\n(27), is obtained\n'\n\nputting\n\nthe\n\nfirst\n\nof equations\n\nfrom (22) by\n\nresults\n\nafter\n\n(27).\n\nof an\n\nx'\n\n/'/\"\n\nand x\"\n\nwill be obtained\n\ntrial\n\nby\n\nfrom\n\n## It should be remarked, further, that in the first determination\n\norbit entirely unknown, the intervals of time between the ob-\n\nf\nservations will generally be small, and hence the value of x derived\nand Q\"\nfrom the assumption of Q f\nwill be sufficiently ap-\n\n## proximate to facilitate the solution of equations (27).\n\nAs soon as the approximate values of r f and r\" have thus been\nfound, those of\n\nWith\n\nand\n\nP\"\n\n## Q and Q\" already obtained, the\n\nmust be completed.\nWhen\n\nfirst\n\n## assumed values of P', P\", Q', and\n\nof these elements.\nThus, from\nr'r\" sin\n\n0\"\n\nQ\"\n\ntO\n\n'\n\nv}\nm\nr\"r sin(v\"'v\")\n\n>\n\nwe\n\nfind n,\n\nn n\", and n\nf\n\nrr' sin (v\n\nt;)\n'\n\n,\n\n_~\n\nv)\n\nrV'sinQ/'\nr'r\"' sin (v\"\n\nv')'\n\nv'}\nf\n\nv'\n\n=322\n=353\n\nv\"=\n\n14\n\n55'\n\n9\".3,\n\n19 26\n\n45\n\ni/\n\n\".\n\ngive\nv\n\nfirst\n\n## should be computed by means\n\n.3,\n\n8.5,\n\nt/\"= 47 23 32\n\n.8,\n\n=0.308327,\nlog/ =0.294225,\nlog/' =0.296088,\n0.317278,\nlog /\"\nlogr\n\nTHEORETICAL ASTRONOMY.\n\n298\nand hence we obtain\n\n= 9.806836,\n= 9.633171.\nlog n'\"\n\n= 9.653052,\n\nlog n\n\nlog n\"\n\nThen, from\n\n\"\n\nwe\n\nget\n\n= 9.846216,\n= 9.807763,\nlog P\"\nlog\n\nThe\n\n= (n'+ n'\"\n\nP'\n\nequations (41)\n\nand\n\nlog\n\n\"\n\nlog\n\nmay\n\n3\n1) r\" ,\n\n= 9.840771,\n== 9.882480.\n\n## computed by means of the\n\nalso be\n\n(42).\n\nNext, from\nC\n\n'\n\nwe\n\nfind\n\nlog\n\nlog\n\n'\n\n-\\-\n\n~\n-i\n\nf = FT?\n\n+ P''\n\nft\n\n-pti\n\n= 0.541344\n= 9.807665\n\n-i\n\np//'\n\n= 0.047658\nlog/\" = 9.889385.\nlog/\n\nn,\n\n;/\n\n'\n\n~~\n\n'\nI\n\nn,\n\nThen we have\n\nif.r^ rf\n,\n\ntan 2'\n\ntan z\"\n\n=\nsin\n\n=~ cos\n\n=\n2/'\n\nof\n\nwe\n\nIn the\n\n= 1r'\n\nThen the\n\nfirst\n\nJ^Bin1 *',\n\n= 0.242737.\n\ngives\nlog a\"\n\nsin4/;\nsin 2\"\n\nfirst place,\n\n## obtain the approximate value\n\nlog x'\n\n= 0.237687.\n\n=~\n\nfrom\n\n^\ncos 3'\"\n\nNUMERICAL EXAMPLE.\n\nFrom\n\nthis\n\nwe\n\n299\n\nget\n\nz\"\n\n= 29\n\n3'\n\n## and then the equation\n\n11\" .7,\n\nfor\n\nlog r\"\n\n= 0.296092\n\n= 0.294249\n\nx gives\nlog\n\nx'\n\n==0.242768.\n\nHence we have\nz'\n\n= 27\n\nlog /\n\n20' 59\".6,\n\nf\nf\nand, repeating the operation, using these results for x and r ,\n\nlog x\"\n\n= 0.237678,\n\nlog x\n\nwe\n\nget\n\n= 0.242757.\n\n## correct value of log x may now be found by means of equation\n\nThus, in units of the sixth decimal place, we have\n(28).\nf\n\nThe\n\n= 242768\n\nand\n\n242737\n\n~ -f 31,\n\n## for the correction to\n\n= 242757\n\nbe applied to the\n\nlast\n\n242768\n\n11,\n\nvalue of log x f , in\n\nand from\n\nThese\n\nthis\n\nlog x'\n\n= 0.242760,\n\nlog s\"\n\n= 0.237681.\n\nderive\n\n2'\n\nz\"\n\nTo\n\nwe\n\nis\n\n=27\n\n= 29\n\n21'\n\n1\".2,\n\n312\n\n.9,\n\n## and x rf and give\n\n,\n\nlog/ =0.294242,\nlog r\" == 0.296087.\n\nwhich give\nr\n\nlog p\n\n= 0.133474,\n\n## Then, by means of the equations\n\nlog p\"\n\n= 0.289918.\n\nTHEORETICAL ASTRONOMY.\n\n300\n\n= cos\n=\np sin\n0')\n= tan\nr\" cos V cos (^ - 0\") = p\" cos\nr\" cos b\" sin\n\") = p\" sin\n\n/ cos V cos (J\nr' cos V sin (f\n/ sin 6'\n\n0')\n\n(A'\n\np'\n\n## find the following heliocentric places\n\n= 37\nr = 58\n\nI'\n\n'),\n\n/S',\n\n(/\"\n\nwe\n\nR,\n\n0')\n\n(A'\n\np'\n\n(A\"\n\n- 0\") - R',\n\n(A\"\n\n\"),\n\n8.182861 n\n\n35' 26\".4,\n\nlog tan\n\n5815.3,\n\nlogtan&\":=8.634209 n\n\n6' ===\n\nlog r'\nlog r\"\n\n= 0.294243,\n= 0.296087.\n\nThe agreement of these values of log r and log r\" with those obtained\ndirectly from x and x\" is a partial proof of the numerical calculaf\n\ntion.\n\nFrom\n\nthe equations\n\n+\n(^ +\n\ntan\n\nsin ( J (I\"\n\ntan\n\ncos (\n\nft )\n\n/')\n\nft )\n\nCOS\n\nwe\n\n= i (tan b\" tan\n\n&')\n\n6')\n\nsec\n\nCOS\n\n(I\"\n\ncosec J\n\nZ'),\n\n(\"\n\nO,\n\nobtain\n\n= 206\n= 190\n\n^\nw'\n\n42' 24\".0,\n\n55\n\nu\"\n\n.6\n\n= 4\n= 212\n\n36' 47\".2,\n\n20 53\n\n.5.\n\nThen, from\n\nwe\n\nget\nlog n\"\nlog w'\n\n= 9.806832,\n= 9.825408,\n\n=9.653048,\nlog w'\"\n\n= 9.633171,\n\n## and the equations\n\nf\n\nr sin ((u\n\nr cos ((u\n\nsin\n\n-u)\n\ntt)\n\n((\"'\n\nw-\n\ncos ((w\n\n+ % (u\" - u'))\n+\n\")\n\nJ-\n\ngn\n\n(u\"\n\n+ (\" _\n'\n\n1*) -f i\n\n(^ _ w\n\n'\n\n'\n\n- ~\n\ncos i (u\"\n\nsn\n\ncos \\ (u\"\n\n- u'\\\n\nNUMERICAL EXAMPLE.\ngive\n\n= 160\nu'\" = 24\nu\n\n=0.308379,\n\nlogr\n\n301\n\nlog/\" = 0.317273,\n\n30' 57\".6,\n\n5932.5.\n\n## Next, by means of the formulae\n\ntan\ntan\n\n&)\n\n(I\n\nSI )\n\n(/'\"\n\n= cos\n= cos\n\n/>cos(A\n\nQ)\n\np sin (A\np tan /?\n\nO)\n\nsin (X\"\n\n//\"tan/3\"'\n\nwe\n\ntan u,\ntan w'\",\n\ntan b\n\ntan V\"\n\n= rcos&cos(J\n=r cos 6 sin (7\n\n= tan\n= tan\n\nsin (\n\nsin (l\"\n\n&),\nr\n\n& ),\n\n0)+-^,\n0),\n\n= r sin 6\nm cos\ncos\n0'\") =\n(r\n=\ncos\nsin (r\n0'\")\n= r\"'8in&'\",\n;\n\n,'\" C os (A'\"\n/>'\"\n\ni\ni\n\nr'\"\n\nr'\"\n\n&'\"\n\n0'\")\n\n+ #\",\n\n'\"),\n\nobtain\n\n=\n=\nb\n+\nI\n\n*==\n\n16' 51\".8,\n\n32 14\n\n.4,\n\n6'\"\n\n=\n=-\n\n59\n\n.0,\n\nA'\"\n\n==\n\nj3'\"\n\n= 3 443.4,\n= 0.449258.\n\n16\n\n= + 2 5640.1,\nlog p = 0.025707,\n/9\n\nThe value of\n\nlog ?\"'\n\n91\n\n37' 40\".0,\n\n4 10 47\n75 23 46\n\n.4,\n.9,\n\nrrr\n\n## thus obtained agrees exactly with that given by\n\n/r\n.4 from the observed value.\nThis differobservation, but / differs\nence does not exceed what may be attributed to the unavoidable\nX\n\nplaces.\n\nThe\n\n## between the computed and the observed values of /9 and\n\nshow that the position of the plane of the orbit, as determined\n\ndifferences\nr/\n\n/9\n\nby means of the second and third places, will not completely satisfy\nthe extreme places.\nThe four curtate distances which are thus obtained enable us, in\nthe case of an orbit entirely unknown, to complete the correction for\naberration according to the equations (40).\n\nThe\n\nr/\n,\n\n;\ncalculation of the quantities which are independent of\n,\nand\nand\nare\nwhich\ntherefore\nthe\nsame\nin\nthe\nsuccessive\n,\nQ\",\n\nhypotheses, should\n\nThe\n\n'\n\nvalue of\n\n->\n\nrequired\n\nin\n\nfinding\n\nx\" from x\n\nmay be computed\n\ndirectly from\nSL\n\nf\nd\nthe values of\n\nP>\n\n^\nh'\n\n4\"*\"\n\nh''\n\nand\n\njj\n\n## being found by means of the equations (29)\n\nTHEORETICAL ASTRONOMY.\n\n302\n\nc\"\nand a similar method may be adopted in the case of ~.\n\nFurther,\n\n## in the computation of x' and x\", it may in some cases be advisable\n\nemploy one or both of the equations (22) for the final trial. Thus,\n\nto\n\nin the present case, x\" is found from the first of equations (27) by\ndifference of two larger numbers, and an error in the\nlast decimal place of the logarithm of either of these numbers affects\n\nmeans of the\n\nBut\n\n## in a greater degree the result obtained.\n\nas soon as\n\nr\"\n\nis\n\nknown\n\nQ\"\nso nearly that the logarithm of the factor 1\n\nremains unchanged,\n\n-^\n\n-f-\n\nthe second of equations (22) gives the value of x\" by means of the\nsum of two smaller numbers. In general, when two or more formula for finding the same quantity are given, of those which are\n\n## otherwise equally accurate and convenient for logarithmic calculation,\n\nwhich the number sought is obtained from the sum of smaller\nnumbers should be preferred instead of that in which it is obtained\nthat in\n\n## difference of larger numbers.\n\nvalues of r, r f , r f>', r f \", and u, u', u n ,\n\nby taking the\n\nThe\n\nfirst\n\nQ',\n\nand Q\".\n\n=\nr\nl~7\ntan* = X -p\nJc\n\nsin r cos\n\n(t\"\n\nr\"\n\ntan/'\n\nO,\n\nG = sin ^ (un\nG = cos i (u\"\n\n= cos J (u\"\n\ncos r\n\nsin\nsin\n\nfrom\n\nresult\n\n=~\n\n\\~r'\n\n1,\n\nG\"\n\ncos /'\n\n/\" cos\n/\" sin\n\nT 2 COS6 /\n\n=\ntan/\" =\nT'\"\n\n0,\n\nJc(t'\n\nu'\\\n\nG\" = cos i (u\"\n\nP'\n\nJc\n\n= sin J (u\n= cos ('\n= cos ^\n\n(H\"\n\nw\"),\nf\n\ntt \")\n\nsin\n\n2/\"\n\n^\"\n\n## r\" COS /'\n\n=r\n\nT'\"\n\n'\"'\n\n-77^\n3\n\n;/\n\nl-f^-Ff\n\n\"\n\n+/'-f^\n\n==\n\nCOS 6 /\"\ncos 3^77>\n/\"\n\nsin iy'\n\n7'\n\ncos/\"'\n\n'?'\"\n\nm\nin connection with Tables\n\nfind\n\n*,\n\n*\"),\n\nw) sin 2/\",\n\n(w'\n\nresults are\n\nu\\\n\nThe\n\nP\"\n\nw) cos 2/',\n\n= cos A (V \"\n\ncos r\"'\n\nwhich\n\nThus, from\n\nsin r sin\n\n## hypothesis, suffice to correct the assumed values of\n\nthe\n\nu\"\n\ns\",\n\nand\n\nNUMERICAL EXAMPLE.\n\n= 9.9759441,\n=\n45\n39\".l,\nx\n= 10 42 55\nlogm = 8.186217,\n= 7.948097,\nlog r\n\nlog\n\n32'\n\n/\"'= 45\n\n1\".4,\n\n13 45\n\n41' 55\".2,\n\nr'\"= 16 22 48\n\n.0,\n\n.5,\n\nlogm\"'= 8.590596,\nlog/\"= 8.325365,\n\n== 0.0085248,\n\nr'\"= 0.1800641,\n\nlog\n\nlog/'= 8.260013,\n\nlog,/\n\nlog\n\n44\n15\n\ny\"=\nlogm\"= 8.516727,\n\n.9,\n\n?-\n\nT\"= 0.1386714,\n\n/\"=\n\n3'\n\n303\n\nlog\n\n\"'== 0.0204063.\n\n~\n\n__\n\nr\"\n\n^\nP--L\n~\n\nr'\n\n77\n\n(%\"\n\nti')\n\ncos\n\n-J\n\nw) cos J\n\n(M\"\n\n(w'\n\nw)'\n\n'\n\nT'\"\n\n'\n\nr'r'\" cos\n\nwe\n\nu\") cos\n\n(it'\"\n\nO'\n\n\"\n\nu)\n\ncos J\n\n(it\"\n\nuj\n\nobtain\nlog P'\n\n= 9.8462100,\n\nlog P\"\n\n'\n\n9.8077615,\n\n## with which the next approximation\n\nWe\n\nnow recompute\nand the\n\nillustrated ;\n\n'\n\nlog\n\n',\n\nCQ\",/',/\n\n=s 0.5413485 n\n\nThen we\n\nV ==\n\nr/\n\n^ r,\n\noj\n\ncompleted.\n\n= 9.8076649\n= 9.8893851,\n= 0.2376752,\nlog\n= 29 14\".09,\nlog r\" = 0.2960826,\nlog/' =0.2899124,\nlog n\" = 9.8068345,\n= 9.6331707.\nlog n'\"\ne\n\n\"\n\nn,\n\naf'\n\n4'\n\nz'\n\n'\n\nmay be\n\nlog/\"\n\na/\n\nlog n\nlog n\n\nlog\n\nlog\n\nn,\n\n'\n\n== 9.8407536,\n\n\" =i\n9.8824728,\n\nresults are\n\nlog/' = 0.0476614\n= 0.2427528,\nlog\n= 27 21' 2\".71,\n=\n/\n0.2942369,\nlog\n\nlog P\n\nlog\n\n=0.1334635,\n\n= 9.6530445,\n= 9.8254092,\n\n3'\n\nobtain\n\n= 8.1828572\n\n37\n\n35' 27\".88,\n\nlog tan\n\nr=58\n\n5816.48,\n\nlogtan6\"=8.6342073 H\n\nThese\n\nfrom\n\nw,\n,\n\nlog /\nlog\n\n= 0.2942369,\n\n/'= 0.2960827.\n\nf\nresults for log r\n\nand\n\n## and log r\" agree with those obtained directly\n\nr\nthus\nz\",\nchecking the calculation of ty and ty and of\n\n## the heliocentric places.\n\nNext, we derive\nft\nu'\n\n= 206\n= 190\n\n42' 25\".89,\n\n55\n\n6 .27,\n\nu\"\n\n= 4\n= 212\n\n36' 47\".20,\n\n20 52\n\n.96,\n\nTHEORETICAL ASTRONOMY.\n\n304\nand from\n\nn and\n\nwe\n\nu\"u',\n\nr',\n\nr\", n, n\",\n\nn'\",\n\n',\n\n=0.3083734,\nlog/\" =0.3172674,\nlogr\n\nobtain\n\n= 160\n\nit'\"\n\n=244 5931.98.\n\n30' 55\".45,\n\n## For the purpose of proving the accuracy of the numerical\n\nwe compute\n\nalso, as in\n\nthe\n\nfirst\n\n1=\n\n716'51\".54,\n\n=+\n\nA=\n\n/?=\nlog p\n\n32 14\n\n16\n\n59\n\n.07,\n\n9 .38,\n\n56 39\n\n= 0.0256960,\n\nresults,\n\napproximation,\n\n.54,\n\nl'\"=\nb'\"=\n*'\"=\n\n91\n\nfi'\"=\nlog p'\"\n\n37' 41\".20,\n\n10 47\n\n75\n\n23 46\n\n.99,\n\n4 43\n\n.33,\n\n= 0.4492539.\n\n.36,\n\n//r\nthus found differ, respectively, only 0\".04\nvalues of A and A\nand 0\".09 from those given by the normal places, and hence the\n\nThe\n\n## accuracy of the entire calculation, both of the quantities which are\n\nand Q ff , and of those which depend on\n',\nindependent of P', P\",\nThis condition,\nthe successive hypotheses, is completely proved.\nhowever, must always be satisfied whatever may be the assumed\nf\nvalues of P', P\", Q , and Q\".\nf\nr\nFrom r, r , u, u , &c., we derive\n\nlog\n\n= 0.0085254,\n\n= 0.0174637,\n\nlog s\"\n\nlogP'\n\n= 9.8462110,\n\ndiffer so little\n\nQ',\n\nlog Q'\n\nThese values\n\nlog\n\n\"\n\nlog\n\nfrom those\n\ns'\"\n\n= 0.0204076,\n\n= 9.8407524,\n= 9.8824726.\n\n## for the second approximation,\n\nthe intervals of time between the observations being very large, that\na further repetition of the calculation is unnecessary, since the results\n\nwhich would thus be obtained can differ but slightly from those\nwhich have been derived. We shall, therefore, complete the determination of the elements of the orbit, using the extreme places.\nThus, from\nr\n\n=k(1f\n\nff\n\nsin YQ cos Cr\nsin YQ sin\n\ncos YQ\n\n(f\n\n_I_\n\ntan/ =\n\n-^t') )\n\n= sin (u\n= cos (u\"\n= cos J (um\n\nw'\"\\* r*c\\<v\n\n-J-\n\n-^\n\n'\n\nu),\n\nu) cos 2/\nu) sin 2/\n?0\n\nNUMEKICAL EXAMPLE.\nwe\n\nget\n.\n\nlog r\nro\n\nlog\n\nsQ\n\n= 0.5838863,\n= 42\n30\".17,\n= 0.2917731,\n=\n(\n\ngives\n\n= 8.0521953\n= 9.7179026,\n= 8.9608397.\n\nG\nlog m\n\nlog tan\n\n14'\n\nThe formula\n\n305\n\nn,\n\nlog x\n\n^\n=\n\n-)\n\n0.3712401;\n\nand\n\nif\n\nwe compute\n\nsr'r\" sin\n\nthe\n\narise\n\nu)\n\ns\"\n\nr'\n\nm sin\n\n(u\"\n\n- u\") V\n\nresults\n\nseparate\n\n0.3712414.\n\n(u\"\n\nby means of\n\n## are, respectively, 0.3712397, 0.3712418,\n\ndifferences between these results are very small,\n\nThe\n\nand\nand\n\n## and Q\" from the\n\ndeviation of the adopted values of P', P\",\n',\nlimit of accuracy attainable with logarithms of seven decimal places.\n\nr/\nf\nu and\nvariation of only O .2 in the values of u\naccordance\nof\nthe\nan\nentire\nresults.\nparticular\nproduce\n\nFrom\n\nu\"\n\nwill\n\nthe equations\nsin\n\ncos\n\nwe\n\nu rrf\n\n<f>\n\n\\(E'\"-E}\n\n= l/x\n\nobtain\nI (E'\"\n\nE)\n\n= 17\n\n35' 42\".12,\n\nlog cos\n\nThe\n\n<p\n\nlog (a cos ?)\n\n= 9.9915518.\n\n= 0.3796883,\n\nformulae\ne sin (a*\n\nA (>'\"\ni\n\ne cos (a,\n\ngive\n\n197\n? s 11\n\nai\n\n(u\"\n\n+ w)) = COS\n\n+ M)) = cos\n\n38' 8\".48,\n\n<p\n\n^\ny Vvf\"\n\nsec 1\n\nMw\n\nu),\n\n## = log sin ? = 9.2907881,\n\n= + ^ = 44 20' 34\".37.\n= 9.9915521, which differs only 3\nTT\n\ntan\n\nYffftt\n\nlog e\n\n15' 52\".22,\n\n^\nft\n\n<p\n\n01\n\n## from the value found from\n\nThen, from\n20\n\nand a cos\n\n<p.\n\nTHEOKETICAL ASTKONOMY.\n\n306\n\n= cos^p\n\nP-\n\n>\n\nr>\n\nal\n\n<?\n\nwe\n\nget\nlog a\n\nFor the\n\n= 0.3881359,\n\nlog fi\n\nIE =ta,n(u\n=tan(w'\n\ntan IE'\n\n## tan JjE\" == tan J (u\"\n\ntan J- (M'\"\ntan lE\"\nf\n\n= 2.9678027.\n\nwe have\n\neccentric anomalies\n\ntan\n\n= 3.5500066,\n\nw) tan (45\n\n^),\n\nw) tan(45\n) tan (45\n\n1\n\n-f\n\n30'\n\n6) sec J (f\"\n6) cosec J (f\"\n\n=4\n\n36' 49\".76.\n\n## latitude the results are\n\n35\".99,\n\nu'\"\n\n= 244\n\n59' 12\".53.\n\nQ,\nI),\n\nTHEORETICAL ASTRONOMY.\n\n308\n\nThe\n\nequations\n\ntan\n\n= tan\n\nb'\n\ngive\nlog tan\n\nb'\n\nsin\n\ntan i sin\n\ntan b\"\n\n= 8.1827129\n\n(I'\n\n(I\"\n\n& ),\n&\n\n),\n\n## log tan b\"\n\nn,\n\n= 8,6342104\n\nn,\n\nand the comparison of these results with those derived directly from\n/r\nl .04 in b\nand of\n0\".06 in\np' and p\" exhibits a difference of -f\nb\".\nHence, the position of the plane of the orbit as determined from\nr\n\n## the extreme places very nearly satisfies the intermediate latitudes.\n\nIf we compute the remaining elements by means of these values\n\nof\n\nr,\n\n## r'\" , and u, u' n', the separate results are\n\n= 8.0522282\n=\n0.2917731,\nlog\nlogjp = 0.3712405,\nlog (a cos ?) = 0.3796884,\nw = 197 37' 47\".72,\nV = 11 15 52\nlog a = 0.3881365,\nlog tan\n\nQ\ns\n\n= 9.7179026,\n= 8.9608397,\n=\n17 35' 42\".12,\nE)\n\\ (E\"\n=\ncos\n9.9915521,\nlog\nlog = 9.2907906,\n= 9:9915520,\nlog cos\nlog\n\nn,\n\nlog x\n\n<p\n\n.46,\n\n=329\n\n<p\n\nlog 11 == 2.9678019,\n\n= 39\n= 32\n=\nM\n\n11' 47\".24,\n\nE'\"\n\n34' 35\".70,\n\nM=3Z4\n\n55 40\n\n.46,\n\nM'\"\n\n26 45\n\n.49,\n\n29 40\n\n.36,\n\n29 40\n\n.37.\n\nM=\nQ\n\nEpoch\n\n## = 1864 Jan. 1.0 Greenwich mean time.\n\nM=\n*=\n\nOi=\n\n29' 40\".36\n\n44 20 32 .95^\n-,..!,\nAT\nEcll\nP tlc and Mean\n206 42 45\nI\n\n36 49-:S/\n\n= 11 15 52\n= 0.3881365\nP = 928\".5427.\n\n-,\n\nE ^-x\n\n1864.0.\n\n.46\n\nlog a\n\nIt appears,\ntherefore, that the principal effect of neglecting the\nextreme latitudes in the determination of an orbit from four observations is on the inclination of the orbit and on the\nof the\n\nlongitude\nThe\nascending node, the other elements being very slightly changed.\nelements thus derived\nrepresent the extreme places exactly, and if\nwe compute the second and third places\ndirectly from these elements,\nwe obtain\n\nM = 355\nE' ^=354\n= 353\nv\n\n33' 43\".88,\n\nM\" =\n\n29 12\n\n.93,\n\nE\"\n\n16 59\n\n.07,\n\n<y\"\n\n= 12\n^ 14\n\n44' 53\".73,\n\n5 34\n42 45\n\n.81,\n.96,\n\nNUMERICAL EXAMPLE.\nlog r\n\nu'\n\n35 27\n\n.75,\n\n52 21\n\n.25,\n\n= 0.2960826,\n= 212 20' 33\".68,\n= 58 58 16\n2 27 59\nb\" = -\n\n17\n\n.35,\n\nA\"=\n\n47\n\n.67,\n\n= 0.2942366,\n= 190 54' 46\".79,\n\nl'=\nV=-\n\n37\n\n309\n\n= 10 14\n15\nP== 0.1334634,\nlog\nA'\n\nlog r\"\n\nu\"\n\nI\"\n\n.50,\n.06,\n\n29\n\n53 21\n\n.99,\n\n57\n\n.62,\n\n= - 2 29\n=\n0.2899122.\nlog p\"\nP'\n\np'\n\nHence, the residuals for the second and third places of the planet\nare\n\nObs.\n\nComp.\n\n=\nAA\" =\nAA'\n\n*P\n\n0\".22,\n\n= + 1\".53,\n.06;\n\nA/5\"\n\n.00,\n\nand the elements very nearly represent the four normal places. Since\nthe interval between the extreme places is 223 days, these elements\n\nmust\n\n## represent, within the limits of the errors of observation, the\n\nIt\nentire series of observations on which the normals are based.\n\nmay\n\nbe observed,\n\ncase of intervals\n\n## same degree of rapidity\n\nas\n\napproximations, in the\n\n## do not converge with the\n\nwhen the intervals are small, and that in\nlarge,\n\n## much abbreviated by the\n\nassumed values of P', P\",\nf\nQ and Q\" by means of approximate elements already known. For\nthe first determination of an unknown orbit, the intervals will gene-\n\nis\n\nvery\n\n,\n\nquantities, as\n\nP\"\n\n_\n-I\n\nff\n\nff\n\nI\nS\n\n\"\n0\" ~ -Ir-'\"\n^\n'\n\nT \"*\n\n}'\n\n'\n\n## much from the correct values, and two or three\n\nor\neven\nBut when the intervals\nless, will be sufficient.\nhypotheses,\nare large, and especially if the eccentricity is also considerable, several\nhypotheses may be required, the last of which will be facilitated by\nwill not differ\n\n## using the equations (82) 4\n\nThe application of the formula? for the determination of an orbit\n.\n\n## from four observations, is not confined to orbits whose inclination to\n\nthe ecliptic is very small, corresponding to the cases in which the\nmethod of finding the elements by means of three observations fails,\n\nTHEORETICAL ASTRONOMY.\n\n310\n\n## On the contrary, these formulas\n\nor at least becomes very uncertain.\norbits\nof any inclination whatever,\nof\nin\ncase\nthe\nwell\napply equally\nand\n\n## when only three observed places\n\nare used, while the results must evidently be more approximate, it\nwill be expedient, in very many cases, to use the formulae given in\ndoes not\n\nmuch exceed\n\nthis chapter\n\nthat required\n\nfirst\n\nCIRCULAR ORBIT.\n\nCHAPTER\n\n311\n\nVI.\n\n## INVESTIGATION OF VARIOUS FORMULAE FOR THE CORRECTION OF THE APPROXIMATE\n\nELEMENTS OF THE ORBIT OF A HEAVENLY BODY.\n\n103.\n\nIN\n\nthe case of the discovery of a planet, it is often convehave been obtained for the determination\n\n## elliptic elements, to compute a system of circular elements, an\n\nephemeris computed from these being sufficient to follow the planet\nfor a brief period, and to identify the comparison stars used in dif-\n\nof\n\n## For this purpose, only two observed places\n\nare required, there being but four elements to be determined, namely,\nAs soon as\n2, i, a, and, for any instant, the longitude in the orbit.\na has been found, the geocentric distances of the planet for the\nferential observations.\n\ninstants of observation\n\nmay\n\n## be obtained by means of the formulae\n\n= R cos 4 + I/a\nA\" = E\" cos 4\" + I/a\nJ\n\nthe values of\n(43) 3\n\nand\n\nff\n\nTV\nD\n\nT~A\n\n~fV\n\ndA\n\n,_,\n\n'\n\n* 8\n\n## and the equations\n\ndo!\n\nTT\ndA\n\n315\n\ndY'^lT'\n\ndl^^W''\n\ngive\n\n(8)\n\nIn the same manner, computing the places for various dates, for\nwhich observed places are given, by means of each of the three systems\nof elements, the equations for the correction of A and A\" as determined by each of the additional observations employed, may be\n,\n\nformed.\n\n## For the purpose of illustrating the application of this method,\n\nus suppose that three observed places are given, referred to the\necliptic as the fundamental plane, and that the corrections for parallax,\naberration, precession, and nutation have all been duly applied.\nBy\n105.\n\nlet\n\n## means of the approximate elements already known, we compute the\n\nvalues of A and A\" for the extreme places, and from these the heliocentric places are obtained by means of the equations (71) 8 and (72) 3\nn\nThe values of\nwriting A cos/9 and A\" cos/9\" in place of p and p\nand\nu\"\nbe\nobtained\nmeans\nthe\nformula\nwill\nof\n& i, u,\nby\n(76) 3 and\nr\"\nand\nu\"\nu\nelements\nof the\nand\nfrom\nthe\nr,\nremaining\n(77) 3\n,\n\n'\n\n## orbit are determined as already illustrated.\n\nThe first system of elements is thus obtained. Then we assign an increment to A, which\nand A\"\nwe denote by D, and with the geocentric distances A -f\n\nwe compute\n\n## same manner a second system of elen\n\nand from A and\nments. Next, we assign to A\" an increment D\nA\" -J- D\" a third system of elements is derived. Let the geocentric\nlongitude and latitude for the date of the middle observation computed from the first system of elements be designated, respectively,\n'\nby ^/ and /9/ from the second system of elements, by X2 and /92\n'\nand from the third system, by ^ 3 and /33 '. Then from\nin precisely the\n\n',\n\na\"\n\nwe compute\nof\n\n'\n\n(A 3\n\n- A/) cos\nf/\n\na,\n\nD and D\"\n\n/?/,\n\nd\"\n\n= ft'\n\nft',\n\n## a\", d, and d , and by means of these and the values\n\nthe equations\n\nwe form\n\nTHEORETICAL ASTRONOMY.\n\n316\n\n^AJ + ~A/I\"\n(10)\n\n## assumed values of A and J\", by means of the differences between\n\nThe observed longitude and latitude\nobservation and computation.\nA' and /9', respectively, we shall have\ndenoted\nby\nbeing\n\nCMp*X = (l\nA/5'\n\nA/) cos/?,\n\n= /5'_/3/,\n\n(10),\n\n## and then by elimination we obtain the values of the corrections A J\n\nand A//\" to be applied to' the assumed values of the distances.\nFinally, we compute a fourth system of elements corresponding to\nthe geocentric distances A -f- AZ/ and A\" -j- A A\" either directly from\nthese values, or by interpolation from the three systems of elements\n\n## already obtained ; and, if the first assumption is not considerably in\n\nIt\nerror, these elements will exactly represent the middle place.\nshould be observed, however, that if the second system of elements\n'\n'\nrepresents the middle place better than the first system, ^ 2 and /32\nshould be used instead of ^/ and /9/ in the equations (11), and, in\nthis case, the final system of elements must be computed with the\n\n+D\n\ndistances A\n-f A A and A\" -f- A A\".\nSimilarly, if the middle\nplace is best represented by the third system of elements, the corrections will be obtained for the distances used in the third hypothesis.\n\nIf the computation of the middle place by means of the final elestill exhibits residuals, on account of the\nneglected terms of\n\nments\n\n## the second order, a repetition of the calculation of the corrections\n\nAJ and AJ\", using these residuals for the values of the second\n\nmembers of the equations (10), will furnish the values of the distances for the extreme places with all the\nThe\nprecision desired.\nincrements _D and\nto be assigned successively to the first assumed\n\nD\"\n\nvalues of\n\ncomputed\n\nand\n\nlittle\n\ndifficulty,\n\nit\n\n## stead of the geocentric distances\n\nthemselves, the logarithms of these\ndistances, and to express the variations of these quantities in units\nof the last decimal place of the\n\nlogarithms.\n\n## These formulae will generally be\n\napplied for the correction of\n\n317\n\n## approximate elements by means of several observed places, which\n\nbe either single observations or normal places, each derived from\n\nmay\n\nseveral observations,\n\n## and the two places selected for the computation\n\nA and A\" should not only be the most accurate\n\n## of the elements from\n\npossible, but they should also be such that the resulting elements are\nnot too much affected by small errors in these geocentric places.\n\nas distant\n\n## other considerations not being overlooked.\n\nWhen the three systems\nof elements have been computed, each of the remaining observed\nplaces will furnish two equations of condition, according to equations\n(10), for the\n\n## determination of the corrections to be applied to the\n\n;\nand, since the number\n\n## of equations will thus exceed the number of unknown quantities,\n\nthe entire group must be combined according to the method of least\nThus, we multiply each equation by the coefficient of AJ\nsquares.\nits proper algebraic sign, and the sum\nthe equations thus formed gives one of the final equations\nThen we multiply each equation by the coefficient of A A\"\nrequired.\nin that equation, taken also with its proper algebraic sign, and the\n\nof\n\nall\n\nsum\n\nof\n\nFrom\n\nthese\n\n## two final equations, by elimination, the most probable values of A J\n\nand A A\" will be obtained and a system of elements computed with\nthe distances thus corrected will exactly represent the two funda;\n\nmental places selected, while the sum of the squares of the residuals\nThe observations are thus\nfor the other places will be a minimum.\nsupposed to be equally good; but if certain observed places are\nentitled to greater influence than the others, the relative precision\nof these places must be taken into account in the combination of the\nequations of condition, the process for which will be fully explained\nin the next chapter.\n\n## of the approximate elements of the orbit of a planet or comet, it will\n\nbe most convenient to adopt the equator as the fundamental plane.\nIn this case the heliocentric places will be computed from the assumed\nvalues of A and A\", and the corresponding geocentric right ascensions\nand declinations by means of the formulae (106) 3 and (107) 3 and the\nposition of the plane of the orb' 5 as determined from these by means\n;\n\nof the equations (76) 3 will be referred to the equator as the fundamental plane. The formation of the equations of condition for the\ncorrections A A and A A\" to be applied to the assumed values of the\ndistances will then be effected precisely as in the case of ^\n\nand\n\n/9,\n\nthe\n\nTHEORETICAL ASTRONOMY.\n\n318\n\nIn a similar manner,\nnecessary changes being made in the notation.\nthe calculation may be effected for any other fundamental plane which\n\nmay be\n\nwhen\n\nthe ecliptic\n\nis\n\ntaken as\n\n## the fundamental plane, the geocentric latitudes should be corrected\n\nby means of the equation (6) 4 , in order that the latitudes of the sun\nshall vanish, otherwise, for\n\nstrict accuracy,\n\n(89),\n\n106.\n\nThe\n\nmeans of\n\n## ordinates with respect to\n\ndifferential\n\nnumerical calculation\n\nis\n\ndirectly\n\nby\n\n## special cases, the\n\nless expeditious than in the case of the indi-\n\nformulae;\n\nbut, except\n\nfor\n\n## method, while the liability of error is much greater. If we\n\nadopt the plane of the orbit as determined by the approximate values\nrect\n\nof A and\n\nA\"\n\nas the\n\n## elements of the orbit, as in the equations (72) 2 the variation of the\n\ngeocentric longitude d measured in this plane, neglecting terms of the\nsecond order, depends on only four elements; and in this case the\n,\n\n## may be applied with facility. Thus, if we exr\n\nv\nand\nin\nterms\nof the elements <p,\nand //, we shall have\npress\nQ,\ndifferential formulae\n\ndr\n\ndr\n\ndr\n\nd(p\n\ndM\n\ndr\n\ndM\n\ndv\n\ndfj-\n\nand\ndv\nJ\n\ndv\n*\n\n~J~T\n\ndv\n\ndy\nJ\n\n'\n\nJ Hf\n~dJL\n\n\"!\"\n\n~dA\n\ndfj.\n'\n\nT J\nJ A\nJ A\n~dT\n^fa'dA'\n'\n\n..\n\nor\n\nd (v\n\n-f-\n\n/)\n\ndA\nIn\n\nlike\n\ndv\n\nd<p\n\ndv\n\ndM\n\ndv\n\ndjj.\n\nd<p\n\ndA\n\ndM\n\ndA\n\ndn\n\ndA'\n\nd%\n\ndA\n\nmanner, we have\n'\n\ndA\n\ndA\n\nd<p\n\n'\n\ndA\n\nAs\n\nd<f>\n\n'\n\ndM\n\n'\n\ndM\n\ndr\n\n~djl\n\ndA\n\nd(v\n,\n\n'\n'\n\n'\n\n'\n\nTT\ndA\n\ndr\"\n~\n\ndA\n\n_ ~\n\ndA\n\ndM\n\ndr\"\n\ndr^_dr^ d^\n\n+ y)\n\n~~^-,\ndA\n\ndp\n^A'\n\n'\n\ndn\n\ndA\n\ndr\"\njj,\ndA\n\nand\n\n,,\ndA\n\nare\n\n## known, the equations necessary for finding the differential coefficients\n\nof the elements ^,\nand p with respect to A are thus provided.\n0)\nIn the case under consideration, when an increment is\nto\n<f>,\n\nassigned\n\nJ,\n\nthe value of\n\n319\n\nand hence\n.\n\ndA\n\nTo\n\nfind\n\n-7-7-\n\nand\n\nA cos f] cos B\nJ cos sin\n>?\n\nin\n\nwhich\n\norbit\n\ndA\n\n=x\n=y\n\n-f- -3T,\n-\\-\n\nY,\n\n## the geocentric latitude in reference to the plane of the\n\ncomputed from A and A\" as the fundamental plane, and y Y\n57\n\nis\n\nget\n\nwe\n\n## = cos y cos d dA,\n\ndx\n\ndA,\nor, substituting for\n\n## dx and dy their values given by\n\ncos\n\n?)\n\ncos\n\ncos\n\nfj\n\nsin 6\n\n= cos u dr\n= sin udr\n\ndA\ndA\n\nEliminating, successively, d (v\ndr\n\n-f-\n\nf)\n\n= -i cos\nTherefore,\n\ndy\n\nwe\n\nshall\n\ndv\nj\n\nd<p\n\nc?r,\n\nCOS (0\n\nT?\n\nu d (v\n\nr cos u\n\nand\n\n^)\n\n= COS\n\nr sin\n-\\-\n\nd (v\n\nwe\n\n(73),,\n\n-f-\n\n/),\n\n-f-\n\n/).\n\nget\n\nU) t\n\nsm (0\n\nit).\n\nhave\n\ndM\n\ndv\n\ndv\n\nTT\ndA\n\nTT^J-\n\nd<p\n\ndr\n\nd<p\n\ndr\n\ndy.\n\n~r-Tdp dA\n\n^7\ndA\n\ndM\n\n7\n(^\nTT\n,\n\ndfj.\n\n^j^T H\n\n~j\n\nw )>\n\n'\n,\n\nj__\n\"\n\n'\n\ndr\"\n'\n\ndip\n\nand\n\nif\n\n.A.\n\n'\n\n'\n\nwe compute\n\n'\n\ndA\n\nd^,dr^_\ndM^\n+\n\ndM\n\ndA\n\n'\n\n~dl~\n\n'\n\ndfi\n\ndr\"\n'\n\n'\n\ndp\n\n''\n\n'\n\ndA\n\n^_\n\n'dA~\n\n## the numerical values of the differential coefficients\n\nthese\nand\nv, and v\" with respect to the elements <p,\nequations will furnish, by elimination, the values of the four un-\n\nof\n\nTJ r'\n\nknown\n\n/Ji,\n\n1\nand -j-rJ-T> -rr\ndJ c?J\ndJ\ndJ\nIn precisely the same manner we derive the following equations\n\nquantities -yp\n\nTHEORETICAL ASTRONOMY.\n\n320\n\nA\"\n\ndv\n\nd%\n'dA\"\n\n^\ndx\n\ndM\n\ndv\n\nd<p\n\ndA\"\n\n~dv\n\n~\"\n\ndv\n\ndMl dA\"\n\n~*~\n\ndr\n\nd<p\n\ndr\n\ndM\n\ndr\n\ndp.\n\ndA\"\n\ndM\n\ndA\"\n\nd/j.\n\ndA\"\n\ndv\"\n\nd<p\n\ndv\"\n\ndM\n\ndv\"\n\nd/j.\n\n+ ^'^ + lMr^ +\n'\n\ndM^\n_d^.dr^_\n+\n\n'\n\ndM\n\ndA\"\n\ndA\"\n\n'\n\ndA\"\n\n~dp.\n\nd<p\n\ndf_\n~d^\n\ndp.\n'\n\n'\n\n'\n\n\"*\n\n_\n,\n\n^ ^\"^\ndr^\n\n'\n\nfa\n\nfa\ndA\"\n\n## Since the geocentric latitude y is affected chiefly by a change of the\n\nposition of the plane of the orbit, while the variation of the longitude\n6 is independent of Q> and i when the squares and products of the\nvariations of the elements are neglected, if\n\nwe determine\n\nthe elements\n\nwhich exactly represent the places to which A and A\" belong, as well\nas the longitudes for two additional places, or, if we determine those\nwhich satisfy the two fundamental places and the longitudes for any\n\nnumber of\n\n## additional observed places, so that the sum of the squares\n\nminimum, the results thus obtained will\n\n## very nearly satisfy the several latitudes.\n\nLet 6 f denote the geocentric longitude of the body, referred to the\nplane of the orbit computed from A and A\" as the fundamental plane,\nfor the date\n\nof any one of the observed places to be used for corThen, to find the partial differential\n\n## recting these assumed distances.\n\ncoefficients of 6' with respect to\n,\n\ndtf\n\nj A\ndA\n\nCOS^\n\n,dO'\nfj\n\ndx\n7\n\nd/\n\ndA\n\nr~\n\nCOS\n\ndp.\n\ndO'\n\ndO'\n\ndy\n\nCOS\n\nif\n\nd<p\n\ndA\n\ndO'\n\nCOS\n\nfl\n\ndM\n\ndtf\n\ndMr\n-.\n\n-.\n\n'\n\ndA\n\ndp.\n\ndA\n\n(15)\ndO'\n\nd<p\n\ndp.\n\nA0'\n\nr~\n\n-,:\n\ndO'\n\n,dO'\nd<p\n\n+ CQ ^,dO'\n/\n\nfl\n\nwe form\n\nif\n\ndA\n\n## fourth observed place will furnish, in the same\n\ntional equation required for\nfinding A J and A A\".\n\nthe equation\n\nAzT.\n\n(16)\n\n321\n\n## observations are used in addition to the fundamental places on which\n\nthe assumed elements as derived from A and A\" are based, the several\nlongitudes will furnish each an equation of condition, and the most\nprobable values of A// and A A\" will be obtained by combining the\nentire group of equations of condition according to the method of\nleast squares.\n\n## of the approximate elements, after all the preliminary corrections\n\nhave been applied to the data, we select the proper observed places\n\ntances\n\nA and\n\nstated,\n\nand\n\ndis-\n\n## A' f j according to the conditions which have already been\n\nfrom these we derive the six elements of the orbit. Since\n\n## the data furnished directly by observation are the right ascensions\n\nand the declinations of the body, the elements will be derived in\nreference to the equator as the plane to\n\nwhich the\n\ninclination\n\nand the\n\n## longitude of the ascending node belong. These elements will exactly\n\nrepresent the two fundamental places, and, if the assumed distances\nA and A\" are not much in error, they will also very nearly satisfy\nthe remaining places.\nnow adopt as the fundamental plane the plane of the approximate orbit thus determined, and by means of the equations (83) 2 and\n\nWe\n\n(85) 2 or\n,\n\n&\n\nand\n\nby means of\n\n## (87) 2 , writing a, d, &',\n\nwe compute\n\ni,\n\nthe values of\n\nrespectively,\ndates of the several places to be employed.\n,\n\nand\n\nin place of\n\n#,\n\nThen\n\n37,\n\nand f\n\n^, ft,\n\nfor the\n\ncos\n\nif)\n\n= sin f A\n= cos Y\n\nA0\n\n-}~\n\nA<5\n\nAT?\n\n## Aa and A for each place being found by subtracting\n\nfrom the observed right ascension and declination, respectively, the\nright ascension and declination computed by means of the elements\n\nthe values of\n\nderived from\n\nA and A\".\n\nThe\n\n## 0, y, and f being required\n\nonly for finding cos r] A#, A\"^, and the differential coefficients of d and\nwith respect to the elements of the orbit, need not be determined\n*y,\n\nvalues of\n\nNext, we compute\n,+\n\nr>\\\n\n-i\n\n## (16) 2 the values of\n\n-r-\n\ndr\n\ndr\"\n,\n\ndtp\n\nand -:\n\ndy>\n\n'\n\nfrom equations\n\ndv\n\ndr\n\ndv\"\n\nd<f>\n\nd(f>\n\ndM\n\n&c,,\n\n(12),\n\nand from\n/\n\nby means of which,\n\nusing the value of u in reference to the equator, we form the equaThe accent is added to % to indicate that it refers to the\ntions (13).\n21\n\nTHEORETICAL ASTRONOMY.\n\n322\n\nelements.\nThus we obtain four\nequator as the plane for defining the\nthe\nvalues\nof the differential\nfrom\nelimination,\nwhich, by\nequations,\ncoefficients\n\nof\n\n#',\n\nthe\n\nfirst,\n\n-Tj,\n\nand\n\n/z\n\nwith respect to\n\nis\n\nquantity -rj\n\nbe obtained.\n\n-J-T.\n\nA may\n\nunknown\n\nthe\n\nwe have\n\n3/ and\n\n<p,\n\nIn the numerical\n\n-j-\n\n## first or from the third equation.\n\nIn the same manner we form the equations\n\nunknown\n\nmay\n\nquantities\n\nbe obtained from\n\nthe\n\nthe values of\n\nmula\nto be\n\nanc\nThen, by means of the for-rp ~jW>\nJj 77\nand (79) 2 we compute for the date of each place\n'\n\n## and thence derive\n\n-T-p\n\n,\n(76) 2 , (78) 2 ,\nemployed in correcting the\n\nand cos if\n\n(14),\n\n,,\n\n&c.,\n\nThe\n\n## assumed distances the values of\n\n(15) the values of COST/-J-\n\nCfr^J\n\n## (17), enable us to form, accordto\n\nthe\nof\ncondition\nfor finding the values of the\ning\n(16),\nequations\ncorrections AZ/ and &A\"\nThe solution of all the equations thus\n.\n\nmethod of\n\n## least squares, will give the most\n\nvalues\nof\nthese quantities, and the system of elements which\nprobable\ncorresponds to the distances thus corrected will very nearly satisfy\n\n## expressed in seconds of arc, the resulting values of A J and A A\" will\n\nalso be expressed in seconds of arc in a circle whose radius is\nequal\nto the\n\nmean\n\nTo express them in\ndivide their values in seconds of\n\narc by 206264.8.\n\nwe must\n\n## The corrections to be applied to the elements computed from A and\n\nA\", in order to satisfy the corrected values A -\\- A A and A\" 4- A A\",\nmay be computed by means of the partial differential coefficients\n\nmay be\n\nobtained.\n\nwe compute\n\nAJ('\n\nIf,\n\n',\n\nwe have\n\n^.t*\ndA\n\ntup,\n\nAlfc and\n\nand\n\n## VARIATION OF TWO GEOCENTRIC DISTANCES.\n\n323\n\nand apply these corrections to the values of v and v\" found from A\nand A\" we obtain the true anomalies corresponding to the distances\nA -f A A and A\" -\\- A J\". The corrections to be applied to the values\nof r and r\" derived from A and A\" are given by\n^\n\ndr\n\nar\"\n\n## If AJ and A A\" are expressed in seconds of arc, the corresponding\n\nThe corrected\nvalues of Ar and Ar /r must be divided by 206264.8.\nwith the values of r and r\" comfrom\ncorrected\nthe\nvalues\nof V, v ff p, and e by means\nputed directly\nof the polar equation of the conic section. Finally, we have\nresults thus obtained should agree\n\ndz\n\nand similarly\n\ndz\"\n\nfor\n\nT sin\n\nand the\nr cos\n\nAi'\n\nV\nto find A^\n\nlast\n\nsin\n\nr\" cos\n\ni'\n\nfrom which\n\n= sin\n\nand A\n\ni'\n\nit\"\n\n',\n\nTJ\n\ndA,\n\n'\n\nit\n\ni'\n\nAw'\nATT'\n\nA &'\n\nand\n\n(?2) 2 ,\n\n= sin A\n= sin V\n17\n\nsin\n\nit\"\n\nJ,\n\n## being the arguments of\n\nWe have also,\n\naccording to\n\n= A/ COS A&',\n= A/ + 2 Sin ^' A &',\ni'\n\n## from which to find the corrections to be applied to co f and TT'. The\n\nelements which refer to the equator may then be converted into those\nfor the ecliptic by means of the formula which may be derived from\nV and i.\n(109)! by interchanging & and &' and 180\n\nThe\n\n## final residuals of the longitudes may be obtained by substithe\n\nadopted values of A A and A A\" in the several equations of\ntuting\ncondition, or, which affords a complete proof of the accuracy of the\n\n## by direct calculation from the corrected elements\n\nand the determination of the remaining errors in the values of y will\nshow IIOAV nearly the position of the plane of the orbit corresponding\nentire calculation,\n\n## to the corrected distances satisfies the intermediate latitudes.\n\nw and //, we may introduce any other elements\nwhich determine the form and magnitude of the orbit, the necessary\n\nTHEORETICAL ASTRONOMY.\n\n324\n\n## changes being made in the formulae. Thus, if we use the elements\n\nT, q, and e, these must be written in place of Jf //, and <p, respectively, in the equations (13), (14), and (15), and the partial differential\n,\n\n## r, r\", v, and v\" with respect to these elements must be\n\nof the various differential formulae which have\nmeans\ncomputed by\n\ncoefficients\n\nof\n\n## Further, in all these cases, the\n\nmust\nbe carefully attended to.\nthe\nformulae\nof\ngeneity\n108.\n\n## The approximate elements of the\n\nhomo-\n\norbit of a heavenly\n\nbody\n\n## by varying the elements which fix the position\n\nmay\nof the plane of the orbit. Thus, if the observed longitude and latitude and the values of\nand i are given, the three equations (91) t\nalso be corrected\n\n&\n\n## will contain only three unknown quantities, namely, J,\n\nthe values of these may be found by elimination.\n\nr,\n\nand\n\nu,\n\nand\n\nlatitude\n\n/9\n\nis\n\ncorrected\n\n## (6) 4 , the latitudes\n\nof\n\nthe sun disappear from these equations, and if we multiply the first\nby sin (O\nQ) sin ft the second (using only the upper sign) by\ncos\n\nthe products,\n\nwe get\n\nthe third by\n\n__\n\ncos\n\nsin\n\nsin\n\nsin\n\n/5\n\ncos\n\n(0\n\n&)\n\n(O\n\nsin (A\n\nO)\n\ncos ft\n\nft)\n\nsin i cos\n\n/?\n\nsin (A\n\nO)'\n\n## from which u may be found. If we multiply the second of these\n\ncos ft sin (A\nequations by sin ft and the third by\nproducts,\n\nwe\n\nfind\n\nThe\n\n= sin u\n\n(sin\n\ncot\n\nft\n\nsin (A\n\nand\n\n-Rsin(Q-ft)\n\n-.\n\nfirst\n\nn sin\nthe formula for\n\nThe\nsame\n\nN=\n\n^)\n\nknown\n\ncos i)\n\nquantities\n\nmay\n\nalso be\n\nIf we put\nsin/3cos(O\ncos\n\n/?\n\nsin (A\n\nfirst\n\n&\nQ),\n\nu becomes\n\n## of equations (91) t shows that sin u and\n\nsin/9 must have the\nsign, and thus the quadrant in which u must be taken is deter-\n\nlast\n\nmined.\n\nPutting, also,\n\nmcos 3f\n\n= sinw,\n\n/5\n\nsin (A\n\n&),\n\n325\n\nwe have\n\n__\n\ncos\ncos\n\nWhen\n\nJf\n\nJt?sin(Q\n\n(M + i)\nis\n\n'\n\nsin\n\n## taken as the fundamental plane, the\n\nlatitude of the sun (which will then refer to this plane) will be retained in the equations (91)! and in the resulting expressions for u\n\nand\n\nr.\n\n## also be obtained by first computing w and ^\n\nof\nthe\nmeans\nequations (42) 3 and then, if z denotes the angle at\nby\nthe planet or comet between the earth and sun, the values of u and\nwill be determined by means of the relaz, as may be readily seen,\ntions of the parts of a spherical triangle of which the sides are\n180\n180\nO &, and u^ the angle opposite to the\n(z + oj/),\nside u being that which we designate by w and the side 180 + O\nSI\n(z + \\^),\nbeing included by this and the inclination i. Let 8= 180\n\n(23)\n\nfrom which\n\n8 and u\n\nz\n\n= 180\n=\n\nfl\n\nThen we have\n\ntS,\n\nC 2 *)\n\nB in4,\n\nsing\nto find r.\n\n## If we assume approximate values of & and *, as given by a system\n\nof elements already known, the equations here given enable us to find\nrf\nand u\" from ^, /? and A\" /9 /; , corresponding to the dates t\nr, u, r\n',\n\nand\ntwo\n\nt\" of the\n\nfundamental places\n\nelements\n\nbody may\n\nmay\n\nselected,\n\nbe derived.\n\nFrom\n\nresults for\n\nlatitude, the\n\nremaining\n\n## these the geocentric place of the\n\nobserved place, and the difference between the computed and the\nobserved place will indicate the degree of precision of the assumed\n\nCOS\n\nA/ ==\n\n/?'\n\nCOS\n\n&\n\nand\n\np -^- A\n\ni.\n\n-f-\n\nCOS\n\nThus\n\nthe equations\n\nAl,\n\n(25)\n\n'\n\n## for finding the corrections A& and A^ to be applied to the assumed\n\nvalues of these elements, will be formed ; and each additional observation or normal place will furnish two equations of condition for\n\n## If the observed right ascensions and declinations are used directly\n\ninstead of the longitudes and latitudes, the elements\nand i must\nbe referred to the equator as the fundamental plane, and the declinations of the sun will appear in the formula for u and r obtained from\n\n## the equations (91)^ thus rendering them more complex.\n\nTheir derivation offers no difficulty, being 'similar in all respects to that of the\nequations (19) and (20), and since they will be rarely, if ever, reit is not\nIn\nnecessary to give the process here in detail.\ngeneral, the equations (23) and (24) will be most convenient for\nquired,\n\nu from the\n\nfinding r and\n\nelements\n\n&\n\nand\n\nit\n\n## geocentric spherical co-ordinates and the\n\nw fr , and o|/ r remain unchanged for the\n\nsince w, ^,\n\nthree hypotheses.\nWhen the equator\n\n## is taken as the fundamental plane, ^ is the\n\ntwo points on the celestial sphere for which the\nand a, d, those of the sun\ngeocentric spherical co-ordinates are A,\ndenoted\nA\nand\nD.\nHence\nwe\nshall\nhave\nbeing\nby\n\ndistance between\n\nsin\n\n4>\n\nsin\n\nsin\n\ncos\n\ncos\n\n4-\n\nB = cos d sin (a\nB = cos D sin d\n\n= sin D sin\n\nA),\nsin\n-j-\n\ncos\n\nD cos 8 cos (a\nD cos d cos (a\n\nA),\n\n(26)\n\nA),\n\nit\n\nwhile for\n\n6\nsin ie\n\n+ &c.,\n\n## is most convenient to use the\n\nand Q ; but, since the cases of\nthose which do occur the eccenf\n\n## greater than that of the parabola, it will be sufficient to tabulate Q\n\nThe same table,\ndirectly with the argument m.\n\ntricity is\n\nusing\n\nvery\n\nlittle\n\n## as the argument, will give the value of Q'.\n\nTable\nfrom\ncorresponding to values of\n\nm = 0.2.\nWhen\nis\n\n% from\n\nr',\n\nthe values of r\n\nrequired,\n\n+r\n\nff\n,\n\nr',\n\nXVI.\n\nm=\n\nto\n\n## and a are given, and the chord Jt\n\nfrom (77), and finally\n(78), r\n\n(73).\n\nIt\n\nmay\n\n+ r\",\n\n## be remarked, also, that the formulae for the relation between\n\nr\"\nx, and a suffice to find by trial the value of a when r\n\n349\n\n## J and J\", the value of x may be computed from r, r\", and\n\nthen a may be found in the manner here indicated.\nand\nu,\nIf we substitute in the equations (84) the values of sin Je and sin\nr\" i x> and a, and then substitute the resulting values\nin terms of r\nvalues of\n\nu\"\n\nof\n\n(65),\n\nwe\n\nobtain\n\n1\n\nwhen u\"\n\nu exceeds 180.\n\nIn the\n\ngreat rapidity.\n\n(85)\n\nWhen\n\nthe\n\nconverges with\n\na must\n\nbe changed.\n119.\n\nThe\n\nand hyperbolic\n\nan\n\n## approximate orbit by varying the semi-transverse axis a and the\n\nof two curtate distances. But since the formula? will generatio\n\n/=-;\n\nfirst place,\n\n## will be expedient to use\n\nand\n\nbe determined.\n\nas the quantities to\n\nIn the\n\nit\n\nwe compute\n\nM and\n\n## and, for the determination of the auxiliary quantities pre-\n\nliminary to the calculation of the values of r, r\", and x, the equations (41) 3 (50) 3 , and (51) 3 will be employed when the ecliptic is the\nfundamental plane. But when the equator is taken as the funda,\n\n## mental plane, we must first compute g, K, and G by means of the\n\nThen, by a process entirely analogous to that by\nequations (96) 3\nwhich the equations (47) 3 and (50) 3 were derived, we obtain\n.\n\nh COS C COS\nh cos C sin\n\n(H\n(H\n\na\")\n\ncos\n\n<p\n\nCOS (a\"\n\nh sin C\n\n=M\n\n= M tan 8\"\n\nand h ; and\n\na),\n\n(86)\n\na),\n\ntan\n\n8,\n\nalso\n\nH)\n\n-f sin C sin\n\nK,\n\n(87)\n\n## will be referred to the\n\nand\nfrom which to find tp. In this case,\nThe angles ^ and ty f will be\nequator as the fundamental plane.\nobtained from the equations (102) 3 or from equations of the form\n,\n\nTHEORETICAL ASTRONOMY.\n\n350\n\n## of (26), and finally the auxiliary quantities A, 5, J5\", &c. will be\n\nobtained from (51) 3 , writing d and d\" in place of /9 and /?\", respectively.\n\n## -As soon as these auxiliary quantities have been determined, by\n\nthe value of K must be found which will exactly\nmeans of\n3\n(52)\n\nand, if\n\nit\n\nTo\n\neffect this,\n\nwe\n\nfirst\n\nfrom\n\ncompute\n\n+ r\"\n\nand\n\nx.\n\nThen we\n\nfind\n\nfrom\n\n## or from (78), the logarithms of the auxiliary\n\n(66), and Ar/ from (76)\nfound by means of Table XV. with the\nand\nB\nbeing\nQ\nquantities\nr ' having been found from (77), the\nof\nThe\nvalue\ne.\nargument\n\n## equations (73) and (74), in connection with Table XI., enable us to\n\nWith this\nobtain a closer approximation to the correct value of K.\n\nwe compute new\ntrials\n\n## values of r and r\", and repeat the determination\n\ngenerally give the correct result, and these\n\nof K.\n\nmay be\n\nfacilitated\n\n## by the use of the formula\n\n(67) 3\n\nIt will be\n\n'\nare very slightly changed by a small\nobserved, also, that Q and Ar\nr\nr\" and tf, so that a repetition of the\nchange in the values of\n\n## calculation of these quantities only becomes necessary for the final\n\ntrial in finding the value of Jf which completely satisfies the equaWhen the value of a is such that the values\ntions (52) 3 and (65).\n\nQ and exceed the limits of Table XV., the equation (61) may\nbe employed, and, in the case of hyperbolic motion, when Q and Q f\nexceed the limits of Table XVI., we may employ the complete exand n as given by (79).\npression for the time r' in terms of\nof\n\nThe\n\nvalues of\n\nr,\n\nr ff ,\n\nP\n\np and p\".\n\nWhen\n\nthe equator\n\nis\n\nwe have\n\n= A cos\n\n\"\n<S,\n\n= A\" cos\n\n3\".\n\n## From /?, p\", and the corresponding geocentric spherical co-ordinates,\n\nthe radii-vectores and the heliocentric spherical co-ordinates I, I\", b,\nand b\" will be obtained, and thence &,\n\ni,\n\nu, u\",\n\n351\n\n## In the case of elliptic\n\nelements of the orbit, as already illustrated.\nif\nwe compute the auxiliary quantities e and d by means of\nmotion,\nthe equations (60), we shall have\n\n(E\" + E)\n\ne cos\n\nfrom which\n\n\\(E\"\nmay\n\nE} = i(e\n\n3),\n\nwe\n\n= cos J +\n(e\n\nd),\n\n## be found, and hence, since\n\nand E\". The values of q and\n\nE} may\n\nderive\n\n## then be found directly\n\nobtained.\nThus, the last of equations (43)! gives\n\ncos\n\n_ cos \\E\n\n-Jv\n\nv\"\n\ncos\n\n'\n\nV~r\n\nV~q\n\n_ cos &E\"\n1/7\n\nV~q\n\n## first of these expressions by sin Ju\", and the second\n\nadding the products, and reducing, we obtain\n\nMultiplying the\n\nby\n\nsin Jv,\n\nsin J\n\nv) sin \\v\n\n(i/'\n\n_ cos\n\nj-\n\n(i/'\n\nV~q\nTherefore,\n\nwe\n\nshall\n\ncos | J?\n\nE\"\n1/7\n\ncos\n\nhave\ncos\n\ncos Av =\n\nVq\n\ni;)\n\nVv\n\n1/r tan\ncos i-E\n\nE\n(u\"\n\ncos\n\nu)\n\nVr\" sin\n\nE\"\n(it\"\n\nw)\n\n1/ r\n\nfrom which q and v may be found as soon as cos \\E and cos \\~E\" are\nknown. In the case of parabolic motion the eccentric anomaly is\nThe\nequal to zero, and these equations become identical with (92) 3\nfrom\nthe\nnode\nbe\nthe\nwill\nof\ndistance\nperihelion\nascending\nangular\n.\n\nobtained from\n\n=u\nae cos E, and q = a(1\nto\n\nSince r = a\n\nv.\n\n1-1\n\ne),\n\nwe have\n\n1-1\n\nand hence\n\n(89)\n\nTHEORETICAL ASTRONOMY.\n\n352\n\n## is nearly equal to unity, the value of q given\n\nwill be sufficient to compute cos^E and\nelements\nby approximate\nthese\nof\nmeans\nequations, and the results thus derived\ncos^E\" by\n\nWhen\n\nthe eccentricity\n\n## from which a new value of\n\nconsiderably from that used in com-\n\n## will be substituted in the equations (88),\n\nIf this should\n\nq results.\n\ndiffer\n\nputing cos \\E and cos \\E\", a repetition of the calculation will give\nthe correct result.\n\n## and E\" are imagiIn the case of hyperbolic motion, although\n\nnumerical\nof\nthe\nvalues\nwe\ncos^E and cos^E\"\nmay compute\nnary,\nfrom the equations (89), regarding a as negative, and the results will\nbe used for the corresponding quantities in (88) in the computation\nof q and v for the hyperbolic orbit.\n\n## Next, we compute a second system of elements from\n\nand /, df and\nand a third system from\n\nM + 8M\n\n'\n\nM and/-f-\n\ndM\n\n8f,\n\ndenoting the\n\nThe comrespectively.\narbitrary increments assigned to / and\nparison of these three systems of elements with additional observed\nplaces of the comet, will enable us to form the equations of condition\nfor the determination of the most probable values of the corrections\n\n&M and\n\nA/ to be applied\n\nto\n\nM and /respectively.\n\nThe formation of\n\n## these equations is effected in precisely the same manner as in the case\n\nof the variation of the geocentric distances or of Q, and i, and it does\n\nThe\n\nfinal\n\nWe\n\nmay\n\n## and/H- A/, either directly or by interpolation.\n\nas\nremark, further, that it will be convenient to use log\n\n## the quantity to be corrected, and to express the variations of log\n\nin units of the last decimal place of the logarithms.\nWhen the orbit differs very little from the parabolic form, it will\n\nbe most expeditious to\nin each case\n\nin regard to\n\nM,\n\nputting\n\n## = 0, and only compute elliptic or hyperbolic elements\n\nwhich we use\nand f=8f.\nand second systems of elements will thus be parabolic.\n\n## in the third hypothesis, for\n\nM and - we\n\nmay\n\nuse\n\nA and -\n\nThe\n\nfirst\n\nas the quantities to\n\n## In this case we assume an approximate value of A by\n\nmeans of elements already known, and by means of (96) 3 (98) 3 (102) 3\nand (103) 3 we compute the auxiliary quantities C, J5, B\", &c., re-\n\nbe corrected.\n\nWe\n\n## quired in the solution of the equations (104) 3\n\nassume, also, an\napproximate value of A\" and compute the corresponding value of r\" 9\n.\n\nthe value of r having been already found from the assumed value of\nA.\nThen, by trial, we find the value of K which, in connection with\n\nEQUATIONS OF CONDITION.\nthe assumed value of\n\n-,\n\nand\n\nA\"\n\n## The corresponding value of\n\n(61).\n\n353\n\n= cVx\n\nA\"\n\nis\n\n(65) or\n\ngiven by\n\nC\\\n\nWhen A\" has thus been determined, the heliocentric places will be\nobtained by means of the equations (106) 3 and (107) 3 and, finally,\nIf the\nthe corresponding elements of the orbit will be computed.\n,\n\n## taken as the fundamental plane, we put\n\nand write A and /9 in place of a and d respectively.\necliptic is\n\nIf\n\nwe now compute\n\nf=-,\n\nD = 0, A = O,\n\nand/+\n\n<?/\",\n\n-f-\n\nd A and\n\n## three systems of elements with additional observed places will furnish\n\nthe equations of condition for the determination of the corrections\n\n## A A and A/ to be applied to A and\n\nrespectively.\n\nWhen the eccentricity is very nearly equal to unity, we may asfor the first and second hypotheses, and only compute\nsume / =\nelliptic or\n\n121.\n\n## hyperbolic elements for the third hypothesis.\n\nThe comparison of\n\n## body with one of the three systems of elements obtained by varying\n\nthe two quantities selected for correction, or, when the required dif-\n\n## known, with any other system of elements\n\nsuch that the squares and products of the corrections may be neglected, gives a series of equations of the form\nferential coefficients are\n\nmx\n\n-f- ny\np,\nm'x -f n'y =p', &c.,\n\n## which x and y denote the final corrections to be applied to the two\n\nassumed quantities respectively. The combination of these equations\nwhich gives the most probable values of the unknown quantities, is\nin\n\n## eifected according to the\n\nThus, we multiply\nof x in that equation, and the sum\n\nmethod of\n\nleast squares.\n\n## each equation by the coefficient\n\nof all the equations thus formed gives the first normal equation.\nThen we multiply each equation of condition by the coefficient of y\nin that equation, and the sum of all the products gives the second\n\nnormal equation.\n\n## Let these equations be expressed thus\n\n[mm] x\n\\_mn~]\n\n-j-}-\n\n= [mp],\n=\n[nn\\ y\n[mri] y\n\n[lip},\n\n23\n\nTHEORETICAL ASTRONOMY.\n\n354\n\n## m2 +ra' -f-m' -f-&c., [mn]=mn+m'n +m\"n\"-\\-&G.,\n\nwhich [mra]\nand similarly for the other terms. These two final equations give,\nby elimination, the most probable values of x and y, namely, those\nin\n\nfor\n\nIt\n\n/2\n\n## which the sum of the squares of the residuals will be a minimum.\n\noften convenient to determine x in terms of y, or y\nis, however,\n\n## we may find the influence of a variation of one\n\nunknown quantities on the differences between computation\nand observation when the most probable value of the other unknown\nin terms of x, so that\n\nof the\n\nis used.\nThus, if it be desired to find x in terms of\nmost probable value of x will be\n\nquantity\n\nwe\n\nthe\n\n__\n\n[mm]\nIf\n\ny,\n\n[mm]\n\n## x in the original equations of condition,\n\nthe remaining differences between computation and observation willbe expressed in terms of the unknown quantity y, or in the form\n\n*0\n\n= m +n\n\n(90)\n\ny.\n\nThen, by assigning different values to yy we may find the corresponding residuals, and thus determine to what extent the correction y may\nbe varied without causing these residuals to surpass the limits of the\nprobable errors of observation.\nIn the determination of the orbit of a comet there must be more\nor less uncertainty in the value of a,\nto be applied to the\n\nand\n\nassumed value of\n\n-,\nCL\n\nif\n\n## we may thus determine\n\nthe\n\nprobable limits within which the true value of the periodic time\nmust be found. In the case of a comet which is identified, by the\n\n## which has previously appeared, if\n\nthe system of elements which will best satisfy the series\nof observations, the supposition being made that the comet has per*brmed but one revolution around the sun during the intervening\n\nwe compute\n\n## interval, it will be easy to determine whether the observations are\n\nbetter satisfied by\nassuming that two or more revolutions have been\n\n## completed during this interval. Thus, let T denote the periodic\n\ntime assumed, and the relation between T and a is\nexpressed by\n\nin\n\ncircle\n\nA COMET.\n\nORBIT OF\n\n355\n\n## Let the periodic time corresponding to &\n\nthen we shall have\n\nunity.\n\nis\n\nby\nJ\n\n+ y be\n\ndenoted\n\n= a1-\n\n33\n\n_,\n\nform\n\n## for the residuals are transformed into the\n\n*0\n\n= (m\n\nnJ)+nJz*.\n\n(91)\n\nIf we now assign to z, successively, the values 1, 2, 3, &c., the residuals thus obtained will indicate the value of z which best satisfies\nthe series of observations, and hence how many revolutions of the\ncomet have taken place during the interval denoted by T.\n122. In the determination of the orbit of a comet from three ob-\n\n## served places, a hypothesis in regard to the semi-transverse axis may\n\nwith facility be introduced simultaneously with the computation of\nThe numerical calculation as far as the formthe parabolic elements.\nation of the equations (52) 3 will be precisely the\nparabolic and the elliptic or hyperbolic elements.\ncase\n\nwe\n\n## find the values of r, r ff ,\n\nand K which\nwe find those which\n\nFrom\n\nThen\n\nin the one\n\n## will satisfy equation\n\nwill satisfy the equa-\n\n## the results thus obtained, the\n\nwill be computed.\n\n## case of the system of parabolic elements\n\nparison of the middle place with these\n\nsame\n\nLet\n\n/=-\n\nthen in the\n\nwe have/=0, and\nand\n\nalso\n\nwith the\n\nthe com-\n\nelliptic or\n\ndO\n\n## which 6 l denotes the geocentric spherical co-ordinate computed\n\nfrom the parabolic elements, and 2 that computed from the other\nsystem of elements. Further, let A# denote the difference between\ncomputation and observation for the middle place, and the correction\nto be applied to /, in order that the computed and the observed\nin\n\nvalues of 6\n\nmay\n\n## agree, will be given\n\nby\n\nr/+^=oHence, the two observed spherical co-ordinates for the middle place\nwill give two equations of condition from which A/ may be found,\n\nTHEORETICAL, ASTRONOMY.\n\n356\n\n## and the corresponding elements will be those which best represent\n\nto be correct.\nthe observations, assuming the adopted value of\n\n123.\n\nThe\n\norbit of a\n\nfirst\n\ncomet\n\nis\n\n## the fundamental plane.\n\n- and\n\nments, by varying\n\nor J,\n\nit\n\nele-\n\n## and the first .assumption in\n\nmade by means of the values of the distances\napproximate elements already known. But if it be\n\n## the equator as the fundamental plane,\n\nregard to\n\nwill be\n\ngiven by the\n\ndesired to compute\ndirectly from three observed places in reference\nto the equator, without converting the right ascensions and declinations into longitudes and latitudes, the requisite formulae may be\n\n## derived by a process entirely analogous to that\n\ncurtate distances refer to the ecliptic.\nThe case\n\n## employed when the\n\nmay occur in which\n\nonly the right ascension for the middle place is given, so that the\nIt will then be necessary\ncorresponding longitude cannot be found.\n\n## fundamental plane in determining a\n\nof\nelements\nsystem\nparabolic\nby means of two complete observations\nand this incomplete middle place. If we substitute the expressions\n\nequator in the\n\n= n (p cos a\n\nwe\n\nshall\n\nRcosD cos A)\n\nhave\n(//\n\ncos\n\na'\n\n-f n\"\n\n0\"\n\ncos\n\nsin a\"\n\nR\"\n\ncos\n\n## = n (p sin a R cos D sin A) (p sin R' cos D' sin A')\n\n(92)\nR\" cos D\" sin A\"),\n-f n\" (p\" sin a\"\n= n(ptsmd~R sin D) (p tan R' sin D')\nr\n\na'\n\nwhich\n\nin\n\n/>,\n\np' t\n\np\" denote\n\nA\"\n\n<5\"\n\n## and D, D', D\"\n\nThese equations correspond to (6) 3 and may be\ntreated in a similar manner.\nFrom the first and second of equations (92) we get\nequator,\nits\n\nA, A',\n\ndeclinations.\n\n= n (p sin\n\n(a'\n\na)\n\nand hence\n\n## + R' cos D' sin\n\nA*)\na') + R\" cosD\" sin\n\n4\")),\n\n~\n\nA\")\n\nA))\n\nA)-\n\n(a'\n\n(a'\n\na')\n\n357\n\n## This formula, being independent of the declination S f , may be used\n\nwhen only the right ascension for the middle place is\ncompute\n\nto\n\nFor the\n\ngiven.\n\nfirst\n\n## assumption in the case of an unknown\n\norbit,\n\nwe take\n\"\n\nM=\n\nsin (a'\n\nif\n\na)\n\n'\n\nsin (a\"\n\nif\n\na')'\n\nand, by means of the results obtained from this hypothesis, the complete expression (93) may be computed.\nBy a process identical with\nthat employed in deriving the equation\nthe expression\n\n_i ^_(\n*?TV*\n\n(36) 3 ,\n\n^x/ 1\n\nwe\n\nderive,\n\n1 \\ -R' cos\n\n;\\/3\n\nD'\n\nfrom\n\nsin (a'\n\nsin (a\"\n\njpjj\n\n(93),\n\nA')\n\na')\n\nand, putting\n\n-..\n\nsin (a'\n\na)\n\n'\n\n77\n\n~~^\n__\n\nSin (a\"\n\na')'\n\n_ ^ 'T\"\nIlV\n1\n\ncos\n\n'\n\n\"-)\n\nD'\n\nsin (a/\n\nA')\n'\n\nsin(a'\n\na)\n\nR_l^\np \\r'\n\n1_\\\nK'*}'\n\nwe have\n\nM= P~==M F.\nThe\n\n## by means of the formula (96) 3\n\nand\n, (86), (87), (102) 3 ,\nheliocentric places for the times t and t\" will be given\nand (107) 3? and from these the elements of the orbit will\n\nwill be effected\n(51) 3 .\n\n(95)\n\nThe\n\nby (106) 3\n\n## be found according to the process already\n\nillustrated.\n\n124. The methods already given for the correction of the approximate elements of the orbit of a heavenly body by means of additional\nobservations or normal places, are those which will generally be\napplied.\n\nThere\n\nare,\n\nand\n\nmay be\n\n## special cases, and which will readily suggest\n\nif\nit\nbe\ndesired to correct approximate elements\nthemselves.\nThus,\ntwo\nr and r fr we may assume an approxiradii-vectores\nby varying\nmate value of each of these, and the three equations (88)! will contain only the three unknown quantities J, 6, and I.\nBy elimination,\nthese unknown quantities may be found, and in like manner the\n\nTHEORETICAL ASTRONOMY.\nIt will be most convenient to compute\nvalues of J\", b\", and I\".\nand $\", and then find z and z\" from the angles sin 2 putting a? or, The r* tan$\n= E sin\nr\n\nsin\n\n2\n^, and x\"\n\n*\n= R sin\nx\n\nsin z\n\n,,\n\n-4\"\n= R\" sin\nT\nTt\n\n= r\"\n\nsin 4/',\n\nfrom\n\ntan z\"\n\nR\"\n\nby the equations\n\n(3),\n\nand the\n\n## heliocentric spherical co-ordinates by means of (4), writing r in place\n\nu may be found, and by means of the values\nof a. From these u\"\n\nof\n\nr,\n\nr ff y and\n\nu\"\n\n## u the determination of the elements of the\n\norbit\n\nmay be completed. Then, assigning to r an increment dr, we compute a second system of elements, and from r and r\" -f dr\" a third\nThe comparison of these three systems of elements with an\nsystem.\nadditional or intermediate observed place will furnish the equations\nfor the determination of the corrections Ar and Ar\" to be applied to\n\n## The comparison of the middle place may be\n\nr and r\", respectively.\nmade with the observed geocentric spherical co-ordinates directly, or\nwith the radius-vector and argument of the latitude computed directly\nfrom the observed co-ordinates; and in the same manner any number\nof additional observed places may be employed in forming the equations of condition for the determination of\n\nInstead of r and r r/ ,\n\nAr and Ar\".\n\nwe may\n\ntake the projections of these radiivectores on the plane of the ecliptic as the quantities to be corrected.\nLet these projected distances of the body from the sun be denoted\n\nby r and r\n\nwe\n\n\", respectively\n\nthen,\n\nriri-.Q =\nfrom which\nI\".\n\nobtain\n\nIf\n\nI\nmay be found\nwe put\n\nwe have\n\nBrin <'- g\n\n(96)\n\n## and in a similar manner we may find\n\n=r\n\n-^sm'(A-0),\n\ntan(;-A)\n\n= * sin X (A\n\n).\n\n(97)\n\nLet S denote the angle at the sun between the earth and the\nplace\nof the planet or comet projected on the\nplane of the ecliptic ; then\n\nwe\n\nshall\n\nhave\n\n=180-}P\n\n359\n\nI,\n\n0)\n\n*(l\n\n(98)\n\nsin (I\n\nand\n\ntenb^-^Z,\nr\n\n(99)\n\n## by means of which the heliocentric latitudes b and b\" may be found.\n\n\" are\ncalculation of the elements and the correction of rQ and r\nthen effected as in the case of the variatioVi of r and r\".\n\nThe\n\n## In the case of parabolic motion, the eccentricity being known, we\n\nmay take q and T as the quantities to be corrected. If we assume\n/r\nand #, v v\" will be\napproximate values of these elements, r, r r\nff\nr\nand\nthe\nobserved spherical\nThen\nfrom\nr, r',\ngiven immediately.\n1'\nu and\nco-ordinates of the body we may compute the values of u\nthe\nmeans\nof\nobserved\nu\nIn the same manner, by\nu.\nplaces, we\ncompute the angles u\" u and u' u corresponding to q-\\-dq and T,\nr\n\n## to q and T -\\- STy dq and dT denoting the arbitrary increments\n\nThe comparison of the helioassigned to q and T, respectively.\nf\nt\nand t'\nthus obtained,\ncentric motion, during the intervals t\"\nt,\nin the case of each of the three systems of elements, from the ob-\n\nand\n\nby which we may\n\nAT to\n\n## be applied to the assumed values of q and T,\n\nff\nu r and u f\nu computed\nin\norder\nthat\nthe values of u\nrespectively,\nwith\nthose\nshall\nmeans\nof\nthe observed places\nagree\ngiven by the\nby\nrections\n\nAg and\n\n## true anomalies computed directly from q and T.\n\nTHEOEETICAL ASTRONOMY.\n\n360\n\nCHAPTER VIL\nMETHOD OF LEAST SQUARES, THEORY OF THE COMBINATION OF OBSERVATIONS, AND\nDETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES\nOF OBSERVATIONS.\n\n125.\n\n## to such a degree of approximation that the squares and products of\n\nthe corrections which should be applied to them may be neglected,\n\nby computing the\n\n## with respect to each of the observed spherical co-ordinates, we may\n\nform, by means of the differences between computation and observation, the equations for the\n\n## determination of these corrections.\n\nThree\n\ncomplete observations will furnish the six equations required for the\ndetermination of the corrections to be applied to the six elements of\nthe orbit; but, if more than three complete places are given, the\n\nnumber of equations\n\n## will exceed the\n\nnumber of unknown\n\nquantities,\n\nIf the observed\n\n## places were absolutely exact, the combination of the equations of\n\ncondition in any manner whatever would furnish the values of these\ncorrections, such that each of these equations\n\nwould be completely\n\nThe\n\n## conditions, however, which present themselves in the\n\nactual correction of the elements of the orbit of a\nheavenly body by\nsatisfied.\n\n## means of given observed places, are entirely different. When the\n\nobservations have been corrected for all known instrumental errors,\nand when all other known corrections have been duly applied, there\nremain those accidental errors which arise from various causes,\nsuch as the abnormal condition of the\natmosphere, the imperfections\nof vision, and the imperfections in the\nperformance of the instrument\nemployed. These accidental and irregular errors of observation cannot\nbe eliminated from the observed data, and the\nequations of condition\nfor the determination of the corrections to be\napplied to the elements\nof an approximate orbit cannot be\ncompletely satisfied by any system\nof values assigned to the unknown\nquantities unless the number of\nequations is the same as the number of these unknown\nstill\n\nquantities.\n\n## becomes an important problem, therefore, to determine the\n\nparticular combination of these\nequations of condition, by means of which\nIt\n\n361\n\n## the resulting values of the unknown quantities will be those which,\n\nwhile they do not completely satisfy the several equations, will afford\nthe highest degree of probability in favor of their accuracy.\nIt will\n\n## be of interest also to determine, as far as it may be possible, the\n\ndegree of accuracy which may be attributed to the separate results.\nBut, in order to simplify the more general problem, in which the\nquantities sought are determined indirectly by observation, it will be\nexpedient to consider first the simpler case, in which a single quantity\nis\n\n## obtained directly by observation.\n\n126. If the accidental errors of observation could be obviated, the\nmagnitude directly by observation would\n\ndifferent determinations of a\n\nbe identical\nprecision\n\nis\n\nsought,\n\nis\n\n## impossible when an extreme limit of\n\nmean or average value to be derived\n\nfrom the separate results obtained. The adopted value may or may\nnot agree with any individual result, since it is only necessary that\nthe residuals obtained by comparing the adopted value with the\nobserved values shall be such as to make this adopted value the most\nIt is evident, from the very nature of the case, that\nprobable value.\nhere the confines of the unknown, and, before we proceed further, something additional must be assumed.\n\nwe approach\nHowever\n\n## and uncertain the law of the accidental errors\n\nmay be, we may at least assume that small errors are\nmore probable than large errors, and that errors surpassing a certain\nWe may also assume that in the case of a large\nlimit will not occur.\nirregular\n\nof observation\n\nnumber of\n\n## observations, errors in excess will occur as frequently as\n\nerrors in defect, so that, in general, positive and negative residuals\nIt appears, therefore,\nof equal absolute value are equally probable.\nthat the relative frequency of the occurrence of an accidental error J\n\n## depend on the magnitude of this error,\n\nThis function will also express the\n<p ( J)."
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https://ncertsolutions.guru/mcq-questions-for-class-8-maths-chapter-13/ | [
"# MCQ Questions for Class 8 Maths Chapter 13 Direct and Indirect Proportions with Answers\n\nStudents can access the NCERT MCQ Questions for Class 8 Maths Chapter 13 Direct and Indirect Proportions with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 8 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Direct and Indirect Proportions Class 8 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 8 Maths Chapter 13 Direct and Indirect Proportions Objective Questions.\n\n## Direct and Indirect Proportions Class 8 MCQs Questions with Answers\n\nStudents are advised to solve the Direct and Indirect Proportions Multiple Choice Questions of Class 8 Maths to know different concepts. Practicing the MCQ Questions on Direct and Indirect Proportions Class 8 with answers will boost your confidence thereby helping you score well in the exam.\n\nExplore numerous MCQ Questions of Direct and Indirect Proportions Class 8 with answers provided with detailed solutions by looking below.\n\nQuestion 1.\nA mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. How many parts of base will be used in mixture by mixing 7 part of red pigment?\n(a) 70\n(b) 56\n(c) 63\n(d) 49\n\nQuestion 2.\nIf 3 men or 6 women can do a piece of work in 16 days, in how many days can 12 men and 8 women do the same piece of work?\n(a) 4 days\n(b) 5 days\n(c) 3 days\n(d) 2 days\n\nQuestion 3.\nIf the cost of 27 bags of paddy is Rs.9450, what is the cost of 36 bags of paddy?\n(a) Rs.12000\n(b) Rs.12600\n(c) Rs.16200\n(d) Rs.10620\n\nQuestion 4.\nRaju earns Rs 1440, if he works for 12 days. If he works for 30 days, he will earn\n(a) Rs 2400\n(b) Rs 3600\n(c) Rs 4800\n(d) None of these\n\nQuestion 5.\nThe scale of a map is 1:3×107 Two cities are 5 cm apart on the map. Find the actual distance between them in kilometres.\n(a) 1500 km\n(b) 1000 km\n(c) 1100 km\n(d) 2000 km\n\nQuestion 6.\nAn electric pole, 14 metres high, casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions.\n(a) 21 metres\n(b) 30 metres\n(c) 25 metres\n(d) None of these\n\nQuestion 7.\nA boy runs 1 km in 10 minutes. How long will he take to ran 600 m ?\n(a) 2 minutes\n(b) 3 minutes\n(c) 4 minutes\n(d) 6 minutes\n\nQuestion 8.\nA van covers 432 km with 36 litres of diesel. How much distance would it cover with 25 litres of diesel?\n(a) 200 km\n(b) 300 km\n(c) 100 km\n(d) 350 km\n\nQuestion 9.\nThe price of 357 mangoes is Rs.1517.25. What will be the approximate price of 49 dozens of such mangoes?\n(a) Rs.3000\n(b) Rs.3500\n(c) Rs.4000\n(d) Rs.2500\n\nQuestion 10.\n72 books are packed in 4 cartons of the same size. How many cartons are required for 360 books?\n(a) 22\n(b) 18\n(c) 20\n(d) None of these\n\nQuestion 11.\nA garrison of 500 persons had provisions for 27 days. After 3 days a reinforcement of 300 persons arrived. For how many more days will the remaining food last now?\n(a) 12 days\n(b) 14 days\n(c) 16 days\n(d) 15 days\n\nQuestion 12.\nTwo quantities x and y are said to be in ___________ if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.\n(a) inverse proportion\n(b) mix proportion\n(c) direct proportion\n(d) None of these\n\nQuestion 13.\nThere is enough food to last for 40 people for 10 days. If 10 more people join them, the food will last for\n(a) 10 days\n(b) 12 days\n(c) 8 days\n(d) None of these\n\nQuestion 14.\n36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work ?\n(a) 24 days\n(b) 28 days\n(c) 34 days\n(d) 35 days\n\nQuestion 15.\nx varies inversely as square of y. Given that y = 3 for x = 1.find the value of x for y = 4.\n(a) 3\n(b) 9\n(c) $$\\frac { 1 }{ 3 }$$\n(d) $$\\frac { 9 }{ 16 }$$\n\nAnswer: (d) $$\\frac { 9 }{ 16 }$$\n\nQuestion 16.\nA journey by bus takes 45 minutes at 40 km/hour. How fast must a car go to undertake the same journey in 25 minutes?\n(a) 36 km/h\n(b) 48 km/h\n(c) 72 km/h\n(d) None of these\n\nQuestion 17.\nA can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 10 days and C alone in 50 days, then B alone could do it in:\n(a) 15 days\n(b) 20 days\n(c) 25 days\n(d) 30 days\n\nQuestion 18.\nThe cost of 5 metres of a particular quality of cloth is Rs 210. Find the cost of 2 metres of cloth of the same type.\n(a) Rs 100\n(b) Rs 84\n(c) Rs 90\n(d) Rs 60\n\nQuestion 19.\n120 copies of a book cost Rs 600. What will 400 copies cost ?\n(a) Rs 1000\n(b) Rs 2000\n(c) Rs 3000\n(d) Rs 2400\n\nQuestion 20.\n6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?\n(a) 56 minutes\n(b) 72 minutes\n(c) 96 minutes\n(d) 80 minutes\n\nQuestion 21.\n8 g of sandal wood cost Rs 40. What will 10 g cost ?\n(a) Rs 30\n(b) Rs 36\n(c) Rs 48\n(d) Rs 50\n\nQuestion 22.\nWhen one quantity is increased, the other quantity is also increased. This proportion is called _______\n(a) Kally proportion\n(b) Direct proportion\n(c) Inverse proportion\n(d) None of these"
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http://docs.ros.org/en/indigo/api/image_geometry/html/c++/classimage__geometry_1_1PinholeCameraModel.html | [
"image_geometry::PinholeCameraModel Class Reference\n\nSimplifies interpreting images geometrically using the parameters from sensor_msgs/CameraInfo. More...\n\n`#include <pinhole_camera_model.h>`\n\nList of all members.\n\nstruct Cache\n\n## Public Member Functions\n\nuint32_t binningX () const\nReturns the number of columns in each bin.\nuint32_t binningY () const\nReturns the number of rows in each bin.\nconst sensor_msgs::CameraInfo & cameraInfo () const\nReturns the camera info message held internally.\ndouble cx () const\nReturns the x coordinate of the optical center.\ndouble cy () const\nReturns the y coordinate of the optical center.\nconst cv::Mat_< double > & distortionCoeffs () const\nReturns the distortion coefficients.\nbool fromCameraInfo (const sensor_msgs::CameraInfo &msg)\nSet the camera parameters from the sensor_msgs/CameraInfo message.\nbool fromCameraInfo (const sensor_msgs::CameraInfoConstPtr &msg)\nSet the camera parameters from the sensor_msgs/CameraInfo message.\nconst cv::Matx33d & fullIntrinsicMatrix () const\nReturns the original camera matrix for full resolution.\nconst cv::Matx34d & fullProjectionMatrix () const\nReturns the projection matrix for full resolution.\ncv::Size fullResolution () const\nThe resolution at which the camera was calibrated.\ndouble fx () const\nReturns the focal length (pixels) in x direction of the rectified image.\ndouble fy () const\nReturns the focal length (pixels) in y direction of the rectified image.\ndouble getDeltaU (double deltaX, double Z) const\nCompute delta u, given Z and delta X in Cartesian space.\ndouble getDeltaV (double deltaY, double Z) const\nCompute delta v, given Z and delta Y in Cartesian space.\ndouble getDeltaX (double deltaU, double Z) const\nCompute delta X, given Z in Cartesian space and delta u in pixels.\ndouble getDeltaY (double deltaV, double Z) const\nCompute delta Y, given Z in Cartesian space and delta v in pixels.\nbool initialized () const\nReturns true if the camera has been initialized.\nconst cv::Matx33d & intrinsicMatrix () const\nReturns the original camera matrix.\nPinholeCameraModeloperator= (const PinholeCameraModel &other)\nPinholeCameraModel ()\nPinholeCameraModel (const PinholeCameraModel &other)\ncv::Point2d project3dToPixel (const cv::Point3d &xyz) const\nProject a 3d point to rectified pixel coordinates.\nconst cv::Matx34d & projectionMatrix () const\nReturns the projection matrix.\ncv::Point3d projectPixelTo3dRay (const cv::Point2d &uv_rect) const\nProject a rectified pixel to a 3d ray.\ncv::Rect rawRoi () const\nThe current raw ROI, as used for capture by the camera driver.\ncv::Rect rectifiedRoi () const\nThe current rectified ROI, which best fits the raw ROI.\nvoid rectifyImage (const cv::Mat &raw, cv::Mat &rectified, int interpolation=cv::INTER_LINEAR) const\nRectify a raw camera image.\ncv::Point2d rectifyPoint (const cv::Point2d &uv_raw) const\nCompute the rectified image coordinates of a pixel in the raw image.\ncv::Rect rectifyRoi (const cv::Rect &roi_raw) const\nCompute the rectified ROI best fitting a raw ROI.\ncv::Size reducedResolution () const\nThe resolution of the current rectified image.\nconst cv::Matx33d & rotationMatrix () const\nReturns the rotation matrix.\nros::Time stamp () const\nGet the time stamp associated with this camera model.\nstd::string tfFrame () const\nGet the name of the camera coordinate frame in tf.\ncv::Point2d toFullResolution (const cv::Point2d &uv_reduced) const\ncv::Rect toFullResolution (const cv::Rect &roi_reduced) const\ncv::Point2d toReducedResolution (const cv::Point2d &uv_full) const\ncv::Rect toReducedResolution (const cv::Rect &roi_full) const\ndouble Tx () const\nReturns the x-translation term of the projection matrix.\ndouble Ty () const\nReturns the y-translation term of the projection matrix.\nvoid unrectifyImage (const cv::Mat &rectified, cv::Mat &raw, int interpolation=cv::INTER_LINEAR) const\nApply camera distortion to a rectified image.\ncv::Point2d unrectifyPoint (const cv::Point2d &uv_rect) const\nCompute the raw image coordinates of a pixel in the rectified image.\ncv::Rect unrectifyRoi (const cv::Rect &roi_rect) const\nCompute the raw ROI best fitting a rectified ROI.\n\n## Protected Member Functions\n\nvoid initRectificationMaps () const\n\n## Protected Attributes\n\nboost::shared_ptr< Cachecache_\nsensor_msgs::CameraInfo cam_info_\ncv::Mat_< double > D_\ncv::Matx33d K_\ncv::Matx33d K_full_\ncv::Matx34d P_\ncv::Matx34d P_full_\ncv::Matx33d R_\n\n## Friends\n\nclass StereoCameraModel\n\n## Detailed Description\n\nSimplifies interpreting images geometrically using the parameters from sensor_msgs/CameraInfo.\n\nDefinition at line 22 of file pinhole_camera_model.h.\n\n## Constructor & Destructor Documentation\n\n image_geometry::PinholeCameraModel::PinholeCameraModel ( )\n\nDefinition at line 32 of file pinhole_camera_model.cpp.\n\n image_geometry::PinholeCameraModel::PinholeCameraModel ( const PinholeCameraModel & other )\n\nDefinition at line 43 of file pinhole_camera_model.cpp.\n\n## Member Function Documentation\n\n uint32_t image_geometry::PinholeCameraModel::binningX ( ) const` [inline]`\n\nReturns the number of columns in each bin.\n\nDefinition at line 309 of file pinhole_camera_model.h.\n\n uint32_t image_geometry::PinholeCameraModel::binningY ( ) const` [inline]`\n\nReturns the number of rows in each bin.\n\nDefinition at line 310 of file pinhole_camera_model.h.\n\n const sensor_msgs::CameraInfo & image_geometry::PinholeCameraModel::cameraInfo ( ) const` [inline]`\n\nReturns the camera info message held internally.\n\nDefinition at line 294 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::cx ( ) const` [inline]`\n\nReturns the x coordinate of the optical center.\n\nDefinition at line 304 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::cy ( ) const` [inline]`\n\nReturns the y coordinate of the optical center.\n\nDefinition at line 305 of file pinhole_camera_model.h.\n\n const cv::Mat_< double > & image_geometry::PinholeCameraModel::distortionCoeffs ( ) const` [inline]`\n\nReturns the distortion coefficients.\n\nDefinition at line 296 of file pinhole_camera_model.h.\n\n bool image_geometry::PinholeCameraModel::fromCameraInfo ( const sensor_msgs::CameraInfo & msg )\n\nSet the camera parameters from the sensor_msgs/CameraInfo message.\n\nTodo:\nCalculate and use rectified ROI\nTodo:\n\nDefinition at line 82 of file pinhole_camera_model.cpp.\n\n bool image_geometry::PinholeCameraModel::fromCameraInfo ( const sensor_msgs::CameraInfoConstPtr & msg )\n\nSet the camera parameters from the sensor_msgs/CameraInfo message.\n\nDefinition at line 188 of file pinhole_camera_model.cpp.\n\n const cv::Matx33d & image_geometry::PinholeCameraModel::fullIntrinsicMatrix ( ) const` [inline]`\n\nReturns the original camera matrix for full resolution.\n\nDefinition at line 299 of file pinhole_camera_model.h.\n\n const cv::Matx34d & image_geometry::PinholeCameraModel::fullProjectionMatrix ( ) const` [inline]`\n\nReturns the projection matrix for full resolution.\n\nDefinition at line 300 of file pinhole_camera_model.h.\n\n cv::Size image_geometry::PinholeCameraModel::fullResolution ( ) const\n\nThe resolution at which the camera was calibrated.\n\nThe maximum resolution at which the camera can be used with the current calibration; normally this is the same as the imager resolution.\n\nDefinition at line 193 of file pinhole_camera_model.cpp.\n\n double image_geometry::PinholeCameraModel::fx ( ) const` [inline]`\n\nReturns the focal length (pixels) in x direction of the rectified image.\n\nDefinition at line 302 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::fy ( ) const` [inline]`\n\nReturns the focal length (pixels) in y direction of the rectified image.\n\nDefinition at line 303 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::getDeltaU ( double deltaX, double Z ) const` [inline]`\n\nCompute delta u, given Z and delta X in Cartesian space.\n\nFor given Z, this is the inverse of getDeltaX().\n\nParameters:\n deltaX Delta X, in Cartesian space Z Z (depth), in Cartesian space\n\nDefinition at line 312 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::getDeltaV ( double deltaY, double Z ) const` [inline]`\n\nCompute delta v, given Z and delta Y in Cartesian space.\n\nFor given Z, this is the inverse of getDeltaY().\n\nParameters:\n deltaY Delta Y, in Cartesian space Z Z (depth), in Cartesian space\n\nDefinition at line 318 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::getDeltaX ( double deltaU, double Z ) const` [inline]`\n\nCompute delta X, given Z in Cartesian space and delta u in pixels.\n\nFor given Z, this is the inverse of getDeltaU().\n\nParameters:\n deltaU Delta u, in pixels Z Z (depth), in Cartesian space\n\nDefinition at line 324 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::getDeltaY ( double deltaV, double Z ) const` [inline]`\n\nCompute delta Y, given Z in Cartesian space and delta v in pixels.\n\nFor given Z, this is the inverse of getDeltaV().\n\nParameters:\n deltaV Delta v, in pixels Z Z (depth), in Cartesian space\n\nDefinition at line 330 of file pinhole_camera_model.h.\n\n bool image_geometry::PinholeCameraModel::initialized ( ) const` [inline]`\n\nReturns true if the camera has been initialized.\n\nDefinition at line 260 of file pinhole_camera_model.h.\n\n void image_geometry::PinholeCameraModel::initRectificationMaps ( ) const` [protected]`\nTodo:\nFor large binning settings, can drop extra rows/cols at bottom/right boundary. Make sure we're handling that 100% correctly.\nTodo:\nShould binned resolution, K, P be part of public API?\nTodo:\nUse rectified ROI\n\nDefinition at line 407 of file pinhole_camera_model.cpp.\n\n const cv::Matx33d & image_geometry::PinholeCameraModel::intrinsicMatrix ( ) const` [inline]`\n\nReturns the original camera matrix.\n\nDefinition at line 295 of file pinhole_camera_model.h.\n\n PinholeCameraModel & image_geometry::PinholeCameraModel::operator= ( const PinholeCameraModel & other )\n\nDefinition at line 36 of file pinhole_camera_model.cpp.\n\n cv::Point2d image_geometry::PinholeCameraModel::project3dToPixel ( const cv::Point3d & xyz ) const\n\nProject a 3d point to rectified pixel coordinates.\n\nThis is the inverse of projectPixelTo3dRay().\n\nParameters:\n xyz 3d point in the camera coordinate frame\nReturns:\n(u,v) in rectified pixel coordinates\n\nDefinition at line 262 of file pinhole_camera_model.cpp.\n\n const cv::Matx34d & image_geometry::PinholeCameraModel::projectionMatrix ( ) const` [inline]`\n\nReturns the projection matrix.\n\nDefinition at line 298 of file pinhole_camera_model.h.\n\n cv::Point3d image_geometry::PinholeCameraModel::projectPixelTo3dRay ( const cv::Point2d & uv_rect ) const\n\nProject a rectified pixel to a 3d ray.\n\nReturns the unit vector in the camera coordinate frame in the direction of rectified pixel (u,v) in the image plane. This is the inverse of project3dToPixel().\n\nIn 1.4.x, the vector has z = 1.0. Previously, this function returned a unit vector.\n\nParameters:\n uv_rect Rectified pixel coordinates\nReturns:\n3d ray passing through (u,v)\n\nDefinition at line 276 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::rawRoi ( ) const\n\nThe current raw ROI, as used for capture by the camera driver.\n\nDefinition at line 239 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::rectifiedRoi ( ) const\n\nThe current rectified ROI, which best fits the raw ROI.\n\nDefinition at line 247 of file pinhole_camera_model.cpp.\n\n void image_geometry::PinholeCameraModel::rectifyImage ( const cv::Mat & raw, cv::Mat & rectified, int interpolation = `cv::INTER_LINEAR` ) const\n\nRectify a raw camera image.\n\nDefinition at line 287 of file pinhole_camera_model.cpp.\n\n cv::Point2d image_geometry::PinholeCameraModel::rectifyPoint ( const cv::Point2d & uv_raw ) const\n\nCompute the rectified image coordinates of a pixel in the raw image.\n\nTodo:\ncv::undistortPoints requires the point data to be float, should allow double\n\nDefinition at line 325 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::rectifyRoi ( const cv::Rect & roi_raw ) const\n\nCompute the rectified ROI best fitting a raw ROI.\n\nTodo:\nActually implement \"best fit\" as described by REP 104.\n\nDefinition at line 365 of file pinhole_camera_model.cpp.\n\n cv::Size image_geometry::PinholeCameraModel::reducedResolution ( ) const\n\nThe resolution of the current rectified image.\n\nThe size of the rectified image associated with the latest CameraInfo, as reduced by binning/ROI and affected by distortion. If binning and ROI are not in use, this is the same as fullResolution().\n\nDefinition at line 199 of file pinhole_camera_model.cpp.\n\n const cv::Matx33d & image_geometry::PinholeCameraModel::rotationMatrix ( ) const` [inline]`\n\nReturns the rotation matrix.\n\nDefinition at line 297 of file pinhole_camera_model.h.\n\n ros::Time image_geometry::PinholeCameraModel::stamp ( ) const` [inline]`\n\nGet the time stamp associated with this camera model.\n\nDefinition at line 288 of file pinhole_camera_model.h.\n\n std::string image_geometry::PinholeCameraModel::tfFrame ( ) const` [inline]`\n\nGet the name of the camera coordinate frame in tf.\n\nDefinition at line 282 of file pinhole_camera_model.h.\n\n cv::Point2d image_geometry::PinholeCameraModel::toFullResolution ( const cv::Point2d & uv_reduced ) const\n\nDefinition at line 207 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::toFullResolution ( const cv::Rect & roi_reduced ) const\n\nDefinition at line 214 of file pinhole_camera_model.cpp.\n\n cv::Point2d image_geometry::PinholeCameraModel::toReducedResolution ( const cv::Point2d & uv_full ) const\n\nDefinition at line 223 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::toReducedResolution ( const cv::Rect & roi_full ) const\n\nDefinition at line 230 of file pinhole_camera_model.cpp.\n\n double image_geometry::PinholeCameraModel::Tx ( ) const` [inline]`\n\nReturns the x-translation term of the projection matrix.\n\nDefinition at line 306 of file pinhole_camera_model.h.\n\n double image_geometry::PinholeCameraModel::Ty ( ) const` [inline]`\n\nReturns the y-translation term of the projection matrix.\n\nDefinition at line 307 of file pinhole_camera_model.h.\n\n void image_geometry::PinholeCameraModel::unrectifyImage ( const cv::Mat & rectified, cv::Mat & raw, int interpolation = `cv::INTER_LINEAR` ) const\n\nApply camera distortion to a rectified image.\n\nTodo:\nImplement unrectifyImage()\n\nDefinition at line 311 of file pinhole_camera_model.cpp.\n\n cv::Point2d image_geometry::PinholeCameraModel::unrectifyPoint ( const cv::Point2d & uv_rect ) const\n\nCompute the raw image coordinates of a pixel in the rectified image.\n\nDefinition at line 343 of file pinhole_camera_model.cpp.\n\n cv::Rect image_geometry::PinholeCameraModel::unrectifyRoi ( const cv::Rect & roi_rect ) const\n\nCompute the raw ROI best fitting a rectified ROI.\n\nTodo:\nActually implement \"best fit\" as described by REP 104.\n\nDefinition at line 386 of file pinhole_camera_model.cpp.\n\n## Friends And Related Function Documentation\n\n friend class StereoCameraModel` [friend]`\n\nDefinition at line 277 of file pinhole_camera_model.h.\n\n## Member Data Documentation\n\n boost::shared_ptr image_geometry::PinholeCameraModel::cache_` [protected]`\n\nDefinition at line 272 of file pinhole_camera_model.h.\n\n sensor_msgs::CameraInfo image_geometry::PinholeCameraModel::cam_info_` [protected]`\n\nDefinition at line 263 of file pinhole_camera_model.h.\n\n cv::Mat_ image_geometry::PinholeCameraModel::D_` [protected]`\n\nDefinition at line 264 of file pinhole_camera_model.h.\n\n cv::Matx33d image_geometry::PinholeCameraModel::K_` [protected]`\n\nDefinition at line 266 of file pinhole_camera_model.h.\n\n cv::Matx33d image_geometry::PinholeCameraModel::K_full_` [protected]`\n\nDefinition at line 268 of file pinhole_camera_model.h.\n\n cv::Matx34d image_geometry::PinholeCameraModel::P_` [protected]`\n\nDefinition at line 267 of file pinhole_camera_model.h.\n\n cv::Matx34d image_geometry::PinholeCameraModel::P_full_` [protected]`\n\nDefinition at line 269 of file pinhole_camera_model.h.\n\n cv::Matx33d image_geometry::PinholeCameraModel::R_` [protected]`\n\nDefinition at line 265 of file pinhole_camera_model.h.\n\nThe documentation for this class was generated from the following files:\n\nimage_geometry\nAuthor(s): Patrick Mihelich\nautogenerated on Thu Jun 6 2019 21:23:35"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.5068215,"math_prob":0.77253973,"size":15337,"snap":"2022-05-2022-21","text_gpt3_token_len":4095,"char_repetition_ratio":0.22663537,"word_repetition_ratio":0.2866276,"special_character_ratio":0.24457195,"punctuation_ratio":0.24619482,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96404266,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-23T17:09:17Z\",\"WARC-Record-ID\":\"<urn:uuid:3613306b-b835-4f1b-bb16-d3b5434f0ec6>\",\"Content-Length\":\"82337\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:16a588fd-5c3e-4adb-9c18-d0913462e964>\",\"WARC-Concurrent-To\":\"<urn:uuid:fb8ac20f-2932-4136-816d-76f62e5e9099>\",\"WARC-IP-Address\":\"140.211.9.98\",\"WARC-Target-URI\":\"http://docs.ros.org/en/indigo/api/image_geometry/html/c++/classimage__geometry_1_1PinholeCameraModel.html\",\"WARC-Payload-Digest\":\"sha1:7VBPFA3H4GUEOHWI5V3FHVRPC2VE2BZ3\",\"WARC-Block-Digest\":\"sha1:3I7454AZYJZOSQ65VCS7YAYPQTK4KBH4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662560022.71_warc_CC-MAIN-20220523163515-20220523193515-00137.warc.gz\"}"} |
http://artint.info/html/ArtInt_152.html | [
"6.4.2 Approximate Inference Through Stochastic Simulation\n\nMany problems are too big for exact inference, and one must resort to approximate inference. One of the most effective methods is based on generating random samples from the (posterior) distribution that the network specifies.\n\nStochastic simulation is based on the idea that a set of samples can be used to compute probabilities. For example, you could interpret the probability P(a)=0.14 as meaning that, out of 1,000 samples, about 140 will have a true. You can go from (enough) samples into probabilities and from probabilities into samples.\n\nWe consider three problems:\n\n• how to generate samples,\n• how to incorporate observations, and\n• how to infer probabilities from samples.\n\nWe examine three methods that use sampling to compute the posterior distribution of a variable: (1) rejection sampling, (2) importance sampling, and (3) particle filtering."
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.8881276,"math_prob":0.96366876,"size":1060,"snap":"2022-05-2022-21","text_gpt3_token_len":214,"char_repetition_ratio":0.12973484,"word_repetition_ratio":0.0,"special_character_ratio":0.20754717,"punctuation_ratio":0.13297872,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9882416,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-21T04:47:21Z\",\"WARC-Record-ID\":\"<urn:uuid:e3c11572-f0de-4449-b404-dd105974039c>\",\"Content-Length\":\"4959\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ec14d5d4-8025-4464-8b14-37e683f6c15b>\",\"WARC-Concurrent-To\":\"<urn:uuid:b05fde45-2138-41df-903b-a4edef06d6c7>\",\"WARC-IP-Address\":\"142.103.6.162\",\"WARC-Target-URI\":\"http://artint.info/html/ArtInt_152.html\",\"WARC-Payload-Digest\":\"sha1:6PXDTILOMHAL3SGF2QXKO4HIKZDFP2FJ\",\"WARC-Block-Digest\":\"sha1:5CQRVR5KVPQJK73DZCGKR6A6JTIUSW5L\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320302723.60_warc_CC-MAIN-20220121040956-20220121070956-00154.warc.gz\"}"} |
https://www.physicsforums.com/threads/rods-with-masses-orbiting-earth.793069/ | [
"# Rods with Masses orbiting Earth\n\n## Homework Statement\n\nTwo small, equal masses are attached by a lightweight rod. This object orbits a planet; the length of the rod is\nsmaller than the radius of the orbit, but not negligible. The rod rotates about its axis in such a way that it remains\nvertical with respect to the planet.\n• Is there a force in the rod? If so, is it tension or compression?\n• Is the equilibrium stable, unstable, or neutral with respect to a small perturbation in the angle of the\nrod? (Assume this perturbation maintains the rate of rotation, so that in the co-rotating frame the rod\nis still stationary but at an angle to the vertical.)\n\n(A) There is no force in the rod; the equilibrium is neutral.\n(B) The rod is in tension; the equilibrium is stable.\n(C) The rod is in compression; the equilibrium is stable.\n(D) The rod is in tension; the equilibrium is unstable.\n(E) The rod is in compression; the equilibrium is unstable.\n\nA picture may be found here http://www.aapt.org/physicsteam/2014/upload/exam1-2013-1-6-unlocked.pdf\npage 6, problem 12\n\n## The Attempt at a Solution\n\nI know that there is a force of tension because the Earth is pulling the black mass (according to diagram) down and so the rod must being pushing the white mass up to keep it in equilibrium as it rotates. I'm just having trouble about whether the equilibrium is stable or not. Any thoughts?\n\n## Answers and Replies\n\nNathanael\nHomework Helper\nI know that there is a force of tension because the Earth is pulling the black mass (according to diagram) down and so the rod must being pushing the white mass up to keep it in equilibrium as it rotates.\nThe force of gravity (Earth) is pulling on both masses...\n\nI'm just having trouble about whether the equilibrium is stable or not. Any thoughts?\nWhat would happen if you slightly displaced the rod so that it wasn't perfectly vertical with respect to the planet?\n\nOK, even though the Earth is pulling on both masses, isn't the rod still pulling up on the white mass to keep it taut as it revolves?\n\nNathanael\nHomework Helper\nisn't the rod still pulling up on the white mass to keep it taut as it revolves?\nI'm not sure what you mean by this.\n\nLike the system isn't collapsing as it revolves.\n\nNathanael\nHomework Helper\nBut why would it collapse?\n\nBecause the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.\n\nNathanael\nHomework Helper\nBecause the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.\nThere we go :)\nBut if there was no tension then the system would not collapse, it would stretch apart! That is why I thought you were misunderstanding.\n\nAbout the stability of the object. I'm not sure if you already figured it out, but what would happen if you slightly displaced the object so it wasn't vertical? Would it start rotating more and more away from being vertical, (unstable) or would it go back to being vertical (stable)?\n\nI think it would go back to vertical, but I can't explain why.\n\nNathanael\nHomework Helper\nIs there any torque on the system?\n\nYes because there is a force (tension) and it acts over the length of a rod, giving it both a force and a distance.\n\nNathanael\nHomework Helper\nYes because there is a force (tension) and it acts over the length of a rod, giving it both a force and a distance.\nTension acts in the direction through the center of mass, therefore it produces no torque. Just like when the object is rotating vertically, gravity acts through the center of mass, so it provides no torque. But what about when we displace the object slightly; does gravity still act through the center of mass?\n\nYes, gravity does acts through the center of mass.\n\nNathanael\nHomework Helper\nThe center of mass is located in the center of rod. But the force of gravity acts on the two masses on the edges of the rod (because it said the rod's weight was negligible). Perhaps draw a free body diagram of the object slightly tilted from it's vertical position.\n\nOk ok ,doing the FBD I see now that gravity does not act through the center of mass.\n\nNathanael\nHomework Helper\nOk ok ,doing the FBD I see now that gravity does not act through the center of mass.\nSo is there a net torque? If so, does it cause the rod to return to a vertical position or to rotate away from being vertical?\n\nYes because the force is acting some distance away perpendicular to the center of mass.\n\nAnd it returns it to a vertical position.\n\nNathanael\nHomework Helper\nYes because the force is acting some distance away perpendicular to the center of mass.\nYour explanation is a bit unclear again. The torque from gravity on the white mass acts to rotate the object away from the vertical, but the torque on the black mass acts to rotate the object back to the vertical. Does one of these torques win? (Is one of them stronger?)\n\nI think that the torques are the same because even though the black mass is experiencing more force ,the white mass has more distance, hence balancing the torques.\n\nNathanael\nHomework Helper\nWhen measuring torque you want to use the distance to the axis of rotation (in this case the center of mass). You also want to only use the component of the for force that is perpendicular to the rod. Equivalently, you can use the full force and the \"Lever arm\"\n\nI honestly don't understand what you're trying to get me to do.\n\nNathanael\nHomework Helper\nI think that the torques are the same because even though the black mass is experiencing more force ,the white mass has more distance, hence balancing the torques.\nThe lever arm is the same for both the white and black masses. Therefore the larger force produces a larger torque, so there is a net torque.\n\nOhh OK, I get it so since the torque is not equal the black mass experiences a torque clockwise making it vertical again and making it stable.\n\nBvU\nScience Advisor\nHomework Helper\nI'm going to make this one more difficult. So far you two looked at gravity (from the planet, it didn't mention Earth !). From the wording of the exercise (co-rotating frame of reference, rod length non-negligible wrt planet radius --either a big stick or a small planet :) ) I get the impression you also want to look at the apparent force that comes with such a frame of reference. There too, the black one gets more than the white one, and the torque is in the other direction !"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.88903123,"math_prob":0.8712279,"size":1382,"snap":"2021-04-2021-17","text_gpt3_token_len":332,"char_repetition_ratio":0.16255443,"word_repetition_ratio":0.06666667,"special_character_ratio":0.2366136,"punctuation_ratio":0.113074206,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9769884,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-21T01:54:25Z\",\"WARC-Record-ID\":\"<urn:uuid:43329ccf-0d72-4960-8f41-35f350c336fc>\",\"Content-Length\":\"152152\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8d73a9b2-cf88-4b82-9442-3348b027a3db>\",\"WARC-Concurrent-To\":\"<urn:uuid:c27d912b-d3dc-4c0b-8bb4-2528164316f7>\",\"WARC-IP-Address\":\"104.26.14.132\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/rods-with-masses-orbiting-earth.793069/\",\"WARC-Payload-Digest\":\"sha1:5DXF2F3L36UARRYHGBYA5X5MNXQM37JQ\",\"WARC-Block-Digest\":\"sha1:JOMKZ735ZYNOL5LIZQGR327IKD4WZQ3J\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618039503725.80_warc_CC-MAIN-20210421004512-20210421034512-00425.warc.gz\"}"} |
https://jifriedman.com/2020/09/20/ | [
"## Moving Average Transformation Calculations\n\nI decided to change the moving average calculation to use natural logs instead of the formula: (Weighted Price - Moving Average) / Weighted Price * 100. The returns are identical but the numbers are easier to understand with natural logs. Once the given moving average is determined, the new formula is: WorksheetFunction.Ln(Arg1:=(WPrice) / Mx(xw) *… Continue reading Moving Average Transformation Calculations"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7911703,"math_prob":0.98512423,"size":411,"snap":"2021-31-2021-39","text_gpt3_token_len":87,"char_repetition_ratio":0.12285012,"word_repetition_ratio":0.0,"special_character_ratio":0.20681265,"punctuation_ratio":0.104477614,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98599887,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-23T03:40:10Z\",\"WARC-Record-ID\":\"<urn:uuid:40017bf0-67a8-4a1c-9aed-af9ce68a0435>\",\"Content-Length\":\"55070\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e647207f-4b82-4c52-99fc-cb29c723fd94>\",\"WARC-Concurrent-To\":\"<urn:uuid:9bd258c9-b7da-493e-8787-f0e778cc4b3c>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://jifriedman.com/2020/09/20/\",\"WARC-Payload-Digest\":\"sha1:UUKMDSAZSFYRXOCMATMUNCIQHPMVOKRU\",\"WARC-Block-Digest\":\"sha1:XNBXBR4M4GPILC4D7RAIFFHBYWIWSQF3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057416.67_warc_CC-MAIN-20210923013955-20210923043955-00271.warc.gz\"}"} |
https://matthew.maennche.com/2014/04/assume-given-three-variables-revenue-expenses-profit-type-money-structured-type-two-int-fields-dollars-cents-assign-profit-result-subtracting-expenses/ | [
"# Assume you are given three variables, revenue, expenses, and profit, all of type Money (a structured type with two int fields, dollars and cents). Assign to profit the result of subtracting expenses from revenue. Let’s make the happy assumption that revenue exceeds expenses. However you still may find that the cents part of expenses exceeds that of revenue. If that is the case you will have to “borrow” 1 from revenue dollars (i.e. subtract 1) and “give” it to revenue’s cents (i.e. add 100!) in order to carry out the subtraction properly.\n\n1\n\n### CHALLENGE:\n\nAssume you are given three variables, revenue, expenses, and profit, all of type Money (a structured type with two int fields, dollars and cents). Assign to profit the result of subtracting expenses from revenue. Let’s make the happy assumption that revenue exceeds expenses. However you still may find that the cents part of expenses exceeds that of revenue. If that is the case you will have to “borrow” 1 from revenue dollars (i.e. subtract 1) and “give” it to revenue’s cents (i.e. add 100!) in order to carry out the subtraction properly.\n\n### SOLUTION:\n\n```\nfloat Income=0.0;\nfloat Expense=0.0;\nfloat Profit=0.0;\n\nIncome = revenue.dollars + revenue.cents/100.0;\nExpense = expenses.dollars + expenses.cents/100.0;\nProfit = Income-Expense;\n\nprofit.dollars = (int) Profit;\nprofit.cents = Profit /100;\n\n```\n\n#### One comment\n\n•",
null,
"booga says:\n\nfloat Income=0.0;\nfloat Expense=0.0;\nfloat Profit=0.0;\n\nIncome = revenue.dollars + revenue.cents/100.0;\nExpense = expenses.dollars + expenses.cents/100.0;\nProfit = Income-Expense;\n\nprofit.dollars = (int) Profit;\nprofit.cents = Profit /100;"
] | [
null,
"https://secure.gravatar.com/avatar/92403dd3746eea8aaf78a3fcb63423b5",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.83429277,"math_prob":0.7500757,"size":1054,"snap":"2020-34-2020-40","text_gpt3_token_len":283,"char_repetition_ratio":0.2,"word_repetition_ratio":0.31506848,"special_character_ratio":0.2751423,"punctuation_ratio":0.24347825,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99792355,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-29T17:10:13Z\",\"WARC-Record-ID\":\"<urn:uuid:4f1faaa6-4453-41fa-9383-e082f96fe697>\",\"Content-Length\":\"70606\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e06e2dd3-0935-4efb-9743-8ae41204373c>\",\"WARC-Concurrent-To\":\"<urn:uuid:0c32fcdd-f97f-4a62-b827-d79703242b7f>\",\"WARC-IP-Address\":\"104.28.6.215\",\"WARC-Target-URI\":\"https://matthew.maennche.com/2014/04/assume-given-three-variables-revenue-expenses-profit-type-money-structured-type-two-int-fields-dollars-cents-assign-profit-result-subtracting-expenses/\",\"WARC-Payload-Digest\":\"sha1:IC3GOOPDXKM43SID3KEGJIAIW3WP662U\",\"WARC-Block-Digest\":\"sha1:KL4N4RAX3ZJRH4IV7ITXP7LLALWOOHY7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400202418.22_warc_CC-MAIN-20200929154729-20200929184729-00306.warc.gz\"}"} |
https://sherpa.readthedocs.io/en/4.10.2/model_classes/api/sherpa.models.basic.Gauss1D.html | [
"# Gauss1D¶\n\nclass sherpa.models.basic.Gauss1D(name='gauss1d')[source]\n\nOne-dimensional gaussian function.\n\nfwhm\n\nThe Full-Width Half Maximum of the gaussian. It is related to the sigma value by: FWHM = sqrt(8 * log(2)) * sigma.\n\npos\n\nThe center of the gaussian.\n\nampl\n\nThe amplitude refers to the maximum peak of the model.\n\nNotes\n\nThe functional form of the model for points is:\n\nf(x) = ampl * exp(-4 * log(2) * (x - pos)^2 / fwhm^2)\n\n\nand for an integrated grid it is the integral of this over the bin.\n\nExamples\n\nCompare the gaussian and normalized gaussian models:\n\n>>> m1 = sherpa.models.basic.Gauss1D()\n>>> m2 = sherpa.models.basic.NormGauss1D()\n>>> m1.pos, m2.pos = 10, 10\n>>> m1.ampl, m2.ampl = 10, 10\n>>> m1.fwhm, m2.fwhm = 5, 5\n>>> m1(10)\n10.0\n>>> m2(10)\n1.8788745573993026\n>>> m1.fwhm, m2.fwhm = 1, 1\n>>> m1(10)\n10.0\n>>> m2(10)\n9.394372786996513\n\n\nThe normalised version will sum to the amplitude when given an integrated grid - i.e. both low and high edges rather than points - that covers all the signal (and with a bin size a lot smaller than the FWHM):\n\n>>> m1.fwhm, m2.fwhm = 12.2, 12.2\n>>> grid = np.arange(-90, 110, 0.01)\n>>> glo, ghi = grid[:-1], grid[1:]\n>>> m1(glo, ghi).sum()\n129.86497637060958\n>>> m2(glo, ghi).sum()\n10.000000000000002\n\n\nAttributes Summary\n\nMethods Summary\n\n apply(outer, *otherargs, **otherkwargs) calc(pars, xlo, *args, **kwargs) get_center() guess(dep, *args, **kwargs) Set an initial guess for the parameter values. regrid(*arrays) reset() set_center(pos, *args, **kwargs) startup() Called before a model may be evaluated multiple times. teardown() Called after a model may be evaluated multiple times.\n\nAttributes Documentation\n\nthawedparhardmaxes\nthawedparhardmins\nthawedparmaxes\nthawedparmins\nthawedpars\n\nMethods Documentation\n\napply(outer, *otherargs, **otherkwargs)\ncalc(pars, xlo, *args, **kwargs)\nget_center()[source]\nguess(dep, *args, **kwargs)[source]\n\nSet an initial guess for the parameter values.\n\nAttempt to set the parameter values, and ranges, for the model to match the data values. This is intended as a rough guess, so it is expected that the model is only evaluated a small number of times, if at all.\n\nregrid(*arrays)\nreset()\nset_center(pos, *args, **kwargs)[source]\nstartup()\n\nCalled before a model may be evaluated multiple times.\n\nteardown()\n\nCalled after a model may be evaluated multiple times.\n\nsetup()"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.5627361,"math_prob":0.9880119,"size":2490,"snap":"2020-24-2020-29","text_gpt3_token_len":817,"char_repetition_ratio":0.11987128,"word_repetition_ratio":0.07310705,"special_character_ratio":0.32730925,"punctuation_ratio":0.19496855,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96053934,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-13T20:41:52Z\",\"WARC-Record-ID\":\"<urn:uuid:0b9f6040-de27-4c93-b635-7712b08dee84>\",\"Content-Length\":\"34441\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5165de81-0cfb-489e-9737-b9d1148477a0>\",\"WARC-Concurrent-To\":\"<urn:uuid:793d4e23-32b7-48d2-ac1a-f75e46ec2db8>\",\"WARC-IP-Address\":\"104.17.33.82\",\"WARC-Target-URI\":\"https://sherpa.readthedocs.io/en/4.10.2/model_classes/api/sherpa.models.basic.Gauss1D.html\",\"WARC-Payload-Digest\":\"sha1:JOWS2DP7HYAPVNHPTCECV6CM6C3ULPNE\",\"WARC-Block-Digest\":\"sha1:WCMLG4DFKZT2N5P3OKTQDP2XXWAPSH4Q\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657146845.98_warc_CC-MAIN-20200713194203-20200713224203-00234.warc.gz\"}"} |
https://developer.apple.com/documentation/simd/1424095-trunc | [
"Function\n\n# trunc(_:)\n\nReturns each element in a vector rounded toward zero to the nearest integer.\n\n## Declaration\n\n### Math Functions\n\n`func ceil(SIMD4<Double>) -> SIMD4<Double>`\n\nReturns the ceiling of each element in a vector.\n\n`func floor(SIMD4<Double>) -> SIMD4<Double>`\n\nReturns the floor of each element in a vector.\n\n`func simd_muladd(simd_double4, simd_double4, simd_double4) -> simd_double4`\n\nReturns the multiply-add result for corresponding elements in three vectors."
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.57805556,"math_prob":0.7408953,"size":615,"snap":"2020-10-2020-16","text_gpt3_token_len":173,"char_repetition_ratio":0.15057284,"word_repetition_ratio":0.27659574,"special_character_ratio":0.2796748,"punctuation_ratio":0.11304348,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9790707,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-18T07:39:42Z\",\"WARC-Record-ID\":\"<urn:uuid:cc4f8b03-6a8c-4f61-96cf-8bb8b0570742>\",\"Content-Length\":\"29540\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cf256cff-8cbf-4165-a05a-587fcc0b26df>\",\"WARC-Concurrent-To\":\"<urn:uuid:1a34f29f-46c4-4286-a019-c7f6f1550e99>\",\"WARC-IP-Address\":\"17.253.21.208\",\"WARC-Target-URI\":\"https://developer.apple.com/documentation/simd/1424095-trunc\",\"WARC-Payload-Digest\":\"sha1:UCZCH7G2KGMCJQD4IBXFV3UBBEJUPO5D\",\"WARC-Block-Digest\":\"sha1:745HTILO2CURCDQ25R3J2BLIT72NR2S3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875143635.54_warc_CC-MAIN-20200218055414-20200218085414-00207.warc.gz\"}"} |
https://byjus.com/question-answer/a-patient-in-a-hospital-is-given-soup-daily-in-a-cylindrical-bowl-of-diameter-7-cm/ | [
"",
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"Question\n\n# A patient in a hospital is given soup daily in a cylindrical bowl of diameter $7cm$. If the bowl is filled with soup to a height of $4cm$, how much soup the hospital have to prepare daily to serve $250$ patients.\n\nOpen in App\nSolution\n\n## Radius of the cylindrical bowl $=\\frac{7}{2}cm$, Height of the cylindrical bowl $=4cm$Volume of the cylindrical bowl $={\\mathrm{\\pi r}}^{2}\\mathrm{h}$ $=\\frac{22}{7}×{\\left(\\frac{7}{2}\\right)}^{2}×4$ $=\\frac{22}{7}×\\frac{7}{2}×\\frac{7}{2}×4$ $=22×7=154c{m}^{3}$Volume of the soup given to $250$ patients $=250×154$ $=38,500c{m}^{3}$ $=\\frac{38500}{1000}=\\frac{385}{10}=38.5litres$ Note -: $1litre=1000c{m}^{3}$Therefore, volume of the soup given to $250$ patients $=38.5litres$",
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https://books.google.com.tw/books?id=X9U3BQAAQBAJ&hl=zh-TW&source=gbs_book_other_versions_r&cad=4 | [
"Mathematical Methods for Physicists\n\nAcademic Press, 2013年10月22日 - 1008 頁\nMathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on the mathematics of theoretical physics.\nThis edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations, numerical application of Chebyshev polynomials, the fast Fourier transform, and transfer functions. Many of the physical examples provided in this book, which are used to illustrate the applications of mathematics, are taken from the fields of electromagnetic theory and quantum mechanics. The Hermitian operators, Hilbert space, and concept of completeness are also deliberated.\nThis book is beneficial to students studying graduate level physics, particularly theoretical physics.\n\n讀者評論 -撰寫評論\n\nLibraryThing Review\n\nAs far as I remember this book was required reading for the “Special Functions” course. You could argue that this is not a pure Physics book. However, it is certainly not a pure Mathematics book ... 閱讀評論全文\n\n內容\n\n CHAPTER 1 VECTOR ANALYSIS 1 CHAPTER 2 COORDINATE SYSTEMS 85 CHAPTER 3 TENSOR ANALYSIS 118 CHAPTER 4 DETERMINANTS MATRICES AND GROUP THEORY 168 CHAPTER 5 INFINITE SERIES 277 CHAPTER 6 FUNCTIONS OF A COMPLEX VARIABLE I 352 CHAPTER 7 FUNCTIONS OF A COMPLEX VARIABLE II 396 CHAPTER 8 DIFFERENTIAL EQUATIONS 437\n CHAPTER 12 LEGENDRE FUNCTIONS 637 CHAPTER 13 SPECIAL FUNCTIONS 712 CHAPTER 14 FOURIER SERIES 760 CHAPTER 15 INTEGRAL TRANSFORMS 794 CHAPTER 16 INTEGRA LEQUATIONS 865 CHAPTER 17 CALCULUS OF VARIATIONS 925 REAL ZEROS OF A FUNCTION 963 GAUSSIAN QUADRATURE 968\n\n CHAPTER 9 STURMLIOUVILLE THEORYORTHOGONAL FUNCTIONS 497 CHAPTER 10 THE GAMMA FUNCTION FACTORIAL FUNCTION 539 CHAPTER 11 BESSEL FUNCTIONS 573\n GENERAL REFERENCES 974 INDEX 975 版權所有"
] | [
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https://bsebsolutions.com/bihar-board-class-10th-maths-solutions-chapter-6-ex-6-2-english-medium/ | [
"# Bihar Board Class 10th Maths Solutions Chapter 6 Triangles Ex 6.2\n\nBihar Board Class 10th Maths Solutions Chapter 6 Triangles Ex 6.2 Textbook Questions and Answers.\n\n## BSEB Bihar Board Class 10th Maths Solutions Chapter 6 Triangles Ex 6.2",
null,
"Question 1.\nIn figures, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).",
null,
"Solution:\n(i) In Fig. (i),\nsince DE || BC, therefore\n$$\\frac { AD }{ DB }$$ = $$\\frac { AE }{ EC }$$ or $$\\frac { 1.5 }{ 3 }$$ = $$\\frac { 1}{ EC }$$\nor EC = $$\\frac { 3}{ 1.5}$$ = $$\\frac{3 \\times 10}{15}$$ Cm = 2 cm\n\n(ii) In Fig. (ii),\nsince DE || BC, therefore\n$$\\frac { AD }{ DB }$$ =$$\\frac { AE }{ EC}$$ or $$\\frac { AD }{7.2 }$$ = $$\\frac { 1.8 }{ 5.4 }$$\nor AD = $$\\frac { 18 }{ 54 }$$ x $$\\frac { 72 }{ 10 }$$ cm = 2.4 cm.\n\nQuestion 2.\nE and F are points on the sides PQ and PR respectively of a A PQR. For each of the following cases, state whether EF || QR :\n(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm\n(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm\n(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm\nSolution:\n(i) We have :\nPE = 3.9 cm, EQ = 3 cm,\nPF = 3.6 cm and FR = 2.4 cm",
null,
"i. e., EF does not divide the sides PQ and PR of A PQR in the same ratio. Therefore, EF is not parallel to QR.\n\n(ii) We have : PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm\nNow, $$\\frac { PE }{ EQ }$$ = $$\\frac { 4 }{ 4.5 }$$ = $$\\frac { 40 }{ 45 }$$ = $$\\frac { 8 }{ 9 }$$\nand $$\\frac { PF }{ FR }$$ = $$\\frac { 8 }{ 9 }$$\nSo, $$\\frac { PE }{ EQ }$$ = $$\\frac { PF }{ FR }$$\nThus, EF divides sides PQ and PR of ∆ PQR in the same ratio. Therefore, by the converse of Basic Proportionality Theorem, we have EF || QR.\n\n(iii) We have :PQ= 1.28 cm, PR = 2.56 cm,\nPE = 0.18 cm and PF 0.36 cm\n∴ EQ = PQ – PE = (1.28 – 0.18) cm = 1.10 cm\nand FR = PR – PF = (2.56 – 0.36) = 2.20 cm\nNow, $$\\frac { PE }{ EQ }$$ = $$\\frac { 0.18 }{ 1.10 }$$ = $$\\frac { 18 }{ 110 }$$ = $$\\frac { 9 }{ 55 }$$\nand $$\\frac { PF }{ FR }$$ = $$\\frac { 0.36 }{ 2.20 }$$ = $$\\frac { 36 }{ 220 }$$ = $$\\frac { 9 }{ 55 }$$\nSo, $$\\frac { PE }{ EQ }$$ = $$\\frac { PF }{ FR }$$\nThus, EF divides sides PQ and PR of ∆ PQR in the same ratio. Therefore, by the converse of Basic Proportionality Theorem, we have EF || QR.",
null,
"Question 3.\nIn the figure, if LM || CB and LN || CD, prove that $$\\frac { AM }{ AB }$$ = $$\\frac { AN }{ AD }$$",
null,
"Solution:\nIn ∆ ABC, we have :\nLM || CB\n∴By a result based on Basic Proportionality Theorem (a corollary), we have :\n$$\\frac { AM }{ AB }$$ = $$\\frac { AL }{ AC }$$ … (1)\nIn ∆ ACD, we have :\nLN || CD [Given]\n∴By a result based on Basic Proportionality Theorem, we have :\n$$\\frac { AL }{ AC }$$ = $$\\frac { AN }{ AD }$$\nFrom (1) and (2), we obtain that\n$$\\frac { AM }{ AB }$$ = $$\\frac { AN }{ AD }$$\n\nQuestion 4.\nIn the figure, DE || AC and DF || AE. Prove that $$\\frac { BF }{ FE }$$ = $$\\frac { BE }{ EC }$$",
null,
"Solution:\nIn ∆ BGA, we have :\nDE || ‘AC\n∴By Basic Proportionality Theorem, we have :\n$$\\frac { BE }{ EC }$$ = $$\\frac { BD }{ DA }$$ …. (1)\nIn ∆ BEA, we have :\nDF || AE [Given]\n∴By Basic Proportionality Theorem, we have\n$$\\frac { BF }{ FE }$$ = $$\\frac { BD }{ DA }$$ …. (2)\nFrom (1) and (2), we obtain that\n$$\\frac { BF }{ FE }$$ = $$\\frac { BE }{ EC }$$.\n\nQuestion 5.\nIn the figure, DE || OQ and DF || OR. Show that EF || QR.",
null,
"Solution:\nIn ∆ PQO, we have :\nDE || OQ [Given]\n∴By Basic Proportionality Theorem, we have :\n$$\\frac { PE }{ EQ }$$ = $$\\frac { PD }{ DO }$$ … (1)\nIn ∆ POR, We have :\nDF || OR [Given]\n∴By Basic Proportionality Theorem, we have\n$$\\frac { PD }{ DO }$$ = $$\\frac { PF }{ FR }$$ … (2)\nFrom (1) and (2), we obtain that\n$$\\frac { PE }{ EQ }$$ = $$\\frac { PF }{ FR }$$\nSo, EF || QR [By the converse of BPT]",
null,
"Question 6.\nIn the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.",
null,
"Solution:\nGiven : O is any point within ∆ PQR, AB || PQ and AC || PR\nTo prove : BC || QR.\nConstruction : Join BC.\nProof:\nIn ∆ OPQ, we have :\nAB || PQ [Given]\n∴By Basic Proportionality Theorem, we have :\n$$\\frac { OA }{ AP }$$ = $$\\frac { OB }{ BQ }$$ … (1)\nIn ∆ OPR, We have :\nAC || PR [Given]\n∴ By Basic Proportionality Theorem, we have\n$$\\frac { OA }{ AP }$$ = $$\\frac { OC }{ CR }$$ … (2)\nFrom (1) and (2), we obtain that\n$$\\frac { OB }{ BQ }$$ = $$\\frac { OC }{ CR }$$\nThus, in ∆ OQR, B and C are points dividing the sides OQ and OR in the same ratio. Therefore, by the converse of Basic Proportionality Theorem, we have :\nBC || QR.\n\nQuestion 7.\nUsing Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).",
null,
"Solution:\nGiven : ∆ ABC, in which D is the mid-point of side AB and the line DE is drawn parallel to BC, meeting AC in E.\nTo prove : AE = EC\nProof:\nSince DE || BC, therefore by Basic Proportionality Theorem, we have :\n$$\\frac { AD }{ DB }$$ = $$\\frac { AE }{ EC }$$ … (1)\nBut AD = DB [∵D is the mid-point of AB]\ni.e., $$\\frac { AD }{ DB }$$ = 1\n∴ From (1), $$\\frac { AE }{ EC }$$ = 1 or AE = EC\nHence, E is the mid-point of the third side AC.",
null,
"Question 8.\nUsing Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).",
null,
"Solution:\nGiven : ∆ ABC, in which D and E are the mid¬points of sides AB and AC respectively.\nTo prove : DE || BC.\nProof :\nSince D and E are the mid-points of AB and AC respectively, therefore\nAD = DB and AE = EC\n∴ $$\\frac { AD }{ DB }$$ = 1\nand AE = EC\n∴ $$\\frac { AE }{ EC }$$ = 1\nThus, in ∆ ABC, D and E are points dividing the sides AB and AC in the same ratio. Therefore, by the converse of Basic Proportionality Theorem (Theorem 6.2), we have :\nDE || BC.\n\nQuestion 9.\nABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that $$\\frac { AO }{ BO }$$ = $$\\frac { CO }{ DO }$$.\nSolution:\nGiven : A trapezium ABCD, in which AB || DC and its diagonals AC and BD intersect each other at O.\nTo prove : $$\\frac { AO }{ BO }$$ = $$\\frac { CO }{ DO }$$.\nConstruction : Through O, draw OE || AB, i.e., OE || DC.\nProof:\nIn ∆ ADC, we have :\nOE || DC [Construction]\n∴ By Basic Proportionality Theorem, we have :",
null,
"$$\\frac { AE }{ ED }$$ = $$\\frac { AO }{ CO }$$ … (1)\nAgain, in ∆ ABD, we have :\nOE || AB [Construction]\n∴By Basic Proportionality Theorem, we have :0\n$$\\frac { ED }{ AE }$$ = $$\\frac { DO }{ BO }$$ or $$\\frac { AE }{ ED }$$ = $$\\frac { BO }{ DO }$$ … (2)\nFrom (1) and (2), we obtain that\n$$\\frac { AO }{ CO }$$ = $$\\frac { BO }{ DO }$$ or $$\\frac { AO }{ BO }$$ = $$\\frac { CO }{ DO }$$.",
null,
"Question 10.\nThe diagonals of a quadrilateral ABCD intersect each other at the point O such that $$\\frac { AO }{ BO }$$ = $$\\frac { CO }{ DO }$$ that ABCD is a trapezium.\nSolution:\nGiven : A quadrilateral ABCD in which its diagonals AC and BD intersect each other at the point O such that $$\\frac { AO }{ BO }$$ = $$\\frac { CO }{ DO }$$, i.e., $$\\frac { AO }{ CO }$$ = $$\\frac { BO }{ DO }$$\nTo prove : Quadrilateral ABCD is a trapezium.\nConstruction : Through O, draw OE || AB meeting AD in E.\nProof:\nIn ∆ ADB, we have :\nOE || AB [Construction]\n∴ By Basic Proportionality Theorem, we have :\n$$\\frac { DE }{ EA }$$ = $$\\frac { OD }{ BO }$$\nor $$\\frac { EA }{ DE }$$ = $$\\frac { BO }{ DO }$$\ni.e., $$\\frac { EA }{ DE }$$ = $$\\frac { BO }{ DO }$$ = $$\\frac { AO }{ CO }$$ [ ∵$$\\frac { AO }{ CO }$$ = $$\\frac { BO }{ DO }$$(given)]\nor $$\\frac { EA }{ DE }$$ = $$\\frac { AO }{ CO }$$\nThus, in ∆ ADC, points E and O are dividing the sides AD and AC in the same ratio. Therefore, by the converse of Basic Proportionality Theorem, we have :\nEO || DC\nBut, EO || AB [construction]\nHence, AB || DC\n∴ Quadrilateral ABCD is a trapezium"
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https://gitlab.science.ru.nl/clean-and-itasks/abc-interpreter/commit/27cc66bf5ca5dfc459994cf439f828775afb3f35 | [
"### Merge branch '81-runtime-error-with-currying-constructors' into 'master'\n\n```Resolve \"Runtime error with currying constructors\"\n\nCloses #81\n\nSee merge request !109```\nparents 01f01dc6 15d1e089\nPipeline #25963 canceled with stages\nin 1 minute and 25 seconds\n ... ... @@ -106,6 +106,8 @@ const char *instruction_type (BC_WORD i) { case Cbuildhr30: return \"l\"; case Cbuildhr31: return \"l\"; case Cbuildhr40: return \"l\"; case Cbuild_node_rtn: return \"n\"; case Cbuild_node2_rtn: return \"\"; case Cbuild_r: return \"nnlnn\"; case Cbuild_ra0: return \"nnl\"; case Cbuild_ra1: return \"nnln\"; ... ...\n ... ... @@ -78,6 +78,8 @@ enum { INSTRUCTION(buildhr30) INSTRUCTION(buildhr31) INSTRUCTION(buildhr40) INSTRUCTION(build_node_rtn) INSTRUCTION(build_node2_rtn) INSTRUCTION(build_r) INSTRUCTION(build_ra0) INSTRUCTION(build_ra1) ... ...\n ... ... @@ -365,19 +365,14 @@ void wprint_node(WINDOW *win, BC_WORD *node, int with_arguments) { } void debugger_update_a_stack(BC_WORD *ptr) { char _tmp; BC_WORD *start = asp + 1; mvwprintw(winh_a, 0, 0, \"A-stack (%d)\\n\", ptr-start+1); wrefresh(winh_a); wmove(win_a, 0, 0); while (start <= ptr) { print_label(_tmp, 256, 0, (BC_WORD*) *start, program, hp, heap_size); wprintw(win_a, \"%3d %s\", ptr-start, _tmp); if (hp <= (BC_WORD*) *start && (BC_WORD*) *start < hp + heap_size) { wprintw(win_a, \" \"); wprint_node(win_a, (BC_WORD*) *start, 0); } wprintw(win_a, \"%3d \",ptr-start); wprint_node(win_a, (BC_WORD*) *start, 0); wprintw(win_a, \"\\n\"); start++; } ... ...\n ... ... @@ -498,6 +498,7 @@ eval_to_hnf_return_failure: pc = (BC_WORD*)program->symbol_table[program->start_symbol_id].offset; } BC_WORD fast_ap_descriptor=0; #ifdef COMPUTED_GOTOS goto **(void**)pc; # include \"interpret_instructions.h\" ... ...\n ... ... @@ -27,4 +27,4 @@ typedef int64_t CleanInt; #define BCGEN_INSTRUCTION_TABLE_SIZE 512 #define ABC_MAGIC_NUMBER 0x2a434241 #define ABC_VERSION 8 #define ABC_VERSION 9\n ... ... @@ -6,6 +6,7 @@ brainfuck cafs chars compress curry e fills fsieve ... ...\n [(Cons 1 (Cons 2 (Cons 3 Nil))),(Cons2 1 2 (Cons2 1 3 Nil)),(Cons3 1 2 3 (Cons3 1 2 4 Nil))] halt at 4 102 524186 524288\n [(Cons 1 (Cons 2 (Cons 3 Nil))),(Cons2 1 2 (Cons2 1 3 Nil)),(Cons3 1 2 3 (Cons3 1 2 4 Nil))] halt at 4 102 262042 262144\ntest/curry.icl 0 → 100644\n module curry :: List = Cons Int List | Cons2 Int Int List | Cons3 Int Int Int List | Nil foldr :: (.a -> .(.b -> .b)) .b ![.a] -> .b foldr op r l = foldr l where foldr [] = r foldr [a:x] = op a (foldr x) Start = [ foldr Cons Nil [1,2,3] , foldr (Cons2 1) Nil [2,3] , foldr (Cons3 1 2) Nil [3,4] ]\n Version: 1.4 Global ProjectRoot: . Target: StdEnv Exec: {Project}/curry ByteCode: {Project}/curry.bc CodeGen CheckStacks: False CheckIndexes: False GenerateByteCode: True OptimiseABC: True Application HeapSize: 2097152 StackSize: 512000 ExtraMemory: 8192 IntialHeapSize: 204800 HeapSizeMultiplier: 4096 ShowExecutionTime: False ShowGC: False ShowStackSize: False MarkingCollector: False DisableRTSFlags: False StandardRuntimeEnv: True Profile Memory: False MemoryMinimumHeapSize: 0 Time: False Stack: False Dynamics: False GenericFusion: False DescExL: True Output Output: ShowConstructors Font: Monaco FontSize: 9 WriteStdErr: False Link LinkMethod: Static GenerateRelocations: False GenerateSymbolTable: False GenerateLinkMap: False LinkResources: False ResourceSource: GenerateDLL: False ExportedNames: StripByteCode: True KeepByteCodeSymbols: True PrelinkByteCode: True Paths Path: {Project} Precompile: Postlink: MainModule Name: curry Dir: {Project} Compiler NeverMemoryProfile: False NeverTimeProfile: False StrictnessAnalysis: True ListTypes: StrictExportTypes ListAttributes: True Warnings: True Verbose: True ReadableABC: False ReuseUniqueNodes: True Fusion: False\n ... ... @@ -23,7 +23,7 @@ lit_char :: !Char -> Expr TChar lit_short :: !Int -> Expr TShort lit_int :: !Int -> Expr TInt instance to_word TWord, TChar, TInt, TShort, (TPtr t), TReal instance to_word TWord, TPtrOffset, TChar, TInt, TShort, (TPtr t), TReal instance to_bool TWord instance to_char TWord instance to_int TWord ... ... @@ -51,7 +51,7 @@ instance ^ (Expr TReal) (&.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (|.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr a) !(Expr a) -> Expr a (>>.) infix 7 :: !(Expr a) !(Expr a) -> Expr a xorI :: !(Expr TWord) !(Expr TWord) -> Expr TWord ~. :: !(Expr TWord) -> Expr TWord ... ... @@ -150,6 +150,7 @@ small_integer :: !(Expr TInt) -> Expr TWord static_character :: !(Expr TChar) -> Expr TWord static_boolean :: !(Expr TWord) -> Expr TWord caf_list :: Expr (TPtr TWord) fast_ap_descriptor :: Expr TWord push_c :: !(Expr (TPtr TWord)) !Target -> Target pop_pc_from_c :: !Target -> Target ... ...\n ... ... @@ -197,12 +197,13 @@ lit_short i = toString i lit_int :: !Int -> Expr TInt lit_int i = toString i instance to_word TWord where to_word e = e instance to_word TChar where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TInt where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TShort where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word (TPtr t) where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TReal where to_word e = \"*(BC_WORD*)&(\"+-+e+-+\")\" instance to_word TWord where to_word e = e instance to_word TPtrOffset where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TChar where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TInt where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TShort where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word (TPtr t) where to_word e = \"(BC_WORD)(\"+-+e+-+\")\" instance to_word TReal where to_word e = \"*(BC_WORD*)&(\"+-+e+-+\")\" instance to_bool TWord where ... ... @@ -264,7 +265,7 @@ instance ^ (Expr TReal) where ^ a b = \"pow(\"+-+a+-+\",\"+-+b+-+\")\" (|.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (|.) a b = \"(\"+-+a+-+\"|\"+-+b+-+\")\" (<<.) infix 7 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr a) !(Expr a) -> Expr a (<<.) a b = \"(\"+-+a+-+\"<<\"+-+b+-+\")\" (>>.) infix 7 :: !(Expr a) !(Expr a) -> Expr a ... ... @@ -495,6 +496,9 @@ static_boolean b = \"(BC_WORD)&static_booleans[(\"+-+b+-+\") ? 2 : 0]\" caf_list :: Expr (TPtr TWord) caf_list = \"(BC_WORD*)caf_list\" fast_ap_descriptor :: Expr TWord fast_ap_descriptor = \"fast_ap_descriptor\"; push_c :: !(Expr (TPtr TWord)) !Target -> Target push_c v t = append (\"\\t*++csp=(BC_WORD)\"+-+v+-+\";\") t ... ...\n ... ... @@ -56,7 +56,7 @@ instance ^ (Expr TReal) (&.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (|.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr a) !(Expr a) -> Expr a (>>.) infix 7 :: !(Expr a) !(Expr a) -> Expr a xorI :: !(Expr TWord) !(Expr TWord) -> Expr TWord ~. :: !(Expr TWord) -> Expr TWord ... ... @@ -153,6 +153,7 @@ small_integer :: !(Expr TInt) -> Expr TWord static_character :: !(Expr TChar) -> Expr TWord static_boolean :: !(Expr TWord) -> Expr TWord caf_list :: Expr (TPtr TWord) fast_ap_descriptor :: Expr TWord push_c :: !(Expr (TPtr TWord)) !Target -> Target pop_pc_from_c :: !Target -> Target ... ...\n ... ... @@ -182,6 +182,8 @@ where [ \"(func (export \\\"get_\"+++{if (c=='-') '_' c \\\\ c <-: v}+++\"\\\") (result i32) (global.get \\$g-\"+++v+++\"))\" \\\\ v <- rt_vars ] ++ [ \"(func (export \\\"set_\"+++{if (c=='-') '_' c \\\\ c <-: v}+++\"\\\") (param i32) (global.set \\$g-\"+++v+++\" (local.get 0)))\" \\\\ v <- rt_vars ] ++ [ \"(global \\$g-fast-ap-descriptor (mut i64) (i64.const 0))\" ] ++ IF_GLOBAL_TEMP_VARS [\"(global \\$vw\"+++toString i+++\" (mut i32) (i32.const 0))\" \\\\ i <- [0..maxList [i.temp_vars.tv_i32 \\\\ i <- is]]] [] ++ IF_GLOBAL_TEMP_VARS [\"(global \\$vq\"+++toString i+++\" (mut i64) (i64.const 0))\" \\\\ i <- [0..maxList [i.temp_vars.tv_i64 \\\\ i <- is]]] [] ++ IF_GLOBAL_TEMP_VARS [\"(global \\$vd\"+++toString i+++\" (mut f64) (f64.const 0))\" \\\\ i <- [0..maxList [i.temp_vars.tv_f64 \\\\ i <- is]]] [] ++ ... ... @@ -269,6 +271,7 @@ where | v==\"hp\" = I32 | v==\"hp-free\" = I32 | v==\"hp-size\" = I32 | v==\"fast-ap-descriptor\" = I64 | otherwise = abort (\"unknown variable \"+++v+++\"\\n\") instr_unimplemented :: !Target -> Target ... ... @@ -397,7 +400,7 @@ instance ^ (Expr TReal) where ^ a b = Ecall \"clean_powR\" [a,b] (|.) infixl 6 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (|.) a b = Eor (type2 a b) a b (<<.) infix 7 :: !(Expr TWord) !(Expr TWord) -> Expr TWord (<<.) infix 7 :: !(Expr a) !(Expr a) -> Expr a (<<.) a b = Eshl (type2 a b) a b (>>.) infix 7 :: !(Expr a) !(Expr a) -> Expr a ... ... @@ -733,6 +736,9 @@ where caf_list :: Expr (TPtr TWord) caf_list = Econst I32 (97*8) fast_ap_descriptor :: Expr TWord fast_ap_descriptor = Ivar (Global \"g-fast-ap-descriptor\") C = rt_var \"csp\" push_c :: !(Expr (TPtr TWord)) !Target -> Target ... ...\n ... ... @@ -3530,6 +3530,7 @@ all_instructions opts t = bootstrap \\$ collect_instructions opts \\$ map (\\i -> i t new_local TWord (n @ 0) \\d -> if_then_else (to_short_ptr d @ 0 ==. lit_short (8*ns)) ( new_local TShort (to_short_ptr d @ -1) \\arity -> fast_ap_descriptor .= d - to_word (arity <<. if_i64_or_i32_expr (lit_short 4) (lit_short 3)) :. Pc .= to_word_ptr (to_word_ptr (d + (lit_word (ns*2-1) * if_i64_or_i32_expr (lit_word 8) (lit_word 4)) - lit_word 2) @ 0) :. rewind_ptr Pc 3 :. if_then_else (arity <=. lit_short 1) ( ... ... @@ -3560,7 +3561,27 @@ all_instructions opts t = bootstrap \\$ collect_instructions opts \\$ map (\\i -> i t Pc .= to_word_ptr (to_word_ptr (d + if_i64_or_i32_expr (lit_word 6) (lit_word 2)) @ 0) )) ] ++ [ instr \"add_arg0\" Nothing \\$ [ instr \"build_node2_rtn\" Nothing \\$ ensure_hp 3 :. pop_pc_from_c :. Hp @ 0 .= fast_ap_descriptor + if_i64_or_i32_expr (lit_word 32) (lit_word 16) :. Hp @ 1 .= A @ 0 :. Hp @ 2 .= A @ -1 :. A @ -1 .= to_word Hp :. shrink_a 1 , instr \"build_node_rtn\" Nothing \\$ new_local TPtrOffset (to_ptr_offset (Pc @ 1)) \\n_args_m_2 -> ensure_hp (n_args_m_2 + lit_hword 3) :. Hp @ 0 .= fast_ap_descriptor + to_word ((n_args_m_2 + lit_hword 2) <<. if_i64_or_i32_expr (lit_hword 4) (lit_hword 3)) :. Hp @ 1 .= A @ 0 :. Hp @ 2 .= to_word (Hp @? 3) :. pop_pc_from_c :. Hp @ 3 .= A @ -1 :. unrolled_loop [1..32] (\\i -> n_args_m_2 <. lit_hword i) (\\i -> Hp @ (i+3) .= A @ (-1-i)) :. shrink_a (n_args_m_2 + lit_hword 1) :. A @ 0 .= to_word Hp :. advance_ptr Hp (n_args_m_2 + lit_hword 3) , instr \"add_arg0\" Nothing \\$ ensure_hp 2 :. new_local (TPtr TWord) (to_word_ptr (A @ 0)) \\n -> pop_pc_from_c :. ... ...\nMarkdown is supported\n0% or\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!"
] | [
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https://math.answers.com/questions/What_does_fractions_and_percents_have_in_common | [
"",
null,
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null,
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"0\n\n# What does fractions and percents have in common?\n\nA percentage is the numerator of a fraction whose denominator is 100.\n\nA percentage is the numerator of a fraction whose denominator is 100.\n\nA percentage is the numerator of a fraction whose denominator is 100.\n\nA percentage is the numerator of a fraction whose denominator is 100.",
null,
"Study guides\n\n20 cards\n\n## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials\n\n➡️\nSee all cards\n3.8\n1991 Reviews\n\nA percentage is the numerator of a fraction whose denominator is 100.",
null,
"",
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"Earn +20 pts",
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"",
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""
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https://electronics.stackexchange.com/questions/255486/propagation-delay-in-4-inverter-w-l-ratio | [
"# Propagation delay in 4 inverter w/l ratio",
null,
"hi guys, I was given a task on optimizing propagation delay of four inverters using cadence software. The circuit connection is shown in above figure. I was asked to change the ratio of w/l for pmos and nmos to get minimum delay. The length of pmos and nmos is 160nm. So, I just need to change the width of pmos and nmos. However, what should I put the value of width for pmos and nmos??\n\nI need tphl=tplh (symmetrical cmos).I understood that the in order for tphl=tplh, the ratio of w/l pmos must greater than 2.5 w/l nmos. or any other suggested value?????????\n\nIncreasing the w of pmos: tplh will reduce Increasing the w of nmos: tphl will reduce\n\n1)What is the suitable width value for each pmos and nmos considering length=160nm ? 2)Can the width of pmos/nmos smaller than length ie >>>> w/l= 48nm/160nm????or the width must be larger than the length of pmos/nmos??? 3)Since I am using cadence software to design,do I need to take in consideration of capacitance or other formula?\n\nps: I am not familiar with VLSI\n\n• Please take the time to properly format, punctuate, and proof-read your question... – Tom Carpenter Sep 1 '16 at 16:02\n• You should look up the concept of \"logical effort\" en.wikipedia.org/wiki/Logical_effort : It is an incredibly useful and important abstraction used to size logic gates. It is also very commonly asked in interview questions. – jbord39 Oct 1 '16 at 16:26\n\nSince you are working with Cadence you could use simulation to find an optimum.\n\nUse a design variable WP for the width of the PMOS and WN for the width of the NMOS. Define a parameter k.\n\nFor the first inverter specify the width of the PMOS as WP and for the NMOS as WN. For the second inverter use k * WP and k * WN, for the third k * k * WP and k * k * WN, and so on.\n\nPerform a parametric sweep for k from say 2 to 10 with an increment of 0.1 and evaluate the delay of your buffer. Plot the result and find the minimum.\n\nFor a more theoretical approach you find some information here.\n\n• In your solution you implicitly assumed that each gate should be sized as k, k^2, k^3, ... This is true, but it comes from the concept of logical effort. Since you did not mention that in your answer, the OP may be curious why you knew to choose these ratios. – jbord39 Oct 1 '16 at 16:28\n• @jbord39 - This was no assumption. The scaling is given in the OP's schematic. – Mario Oct 1 '16 at 16:52\n• Ah okay, I didn't see that. – jbord39 Oct 1 '16 at 16:57\n• I would like to add @Mario, the last inverter can also be designed first i.e. the one which is driving $C_L$. As you said $W_p = 2W_n = 2W$. Then we just keep decreasing the size of inverter as $W_p' = 2W_n' = \\frac{2W}{a}$, $W_p'' = 2W_n'' = \\frac{2W}{a^2}$ and so on.. – rsg1710 Nov 9 '17 at 9:12\n• @Mario, I think both analogy are equally used, in my experience I have used the methodology I have mentioned. Last to first ; but I have seen the other way round as well – rsg1710 Nov 9 '17 at 9:15"
] | [
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http://www.openmx-square.org/forum/patio.cgi?mode=view&no=2728 | [
"",
null,
"This thread is locked.Only browsing is available.\n Top Page > Browsing",
null,
"Wrong calculation of chemical potential in some cases Date: 2020/12/03 13:56 Name: Seungjin Kang I have encountered some cases where the chemical potential is not calculated correctly.- 1. Input fileI used the the input file for Fe-bcc in openmx work directory, openmx/work/large_example/FeBCC.datThen I changed some parameters as follows.scf.SpinPolarization nc scf.SpinOrbit.Coupling onscf.maxIter 1This turns on non-collinear calculation, and scf.maxIter is just set to be 1 since the only purpose is to check whether chemical potential is correctly calculated or not.- 2. Test caseThis system contains 16 Fe atoms.For test calculation, I varied the number of Fe atoms by deleting some lines in Atoms.SpeciesAndCoordinates section.With 15 Fe system, for example, I just deleted[16 Fe 0.75 0.75 0.75 8.0 6.0]line in Atoms.SpeciesAndCoordinates, and for 14 Fe system, deleted[15 Fe 0.75 0.75 0.25 8.0 6.0][16 Fe 0.75 0.75 0.75 8.0 6.0]and so on. Here's the result of the chemical potential value from the output file.[11 Fe atoms case]Chemical Potential (Hatree) = -0.15186311662247total= 154.00000 ideal(neutral)= 154.00000[12 Fe atoms case]Chemical Potential (Hatree) = 20.00000000000000total= 146.00000 ideal(neutral)= 168.00000[13 Fe atoms case] Chemical Potential (Hatree) = 20.00000000000000 total= 158.00000 ideal(neutral)= 182.00000[16 Fe atoms case]Chemical Potential (Hatree) = 20.00000000000000total= 195.00000 ideal(neutral)= 224.00000So, up to 11 Fe atoms, the chemical potential is calculated correctly, so the number of electrons is also correctly calculated.But if we add more Fe atoms, we see that the chemical potential and the number of electrons is not calculated correctly.In fact, the number 20.0000 for chemical potential is the default ChemP_MAX value.- 3. Origin of the ProblemI guess that the origin of this problem is the wrong estimation of the number of state to be solved in the calculation.For non-collinear case, this is determined between the line 248~251 in Band_DFT_Noncol.c file.[Band_DFT_NonCol.c]248 lumos = (double)n2*0.200;249 if (lumos<60.0) lumos = 400.0;250 MaxN = (TZ-system_charge)/2 + (int)lumos;251 if (n2\n Page:",
null,
"Re: Wrong calculation of chemical potential in some cases ( No.1 ) Date: 2020/12/10 18:38 Name: Naoya Yamaguchi Hi,I think that your point is right.You can find the following in \"Band_DFT_Col.c\" in OpenMX 3.9.2. lumos = (double)n*0.200; if (lumos<60.0) lumos = 400.0; MaxN = (TZ-system_charge)/2 + (int)lumos; if (n\n\n Page:"
] | [
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"http://www.openmx-square.org/forum/img/alarm.gif",
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"http://www.openmx-square.org/forum/img/fold3.gif",
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"http://www.openmx-square.org/forum/img/file.gif",
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.78385407,"math_prob":0.9812347,"size":3160,"snap":"2021-43-2021-49","text_gpt3_token_len":963,"char_repetition_ratio":0.15114069,"word_repetition_ratio":0.0527307,"special_character_ratio":0.33892405,"punctuation_ratio":0.14798206,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9886254,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T08:20:00Z\",\"WARC-Record-ID\":\"<urn:uuid:fb0a0ddd-0ac5-4f2d-88a7-4d09af9baff7>\",\"Content-Length\":\"7353\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6373eeda-647b-4df0-9544-a95bf1f9e8ee>\",\"WARC-Concurrent-To\":\"<urn:uuid:734f4b11-cc04-46d2-8ed9-1e528fbc3b7e>\",\"WARC-IP-Address\":\"153.127.139.222\",\"WARC-Target-URI\":\"http://www.openmx-square.org/forum/patio.cgi?mode=view&no=2728\",\"WARC-Payload-Digest\":\"sha1:RLWFI6LPT32VKTTLP3NYRVRUPEMJBEJL\",\"WARC-Block-Digest\":\"sha1:33TJDD4IV7DAYFQ3SBNHHCKO5OTSWYUA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363337.27_warc_CC-MAIN-20211207075308-20211207105308-00359.warc.gz\"}"} |
https://www.colorhexa.com/01e9d5 | [
"# #01e9d5 Color Information\n\nIn a RGB color space, hex #01e9d5 is composed of 0.4% red, 91.4% green and 83.5% blue. Whereas in a CMYK color space, it is composed of 99.6% cyan, 0% magenta, 8.6% yellow and 8.6% black. It has a hue angle of 174.8 degrees, a saturation of 99.1% and a lightness of 45.9%. #01e9d5 color hex could be obtained by blending #02ffff with #00d3ab. Closest websafe color is: #00ffcc.\n\n• R 0\n• G 91\n• B 84\nRGB color chart\n• C 100\n• M 0\n• Y 9\n• K 9\nCMYK color chart\n\n#01e9d5 color description : Vivid cyan.\n\n# #01e9d5 Color Conversion\n\nThe hexadecimal color #01e9d5 has RGB values of R:1, G:233, B:213 and CMYK values of C:1, M:0, Y:0.09, K:0.09. Its decimal value is 125397.\n\nHex triplet RGB Decimal 01e9d5 `#01e9d5` 1, 233, 213 `rgb(1,233,213)` 0.4, 91.4, 83.5 `rgb(0.4%,91.4%,83.5%)` 100, 0, 9, 9 174.8°, 99.1, 45.9 `hsl(174.8,99.1%,45.9%)` 174.8°, 99.6, 91.4 00ffcc `#00ffcc`\nCIE-LAB 83.487, -50.547, -3.48 41.158, 63.084, 72.954 0.232, 0.356, 63.084 83.487, 50.667, 183.939 83.487, -66.598, 2.529 79.425, -46.497, 1.138 00000001, 11101001, 11010101\n\n# Color Schemes with #01e9d5\n\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #e90115\n``#e90115` `rgb(233,1,21)``\nComplementary Color\n• #01e961\n``#01e961` `rgb(1,233,97)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #0189e9\n``#0189e9` `rgb(1,137,233)``\nAnalogous Color\n• #e96101\n``#e96101` `rgb(233,97,1)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #e90189\n``#e90189` `rgb(233,1,137)``\nSplit Complementary Color\n• #e9d501\n``#e9d501` `rgb(233,213,1)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #d501e9\n``#d501e9` `rgb(213,1,233)``\n• #15e901\n``#15e901` `rgb(21,233,1)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #d501e9\n``#d501e9` `rgb(213,1,233)``\n• #e90115\n``#e90115` `rgb(233,1,21)``\n• #019d8f\n``#019d8f` `rgb(1,157,143)``\n• #01b6a7\n``#01b6a7` `rgb(1,182,167)``\n• #01d0be\n``#01d0be` `rgb(1,208,190)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #06fee9\n``#06fee9` `rgb(6,254,233)``\n• #1ffeeb\n``#1ffeeb` `rgb(31,254,235)``\n• #38feed\n``#38feed` `rgb(56,254,237)``\nMonochromatic Color\n\n# Alternatives to #01e9d5\n\nBelow, you can see some colors close to #01e9d5. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #01e99b\n``#01e99b` `rgb(1,233,155)``\n• #01e9ae\n``#01e9ae` `rgb(1,233,174)``\n• #01e9c2\n``#01e9c2` `rgb(1,233,194)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #01e9e8\n``#01e9e8` `rgb(1,233,232)``\n• #01d6e9\n``#01d6e9` `rgb(1,214,233)``\n• #01c3e9\n``#01c3e9` `rgb(1,195,233)``\nSimilar Colors\n\n# #01e9d5 Preview\n\nThis text has a font color of #01e9d5.\n\n``<span style=\"color:#01e9d5;\">Text here</span>``\n#01e9d5 background color\n\nThis paragraph has a background color of #01e9d5.\n\n``<p style=\"background-color:#01e9d5;\">Content here</p>``\n#01e9d5 border color\n\nThis element has a border color of #01e9d5.\n\n``<div style=\"border:1px solid #01e9d5;\">Content here</div>``\nCSS codes\n``.text {color:#01e9d5;}``\n``.background {background-color:#01e9d5;}``\n``.border {border:1px solid #01e9d5;}``\n\n# Shades and Tints of #01e9d5\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, #001211 is the darkest color, while #feffff is the lightest one.\n\n• #001211\n``#001211` `rgb(0,18,17)``\n• #002622\n``#002622` `rgb(0,38,34)``\n• #003934\n``#003934` `rgb(0,57,52)``\n• #004d46\n``#004d46` `rgb(0,77,70)``\n• #006058\n``#006058` `rgb(0,96,88)``\n• #00746a\n``#00746a` `rgb(0,116,106)``\n• #01877c\n``#01877c` `rgb(1,135,124)``\n• #019b8e\n``#019b8e` `rgb(1,155,142)``\n• #01ae9f\n``#01ae9f` `rgb(1,174,159)``\n• #01c2b1\n``#01c2b1` `rgb(1,194,177)``\n• #01d5c3\n``#01d5c3` `rgb(1,213,195)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\n• #01fde7\n``#01fde7` `rgb(1,253,231)``\n• #13feea\n``#13feea` `rgb(19,254,234)``\n• #27feec\n``#27feec` `rgb(39,254,236)``\n• #3afeed\n``#3afeed` `rgb(58,254,237)``\n• #4efeef\n``#4efeef` `rgb(78,254,239)``\n• #61fef1\n``#61fef1` `rgb(97,254,241)``\n• #75fef3\n``#75fef3` `rgb(117,254,243)``\n• #88fef4\n``#88fef4` `rgb(136,254,244)``\n• #9cfff6\n``#9cfff6` `rgb(156,255,246)``\n• #affff8\n``#affff8` `rgb(175,255,248)``\n• #c3fffa\n``#c3fffa` `rgb(195,255,250)``\n• #d7fffb\n``#d7fffb` `rgb(215,255,251)``\n• #eafffd\n``#eafffd` `rgb(234,255,253)``\n• #feffff\n``#feffff` `rgb(254,255,255)``\nTint Color Variation\n\n# Tones of #01e9d5\n\nA tone is produced by adding gray to any pure hue. In this case, #6d7d7c is the less saturated color, while #01e9d5 is the most saturated one.\n\n• #6d7d7c\n``#6d7d7c` `rgb(109,125,124)``\n• #648683\n``#648683` `rgb(100,134,131)``\n• #5b8f8b\n``#5b8f8b` `rgb(91,143,139)``\n• #529892\n``#529892` `rgb(82,152,146)``\n• #49a199\n``#49a199` `rgb(73,161,153)``\n• #40aaa1\n``#40aaa1` `rgb(64,170,161)``\n• #37b3a8\n``#37b3a8` `rgb(55,179,168)``\n• #2ebcb0\n``#2ebcb0` `rgb(46,188,176)``\n• #25c5b7\n``#25c5b7` `rgb(37,197,183)``\n• #1ccebf\n``#1ccebf` `rgb(28,206,191)``\n• #13d7c6\n``#13d7c6` `rgb(19,215,198)``\n• #0ae0ce\n``#0ae0ce` `rgb(10,224,206)``\n• #01e9d5\n``#01e9d5` `rgb(1,233,213)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #01e9d5 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"
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https://www.jiskha.com/questions/561679/state-4-reason-why-ais-is-classified-as-a-mixture | [
"# chemistry\n\nState 4 reason why ais is classified as a mixture.\n\n1. 👍 0\n2. 👎 0\n3. 👁 149\n1. Why air ir classified as a mixture\n\n1. 👍 0\n2. 👎 0\n2. The best reason I can think of for why we believe that air is a mixture is that freezing air slowly yields different liquids at different temperatures. Liquid nitrogen has a different boiling point than liquid oxygen. They also freeze at different temperatures. If air were only 1 compound, then air in its entirety would have a single boiling point and a single freezing point.\n\n1. 👍 0\n2. 👎 0\n\n## Similar Questions\n\n1. ### Another Physics\n\nAt some point in the combustion cycle of an engine, the temperature of a combusted air-fuel mixture is 582 degrees Celsius. Given that the pressure in the mixture is 6.2 bar, determine the density of the mixture (in kilogram per\n\nasked by Nick on March 28, 2014\n2. ### CHEMISTRY\n\nA gaseous mixture was prepared by taking equal mole of CO and N2 .if the total pressure of the mixture was found 1 atmosphere ,the partial pressure of nitrogen (N2) in the mixture is.\n\nasked by Tanya on March 22, 2015\n3. ### Chemistry - Quantitative analysis\n\nA mixture weighing 7.290 mg contained only cyclohexane, C6H12 (FM 84.159), and oxirane, C2H4O (FM 44.053). When the mixture was analyzed by combustion analysis, 21.999 mg of CO2 (FM 44.010) were produced. Find the weight percent\n\nasked by RJ on January 16, 2012\n4. ### Biology\n\nstate one reason why the concentration of glucose in the blood increases.\n\nasked by Eman on November 16, 2015\n1. ### Chemistry\n\nA dry mixture of kno3 and sand could be separated by 1. adding water to the mixture and filtering 2. adding water to the mixture and evaporating 3. heating the mixture to a high temperature 4. cooling the mixture to a low\n\nasked by . on April 13, 2009\n2. ### Chemistry\n\nState whether each of the following aqueous solutions would be acidic, basic, or neutral. Include appropriate net-ionic equations to show why a given solution is acidic or basic. a) NaNO3 b)KC6H5CO2 c)50:50 mixture of\n\nasked by Julie on March 13, 2010\n3. ### Chemistry\n\nA mixture contains 0.250 mol of Mn2O3 and 20.0 g of MnO2. 1.How many atoms of oxygen are present in the mixture? 2.How many grams of manganese are in the mixture?\n\nasked by Dr.Aven on April 4, 2020\n4. ### Science\n\nWhich of these represents a chemical change? A) two compounds combining to form a mixture B) a mixture being separated into two elements C) a compound being separated into two elements D) an element changing state from solid to\n\nasked by Help Needed on October 12, 2016\n1. ### Ap Chemistry\n\na mixture contains only sodium chloride and potassium chloride. a .1586-g sample of the mixture was dissolved in water. it took 22.90 mL of 0.1000 M AgNO3 to completely precipitate all the chloride present. what is the\n\nasked by Blanca on February 2, 2014\n2. ### biology\n\nstate one reason why the concentration of glucose in the blood decreases.\n\nasked by Eman on November 16, 2015\n3. ### Physical Chemistry\n\nA mixture of ethane and ethene occupied 35.5 L at 1.000 bar and 405 K. This mixture reacted completely with 110.3 g of O2 to produce CO2 and H2O. What was the composition of the original mixture? Assume ideal gas behavior\n\nasked by Dia B on September 12, 2012\n4. ### Math\n\nm∠2 + m∠3 + m∠4 = 180°/ reason: Given? m∠1 + m∠2 = 180°/reason: Supplementary Angle definition m∠1 + m∠2 = m∠2 + m∠3 + m∠4/reason: Substitution property equality m∠1 + m∠2 - m∠2 = m∠2 + m∠3 + m∠4\n\nasked by Sasha on October 24, 2016"
] | [
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https://grahapada.com/5th-grade-place-value-worksheets/ | [
"# 21 New 5th Grade Place Value Worksheets Images\n\nPosted on\n\n3rd grade worksheet place value save attractive free place value writing numbers in digits worksheets refrence collection math 3rd grade worksheet place value refrence math about worksheets 2nd grade math worksheets ordering numbers myscres 12 fabulous place value a number worksheets charming collection of free printable math worksheets decimals rounding numbers worksheet new math worksheets worksheet place standard form of decimals worksheet education pinterest renaming numbers place value worksheets new math worksheets 5th renaming numbers place value worksheets refrence place value",
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"https://grahapada.com/wp-content/uploads/2018/09/5th-grade-place-value-worksheets-lovely-3rd-grade-worksheet-place-value-save-attractive-free-place-value-of-5th-grade-place-value-worksheets.jpg",
null,
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null,
"https://grahapada.com/wp-content/uploads/2018/09/5th-grade-place-value-worksheets-inspirational-3rd-grade-worksheet-place-value-refrence-math-about-worksheets-of-5th-grade-place-value-worksheets.jpg",
null,
"https://grahapada.com/wp-content/uploads/2018/09/5th-grade-place-value-worksheets-beautiful-2nd-grade-math-worksheets-ordering-numbers-myscres-of-5th-grade-place-value-worksheets.jpg",
null,
"https://grahapada.com/wp-content/uploads/2018/09/5th-grade-place-value-worksheets-best-of-12-fabulous-place-value-a-number-worksheets-charming-of-5th-grade-place-value-worksheets.jpg",
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https://gitlab.mpi-sws.org/lczch/iris/-/commit/6cdc15f5cbc9c1e95d7a402f1320c04480243dba | [
"### docs: DC logic, be gone\n\nparent c2cbcda8\n \\section{Extensions of the Base Logic} In this section we discuss some additional constructions that we will within and on top of the base logic. In this section we discuss some additional constructions that we define within and on top of the base logic. These are not extensions'' in the sense that they change the proof power of the logic, they just form useful derived principles. \\subsection{Derived rules about base connectives} ... ... @@ -134,101 +134,6 @@ The following rules identify the class of timeless assertions: {\\timeless{\\mval(\\melt)}} \\end{mathparpagebreakable} \\subsection{DC logic: Dynamic Composeable Resources} \\label{sec:dc-logic} The base logic described in \\Sref{sec:base-logic} works over an arbitrary CMRA $\\monoid$ defining the structure of the resources. It turns out that we can generalize this further and permit picking CMRAs $\\iFunc(\\Prop)$'' that depend on the structure of assertions themselves. Of course, $\\Prop$ is just the syntactic type of assertions; for this to make sense we have to look at the semantics. Furthermore, there is a composeability problem with the given logic: if we have one proof performed with CMRA $\\monoid_1$, and another proof carried out with a \\emph{different} CMRA $\\monoid_2$, then the two proofs are actually carried out in two \\emph{entirely separate logics} and hence cannot be combined. Finally, in many cases just having a single instance'' of a CMRA available for reasoning is not enough. For example, when reasoning about a dynamically allocated data structure, every time a new instance of that data structure is created, we will want a fresh resource governing the state of this particular instance. While it would be possible to handle this problem whenever it comes up, it turns out to be useful to provide a general solution. The purpose of this section is to describe how we solve these issues. \\paragraph{Picking the resources.} The key ingredient that we will employ on top of the base logic is to give some more fixed structure to the resources. To instantiate the DC logic (base logic with dynamic composeable resources), the user picks a family of locally contractive bifunctors $(\\iFunc_i : \\COFEs \\to \\CMRAs)_{i \\in \\mathcal{I}}$. (This is in contrast to the base logic, where the user picks a single, fixed CMRA that has a unit.) From this, we construct the bifunctor defining the overall resources as follows: \\begin{align*} \\textdom{ResF}(\\cofe^\\op, \\cofe) \\eqdef{}& \\prod_{i \\in \\mathcal I} \\nat \\fpfn \\iFunc_i(\\cofe^\\op, \\cofe) \\end{align*} (We will motivate both the use of a product and the finite partial function below.) $\\textdom{ResF}(\\cofe^\\op, \\cofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). Furthermore, since the $\\iFunc_i$ are locally contractive, so is $\\textdom{ResF}$. Now we can write down the recursive domain equation: $\\iPreProp \\cong \\UPred(\\textdom{ResF}(\\iPreProp, \\iPreProp))$ $\\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor. This fixed-point exists and is unique by America and Rutten's theorem~\\cite{America-Rutten:JCSS89,birkedal:metric-space}. We do not need to consider how the object is constructed. We only need the isomorphism, given by \\begin{align*} \\Res &\\eqdef \\textdom{ResF}(\\iPreProp, \\iPreProp) \\\\ \\iProp &\\eqdef \\UPred(\\Res) \\\\ \\wIso &: \\iProp \\nfn \\iPreProp \\\\ \\wIso^{-1} &: \\iPreProp \\nfn \\iProp \\end{align*} Notice that $\\iProp$ is the semantic model of assertions for the base logic described in \\Sref{sec:base-logic} with $\\Res$: $\\Sem{\\Prop} \\eqdef \\iProp = \\UPred(\\Res)$ Effectively, we just defined a way to instantiate the base logic with $\\Res$ as the CMRA of resources, while providing a way for $\\Res$ to depend on $\\iPreProp$, which is isomorphic to $\\Sem\\Prop$. We thus obtain all the rules of \\Sref{sec:base-logic}, and furthermore, we can use the maps $\\wIso$ and $\\wIso^{-1}$ \\emph{in the logic} to convert between logical assertions $\\Sem\\Prop$ and the domain $\\iPreProp$ which is used in the construction of $\\Res$ -- so from elements of $\\iPreProp$, we can construct elements of $\\Sem{\\textlog M}$, which are the elements that can be owned in our logic. \\paragraph{Proof composeability.} To make our proofs composeable, we \\emph{generalize} our proofs over the family of functors. This is possible because we made $\\Res$ a \\emph{product} of all the CMRAs picked by the user, and because we can actually work with that product pointwise''. So instead of picking a \\emph{concrete} family, proofs will assume to be given an \\emph{arbitrary} family of functors, plus a proof that this family \\emph{contains the functors they need}. Composing two proofs is then merely a matter of conjoining the assumptions they make about the functors. Since the logic is entirely parametric in the choice of functors, there is no trouble reasoning without full knowledge of the family of functors. Only when the top-level proof is completed we will close'' the proof by picking a concrete family that contains exactly those functors the proof needs. \\paragraph{Dynamic resources.} Finally, the use of finite partial functions lets us have as many instances of any CMRA as we could wish for: Because there can only ever be finitely many instances already allocated, it is always possible to create a fresh instance with any desired (valid) starting state. This is best demonstrated by giving some proof rules. So let us first define the notion of ghost ownership that we use in this logic. Assuming that the family of functors contains the functor $\\Sigma_i$ at index $i$, and furthermore assuming that $\\monoid_i = \\Sigma_i(\\iPreProp, \\iPreProp)$, given some $\\melt \\in \\monoid_i$ we define: $\\ownGhost\\gname{\\melt:\\monoid_i} \\eqdef \\ownM{(\\ldots, \\emptyset, i:\\mapsingleton \\gname \\melt, \\emptyset, \\ldots)}$ This is ownership of the pair (element of the product over all the functors) that has the empty finite partial function in all components \\emph{except for} the component corresponding to index $i$, where we own the element $\\melt$ at index $\\gname$ in the finite partial function. We can show the following properties for this form of ownership: \\begin{mathparpagebreakable} \\inferH{res-alloc}{\\text{$G$ infinite} \\and \\melt \\in \\mval_{M_i}} { \\TRUE \\proves \\upd \\Exists\\gname\\in G. \\ownGhost\\gname{\\melt : M_i} } \\and \\inferH{res-update} {\\melt \\mupd_{M_i} B} {\\ownGhost\\gname{\\melt : M_i} \\proves \\upd \\Exists \\meltB\\in B. \\ownGhost\\gname{\\meltB : M_i}} \\inferH{res-empty} {\\text{$\\munit$ is a unit of $M_i$}} {\\TRUE \\proves \\upd \\ownGhost\\gname\\munit} \\axiomH{res-op} {\\ownGhost\\gname{\\melt : M_i} * \\ownGhost\\gname{\\meltB : M_i} \\provesIff \\ownGhost\\gname{\\melt\\mtimes\\meltB : M_i}} \\axiomH{res-valid} {\\ownGhost\\gname{\\melt : M_i} \\Ra \\mval_{M_i}(\\melt)} \\inferH{res-timeless} {\\text{$\\melt$ is a discrete COFE element}} {\\timeless{\\ownGhost\\gname{\\melt : M_i}}} \\end{mathparpagebreakable} Below, we will always work within (an instance of) the DC logic. Whenever a CMRA is used in a proof, we implicitly assume it to be available in the global family of functors. We will typically leave the $M_i$ implicit when asserting ghost ownership, as the type of $\\melt$ will be clear from the context. %%% Local Variables: ... ...\n ... ... @@ -2,9 +2,106 @@ \\section{Program Logic} \\label{sec:program-logic} This section describes how to build a program logic for an arbitrary language (\\cf \\Sref{sec:language}) on top of the logic described in \\Sref{sec:dc-logic}. This section describes how to build a program logic for an arbitrary language (\\cf \\Sref{sec:language}) on top of the base logic. So in the following, we assume that some language $\\Lang$ was fixed. \\subsection{Dynamic Composeable Resources} \\label{sec:composeable-resources} The base logic described in \\Sref{sec:base-logic} works over an arbitrary CMRA $\\monoid$ defining the structure of the resources. It turns out that we can generalize this further and permit picking CMRAs $\\iFunc(\\Prop)$'' that depend on the structure of assertions themselves. Of course, $\\Prop$ is just the syntactic type of assertions; for this to make sense we have to look at the semantics. Furthermore, there is a composeability problem with the given logic: if we have one proof performed with CMRA $\\monoid_1$, and another proof carried out with a \\emph{different} CMRA $\\monoid_2$, then the two proofs are actually carried out in two \\emph{entirely separate logics} and hence cannot be combined. Finally, in many cases just having a single instance'' of a CMRA available for reasoning is not enough. For example, when reasoning about a dynamically allocated data structure, every time a new instance of that data structure is created, we will want a fresh resource governing the state of this particular instance. While it would be possible to handle this problem whenever it comes up, it turns out to be useful to provide a general solution. The purpose of this section is to describe how we solve these issues. \\paragraph{Picking the resources.} The key ingredient that we will employ on top of the base logic is to give some more fixed structure to the resources. To instantiate the program logic, the user picks a family of locally contractive bifunctors $(\\iFunc_i : \\COFEs \\to \\CMRAs)_{i \\in \\mathcal{I}}$. (This is in contrast to the base logic, where the user picks a single, fixed CMRA that has a unit.) From this, we construct the bifunctor defining the overall resources as follows: \\begin{align*} \\textdom{ResF}(\\cofe^\\op, \\cofe) \\eqdef{}& \\prod_{i \\in \\mathcal I} \\nat \\fpfn \\iFunc_i(\\cofe^\\op, \\cofe) \\end{align*} We will motivate both the use of a product and the finite partial function below. $\\textdom{ResF}(\\cofe^\\op, \\cofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). Furthermore, since the $\\iFunc_i$ are locally contractive, so is $\\textdom{ResF}$. Now we can write down the recursive domain equation: $\\iPreProp \\cong \\UPred(\\textdom{ResF}(\\iPreProp, \\iPreProp))$ $\\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor. This fixed-point exists and is unique\\footnote{We have not proven uniqueness in Coq.} by America and Rutten's theorem~\\cite{America-Rutten:JCSS89,birkedal:metric-space}. We do not need to consider how the object is constructed. We only need the isomorphism, given by \\begin{align*} \\Res &\\eqdef \\textdom{ResF}(\\iPreProp, \\iPreProp) \\\\ \\iProp &\\eqdef \\UPred(\\Res) \\\\ \\wIso &: \\iProp \\nfn \\iPreProp \\\\ \\wIso^{-1} &: \\iPreProp \\nfn \\iProp \\end{align*} Notice that $\\iProp$ is the semantic model of assertions for the base logic described in \\Sref{sec:base-logic} with $\\Res$: $\\Sem{\\Prop} \\eqdef \\iProp = \\UPred(\\Res)$ Effectively, we just defined a way to instantiate the base logic with $\\Res$ as the CMRA of resources, while providing a way for $\\Res$ to depend on $\\iPreProp$, which is isomorphic to $\\Sem\\Prop$. We thus obtain all the rules of \\Sref{sec:base-logic}, and furthermore, we can use the maps $\\wIso$ and $\\wIso^{-1}$ \\emph{in the logic} to convert between logical assertions $\\Sem\\Prop$ and the domain $\\iPreProp$ which is used in the construction of $\\Res$ -- so from elements of $\\iPreProp$, we can construct elements of $\\Sem{\\textlog M}$, which are the elements that can be owned in our logic. \\paragraph{Proof composeability.} To make our proofs composeable, we \\emph{generalize} our proofs over the family of functors. This is possible because we made $\\Res$ a \\emph{product} of all the CMRAs picked by the user, and because we can actually work with that product pointwise''. So instead of picking a \\emph{concrete} family, proofs will assume to be given an \\emph{arbitrary} family of functors, plus a proof that this family \\emph{contains the functors they need}. Composing two proofs is then merely a matter of conjoining the assumptions they make about the functors. Since the logic is entirely parametric in the choice of functors, there is no trouble reasoning without full knowledge of the family of functors. Only when the top-level proof is completed we will close'' the proof by picking a concrete family that contains exactly those functors the proof needs. \\paragraph{Dynamic resources.} Finally, the use of finite partial functions lets us have as many instances of any CMRA as we could wish for: Because there can only ever be finitely many instances already allocated, it is always possible to create a fresh instance with any desired (valid) starting state. This is best demonstrated by giving some proof rules. So let us first define the notion of ghost ownership that we use in this logic. Assuming that the family of functors contains the functor $\\Sigma_i$ at index $i$, and furthermore assuming that $\\monoid_i = \\Sigma_i(\\iPreProp, \\iPreProp)$, given some $\\melt \\in \\monoid_i$ we define: $\\ownGhost\\gname{\\melt:\\monoid_i} \\eqdef \\ownM{(\\ldots, \\emptyset, i:\\mapsingleton \\gname \\melt, \\emptyset, \\ldots)}$ This is ownership of the pair (element of the product over all the functors) that has the empty finite partial function in all components \\emph{except for} the component corresponding to index $i$, where we own the element $\\melt$ at index $\\gname$ in the finite partial function. We can show the following properties for this form of ownership: \\begin{mathparpagebreakable} \\inferH{res-alloc}{\\text{$G$ infinite} \\and \\melt \\in \\mval_{M_i}} { \\TRUE \\proves \\upd \\Exists\\gname\\in G. \\ownGhost\\gname{\\melt : M_i} } \\and \\inferH{res-update} {\\melt \\mupd_{M_i} B} {\\ownGhost\\gname{\\melt : M_i} \\proves \\upd \\Exists \\meltB\\in B. \\ownGhost\\gname{\\meltB : M_i}} \\inferH{res-empty} {\\text{$\\munit$ is a unit of $M_i$}} {\\TRUE \\proves \\upd \\ownGhost\\gname\\munit} \\axiomH{res-op} {\\ownGhost\\gname{\\melt : M_i} * \\ownGhost\\gname{\\meltB : M_i} \\provesIff \\ownGhost\\gname{\\melt\\mtimes\\meltB : M_i}} \\axiomH{res-valid} {\\ownGhost\\gname{\\melt : M_i} \\Ra \\mval_{M_i}(\\melt)} \\inferH{res-timeless} {\\text{$\\melt$ is a discrete COFE element}} {\\timeless{\\ownGhost\\gname{\\melt : M_i}}} \\end{mathparpagebreakable} Below, we will always work within (an instance of) the logic as described here. Whenever a CMRA is used in a proof, we implicitly assume it to be available in the global family of functors. We will typically leave the $M_i$ implicit when asserting ghost ownership, as the type of $\\melt$ will be clear from the context. \\subsection{World Satisfaction, Invariants, Fancy Updates} \\label{sec:invariants} ... ... @@ -172,7 +269,7 @@ The fragment will then be available to the user of the logic, as their way of ta $\\ownPhys\\state \\eqdef \\ownGhost{\\gname_{\\textmon{State}}}{\\authfrag \\state}$ \\paragraph{Laws of weakest precondition.} The following rules can all be derived inside the DC logic: The following rules can all be derived: \\begin{mathpar} \\infer[wp-value] {}{\\prop[\\val/\\var] \\proves \\wpre{\\val}[\\mask]{\\Ret\\var.\\prop}} ... ...\nMarkdown is supported\n0% or .\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!\nPlease register or to comment"
] | [
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https://byjus.com/maths/trigonometric-ratios/ | [
"",
null,
"# Trigonometric Ratios\n\nIn trigonometry, trigonometric ratios are derived from the sides of a right-angled triangle. There are six 6 ratios such as sine, cosine, tangent, cotangent, cosecant, and secant. You will learn here to build a trigonometry table for these ratios for some particular angles, such as 0 °, 30 °, 45 °, 60 °, 90°. There are many trigonometry formulas and trigonometric identities, which are used to solve complex equations in geometry. Here, the concept of ratios of trigonometry is covered along with its definitions and applications.\n\n## Trigonometric Ratios Definition\n\nIt is defined as the values of all the trigonometric function based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle. Consider a right-angled triangle, right-angled at B.",
null,
"With respect to ∠C, the ratios of trigonometry are given as:\n\n• sine: Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.\n• cosine: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.\n• tangent: Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.\n• cosecant: Cosecant is a multiplicative inverse of sine.\n• secant: Secant is a multiplicative inverse of cosine.\n• cotangent: Cotangent is the multiplicative inverse of the tangent.\n\nThe above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ ABC, the ratios are defined as:\n\nsin ∠C = (Side opposite to ∠C)/(Hypotenuse) = AB/AC\n\ncos ∠C = (Side adjacent to ∠C)/(Hypotenuse) = BC/AC\n\ntan ∠C = (Side opposite to ∠C)/(Side adjacent to ∠C) = AB/AC = sin ∠C/cos ∠C\n\ncosec ∠C= 1/sin ∠C = (Hypotenuse)/ (Side Opposite to ∠C) AC/AB\n\nsec∠A = 1/cos ∠C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/BC\n\ncot ∠C = 1/tan ∠C = (Side adjacent to ∠C)/(Side opposite to ∠C)= BC/AB\n\nIn right Δ ABC, if ∠and ∠C are assumed as 30° and 60°, then there can be infinite right triangles with those specifications but all the ratios written above for ∠C in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠A or ∠C) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle. Also, check out trigonometric functions to learn about each of these ratios or functions in detail.\n\n## Trigonometric Ratios Table\n\nBelow is the table where each ratios values are given with respect to different angles, particularly used in calculations.\n\n Angle 0° 30° 45° 60° 90° Sin∠C 0 1/2 1/√2 √3/2 1 Cos∠C 1 √3/2 1/√2 1/2 0 Tan∠C 0 1/√3 1 √3 ∞ Cot∠C ∞ √3 1 1/√3 0 Sec∠C 1 2/√3 √2 2 ∞ Cosec∠C ∞ 2 √2 2/√3 1\n\n### Trigonometry Applications\n\nTrigonometry is one of the most important branches of mathematics. Some of the applications of trigonometry are:\n\n• Measuring the heights of towers or big mountains\n• Determining the distance of the shore from the sea\n• Finding the distance between two celestial bodies\n• Determining the power output of solar cell panels at different inclinations\n• Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.\n\nIt is evident from the above examples that trigonometry has its involvement in a major part of our day-to-day life and much more. In most of the applications listed above, something was being measured and that is what trigonometry is all about.\n\nDownload BYJU’S App and learn thousands of concepts here through interesting and personalised videos.\n\n Trigonometry Related Articles Trigonometric Table Trigonometric Identities Trigonometry Angles Trigonometric Applications Trigonometric Ratios of Standard Angles Inverse Trigonometric Functions Range of Inverse Functions Domain of Inverse functions Cosine Rule Direction Cosines"
] | [
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"https://www.facebook.com/tr",
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"https://cdn1.byjus.com/wp-content/uploads/2018/11/maths/2017/08/09065011/0-11-300x2861.png",
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.9064416,"math_prob":0.97384006,"size":3697,"snap":"2019-51-2020-05","text_gpt3_token_len":1040,"char_repetition_ratio":0.15272136,"word_repetition_ratio":0.06962025,"special_character_ratio":0.24857993,"punctuation_ratio":0.08923513,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9987444,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T13:22:45Z\",\"WARC-Record-ID\":\"<urn:uuid:7e1b3a21-3e5d-426b-87d6-cbe5b59d2ba1>\",\"Content-Length\":\"502979\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9e4fe401-156f-4489-8481-140899353ba5>\",\"WARC-Concurrent-To\":\"<urn:uuid:23cc049d-af04-42fc-b62f-57f9ee4da33d>\",\"WARC-IP-Address\":\"52.77.80.199\",\"WARC-Target-URI\":\"https://byjus.com/maths/trigonometric-ratios/\",\"WARC-Payload-Digest\":\"sha1:2XE5RJN3QR6EITADSCFG6BROFZYWY2MH\",\"WARC-Block-Digest\":\"sha1:EHB2QMDF63Z2OG5DCTVYO4Z6HSFRFTJ7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540543850.90_warc_CC-MAIN-20191212130009-20191212154009-00543.warc.gz\"}"} |
https://studiobuffery.com/buy-a-custom-paper/write-a-hypothesis-based-on-your-observations-rank-5487ri.php | [
"# Write A Hypothesis Based On Your Observations Rank\n\nWhen you write your hypothesis, it should be based on your \"educated take measurements or make observations to see how two things (your. Learn about the scientific method, with these explanations of each step of the process, Design and perform an experiment to test your hypothesis. In this model, the first step of the scientific method is to make observations.\n\n## SCIENTIFIC METHOD\n\nThe scientific method is an empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. The scientific method is a process for experimentation that is used to explore observations and answer questions. Does this mean all scientists follow exactly this.\n\n### HYPOTHESIS EXAMPLES FOR KIDS\n\nA hypothesis has classical been referred to as an educated guess. In the context of the scientific method, this description is somewhat correct. After a problem is. A hypothesis is a tentative, testable answer to a scientific question. Once a scientist has a scientific question she is Examples of Hypotheses and Predictions.\n\n## HYPOTHESIS EXAMPLES BIOLOGY\n\nHere are examples of a scientific hypothesis and how to improve a hypothesis to use it for an experiment. Experiments and further observations are often used to test the hypotheses. For example, a scientist studying the mating behaviors of ladybugs might begin.\n\n## RESEARCH HYPOTHESIS PDF\n\nPDF | Definition of Hypothesis; Assumption, Postulate and Hypothesis; of Hypothesis; Types of Research Hypothesis; Uses of Hypotheses in. Describe how the research question and hypothesis relate to the other components of Compare and contrast the use of statistical versus research hypotheses.\n\n### HYPOTHESIS IN RESEARCH\n\nFor example, a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states. Hypotheses. An hypothesis is a specific statement of prediction. It describes in concrete (rather than theoretical) terms what you expect will happen in your study .\n\nstudiobuffery.com 2019. buy a custom paper"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.9416082,"math_prob":0.4256388,"size":1868,"snap":"2019-51-2020-05","text_gpt3_token_len":350,"char_repetition_ratio":0.16791846,"word_repetition_ratio":0.0,"special_character_ratio":0.17719486,"punctuation_ratio":0.10216718,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9852269,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-08T19:28:58Z\",\"WARC-Record-ID\":\"<urn:uuid:ecffa8d5-1284-4110-90ce-ed8be9a295d6>\",\"Content-Length\":\"6119\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c5ebf271-b37f-4694-8c56-dd976d645b92>\",\"WARC-Concurrent-To\":\"<urn:uuid:f47e469b-2031-4261-bf42-e6fb3c6e0d9e>\",\"WARC-IP-Address\":\"104.31.76.122\",\"WARC-Target-URI\":\"https://studiobuffery.com/buy-a-custom-paper/write-a-hypothesis-based-on-your-observations-rank-5487ri.php\",\"WARC-Payload-Digest\":\"sha1:FM7QF74AIP5ZA7LPCZZR63XNLMBSCKGN\",\"WARC-Block-Digest\":\"sha1:GQQ2JYBGJBVKAVMKPJ5MEOUOIBXTSCRL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540514475.44_warc_CC-MAIN-20191208174645-20191208202645-00420.warc.gz\"}"} |
https://iasj.net/iasj/export/154783?format=bibTex | [
"@Article{, title={Equivalence Between System of Volterra Integro-Fractional Differential Equations and Volterra Integral Equations}, author={Dr. Omar M. Al-Faour¬ and Dr. Suha N. Al-Rawi and Dr. Shazad S. Ahmed}, journal={journal of the college of basic education مجلة كلية التربية الاساسية}, volume={18}, number={74 / ملحق}, pages={159-176}, year={2012}, abstract={In this paper, an important theorem of equivalence between system of linear Volterra integro-fractional differential equations (LVIFDE's) with constant coefficients in Caputo derivative scene and the corresponding well-known Volterra integral equation system is demonstrate.\n\n} }"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.69556797,"math_prob":0.9529774,"size":644,"snap":"2021-43-2021-49","text_gpt3_token_len":163,"char_repetition_ratio":0.115625,"word_repetition_ratio":0.0,"special_character_ratio":0.22670807,"punctuation_ratio":0.1495327,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95258796,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-29T00:33:36Z\",\"WARC-Record-ID\":\"<urn:uuid:3397b856-f882-4834-bf8a-d9bd6c2ccb99>\",\"Content-Length\":\"1049\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d2317d6d-4883-4d15-aaf8-aca01912b960>\",\"WARC-Concurrent-To\":\"<urn:uuid:186dd9f4-869e-43c7-8109-f67c0ffa438b>\",\"WARC-IP-Address\":\"176.222.236.230\",\"WARC-Target-URI\":\"https://iasj.net/iasj/export/154783?format=bibTex\",\"WARC-Payload-Digest\":\"sha1:N5QZ2JZX4HPLO5ONKSLGOBX47WPRVDBB\",\"WARC-Block-Digest\":\"sha1:BXF4QTFIPGX6GLAEWKVZQ3MHSSIXLPUT\",\"WARC-Identified-Payload-Type\":\"application/x-bibtex-text-file\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358673.74_warc_CC-MAIN-20211128224316-20211129014316-00049.warc.gz\"}"} |
http://samskivert.com/blog/2008/01/euler015/ | [
"# samskivert: Euler 015\n\n## 17 January 2008\n\n``````object Euler15 extends Application {\nval size = 20;\ndef fact (n: BigInt): BigInt = if (n == 0) 1 else n * fact(n - 1)\nprintln(fact(size * 2) / (fact(size) * fact(size)))\n}``````\n\nI first started thinking about this as a caching problem, having just finished problem 14 which was reduced to a reasonable runtime by caching. At any coordinate in the grid, there are some number of unique paths from that position to the destination, so one could start in the bottom right and traverse the graph breadth first from the end to the start, caching the number of paths at each intersection and using that to efficiently compute the number of paths as you made your way to the start. I implemented this solution in Java and indeed 841 calls to a recursive function later I had my solution.\n\nThen I realized that every path from the start to the finish is the same length and a series of an equal number of horizontal and vertical segments (20 horizontal and 20 vertical). Thus each unique path was a unique arrangement of those horizontal and vertical segments. If we consider the 40 positions in our path to be distinct elements then the problem becomes how many ways can we choose 20 of our positions in the path to contain horizontal segments (or vertical segments if you swing that way), which any good computer scientist knows is 40 choose 20: 40! / 20! (40 - 20)!.\n\nThe funny thing is that the caching solution in Java runs in less than 1 millisecond whereas computing the factorial in Scala above requires 2 milliseconds. It isn't a tail recursive factorial function though, so that and the BigInt math probably take up all the time."
] | [
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https://in.mathworks.com/matlabcentral/profile/authors/98150 | [
"Community Profile",
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"# J-G van der Toorn\n\nLast seen: 2 months ago Active since 2012\n\n#### Statistics\n\nAll\n•",
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"#### Content Feed\n\nView by\n\nSolved\n\nOperate on matrices of unequal, yet similar, size\nYou may want to add a vector to a matrix, implying that the vector is added to each column of the matrix. Or multiply a 3x4x5 ma...\n\n3 years ago\n\nSolved\n\nNumbers on 7-segment\nThis is a 7-segment: _ |_| |_| It's a 3-by-3 char matrix.It has made by 3 characters: '_' , '|' and ' ' (space...\n\n3 years ago\n\nSolved\n\nCalculate time taken by light to reach earth surface\nWe know the time(seconds) taken by light to reach surface of earth. What if the distance varies yearly or source of light moves ...\n\n3 years ago\n\nSolved\n\nDraw 'H'\nDraw a x-by-x matrix 'H' using 1 and 0. (x is odd and bigger than 2) Example: x=5 ans= [1 0 0 0 1 1 0 0 0 1 ...\n\n3 years ago\n\nSolved\n\nDraw 'F'\nDraw a x-by-x matrix 'F' using 1 and 0. (x is odd and bigger than 4) Example: x=5 ans= [1 1 1 1 1 1 0 0 0 0 ...\n\n3 years ago\n\nSolved\n\nDraw 'E'\nDraw a x-by-x matrix 'E' using 1 and 0. (x is odd and bigger than 4) Example: x=5 ans= [1 1 1 1 1 1 0 0 0 0 ...\n\n3 years ago\n\nSolved\n\nMean = Standard Deviation\nCreate a series with following properties; # All of the members should be positive integer # Mean of the series should be in...\n\n3 years ago\n\nSolved\n\nFind prime number couples\nGiven a vector a, which will always contain at least one pair of prime numbers couple, return a matrix called 'couple' in which ...\n\n3 years ago\n\nSolved\n\nAngle between Two Vectors\nThe dot product relationship, a dot b = | a | | b | cos(theta), can be used to determine the acute angle between vector a and ve...\n\n3 years ago\n\nSolved\n\nClockwise or Counterclockwise\nGiven a list of 2-d points defining the vertices of a polygon, determine whether these points are sorted clockwise. The input...\n\n3 years ago\n\nSolved\n\nCalculate the area of a triangle between three points\nCalculate the area of a triangle between three points: P1(X1,Y1) P2(X2,Y2) P3(X3,Y3) these three points are the vert...\n\n3 years ago\n\nSolved\n\nTelevision Screen Dimensions\nGiven a width to height ratio of a TV screen given as _w_ and _h_ as well as the diagonal length of the television _l_, return t...\n\n3 years ago\n\nSolved\n\nSecond Diagonal\nTranspose the matrix from it's second diagonal.\n\n3 years ago\n\nSolved\n\nfind number of times of occurrence of the most frequent number in a row vector\nIn a given row vector, find the number of times a mode of a row vector has occurred example: in [2 5 5 5 5 3], output is 4 ...\n\n3 years ago\n\nSolved\n\nDraw 'D'.\nDraw a x-by-x matrix 'D' using 0 and 1. example: x=4 ans= [1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0]\n\n3 years ago\n\nSolved\n\nCreate the equation: y=(3x)^2+(5x)+35 and compute y for various values of x\n\n3 years ago\n\nSolved\n\nfunction to compute root mean square of first nn positive odd integers\nWrite a function called odd_rms that returns orms, which is the square root of the mean of the squares of the first nn positive ...\n\n3 years ago\n\nSolved\n\nPercentage of zeros in a matrix of only 1s and 0s\nWrite a function called _zero_stat_ that takes a matrix as an input that only has 0 and 1 elements. The function needs to comput...\n\n3 years ago\n\nSolved\n\nDatetime basics\nGenerate the datetime scalar representing the current date\n\n3 years ago\n\nSolved\n\nCheck if there are white spaces in the input string\nIf there are white spaces in the input string, output=1 else 0\n\n3 years ago\n\nSolved\n\nRemove white spaces at the end of the input string\nRemove all trailing white spaces at the end of the input strings\n\n3 years ago\n\nSolved\n\nPerimeter of a semicircle\nGiven the diameter d, find the perimeter of a semicircle\n\n3 years ago\n\nSolved\n\nNth root of a number\nGiven an input and a number N, find the Nth root of the number(s)\n\n3 years ago\n\nSolved\n\nDial Up\nEach number on telephone keypads, except 0 and 1, corresponds to a set of uppercase letters as shown in this list: 2 ABC, 3 DEF...\n\n3 years ago\n\nSolved\n\nDraw 'O' !\nGiven n as input, generate a n-by-n matrix 'O' using 0 and 1 . example: n=4 ans= [1 1 1 1 1 0 0 1 ...\n\n3 years ago\n\nSolved\n\nDraw 'I'\nGiven n as input, draw a n-by-n matrix 'I' using 0 and 1. example: n=3 ans= [0 1 0 0 1 0 0 1 0] n=...\n\n3 years ago\n\nSolved\n\nDay counter function\nWrite a function called _day_counter_ that returns the number of Mondays that fell on the first day of the month in a given year...\n\n3 years ago\n\nSolved\n\nDraw a 'N'!\nGiven n as input, generate a n-by-n matrix 'N' using 0 and 1 . Example: n=5 ans= [1 0 0 0 1 1 1 0 0 1 1 0 ...\n\n3 years ago\n\nSolved\n\nComputational power of Cody servers\nIt has been <https://en.wikipedia.org/wiki/Moore%27s_law#History predicted> that the performance of integrated circuits would _d...\n\n3 years ago\n\nSolved\n\nBasics: counting digits of a number irrespective of the sign\npublish the number of digits in any input integer example: -23---->2\n\n3 years ago"
] | [
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http://mat.utfsm.cl/investigacion/publicaciones/ | [
"# Publicaciones del año 2019\n\nAutores Artículo Revista\n\nThe Poisson and Stokes problems on weighted spaces in Lipschitz domains and under singular forcing\n\nJournal of Mathematical Analysis and Applications; Vol. 471(1-2), pp. 599-612; Mar. 2019\n\nP. Gajardo, C. Hermosilla\n\nThe viability kernel of dynamical systems with mixed constraints: A level-set approach\n\nSystems & Control Letters; Vol. 127, pp. 6-12; May. 2019.\n\nA. Allendes, E. Otarola, A. J. Salgado\n\nA posteriori error estimates for the Stokes problem with singular sources\n\nComputer Methods in Applied Mechanics and Engineering; Vol. 345, pp. 1007 – 1032; Mar. 2019.\n\nE. Otarola, T.N.T Quyen\n\nA reaction coefficient identification problem for fractional diffusion\n\nInverse Problems; Vol. 35(4), pp. 45010; Mar. 2019.\n\nF. Fuica, E. Otarola, A. J. Salgado\n\nAn a posteriori error analysis for an elliptic optimal control problem in measure space\n\nComputers and Mathematics with Applications; Vol. 77(10), pp. 2659 – 2675; May 2019.\n\nN. Martínez-Jeraldo, P. Aguirre\n\nAllee effect acting on the prey species in a Leslie-Gower predation model\n\nNonlinear Analysis: Real World Applications; Vol 45, pp. 895-917; Feb. 2019.\n\nE. Porcu, S. Castruccio, A. Alegría, P. Crippa\n\nAxially symmetric models for global data: A journey between geostatistics and stochastic generators\n\nEnvironmetrics; Vol 30(1), pp. e255; Jan. 2019.\n\nN. Carreño, M. C. Santos\n\nStackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation\n\nJournal of Differential Equations; Vol. 266 (9), pp. 6068-6108; Apr. 2019.\n\nL. Briceño, S. López\n\nA projected primal-dual method for solving constrained monotone inclusions\n\nJournal of Optimization Theory and Applications; Vol. 180(3), pp. 907–924; Mar. 2019.\n\nL. Briceño, G. Chierchia, E. Chouzenoux, J.-C. Pesquet,\n\nA random block-coordinate Douglas-Rachford splitting method with low computational complexity for binary logistic regression\n\nComputational Optimization and Applications; Vol. 72(3), pp. 707–726; Apr. 2019.\n\nA. Donoso-Bravo, P. Gajardo, M. Sebbah, D. Vicencio\n\nComparison of performance in an anaerobic digestion process: one-reactor vs two-reactor configurations\n\nMathematical Biosciences and Engineering; Vol. 16(4), pp. 2447-2465; Apr. 2019.\n\nT. Bayen, P. Gajardo\n\nOn the steady state optimization of the biogas production in a two-stage anaerobic digestion model\n\nJournal of Mathematical Biology; Vol. 78(4), pp. 1067-1087; Mar. 2019.\n\nG. Dávila, A. Quaas, E. Topp\n\nExistence, nonexistence and multiplicity results for nonlocal Dirichlet problems\n\nJournal of Differential Equations; Vol. 266(9), pp. 5971-5997; Apr. 2019.\n\nJ. García-Melián, L. Iturriaga, A. Quaas\n\nLiouville theorems for radial solutions of semilinear elliptic equations\n\nComplex Variables and Elliptic Equations; Vol 64(6), pp. 933-949; 2019.\n\nG. Dávila, A. Quaas, E. Topp\n\nAn ODE approach for fractional Dirichlet problems with gradient nonlinearity\n\nMathematische Zeitschrift; Vol. 291(1–2), pp. 85–111; Feb. 2019."
] | [
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https://pt.b-ok.org/book/556662/c2b4f4?dsource=recommend | [
"Principal PID Controllers\n\n# PID Controllers\n\n,\nThis new edition takes stock of new developments in digital PID Controllers. Modeling methods, implementation details, and problem-solving techniques are presented to help you improve loop performance and product quality. The book examines the auto-tuning and adaptation features of several commercial controllers and It discusses measures for dealing with specific challenges such as reset windup, long process dead times, and oscillatory systems. Design methods and tuning rules that consider factors such as load disturbances, measurement noise, model uncertainty, and set point response are also recommended.\nAno: 1995\nEdição: 2 Sub\nEditora: International Society for Measurement and Con\nLíngua: english\nPáginas: 354\nISBN 10: 1556175167\nISBN 13: 9781556175169\nFile: PDF, 13.47 MB\n\n## Most frequently terms\n\nYou can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.\n1\n\n### Computer Controlled Systems\n\nYear: 1996\nLanguage: english\nFile: PDF, 13.20 MB\n2\n\n### Introduction to stochastic control theory\n\nLanguage: english\nFile: PDF, 2.29 MB\n```2843\n01,\n\n01999018\"\n\n3*. O\nco ^\nO: O\n>\nZ\n\no\n\nCD\n\nm §\n\ni\n\n1\n\nISBN 1-55617-516-7\n\n90000>\n\n9\"781556\"175169\n\n: PID Controllers:\n: Theory,\n• Design,\nTuning\n\n2ND\n\nE 0 I T I 0\n\nPID Controllers, 2nd Edition\nby\nKarlJ. Astrom\nand\nTore Hagglund\n\nc\n\n2843\n01\n\ni mill\n019990103\n\n67 Alexander Drive\nP.O. Box 12277\nResearch Triangle Park, NC 27709\nPrinted in the United States of America.\n10 9 8 7 6 5 4 3\nISBN 1-55617-516-7\nNo part of this work may be reproduced, stored in a retrieval system, or\ntransmitted in any form or by any means, electronic, mechanical, photocopying,\nrecording or otherwise, without the prior written permission of the publisher.\n\nAstrom, Karl J. (Karl Johan), 1934PID controllers: theory, design, and tuning/Karl Johan Astrom and Tore Hagglund. — 2nd ed.\np. cm.\nRev. ed. of: Automatic tuning of PID controllers. cl988.\nIncludes bibliographical references and index.\nISBN 1-55617-516-7\n1. PID controllers. I. Hagglund, Tore. II. Astrom, Karl J. (Karl Johan), 1934Automatic tuning of PID controllers. HI. Title\nTJ223.P55A87 1994\n94-10795\n629.8-dc20\nCIP\n\nPreface\n\nIn 1988 we published the book Automatic Tuning of PID Controllers,\nwhich summarized experiences gained in the development of an au¬\ntomatic tuner for a PID controller. The present book may be regarded\nas a continuation of that book, although it has been significantly ex¬\npanded. Since 1988 we have learned much more about PID control as\na result of our involvement in research and industrial development of\nPID controllers. Because of this we strongly believe that the practice\nof PID can be improved considerably, and that this will contribute\nsignificantly to improved quality of manufacturing. This belief has\nbeen strongly reinforced by recent publications of the industrial state\nof the art, which are referenced in Chapter 1.\nThe main reason for writing this book is to contribute to a bet¬\nter understanding of PI; D control. Another reason is that information\nabout PID control is scattered in the control literature. The PID con¬\ntroller has not attracted much attention from the research community\nduring the past decades, and it is often covered inadequately in stan¬\ndard textbooks in control. We believe that this book will be useful to\nusers and manufacturers of PID controllers as well as educators. It\nis important to teach PID control in introductory courses on feedback\ncontrol at universities, and we hope that this book can give useful\nbackground for such courses.\nIt is assumed that the reader has a control background. A reader\nshould be familiar with concepts such as transfer functions, poles,\nand zeros. Even so, the explanations are elementary. Occasionally,\nwe have stated facts without supporting detailed arguments, when\nthey have seemed unnecessary, in an effort to focus on the practical\naspects rather than the theory. A reader who finds that he needs som\nspecific background in process control is strongly advised to consult\na text in process control such as Seborg et al. (1989).\nCompared to the earlier book we have expanded the material\nsubstantially. The chapters on modeling, PID control, and design\nof PID controllers have been more than doubled. The chapter on\nautomatic tuning has been completely rewritten to account for the\ndynamic product development that has taken place in the last years.\nThere are two new chapters. One describes new tuning methods. This\n\nvi\n\nPreface\n\nmaterial has not been published before. There is also a new chapter\non control paradigms that describes how complex systems can be\nobtained by combining PID controllers with other components.\nWe would like to express our gratitude to several persons who\nhave provided support and inspiration. Our original interest in PID\ncontrol was stimulated by Axel Westrenius and Mike Sommerville\nof Eurotherm who shared their experience of design and of PID\ncontrollers with us. We have also benefited from discussions with\nManfred Morari of Caltech, Edgar Bristol of Foxboro, Ken Goff for¬\nmerly of Leeds and Northrup, Terry Blevins of Fisher-Rosemount\nControl, Gregory McMillan of Monsanto. Particular thanks are due to\nSune Larsson who initiated our first autotuner experiments and Lars\nBaath with whom we shared the pleasures and perils of developing\nour first industrial auto-tuner. We are also grateful to many instru¬\nment engineers who participated in experiments and who generously\nshared their experiences with us. Among our research colleagues we\nhave learned much from Professor C. C. Hang of Singapore National\nUniversity with whom we have done joint research in the field over\na long period of time. We are also grateful to Per Persson, who devel¬\noped the dominant pole design method.\nSeveral persons have read the manuscript of the book. Willy\nWojsznis of Fisher-Rosemount gave many valuable suggestions for im¬\nprovements. Many present and former colleagues at our department\nhave provided much help. Special thanks are due to Eva Dagnegard\nand Leif Andersson who made the layout for the final version and\nBritt-Marie Martensson who drew many of the figures. Ulf Holmberg, Karl-Erik Arzen and Mikael Johansson gave very useful input\non several versions of the manuscript.\nFinally we would like to express our deep gratitude to the Swedish\nNational Board of Industrial and Technical Development (NUTEK)\nwho have supported our research.\nKARL JOHAN ASTROM\nTORE HAGGLUND\n\nDepartment of Automatic Control\nLund Institute of Technology\nBox 118, S-221 00 Lund, Sweden\nkarl_j ohan. astromQcontrol. lth. se\ntore.hagglundQcontrol.lth. se\n\n1. Introduction 1\n2. Process Models 5\n2.1 Introduction 5\n2.2 Static Models 6\n2.3 Dynamic Models 8\n2.4 Step Response Methods 11\n2.5 Methods of Moments 24\n2.6 Frequency Responses 34\n2.7 Parameter Estimation 43\n2.8 Disturbance Models 46\n2.9 Approximate Models and Unmodeled Dynamics 51\n2.10 Conclusions 57\n2.11 References 58\n3. PID Control 59\n3.1 Introduction 59\n3.2 The Feedback Principle 60\n3.3 PID Control 64\n3.4 Modifications of the PID Algorithm 70\n3.5 Integrator Windup 80\n3.6 Digital Implementation 93\n3.7 Operational Aspects 103\n3.8 Commercial Controllers 108\n3.9 When Can PID Control Be Used? 109\n3.10 Conclusions 116\n3.11 References 117\n4. Controller Design 120\n4.1 Introduction 120\n4.2 Specifications 121\n4.3 Ziegler-Nichols' and Related Methods 134\n4.4 Loop Shaping 151\n4.5 Analytical Tuning Methods 156\nVII\n\nviii\n\n4.6 Optimization Methods 164\n4.7 Pole Placement 173\n4.8 Dominant Pole Design 179\n4.9 Design for Disturbance Rejection 193\n4.10 Conclusions 196\n4.11 References 197\n5. New Tuning Methods 200\n5.1 Introduction 200\n5.2 A Spectrum of Tools 201\n5.3 Step-Response Methods 203\n5.4 Frequency-Response Methods 212\n5.5 Complete Process Knowledge 218\n5.6 Assessment of Performance 220\n5.7 Examples 225\n5.8 Conclusions 227\n5.9 References 229\n6. Automatic Tuning and Adaptation 230\n6.1 Introduction 230\n6.2 Process Knowledge 232\n6.4 Model-Based Methods 237\n6.5 Rule-Based Methods 241\n6.6 Commercial Products 243\n6.7 Integrated Tuning and Diagnosis 262\n6.8 Conclusions 270\n6.9 References 270\n7.1 Introduction 273\n7.3 Feedforward Control 281\n7.4 Model Following 284\n7.5 Nonlinear Elements 287\n7.6 Neural Network Control 295\n7.7 Fuzzy Control 298\n7.8 Interacting Loops 304\n7.9 System Structuring 313\n7.10 Conclusions 321\n7.11 References 321\nBibliography 323\nIndex 339\n\nCHAPTER\n\n1\n\nIntroduction\n\nThe PID controller has several important functions: it provides feed¬\nback; it has the ability to eliminate steady state offsets through in¬\ntegral action; it can anticipate the future through derivative action.\nPID controllers are sufficient for many control problems, particularly\nwhen process dynamics are benign and the performance requirements\nare modest. PID controllers are found in large numbers in all indus¬\ntries. The controllers come in many different forms. There are stand¬\nalone systems in boxes for one or a few loops, which are manufac¬\ntured by the hundred thousands yearly. PID control is an important\ningredient of a distributed control system. The controllers are also\nembedded in many special-purpose control systems. In process con¬\ntrol, more than 95% of the control loops are of PID type, most loops\nare actually PI control. Many useful features of PID control have not\nbeen widely disseminated because they have been considered trade\nsecrets. Typical examples are techniques for mode switches and antiwindup.\nPID control is often combined with logic, sequential machines, se¬\nlectors, and simple function blocks to build the complicated automa¬\ntion systems used for energy production, transportation, and manu¬\nfacturing. Many sophisticated control strategies, such as model pre¬\ndictive control, are also organized hierarchically. PID control is used\nat the lowest level; the multivariable controller gives the setpoints to\nthe controllers at the lower level. The PID controller can thus be said\nto be the \"bread and butter\" of control engineering. It is an important\ncomponent in every control engineer's toolbox.\nPID controllers have survived many changes in technology rang¬\ning from pneumatics to microprocessors via electronic tubes, tran¬\nsistors, integrated circuits. The microprocessor has had a dramatic\n\n2\n\nChapter 1 Introduction\n\ninfluence on the PID controller. Practically all PID controllers made\ntoday are based on microprocessors. This has given opportunities to\nprovide additional features like automatic tuning, gain scheduling,\nand continuous adaptation. The terminology in these areas is not\nwell-established. For purposes of this book, auto-tuning means that\nthe controller parameters are tuned automatically on demand from\nan operator or an external signal, and adaptation means that the\nparameters of a controller are continuously updated. Practically all\nnew PID controllers that are announced today have some capability\nfor automatic tuning. Tuning and adaptation can be done in many\ndifferent ways. The simple controller has in fact become a test bench\nfor many new ideas in control.\nThe emergence of the fieldbus is another important development.\nThis will drastically influence the architecture of future distributed\ncontrol systems. The PID controller is an important ingredient of\nthe fieldbus concept. It may also be standardized as a result of the\nfieldbus development.\nA large cadre of instrument and process engineers are familiar\nwith PID control. There is a well-established practice of installing,\ntuning, and using the controllers. In spite of this there are substantial\npotentials for improving PID control. Evidence for this can be found\nin the control rooms of any industry. Many controllers are put in man¬\nual mode, and among those controllers that are in automatic mode,\nderivative action is frequently switched off for the simple reason that\nit is difficult to tune properly. The key reasons for poor performance is\nequipment problems in valves and sensors, and bad tuning practice.\nThe valve problems include wrong sizing, hysteresis, and stiction.\nThe measurement problems include: poor or no anti-aliasing filters;\nexcessive filtering in \"smart\" sensors, excessive noise and improper\ncalibration. Substantial improvements can be made. The incentive for\nimprovement is emphasized by demands for improved quality, which\nis manifested by standards such as ISO 9000. Knowledge and un¬\nderstanding are the key elements for improving performance of the\ncontrol loop. Specific process knowledge is required as well as knowl¬\nBased on our experience, we believe that a new era of PID control\nis emerging. This book will take stock of the development, assess its\npotential, and try to speed up the development by sharing our expe¬\nriences in this exciting and useful field of automatic control. The goal\nof the book is to provide the technical background for understanding\nPID control. Such knowledge can directly contribute to better product\nquality.\nProcess dynamics is a key for understanding any control problem.\nChapter 2 presents different ways to model process dynamics that\nare useful for PID control. Methods based on step tests are discussed\n\nChapter 1 Introduction\n\n3\n\ntogether with techniques based on frequency response. It is attempted\nto provide a good understanding of the relations between the different\napproaches. Different ways to obtain parameters in simple transfer\nfunction models based on the tests are also given. Two dimensionfree parameters are introduced: the normalized dead time and the\ngain ratio are useful to characterize dynamic properties of systems\ncommonly found in process control. Methods for parameter estimation\nare also discussed. A brief description of disturbance modeling is also\ngiven.\nAn in depth presentation of the PID controller is given in Chap¬\nter 3. This includes principles as well as many implementation de¬\ntails, such as limitation of derivative gain, anti-windup, improvement\nof set point response, etc. The PID controller can be structured in dif¬\nferent ways. Commonly used forms are the series and the parallel\nforms. The differences between these and the controller parameters\nused in the different structures are treated in detail. Implementation\nof PID controllers using digital computers is also discussed. The un¬\nderlying concepts of sampling, choice of sampling intervals, and anti¬\naliasing niters are treated thoroughly. The limitations of PID control\nare also described. Typical cases where more complex controllers are\nworthwhile are systems with long dead time and oscillatory systems.\nExtensions of PID control to deal with such systems are discussed\nbriefly.\nChapter 4 describes methods for the design of PID controllers.\nSpecifications are discussed in detail. Particular attention is given to\nthe information required to use the methods. Many different meth¬\nods for tuning PID controllers that have been developed over the\nyears are then presented. Their properties are discussed thoroughly.\nA reasonable design method should consider load disturbances, model\nuncertainty, measurement noise, and set-point response. A drawback\nof many of the traditional tuning rules for PID control is that such\nrules do not consider all these aspects in a balanced way. New tuning\ntechniques that do consider all these criteria are also presented.\nThe authors believe strongly that nothing can replace under¬\nstanding and insight. In view of the large number of controllers used\nin industry there is a need for simple tuning methods. Such rules will\nat least be much better than \"factory tuning,\" but they can always be\nimproved by process modeling and control design. In Chapter 5 we\npresent a collection of new tuning rules that give significant improve¬\nment over previously used rules.\nIn Chapter 6 we discuss some techniques for adaptation and au¬\ntomatic tuning of PID controllers. This includes methods based on\nparametric models and nonparametric techniques. A number of com¬\nmercial controllers are also described to illustrate the different tech¬\nniques. The possibilities of incorporating diagnosis and fault detection\n\n4\n\nChapter 1 Introduction\n\nin the primary control loop is also discussed.\nIn Chapter 7 it is shown how complex control problems can be\nsolved by combining simple controllers in different ways. The control\nratio control, split range control, and control with selectors are dis¬\ncussed. Use of currently popular techniques such as neural networks\nand fuzzy control are also covered briefly.\nReferences\nA treatment of PID control with many practical hints is given in\nShinskey (1988). There is a Japanese text entirely devoted to PID\ncontrol by Suda et al. (1992). Among the books on tuning of PID\ncontrollers, we can mention McMillan (1983) and Corripio (1990),\nThere are several studies that indicate the state of the art of in¬\ndustrial practice of control. The Japan Electric Measuring Instrument\nManufacturers'Association conducted a survey of the state of process\ncontrol systems in 1989, see Yamamoto and Hashimoto (1991). Ac¬\ncording to the survey more than than 90% of the control loops were\nof the PID type.\nThe paper, Bialkowski (1993), which describes audits of paper\nmills in Canada, shows that a typical mill has more than 2000 control\nloops and that 97% use PI control. Only 20% of the control loops were\nfound to work well and decrease process variability. Reasons for poor\nperformance were poor tuning (30%) and valve problems (30%). The\nremaining 20% of the controllers functioned poorly for a variety of\nreasons such as: sensor problems, bad choice of sampling rates, and\nanti-aliasing filters. Similar observations are given in Ender (1993),\nwhere it is claimed that 30% of installed process controllers operate\nin manual, that 20% of the loops use \"factory tuning,\" i.e., default\nparameters set by the controller manufacturer, and that 30% of the\nloops function poorly because of equipment problems in valves and\nsensors.\n\nCHAPTER\n\n2\n\nProcess Models\n\n2.1\n\nIntroduction\n\nA block diagram of a simple control loop is shown in Figure 2.1. The\nsystem has two major components, the process and the controller, rep¬\nresented as boxes with arrows denoting the causal relation between\ninputs and outputs. The process has one input, the manipulated vari¬\nable, also called the control variable. It is denoted by u. The process\noutput is called process variable (PV) and is denoted by y. This vari¬\nable is measured by a sensor. The desired value of the process variable\nis called the setpoint (SP) or the reference value. It is denoted by ysp.\nThe control error e is the difference between the setpoint and the\nprocess variable, i.e., e = ysp — y. The controller in Figure 2.1 has\none input, the error, and one output, the control variable. The figure\nshows that the process and the controller are connected in a closed\nfeedback loop.\nThe purpose of the system is to keep the process variable close\nto the desired value in spite of disturbances. This is achieved by the\nfeedback loop, which works as follows. Assume that the system is in\nequilibrium and that a disturbance occurs so that the process variable\nbecomes larger than the setpoint. The error is then negative and the\ncontroller output decreases which in turn causes the process output\nto decrease. This type of feedback is called negative feedback, because\nthe manipulated variable moves in direction opposite to the process\nvariable.\nThe controller has several parameters that can be adjusted. The\ncontrol loop performs well if the parameters are chosen properly. It\nperforms poorly otherwise, e.g., the system may become unstable.\nThe procedure of finding the controller parameters is called tuning.\n\n6\n\nChapter 2 Process Models\nController!\n\nProcess\n\n- 1 -*\nFigure 2.1 Block diagram of a simple feedback system.\n\ni\n\n)\n\nThis can be done in two different ways. One approach is to choose\nsome controller parameters, to observe the behavior of the feedback\nsystem, and to modify the parameters until the desired behavior is\nobtained. Another approach is to first develop a mathematical model\nthat describes the behavior of the process. The parameters of the\ncontroller are then determined using some method for control design.\nAn understanding of techniques for determining process dynamics is\na necessary background for both methods for controller tuning. This\nchapter will present such techniques.\nStatic models are discussed in the next section. Dynamic models\nare discussed in Section 2.3. Transient response methods, which are\nuseful for determining simple dynamic models of the process, are pre¬\nsented in Section 2.4. Section 2.5 treats methods based on moments.\nThese methods are less sensitive to measurement noise and, further¬\nmore, are not restricted to any specific input signal. The frequency\nresponse methods, described in Section 2.6, can be used to obtain\nboth simple models and more detailed descriptions. Methods based\non estimation of parametric models are more complex methods that\nrequire more computations but less restrictions on the experiments.\nThese methods are presented in Section 2.7. The models discussed so\nfar describe the relation between the process input and output. It is\nalso important to model the disturbances acting on the system. This\nis discussed in Section 2.8. Section 2.9 treats methods to simplify a\ncomplex model and the problem of unmodeled dynamics and mod¬\neling errors. Conclusions and references are given in Sections 2.10\nand 2.11.\n\n2.2\n\nStatic Models\n\nThe static process characteristic is a curve that gives the steady state\nrelation between process input signal u and process output y. See\nFigure 2.2. Notice that the curve has a physical interpretation only\nfor a stable process.\n\n2.2 Static Models\n\nFigure 2.2 Static process characteristic. Shows process output y\nas a function of process input u under static conditions.\n\nAll process investigations should start by a determination of the\nstatic process model. It can be used to determine the range of control\nsignals required to change the process output over the desired range,\nto size actuators, and to select sensor resolution. It can also be used\nto assess whether static gain variations are so large that they have\nto be accounted for in the control design.\nThe static model can be obtained in several ways. It can be de¬\ntermined by an open-loop experiment where the input signal is set\nto a constant value and the process output is measured when it has\nreached steady state. This gives one point on the process characteris¬\ntics. The experiment is then repeated to cover the full range of inputs.\nAn alternative procedure is to make a closed-loop experiment.\nThe setpoint is then given a constant value and the corresponding\ncontrol variable is measured in steady state. The experiment is then\nrepeated to cover the full range of setpoints.\nThe experiments required to determine the static process model\noften give a good intuitive feel for how easy it is to control the process,\nif it is stable, and if there are many disturbances.\nSometimes process operations do not permit the experiments to be\ndone as described above. Small perturbations are normally permitted,\nbut it may not be possible to move the process over the full operating\nrange. In such a case the experiment must be done over a long period\nof time.\n\nProcess Noise\nProcess disturbances are easily determined by logging the process\noutput when the control signal is constant. Such a measurement\n\n8\n\nChapter 2 Process Models\n\nwill give a combination of measurement and load disturbances. There\nare many sophisticated techniques such as time-series analysis and\nspectral analysis that can be used to determine the characteristics\nof the process noise. Crude estimates of the noise characteristics\nare obtained simply by measuring the peak-to-peak value and by\ndetermining the average time between zero crossings of the error\nsignal. This is discussed further in Section 2.8.\n\n2.3\n\nDynamic Models\n\nA static process model like the one discussed in the previous section\ntells the steady state relation between the input and the output sig¬\nnal. A dynamic model should give the relation between the input and\nthe output signal during transients. It is naturally much more diffi¬\ncult to capture dynamic behavior. This is, however, very significant\nwhen discussing control problems.\nFortunately there is a restricted class of models that can often be\nused. This applies to linear time-invariant systems. Such models can\noften be used to describe the behavior of control systems when there\nare small deviations from an equilibrium. The fact that a system is\nlinear implies that the superposition principle holds. This means that\nif the input u\\ gives the output yi and the input ui gives the output\nj2 it then follows that the input au\\ + bui gives the output ay\\ + by 2A system is time-invariant if its behavior does not change with time.\nA very nice property of linear time-invariant systems is that their\nresponse to an arbitrary input can be completely characterized in\nterms of the response to a simple signal. Many different signals can be\nused to characterize a system. Broadly speaking we can differentiate\nbetween transient and frequency responses.\nIn a control system we typically have to deal with two signals\nonly, the control signal and the measured variable. Process dynamics\nas we have discussed here only deals with the relation between those\nsignals. The measured variable should ideally be closely related to the\nphysical process variable that we are interested in. Since it is difficult\nto construct sensors it happens that there is considerable dynamics\nin the relation between the true process variable and the sensor. For\nexample, it is very common that there are substantial time constants\nin temperature sensors. There may also be measurement noise and\nother imperfections. There may also be significant dynamics in the\nactuators. To do a good job of control, it is necessary to be aware of\nthe physical origin the process dynamics to judge if a good response\nin the measured variable actually corresponds to a good response in\nthe physical process variable.\n\n2.3 Dynamic Models\n\nTransient Responses\nIn transient response analysis the system dynamics are character¬\nized in terms of the response to a simple signal. The particular signal\nis often chosen so that it is easy to generate experimentally. Typical\nexamples are steps, pulses, and impulses. Because of the superpo¬\nsition principle the amplitude of the signals can be. normalized. For\nexample, it is sufficient to consider the response to a step with unit\namplitude. If s(t) is the response to a unit step, the output y{i) to an\narbitrary input signal u(t) is given by\n\nal\n\nu(T)h(t-T)dT\n\n(2.1)\n\nwhere the impulse response h{t) is introduced as the time derivative\nof the step response.\nIn early process control literature the step response was also\ncalled the reaction curve.\nPulse response analysis is common in medical and biological ap¬\nplications, but rather uncommon in process control. Ramp response\nanalysis is less common. One application is the determination of the\nderivative part of a PID controller. In process control, the step re¬\nsponse is the most common transient used for process identification.\nThis is primarily because this is the type of disturbance that is easi¬\nest to generate manually. Step response methods are treated in detail\nin Section 2.4.\n\nFrequency Response\nAnother way to characterize the dynamics of a linear time-invariant\nsystem is to use sine waves as a test signal. This idea goes back\nto Fourier. The idea is that the dynamics can be characterized by\ninvestigating how sine waves propagate through a system.\nConsider a stable linear system. If the input signal to the system\nis a sinusoid, then the output signal will also be a sinusoid after a\ntransient (see Figure 2.3). The output will have the same frequency as\nthe input signal. Only the phase and the amplitude are different. This\nmeans that under stationary conditions, the relationship between the\ninput and the output can be described by two numbers: the quotient\n(a) between the input and the output amplitude, and the phase shift\n(<p) between the input and the output signals. The functions a((o)\nand (p(co) describe a and <p for all frequencies (co). It is convenient\nto view a and (p as the magnitude and the argument of a complex\nnumber\nG(ia>) = aiwyv^\n\n(2.2)\n\n10\n\nChapter 2\n\no.i T\n\n'x\n\nProcess Models\n\ny\n\n10\n\n15\n\nFigure 2.3 Input signal u is a sinusoid and output signal y be¬\ncomes sinusoidal after a transient.\nThe function G(iw) is called the frequency response function of the\nsystem. The function a(co) - \\G(ico)\\ is called the amplitude function,\nand the function (p{co) ~ arg(G(ico)) is called the phase function.\nThe complex number G{ico) can be represented by a vector with\nlength a(ico) that forms angle <p(ico) with the real axis (see Figure\n2.4). When the frequency goesfrom0 to oo, the endpoint of the vector\ndescribes a curve in the plane, which is called the frequency curve\nor the Nyquist curve. The Nyquist curve gives a complete description\nof the system. It can be determined experimentally by sending sinu¬\nsoids of different frequencies through the system. This may be time\nconsuming. Normally, it suffices to know only parts of the Nyquist\ncurve. For controller tuning there are some parts that are of particu¬\nlar interest. The lowest frequency where the phase is -180° is called\nthe ultimate frequency (cou). The corresponding point on the Nyquist\ncurve is called the ultimate point. The value of G(icou) is all that is\nneeded for the tuning methods developed by Ziegler and Nichols.\nThefrequencyresponse is intimately related to the Laplace trans¬\nform. Let f(t) be a signal. The Laplace transform of the signal, F(s),\nis then defined by\n\nr\n\nF(s) = /\n\n*f{t)dt\n\n(2.3)\n\nJo\nLet U{s) and Y(s) be the Laplace\ntransforms of the input and the\noutput of a linear time-invariant dynamical system. Assume that the\nsystem is at rest at time t = 0. The following relation then holds\n\nY(s) = G(s)U(s)\n\n(2-4)\n\nwhere G(s) is the transfer function of the system.\nIt follows from Equation (2.3) that the Laplace transform of an\nimpulse is 1. From Equation (2.4) we can conclude that G(s) is the\n\n2.4\n\nStep Response Methods\n\n11\n\nk Im Giico)\n\nUltimate point\n\nReGdco)\n\nFigure 2.4 The Nyquist curve of a system.\nLaplace transform of the impulse response. The frequency response\nis simply G{ico).\nIn the following sections we will show how linear system dynam¬\nics can be obtained experimentally. We will illustrate both transient\nand frequency response methods.\n\n2.4\n\nStep Response Methods\n\nThe dynamics of a process can be determined from the response of\nthe process to pulses, steps, ramps, or other deterministic signals.\nThe dynamics of a linear system is, in principle, uniquely given from\nsuch a transient response experiment. This requires, however, that\nthe system is at rest before the input is applied, and that there are no\nmeasurement errors. In practice, however, it is difficult to ensure that\nthe system is at rest. There will also be measurement errors, so the\ntransient response method, in practice, is limited to the determination\nof simple models. Models obtained from a transient experiment are,\nhowever, often sufficient for PID controller tuning. The methods are\nalso very simple to use. This section focuses on the step response\nmethod.\n\nThe Step Response\nAssuming a control loop with a controller, the step response experi¬\nment can be determined as follows. Wait until the process is at rest.\n\n12\n\nChapter 2 Process Models\nB\n\n0\n\n2\n\n4\n\n6\n\nD\n2\n0\n0\n\nF\n0.5\n-0.5-\n\nFigure 2.5 Open-loop step responses.\nSet the controller to manual. Change the control variable rapidly, e.g.,\nthrough the use of increase/decrease buttons. Record the process vari¬\nable and scale it by dividing by the change in the control variable.\nThe change in control variable should be as large as possible in order\nto get a maximum signal to noise ratio. The limit is set by permissible\nprocess operation. It is also useful to record the fluctuations in the\nmeasurement signal when the control signal is constant. This gives\nIt is good practice to repeat the experiment for different ampli¬\ntudes of the input signal and at different operating conditions. This\ngives an indication of the signal ranges when the model is linear. It\nalso indicates if the process changes with the operating conditions.\nExamples of open-loop step responses are shown in Figure 2.5.\nMany properties of the system can be read directly from the step re¬\nsponse. In Figure 2.5A, the process output is monotonically changed\nto a new stationary value. This is the most common type of step re¬\nsponse encountered in process control. In Figure 2.5B, the process\noutput oscillates around its final stationary value. This type of pro¬\ncess is uncommon in process control. One case where it occurs is in\nconcentration control of recirculation fluids. In mechanical designs,\nhowever, oscillating processes are common where elastic materials\nare used, e.g., weak axles in servos, spring constructions, etc. The sys¬\ntems in Figures 2.5A and B are stable, whereas the systems shown\nin Figures 2.5C and 2.5D are unstable. The system in Figure 2.5C\n\n2.4 Step Response Methods\n\n13\n\nshows the output of an integrating process. Examples of integrating\nprocesses are level control, pressure control in a closed vessel, concen¬\ntration control in batches, and temperature control in well isolated\nchambers. The common factor in all these processes is that some kind\nof storage occurs in them. In level, pressure and concentration control\nstorage of mass occurs, while in the case of temperature control there\nis a storage of energy. The system in Figure 2.5E has a long dead\ntime. The dead time occurs when there are transportation delays in\nthe process. The system in Figure 2.5F is a non-minimum phase sys¬\ntem, where the measurement signal initially moves in the \"wrong\"\ndirection. The water level in boilers often reacts like this after a step\nchange in feed water flow.\nIf the system is linear, all step responses are proportional to\nthe size of the step in the input signal. It is then convenient to\nnormalize the responses by dividing the measurement signal by the\nstep size of the control signal. Throughout this book we assume that\nthis normalization is done.\nThe step response is a convenient way to characterize process\ndynamics because of its simple physical interpretation. Many tuning\nmethods are based on it. A formal mathematical model can also be\nobtained from the step response. General methods for the design of\ncontrol systems can then be used.\nFor small perturbations the static process model can be described\nby one parameter called the process gain. This is simply the ratio of\nthe steady state changes of process output and process input. The gain\ncan be obtained as the slope of the curve in Figure 2.2. It can also\nbe obtained directly from a step response. For nonlinear systems the\nprocess gain will depend on the operating conditions. It is, however,\nconstant for linear systems. For such systems the static properties are\nthus described by one parameter. Additional parameters are needed\nto also capture dynamics. Some simple parametric models will be\ndescribed below. Stable processes with a monotone step response, as\nshown in Figure 2.5A, are quite common. Many methods to obtain\nparametric models from such a step response have been presented in\nthe literature over the years. We will present here models with two,\nthree, and, four parameters respectively.\n\nTwo-Parameter Models\nThe simplest parametric models of process dynamics have two param¬\neters. One parameter can be process gain. The other has to capture\nthe time behavior. The average residence time Tar is a useful param¬\neter. This is obtained as\n\n14\n\nChapter 2\n\nProcess Models\n\nwhere K is the static process gain and Ao is defined as\noo\n\nAo = /\"(s(oo) - s{t))dt\no\nwhere s{t) is the step response. Notice that K = s(oo) and that Ao is\nthe shaded area in Figure 2.6.\nThe time Tar is a rough measure of the time it takes for the step\nresponse to settle. Using the static gain and the average residence\ntime, the process can be approximated by the model\n\nWe call this model the residence time approximation.\nAnother approximation to the step response that also has two\nparameters is given by the transfer function\n,-»£\n\n(2.6)\n\nThis model corresponds to an integrator with dead time. This model\nis characterized by the two parameters, a and L, that are easily de¬\ntermined graphically from the step response (see Figure 2.6). The\ntangent to the step response s(t) that has the largest slope is drawn,\nand the intersections of this tangent with the vertical and horizontal\naxes give a and L, respectively. The model given by Equation (2.6) is\nthe basis for the Ziegler-Nichols tuning procedure discussed in Chap¬\nter 4. Notice that the model can also be fitted to unstable processes.\n\nK\n\nFigure 2.6 Graphical determination of a two-parameter model\nfrom a step response for a stable system with a monotone step\nresponse.\n\n2.4 Step Response Methods\n\n15\n\nThe properties of the approximations (2.5) and (2.6) are illus¬\ntrated by an example.\nEXAMPLE 2.1\n\nThe two-parameter models (2.5) and (2.6) have been fitted to the\nprocess model\n\nThe following models were obtained\nG2a(s) =\n\nG2b(s) =\n\n0.64 ,-4.3s\n4.3s'\n\n1 + 8.0s\nFigure 2.7 shows the step responses and the Nyquist curves of the\ntransfer functions.\nNotice that the model G2a gives a good description of the step\nresponse for long times. The static gain is correct and the step re¬\nsponse is very close to the correct one for large t. There are, however,\nlarge discrepances for small t. The system given by G2a has, for ex¬\nample, a significant response at time t = 2, but the system (2.7)\nhas barely responded at that time. The model G2b has the opposite\nproperties. It approximates the true step response very well in the\ninterval 5 < t < 9, but the approximation is very poor for large t.\nThese properties are also reflected in the Nyquist curves. They\nshow that the average residence time approximation is quite good at\nlow frequencies but very poor at high frequencies. The model G2b, on\nthe other hand, is poor at low frequencies but reasonable at middle\nrange frequencies.\n•\n\nFigure 2.7 Step responses and Nyquist curves of the process\nG{s) = l/(s + I) 8 (solid line) and the two-parameter models G^s)\n(dotted line) and G2b(s) (dashed line).\n\n16\n\nChapter 2\n\nProcess Models\n\nThree-Parameter Models\nBetter approximations are obtained by increasing the number of pa¬\nrameters. The model\n\nis characterized by three parameters: the static gain K, the time\nconstant T, and the dead time L. This is the most common process\nmodel used in papers on PID controller tuning. The parameters L\nand T are often called the apparent dead time and the apparent time\nconstant, respectively. The step response of the model (2.8) is\n\nFrom this equation, it follows that the average residence time is\noo\n\nTar\n\n/(s(oo) - s(t))dt\n=\"\n^\n=L+T\n\nThe ratio\n\nr\n\n= LTT = h\n\n(2 9)\n\n\"\n\nwhich has the property 0 < r < 1, is called the normalized dead time.\nThis quantity can be used to characterize the difficulty of controlling a\nprocess. It is sometimes also called the controllability ratio. Roughly\nspeaking, it has been found that processes with small t are easy\nto control and that the difficulty in controlling the system increases\nas t increases. Systems with r = 1 correspond to pure dead-time\nprocesses, which are indeed difficult to control well.\nThe parameters in the model (2.8) can be determined graphically.\nThe static gain (K) is obtained from the final steady-state level of the\nprocess output. Remember that the process output must be scaled\nwith the change in the control variable. The intercept of the tangent\nto the step response that has the largest slope with the horizontal\naxes gives L (see Figure 2.8). The dead time L can also be obtained\nas the time between the onset of the step and the time s(i) has\nreached a few percent of its final value. There are different ways\nto determine T. One method determines T from the distance AC in\nFigure 2.8, where the point C is the time when the tangent intersects\nthe line s(t) = K. Another method determines T from the distance\nAB in Figure 2.8, where B is the time when the step response has\nreached the value 0.63^. Both methods give identical results if the\nprocess dynamics are given by Equation (2.8), but they may differ\nsignificantly in other cases. The method based on the point B gives\nnormally better approximations. The other method tends to give a too\nlarge value of T.\n\n2.4 Step Response Methods\n\n17\n\nK\n\n0.63 K\n\nB\n\nC\n\nFigure 2.8 Graphical determination of three-parameter models\nfor systems with a monotone step response.\nEXAMPLE 2.2\nThe three-parameter models of the process model (2.7) are\nG3a(s) =\n\n1 + 6.7s\n\nG3b(s) =\n\no -4.3s\n\n1 + 4.3s\n\nwhere the time constant T is determined from the point C in model\nG^a, and from the point B in the model G31,. Figure 2.9 shows the\nstep responses of the true process and the models, as well as the\nNyquist curves of the transfer functions. The figure shows that the\ntime constant T is overestimated in the model G3a. This overestimation is unfortunately common in this method, since most process\ncontrol plants have an S-shaped step response similar to the model\n(2.7). Notice that the true step response and the step response of the\nmodel Gzb coincide at the 63% point.\n•\nAnother Model Structure\nThe model (2.8) is by far the most commonly used model in the papers\nof PID controller tuning. In spite of this, it is not a representative\nmodel. In fact, the conclusions drawn based on this model may often\nbe misleading when applied to real processes. This will be illustrated\nby several examples in Chapter 4. One reason for this is that the step\nresponse of the model (2.8) is not S-shaped, or equivalently, that the\nfrequency response of the model does not decay fast enough for high\nfrequencies.\n\n18\n\nChapter 2 • Process Models\n\nFigure 2.9 Step responses and Nyquist curves of the process\nG(s) — l/(s + l) 8 (solid line) and the three-parameter models G^s)\n(dashed line) and G3b(s) (dotted line).\n\nAnother three-parameter model is\nK\nsT)2\n\nG(s) =\n\n(2.10)\n\nThe step response of this model is\n(2.H)\n\n*(') =\n\nThis model has an S-shaped step response and often gives a better\napproximation than the first-order plus dead-time model (2.8). Static\ngain K and dead time L can be determined in the same way as\nfor the model (2.8). Time constant T can then be determined from\nEquation 2.11 if the value of the step response at one time is known.\nThe equation obtained must be solved numerically.\nEXAMPLE 2.3\nFitting the model (2.10) to the process model (2.7) gives\nG3c(s) =\n\n.-4.3s\n\n(1 + 2.0s)2\n\nThe gain K = 1 is obtained from the steady-state value of the signal,\nand the dead time L = 4.3 is obtained from the intersection of the\ntangent with the largest slope and the horizontal axis as in the\nprevious examples. The two time constants T = 2.0 are obtained\nby numerical solution of Equation (2.11). The point s(8.6) = 0.63 is\nused to obtain the additional condition. Figure 2.10 shows the step\n\n2.4 Step Response Methods\n\n19\n\nFigure 2.10 Step\nresponses and Nyquist curves of the process\nG(s) = l/(s + I)8 (solid line) and the three-parameter model G^^)\n(dashed line).\nresponses of the true process model and G3C(s), as well as the Nyquist\ncurves of the two transfer functions. The two step responses coincide\nat the 63% point. The model now has the S-shaped form because of\nthe second-order model, and the fit is much better than the previous\nfirst-order models.\nO\nFour-Parameter Models\nAn even better approximation may be obtained by the transfer func¬\ntion\nG s\n\n( ) = „ . .„?„ . ^ e\"sL\n\n(2.12)\n\nThis model has four parameters: the gain K, the time constants 2\\\nand Ti, and the dead time L. The gain K can be determined from the\nsteady-state value of the step response. The dead time L can also be\nobtained in the same way as for the three-parameter models either\nby drawing the tangent of maximum slope of s(t) or by determining\nthe time between the onset of the step and the time s(t) has reached\na few percent of its final value. The step response of the model (2.12)\nis\n\ns(t) = K(l +\n\nT1-T2\n\nT2\n\n(2.13)\n\nThe time constants (T{) and (T2) can be calculated from this expres¬\nsion by determining two points of the step response. The calculation\ndoes involve solution of transcendental equations. This must be done\nnumerically.\n\n20\n\nChapter 2 Process Models\n\n1\n\nFigure 2.11 Step responses and Nyquist curves of the process\nG(s) = l/(s + I) 8 (solid line) and the four-parameter model G^s)\n(dashed line).\n\nEXAMPLE 2.4\n\nA four-parameter model (2.12) of the process model (2.7) has been\nobtained in the following way. The gain K = 1 is determined from\nthe steady-state values, and the dead time L = 4.3 is obtained from\nthe largest slope, as in the previous examples. The time constants 7\\\nand Ti are then obtained by numerically fitting the equation for the\nstep response (2.13) to the values of the true step response at the 33%\npoint and the 67% point. With s(6.5) = 0.33 and s(8.9) = 0.67, the\ntime constants become 7\\ = 0.93 and T2 = 3.2. The transfer function\nis thus\no -4.3s\n\n0.93s)(l + 3.2s)\nFigure 2.11 shows the step responses of the true process model and\nG4a(s), as well as the Nyquist curves of the two transfer functions.\nNotice that the two step responses coincide at the 33% point and at\nthe 67% point.\n•\nIn the previous example, gain K and dead time L were deter¬\nmined graphically from the step response, whereas time constants T\\\nand 72 were determined by numerical solution of the equation for the\nstep response. There are several methods presented for a graphical\ndetermination of all four parameters of the model (2.12). These meth¬\nods are useful when no computers are available for numerical solu¬\ntions. Using computer optimization programs, however, often gives a\nbetter approximation than the graphical methods. This is illustrated\nin the following example.\n\n2.4 Step Response Methods\n\n21\n\nFigure 2.12 Step responses and Nyquist curves of the process\nG(s) = l/(s + I) 8 (solid line) and the four-parameter model Gib\n(dashed line).\nEXAMPLE 2.5\n\nThe four-parameter model (2.12) has been fitted to the process model\n(2.7) using least squares optimization, where the aim was to obtain\nan accurate model in the third quadrant, i.e., where the phase shift\nis between —90° and —180°. The following model was obtained.\nG4b(s) =\n\n1.05\n,-3.75s\n(1 + 2.39s)2\n\nFigure 2.12 shows the step responses of the true process model and\nG4j(s), as well as the Nyquist curves of the transfer functions.\nD\nModels for Integrating Systems\nThere are some process control systems where the dynamics contain\nintegration or very long time constants. Such systems will not reach\na steady state under open-loop conditions. They are sometimes called\nsystems without self regulation. For PID tuning it is useful to treat\nsuch systems separately.\nImpulse Responses\nFor a system with integral action a steady state will not be achieved\nwhen the input signal is a step, since the output will asymptotically\nchange at a constant rate. There will be, however, a steady state when\nthe input is an impulse. To determine the dynamics we can, therefore,\napply a short pulse to the process. After normalizing the response by\ndividing with the pulse area, we then get a step response that can\n\n22\n\nChapter 2 Process Models\n\nbe modeled using the methods we have just discussed. The transfer\nfunction of a system with integral action is then obtained simply by\nmultiplying the transfer function by 1/s. We illustrate the procedure\nwith an example.\nEXAMPLE 2.6\nAssume that a square pulse with unit height and duration r has been\napplied to a process and that the model\n\nhas been fitted to the response as described in Example 2.2. The\ntransfer function of the process is then\n\nStep Responses\nModels based on step responses can also be applied to processes with\nintegral action. One possibility is to calculate the derivative of the\nstep response and apply the impulse response method that was just\ndiscussed.\nThe two-parameter model\n\nthat was used to model stable processes previously in this section\ncan also be applied to integrating processes. This model gives a bad\ndescription of stable processes at high frequencies, but for integrating\nprocesses the low frequency behavior is well captured by the model.\nA more sophisticated model that gives a better approximation at\nhigher frequencies is given by the transfer function\n\nThe model is characterized by three parameters: the velocity gain K,\nthe time constant T, and the dead time L. The step response of the\nmodel (2.14) is\n= K(t-L-T(l-\n\ne-C-^) )\n\n(2.15)\n\nThe gain K and the average residence time Tar = L + T can be\ndetermined graphically as shown in Figure 2.13.\n\n2.4 Step Response Methods\n\n23\n\nK(L+T)\n\nL+T\n\nFigure 2.13 Graphical determination of a three-parameter model\nfor an integrating process.\nThe dead time L and the time constant T can be determined by\nfitting Equation (2.15) to one point of the step response. A suitable\npoint is\ns(L + T) =\nwhich gives\nT=\n\nK\n\nModels for Oscillatory Systems\nOscillatory systems with step responses, as shown in Figures 2.5B\nand D, can be crudely approximated by the two-parameter model\n(2.6), but this model will not capture the oscillations. None of the\nthree- or four-parameter models presented above is suitable either. A\nthree-parameter model that describes the oscillations is given by the\ntransfer function\nKm2\nG\n(216)\n^\nThis model has three parameters: the static gain K, the natural\nfrequency co, and the relative damping £. These parameters can be\ndetermined approximately from the step response as indicated in\nFigure 2.14. The period of the oscillation Tp and the decay ratio d\np\nare first determined. Parameters ft) and\nare\nrelated to Tp and d as\nfollows.\nd=\n\n24\n\nChapter 2\n\nProcess Models\n\nK\n\nFigure 2.14 Graphical determination of mathematical models for\nsystems with an oscillatory step response.\n\nor\n(O =\n\n2nl log df\n\n(2.17)\n\nA time delay can also be added to the model (2.16) and determined in\nthe same way as for the previous models, e.g., by drawing the tangent\nof maximum slope or determining the time between the onset of the\nstep and the time the step response has reached a few percent of its\nfinal value.\n\n2.5\n\nMethods of Moments\n\nAll average residence time was determined based on calculation of\nan area. All other methods discussed in Section 2.4 were based on\nevaluation of the step response at single points only. Such methods\nare quite sensitive to measurement noise. In this section we will\ndiscuss methods that are based on integrals of the step response.\n\nArea Methods\nWe will first discuss a method that is based on area calculations.\nStatic gain K and average residence time Tar are first determined as\nin Figure 2.6. The area A\\ under the step response up to time Tar is\nthen determined. For a system having the transfer function\nK\nST'\n\n--si\n\n2.5 Methods of Moments\n\n25\n\nwe have\nAx = I s{t)dt= [ K{l-e-tlT)dt\no\no\nThe time constant is thus given by\n\n= KTe'1\n\nThe dead time is then given by\n\nWith this method parameters L and T are both determined from\ncomputations of areas. The method is illustrated by the following\nexample.\nEXAMPLE 2.7\nThe method based on area determination has been applied to the\nprocess model (2.7). Static gain K is first determined from the sta¬\ntionary values to K = 1. Area Ao is then determined to 8.0 providing\nthe average residence time Tar = 8. Area Ai can be determined by\nintegrating the step response up to time Tar to Aj = 1.1. From Equa¬\ntion (2.18), time T can be calculated to T = 3.0, and finally Equation\n(2.19) gives L = 5.0. To summarize, the method based on area deter¬\nmination gives the following three-parameter model\nn\n\nJ<,\\\n\n_\n\n•*-\n\n,,-5-Os\n\nFigure 2.15 shows the step responses of the true process model and\nG3d(s), as well as the Nyquist curves of the two transfer functions.\nD\n\nThe same idea can easily be applied to a system with the transfer\nfunction\nParameters K and residence time Tar are determined as before. In\nthis case we have\nTar = L + 2T\n\nThe area Ai under the step response up to time Tar is then\ndetermined. For a system having transfer function (2.20) we have\nT^\n\nIT\n\nAi = [s{t)dt = [K (1 - e~t/T - ^e-tlT\\\n0\n\n0\n\ndt = AKTe'2\n\n26\n\nChapter 2 Process Models\n\nFigure 2.15 Step responses and Nyquist curves of the process\nG(s) = l/(s +1)8 (solid line) and the three-parameter model\n(dashed line).\n\nThe time constant is thus given by\nT=\n\n(2.21)\n\nAK\n\nLi\n\n—\n\n± a r — 61\n\n—\n\n—\n\nAxe2\n\n(2.22)\n\nK\n2K\nThe following example illustrates the properties of the method.\nEXAMPLE 2.8\n\nThe three-parameter model (2.20) has been fitted to the process model\n(2.7) using the method of area determination. Static gain K is deter¬\nmined from the stationary values to K = 1. The area Ao is 8.0, which\ngives the average residence time Tar = 8.0. Furthermore the area A\\\nis 1.1 and Equation (2.21) then gives T ~ 2.0. Equation (2.22) finally\ngives L ~ 4.0 and the model becomes\nG3e(s) =\n\n,-4.0s\n\n(1 + 2.0s)2\nFigure 2.16 shows the step responses of the true process model and\nG3e(s), as well as the Nyquist curves of the two transfer functions.\n\n•\nThe methods based on area determination are less sensitive to\nhigh-frequency disturbances than the previous methods, where the\nmodel is determined from only a few values of the step response. On\nthe other hand, they are more sensitive to low-frequency disturbances\nsuch as a change in static load.\n\n2.5 Methods of Moments\n\n27\n\nFigure 2.16 Step responses and Nyquist curves of the process\nG(s) = l/(s + I) 8 (solid line) and the three-parameter model\nG&,(dashed line).\n\nThe Method of Moments\nA drawback with the area methods is that they require a storage\nof the step response. Area A\\ cannot be computed until area AQ is\ndetermined. Therefore, some alternative methods that are also based\non integration will be considered.\nLet h(t) be an impulse response and G(s) the corresponding\ntransfer function. The functions are related through\noo\n\nG(s) = f e-sth(t)dt\no\nTaking derivatives with respect to s gives\noo\n\ndnG{s) = G{n\\s) = (-1)\" f e-s'tnh(t)dt\ndsn\nHence,\noo\n\nG(n)(0) = (-1)\"\n\nftnh(t)dt\n\n(2.23)\n\nThe values of the transfer function and its derivatives at s = 0 can\nthus be determined from integrals of the impulse response.\nThe Average Residence Time\nThe impulse response is positive for systems with monotone step re¬\nsponses. It can be interpreted as the density function of a probability\n\n28\n\nChapter 2 Process Models\n\ndistribution if it is normalized as follows:\nfh(t)dt\n0\n\nThe quantity f(t)dt can then be interpreted as the probability that\nan impulse entering the system at time 0 will leave at time t. The\naverage residence time is then\noo\n\noo\n\nJth(t)dt\n\nTor= [tf(t)dt=^\n\n(2.24)\n\ni\n\nfh(t)dt\no\n\nIntroduce\ng(t) = «(oo) - s(t)\nwhere s(t) is the unit step response. Then\ndg{t)\n\ndt\n\n= -h(t)\n\nIt follows that\noo\n\nth(t)dt= \\-tg(t)\\ + I g{t)dt\no\no\nThe first term of the right-hand side is zero if g(t) goes to zero at\nleast as fast as t1+e for large t. The average residence time can thus\nalso be written as\noo\n\nf(8(co)-8(t))dt\nrp\n\n_ 0\n\ns(oo)\n\nwhich is the definition used previously.\nEquation (2.23) gives a convenient way to determine parameters\nof different models by computing the moments. This will be illustrated\nby some examples.\nA Three-Parameter Model\nConsider the transfer function\ne~sL\n\n(2.25)\n\nK = G(0) = I h{t)dt\no\n\n(2.26)\n\nG(s) =\n\nK\n\nIt follows that\noo\n\n2.5 Methods of Moments\n\n29\n\nTaking logarithms of Equation (2.25) gives\nlogG(s) = log tf - sL-log (1 + sT)\nDifferentiating this expression gives\nG\n\n'( s )\nG(s)\n\nT\nL\nu\n\n1\n\n/'G'(s)'\\\n\\, G(«),/\n\nG\"(s)\nG(s)\n\n2\n\nr2\n{1 + sTf\n\nHence\n\nOO\n\nJth(t)dt\n\nG'(o;\nG(0)\n\nOO\n\nr h(t)dt\n0\n\n(2.27)\n\noo\n\n2\n\nJt h(t)dt\n- °oo\n\nG\"(0)\n\n3\n\nJh(t)dt\n0\n\nGain K is thus given by Equation (2.26) and average residence time\nTar and time constant T by Equation (2.27). The dead time L can\nthen be computed to\nL = Tar - T\n\nIt has thus been shown that the parameters of the model can be\nobtained from the first two moments of the impulse response. We\nillustrate the procedure with an example.\nEXAMPLE 2.9\nConsider the process model\nG(s) =\n\n(s + 1)8\n\nThe first two derivatives with respect to s become\n- w\n\n(s+1)9\n\nG\"(S) =\n- v-,\n\n72\nMO\n(s+1)\n\nHence G(0) = 1, G'(0) = - 8 , and G\"(0) = 72. Equations (2.26) and\n(2.27) now give\nK =1\nTar = 8\n712 = 72 - 64 = 8\n\n30\n\nChapter 2 Process Models\n\nWe thus find T = 2\\/2 « 2.8 and L = 8 - 2\\/5 « 5.2. This result can\nbe compared with the previous methods in Examples 2.2 and 2.7. •\nAnother Three-Parameter Model\nThe method of moments will now be applied to determine the param¬\neters of the transfer function\n\nWe have\n\nlog G(s) = log K - sL- 2 log (1 + ;\n\nHence\n\nG'(s)\nG{8)\nG\"(s)\nG(S)\n\nHence\n\n2T\n\n\"\n\n1+sT\n\nfG'(s)\\2\n\\G(s)J\n\n2T2\n(1 + sT)2\n\noo\n\nK = G(0) = f h(t)dt\n0\n00\n\nfth(t)dt\n\nG\n\nTar = - J ° =L\n\n2=\n•*\n\nG^0)_l\nO/nr/^\\^\n\n2G(0)\n\no\n\n2\n\n+\n\n2\nor\n\n2T=°\n\n(2 28)\n\n-\n\n^\n\n1\noo\n\nn\n\n2\nar\n\n2/A(0*\n0\n\nWe illustrate the method with an example.\nEXAMPLE 2.10\n\nConsider the process model (2.7). It followsfromthe previous example\nthat G(0) = 1, G'(0) = - 8 , and G\"(0) = 72. We thus find K = 1,\nTar = 8, T = 2 and L = 4. This is the same model as the one obtained\nin Example 2.8.\n•\nOther Input Signals\nFrom a practical point of view it is a drawback to have methods that\nrequire special input signals. The method of moments can be applied\nto any signal provided that the system is initially at rest.\n\n2.5 Methods of Moments\n\n31\n\nLet U(s) and Y(s) be the Laplace transforms of an arbitrary input\nand the corresponding output, respectively. Taking derivatives we get\nY(s) = G(s)U(s)\nY'(s) = G'(s)U{s)+G(s)U'(s)\nY\"{s) = G\"(s)U(s) + 2G'(s)U'(s) + G(s)U\"(s)\netc.\nHence,\nY(0) = G(O)t/(O)\nF'(0) = G'(O)C7(O) + G(O)t/'(O)\nY\"(0) = G\"(O)C7(O) ¥ 2G'(0)f/'(0) + G(O)f/\"(O)\netc.\n\n(2 29)\n\nThe transfer function G(0) and its derivatives can thus be calculated\nfrom experiments with arbitrary inputs by calculating the following\nmoments of the input and output\noo\n\nC/(«)(0) = (-1)\" f tnu(t)dt\n0\noo\n\n7 w (0) = (-l) n ftny(t)dt\nand using Equation (2.29).\nBy using these formulas it is possible to calculate G^(0) for any\nsignals for which the moments\noo\n\nun = I tnu{t)dt\n0\n\nand\noo\n\nyn =\n\nftny(t)dt\n\nexist. This means that the signals must decay sufficiently fast.\nA typical case where the method can be used is when an exper¬\niment is performed in a closed loop with a pulse-like perturbation\nsignal on the process input.\n\n32\n\nChapter 2 Process Models\n\nWeighted Moments\nThe method just discussed cannot be used if the signals do not go\nto zero or, equivalently, to a priori known mean values that can be\nsubtracted in the calculations of moments, because the moments will\nthen be infinite. There is, however, a simple modification that can\nbe used in this case. It follows from the definition of the Laplace\ntransform that\noo\n\nFW(S)\n\n1\n\n= (- )\" /e-stt\"y(t)dt\n\n= ^Sr-\n\no\nThe weighted moments\noo\n\nyn = Jtne-aty(t)dt=(-l)\"Y^(a)\no\nwill exist provided that y(t) does not grow faster than eat for large\nt. By computing yn and the analogously defined moment un, we can\ncompute YW(a) and £/(\">(«), and thus also G' n '(a).\nA Three-Parameter Model\nConsider a system with the transfer function\n\nG{s) =\n\nihfe'sL\n\n(2 30)\n\n-\n\nWe have\nlogG(s) = logK -sL-\n\nlog (1 + sT)\n\nHence\nG'(s)\nG(s)\n\n= L\n\nT\n1 + sT\n\nG\"(s) _ (G\\s)\\2\nG(s)\n\n_\n\nT2\n(1 + sT)\n\nThus we get\n{1 + aT)\n\n2\n\ni\n\nG\"(a)\nG(a)\n\n(G'(a)\\2\n\n[G(a)J\n\nHence,\n1 - aa\nG'(a)\n\n,.2\n\n(2.31)\n\n2.5 Methods of Moments\n\n33\n\nThe average residence time thus becomes\nG(a)\n\n1 - aa\n\nFurthermore the static gain is given by\nK = (1 + aT)G(a)eaL\n\n(2.33)\n\nThe formulas are illustrated by an example.\nEXAMPLE 2.11\n\nConsider a system with the transfer function\nG(s) =\n\n1\n(s + I) 8\n\nWe have\nG a\n\n( )\n\nG\n\n'(va)\n\n= (1\n,-, +* a)\nM8I8\n\n(1 + a)\n\n'\n\n=\n\n^ 9\n(1+a)\n\nv\n\n'\n\n(1 + a) 10\n\nComputing the derivatives at the origin from the first terms in the\nTaylor series expansion gives\n1\n\nG(0)\nG'(0) :\n\n1\n\nQ^v\n\n8\n\ni\n\nO.™\n\n9\n\na)\n(1 + a)\n(1 + a) 9\n8\n72a\n8(1 + 10a)\n(1 + a) 9 (1 + a) 10\n(1 + a) 10\n\nThe estimate of the average residence time becomes\nTnr= -\n\nG'(0)\nG(0)\n\n8(1+10a)\n(l + a)(l + 9a)\n\n8(1+10a)\n1 + 10a + 9a 2\n\nFrom these expressions it follows that a must be small in order to\ngive reasonably good approximations. To discuss the values of a, it is\nreasonable to normalize and consider aTar. In this case, Tar = 8. With\naTar = 1 we get G(0) = 0.74, G'(0) = -5.54, and tar = 7.53. With\naTar = 0.5 we get G(0) = 0.91, G'(0) = -7.1, and far = 7.83, giving\nerrors in the range of 10%. With aTar = 0.2 we get G(0) = 0.98,\nG'(0) = -7.81, and far = 7.96.\nIt follows from Equation (2.31) that\n72\n64\n2V2\n+ a)2 ~ (1 + a) 2 ~ 1 + a\n\n34\n\nChapter 2 Process Models\n\nIt follows from Equations (2.32) and (2.33) that\nT =\n\n1 + (1 - 2y/2)a\n\nL = —\nK\n\ne\n7\n\n(l + a) (l + ( l - 2 \\ / 2 ) a )\nThe average residence time becomes\n1 + (2 - 2y/2)a\nTar = T + L = 8\n1 + (2 - 2\\/2)a + (1 - 2v/2)a2\nWith aTar = 1, 0.5, and 0.2, we get the estimates far = 8.26,\ntar = 8.06, and far - 8.01, respectively. This method of estimat¬\ning the average residence time gives slightly better results than the\nextrapolation method.\n•\nThe example shows that we can obtain reasonable estimates\nof the model parameters and the average residence time by using\nweighted moments. It also seems reasonable to choose parameter a\nso that ccTar is in the range of 0.2 to 1. The best results are obtained\nfor a small value of a. There is, however, an advantage in using larger\nvalues of a because there is then a less risk for disturbances to enter\nthe system.\n\n2.6\n\nFrequency Responses\n\nTwo methods for determining interesting points on the Nyquist curve\nare presented below. Both are based on the idea of using feedback to\ngenerate sinusoids having the appropriate frequency.\n\nThe Ziegler-Nichols Frequency Response Method\nZiegler and Nichols have provided a method for determining the\nultimate point on the Nyquist curve experimentally. The method is\nbased on the observation that many systems can be made unstable\nunder proportional feedback by choosing sufficiently high gain in the\nproportional feedback (see Figure 2.17). Assume that the gain is\nadjusted so that the process is at the stability boundary. The control\nsignal and the process output are then sinusoids with a phase shift\n\n2.6 Frequency Responses\n\n35\n\nK = 0.5\n\n15\n\n20\n\n10\n\n15\n\n20\n\n10\n\n15\n\n20\n\nK = 2.0\nysP\n\n5\n\nK = 2.5\ny\n\nysP\n\n/\n\nFigure 2.17 Setpoint ysp and process output y for a closed-loop\nsystem with proportional feedback. The figure shows responses for\nthree values of controller gain K.\n\nof -180° (see Figure 2.18). Because of the proportional feedback they\nare related by\nu = -Ky\nFor simplicity it has been assumed that the setpoint is ysp = 0. Since\nthe gain around the loop must be unity to maintain an oscillation, we\nhave\nKuG{itau) = - 1\n\nwhere the gain, which brings the system to the stability limit, is called\nController\nysp\n\n~N\n\ne\n\nK\n\nProcess\nu\n\nG(s)\n\ny\n\n-1\nFigure 2.18 Block diagram of a closed-loop system under propor¬\ntional feedback.\n\n36\n\nChapter 2 Process Models\nTable 2.1 Relations between gain ratio K and normalized dead\ntime x for processes with the transfer functions G(s) = l/(s + 1)\".\n\nr\n\n0.15\n\n0.25\n\n0.35\n\n0.55\n\nK\n\n0\n\n0.125\n\n0.25\n\n0.53\n\nthe ultimate gain (Ku). It follows from the above equation that\nG(iatu) = ~\n\n(2.34)\n\nSeveral design methods based only on the knowledge of G(ia>u) are\ngiven in Chapter 4. It is convenient to introduce the gain ratio,\n\nr- ^ f\n\n(2-35)\n\ni.e., the gain at the ultimate frequency divided by the static gain. This\nparameter is an indicator of how difficult it is to control the process.\nProcesses with a small K are easy to control. The difficulty increases\nwith increasing K.\nParameter K is also related to the normalized dead time r, which\nwas defined in Equation (2.9). For processes described by the transfer\nfunction (2.8) parameters r and K are related in the following way:\n\nit - arctan \\/l/ic2 - 1\n~ n- arctan S/I/K2 - 1 + y/l/ic2 - 1\nThis relation is close to linear, it gives r = 0 for K - 0 and r = 1\nfor K = 1. For small values of K it can be approximated by r = 1.6ic.\nThis is illustrated in the following example.\nEXAMPLE 2.12\nTo illustrate the relation between the parameters K and t, we give\ntheir values for systems with the transfer functions\n\nThe results are presented in Table 2.1. For small values of n, both K\nand T are small. These processes are easy to control. For large values\nof n, both K and x approach 1. These processes are difficult to control.\n\n•\nThe Ziegler-Nichols frequency response method has some advan¬\ntages. It is based on a simple experiment, and the process itself is\n\n2.6 Frequency Responses 37\nRelay\n\nProcess\n\nFigure 2.19 Block diagram of a process under relay feedback.\nused to find the ultimate frequency. It is, however, difficult to auto¬\nmate this experiment or perform it in such a way that the amplitude\nof the oscillation is kept under control. Operating the process near in¬\nstability is also dangerous and may need management authorization\nin an industrial plant. It is difficult to use this method for automatic\ntuning. An alternative method for automatic determination of specific\npoints on the Nyquist curve is suggested below.\n\nRelay Feedback\nAn alternative method to determine interesting points on the Nyquist\ncurve is based on the observation that the appropriate oscillation\ncan be generated by relay feedback. The system is thus connected\nas shown in Figure 2.19. For many systems there will then be an\noscillation (as shown in Figure 2.20) where the control signal is a\nsquare wave and the process output is close to a sinusoid. Notice\nthat the process input and output have opposite phase.\nTo explain how the system works, assume that the relay output is\nexpanded in a Fourier series and that the process attenuates higher\n\n-2\n\nu\n\n10\n\n20\n\n30\n\n40\n\n10\n\n20\n\n30\n\n40\n\n\\\n\n-1\n\nFigure 2.20 Relay output u and process output y for a system\nunder relay feedback.\n\n38\n\nChapter 2 Process Models\n\nharmonics effectively. It is then sufficient to consider the first har¬\nmonic component of the input only. The input and the output then\nhave opposite phase, which means that the frequency of the oscilla¬\ntion is the ultimate frequency. If d is the relay amplitude, the first\nharmonic of the square wave has amplitude 4d/n. Let a be the am¬\nplitude of the oscillation in the process output. Then,\n(2.36)\n\nNotice that the relay experiment is easily automated. Since the am¬\nplitude of the oscillation is proportional to the relay output, it is easy\nto control it by adjusting the relay output. Also notice in Figure 2.20\nthat a stable oscillation is established very quickly. The amplitude\nand the period can be determined after about 20 s only, in spite of\nthe fact that the system is started so far from the equilibrium that it\ntakes about 8 s to reach the correct level. The average residence time\nof the system is 12 s, which means that it would take about 40 s for\na step response to reach steady state.\nDescribing Function Analysis\nThe intuitive discussion about relay oscillations can be dealt with\nmore quantitatively using a technique called the describing function\nmethod. This is an approximate method that can be used to deter¬\nmine if there will be an oscillation in a nonlinear feedback system\nthat is composed of a linear element and a static nonlinearity. To\ndetermine conditions for oscillation, the nonlinear block is described\nby a gain, N(a), which depends on signal amplitude a at the in¬\nput of the nonlinearity. This gain, which describes how a sinusoid of\namplitude a propagates through the system, is called the describing\nfunction. If the process has the transfer function G{ico), the condition\ni i Im Giico)\n\nRe G(i(o)\nyquist curve G(im)\nDescribing function -\n\nN(a)\n\nFigure 2.21 Determination of possible oscillations using the de¬\nscribing function method.\n\n2.6\n\nFrequency Responses\n\n39\n\nfor oscillation is simply given by\nN(a)G(i(o) = - 1\n\n(2.37)\n\nThis equation is obtained by requiring that a sine wave with fre¬\nquency co should propagate around the feedback loop with the same\namplitude and phase. The equation gives two equations for determin¬\ning a and co, since N and G may be complex numbers. The equation\ncan be solved graphically by plotting -l/N(a) in the Nyquist dia¬\ngram (as in Figure 2.21) together with the Nyquist curve G(ia>) of\nthe linear system. An oscillation may occur if there is an intersec¬\ntion between the two curves. The amplitude and the frequency of the\noscillation are the same as the parameters of the two curves at the\nintersection point. Therefore, measuring the amplitude and the pe¬\nriod of the oscillation, the position of one point of the Nyquist curve\ncan be determined.\nThe describing function, N(a), for a relay is given by\nN(a) =\n\n^\n\n(2.38)\n\nSince this function is real, an oscillation may occur if the Nyquist\ncurve intersects the negative real axis. This explains why the exper¬\niment with relay feedback gives the point where the Nyquist; curve\nintersects the negative real axis.\nA Relay with Hysteresis\nThere are advantages in having a relay with hysteresis instead of a\npure relay. With an ordinary relay, a small amount of noise can make\nthe relay switch randomly. By introducing hysteresis, the noise must\nbe larger than the hysteresis width to make the relay switch. See Fig¬\nure 2.22. The describing function approach will be used to investigate\n\nd\n1\n\nt\n\nu\n\nz\n\nFigure 2.22 Output y from a relay with hysteresis with input u.\n\n40\n\nChapter 2 Process Models\n\nN(a)\n\nFigure 2.23 The negative reciprocal of the describing function\nN(a) for a relay with hysteresis.\n\nthe oscillations obtained. The negative inverse of the describing func¬\ntion of such a relay is\nN(a)\n\n(2.39)\n\nwhere d is the relay amplitude and e is the hysteresis width. This\nfunction can be represented as a straight line parallel to the real axis,\nin the complex plane (see Figure 2.23).\nBy choosing the relation between e and d, it is therefore possible\nto determine a point on the Nyquist curve with a specified imaginary\npart. Several points on the Nyquist curve can be obtained by repeat¬\ning the experiment with different relations between £ and d. It is\neasy to control the amplitude of the limit cycle to a desired level by\na proper choice of the relay amplitude.\nOther Uses of Relay Feedback\nA slight modification of the experiment shown in Figure 2.19 gives\nother frequencies of interest. Figure 2.24 shows an experiment that\ngives the frequency ago, i.e. the frequency where the process has a\nphase lag of 90°. Notice that there are two different versions of the\nexperiment depending on the order in which the integrator and the\nrelay are connected.\nClosed Loop Experiments\nRelay feedback can also be applied to closed-loop systems. Figure 2.25\nshows an experiment that can be used to determine the amplitude\nmargin on-line. Let G( be the loop transfer function, i.e., the combined\ntransfer function of the controller and the process. The closed-loop\n\n2.6 Frequency Responses\ns\n\n* * •\n\n41\n\nProcess —\n\nj\n\n-1\n\nFigure 2.24 Using relay feedback to determine thefrequencya>90.\n\ntransfer function is then\nGt{s)\n1 + G,(s\n\n(2.40)\n\nThe experiment with relay feedback then gives an oscillation with the\nfrequency such that the phase lag of Gci{i(o) is 180°. It then follows\nfrom Equation (2.40) that this is also thefrequencywhere Ge(iO)) has\na phase lag of 180°, i.e., the ultimatefrequency.If m is the magnitude\nof Gci at that frequency, we find that an estimate of the amplitude\nmargin of the closed-loop system is given by\nt\n\nm\n\n1-m\nIf the relay has hysteresis, a conformal mapping argument shows that\nthe experiment gives thefrequency,where the loop transfer function\n\nFigure 2.25 Using relay feedback to determine the amplitude\nmargin of the closed-loop system.\n\n42\n\nChapter 2 Process Models\n\nFigure 2.26 Experiments with relay feedback give the points\nwhere the curve Gi(ico) intersects the circles.\n\nintersects part of the circle,\nGt{ico)-l\n\n+\n\ni -\n\n1\n2a\n\nwhich is shown as curve A in in Figure 2.26. By introducing an\nintegrator in series with the relay, the frequency where Gci(ia>) has\na phase lag of 90° is obtained. This occurs for loop transfer functions\nGe with the property\narg 1 +\n\nLr(\n\n= argGt - arg(l +\n\nZ\n\nThis corresponds to the circle,\n1\n(2.41)\n2\nwhich is shown as curve B in Figure 2.26. The experiment will thus\ngive the point where the loop transfer function Gi of the closed-loop\nsystem intersects the circle given by Equation (2.41). Combining this\nresult with the result from the experiment in Figure 2.24, it is also\npossible to approximately determine the maximum sensitivity Ms.\nMany controllers use a two-degree-of-freedom configuration in¬\nstead of pure error feedback. This is discussed in Chapter 3. This\nmeans that the control law is given by\nGe(ico) + -\n\nU(s) = Gfr(s)Ysp{s)-Gfb(s)Y{s)\nThe experiment shown in Figure 2.25 must then be modified by\nintroducing a block with the transfer function Gft,jGff in series with\nthe relay.\nIt has thus been demonstrated that several of the quantities\nneeded to make an assessment of control performance can be obtained\nfrom experiments with relay feedback.\n\n2.7 Parameter Estimation . 43\n\n2.7\n\nParameter Estimation\n\nA mathematical model of the process can also be obtained by fitting\ntheparameters of a model to experimental data. For example, a model\nof the type given by Equation 2.8 can be obtained by adjusting the\nparameters so that they match observed input/output data. The ad¬\nvantage of such an approach is that any type of input/output data can\nbe used. However, parameter estimation requires more computations\nthan the methods discussed previously.\n\nParametric Models\nSince the calculations will typically be made using a digital computer,\nthe input/output data will typically be sampled. It is then convenient\nto operate with a discrete time model based on signals that are sam¬\npled periodically. Moreover, if the experimental data is also computergenerated, it is reasonable to assume that the input to the process is\nconstant between the sampling instants. Let the sampling period be\nh. Assume that time delay L is less than h. The model (2.8) can then\nbe described as\ny(kh) = ay(kh - h) + blU(kh - h) + b2u(kh - 2h)\nwhere\n\n(2.42)\n\na = e-h'T\n\nb2 = Ke-h'T (eL'T\n\n-\n\nFor arbitrary time delays L, the model becomes instead\n\ny(kh) = ay(kh - h) + biu(kh - nh) + b2u{kh -nh-h)\n\n(2.43)\n\nwhere parameters a, b\\, and b2 are given as above with n = Ldivh\nand r = L mod h replacing L. The model can be given a convenient\nrepresentation by introducing a shift operator q, defined by\n\nqy(kh) =\nThe model (2.43) can then be written as\nqn(q - a)y{kh) = (blQ + b2)u{kh)\nIf the complex variable z (similar to the Laplace transform variable s)\nis introduced, the process can also be described by the pulse transfer\nfunction:\n\n44\n\nChapter 2 Process Models\n\nNotice that the transfer function is a ratio of two polynomials even if\nthe corresponding physical process has time delays.\nThe discussion can be extended to systems of higher order, and\nthe result is then an input/output relation of the form:\ny{kh) + a\\y{kh - h) + • • • + any(kh - nh)\n= biu{kh - h) + ••• + bnu(kh - nh)\nThis equation can be written compactly as\nA(q)y(kh) = B(q)u(kh)\n\n(2.45)\n\nwhere A(q) and B(q) are polynomials:\n\nB(q) = b1qn-1 + b2qn-2 + The corresponding transfer function is then\nH{z) =\n\nB{z)\nA(z)\n\nb1zn-1 + b2zn-\n\nParameter Estimation\nThere are many ways to estimate the parameters of the discrete time\nmodel (Equation 2.45). A simple method is as follows. Assume that a\nsequence of input/output pairs ({u(kh),y(kh),k = 1,2, ...,2V}) have\nbeen observed. The parameters can then be determined in such a\nway that Equation (2.45) fits the data as well as possible in the least\nsquares sense. The sum of the squares of the errors is\nN\n\nV{6) = J2 e2(kh)\n\n( 2 - 46 )\n\nk=n + l\n\nwhere\ne{kh + nh) = A(q)y(kh)-\n\nB(q)u(kh),\n\nk =\n\nl,---,N-n\n\nNotice that the error is linear in parameters a, and 6; of the model\nand that the sum of squares of the errors is a quadratic function.\nThis means that the minimum of the loss function can be computed\nanalytically. Rather than showing the solution to the optimization\nproblem, a convenient way of computing the parameters recursively\nis presented below.\n\n2.7 Parameter Estimation\n\n45\n\nRecursive Computations\nIn a tuning experiment, a new input/output pair is normally obtained\nin each sampling. It is then convenient to compute the parameter\nestimates recursively. All parameters are grouped together in the\nvector:\nIntroduce the regression vector denned by\n<Pk-i = (-y(kh-\n\nh)... - y(kh - nh) u(kh- h)...u(kh - nh))T\n\nThe estimate can then be calculated recursively by\ne* = y{kh) - <Pk-i6k-i\n\n(2.47A)\nP\n\n_ p\nPk-i(Pk-i<Pk-i k-i\np\n*A — \"k-\\ —Z\nf—p\n\n,„ . „ „ .\n(zAtiS)\n\nOk = 0k-i + Pk<Pk-iek\n\n(2.47C)\n\nThese equations have good physical interpretations. The new esti¬\nmate Ok is obtained by adding a correction term P<pe to the old es¬\ntimate &k-i- Th e correction term is a product of three quantities: P,\nq>, and e. The error e is the difference between the last measurement\ny(kh) and the prediction <pT6 of this measurement based on old esti¬\nmates. Regression vector <p can be interpreted as the gradient of the\nerror with respect to the parameters. This vector tells how the scalar\nerror is distributed to give corrections in all parameters.\nEquation (2.47B) may be interpreted as follows. Matrix P^ is\nproportional to the covariance matrix of the estimates; the last term\nin Equation (2.47B) is the reduction in uncertainty due to the last\nmeasurement.\nThe equations have to be initialized. The initial value of param¬\neter vector 6 can be chosen as the best initial guesses of the parame¬\nters. The initial value of matrix P is typically chosen as the identity\nmatrix multiplied by a large number.\nComputer Code\nRecursive least squares estimation is an essential part of many\nschemes for automatic tuning. The following is a computer code that\nimplements the algorithm.\n{The recursive least squares algorithm}\ne=y\nfor i=l to 2*n do e=e-^> [i]*#[i]\n{Compute estimator gain}\nfor i=l to 2*n do\n\n46\n\nChapter 2 Process Models\nbegin\ns=0\nd=l\nfor j=l to 2*n do\nbegin\ns=s+P[i,j]*<p[j]\nd=d+s*«p[j]\nend\nr[i]=s\nend\n{Update estimates}\nfor i=l to 2*n do 0[i]=0[i]+r[i]*e/d\n{Update P matrix}\nfor i=l to 2*n do\nbegin\nfor j=i to 2*n do P[i, j]=P[i, j ] - r [i] *r [j]/d\nfor j=i+l to 2*n do P[j ,i]=P[i, j]\nend\n{Update ^-vector}\nfor i=l to 2*n-l do g> [2*n-i+l]=#> [2*n-i]\n<p[l]=-y\n\nThe code description is given in \"pidgin\" Pascal, and it is assumed\nthat the variables have been properly declared. There are many re¬\nfinements to the algorithm; for instance, its numerical properties can\nbe improved by using a so-called square root algorithm. It is also com¬\nmon practice to bandpass filter the signals before introducing them\ninto the algorithm to get rid of static levels and high frequency distur¬\nbances. There are also many variations of the algorithm to discount\npast data. The code gives an indication of the type of algorithms that\nare used in recursive parameter estimation.\n\n2.8\n\nDisturbance Models\n\nSo far, we have only discussed modeling of process dynamics. Dis¬\nturbances is another important side of the control problem. In fact,\nwithout disturbances and process uncertainty there would be no need\nfor feedback. There is a special branch of control, stochastic control\ntheory, that deals explicitly with disturbances. This has had little\nimpact on tuning and design of PID controllers. For PID control, dis¬\nturbances have mostly been considered indirectly, e.g., by introducing\nintegral action. As our ambitions increase and we strive for control\nsystems with improved performances it will be useful to consider dis¬\nturbances explicitly. In this section, therefore, we will present some\nmodels that can be used for this purpose.\n\n2.8\nA\n\nDisturbance Models\n\n47\n\nB\n\n0.8\n0.4 ^\n0\n0\n\nD\n0\n-2\n\n0\n\n1\n\n2\n\n3\n\n4\n\n5\n\n0\n\n1\n\n2\n\n3\n\n4\n\n5\n\nFigure 2.27 Prototype disturbances, A impulse, B step, C ramp,\nand D sinusoid.\n\nThere are some fundamental problems in dealing with distur¬\nbances. An inherent property of disturbances is that they cannot be\npredicted exactly. Most mathematical models, however, do have the\nproperty that they give signals that can be predicted exactly. Some\ncare must be exercised when interpreting models and results.\nSimple Models\nSimple mathematical models were found to be very useful when deal¬\ning with process dynamics in the previous sections. We will now try\nto make models that also can be used to characterize disturbances.\nExamples of some simple disturbances—impulse, step, ramp, and\nsinusoid—are given in Figure 2.27. The impulse is a mathematical\nidealization of a pulse whose duration is short in comparison with\nthe time scale. The signals are essentially deterministic. The only\nuncertain elements in the impulse, step, and ramp are the times\nwhen they start and the signal amplitude. The uncertain elements of\nthe sinusoid are frequency, amplitude, and phase.\nMore complex disturbances can be obtained by combining the\nsimple disturbances, as shown in Figure 2.28. These disturbances are\nobtained by repeating a number of impulses, by combining steps and\nramps, and by changing the amplitude and phase of the sinusoid.\nNoise\nThere are also other types of disturbances that are much more ir¬\nregular than the signals shown in Figure 2.28. Some examples are\ngiven in Figure 2.29. To characterize signals like the ones shown\n\n48\n\nChapter 2 Process Models\n\nA\n\nB\n1\n\n0\n\n1\n\n-4()\n\n5\n\n,—1\n\n0\n\n10\n\n15\n\n-A()\n\n20\n\n5\n\n10\n\n15\n\nD\n\nc\n\n20\n\n1\n\n10-\n\n0/\n\n\" \\\n\n-10\n\n010\n\n15\n\n20\n\nmif m\n10\n\n15\n\n20\n\nFigure 2.28 Disturbances that are obtained by combining the\nsimple prototype disturbances.\n\nin Figure 2.29, it is necessary to describe both the amplitude and\nthe time characteristics. A distinction between stationary and nonstationary behavior must first be made. A signal is stationary if its\nbehavior is essentially the same for all times. The amplitude prop¬\nerties of a stationary signal can be described by giving a histogram\nthat tells the fraction of time when the signal has a given ampli¬\ntude.\nThe mean value, the standard deviation or the variance, and the\npeak-to-peak values are simple ways to characterize the amplitude\ndistribution. If the amplitude distribution is normal, the distribution\nis uniquely given by the mean value m and the standard deviation\na. The probability for the signal to be outside the 3<7 limits is about\n0.0026.\nThe time behavior of a stationary signal can be described by\nthe spectral density function <p{(o). This function characterizes the\nfrequency content of a signal. The value\n— <p{co)Aa>\n\nis the average energy of a signal in a narrow band of width Aco\ncentered around a. The average energy is then\noo\n\n=\n\n2^7\n\n<p{co)do)\n\nA signal where <j> {a) is constant is called white noise. Such a signal\nhas its energy equally distributed among all frequencies.\n\n2.8 Disturbance Models\n\n49\n\nB\n0\n\n0\n-2\n\n-0.4\n\n0\n\n10\n\n20\n\n30\n\n40\n\n0\n\n10\n\n20\n\n30\n\n40\n\n0\n\n10\n\n20\n\n30\n\n40\n\nD\n\n-1\n\n0\n\n10\n\n20\n\n30\n\n40\n\nFigure 2.29 Examples of noise signals.\n\nMeasuring Noise Characteristics\nThe noise characteristics can be determined in several different ways.\nThere are simple methods that can be used for crude estimates and\nmore sophisticated methods that give more precise descriptions.\nA simple way to estimate the amplitude characteristics is to mea¬\nsure the average\nm\n\n=^J\n\nand the mean square error\n\nTo evaluate the integrals it is necessary to know a reasonable value\nof T, which requires knowledge about the time scale. An alternative\nis to compute the peak-to-peak value ypp. The standard deviation can\nthen be estimated as\nNotice that it is also necessary to know the time scales in order to\ndetermine the time interval over which the peak-to-peak value is\ncomputed.\nThe energy in a given frequency band can be determined by\ncomputing the mean square average or the peak-to-peak value of a\nfiltered signal.\nUseful information about the frequency content in a signal can\nalso be determined from the zero crossings. For a stationary signal\n\n.50\n\nChapter 2 Process Models\n\nthe average number of zero crossings per second can be determined\nfrom\nco2(p(co)dco\n\nN=±\n\n1/2\n\n( J\nj\n'! j\n|\nI\n!\n'1\n\nwhere <f>(co) is t h e spectral density. Notice t h a t this formula h a s\nsimilarities with t h e formula for determining t h e average residence\ntime, Equation (2.24).\nFor a spectral density t h a t is uniform over t h e interval (a>i,0)2),\nwe get\n'3 N\\V1 2/ 2 _ 11/O)\\\n_ 1 / co\\ - w\\\nrco\\++O)iO)2\ncoia)2+ + O)\\\\m\n(o\\\\\n~ K\nn \\Z(O)2\n{3((0o-co,)\n~ COi)I)\n~ n\\\n3\n/\n\nFor an ideal low-pass filter we have O)i = 0 and 0)2 = coi,, where O)f,\nis the band width. In this case we get\n\nThe average rate of zero crossings per second is thus approximately\nequal to the bandwidth measured in Hz. Measurement of zero cross¬\nings can easily be combined with computation of the peak-to-peak\nvalue. More accurate determination of the spectral characteristics\ncan be done by using a spectral analyzer or by recording a data set\nand computing the spectrum numerically.\n\nRepresentation of Disturbances\nIt is often very convenient to consider signals as generated from a\ndynamic system with simple inputs as shown in Figure 2.30. For\nexample, the signals shown in Figure 2.27 can be represented by\nsending an impulse through a dynamic system. The transfer function\nof the systems for the different signals are\nImpulse G(s) = 1\nStep\n\nG(s) = s\n\nRamp\n\nG(s) = -?\n\nSinusoid G(s) =\n\na2\nCO2\n\n2.9 Approximate Models and Unmodeled Dynamics\nsimple signal\n\n51\n\nsignal\n\nFigure 2.30 Signals represented as outputs of dynamic systems.\nSimilarly the signals in Figure 2.28 can be generated by sending a\nsequence of pulses through the same systems.\nThe same idea can be extended to describe noise. In this case the\ngenerating signal is white noise. We illustrate the idea.\nEXAMPLE 2.13\n\nThe so called RC noise has the spectral density\n\nIt can be represented by sending white noise through a system with\nthe transfer function\nO(s) = -5Ls +a\n\n•\n\nThe possibility of representing signals in this way also gives a pos¬\nsibility of dealing with nonstationary signals. The process obtained by\nsending white noise through an integrator, for example, is a drifting\nprocess that is called a random walk or a Wiener process.\nThe representation of signals in this way also makes it possible\nto see similarities between signals of different type. It turns out\nthat signals that are generated from the same system have many\nsimilarities. For example, a step signal (Figure 2.27B), a piece-wise\nconstant signal (Figure 2.28B), and a random walk are all generated\nby sending primitive signals through an integrator. The primitive\nsignals are an impulse, a sequence of impulses for the piece-wise\nconstant signals, or white noise for a random walk. A consequence of\nthis is that a controller that is designed to work well for one of these\nsignals will work well for all of them. A step disturbance is thus not\nas special as it may first appear.\n\n2.9\n\nApproximate Models and Unmodeled\nDynamics\n\nIn this chapter, we have discussed various ways to model the process\nto be controlled. We end the chapter with a discussion about what is\nnot captured by the models. Typical examples are nonlinearities and\n\n52\n\nChapter 2\n\nProcess Models\n\nprocess variations. Furthermore, the parametric low-order models\ngive a satisfactory description of the behavior of the true system for\nsignals with a limited frequency range only.\nThe process models describe the relation between the process in¬\nput signal and the process output signal only, but the control system\nconsists of other signals that influence the control performance. The\ncharacteristics of the setpoint changes, load disturbances, and mea¬\nsurement noise must also be taken into account in the design of the\ncontrol system.\nMany design methods for PID controllers require that the process\nmodel be of low order. Some methods to approximate higher-order pro¬\ncess models with low-order models are also presented in this section.\nNonlinearities\nAll dynamic models presented in this chapter are linear, whereas\nmost processes in practice are nonlinear. Nonlinear valves, actuators\nor sensors result in the process having different dynamics at different\noperating points. A linear model, obtained by transient or frequency\nresponse analysis of a nonlinear process, is only valid at the actual\noperating point. This means that a controller that is tuned based on\nthis model may work well only at this operating point.\nThere are several ways to overcome the problem. A simple way\nis to tune the controller for the worst case and accept degraded\nperformance at other operating conditions. If the characteristics of\nthe nonlinearity are known, it can be compensated by feeding signals\nthrough a function module that forms the inverse of the nonlinearity.\nAn example is a flow meter based on measurement of difference\npressure. The flow is proportional to the square of the difference\npressure. A linear relation between the flow and the output signal\nfrom the flow meter can be obtained by feeding the measured signal\nthrough a square root function.\nAnother way to compensate for nonlinearities is to divide the op¬\nerating range into several smaller ranges where the process can be\nwell approximated by linear models. A controller with satisfactory\nbehavior for the full operating range can be obtained by determining\none model for each operating range and changing the controller pa¬\nrameters with the operating condition. This approach is called gain\nscheduling.\n\nParametric Models\nIf the process is linear, a step response reveals all information about\nthe process dynamics. In the same way, a Nyquist curve or frequency\n\n2.9 Approximate Models and Unmodeled Dynamics\n\n53\n\nresponse gives a complete description of the process dynamics. In¬\nformation, is lost when going from these graphical descriptions to\nparametric models. The parametric models derived for PID controller\ntuning are normally of low order. This means that quite a lot of in¬\nformation is lost. It is, therefore, particularly important that these\nsimple models are derived properly and that their limitations are\nkept in mind when using them for controller tuning.\nThe parametric models based on step response analysis are often\naccurate at low frequencies, whereas they become more uncertain at\nhigher frequencies. The simple models based on frequency response\nanalysis, described in Section 2.6, are accurate at the frequencies of\nthe input signals, but not at other frequencies. The basic relay method\nthus is accurate around the ultimate frequency (ou, but not for higher\nand lower frequencies.\n\nProcess Variations\nThe model is valid at the time the experiment is performed. If the\nprocess dynamics change with time, it may not be valid at a later time.\nThis problem can be handled in the same way as the nonlinearities\ndescribed above.\nA simple solution is to base the controller tuning on the model\nthat describes the worst case. Gain scheduling can be used if the time\nvariations can be related to some measurable variable. Adaptive con¬\ntrol can be used if the process variations are random in the sense that\nthey cannot be related to any measurable variable. Such a controller\nwill adapt itself automatically to the actual process dynamics.\n\nDisturbances\nThere are always disturbances acting on a control system. We distin¬\nguish between three types of disturbances, namely, setpoint changes\n(ySp), load disturbances (I), and measurement noise (n) (see Figure\n2.31).\nSetpoint Changes\nIn process control, most control loops have a constant setpoint. (An\nexception is the controller sitting in the inner loop in cascade con¬\ntrol.) The setpoint may change at certain time instances because of\ndesires to change operating conditions such as production rates. The\nsetpoint is, as a result, typically piece-wise constant with changes\noccurring rarely. It is, therefore, suitable to model the setpoint as a\nstep function.\n\n54\n\nChapter 2 Process Models\n\nSince the setpoint is a disturbance that we have access to, it\nis possible to feed it through a low-pass filter or a ramping module\nbefore it enters the PID controller. In this way, the step function can\nbe made smoother. This property is useful, since many control design\nmethods giving a good rejection of load disturbances give too large\novershoots after a sudden change in the setpoint.\nLoad disturbances are disturbances that enters the control loop some¬\nwhere in the process and drive the system awayfromits desired oper¬\nating point. They may be caused by quality variations in a feed flow\nor variations in the demand flow, for example. These disturbances\nare the most common and the most important disturbances in pro¬\ntherefore, focus on the behavior with respect to these disturbances.\nThe load disturbance is typically a low-frequency disturbance,\nand it will, furthermore, be more or less low-pass filtered by the\nprocess depending on where in the process it enters. Consequently, it\nusually results in a low-frequency disturbance in the process output.\nTo obtain this characteristic in the process output, we model the\nload disturbance as a step function added to the control signal at\nthe process input (see Figure 2.31).\nMeasurement Noise\nMeasurement noise represents disturbances that distort the informa¬\ntion about the process variables obtained from the sensors. Measure¬\nment noise may be of different character. It may be high-frequency\nfluctuations and it may be low-frequency calibration errors. With sev¬\neral sensors it is possible to reduce calibration errors. With only one\nsensor nothing can be done about calibration errors; we, therefore,\n\nFigure 2.31 Block diagram of a simple feedback loop with three\ntypes of disturbances: Setpoint changes (ysp), load disturbances (I)\nand measurement noise (n).\n\n2.9 Approximate Models and Unmodeled Dynamics\n\n55\n\nwill model measurement noise as a high-frequency signal added to\nthe process output.\nSince measurement noise does not contain any information about\nthe status of the process, it should be filtered out. Furthermore, highfrequency components in the measurement signal might be amplified\nby the controller and cause wear on the actuator. Filtering does intro¬\nduce additional dynamics. It is therefore important to take the filter\ndynamics into account in the controlling design. We will model the\nmeasurement noise as an impulse function.\n\nApproximating Complex Models\nIn modeling it is often convenient to split a system into intercon¬\nnected subsystems. An example could be to divide a system into ac¬\ntuator, process, and sensor. Another example occurs when general\ncontrol loops are cascaded. We may be interested then in obtaining a\nsimplified description of the closed loop. Even if the model for each\npart is simple, the complete model may then be quite complex. Since\nmany of the design methods for PID controllers are based on simple\nmodels, we need a procedure to simplify a complex model. Some ways\nto make such approximations are discussed below.\nTb perform the approximations it is necessary to determine the\nfrequency range where the approximation should be valid. We do this\nsimply by saying that the models should describe the system well\naround the frequency (Oo- This frequency should be approximately\nthe same as the frequency of the dominant closed-loop poles of the\ndesired system or the desired bandwidth of the closed-loop system.\n(The notion of dominant poles is discussed in Chapter 4.) Having\nrestricted the modeling to a rather narrowfrequencyrange, low-order\nmodels can now be determined by fitting them to experimental data,\nas described previously in this section.\nAnother possibility is to start with a complex model of the form\nG p ( .)\n\nK\n\ne\n\n1 + ens + a,2Sz + • • • + ansn\nand approximate it. The approximation is done in the following way.\nPoles and zeros that are much slower than (OQ are approximated by\nintegrators, poles, and zeros of the same order as WQ are retained,\nand poles and zeros that are much faster than a>o are neglected or\napproximated by a small time lag. A dead time such that COQL <K 1\nis neglected or approximated by a time constant. The approximation\nof fast poles and zeros by a first order system is illustrated by an\nexample.\n\n56\n\nChapter 2\n\nProcess Models\n\nEXAMPLE 2.14 Approximation of fast modes\nConsider the transfer function\ntL\n\nwhere\n\n-fT = T3 + T4 + T5 + TQ - Ti - T2 - L > 0\n\nand it is assumed that L «. T. The transfer function G can be\napproximated by\n\nEXAMPLE 2.15 Approximation of fast and slow modes\nConsider the same system as in Example 2.14. Assume that\nT3>T4>TS>\n\nT6\n\nand that\nT5 > max (Ti, T2, L)\n\nFurthermore, let it be desired to obtain a model that describes the\nprocess well in the frequency range\n1\n\n1\n<\n\nCOQ\n\n<\n\nThe time constant T3 is slower than T4 and T\\$, and it will therefore\nbe approximated by an integrator, i.e.,\n1\n\n~ 1\n1 + sT3 ^ sfl\nThe time constants T\\, T2, T§, and the time delay L are all smaller\nthan T5. They will be approximated by a single time constant\nT = T6 - Ti - T2 - L\nIf T is positive the system is then approximated by\nG(8)\n\n=\n\nK\n\nIf T is negative the transfer function is instead approximated by\n\n2.10 Conclusions\n\n57\n\nSummary\nTo summarize: When deriving a simple model to be used for PID\ncontroller tuning, it is important to ensure that the model describes\nthe process well for the typical input signals obtained during the\nprocess operations. The amplitude and frequency distribution of the\nsignal is of importance. Model accuracy may be poor if the process is\nnonlinear or time varying. Control quality can be improved by gain\nscheduling or adaptive control. It is also important to know what kind\nof disturbances are acting on the system and which limitation they\nimpose.\n\n2.10\n\nConclusions\n\nModeling is an important aspect of controller tuning. The models we\nneed should describe how the process reacts to control signals. They\nshould also describe the properties of the disturbances that enter the\nsystem. Most work on tuning of PID controllers have focused on the\nprocess dynamics, which is also reflected in the presentation in this\nchapter.\nA number of methods for determining the dynamics of a process\nhave been presented in this chapter. Some are very simple: they\nare based on a direct measurement of the step response and simple\ngraphical constructions. Others are based on the frequency response.\nIt has been shown that very useful information can be generated from\nrelay feedback experiments. Such experiments are particularly useful\nbecause the process is brought into self-oscillation at the ultimate\nfrequency, which is of considerable interest for design of controllers.\nThe simple methods are useful in field work when a controller\nhas to be tuned and few tools are available. The methods are also\nuseful to provide understanding as well as being references when\nmore complicated methods are assessed. We have also presented more\ncomplicated methods that require significant computations.\nModels of different complexity have been presented. Many models\nwere characterized by a few parameters. Such models are useful for\nmany purposes and are discussed in Chapter 4. When using such\nmodels it should be kept in mind that they are approximations.\nWhen deriving the models we also introduced two dimensionfree quantities, the normalized dead time r and the gain ratio K.\nThese parameters make it possible to make a crude assessment of\nthe difficulty of controlling the process. Processes with small values\nare easy to control. The difficulty increases as the values approach 1.\nTuning rules based on T and K are provided in Chapter 5.\n\n58\n\n2.11\n\nChapter 2 Process Models\n\nReferences\n\nProcess modeling is a key element in understanding and solving a con¬\ntrol problem. Good presentations of modeling are found in standard\ntextbooks on control, such as Buckley (1964), Smith (1972), Seborg\net al. (1989), and Luyben (1990). These books have much material on\nmany different modeling techniques. Similar presentations are given\nin Gille et al. (1959), Harriott (1964), Oppelt (1964), Takahashi et al.\n(1972),Deshpande and Ash (1981),Shinskey (1988),Stephanopoulos\n(1984), and Hagglund (1991). There are also books that specialize in\nmodeling for control system design, see Wellstead (1979), Nicholson\n(1980), Nicholson (1981), and Close and Frederick (1993).\nIn the early work much effort was devoted to characterize dynam¬\nics by the step response, which at that time was called the response\ncurve. See Ziegler et al. (1943) and the books Tucker and Wills (1960)\nand Lloyd and Anderson (1971), which were written by practitioners\nin control companies. A nice overview of step andfrequencyresponse\nmethods is given in the paper Rake (1980). Additional details are\ngiven in Strejc (1959) and Anderssen and White (1971). Frequency\nresponse methods are presented in Anderssen and White (1970).\nThe relay method is treated in Astrom and Hagglund (1984),\nHagglund and Astrom (1991), and Schei (1992). The describing func¬\ntion method is well documented in Atherton (1975) and Gelb and\nVelde (1968). There are many books on parameter estimation, the\nbook Johansson (1993) is quite accessible. More mathematical de¬\ntails are found in Ljung (1987), Ljung and Soderstrom (1983), and\nSoderstrbm and Stoica (1988). Many useful practical aspects on sys¬\ntem identification are given in Isermann (1980).\n\nCHAPTER\n\n3\n\nPID Control\n\n3.1\n\nIntroduction\n\nThe PID controller is by far the most common control algorithm. Most\nfeedback loops are controlled by this algorithm or minor variations\nof it. It is implemented in many different forms, as a stand-alone\ncontroller or as a part of a DDC (Direct Digital Control) package or\na hierarchical distributed process control system. Many thousands\nof instrument and control engineers worldwide are using such con¬\ntrollers in their daily work. The PID algorithm can be approached\nfrom many different directions. It can be viewed as a device that can\nbe operated with a few rules of thumb, but it can also be approached\nanalytically.\nThis chapter gives an introduction to PID control. The basic al¬\ngorithm and various representations are presented in detail. A de¬\nscription of the properties of the controller in a closed loop based on\nintuitive arguments is given. The phenomenon of reset windup, which\noccurs when a controller with integral action is connected to a process\nwith a saturating actuator, is discussed, including several methods to\navoid it.\nSome important aspects of digital computer implementation of\nPID controllers are given: issues such as prefiltering, different digi¬\ntal approximations, noise filtering, and computer code for good imple¬\nmentation. Operational aspects, such as bumpless transfer between\nmanual and automatic mode and between different paramete```"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.91475767,"math_prob":0.9534438,"size":103789,"snap":"2020-10-2020-16","text_gpt3_token_len":24169,"char_repetition_ratio":0.18655875,"word_repetition_ratio":0.0428539,"special_character_ratio":0.23462024,"punctuation_ratio":0.0964016,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99155986,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-29T07:03:28Z\",\"WARC-Record-ID\":\"<urn:uuid:7b69a1a0-d435-4bfd-bca4-3a2a3a9cba3f>\",\"Content-Length\":\"163118\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b0b12b6b-b274-4ad4-a649-7fe68abe1fd7>\",\"WARC-Concurrent-To\":\"<urn:uuid:20824f95-0ffa-4bef-8f41-98e9e55af087>\",\"WARC-IP-Address\":\"179.43.147.124\",\"WARC-Target-URI\":\"https://pt.b-ok.org/book/556662/c2b4f4?dsource=recommend\",\"WARC-Payload-Digest\":\"sha1:L5APFJJRLMU4D77E7AVVDOYR2GZ2EYVR\",\"WARC-Block-Digest\":\"sha1:36R7IZ75I7E5Q5K7DJSTX6KI7MIAMZGL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370493818.32_warc_CC-MAIN-20200329045008-20200329075008-00227.warc.gz\"}"} |
https://www.r-bloggers.com/2019/02/if-you-were-an-r-function-what-function-would-you-be/ | [
"Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nWe’ve been getting some good uptake on our piping in `R` article announcement.\n\nThe article is necessarily a bit technical. But one of its key points comes from the observation that piping into names is a special opportunity to give general objects the following personality quiz: “If you were an `R` function, what function would you be?”\n\n• Everything that exists is an object.\n• Everything that happens is a function call.\n\nSo our question is: can we add a meaningful association between the two deepest concepts in `R` objects (or references to them) and functions?\n\nWe think the answer is a resounding “yes!”\n\nThe following example (adapted from the paper) should help illustrate the idea.\n\nSuppose we had simple linear model.\n\n```set.seed(2019)\ndata_use <- base::sample(c(\"train\", \"test\"),\nnrow(mtcars), replace = TRUE)\nmtcars_train <- mtcars[data_use == \"train\", , drop = FALSE]\nmtcars_test <- mtcars[data_use == \"test\", , drop = FALSE]\nmodel <- lm(mpg ~ disp + wt, data = mtcars_train)\n```\n\nNow if “`model`” were an `R` function, what function would it be? One possible answer is: it would be `predict.lm()`. It would be nice if “`model(mtcars_test)`” meant “`predict(model, data = mtcars_test)`“. Or, if we accept the pipe notation “`mtcars_test %.>% model`” as an approximate substitute for (note: not an equivalent of) “`model(mtcars_test)`” we can make that happen.\n\nThe “`%.>%`” is the `wrapr` dot arrow pipe. It can be made to ask the question “If you were an `R` function, what function would you be?” as follows.\n\nFirst a bit of preparation, we tell R‘s `S3` class system how to answer the question.\n\n```apply_right.lm <-\nfunction(pipe_left_arg,\npipe_right_arg,\npipe_environment,\nleft_arg_name,\npipe_string,\nright_arg_name) {\npredict(pipe_right_arg,\nnewdata = pipe_left_arg)\n}\n```\n\nAnd now we can treat any reference to an object of class “`lm`” as a pipe destination or function.\n\n```mtcars_test %.>% model\n```\n\nAnd we see our results.\n\n```# Mazda RX4 Mazda RX4 Wag Hornet 4 Drive Duster 360 Merc 280\n# 23.606199 22.518582 20.477232 18.347774 20.062914\n# Merc 280C Merc 450SE Cadillac Fleetwood Lincoln Continental Fiat 128\n# 20.062914 16.723133 10.506642 9.836894 25.888019\n# Dodge Challenger AMC Javelin Porsche 914-2 Lotus Europa Ford Pantera L\n# 18.814401 19.261396 25.892974 28.719255 20.108134\n# Maserati Bora\n# 18.703696\n```\n\nNotice we didn’t have to alter `model` or wrap it in a function. This solution can be used again and again in many different circumstances."
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.8357208,"math_prob":0.8477156,"size":2841,"snap":"2022-27-2022-33","text_gpt3_token_len":751,"char_repetition_ratio":0.10715544,"word_repetition_ratio":0.10085837,"special_character_ratio":0.29778248,"punctuation_ratio":0.14466546,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96907663,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-18T10:54:59Z\",\"WARC-Record-ID\":\"<urn:uuid:12c61af4-74f3-46cc-8283-6851b118facd>\",\"Content-Length\":\"91022\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9bc7efbd-2a7f-4a10-9b1a-578886193686>\",\"WARC-Concurrent-To\":\"<urn:uuid:5e7dd41e-5dc5-41ba-8b93-1a373190c522>\",\"WARC-IP-Address\":\"172.64.105.18\",\"WARC-Target-URI\":\"https://www.r-bloggers.com/2019/02/if-you-were-an-r-function-what-function-would-you-be/\",\"WARC-Payload-Digest\":\"sha1:VMTIKVGWB3KUXFCRDZKQ6MCKXKISC63D\",\"WARC-Block-Digest\":\"sha1:AWCKIPVBCDUHV65VDT3K425AA3ELFJSU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573193.35_warc_CC-MAIN-20220818094131-20220818124131-00693.warc.gz\"}"} |
https://scirp.org/journal/paperinformation.aspx?paperid=81916 | [
"A Geometric Proof of Fermat’s Little Theorem\n\nWe present an intuitively satisfying geometric proof of Fermat's result for positive integers that",
null,
"for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.\n\nKEYWORDS\n\n1. Historical Background\n\nPierre de Fermat wrote his friend Frénicle de Bessy in 1640 stating that he had discovered that ${a}^{p-1}\\equiv 1$ for prime moduli p, provided p did not divide a, but his proof was overlong, so he did not bother to include the details. One might wish that Fermat had been more generous in recording his notes both in this instance and that famous “margin too small to contain∙∙∙ (his proof of Fermat’s Last Theorem)”. Leibniz appears to have proved the theorem prior to 1683 without publishing it, and then Euler reprised Leibniz’ work in a published version. This result, christened Fermat’s Little Theorem by Kurt Hensel in 1913, is the basis for a convenient method for detecting primality, or more correctly, compositeness . If p does not divide a and ${a}^{p-1}$ is not congruent to $1\\mathrm{mod}p$ , then p must be composite. Modular arithmetic, particularly with the aid of a computer, makes short work of calculating the residues of high powers of a needed to check this condition. Unfortunately, the invalidity of the converse to Fermat’s Little Theorem (if ${a}^{n-1}\\equiv 1\\mathrm{mod}n$ with a and n coprime, then n is prime) forces it to be used in a probabilistic way for detecting primality. If ${a}^{n-1}\\equiv 1\\mathrm{mod}n$ for lots of different admissible choices of n, then it looks more and more like n is probably prime. But there are many a for which ${a}^{n-1}\\equiv 1\\mathrm{mod}n$ and yet n is composite. Such a are called “Fermat Liars” and the n that go with them are termed pseudoprimes to the base a. A pseudoprime to every base, and they do exist but are relatively rare, is called a Carmichael number. Carmichael numbers completely defeat the usefulness of the theorem as a primality test. There is certainly no shortage of simple proofs of Fermat’s Little Theorem. It may be proved with a straightforward induction on the base a to show that ${a}^{p-1}-1$ is divisible by p, or by using a modular arithmetic argument. We present a proof of this useful theorem from an intuitively appealing direc- tion based on coloring the vertices of regular polygons with prime numbers of sides.\n\n2. Burnside’s Lemma \n\nIf G is a group of permutations acting on a set S, then for a particular $\\pi \\in G$ , the invariant set of $\\pi$ is the collection of all elements of S that are fixed points of $\\pi$ , i.e. $\\pi \\left(s\\right)=s$ . The orbit of some $s\\in S$ is the collection of elements obtained by letting every permutation in G act on that s. Intuitively, there is a broad inverse size relationship between orbits and invariant sets. If most elements are not moved by most permutations, invariant sets will be large and orbits will be small, but more numerous. In 1897, the British mathematician William Burnside published the result, with attribution to Frobenius , that if G is a group of permutations acting on the finite set S, then the number of orbits\n\nof S under G is given by $\\frac{1}{|G|}{\\sum }_{\\pi \\in G}|inv\\left(\\pi \\right)|$ , where $|inv\\left(\\pi \\right)|$ is the size of the\n\ninvariant set of the permutation $\\pi \\in G$ . Burnside’s Lemma is a direct conse- quence of the Orbit/Stabilizer Theorem and was known at least as early as Cauchy, hence it is sometimes called “the lemma that is not Burnside’s” . We will color the elements of S, specifically the vertices of a regular polygon with a prime number of sides, and adapt Burnside’s/Not Burnside’s Lemma to the problem of enumerating distinct colorings.\n\n3. Colorings\n\nA coloring $\\chi$ is a mapping from the finite set S of objects to be colored to the finite set P, consisting of the “pallette” of colors. This is just a fanciful way of thinking about the rather dry notion of an arbitrary function between two finite sets. Colorings had their origin in the effort to establish the Four-Color Theorem, and they pop up in many seemingly unrelated combinatorial settings . If the group G of permutations shuffles the objects of S, then those permutations will likewise shuffle the possible colorings of S. So a given $\\pi \\in G$ can be regarded as a mapping from the set of P-colorings of S back to itself. More mathematically, there is a group of induced maps ${G}^{\\ast }$ that form an action on the set of colorings ${P}^{S}$ . Each $\\pi \\in G$ gives rise to a companion mapping ${\\pi }^{\\ast }\\in {G}^{\\ast }$ that determines what happens to the available colorings whenever the underlying objects in S are permuted by $\\pi$ . It is clear that ${G}^{\\ast }$ is a group with the same order as G.\n\nJust as the elements of S can be put into equivalence classes on the basis of whether they are in the same orbit under the action of G or not, colorings can be put into equivalence classes depending on whether they are in the same orbit under the action of ${G}^{\\ast }$ or not. Orbits of colorings are called patterns. Two colorings ${\\chi }_{1}$ and ${\\chi }_{2}$ are equivalent, or represent the same pattern, if there is a $\\pi \\in G$ such that for all $s\\in S$ we have ${\\chi }_{1}\\left(s\\right)={\\chi }_{2}\\left(\\pi \\left(s\\right)\\right)$ . For example, if ${\\chi }_{1}\\left(a\\right)=\\text{red}$ , ${\\chi }_{1}\\left(b\\right)=\\text{white}$ , and ${\\chi }_{1}\\left(c\\right)=\\text{blue}$ , then if the permutation $\\pi =\\left(a\\text{ }b\\text{ }c\\right)\\in G$ , ${\\chi }_{1}$ and ${\\chi }_{2}$ are equivalent provided ${\\chi }_{2}\\left(a\\right)=\\text{blue}$ , ${\\chi }_{2}\\left(b\\right)=\\text{red}$ , and ${\\chi }_{2}\\left(c\\right)=\\text{white}$ . You can see that $\\text{red}={\\chi }_{1}\\left(a\\right)={\\chi }_{2}\\left(\\pi \\left(a\\right)\\right)={\\chi }_{2}\\left(b\\right)$ , and so forth. We would write ${\\pi }^{\\ast }\\left({\\chi }_{1}\\right)={\\chi }_{2}$ to signify this situation.\n\n4. Proof\n\nIf we have a symmetrical object with a coloring, we can apply Burnside’s Lemma to enumerate the number of distinct patterns possible for the object. Let us consider a regular p-gon, where p is prime, and color the vertices with a pallette consisting of a colors, where p does not divide a. We will admit the digon to handle the case $p=2$ . Now we are not going to tear or fold the polygon, so the only permutations of colorings allowed are going to be those induced by the rotation group of the polygon, namely the cyclic group ${ℤ}_{p}$ . The invariant set of a particular ${\\pi }^{\\ast }$ induced by an element of this group is easy to characterize. The zero rotation has ${a}^{p}$ colorings fixed by it, namely the total number of possible paint jobs or mappings from the vertices to the pallette. Every non-zero element of ${ℤ}_{p}$ carries each pure coloring (every vertex the same color) into itself, so there would be a such colorings fixed by the action of those rotations. But none of these $p-1$ non-zero rotations can carry a non-pure coloring back to itself due to the indivisibility of p. Burnside’s Lemma then gives us the total count of possible distinct patterns as\n\n$\\frac{1}{|{G}^{\\ast }|}{\\sum }_{{\\pi }^{\\ast }\\in {G}^{\\ast }}|inv\\left({\\pi }^{\\ast }\\right)|=\\frac{1}{p}\\left({a}^{p}+a\\left(p-1\\right)\\right)=\\frac{a}{p}\\left({a}^{p-1}+\\left(p-1\\right)\\right)$ . Evidently this is a\n\npositive integer, and since p does not divide a, it must divide $\\left({a}^{p-1}+\\left(p-1\\right)\\right)$ . But this means $\\left({a}^{p-1}+\\left(p-1\\right)\\right)\\equiv {a}^{p-1}-1\\equiv 0\\mathrm{mod}p$ , which is Fermat’s Little Theorem.\n\n5. Conclusion\n\nWe have established Fermat’s Little Theorem by coloring the vertices of a regular polygon and then finding the patterns that are stable under various rotations of the polygon. When the number of vertices is prime, the set of such invariant patterns is necessarily limited. This process lends itself to intuitively satisfying visualization. Burnside’s Lemma then enumerates the relatively sparse number of invariant patterns and gives a formula that is equivalent to the modular expression of Fermat’s Little Theorem. An interesting further application of this idea would be to search for other number-theoretic results using colorings of more complicated geometric objects and more general pattern enumeration methods, for example Polya’s Counting Theorem.\n\nConflicts of Interest\n\nThe authors declare no conflicts of interest.\n\nCite this paper\n\nBeatty, T. , Barry, M. and Orsini, A. (2018) A Geometric Proof of Fermat’s Little Theorem. Advances in Pure Mathematics, 8, 41-44. doi: 10.4236/apm.2018.81004.\n\n Dummitt, D.S. and Foote, R.M. (2004) Abstract Algebra. 3rd Edition, John Wiley & Sons, Inc. Gallian, J.A. (2010) Contemporary Abstract Algebra. 7th Edition, Brooks-Cole. Rotman, J. (1995) An Introduction to the Theory of Groups. Springer-Verlag. https://doi.org/10.1007/978-1-4612-4176-8 Burnside, W. (1897) Theory of Groups of Finite Order. Cambridge University Press, Cambridge. Frobenius, F.G. (1887) über die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul. Crelle’s Journal, 288. Neumann, P.M. (1979) A Lemma That Is Not Burnside’s. The Mathematical Scientist, 4, 133-141. Rosyida, I., Peng, J., Chen, L., Widodo, W., Indrati, Ch.R. and Sugeng, K.A. (2016) An Uncertain Chromatic Number of an Uncertain Graph Based on Alpha-Cut Coloring. Fuzzy Optimization and Decision Making, 1-21. https://doi.org/10.1007/s10700-016-9260-x Chen, L., Peng, J. and Ralescu, D.A. (2017) Uncertain Vertex Coloring Problem. Soft Computing. https://doi.org/10.1007/s00500-017-2861-7",
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""
] | [
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"https://file.scirp.org/image/Edit_0b50006d-6c00-4238-bfe4-e54f91b8720a.bmp",
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"https://scirp.org/Images/ccby.png",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.9166124,"math_prob":0.993582,"size":6766,"snap":"2019-43-2019-47","text_gpt3_token_len":1543,"char_repetition_ratio":0.11860396,"word_repetition_ratio":0.025125628,"special_character_ratio":0.20854272,"punctuation_ratio":0.08966565,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991542,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,3,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T13:11:10Z\",\"WARC-Record-ID\":\"<urn:uuid:dc1f0eeb-4459-41b8-b02a-406a09e0f183>\",\"Content-Length\":\"94465\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:14dc3894-0c94-42a8-a7fb-e0e248b142d7>\",\"WARC-Concurrent-To\":\"<urn:uuid:b9a96b63-1534-4111-a469-a9bbce8635e8>\",\"WARC-IP-Address\":\"198.204.224.91\",\"WARC-Target-URI\":\"https://scirp.org/journal/paperinformation.aspx?paperid=81916\",\"WARC-Payload-Digest\":\"sha1:EXBAVCX45LULHGTCGUSN574OFNMXUP7R\",\"WARC-Block-Digest\":\"sha1:RRAXNO5NLMNGZYYZVYJV22OMGFTFFXCP\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670821.55_warc_CC-MAIN-20191121125509-20191121153509-00302.warc.gz\"}"} |
https://www.premiumessaywritingservice.com/determine-the-percentage-relationship-between-net-income-the-following-data-in-milli-3268047-4/ | [
"## determine the percentage relationship between net income the following data in milli 3268047\n\nDetermine the percentage relationship between net income\n\nThe following data (in millions) is taken from the financial statements of Williams- Sonoma for the years ending 2007 and 2006:\n\n(a) Determine the amount of change (in millions) and percent of change in net income for 2007.\n(b) Determine the percentage relationship between net income and net sales (net income divided by net sales) for 2007 and 2008.\n(c) What conclusions can you draw from your analysis? Â\n\nDetermine the percentage relationship between net income\nAccounting Basics\n\ns"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.9068404,"math_prob":0.8653734,"size":541,"snap":"2021-31-2021-39","text_gpt3_token_len":119,"char_repetition_ratio":0.15270019,"word_repetition_ratio":0.1097561,"special_character_ratio":0.22550832,"punctuation_ratio":0.043956045,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9843652,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-03T08:11:43Z\",\"WARC-Record-ID\":\"<urn:uuid:e2348905-fd7a-4287-991b-f744c431dd08>\",\"Content-Length\":\"35814\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f5b8039d-d0f7-4499-881a-2c3a118e8273>\",\"WARC-Concurrent-To\":\"<urn:uuid:b868b128-5e8a-42af-9654-1b849e7e246a>\",\"WARC-IP-Address\":\"198.187.29.94\",\"WARC-Target-URI\":\"https://www.premiumessaywritingservice.com/determine-the-percentage-relationship-between-net-income-the-following-data-in-milli-3268047-4/\",\"WARC-Payload-Digest\":\"sha1:XSBLC72CSMV64WOCI4FWXE4KQIB6W5FG\",\"WARC-Block-Digest\":\"sha1:VAKCTZXXNPHX65ED22RZJ7RGGHKR7DQS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154432.2_warc_CC-MAIN-20210803061431-20210803091431-00068.warc.gz\"}"} |
https://www.stata.com/statalist/archive/2006-03/msg00358.html | [
"",
null,
"",
null,
"",
null,
"",
null,
"# RE: st: qvf command for count data\n\n From \"Hugh Colaco\" To Subject RE: st: qvf command for count data Date Fri, 10 Mar 2006 15:12:02 -0500\n\n```Sorry - this was a bad example. There was no \"identification\" issue in\nmy actual regression. My basic issue is with the syntax of the qvf\ncommand.\n\n-----Original Message-----\nFrom: [email protected]\n[mailto:[email protected]] On Behalf Of Austin\nNichols\nSent: Friday, March 10, 2006 3:01 PM\nTo: [email protected]\nSubject: Re: st: qvf command for count data\n\nOn 3/10/06, Hugh Colaco <[email protected]> wrote:\n> qvf y x1 x2 x3 x4 (z1 x3 x4), family(nbinomial) robust cluster (A);\n> ivreg y x1 x2 x3 x4 (z1 = x3 x4), robust cluster (A);\n\nYou seem to be misspecifying both -ivreg- and -qvf- calls at a very\nbasic level--which variables are included and excluded instruments?\nDo you mean z1 to be an excluded instrument for two endogenous variables\nx3 and x4? If so, your equation is not identified. Note your -ivreg-\nsyntax is regressing y on x1 and x2 and z1 (where z1 is instrumented by\nx3 and x4) though I don't think it will run exactly as\nwritten:\n\n. net from http://www.stata-journal.com/software/sj3-4\n. net inst st0049\n. clear\n. set obs 1000\n. gen x1 = uniform()\n. gen x2 = uniform()\n. gen x3 = uniform()\n. gen err = invnorm(uniform())\n. gen y = 1+2*x1+3*x2+4*x3+err\n. gen x4 = uniform()\n. gen t3 = .8*x3 + .6*invnorm(uniform()) . ivreg y x1 x2 x3 x4 (z1 = x3\nx4) equation not identified; must have at least as many instruments not\nin the regression as there are instrumented variables r(481);\n\n. qvf y x1 x2 x3 x4 (x1 x2 x4 t3)\n\nIV Generalized linear models No. of obs =\n1000\nOptimization : MQL Fisher scoring Residual df =\n995\n(IRLS EIM) Scale param =\n2.137276\nDeviance = 2126.589444 (1/df) Deviance =\n2.137276\nPearson = 2126.58962 (1/df) Pearson =\n2.137276\n\nVariance Function: V(u) = 1 [Gaussian]\nLink Function : g(u) = u [Identity]\nStandard Errors : OIM Sandwich\n\n------------------------------------------------------------------------\n------\ny | Coef. Std. Err. z P>|z| [95% Conf.\nInterval]\n-------------+----------------------------------------------------------\n-------------+------\nx1 | 1.914558 .106234 18.02 0.000 1.706343\n2.122773\nx2 | 2.912829 .1086845 26.80 0.000 2.699811\n3.125846\nx4 | .2775132 .1095646 2.53 0.011 .0627706\n.4922558\nx3 | 4.106679 .3455157 11.89 0.000 3.429481\n4.783877\n_cons | .93817 .193988 4.84 0.000 .5579605\n1.31838\n------------------------------------------------------------------------\n------\n\nTry using -ivreg2- instead. It's got good first-stage diagnostics, and\nthe fact that your endogenous variable is a count variable does not\nimply the standard IV estimator is not consistent--just that you lose a\ntiny bit of efficiency by disregarding that fact. Note that many of the\nclassic RHS endogenous variables are counts, e.g. educational\nattainment, and most researchers would use -ivreg2- on these models.\n\n. ssc install ivreg2\n. ivreg2 y x1 x2 x4 (x3=z1), ffirst\n\nSummary results for first-stage regressions\n-------------------------------------------\n\nShea\nVariable | Partial R2 | Partial R2 F( 1, 995) P-value\nx3 | 0.1009 | 0.1009 111.65 0.0000\n\nUnderidentification tests:\nChi-sq(1) P-value\nAnderson canon. corr. likelihood ratio stat. 106.35 0.0000\nCragg-Donald N*minEval stat. 112.21 0.0000\nHo: matrix of reduced form coefficients has rank=K-1 (underidentified)\nHa: matrix has rank>=K (identified)\n\nWeak identification statistics:\nCragg-Donald (N-L)*minEval/L2 F-stat 111.65\n\nAnderson-Rubin test of joint significance of endogenous regressors B1 in\nmain equation, Ho:B1=0\nF(1,995)= 67.79 P-val=0.0000\nChi-sq(1)= 68.13 P-val=0.0000\n\nNumber of observations N = 1000\nNumber of regressors K = 5\nNumber of instruments L = 5\nNumber of excluded instruments L2 = 1\n\nInstrumental variables (2SLS) regression\n----------------------------------------\n\nNumber of obs =\n1000\nF( 4, 995) =\n292.31\nProb > F =\n0.0000\nTotal (centered) SS = 3276.986562 Centered R2 =\n0.7013\nTotal (uncentered) SS = 33323.59494 Uncentered R2 =\n0.9706\nResidual SS = 978.9706817 Root MSE =\n.9894\n\n------------------------------------------------------------------------\n------\ny | Coef. Std. Err. z P>|z| [95% Conf.\nInterval]\n-------------+----------------------------------------------------------\n-------------+------\nx3 | 4.106679 .337572 12.17 0.000 3.44505\n4.768308\nx1 | 1.914558 .1075383 17.80 0.000 1.703787\n2.125329\nx2 | 2.912829 .1075683 27.08 0.000 2.701999\n3.123658\nx4 | .2775132 .1073605 2.58 0.010 .0670905\n.4879358\n_cons | .93817 .1888342 4.97 0.000 .5680617\n1.308278\n------------------------------------------------------------------------\n------\nAnderson canon. corr. LR statistic (identification/IV relevance test):\n106.350\nChi-sq(1) P-val =\n0.0000\n------------------------------------------------------------------------\n------\nSargan statistic (overidentification test of all instruments):\n0.000\n(equation exactly\nidentified)\n------------------------------------------------------------------------\n------\nInstrumented: x3\nIncluded instruments: x1 x2 x4\nExcluded instruments: z1\n------------------------------------------------------------------------\n------\n\n*\n* For searches and help try:\n* http://www.stata.com/support/faqs/res/findit.html\n* http://www.stata.com/support/statalist/faq\n* http://www.ats.ucla.edu/stat/stata/\n\n*\n* For searches and help try:\n* http://www.stata.com/support/faqs/res/findit.html\n* http://www.stata.com/support/statalist/faq\n* http://www.ats.ucla.edu/stat/stata/\n```"
] | [
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"https://www.stata.com/includes/images/statalist_middle.gif",
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.63607097,"math_prob":0.8395614,"size":5480,"snap":"2021-31-2021-39","text_gpt3_token_len":1677,"char_repetition_ratio":0.22041637,"word_repetition_ratio":0.07786885,"special_character_ratio":0.49726278,"punctuation_ratio":0.20423892,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98343045,"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\":\"2021-07-31T03:49:31Z\",\"WARC-Record-ID\":\"<urn:uuid:8542a02b-3f78-4513-8e96-74e1fd35d51c>\",\"Content-Length\":\"12981\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d2a32b9d-6918-4c29-81d4-720292d52db5>\",\"WARC-Concurrent-To\":\"<urn:uuid:bf00b59b-c406-4cc4-bbdd-97bf45bbd51d>\",\"WARC-IP-Address\":\"66.76.6.5\",\"WARC-Target-URI\":\"https://www.stata.com/statalist/archive/2006-03/msg00358.html\",\"WARC-Payload-Digest\":\"sha1:SOW6OXIJBWRZ6TYDD2ALZZZMJHTS2OXX\",\"WARC-Block-Digest\":\"sha1:JKSURI5PCCAC4Q75TQUKNSNIY5ZZ3C75\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154042.23_warc_CC-MAIN-20210731011529-20210731041529-00298.warc.gz\"}"} |
https://www.encyclopediaofmath.org/index.php?title=Routh-Hurwitz_criterion&oldid=33371 | [
"# Routh-Hurwitz criterion\n\n(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)\n\nHurwitz criterion\n\nA necessary and sufficient condition for all the roots of a polynomial\n\n$$f(x)=a_0x^n+a_1x^{n-1}+\\ldots+a_n,$$\n\nwith real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\\Delta_i$, $i=1,\\dots,n$, of the Hurwitz matrix $H$ are positive (cf. Minor). Here $H$ is the matrix of order $n$ whose $i$-th row has the form\n\n$$a_{2-i},a_{4-i},\\dots,a_{2n-i},$$\n\nwhere, by definition, $a_k=0$ if $k<0$ or $k>n$ (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz and is a generalization of the work of E.J. Routh (see Routh theorem).\n\nA polynomial $f(x)$ satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the Liénard–Chipart criterion, and methods for determining the number of real roots of a polynomial are also known."
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7844021,"math_prob":0.9970364,"size":1546,"snap":"2019-43-2019-47","text_gpt3_token_len":466,"char_repetition_ratio":0.14915694,"word_repetition_ratio":0.0,"special_character_ratio":0.26778784,"punctuation_ratio":0.17207792,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9992067,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T15:25:54Z\",\"WARC-Record-ID\":\"<urn:uuid:800c2e9b-431b-43df-985c-f1e5c4d633fa>\",\"Content-Length\":\"16860\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c51df693-8a04-4d80-8dd2-bdeb6c355fe2>\",\"WARC-Concurrent-To\":\"<urn:uuid:7c6665fc-35f2-4ea5-a1b8-5be59bb1f35a>\",\"WARC-IP-Address\":\"80.242.138.72\",\"WARC-Target-URI\":\"https://www.encyclopediaofmath.org/index.php?title=Routh-Hurwitz_criterion&oldid=33371\",\"WARC-Payload-Digest\":\"sha1:FUPJITEWQC2VYRPFJ2OHSPCOBCJ2YZSF\",\"WARC-Block-Digest\":\"sha1:52QJB4ZVHB5UQDWTNXVOMNWBKAZKAXEM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496665575.34_warc_CC-MAIN-20191112151954-20191112175954-00543.warc.gz\"}"} |
http://www.ikompass.edu.sg/parametric-versus-non-parametric-estimating-methods/ | [
"435 Orchard Road, 11th Floor Wisma Atria, Singapore 238877\n+ 65 66381203\nHome»Data Science»Parametric Versus Non-Parametric Estimating methods\n\n# Parametric Versus Non-Parametric Estimating methods\n\nA common hurdle that every data scientist will come at some point need to address is what machine learning model to use? At one end of the spectrum, you have simple models which are easy to interpret but less accurate and at the other end, you have models that are complex and difficult to read but provides higher accuracy.\n\nModels that are easy to interpret are generally grouped as Parametric methods. These include models that make an assumption about the relationship between the features and the output as having a functional form. Regression models are generally considered as parametric methods. Example, linear regression. Other examples of commonly used parametric models include logistic regression, polynomial regression, linear discriminant analysis, quadratic discriminant analysis, (parametric) mixture models, and naïve Bayes (when parametric density estimation is used). Approaches often used in conjunction with parametric models for model selection purposes include ridge regression, lasso, and principal components regression.\n\nThere are models which seem like complex, black boxes which provide high accuracy and they don’t make assumptions about the functional relationship between features and predictors. These are called Non-parametric methods. A simple example of a nonparametric model is a classification tree. A classification tree is a series of recursive binary decisions on the input features. The classification tree learning algorithm uses the target variable to learn the optimal series of splits such that the terminal leaf nodes of the tree contain instances with similar values of the target.\n\nOther examples of nonparametric approaches to machine learning include k-nearest neighbors, splines, basis expansion methods, kernel smoothing, generalized additive models, neural nets, bagging, boosting, random forests, and support vector machines.\n\nMost machine learning applications tend to use non-parametric methods to reflect the underlying complexity of relationships."
] | [
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https://zbmath.org/?q=ai%3Aasok.aravind+se%3A00002722 | [
"# zbMATH — the first resource for mathematics\n\nComparing Euler classes. (English) Zbl 1372.14013\nLet $$k$$ be a perfect field of characteristic different than 2, $$X$$ a $$d$$-dimensional smooth $$k$$ scheme and $$\\xi :{\\mathcal E}\\rightarrow X$$ a rank $$r$$ vector bundle on $$X.$$ There are two ways to define an Euler class $$e(\\xi)$$ of $$\\xi$$. One is via “characteristic class” approach and is as follows. Let $$s_{0}: X\\rightarrow \\mathcal E$$ be the zero section of $$\\xi$$ then one can form a pullback and the Gysin pushforward homomorphism in Chow-Witt groups. The Euler class is then defined as $e_{cw}(\\xi)=({\\xi}^*)^{-1}(s_0)_*<1>\\, \\in \\widetilde{\\mathrm{CH}}^{r}(X,{\\det}({\\xi})^{\\vee}).$ The second definition was given in [F. Morel, $$\\mathbb A^1$$-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)] and relies on a construction of an Euler class as the primary obstruction to existence of a non-vavanishing section of $$\\xi .$$ The first non-trivial stage of a Moore-Postnikov factorization in $${\\mathbb A}^1$$-homotopy theory of the map $$\\mathrm{Gr}_{r-1}\\rightarrow \\mathrm{Gr}_{r},$$ where $$\\mathrm{Gr}_r$$ is the infinite Grassmanian with the universal rank $$r$$ vector bundle $${\\gamma}_r$$ yields a canonical equivariant cohomology class $${o}_r \\in H^r_{\\text{Nis}}(\\mathrm{Gr}_r, {{\\mathbf K}}_r^{MW}(\\det\\,{\\gamma}_r^{\\vee}))$$ [loc. cit., Appendix B]. Then $e_{ob}(\\xi)={\\xi}^*(o_r).$ But one can construct an isomorphism $H^r_{\\text{Nis}}(\\mathrm{Gr}_r, {\\mathbf{K}}_r^{MW}(\\det\\,{\\gamma}_r^{\\vee})) \\,\\rightarrow \\,\\widetilde{\\mathrm{CH}}^{r}(X,{\\det}({\\xi})^{\\vee})$ and this makes possible to compare both definitions. The main theorem of the paper is that if $${\\xi}$$ is oriented then both classes differ by a unit $$u\\in GW(k)^{\\times}.$$\n\n##### MSC:\n 14F42 Motivic cohomology; motivic homotopy theory 14C25 Algebraic cycles 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry\nFull Text:"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7173259,"math_prob":0.9999465,"size":2353,"snap":"2021-04-2021-17","text_gpt3_token_len":803,"char_repetition_ratio":0.09280545,"word_repetition_ratio":0.013201321,"special_character_ratio":0.34891626,"punctuation_ratio":0.1459695,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000066,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-20T14:34:36Z\",\"WARC-Record-ID\":\"<urn:uuid:de594a38-9d27-4deb-8f22-a2f3c8a7af4b>\",\"Content-Length\":\"48113\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f75ea0b2-ea9e-4f58-885d-31a07e28895c>\",\"WARC-Concurrent-To\":\"<urn:uuid:ace2f56a-0594-442a-9fe5-9197019981d0>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/?q=ai%3Aasok.aravind+se%3A00002722\",\"WARC-Payload-Digest\":\"sha1:XR3EVJ47IIAJ6YR6XJE36ZQFFQMKNX5J\",\"WARC-Block-Digest\":\"sha1:ZID3P7YPDYLMPBGN63E6QJENGJIWINUG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703520883.15_warc_CC-MAIN-20210120120242-20210120150242-00781.warc.gz\"}"} |
https://mathswithdavid.com/ks3-number-12-adding-subtracting-decimals/ | [
"# KS3. Number. 12. Adding & Subtracting Decimals\n\nWhen we want to add or subtract decimals, we follow exactly the same procedure as adding and subtracting integers. The important thing is to write the numbers so that the corresponding digits are lined up in columns.\n\nThis video demonstrates the process for addition:\n\nAnd this video demonstrates the process for subtraction:\n\nExamples\n\nWe can practice these with our teacher:\n\nExercise\n\nNow let’s try the following exercise (without using our calculators, other than to check answers afterwards) from exercise 10B in our textbooks:"
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.84909993,"math_prob":0.7463454,"size":548,"snap":"2023-14-2023-23","text_gpt3_token_len":105,"char_repetition_ratio":0.112132356,"word_repetition_ratio":0.024096385,"special_character_ratio":0.18248175,"punctuation_ratio":0.0927835,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9979454,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-21T10:16:35Z\",\"WARC-Record-ID\":\"<urn:uuid:714a4474-cfbe-4c91-83f2-956fc796c4d4>\",\"Content-Length\":\"108440\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cfdbf778-2999-4c22-b7c2-2cd1594b401d>\",\"WARC-Concurrent-To\":\"<urn:uuid:bab136a0-7d37-45c6-895f-184a07bdbb34>\",\"WARC-IP-Address\":\"192.0.78.25\",\"WARC-Target-URI\":\"https://mathswithdavid.com/ks3-number-12-adding-subtracting-decimals/\",\"WARC-Payload-Digest\":\"sha1:MU2I5QMSN6R7RMU6WSWZR5RUOK5DI762\",\"WARC-Block-Digest\":\"sha1:7TJZ45EJTAGT5P2S7ZQ3IW4C7JBTP2DO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296943695.23_warc_CC-MAIN-20230321095704-20230321125704-00117.warc.gz\"}"} |
https://codezup.com/comparable-and-comparator-in-java-with-examples/ | [
"# Comparable and Comparator in Java with Examples\n\nBy | March 13, 2021\n\nIn this tutorial, we will learn about the Comparable and Comparator in Java. First, we will understand the concept of Comparable and Comparator, and then we will see how we can use them in our code with the help of an example.\n\nLet’s start the tutorial.\n\n## Comparable Interface\n\nThe first important concept to know is that Comparable is an interface and it present in java.lang package. Comparable interface contains only one method compareTo( ).\n\n``public int compareTo(Object o);``\n\nFor Example:\n\n``Obj1.compareTo(Obj2);``\n\nSo, there are 3 possible value that is return by compareTo() method:\n\n1. Return -ve if and only if Obj1 has to come before obj2\n2. Return +ve if and only if obj1 has to come after obj2\n3. Return 0 (zero) if and only if obj1 and obj2 are equal\n\nLet’s understand this with the help of an example:\n\n``````package com.comparablecomparatordemo;\n\npublic class Test {\n\npublic static void main(String args[]) {\n\nSystem.out.println(\"C\".compareTo(\"Z\"));// some -ve value\nSystem.out.println(\"Z\".compareTo(\"E\")); // some +ve value\nSystem.out.println(\"C\".compareTo(\"C\")); // 0\n}\n}\n``````\n\nIt is important to know that if we are depending on the default natural sorting order then internally JVM will use compareTo( ) method to arrange objects in sorting order.\n\nLet’s understand above lines with the help of code and its analysis:\n\n``````package com.comparablecomparatordemo;\n\nimport java.util.TreeSet;\n\npublic class TestOne {\npublic static void main(String[] args) {\n\nTreeSet treeSet = new TreeSet();\nSystem.out.println(treeSet);\n}\n}\n``````\n\nOutput\n\n``[0, 10, 15]``\n\nSo, this is the basic information about the comparable interface in Java.\n\nIf we are not satisfied with the default natural sorting order or simply, we can say that if the default natural sorting order is not available then we can define our own customized sorting by Comparator object.\n\nIf we want to give a single line explanation about Comparable and Comparator, then below or these:\n\n• Comparable meant for default natural sorting order.\n• Comparator meant for customized natural sorting order.\n\nIn next section we will learn about the Comparator.\n\n### Comparator Interface in Java\n\nAfter learning about the comparable next step is Comparator. Comparator is also an interface and it is present in java.util package.\n\nComparator interface defines the following 2 methods:\n\n``public int compare( Object obj1, Object obj2);``\n\nBelow are the three possible values return by this method:\n\n1. Return -ve if and only if obj1 has to come before obj2\n2. Return +ve if and only if obj1 has to come after obj2\n3. Return 0( Zero) if an only if obj1 and obj2 are equal.\n\nSecond method which is present inside Comparator interface is:\n\n``public boolean equals(Object obj);``\n\nBelow are some things which we need to know:\n\n1. Whenever we are implementing the Comparator interface, we have to provide implementation only for the compare() method.\n2. Implementing the equals() method is optional because it is already available from the Object class through the concept of inheritance.\n\nLet’s understand this with the help of an example. Suppose we have a requirement to store the integer object into the TreeSet where the sorting order is descending order. Below is the code for this:\n\n``````package com.comparablecomparatordemo;\n\nimport java.util.Comparator;\nimport java.util.TreeSet;\n\npublic class ComparatorDemo {\npublic static void main(String[] args) {\n\nTreeSet treeSet = new TreeSet(new MyComparator());\nSystem.out.println(treeSet);\n}\n}\n\nclass MyComparator implements Comparator {\n\n@Override\npublic int compare(Object o1, Object o2) {\n\nInteger a = (Integer) o1;\nInteger b = (Integer) o2;\nif (a > b) {\nreturn -100;\n} else if (a < b) {\nreturn +1;\n} else\nreturn 0;\n\n}\n\n}``````\n\nOutput\n\n``[20, 15, 10, 5, 0]``\n\nSome important points to know about Comparator given below:\n\n1. If we are not passing the Comparator object, then JVM will always call the compareTo() method which is meant for default natural sorting order ( ascending order).\n2. If we are passing the Comparator object, then JVM will always call the compare() method of MyComparator class which is meant for customized sorting order ( ascending order).\n\nThis is all about the Comparator. Now let’s understand the difference between Comparable and Comparator in Java.\n\n#### Difference between Comparable and Comparator\n\nThat’s all for this tutorial. Hope you like the tutorial.\n\nIt is also recommended that you should learn about Collections in Java.\n\nPlease comment on your thoughts in the comment section down below. And also share this tutorial with others if you find it informative."
] | [
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https://cadcamengineering.net/using-relative-coordinates-autocad/ | [
"Once you understand the coordinate system used by AutoCAD, you can draw\nlines to any length and in any direction. Look at the shape shown previously\nin Figure 2.1. Because you know the dimensions, you can calculate (by adding and subtracting) the absolute coordinates for each vertex—the connecting point between two line segments—and then use the LINE command to draw the shape by entering these coordinates from the keyboard. However, AutoCAD offers you several tools for drawing this box much more easily. Two of these tools are the relative Cartesian and relative polar coordinate systems.\n\nWhen you’re drawing lines, these coordinate systems use a set of new points\nbased on the last point designated rather than on the 0,0 point of the drawing\narea. They’re called relative systems because the coordinates used are relative to the last point specified. If the first point of a line is located at the coordinate 4,6 and you want the line to extend 8 units to the right, the coordinate that is relative to the first point is 8,0 (8 units in the positive X direction and 0 units in the positive Y direction), whereas the actual—or absolute—coordinate of the second point is 12,6.\nThe relative Cartesian coordinate system uses relative x- and y-coordinates in\nthe manner shown, and the relative polar coordinate system relies on a distance and an angle relative to the last point specified. You’ll probably favor one system over the other, but you need to know both systems because you’ll sometimes find that, given the information you have at hand, one will work better than the other. A limitation of this nature is illustrated in Chapter 4, “Developing\nDrawing Strategies: Part 1.”\nTo designate the usage of relative coordinates, you’ll need to prefix each\ncoordinate you enter with an at symbol (@). In the previous example, you would enter the relative Cartesian coordinates as @8,0. The @ lets AutoCAD know that the numbers following it represent coordinates that are relative to the last point designated.\n\n#### Incoming search terms:\n\n• relative coordinate and absoulute in autcad"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.899183,"math_prob":0.94287634,"size":2082,"snap":"2022-27-2022-33","text_gpt3_token_len":428,"char_repetition_ratio":0.18864292,"word_repetition_ratio":0.02052786,"special_character_ratio":0.2017291,"punctuation_ratio":0.089058526,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9672242,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-26T01:54:04Z\",\"WARC-Record-ID\":\"<urn:uuid:24f9b335-417e-4d9d-8c0c-aa19866460c7>\",\"Content-Length\":\"51751\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9eed7c2d-a7ff-4edc-8859-3667a8cf4844>\",\"WARC-Concurrent-To\":\"<urn:uuid:7e6aaf65-a569-47e8-b31d-d0595bdcd399>\",\"WARC-IP-Address\":\"66.42.48.25\",\"WARC-Target-URI\":\"https://cadcamengineering.net/using-relative-coordinates-autocad/\",\"WARC-Payload-Digest\":\"sha1:4TTPAXVUMQF6SWA7FWRFDKDPRQJHFUX5\",\"WARC-Block-Digest\":\"sha1:XPVIFBPGIGIVHR3UFIHVBKPJLK3C56TW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103036363.5_warc_CC-MAIN-20220626010644-20220626040644-00668.warc.gz\"}"} |
https://datascience.stackexchange.com/questions/17676/how-is-the-modulo-number-selected-to-build-the-hash-table-in-dhp-algorithm | [
"# How is the modulo number selected to build the hash table in DHP algorithm?\n\nI'm trying to understand the DHP(Direct Hashing and Pruning) algorithm and I got stuck at explaining the selection of modulo number. The paper shows an example of using the hash function at page 7: h{{x y}) = ((order of x)*10 + (order of y)) mod 7\n\nMy questions are:\n\n1. What are the basis to define the function this way?\n2. How is the modulo number selected(7, in this example)?",
null,
""
] | [
null,
"https://i.stack.imgur.com/kLjpI.png",
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.822899,"math_prob":0.887321,"size":376,"snap":"2021-43-2021-49","text_gpt3_token_len":96,"char_repetition_ratio":0.1155914,"word_repetition_ratio":0.0,"special_character_ratio":0.26595744,"punctuation_ratio":0.078947365,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9967803,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-25T21:18:17Z\",\"WARC-Record-ID\":\"<urn:uuid:5ca0724d-b65a-47e3-a9ce-d68c97c5fb4e>\",\"Content-Length\":\"166535\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:04002c20-3487-408b-8674-3343ecd1a47d>\",\"WARC-Concurrent-To\":\"<urn:uuid:db6118b1-3e21-4cb0-ac50-28ea8e301bc9>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://datascience.stackexchange.com/questions/17676/how-is-the-modulo-number-selected-to-build-the-hash-table-in-dhp-algorithm\",\"WARC-Payload-Digest\":\"sha1:6MMNEM2OAAUPSLIPXQH6ZOB4BAGORHT2\",\"WARC-Block-Digest\":\"sha1:JYODH6TYYAA62T74ENKVNETEZXVFY5EG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587767.18_warc_CC-MAIN-20211025185311-20211025215311-00209.warc.gz\"}"} |
https://file.scirp.org/Html/1-4900301_50450.htm | [
"World Journal of Mechanics\nVol.04 No.10(2014), Article ID:50450,5 pages\n10.4236/wjm.2014.410030\n\nGeneralized Orbitals under the Influence of 2D Central and Noncentral Forces\n\nHaiduke Sarafian\n\nThe Pennsylvania State University, University College, York, USA\n\nEmail: [email protected]",
null,
"",
null,
"",
null,
"Received 22 July 2014; revised 18 August 2014; accepted 15 September 2014\n\nABSTRACT\n\nIn polar coordinate system, we consider fifteen classes of forces resulting in unlimited undiscovered orbitals. The classic conic orbits are one of the special subclasses of the fifteen classes. Among the rest of the forces, we show a few instances displaying typical fresh orbitals. Aside from the common theoretical foundation, the specifics of the orbitals are given by the solution of corresponding equations of motion. These are coupled nonlinear differential equations. Solving these equations numerically, utilizing a Computer Algebra System such as Mathematica is conducive to the orbits. Simulation of the orbitals provides a visual understanding about the motion under the influence of the generalized noncentral forces.\n\nKeywords:\n\nGeneralized Orbitals, Noncentral Forces, Mathematica",
null,
"1. Introduction\n\nIn our previous work, we investigated the motion of a massive point-like particle under the influence of semi generalized central forces . In a polar coordinate system, we consider that forces are merely radial and distance dependent. The scope of the investigation is",
null,
"where n is within",
null,
". This specific range includes two particular instances, namely n = ‒2 and 1. The former specifies the gravity and electrostatic forces i.e. the Keplerian forces, and the latter is merely a linear force. Consequently, the former gives the classic conic orbitals and the latter induces the Elliptic Harmonic orbits. The rest of the suggested n values give peculiar orbits discussed in . In our approach, we outline the general theoretical foundation; as such one has the option of selecting unrestricted values for n including reals. Motivated with the outcome of our study, we craft our current analysis. This augments our previous work in three major frontiers. First, we consider radial forces that are not merely distance dependent. Forces such as,",
null,
"and",
null,
", here",
null,
"is the polar angle. The f’s are arbitrary functions; consequently there are unlimited corresponding orbitals. Secondly, in a polar coordinate system, we consider forces that are merely azimuthal such as,",
null,
",",
null,
"and",
null,
". Similar to the previous case g’s are arbitrary functions, therefore there are unlimited orbitals. Thirdly, we envision two-component forces, i.e. forces with radial and azimuthal components. A complete set of such forces are tabulated in Table 1. The table includes fifteen classes. These are called classes because each one embodies countless forces; these are labeled fij and gij. The main objective of our investigation is to apply the classical mechanics analyzing the orbitals of a massive point-like particle undergoing the influence of tabulated forces in Table 1. This work is composed of four sections. In addition to Motivation and Goals, in Section 2 we outline the general foundation of our analysis. In Section 3 for a few random cases, we simulate the orbitals. We close the work with concluding remarks.\n\n2. Analysis of the Physics Problem\n\nFollowing the objectives outlined in the previous section we consider the kinematics of a mobile massive point- like object of mass m in a two-dimensional space. Utilizing the polar coordinate system the acceleration is ,",
null,
"(1)\n\nwhere according to the standard convention a single-dot and a double-dot are the first and the second derivatives with respect to time, respectively. Applying (1) the equation of motion is,",
null,
"(2)\n\nThe RHS of (2) is one of the fifteen cells of Table 1. Accordingly, (2) is a representative equation of motion. For instance in our previous work we consider the impact of the content of cell12; namely",
null,
". More specifically, we consider",
null,
". For this class of forces the absence of the azimuthal component of the force assists decoupling the associated equations results a single differential equation. Solution of the latter gives the desired orbitals. In general applying the latter procedure yields (2),",
null,
"(3)\n\nThe set of equations given in (3) are coupled ODEs. For arbitrary functions such as",
null,
"and",
null,
"most likely (3) is nonlinear. Unless otherwise for appealing cases one doesn’t seek for their analytic solutions. Utilizing Mathematica we solve (3) numerically.\n\n3. Results\n\nIn this section according to Table 1 for a hand full of cases we investigate the specifics of the motion. Generally speaking, (3) is a set of coupled differential equations, aside from the force constant (see next paragraph) their numeric solutions require a set of four initial conditions. As one expects, the solutions are sensitive to the force strength and the initial conditions. Solutions are expressed as",
null,
". Utilizing these we identify the orbitals and quantities of interest. We give examples highlighting the specifics.\n\nExample 1. Consider the well-known gravity and electrostatic force i.e. the Keplerian forces. The force falls\n\nTable 1. Symbolic representation of fifteen possible classes of forces with components along the radial and azimuthal directions in a polar coordinate system.\n\nin the category of cell12 of Table 1. We substitute with; here is the force constant. For in-\n\nstance, in the case of gravity the value of is independent of m and is G M, where G is the universal gravity\n\nconstant and M is the central mass. In the case of charge-charge interaction is where K is the elec-\n\ntric coupling constant and Q, q are the charges of the point-like charges. By trial and error the initial conditions and the force constant are adjusted so that the orbital is a perfect stable circle. One such set of parameters is given in the figure caption of Figure 1(a). It is reassuring that the output of our current approach is the same as . It is noteworthy mentioning that one of the objectives of our current investigation is to numerically solve more challenging issues, see examples 2, 3 and 4; where in compatible with its objectives the emphasis was on less generalized case studies.\n\nFigure 1(a) is comprised of three panels. The left plot displays the Cartesian components of the particle position. These are. The plot shows shifted but identical oscillations of the individual components; the red curve is the x(t) and the blue curve is the y(t), respectively. The middle graph is the display of the radial distance of the particle as a function of time, t. It shows the distance of the particle from the origin is constant. Consequently its polar plot shown in the right panel exhibits a stable circular orbit. This is a classic and well-known result. Its confirmation within the body of our current work creates the forum for the rest of the investigation.\n\nBy adjusting the initial radial velocity the stable circular orbit of Figure 1(a) becomes a perfect, stable elliptical orbit. Figure caption of Figure 1(b) includes the specifics of the force strength and the initial conditions.\n\n(a)(b)\n\nFigure 1. (a) The orbital is a perfect stable circle. The force strength is x = 4.5, and the initial values are ; (b) A perfect stable ellipse. The force strength is x = 4.5 and initial values are.\n\nThe left plot of Figure 1(b) seemingly is similar to the corresponding Figure 1(a). However, there are minor, hard to see differences. These are shown in the middle plot of Figure 1(b). Its mildly wobbling behavior warrants non-circular, elliptic orbit shown in the right panel of Figure 1(b).\n\nIn this case too our current output coincides with our previous work .\n\nExample 2. Here we consider an example associated with the cell13. We equate; where\n\nis the force constant. Aside from given aforementioned comments concerning the value of m along with the other force related quantities, similar to the ones in the Keplerian case is being absorbed in. This example is similar to the previous examples; its pure radial oriented character conserves the angular momentum of the particle. It is noteworthy mentioning that according to one of our objectives, the intent is to augment the body of knowledge concerning the motion under the influence of noncentral forces. Theoretically speaking any reasonable functional form for such forces should work. One such force e.g. is the subject of example 2. There is nothing special about this suggested function, as such any function sustaining azimuthal characteristics would work as well. Specifically speaking, our suggested approach concerning augmenting the scope of the motion under the noncentral forces is a fresh theoretical idea. As such, we do not necessarily need to seek for a real-life evidence for its existence. As a result our suggested proposal and its solution pave the road for the “what-if scenarios.”\n\nFor a set of parameters specified in the figure caption of Figure 2 we display graphic information including the orbital.\n\nDescription of the individual panel is the same as in the previous examples. It is worthwhile noting unlike the previous examples the orbital is not stable. The character of the pure radial force makes the particle orbiting about the center and then wandering away, tracing a non-returnable trajectory. For a better descriptive word maybe in this case “orbital” should be called “trajectory.”\n\nExample 3. Here we consider an example associated with the cell14. We equate; where\n\nis the force constant. The spirit of this theoretical suggested force is similar to the previous example. Meaning, no such force has been observed in nature, yet! However, as mentioned before, the analysis paves the road for the “what-if scenarios.” Similar to the previous examples the angular momentum of the particle is conserved. For a set of parameters specified in the figure caption of Figure 3 the graphic information includes the orbital.\n\nAccording to the plot of the middle panel the radial distance of the particle is a diminishing oscillatory function with respect to time, t. Its polar plot shown in the right panel is an interesting unstable orbital. Here the particle interestingly orbits about the center and contrary to example 2 stays in sight.\n\nExample 4. As a last example we consider a case associated with cell32. We equate and; here x is the force constant. Our approach is quite general such that one may even adjust the\n\nindividual force strength x along the radial and the azimuthal directions. This example, unlike the previous ex-\n\nFigure 2. The force strength is x = 2.25, and initial values are.\n\nFigure 3. The force strength is x = 4.5,and initial values are.\n\nFigure 4. The common force strength is x = 5.15 and initial values are.\n\namples, doesn’t conserve the angular momentum. For a set of parameters specified in the figure caption of Figure 4 the graphic information includes the orbital.\n\nThe orbital shown in the right panel is somewhat interesting. The rest of the plots are self-explanatory.\n\n4. Conclusions\n\nMotion of a particle under the influence of conventional forces such as gravity and electrostatic is known . These forces are purely radial and do conserve the angular momentum. The corresponding equations of motion are solvable analytic differential equations. The orbitals are stable conic sections. Advances in CAS make it possible to augment the area of the study. As such, our current work investigates the impact of generalized forces on the orbitals. Our work even augments our previous work . We have introduced fifteen classes of forces. We show the details of only a handful of cases. However, according to what we have introduced in the text, our approach recognizes no limitation. As we mentioned in the Motivation and Goals section and in the body of the text of example 2, the freedom of choosing the force functions in any of the fifteen classes puts no limit on our theoretical investigation. To the author’s knowledge, there are no other comparably investigated reports in this area.\n\nOur approach is purely numeric and is based on numeric solution of differential equations provided by Mathematica. We include a Mathematica code so that the interested reader is able to apply the code investigating the orbitals/trajectories of a particular force.\n\nAcknowledgements\n\nThe author appreciates the referee’s comments.\n\nReferences\n\n1. Sarafian, H. (2014) Central Conservative Forces and Orbits beyond Conic Sections, Progress on Difference Equations. Abstract Book, 58, Izmir University of Economics, Turkey.\n2. Sarafian, H., Takato, S., and Kaneko, M. (2014) Central Conservative Forces and Orbits beyond Conic Sections. The Journal of Mathematics and System Sciences, 4, 579-585. www.davidpublishing.org/journals_info.asp?jId=2039\n3. Thronton, S. and Marion, J. (2003) Classical Dynamics of Particles and Systems. 5th Edition, Cengage Learning, Boston.\n4. Symon, K.R. (1965) Mechanics. 2nd Edition, Addison-Wesley, New York.\n5. Wolfram, S., (2012) Mathematica: A Computational Software Program Based on Symbolic Mathematics, V9.0.\n6. Goldstein, H., Safko, J. and Poole, C. (2001) Classical Mechanics. 3rd Edition, Addison-Wesley, New York.\n7. Arya, A.P. (1990) Introduction to Classical Mechanics. Allyn and Bacon, London.\n\nAppendix\n\nMathematica code. Here is the code that runs Example 4 in the text. In this example the right hand sides of (3) are −xr(t) and −xq(t), respectively. However, the given code can run for any desired case. To do so in the first line of the code one needs to replace the right hand side of the corresponding equations with the appropriate functions."
] | [
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https://www.intechopen.com/books/chaos-theory/chaos-on-set-valued-dynamics-and-control-sets | [
"Open access peer-reviewed chapter\n\n# Chaos on Set-Valued Dynamics and Control Sets\n\nBy Heriberto Román-Flores and Víctor Ayala\n\nSubmitted: April 25th 2017Reviewed: November 6th 2017Published: March 28th 2018\n\nDOI: 10.5772/intechopen.72232\n\nDownloaded: 286\n\n## Abstract\n\nThe aim of this chapter is threefold. First, we show some advances in complexity dynamics of set-valued discrete systems in connection with the Devaney’s notion of chaos. Secondly, we start to explore some relationships between control sets for the class of linear control systems on Lie groups with chaotic sets. Finally, through several open problems, we invite the readers to give a contribution to this beauty theory.\n\n### Keywords\n\n• chaos\n• set-valued maps\n• dynamic\n• Devaney\n• control sets\n\n## 1. Introduction\n\nRelevant classes of real problems are modelled by a discrete dynamical system\n\nxn+1=fxn,n=0,1,2,E1\n\nwhere Xdis a metric space and f:XXis a continuous function. The basic goal of this theory is to understand the nature of the orbit Oxf=fnx/n=012for any state xX,as nbecomes large and, in general this is a hard task. The study of orbits says us how the initial states are moving in the base space Xand, in many cases, these orbits present a chaotic structure. In 1989 in , Devaney isolates three main conditions which determine the essential features of chaos.\n\nDefinition 1 Let Xbe a metric space and f:XXa continuous map. Hence, f.\n\n1. is transitive if for any couple of non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fkUV.\n\n2. is periodically dense if the set of periodic points of fis a dense subset of X.\n\n3. has sensitive dependence on initial conditions if there is a positive number δ(a sensitivity constant) such that for every point xXand every neighbourhood Nof xthere exists a point yNand a non-negative integer number nsuch that dfnxfnyδ.\n\nNext, we mention a remarkable characterisation of transitive maps. In fact, as a consequence of the Birkhoff Transitivity Theorem (see for details), it is possible to prove.\n\nProposition 2 Let Xbe a complete metric space which is also perfect (closed and without isolated points). If f:XXis continuous, then fis transitive if and only if there exists at least one orbit Oxfdense in X.\n\nRemark 3 Also, other concepts very useful in this work are the following: i) fis weakly mixing iff for any non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fkUVand fkVV. ii) fis mixing iff given two non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fnUVfor all nk. iii) fis exact iff given a non-empty open subsets Uthere exists a natural number ksuch that fkU=X. It is clear that fexact fmixing fweakly mixing ftransitive.\n\nIt is worth to point out that sensitivity dependence on initial conditions was widely understood as being the central idea in chaos for many years. However, in a surprising way, Banks et al. has proved that transitivity and periodically density imply sensitivity dependence (for details see ). Furthermore, for continuous functions on real intervals, Vellekoop and Berglund in show that transitivity by itself is sufficient to get chaos. This last result is not necessarily true in other type of metric spaces (see Example 4.1 in ).\n\nHowever, sometimes we need to know information about the collective dynamics, i.e. how are moved subsets of Xvia iteration or dynamics induced by f. For example, if Xdenotes an ecosystem and xX, then, by using radio telemetry elements, we can obtain information about the movement of xin the ecosystem X. In this form, it is possible to build an individual displacement function f:XX. Of course, this function could be chaotic or not. Eventually, we could also be interested to get information about the collective dynamics induced by f, means, to follow the dynamics of a group of individuals. Thus, in a natural way the following question appears: what is the relationship between individual and collective dynamics? This is the main topic of this chapter.\n\nGiven the system (1), consider the set-valued discrete system associated to fdefined by\n\nAn+1=f¯An,n=0,1,2,E2\n\nwhere f¯is the natural extension of fto the metric space KXHof the non-empty compact subsets of Xendowed with the Hausdorff metric Hinduced by the original distance dof X.\n\nIn a more general set up, this work is strictly related with the following fundamental question: what is the relationship between individual and collective chaos?\n\nAs a partial response to this question, in this chapter we search the transitivity of a continuous function fon Xin relation to the transitivity of its extension f¯to KX.Our main result here establishes that f¯transitive implies ftransitive. That is to say, collective chaos implies individual chaos under the dynamics of f¯.\n\nOn the other hand, we propose a new approach to this problem: to study the dynamics induced by fon the subextension KcXof KX.Precisely, on the class of non-empty compact-convex subsets of X. We prove that the induced dynamics is less chaotic than the original one!\n\nFinally, we mention that some relevant problems in the theory of control systems can be also approached by the theory of set-valuated map. In fact, to any initial state xof the system, one can associate its reachable set Ax. In other words, Axcontains all the possible states of the manifold that starting from xyou can reach in non-negative time by using the admissible control functions Uof the system. The aim of this section is twofold. First of all, to apply to the class of linear control systems on Lie groups, the existent relationship between control sets of an affine control system Σon a Riemannian manifold Mwith chaotic sets of the shift flow induced by Σon M×U,. In particular, we are looking for the consequences of this relation on the controllability property. At the very end, we propose a challenge to the readers to motivate the research on this topic through some open problem relatives to the mentioned relationship.\n\n## 2. Preliminaries\n\nIn this section, we mention some notions and fundamental results we use through the chapter.\n\n### 2.1. Extensions\n\nIf Xdis a metric space and f:XXcontinuous, then we can consider the space KXHof all non-empty and compact subsets of Xendowed with the Hausdorff metric induced by dand f¯:KXKX,f¯A=fA}, the natural extension of fto KX. Also, we denote by KcX=AKX/Aisconvex. If AKXwe define the “ϵ-dilatation of A” as the set NAϵ=xX/dxA<ϵ, where dxA=infaAdxa.\n\nThe Hausdorff metric on KXis given by\n\nHAB=infϵ>0/ANBϵandBNAϵ.\n\nWe know that KXHis a complete (separable, compact) metric space if and only if Xdis a complete (separable, compact) metric space, respectively, (see [3, 7, 8]).\n\nAlso, if AKX, the set BAϵ=BKX/H(AB)<ϵdenotes the ball centred in Aand radius ϵin the space KXH.\n\nFurthermore, given a continuous function IdfIdon a real interval I, we also consider the extension KcIHf¯cKcIH, where f¯cis the restriction f¯KcI.\n\n### 2.2. Baire spaces\n\nIn this section, we review some properties of Baire spaces.\n\nDefinition 4 A topological space Xis a Baire space if for any given countable family of closed sets An:nNcovering X, then intAnfor at least one n.\n\nDefinition 5 In any Baire space X,\n\n1. DXis called nowhere dense if intclD=.\n\n2. Any countable union of nowhere dense sets is called a set of first category.\n\n3. Any set not of first category is said to be of second category.\n\n4. The complement of a set of first category is called a residual set.\n\nRemark 6 It is important to note that:\n\n1. Any complete metric space is a Baire space.\n\n2. Every residual set is of second category in X.\n\n3. Every residual set is dense in X.\n\n4. The complement of a residual set is of first category.\n\n5. If Bis of first category and AB, then Ais of first category.\n\n(For details, see [8, 9, 10])\n\nIn particular, if X=Iis an interval, then CXand CXR, endowed with the respective supremum metrics, are Baire spaces.\n\nIn a Baire space X, we say that “most elements of X” verify the property (P) if the set of all xXthat do not verify property (P) is of first category in X. In this form, sets of second category can be regarded as “big” sets. A relevant area of the real analysis is to estimate the “size” of some sets associated to a continuous interval function fsuch as the set Pfof periodic points of f, or the set Ffof fixed points of f. Typically, continuous interval functions have a first category set of periodic points (see ) and, in particular, a first category set of fixed points. It has also been recently proved that a typical continuously differentiable interval function has a finite set of fixed points and a countable set of periodic points (see and references therein). It is also well-known that the class of nowhere differentiable functions NDIis a residual set in CI(see [13, 14]). Also, a special class of functions in CIis the class CNLIof all continuous functions whose graphs “cross no lines” defined in a negative way as follows (see ):\n\nDefinition 7 Let f:ababa continuous map and L:RRa function whose graph is a straight line. We say that Lcrosses f(or fcrosses L) if there exists x0aband δ>0such that fx0=Lx0and either.\n\n(a) Lxfxfor all xx0δx0aband Lxfxfor all xx0x0+δab; or.\n\n(b) Lxfxfor all xx0δx0aband new Lxfxfor all xx0x0+δab.\n\nThe following result can be found in :\n\nTheorem 8 () The set CNLI=fCI/fcrossesnolinesis residual in CI.\n\nThe set CNLIwill play an important role in the next sections.\n\n### 2.3. The dynamics of control theory\n\nIn Section 7, we propose some challenges through the relationship between the notion of chaotic sets in the Devaney sense and control sets for the class of Linear Control Systems on Lie Groups, . In particular, we explicitly show some results concerning the controllability property in terms of chaotic dynamics.\n\nIn the sequel, we follow the relevant book The Dynamics of Control by Colonius and Kliemann, . Let Mbe a ddimensional smooth manifold. By an affine control system Σin M, we understand the family of ordinary differential equations:\n\nΣ:ẋt=Xxt+j=1mujtYjxt,u=u1umUE3\n\nwhere X,Yj, j=0,1,,mare arbitrary Cvector fields on M.The set ULRΩRmis the class of restricted admissible control functions where ΩRmwith 0intΩ,is a compact and convex set.\n\nAssume Σsatisfy the Lie algebra rank condition, i.e.\n\nforanyxMSpanLAXY1Ymx=d.\n\nOf course, LAmeans the Lie algebra generated by the vector fields through the usual notion of Lie bracket. Furthermore, the ad-rank condition for Σis defined as follows:\n\nforanyxMSpanadiYj:j=1mandi=01x=d.\n\nFor each uUand each initial value xM, there exists an unique solution φtxudefined on an open interval containing t=0,satisfying φ0xu=x. Since we are concerned with dynamics on Lie Groups, without loss of generality we assume that the vector fields X,Y1,,Ymare completes. Then, we obtain a mapping Φsatisfying the cocycle property\n\nΦ:R×M×UM,txuΦtxuandΦt+sxu=ΦtΦsxuΘsu\n\nfor all t,sR, xM, uU.Where, for any tR, the map Θtis the shift flow on Udefined by Θsutut+s.Hence, Φis a skew-product flow. The topology here is given by the product topology between the topology of the manifold and the weak* topology on U.\n\nIt turns out the following results.\n\nLemma 9 Consider the set Uequipped with the weak* topology associated to LRRm=(L1RRmas a dual vector space. Therefore,\n\n1. Udis a compact, complete and separable metric space with the distance given by\n\ndu1u2=n=112nR<u1tu2tvnt>dt1+R<u1tu2tvnt>dt.\n\nHere, vn:nNL1RRmis a dense set of Lebesgue integrable functions.\n\n• The map Θ:R×UUdefines a continuous dynamical systems on U. Its periodic points are dense and the shift is topologically mixing (and then topologically transitive).\n\n• The map Φdefines a continuous dynamical system on M×U.\n\n• On the other hand, the completely controllable property of Σ,i.e. the possibility to connect any two arbitrary points of Mthrough a Σ-trajectory in positive time, is one of the most relevant issue for any control system. But, few systems have this property. A more realistic approach comes from a Kliemann notion introduced in .\n\nDefinition 10 A non-empty set CMis called a control set of (3) if.\n\n1. for every xMthere exists uUsuch that φtxu:t0C\n\n2. for every xC, CclAx\n\n3. Cis maximal with respect to the properties iand ii.\n\nAxdenotes the states that can be reached from xby Σin positive time and clits closure\n\nAx=yM:uUandt>0withy=φtxu.\n\nMoreover, for an element xM, the set of points that can be steered to xthrough a Σ-trajectory in positive time is denoted by\n\nAx=τ>0yM:uUe=φτ,ux.\n\nFinally, we mention that the Lie algebra rank condition warranty that the system is locally accessible, which means that for every τ>0,\n\nintAτxandintAτxarenon empty,foranyxM.\n\n## 3. f¯transitive implies ftransitive\n\nAs we explain, in terms of the original dynamics and its extensions a natural question arises: what are the relations between individual and collective chaos? As a partial response to this question, in the sequel, we show that the transitivity of the extension f¯implies the transitivity of f.For that, we need to describe some previous results.\n\nLemma 11 Let Abe a non-empty open subset of X. If KKXand KA,then there exists ϵ>0such that NKϵA..\n\nDefinition 12 Let AXbe. Then the extension of Ato KXis given by eA=KKX/KA.\n\nRemark 13 eA=A=..\n\nLemma 14 Let AXbe, A, an open subset of X. Then, eAis a non-empty open subset of KX.\n\nLemma 15 If A,BX, then: i) eAB=eAeB, ii) f¯eAefA, and iii) f¯p=fp¯,for every pN.\n\nNow, we are in a position to prove the following results\n\nTheorem 16 Let f:XXbe a continuous function. Then, f¯transitive implies ftransitive.\n\nProof: Let A,Bbe two non-empty open sets in X. Due to Lemma 13, eAand eBare non-empty open sets in KX. Thus, by transitivity of f¯, there exists some kNsuch that\n\nf¯keAeB=fk¯eAeB\n\nand, from Lemma 14, we obtain\n\nefkAeB=efkAB\n\nwhich implies fkABand, consequently, fis a transitive function.\n\n## 4. Two examples\n\nNow we show that, in general, the converse of Theorem 15 is not true.\n\nExample 4.1 (Translations of the circle). If λRis an irrational number and we define Tλ:S1S1by Tλe=eiθ+2πλ, then it was shown by Devaney that each orbit Tλne/nNis dense in S1and, due Proposition 2, Tλis transitive. Nevertheless, Tλhas no periodic points and, because Tλis isometric, it does not exhibit sensitive dependence on initial conditions either.\n\nIf KKS1, because Tλ¯preserves diameter, then diamK=diamTλ¯nK, for all nN.\n\nNow, let KKS1such that diamK=1,and let ϵ>0sufficiently small. Then\n\nFU=BKϵdiamF1GV=B1ϵdiamG0.\n\nThus, diamTλ¯nF1nNand, consequently, Tλ¯nUV=for all nN, which implies that Tλ¯is not transitive on KS1.\n\nExample 4.2 Define the “tent” function f:0101as fx=2xif0x1/2and fx=21xif1/2x1.\n\nIt is not difficult to show that fis an exact function on [0,1]. In fact, intuitively we can see that, after each iteration, the number of tent in the graphics is increasing, whereas the base of each tent is decreasing and they are uniformly distributed over the interval 01.\n\nThus, if Uis an arbitrary non-empty open subset of 01, then Ucontains an open interval Jand, after certain number of iterations, there exists a tent, with height equal to one, whose base is contained in J, which implies that fU=01and, according to Remark 3, fis an exact mapping and, consequently, fis a mixing function.\n\nThe conclusions in Examples 4.1 and 4.2 come from the next result, Banks in 2005.\n\nTheorem 17 If f:XXis continuous, then the following conditions are equivalent:\n\ni) fis weakly mixing, ii) f¯is weakly mixing, iii) f¯is transitive.\n\nHitherto, we have used the strong topology induced by the H-metric on KX. However, considering the we-topology on KXgenerated by the sets eAwith Aan open set in X, we obtain the following complementary result, see :\n\nTheorem 18 For a continuous map f:XXthe following conditions are equivalent:\n\ni) fis transitive in Xd, ii) f¯is transitive in the we-topology.\n\n## 5. Sensitivity and periodic density of f¯\n\nLet f:XXbe a continuous function and let f¯be its corresponding extension to the hyperspace KX. Then, the study of sensitivity of fin the base space in relation to the sensitivity of f¯on KXhas been very exhaustively analysed in the last years. Román and Chalco published the first result in this direction in 2005, where the authors prove\n\nTheorem 19 f¯sensitively dependent implies fsensitively dependent.\n\nProof: If f¯has sensitive dependence, then there exists a constant δ>0such that for every KKXand every ϵ>0there exists GBKϵand nNsuch that HfnKfnGδ.\n\nNow, let xXbe and ϵ>0. Then, taking K=xKX, we have that there exists GBxϵand nNsuch that HfnxfnG=HfnxfnGδ.\n\nThus, HfnxfnG=supyGdfnxfnyδand, due to the compactness of Gand the continuity of f, there exists y0Gsuch that HfnxfnG=dfnxfny0δ.\n\nBut, GBxϵimplies GBxϵand, consequently, y0Bxϵ. This proves that fis sensitively dependent (with constant δ).\n\nThe reverse of this theorem is not true. In fact, recently Sharma and Nagar show an example where Xdis sensitive but KXHis not. Now, in order to overcome that shortcoming, the authors in introduce the following notion of sensitivity:\n\nDefinition 20 (Stronger sensitivity ). Let f:XXbe a continuous function. Then fis strongly sensitive if there exists δ>0such that for each xXand each ϵ>0, there exists n0Nsuch that for every nn0, there is a yXwith dxy<ϵand dfnxfny>δ.\n\nObviously, the notion of stronger sensitivity is more restrictive than sensitivity, and the authors in obtain the following results:\n\nTheorem 21 If f:XXis a continuous function and KXHf¯is strongly sensitive then Xdfis strongly sensitive.\n\nIn the compact case, it is possible to obtain a characterization as follows.\n\nTheorem 22 Let Xdbe a compact metric space and f:XXa continuous function. Then KXHf¯is strongly sensitive if and only if Xdfis strongly sensitive.\n\nIn connection with these results, recently Subrahmomian (, 2007) has been shown that most of the important sensitive dynamical systems are all strongly sensitive (the author here calls them cofinitely sensitive). Hence, we can say that for most cases, sensitivity is equivalent in both cases Xdand KXH. It turns out that, strongly sensitivity and sensitivity are equivalent on the class of interval functions, which implies that\n\nTheorem 23 If f:IIis a continuous function, the following conditions are equivalent.\n\na) Idfis sensitive, b) KIHf¯is sensitive.\n\nWe finish this section assuming the existence of a dense set of periodic points for f¯, we have\n\nTheorem 24 Let Xdbe a compact metric space and f:XXa continuous function. If f:XXhas a dense set of periodic points then f¯:KXKXhas the same property.\n\nProof: Let KKXand ϵ>0. Then there exists a ϵ/2-net covering K, That is to say, there are x1,,xpin Ksuch that KBx1ϵ/2Bxpϵ/2.Because fhas periodic density, there are yiXand niNsuch that:\n\nyiBxiϵ/2,i=1,,pandfniyi=yi,i=1,,p.\n\nNow, take G=y1yp.By construction, we have HKG<ϵand, moreover, fn1n2npyi=yi, for all i=1,,p. Therefore, fn1n2npG=G, which implies that f¯has periodic density.\n\nThe converse of this theorem is no longer true (for a counterexample, see Banks ). However, to find conditions on f¯warranting the existence of a dense set of periodic points for fis a very hard problem which still remains open.\n\n## 6. The dynamics on the KcIHextension\n\nIn the previous sections, we have studied the diagram\n\nKXHf¯KXHXdfXdE4\n\nand the chaotic relationships between fand f¯. However, in the setting of mathematical modelling of many real-world applications, it is necessary to take into account additional considerations such as vagueness or uncertainty on the variables. This implies the use of interval parameters and, consequently, to deal with interval systems. That is, it is necessary to consider an interval X=Iand to study the following new diagram:\n\nKcIHf¯cKcIHIdfIdE5\n\nalong with the analysis of the connection between their respective dynamical relationships. Here f¯cdenotes the restriction of f¯to KcI, the class of all compact subintervals of I. For A=ab,B=cdKcI, the Hausdorff metric can be explicitly computed as\n\nHAB=maxacbd.E6\n\nThe aim of this section is to show that the Devaney complexity of the extension f¯con KcIis less or equal than the complexity of fon the base space I. More precisely, f¯cis never transitive for any continuous function fCI. Also, we will show that f¯chas no dense set of periodic points for most functions fCI.Finally, we prove that f¯chas no sensitive dependence for most functions fCI.\n\nAs a motivation, we present the following examples.\n\nExample 6.1 Consider the “tent” function f:0101defined by\n\nfx=2xif0x1221xif12x1.\n\nThen it is well known that fis D-chaotic on 01(see ). Moreover, because fis a mixing function on 01, then f¯is transitive on K01(see ). Also, we observe that x=23is a fixed point of f. On the other hand, it is clear that if Kis a compact and convex subset of X=01, then f¯Kis also a compact and convex subset of X. Consequently, if we let Kc01denote the class of all closed subintervals of 01, then we can consider f¯cas a mapping f¯c:Kc01Kc01. We recall that Kc01is a closed subspace of K01(see ). Now, considering the open balls B01110and B0110in Kc01, we have.\n\nKB0111023Kwhich implies 23f¯cpK,pN..\n\nOn the other hand, if FB0110, then F01/10. Consequently, Hf¯cpKF1730for every KB01110and FB0110.\n\nTherefore,\n\nf¯cpB01110B0110=,pN.\n\nThus, f¯cis not transitive on Kc01.\n\nExample 6.1 shows a function fwhich is transitive on the base space X=01and f¯is also transitive on the total extension K01, but f¯cis not transitive on the subextension Kc01.\n\nThe following example shows a function f:010,1]with a dense set of periodic points, and where the total extension of fto K01also has a dense set of periodic points, whereas f¯cdoes not have a dense set of periodic points on Kc01.\n\nExample 6.2. Let X=01and consider the “logistic” function f:0101defined by fx=4x1x. It is well known that fis D-chaotic on 01(see ). Moreover, fis a mixing function. Thus, in particular, fhas a dense set of periodic points and, therefore, f¯also has a dense set of periodic points on the total extension K01) (see Theorem 24).\n\nHowever, f¯chas no a dense set of periodic points on KcX.\n\nIn order to see this, we claim that the open ball B183818in Kc01Hdoes not contain periodic points of f¯c.\n\nIn fact, if K=cdB183818, then c18<18and d38<18, which implies that 0<c<14and 14<d<12.\n\nThus, we obtain that 14K34fKfKK.\n\nOn the other hand,\n\n34fK34fnK,n2fnKK,n1\n\nand, consequently, f¯chas no periodic points in the ball B183814Kc01H, which implies that f¯chas no dense set of periodic points on Kc01H.\n\nLemma 25 f¯ctransitive on Kcabimplies ftransitive on ab.\n\nProof. Let U,Vnon-empty open subsets of X=ab. We can choose xU, yVand ϵ>0such that BxϵUand ByϵV. Now, in Kcabconsider the open balls Bxϵand Byϵwith respect to the H-metric. Due to the transitivity of f¯con Kcab, there exists nNsuch that f¯cnBxϵByϵ.\n\nTherefore, there exists an interval JBxϵsuch that f¯cnJ=fnJByϵ. However, JBxϵand, analogously, fnJByϵ, which implies that fnBxϵByϵand, consequently, fnUV. And fis a transitive function on ab.\n\nIt is well-known that if X=Iis an interval, then most functions fCIhas no dense orbits, that is to say, there exists a residual set DCIsuch that every function fDhas no point whose orbit is dense in I(see ) and, consequently, most functions fCIare not transitive. From Lemma 24, we can conclude that f¯cis not transitive for most functions fCI.\n\nThe next theorem provides a stronger result.\n\nTheorem 26 Let f:ababbe continuous. Then f¯cis not transitive on Kcab.\n\nProof. By Schauder Theorem, fhas at least one fixed point pab.\n\nCase 1. Suppose that paband let r=maxpabp. Without loss of generality, we can suppose that r=paand, because a<b, it is clear that r>0.\n\nNow, let r=bp>0and let ϵ=r2. If we consider the open balls Babϵ,BaϵKcab, it follows that KBabϵpKpf¯nKfor any nN.\n\nOn the other hand,\n\nFBaϵHFa<ϵFa,a+ϵ].\n\nBecause r<rwe get\n\nHf¯nKFpaϵ=rr2>0\n\nfor each KBabϵ,FBaϵand for any nN. Thus,\n\nf¯nB(abϵ)Baϵ=,nN.\n\nConsequently, f¯is not transitive on Kcab.\n\nCase 2. Suppose that fhas no fixed points in ab. From the continuity of f, we have that fx>xfor all xabor fx<xfor all xab. This clearly implies that fis not a transitive function, and consequently, due to Lemma 24, f¯cis not transitive on Kcab.\n\nAn important question to answer is what about the size of the set of periodic points of f¯c. It is clear that there are some functions fCIwith a dense set of periodic points on I, and such that their extensions f¯calso has a dense set of periodic points on KcI(for instance, fx=x). Therefore, an analogous result to Theorem 26, but for periodic density of f¯c, cannot be obtained. However, as we will see, most functions fCIdo not have an extension f¯cwith a dense set of periodic points on KcI. To prove it, we need the following lemma.\n\nLemma 27 Let Ibe a compact interval in R, and f:IIbe a continuous function. If we suppose that f¯chas periodic density on KcI, then fhas periodic density on I.\n\nProof. If x0Iand ϵ>0then x0KcIand, consequently, there exists KKcIand nNsuch that\n\n1. Hx0K<ϵ\n\n2. f¯cnK=K.\n\nCombining a. and b. we get\n\ndx0fnx<ϵ,forallxK.E7\n\nBecause f¯nK=fn¯K=fnK=Kand fnis continuous on Kthen, by the Schauder’s Fixed Point Theorem, there exists xpKsuch that fnxp=xp. Thus, xpis a periodic point of fand, due to (7), we obtain dx0xp<ϵ. Hence, fhas periodic density on I.\n\nTheorem 28 Let I=abbe a compact interval in R. Then f¯cdoes not have a dense set of periodic points in KcI, for most functions fCI.\n\nProof. The proof is based on an exhaustive analysis of the behaviour of the fixed points of f. We connect this analysis with an adequate residual set in CI. The analysis of each fixed point of fis fundamental to decide whether the function fallows or not an extension f¯cthat has a dense set of periodic points. More precisely, the behaviour of each fixed point will imply only two (mutually exclusive) options:\n\n1. f¯cdoes not have a dense set of periodic points, or.\n\n2. fCNLIc, which is a set of first category in CI.\n\nTowards this end, let f:ababbe a continuous function. By the Schauder’s Fixed Point Theorem, fhas at least one fixed point pab. The proof is divided in.\n\nCase1.fhas no fixed points in ab.\n\nIn this case, we have the following three subcases:\n\n1i)p=ais the unique fixed point of f.\n\nWe have, either\n\nfx>x,xabx<fx<f2x<<fnx<,or\nfx<x,xabx>fx>f2x>>fnx>.\n\nIn both cases it follows that fhas no periodic points in ab.\n\n1ii)p=bis the unique fixed point of f.\n\nThis case is analogous to the case 1i).\n\n1iii)p=aand p=bare the unique fixed points of f.\n\nThis case is also analogous to the cases 1i)and 1ii).\n\nTherefore, in case 1the function fdoes not have a dense set of periodic points in ab. Due to Lemma 24, f¯cdoes not have a dense set of periodic points in Kcab.\n\nCase2.fhas at least one fixed point pab.\n\nWe have the following subcases:\n\n2i)qab,qpsuch that fq=p.\n\nWithout loss of generality, suppose that qap. Then, taking 0<ϵ<minqa2pq2, we can consider the open ball Bqϵq+ϵϵin the space Kcab. If J=cdBqϵq+ϵϵ, from (6) we have\n\ncqϵ<ϵanddq+ϵ<ϵ\n\nwhich implies that a<c<qand q<d<pand, consequently, qJwhereas pJ. Thus,\n\nqJfq=pfJfJJ.E8\n\nOn the other hand, pfJimplies that\n\npfnJ,n2fnJJ,n2,E9\n\nand, consequently, f¯chas no periodic points in the ball Bqϵq+ϵϵKcabH, which implies that f¯cdoes not have a dense set of periodic points on KcabH.\n\n2ii)q=a,qp, is the unique point such that fa=p.\n\nWithout loss of generality, we can suppose that fx>p, for all xap.\n\nNow, in addition to hypothesis 2ii), we have two subcases:\n\n2iia1)fdoes not cross the line y=pand fx>pfor all xap.\n\nIn this situation, fxpfor all xa,b]. Thus, choosing qapand 0<ϵ<maxqa2pq2, we can consider the open ball Bqϵto have\n\nK=cdBqϵKap.E10\n\nFrom our hypothesis, we obtain\n\nfnz>p,zK,nN,E11\n\nwhich implies that fnKK, nN. Consequently, f¯chas no periodic points in the ball Bqϵ. In other words, f¯cdoes not have a dense set of periodic points in KcI.\n\n2iia2)fdoes not cross the line y=pand fx<pfor all xap.\n\nIn this case, fxpfor all xab. Thus, choosing qpband 0<ϵ<maxqp2bq2, we can consider the open ball Bqϵto obtain\n\nK=cdBqϵKpb.E12\n\nAgain, from our hypothesis, we get\n\nfnz<p,zK,nN,E13\n\nwhich implies that fnKK, nNand, consequently, f¯chas no periodic points in the ball Bqϵ. In other words, f¯cdoes not have a dense set of periodic points in KcI.\n\n2iib)fcrosses the line y=p.\n\nIt is clear that, in this case, fCNLIcwhich, due to Theorem 8 and Remark 6, is a set of first category in Cab.\n\n2iii)q=b,qp, is the unique point such that fb=p.\n\nThis case is analogous to case 2ii)and, consequently, if fdoes not cross the line y=pthen f¯cdoes not have a dense set of periodic points in KcI, whereas if fcrosses the line y=p, then fCNLIc.\n\n2iv)q1=aand q2=b, q1,q2p, are the unique points such that fa=fb=p.\n\nIn this case, we have the following subcases:\n\n2iva1)fdoes not cross the line y=pand fx>pand fx>pfor all xab\\p.\n\nThis case is analogous to the case 2iia1)and the same is true for 2iva2)when fdoes not cross the line y=pand fx<pfor all xab\\pwhich is analogous to the case 2iia2)Finally, there only remains two subcases:\n\n2ivb1)fcrosses the line y=pand fx>pin apand fx<pin pb, and.\n\n2ivb2)fcrosses the line y=pand fx<pin apand fx>pin pb.\n\nIt is clear that in both cases fCNLIc.\n\nThus, as a direct consequence of the analysis of the behaviour of the set of fixed points of f, it turns out that the unique cases in which fcould have an extension f¯cwith a dense set of periodic points on KcIare when there exists a fixed point pof fsuch that fcrosses the line y=pat x=p. In other words, we obtain\n\nHDSI=fCI/f¯chasadensesetofperiodicpointsinKcIHDSICNLIc,\n\nBut, CNLIis a residual set in CI,therefore from Remark 6, we conclude that HDSIis of first category in CI. Equivalently, f¯cdoes not have a dense set of periodic points, for most functions fCI, which ends the proof.\n\nFinally based on the following result,\n\nTheorem 29 () For most functions fCI, the set of all points where fis sensitive is dense in the set of all periodic points of f.\n\nwe show an analogous result for the sensitivity property, as follows.\n\nTheorem 30 For most functions fCI, the extension f¯cCKcIis not sensitive.\n\nProof. This is a direct consequence of Theorem 28 and Theorem 29.\n\n## 7. Control sets of linear systems and chaotic dynamics\n\nThe aim of this section is twofold. First of all, to start to apply to the class of linear control systems on Lie groups, the existent relationship between control sets of an affine control system Σon a Riemannian manifold Mwith chaotic sets of the shift flow induced by Σon M×U,. In particular, we are looking for the consequences of this relation on the controllability property The second part is intended to motivate the research on this topic to writing down some open problems relatives to this relationship.\n\n### 7.1. Linear control systems on lie groups\n\nLet Gbe a connected ddimensional Lie group with Lie algebra g. A linear control system ΣLon Gis an affine system determined by\n\nΣL:ẋt=Xxt+j=1mujtYjxt,u=u1umUE14\n\nwhere Xis linear, that is, its flow XttRis a one-parameter group of G-automorphism, the control vectors Yj,j=1,,mare invariant vector fields, as elements of g. The restricted class of admissible control Uis the same as before.\n\nCertainly, the drift vector field Xis complete and the same is true for every invariant vector field Yj,j=1,,m. As usual, we assume that ΣLsatisfy the Lie algebra rank condition, i.e.\n\nforanyxMSpanLAXY1Ymx=d.\n\nThe system is said to be controllable if Ae=Ais G.\n\nThe class of systems ΣLis huge and contains many relevant algebraic systems as the classical linear and bilinear systems on Euclidean spaces , and the class of invariant systems on Lie groups, . Furthermore, according to the Jouan Equivalence Theorem , ΣLis also relevant in applications. It approaches globally any affine non-linear control system Σon a Riemannian manifold when the Lie algebra of the dynamics of Σis finite dimensional.\n\nOne can associate to Xa derivation Dof gdefined by DY=XYe,Yg.Indeed, the Jacobi identity shows DXY=DXY+XDYis in fact a derivation. The relation between φtand Dis given by the formula\n\nφtexpY=expetDY,foralltR,Yg.\n\nConsider the generalised eigenspaces of Ddefined by\n\ngα=Xg:DαnX=0forsomen1\n\nwhere αSpecD. Then, gαgβgα+βwhen α+βis an eigenvalue of Dand zero otherwise. Therefore, it is possible to decompose gas g=g+g0g, where\n\ng=g+g0g,where\ng+=α:Reα>0gα,g0=α:Reα=0gαandg=α:Reα<0gα.\n\nActually, g+,g0,gare Lie algebras and g+, gare nilpotent. Denote by G+, Gand G0the connected and closed Lie subgroups of Gwith Lie algebras g+, gand g0respectively.\n\nDespite the fact that for an invariant system the global controllability property is local, this class has been studied for more than 50 years, see and the references there in. The important point to note here is: for an invariant system the reachable set from the identity is a semigroup. However, in the authors show that this is not the case for a linear system which turns the problem more complicated. Therefore, we would like to explore the mentioned connection between control sets and the Devaney and Colonius-Kliemann ideas. This section is the starting point for the ΣLclass. We begin with a fundamental result.\n\nTheorem 31 Assume the system ΣLsatisfy the Lie algebra rank condition. Therefore, there exists a control set\n\nCe=clAeAe\n\nwhich contains the identity element ein its interior. Here, Aeis the set of states of Gthat can be sent by ΣLto ein positive time.\n\nFor a proof in a more general set up, see .\n\nRecently, we were able to establish some algebraic, topological, and dynamical conditions on ΣLto study uniqueness and boundness of control sets and it consequences on controllability .But, the state of arts is really far from being complete. In order to approach this problem for ΣL, as in we assume here that Ghas finite semisimple centre, i.e. all semisimple Lie subgroups of Ghave finite center. We notice that any nilpotent and solvable Lie group, and any semisimple Lie group with finite centre has the finite semisimple centre property. But also, the product between groups with finite semisimple centre have the same property. We also assume that Ais open. This is true if for example, the system satisfy the ad-rank condition. About the uniqueness and boundness of control sets of a linear systems, we know few things .\n\nTheorem 32 Let ΣLa linear control system on the Lie group G.\n\n1. If G=GG0G+is decomposable, Ceis the only control set with non-empty interior. In particular, this is true for any solvable Lie group.\n\n2. Suppose that Gis semisimple or nilpotent, it turns out that\n\nifclAG,clAG+andG0arecompact setsCis bounded.\n\n3. If Gis a nilpotent simply connected Lie group, it follows that\n\nCis boundedclAGandclAG+arecompact sets andDis hyperbolic.\n\nFurthermore, it is possible to determine algebraic sufficient conditions to decide when Cis bounded. Actually, in a forthcoming paper we show that\n\nTheorem 33 Let ΣLbe a linear control system on the Lie group G.Assume that Gis decomposable and G+,0is a normal subgroup of G. Hence, clGAis compact.\n\nA analogous result is obtained for G+Aassuming that G,0is normal. Of course, G+,0is a normal subgroup of Gif and only if g+g0is an ideal of g. On the other hand,\n\ng+g0andg+g0areideals ofgg+g0=0andg+gg0.\n\n### 7.2. Chaos and control sets\n\nWe start with an explicitly relationship between chaotic subsets of M×Uand the Σ-control sets.\n\nTheorem 34 Let M×Uand the canonical projection πM:M×UM.Hence,\n\nπM=xM:there existsuUwithxu\n\nis compact and its non-void interior consists of locally accessible points. Then,\n\n1. is a maximal topologically mixing set if and only if there exists a control Csuch that\n\n=clxuM×U:φ(txu)intCfor everytR\n\nIn this case, Cis unique and intC=intπM,clC=clπM.\n\n• The periodic points of Φare dense in .\n\n• Φrestrict to is topologically mixing, topologically transitive and has sensitive dependence on initial conditions.\n\n• In order to apply this fundamental result for a non-controllable linear control system, the boundness property of its control set is crucial. Let us assume that Cis a bounded control set with non-empty interior of ΣLand define =πM1C=clC×UCwhere\n\nUC=uU:existxCwithφtxuintCfor everytR.\n\nThe Lie group Gis finite dimensional and UCis a closed subset of the compact class of admissible control ULRΩRmwith the weak* topology. Since the projection is a continuous map, it turns out that πMis compact and ,Care uniquely defined.\n\nOn the other hand, we are assuming that ΣLsatisfy the Lie algebra rank condition, hence the system is locally accessible at any point of the state space. Therefore, we are in a position to apply Theorem 32, first, for some classes of controllable linear systems, as follows.\n\nTheorem 35 Let ΣLbe a linear control system on a Lie group G. Any condition.\n\n1. Gis compact, or\n\n2. Gis Abelian, or\n\n3. Ghas the finite semisimple centre property and the Lyapunov spectrum of Dis 0implies that the skew flow Φis chaotic in G×U.\n\nProof. Under the hypothesis in 1, any control set is bounded. Furthermore, if Gis compact, the Lie algebra rank condition assures that the linear control system ΣLis controllable on G,see . Hence, Φis topologically mixing, topologically transitive and the periodic points of Φare dense in G×U, which give us the desired conclusion.\n\nIt is well known that any Abelian Lie group is a product G=Rm×Tnbetween the Euclidean space Rmand the torus Tn=S1××S1(ntimes), for some m,nN.In this case, ΣLis also controllable . Indeed, since the automorphism group of Tnis discrete, any linear vector field on the torus is trivial. But, we are assuming the Lie algebra condition on Gwhich coincides with the Kalman rank condition in Rm.And, on the compact part, we apply 1.Hence, the skew flow Φis chaotic in G×U. In fact, πM1C=G×Uand the hypothesis of the compacity on the projection in Theorem 32is not necessary for the lifting, see Proposition 4.3.3 in . The same is true for 3.Actually, for this more general set up, we recently prove that the system is also controllable, [28, 29].\n\nIn the sequel, we use some topological properties of Ceto translate these properties to its associated chaotic set ,as follows.\n\nTheorem 36 Let ΣLbe a linear control system on a Lie group G.It holds.\n\n1. If G=GG0G+there exists one and only one chaotic set =πM1Cein G×Ugiven by\n\n=clxu:φ(txu)intCefor everytRM×U\n\n• If Gis nilpotent and Dhas only eigenvalues with non-positive real parts, then the only chaotic set =πM1Cin G×Uis closed\n\n• If Gis nilpotent and Dhas only eigenvalues with non-negative real parts then the only chaotic set =πM1Cin G×Uis open\n\n• Proof. If Gis decomposable, we know that there exists just one control set: the one which contains the identity element. Hence, =πM1Ceis the only chaotic set of Φon G×Uwhich proves 1.To prove 2and 3, we observe that the Lyapunov spectrum condition on the derivation Dassociated to the drift vector field Xis equivalent to the control set Cebe closed or open, respectively. Since the projection πG:G×UGis a continuous map with the weak* topology on U,the lifting πG1Ceis both closed and open, respectively.\n\n### 7.3. Challenge\n\nIn this very short section, we would like to invite the readers to work on the relationship between chaotic and control sets. We suggest to go further in this research through some specific examples on low-dimensional Lie groups. For that, we give some relevant information about two groups of dimension three: the simply connected nilpotent Heisenberg Lie group Hand the special linear group SL2R. We finish by computing an example on H.\n\n1. The nilpotent Lie algebra h=R3+,has the basis E12E23E13with E12E23=E13.Here, Eijdenotes the real matrix of order 3with zero everywhere except 1in the position ij.The associated Heisenberg Lie group has the matrix representation\n\nG=g=1xz01y001:xyzRφ:gxyzR3.\n\nAs invariant vector fields, the basis elements of ghas the following description\n\nE12=x,E23=y+xzandE13=z.\n\nThe canonical form of any g-derivation is given by\n\nD=ad0be0cfa+e:a,b,c,d,e,fR.\n\nAny linear vector field Xreads as\n\nXxyz=ax+dyx+bx+eyy+b2x2+d2y2+cx+fy+a+ezz.\n\n2. The vector space g=sl2Rof all real matrices of order three and trace zero is the Lie algebra of the Lie group G=SL2R=det11. Let us consider the following generators of g: Y1=0110,Y2=0100and Y3=1001.The Lie group G is semisimple, then any gderivation is inner which means that there exists an invariant vector field Ysuch that adYrepresents .Thus, a general form of a derivation reads as\n\nαadY1+βadY2+γadY3.\n\nExample 7.1 On the Heisenberg Lie group, consider the system\n\nΣL:gt=Xgt+u1tE12gt+u2tE23gt,u=u1u2UE15\n\nwhere Xis determined by the derivation D=adE12=E32.Since the group is nilpotent, it has the semisimple finite centre property. The Lyapunov spectrum of Dreduces to zero. Finally, the reachable set from the identity Ais open. In fact, the ad-rank condition is obviously true because DE12=E13. It turns out that the skew flow Φis chaotic in H×U.\n\n## Notes\n\n• Partially supported by Conicyt, Chile through Regular Fondecyt Projects no. 1151159 and no. 1150292 respectively.\n\n## How to cite and reference\n\n### Cite this chapter Copy to clipboard\n\nHeriberto Román-Flores and Víctor Ayala (March 28th 2018). Chaos on Set-Valued Dynamics and Control Sets, Chaos Theory, Kais A. Mohamedamen Al Naimee, IntechOpen, DOI: 10.5772/intechopen.72232. Available from:\n\n### chapter statistics\n\n286total chapter downloads\n\n### More statistics for editors and authors\n\nLogin to your personal dashboard for more detailed statistics on your publications.\n\n### Related Content\n\nNext chapter\n\n#### Witnesses of Quantum Chaos and Nonlinear Kerr-Like Oscillator Model\n\nBy Joanna K. Kalaga, Marcin W. Jarosik, Wiesław Leoński and Radosław Szczęśniak\n\n#### Numerical Simulations of Physical and Engineering Processes\n\nEdited by Jan Awrejcewicz\n\nFirst chapter\n\n#### Numerical Solution of Many-Body Wave Scattering Problem for Small Particles and Creating Materials with Desired Refraction Coefficient\n\nBy M. I. Andriychuk and A. G. Ramm\n\nWe are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.\n\n## Book subject areas\n\nView all books"
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https://www.shaalaa.com/question-bank-solutions/tell-what-profit-or-loss-following-transactions-also-find-profit-percent-or-loss-percent-given-case-gardening-shears-bought-rs-250-sold-rs-325-profit-or-loss-percentage_17134 | [
"Share\n\n# Tell What is the Profit Or Loss in the Following Transactions. Also Find Profit Percent Or Loss Percent in Given Case. Gardening Shears Bought for Rs 250 and Sold for Rs 325. - CBSE Class 7 - Mathematics\n\nConceptProfit Or Loss as a Percentage\n\n#### Question\n\nTell what is the profit or loss in the following transactions. Also find profit\n\npercent or loss percent in given case.\n\nGardening shears bought for Rs 250 and sold for Rs 325.\n\n#### Solution\n\nCost price = Rs 250\n\nSelling price = Rs 325\n\nProfit = 325 − 250 = Rs 75\n\nProfit% = \"Profit\"/\"CP\" xx 100\n\n= 75/250 xx 100 = 30%\n\nIs there an error in this question or solution?\n\n#### APPEARS IN\n\nNCERT Solution for Mathematics for Class 7 (2018 to Current)\nChapter 8: Comparing Quantities\nEx. 8.30 | Q: 1.1 | Page no. 171\nSolution Tell What is the Profit Or Loss in the Following Transactions. Also Find Profit Percent Or Loss Percent in Given Case. Gardening Shears Bought for Rs 250 and Sold for Rs 325. Concept: Profit Or Loss as a Percentage.\nS"
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http://dev.loci.wisc.edu/trac/software/wiki/Matlab?version=1 | [
"Version 1 (modified by melissa, 10 years ago) (diff)\n\n--\n\nThis section assumes that you have installed the bfopen.m script and loci_tools.jar, as instructed here.\n\nThe first thing to do is initialize a file:\n\n``` data = bfopen('/path/to/data/file');\n```\n\n'data' is an array whose structure is a bit complicated. It is an n-by-3 array, where n is the number of series in the dataset. The [s, 1] element (if s is the series index between 1 and n) is an m-by-2 array, where m is the number of planes in the series. The [s, 1, t, 1] element (where t is the image index between 1 and m) contains the pixel data for the t-th image in the s-th series. The [s, 1, t, 2] element contains the label for said image.\n\nThe [s, 2] element of 'data' contains metadata key/value pairs that apply to the s-th series. The [s, 3] element of 'data' contains color lookup tables for each image in the series.\n\n## Displaying images\n\nIf you want to display one of the images, you can do so as follows:\n\n``` series1 = data{1, 1};\nseries1_plane1 = series1{1, 1};\nseries1_label1 = series1{1, 2};\nseries1_colorMaps = data{1, 3};\nfigure('Name', series1_label1);\nif (isempty(series1_colorMaps{1})\ncolorMap(gray);\nelse\ncolorMap(series1_colorMaps{1});\nend\nimagesc(series1_plane1);\n```\n\nThis will display the first image of the first series with its associated color map (if present). If you would prefer not to apply the color maps associated with each image, simply comment out the calls to 'colorMap'.\n\nTo retrieve the metadata value for a specific key:\n\n``` metadata = data{1, 2};\n``` metadata = data{1, 2};"
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https://mathematica.stackexchange.com/questions/tagged/wolfram-alpha-queries | [
"# Questions tagged [wolfram-alpha-queries]\n\nUse this for questions about calling Wolfram Alpha from Mathematica. Questions about Wolfram Alpha itself are off-topic.\n\n166 questions\nFilter by\nSorted by\nTagged with\n44 views\n\n### General: 2456 is not a valid variable [closed]\n\n... is not a valid variable i dont know whats happend\n47 views\n\n### How to make this expression?\n\nHow to code this expression in WolframAlpha. I am interested in how to code the second sum, because it contains a boolean expression: $$\\sum_{i=1}^{100}{\\sum_{j>n}}{x_{i}x_{j}}$$\n48 views\n\n### Why doesn't Mathematica provide an answer while Wolfram|Alpha does, concerning a series convergence?\n\nAmong other series I've been working on, I was asked to find whether $$\\sum_n 1-\\cos(\\frac{\\pi}{n})$$ converged, and Mathematica's output to ...\n142 views\n\n### How can one expand an arbitrary boolean combination into the $2^n$ atoms of the associated boolean algebra of size $2^{2^n}$?\n\nThe answer of user250938 to Can one usefully apply the Boolean functions of Mathematica to measurable Boolean sets? and the second comment of Monroe Eskew to the answer to https://mathoverflow.net/...\n67 views\n\n### The Branch Cut of the Complex Logarithm Used by Default in Wolfram Alpha\n\nThe title says it all: what branch cut (like which value of $k$ in $\\ln(z)=\\ln(r)+i(\\theta+2\\pi k))$ does Wolfram Alpha use by default in calculating the complex logarithm. I would say \"principal ...\n105 views\n\n### step-by-step solution unavailable (( possible intermediate steps ))\n\nWhen using the (( WolframAlpha[\"integrate 1/(2+Sqrt[1-x]+Sqrt[1+x])\"])) function to find integration steps, this message appears to me what is the explanation and ...\n42 views\n\n### What is the current WolframAlpha server capacity?\n\nConsidering the increasing workload WolframAlpha is now receiving as it is getting more tightly integrated into the Wolfram language. With things like Entities even ...\n98 views\n\n### How to change the Wolfram Alpha query timeout for Ctrl+=?\n\nSometimes retrieving results from Wolfram Alpha takes time and the default query timeout is not sufficient. For engagement with Wolfram Alpha implicitly for expr ...\n44 views\n\n### How can I price of 100 EUR in RON on 1st of every month back to 4 years\n\nI would like to see the difference between 450 RON and 100EUR converted to RON on 1st of every month back to 4 years. possibly with a cumulated chart...\n168 views\n\n### Difference between free-form input via WolframAlpha[] and CTRL+=\n\nWolframAlpha[\"query\"] - https://reference.wolfram.com/language/ref/WolframAlpha.html Free-form input - https://reference.wolfram.com/language/workflow/EnterFreeFormInput.html For example: ...\n167 views\n\n### Step by step solution of a limit of a sequence - Calling Wolfram|Alpha in Mathematica\n\nI'd like to visualiza the step-by-step solution of the following limit by calling Wolfram|Alpha in Mathematica, but I don't know how to specify that n is a positive integer variable: ...\n71 views\n\n114 views\n\n### CloudDeploy fails when using WolframAlpha as APIFunction\n\nI am trying to create a Wolfram API that returns, for an input expression, how Wolfram Alpha interprets that input. For example, given the string \"1 plus 2\", the API should return \"1 + 2\". Locally, ...\n114 views\n\n### Plot with Wolfram Alpha Scientific Data\n\nNow that Mathematica has already embedded Wolfram Alpha, so I'd like to plot a graph: \"Air Density\" - \"Elevation\". I know that yields But I can't do that 10000 ...\n35 views\n\n### How I can define and arrange following linear equation?\n\nWe have: Theta * (2L-K) = 0 Theta cannot be zero because it's angle parameters. (2L-K)=0 then We can calculate L=K/2 How we can define this constraint in Wolfram|...\n238 views\n\n### Conditional Summations\n\nI am not a math major, but for a networking class, I am taking I am required to do summations for probability. I know the logic but I don't know the mathematical theory to make this work. Using ...\n120 views\n\n### Confusion in finding the definite integral\n\nSo, I'm new to Mathematica. I tried evaluating the following definite integral. However the answer i get from wolfram alpha & mathematica are different. Can someone please point out what I'm ...\n62 views\n\n### How can I compute how many primes of this kind are up to $N$?\n\nIn the following link: https://oeis.org/A079796/b079796.txt we can see the first 10,000 prime numbers $p$ with the property that both $(3p)^2 + p^2 + 3^2$ and $(3p)^2 - p^2 - 3^2$ are primes ...\n827 views\n\n### Step-by-step solution of a system of symbolic equations - Call Wolfram Alpha in Mathematica\n\nI'm trying to get a step-by-step solution of a system of symbolic equations: ...\n321 views\n\n### How to query Wolfram Alpha to find projection of point on plane? [closed]\n\nFor example, the projection of the point $P=(1,2,3)$ on the plane $x+y+z=3$ is the point $Q=(0,1,2)$. How do I ask Wolfram Alpha to compute this? I've tried \"projection of point (1,2,3) on plane x+y+...\n2k views\n\n### How to ask WolframAlpha to find 'x such that p=32*x+1 is prime'\n\nI want to find prime $p$ in form of $p=32 x+1$ How can I ask WolframAlpha for such $x$ that will fulfil this equation? EDIT: let's say that $100$ or $n$ solutions is enough for me EDIT2: I need x's ...\n441 views\n\n### Can I make Wolfram Alpha use a specific substitution when integrating? [closed]\n\nCan I define the '$u = \\dots$' for an integral? For example when I integrate $\\frac{x^3}{\\sqrt{6 - x^2}}$ with respect to $x$, the program automatically sets $u = x^2$, how can I change this so it ...\n112 views\n\n### how to get historical data from wolfram alpha\n\nThe following command give historical data from 21th April 2017 to 19th April 2018. Which is about 1 year. How do i get this result for past three years from wolfram alpha. ...\n60 views\n\n### WolframAlpha robustness\n\nI am experiencing return values of Missing[NotAvailable] from WolframAlpha[], about 10% of the time. For some reason, this seems ...\n190 views\n\n### Find the average GDP of multiple countries at once\n\nHow would I find the average GDP of a list of countries between set years? I know how to do it for one but I'm not sure about multiple values. I also wish to find the average population size. An ...\n146 views\n\n### list of dates for vernal equinox\n\nI'm trying to get a list of dates and times of the vernal equinox for the last 30 years (or some other time interval). I'm flailing about and really have gotten nowhere. When I've tried typing ...\n355 views\n\n### Find optimal integer solution to linear equation (Wolfram Alpha)\n\nI'm sorry if this is potentially off topic, but I posted this question at the Math SE and they pointed me here. Both SE sites have a tag for Wolfram Alpha, but maybe you guys are better equipped to ...\n101 views\n\n### WolframAlpha function does not return structure diagram for chemicals\n\nNeither one of WolframAlpha[\"Caffeine\"] WolframAlpha[\"Caffeine\",\"FullOutput\"] returns a structure diagram for the molecule. It's consistent since ...\n142 views\n\n### how to get a relation between two parameters from a single equation\n\n2 (ϕ'[r])^2 - 2 u'[r]/r - (u'[r])^2 - u''[r] == 0 I need to get a relationship between u and ..."
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http://forums.parallax.com/discussion/comment/1496489/ | [
"Welcome to the Parallax Discussion Forums, sign-up to participate.\n\n# Flip and propellerC language; RESOLVED\n\n• Here is the updated code that will process the serial data coming from the Arduino. The problem is that the values from the Arduino are not scaled.\n0 = 0/255 = 0%\n100 = 100/255 = 39%\n150 = 150/255 = 59%\n200 = 200/255 = 78%\n250 = 250/255 = 98%\n\nOn the Propeller these translate to\n0 = 0/600 = 0%\n100 = 100/600 = 17%\n150 = 150/600 = 25%\n200 = 200/600 = 33%\n250 = 250/600 = 42%\n```/*\nBlank Simple Project.c\nhttp://learn.parallax.com/propeller-c-tutorials\n*/\n#include \"simpletools.h\"\n#include \"propeller.h\" // Include simple tools\n#include \"fdserial.h\"\n\n#define RmotorPin 17\n#define DmotorPin 19\n#define RdirPin 18\n#define DdirPin 20\n#define FDRX 0\n#define FDTX 1\n\nfdserial *fd;\nint b1, b2, b3, b4;\n\nint ck;\nint data;\nchar Buffer;\n\nint i;\n\nint main()\n{\nlow(RmotorPin); //<-- set pin as output\nlow(RdirPin); //<-- default direction\nlow(DmotorPin);\nlow(DdirPin);\n\nfd = fdserial_open(FDRX, FDTX, 0, 115200);\n\nclkset(0x6b,5000000); //<-- Not needed, does nothing.\n\npwm_start(600); //<-- generate a pulse every 600 milliseconds\n//pwm_start(20000);\n\ni = 0;\nwhile(1)\n{\nck = fdserial_rxCheck(fd); //<-- if we have serial data use it\nwhile(fdserial_rxCheck(fd) > 0)\n{\nBuffer[i++] = fdserial_rxChar(fd); //<-- Capture one of the four bytes\n}\n\nif (i > 0) //<-- we have serial data\n{\nif (i == 4) //<-- valid serial packet?\ndata = 'A';\nelse\ndata = ' ';\ni = 0;\nprint(\"%d,%d,%d,%d\\n\", Buffer, Buffer, Buffer, Buffer);\n}\n\nb1 = input(3); //<-- get button input which overrides serial\nb2 = input(4);\nb3 = input(5);\nb4 = input(6);\n\nif (b1 == 1)\ndata = 'W';\n\nif (b2 == 1)\ndata = 'N';\n\nif (b3 == 1)\ndata = 'E';\n\nif(b4 == 1)\ndata = 'S';\n\nswitch(data)\n{\ncase 'W':\nhigh(RdirPin); //<-- set direction before starting motor\npwm_set(RmotorPin,0,500); //<-- turn on motor to 83% - 500/600\nif (b1 == 0) //<-- button released\ndata = ' ';\nbreak;\n\ncase 'N':\nhigh(DdirPin);\npwm_set(DmotorPin,1,500);\nif (b2 == 0)\ndata = ' ';\nbreak;\n\ncase 'E':\nlow(RdirPin);\npwm_set(RmotorPin,0,500);\nif (b3 == 0)\ndata = ' ';\nbreak;\n\ncase 'S':\nlow(DdirPin);\npwm_set(DmotorPin,1,500);\nif (b4 == 0)\ndata = ' ';\nbreak;\n\ncase 'A':\nset_output(RdirPin, Buffer);\nset_output(DdirPin, Buffer);\npwm_set(RmotorPin, 0, Buffer);\npwm_set(DmotorPin, 1, Buffer);\nbreak;\n\ndefault: //<-- no buttons or serial data all stop\npwm_set(RmotorPin,0,50); //<-- set motor speed to 8% - 50/600\npwm_set(DmotorPin,1, 0); //<-- set motor to stop 0% - 0/600\nlow(RdirPin); //<-- set direction to default\nlow(DdirPin);\nbreak;\n}\npause(250); //<-- wait 250 milliseconds or 1/4 of a second.\n}\n}\n```\n\nMike\n• I tried the new program but I don't see anything in the IDE monitor and n movement in motors when I press buttons on the hand controller but the switches on the propeller do illicit reaction on the motors as needed.\n• There could be few things wrong with serial data. This is where a Serial Dongle comes in handy. You can test with hardware that works.\n\n1) The Arduino is 5v and the Propeller is 3.3v which means you need a level shifter otherwise you will damage the Propeller.\n2) The wrong pins are being used. What pin is 2 and 3 now?\n3) Tx and Rx are reversed. Are they saying this is the Tx pin or that the Tx pin goes here or what?\n\nI see you are using soft serial to output to the Propeller. The documentation shows SoftwareSerial softserial(Rx, Tx), but your code is showing just the opposite. \"SoftwareSerial serial_flip(txpin,rxpin);\".\n\nMaybe you need another flip board to replace the Nano board you have. The code would be easy enough to port over to it and would remove the 5v to 3.3v issue.\n\nMike\n• I am using a 3k ohm resistor in series with the arduino tx pin, The pin D3 is the softserial tx pin on the arduino. I rechecked and the rx=2 and tx = 3 and softwareserial serial_flip(rx,tx). This is my current sketch.\n• Your soft serial config looks wrong to me.\n```const byte rxpin = 2 ;\nconst byte txpin = 3 ;\nbyte directionFlag = 0 ;\nbyte westPinState = 0 ;\nbyte lastwestPinState = 0 ;\nbyte northPinState = 0 ;\nbyte lastnorthPinState = 0 ;\nbyte eastPinState = 0 ;\nbyte lasteastPinState = 0 ;\nbyte southPinState = 0 ;\nbyte lastsouthPinState = 0 ;\nbyte currentModeSwitchState = 0 ;\nbyte lastModeSwitchState = 0 ;\nbyte mode = 0 ;\nunsigned long currentMillis = 0 ;\nunsigned long lastMillis = 0 ;\nvolatile byte RmotorPin = 0 ;\nvolatile byte DmotorPin = 0 ;\nvolatile byte RdirPin = 0 ;\nvolatile byte DdirPin = 0 ;\nSoftwareSerial serial_flip(txpin,rxpin); <-- Is this backwards?\n```\n\nMike\n• look at my latest version of the program mar 09 a(3) zip.( My last post, There is no (txPin,rxPin) It reads serial_flip(rx,tx)\n• For some reason it didn't update my local copy. I see where you fixed it. That should work.\n\nMike\n• But as before I receive no information from the rx pin to the serial monitor or to the motors and I do have some readable data to the rx pin from the arduino. perhaps it is time to convert the arduino sketch to a propeller but I may need some help.\n• Lets backup up a little here. We need to asses the serial problem first.\n\nHere is a simple program that will take data in from a serial pin and output it to the monitor on the flip. It will also take data from the flip and send it out the serial pin.\n```#include \"simpletools.h\"\n#include \"fdserial.h\"\n\nfdserial *trm;\nterminal *dft;\nint c;\n\nint main()\n{\nsimpleterm_close();\n\ntrm = fdserial_open(31, 30, 0, 115200);\ndft = fdserial_open(0, 1, 0, 115200);\n\npause(1000);\n\nwhile (1)\n{\nc = fdserial_rxCheck(trm);\nif (c > 0)\nfdserial_txChar(dft, c);\nc = fdserial_rxCheck(dft);\nif (c > 0)\nfdserial_txChar(trm, c);\n}\nfdserial_close(trm);\nfdserial_close(dft);\n}\n```\n\nI don't have a nano board and there seems to be a couple of suppliers that are different out there but here is a diagram of one.\n\nIf you tied pin 1 of the nano to pin 0 of the flip with the resistor it should mirror everything that you send from the nano.\n\nAlso you are using pin 2 for receive and 3 for data out which would be connect to pin 0 on the flip. This should work.\n\nAlso here is the Arduino program converted to Propeller. The Propeller does not support pull-up on pins but I believe you are using resistors on the switches so this is not needed anyway.\n\nMike\n\n• edited 2020-05-12 - 06:37:01\nI have redone the system to two flips. I am trying the program you sent, will this go without using set_pwm or should change it to include set_pwm.\n• Not following, In which program are you referring to.\n\nMike\n• The one for the flip to replace the arduino.\n• I see no pwm output in this progam so no don't see a need for it.\n\nMike\n• How do I scale the values to the pwm values in the output of the mount.\n• You need to decide what ratio you want to output. Based on the motors you would say 25% power for low, 50% power for medium, and 75% for fast and 100% for all out move.\n\nIf your PWM value is say 2000 microseconds then 1000 microseconds is 50% = 1000/2000.\n\nHere is a sample program to drive one motor at the different values:\n```#include \"simpletools.h\"\n\n#define PWM1 15\n\nint i;\n\nint main()\n{\npwm_start(2000); //<-- 2 milliseconds\n\ni = 4; //<-- set speed\n\nswitch (i)\n{\ncase 1:\npwm_set(PWM1, 0, 500); //<-- .5 milliseconds 25%\nbreak;\ncase 2:\npwm_set(PWM1, 0, 1000); //<-- 1 milliseconds 50%\nbreak;\ncase 3:\npwm_set(PWM1, 0, 1500); //<-- 1.5 milliseconds 75%\nbreak;\ncase 4:\npwm_set(PWM1, 0, 2000); //<-- 2 milliseconds 100%\nbreak;\ndefault:\nbreak;\n}\n\nwhile(1)\n{\npause(1000);\n\n}\n}\n```\n\nSetting \"i\" to the 4 values generates the follow output shown on the scope. You can see at 1 the pulse is high only 25% of the time and 2 it is 50% and so on.\n\nThis only works if you are using the UDN2993 chips.\n\nMike\n• I will try to add or replace parts of the program(LX50 ver 3) but I still have no transmission to the second flip. I tried to use the serial monitor on the hand controller, I get \"value1: 200, value2:0, value3: 0, value4:\nvalue1: 0, value2: 0, value3: 0, value4: 0\" after I press the 'w' button on the hand controller. When I press the button the value 1 on the second line reads 200 and rest are zeros. I would like to print the input values of the mount flip.\n• I just looked at the last version of the flip program and there is print statements:\n```print(\"%d,%d,%d,%d\\n\", Buffer, Buffer, Buffer, Buffer);\n```\n\nIf you don't see anything then the serial isn't working.\n\nMike\n• That's what I'm thinking, maybe I need a new Flip board. Thanks.\n• I am lost. about your post on5/13 4:10:37 I looked at the sketch but I don't know how it works with the serial input which uses a group of buffers. I need to convert the variables to the (pwm_set (600)) since that is the level I am using.\n• What about something similar to arduino's mapping function?\n• It's not that complicated. You want to drive the motors at some percentage of full power to make the telescope move at the rate you want. The problem is I don't know how fast the motors move at each of those percentages.\n\nSo if we drive the UDN2993 at 50% what speed does that look like on the telescope. Is it too fast too slow or just right.\n\nOn the flip you are using x/600 for the percentage. On the Arduino they are using x/255 for the percentages.\n\nYou can see from the graph of the scope output that the pulse last x percent based on the value you pick.\n\nOn the Arduino you picked these values\n\nMode 1: --> R 100/255 = 39%\nMode 1: --> D 75/255 = 29%\nMode 2: --> R 150/255 = 59%\nMode 2: --> D 100/255 = 39%\nMode 3: --> R 200/255 = 78%\nMode 3: --> D 150/255 = 59%\nMode 4: --> R 250/255 = 98%\nMode 4: --> D 200/255 = 78%\n\nIf these percentage of full power are what you want then you just build the same percentages on the flip.\n\nTo match the Arduino program you would start pwm at 255 for your value.\n\npwm_start(255); //<-- repeat every 255 microseconds.\n\nNow the output from the pwm_set(x, 0, 100); //<-- 100/255 or 39% of full power.\n\nNow 0 - 255 doesn't give you a lot of choices for the percentage you want to drive the motors so if you want to use 600 then you just need to multiply to get the percentage you want.\n\n100 = 39% = .39 X 600 = 234\n75 = 29% = .29 X 600 = 174\n\nWe can see that the difference is 255/600 = 2.35 which would be the factor you need to multiply by to get the same percentages.\n\n100 X 2.35 = 235 = 235/600 = 39%\n250 X 2.35 = 587 = 587/600 = 98%\n\nNow since we don't want to use floating point math just use 100 times the value or 235 and divide by 100 before you use it.\n\n100 X 235 = 23500/100 = 235.\n250 X 235 = 58750/100 = 587.\n\nMike\n\n• You missed the second question about a mapping function or was the multiplication factors a mapping means.\n• Right, no mapping needed just multiple or use 255 for set frequency and it will work like the Arduino.\n\nMike\n• I am running two flips now but the transmitted signal can be reset in the hand controller to higher values . the pwm_set can be used in the modes, right.\n• Please give an example of what you want to do with pwm_set.\n\nIs serial data now working?\n\nMike\n• // mode 1\ncase 1:\nswitch(directionFlag)\n{\n// mode 1\ncase 1:\n{\npwm_set(RmotorPin, 0, 100);\npwm_set(DmotorPin, 1, 0);\nbreak;\n\nNo function yet and the mount 'flip' is a new one and I have a signal (high going low and back to high, one long then two small and another long) on the P1 pin which is tx pin.\n• So there is going to be a motor attached to the Arduino flip? You would need a pwm_start(255) at the beginning of the program to do that.\n\nMike\n• No arduino's used in new set-up only propellers and no motors used on hand controller. hand controller talks to mount.\n• Great, then you don't need any pwm_set in the program.\n\nMike\n• Concerning your post dated2020/05/10 05:29:24 can you identify these terms fdserial *trm ; terminal *dft; and simpleterm_close. and what do they do."
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https://en.wikibooks.org/wiki/Data_Science:_An_Introduction/Definitions_of_Data | [
"# Data Science: An Introduction/Definitions of Data\n\nChapter 03: Definitions of Data\n\n## Chapter Summary\n\nThe word \"data\" is a general purpose word denoting a collection of measurements. \"Data points\" refer to individual instances of data. A \"data set\" is a well-structured set of data points. Data points can be of several \"data types,\" such as numbers, or text, or date-times. When we collect data on similar objects in similar formats, we bundle the data points into a \"variable.\" We could give a variable a name such as 'age,' which could represents the list of ages of everyone in a room. The data points associated with a variable are called the \"values\" of the variable. These concepts are foundational to understanding data science. There is some quirkiness in the way variables are treated in the R programming language.\n\n## Discussion\n\n#### What is Data?\n\nThe Wiktionary defines data as the plural form of datum; as pieces of information; and as a collection of object-units that are distinct from one another.\n\nThe Wiktionary defines datum as a measurement of something on a scale understood by both the recorder (a person or device) and the reader (another person or device). The scale is arbitrarily defined, such as from 1 to 10 by ones, 1 to 100 by 0.1, or simply true or false, on or off, yes, no, or maybe, etc.; and as a fact known from direct observation.\n\nFor our purposes, the key components of these definitions are that data are observations that are measured and communicated in such a way as to be intelligible to both the recorder and the reader. So, you as a person are not data, but recorded observations about you are data. For example, your name when written down is data; or the digital recording you speaking your name is data; or a digital photograph of your face or video of you dancing are data.\n\n#### What is a Data Point?\n\nRather than call a single measurement by the formal word '\"datum,\" we will use what the Wikipedia calls a data point. We may talk about a single data point or several data points. Just remember that when we talk of \"data,\" what we mean is a set of aggregated data points.\n\n#### What is a Data Set?\n\nThe Wiktionary, unhelpfully, defines a data set as a \"set of data.\" Let us define a data set as a collection of data points that has been observed on similar objects and formatted in similar ways. Thus, a compilation of the written names and the written ages of a room full of people is a data set. In computing, a data set is stored in a file on a disk. Storing the data set in a file makes it accessible to analysis.\n\n#### What are Data Types?\n\nAs illustrated earlier, data can exist in many forms, such as text, numbers, images, audio, and video. People who work with data have taken great care to very specifically define different data types. They do this because they want to compute various operations on the data, and those operations only make sense for particular data types. For example, addition is an operation we can compute on integer data types (2+2=4), but not on text data types (\"two\"+\"two\"=???). Concatenation is an operation we can compute on text. To concatenate means to put together, so: `concatenate(two, two) = twotwo`. For the purposes of this introduction, we will just concern ourselves with simple numeric and simple text data types and leave more complex data types—like images, audio, and video—to more advanced courses. Data scientists use the various data types from mathematics, statistics, and computer science to communicate with each other.\n\n##### Data Types in Mathematics\n\nWe will introduce just the most commonly used data types in Mathematics. There are many more, but we'll save those for more advanced courses.\n\n1. Integers - According to the Wikipedia, integers are numbers that can be written without a fractional or decimal component, and fall within the set {..., −2, −1, 0, 1, 2, ...}. For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers.\n2. Rational Numbers - According to the Wikipedia, rational numbers are those that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. For example, 9.75 2/3, and 5.8144144144… are rational numbers.\n3. Real Numbers - According to the Wikipedia, real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, plus all the irrational numbers such as √2 (1.41421356... the square root of two), π (3.14159265...), and e (2.71828...).\n4. Imaginary Numbers - According to the Wikipedia, imaginary numbers are those whose square is less than or equal to zero. For example, √-25 is an imaginary number and its square is -25. An imaginary number can be written as a real number multiplied by the imaginary unit i, which is defined by its property i 2 = −1. Thus, √-25 = 5i.\n\nData scientists understand that the kind of mathematical operations they may perform depends on the data types reflected in their data.\n\n##### Data Types in Statistics\n\nWe will introduce just the most commonly used data types in statistics, as defined in the Wikipedia. There are a few more data types in statistics, but we'll save those for more advanced courses.\n\n1. Nominal - Nominal data are recorded as categories. For this reason, nominal data is also known as categorical data. For example, rocks can be generally categorized as igneous, sedimentary and metamorphic.\n2. Ordinal - Ordinal data are recorded as the rank order of scores (1st, 2nd, 3rd, etc.). An example of ordinal data is the result of a horse race, which says only which horses arrived first, second, or third but include no information about race times.\n3. Interval - Interval data are recorded not just about the order of the data points, but also the size of the intervals in between data points. A highly familiar example of interval scale measurement is temperature with the Celsius scale. In this particular scale, the unit of measurement is 1/100 of the temperature difference between the freezing and boiling points of water. The zero point, however is arbitrary.\n4. Ratio - Ratio data are recorded on an interval scale with a true zero point. Mass, length, time, plane angle, energy and electric charge are examples of physical measures that are ratio scales. Informally, the distinguishing feature of a ratio scale is the possession of a zero value. For example, the Kelvin temperature scale has a non-arbitrary zero point of absolute zero.\n\nData scientists know that the kind of statistical analysis they will perform is determined by the kinds of data types they will be analyzing.\n\n##### Data Types in Computer Science\n\nWe will introduce just the most commonly used data types in Computer Science, as defined in the Wikipedia. There are many more, but we'll save those for more advanced courses.\n\n1. Bit - A bit (a contraction of binary digit) is the basic unit of information in computing and telecommunications; a bit represents either 1 or 0 (one or zero) only. This kind of data is sometimes also called binary data. When 8 bits are grouped together we call that a byte. A byte can have values in the range 0-255 (00000000-11111111). For example, the byte 10110100 = 180.\n• Hexadecimal - Bytes are often represented as Base 16 numbers. Base 16 is known as Hexadecimal (commonly shortened to Hex). Hex uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a–f) to represent values ten to fifteen. Each hexadecimal digit represents four bits, thus two hex digits fully represent one byte. As we mentioned, byte values can range from 0 to 255 (decimal), but may be more conveniently represented as two hexadecimal digits in the range 00 to FF. A two-byte number would also be called a 16-bit number. Rather than representing a number as 16 bits (10101011110011), we would represent it as 2AF3 (hex) or 10995 (decimal). With practice, computer scientists become proficient in reading and thinking in hex. Data scientists must understand and recognize hex numbers. There are many websites that will translate numbers from binary to decimal to hexadecimal and back.\n2. Boolean - The Boolean data type encodes logical data, which has just two values (usually denoted \"true\" and \"false\"). It is intended to represent the truth values of logic and Boolean algebra. It is used to store the evaluation of the logical truth of an expression. Typically, two values are compared using logical operators such as .eq. (equal to), .gt. (greater than), and .le. (less than or equal to). For example, `b = (x .eq. y)` would assign the boolean value of \"true\" to \"b\" if the value of \"x\" was the same as the value of \"y,\" otherwise it would assign the logical value of \"false\" to \"b.\"\n3. Alphanumeric - This data type stores sequences of characters (a-z, A-Z, 0-9, special digits) in a string--from a character set such as ASCII for western languages or Unicode for Middle Eastern and Asian languages. Because most character sets include the numeric digits, it is possible to have a string such as \"1234\". However, this would still be an alphanumeric value, not the integer value 1234.\n4. Integers - This data type has essentially the same definition as the mathematical data type of the same name. In computer science, however, an integer can either be signed or unsigned. Let us consider a 16-bit (two byte) integer. In its unsigned form it can have values from 0 to 65535 (216-1). However, if we reserve one bit for a (negative) sign, then the range becomes -32767 to +32768 (-7FFF to +8000 in hex).\n5. Floating Point - This data type is a method of representing real numbers in a way that can support a wide range of values. The term floating point refers to the fact that the decimal point can \"float\"; that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated separately in the internal representation, and floating-point representation can thus be thought of as a computer realization of scientific notation. In scientific notation, the given number is scaled by a power of 10 so that it lies within a certain range—typically between 1 and 10, with the decimal point appearing immediately after the first digit. The scaling factor, as a power of ten, is then indicated separately at the end of the number. For example, the revolution period of Jupiter's moon Io is 152853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047×105 seconds. Floating-point representation is similar in concept to scientific notation. The base part of the number is called the significand (or sometimes the mantissa) and the exponent part of the number is unsurprisingly called the exponent.\n• The two most common ways in which floating point numbers are represented are either in 32-bit (4 byte) single precision, or in 64-bit (8 byte) double precision. Single precision devotes 24 bits (about 7 decimal digits) to its significand. Double precision devotes 53 bits (about 16 decimal digits) to its significand.\n6. List - This data type is used to represent complex data structures. In its most simple form, it has a key-value pair structure. For example, think of a to-do list:\nKey Value\n1 Get haircut\n3 Take shower\nLists can become and often do become very complex. The keys do not have to be numeric, but could be words, such as \"one,\" \"two,\" and \"three.\" The values do not have to be a single data point. The value could be a series of numbers, or a matrix of numbers, or a paragraph. For example the first key in a list could be \"Romeo and Juliet,\" and the first value in the list could be the entire play of Romeo and Juliet. The second key in the list could be \"Macbeth,\" and the second value in the list could be the entire play of Macbeth. Finally, a value in a list could even be another list. At this point do not go down the rabbit hole of \"a list within a list within a list . . .\" We will leave that to graduate students in computer science.\n\nData scientists understand the importance of how data is represented in computer science, because it affects the results they are generating. This is especially true when small rounding errors accumulate over a large number of iterations.\n\n##### Data Types in R\n\nThere are at least 24 data types in the R language. We will just introduce you to the 9 most commonly used data types. As you will see they are a blend of the data types that exist in Mathematics, Statistics, and Computer Science. Just what a Data Scientist would expect. The nine are:\n\n1. NULL - for something that is nothing\n2. logical - for something that is either TRUE or FALSE (on or off; 1 or 0)\n3. character - for alphanumeric strings\n4. integer - for positive, negative, and zero whole numbers (no decimal place)\n5. double - for real numbers (with a decimal place)\n6. complex - for complex numbers that have both real and imaginary parts (e.g., square root of -1)\n7. date - for dates only\n8. POSIX - for dates and times (dates are internally represented as the number of days since 1970-01-01, with negative values for earlier dates)\n9. list - for storing complex data structures, including the output of most of the built-in R functions\n\nYou can get R to tell you what type a particular data object is by using the `typeof()` command. If you want to know what a particular data object was called in the original definition of the S language you can use the `mode()` command. If you want to know what object class a particular data object is in the C programming language that was used to write R, you can use the `class()` command. For the purposes of this book, we will mostly use the `typeof()` command.\n\n• Just a note about lists in R. R likes to use the list data type to store the output of various procedures. We generally do not perform statistical procedures on data stored in list data types--with one big exception. In order to do statistical analysis on lists, we need to convert them to tables with rows and columns. R has a number of functions to move data back and forth between table-like structures and list data types. The exception we just referred to, is called the data.frame list object. List objects of the class data.frame store rows and columns of data in such a specifically defined way as to facilitate statistical analysis. We will explain data frames in more detail below.\n\nData scientists must know exactly how their data are being represented in the analysis package, so they can apply the correct mathematical operations and statistical analysis.\n\n#### What are Variables and Values?\n\nLet us start by noting the opposite of a variable is a constant. If we declare that the symbol \"X\" is a constant and assign it a value of 5, then X=5. It does not change; X will always be equal to 5. Now, if we declare the symbol \"Y\" to be a variable, that means Y can have more than one value (see the Wiktionary entry for \"variable\"). For example, in the mathematical equation, Y^^2=4 (Y squared equals 4), the variable Y can either have the value of 2 or -2 and satisfy the equation.\n\nImagine we take a piece of paper and make two columns. At the top of the first column we put the label \"name\" and the top of the second column we put the label \"age.\" We then ask a room full of 20 people to each write down their name and age on the sheet of paper in the appropriate columns. We will end up with a list of 20 names and 20 ages. Let us use the label \"name\" to represent the entire list of 20 names and the label \"age\" to represent the entire list of 20 ages. This is what we mean by the term \"variable.\" The variable \"name\" has 20 data points (the list of 20 names), and the variable \"age\" has 20 data points (the list of 20 ages). A variable is a symbol that represents multiple data points which we also call values. Other words that have approximately the same meaning as \"value\" are measurement and observation. Data scientists use these four terms (data point, value, measurement, and observation) interchangeably when they communicate with each other.\n\nThe word \"variable\" is a general purpose word used in many disciplines. However, various disciplines also use more technical terms that mean approximately the same thing. In mathematics another word that approximates the meaning of the term \"variable\" is vector. In computer science, another word that approximates the meaning of the term \"variable\" is array. In statistics, another word that approximates the meaning of the term \"variable\" is distribution. Data scientists will often use these four words (variable, vector, array, and distribution) interchangeably when they communicate with each other.\n\nLet us think again of the term data set (defined above). A data set is usually two or more variables (and their associated values) combined together. Once our data is organized into variables, combined into a data set, and stored in a file on a disk, it is ready to be analyzed.\n\nThe R programming language is a little quirky when it comes to data types, variables, and data sets. In R we sometimes use the term \"vector\" instead of \"variable.\" When we combine and store multiple vectors (variables) into a data set in R, we call it a data frame. When R stores vectors into a data frame, it assigns a role to indicate how the data will be used in subsequent statistical analyses. So in R data frames, for example, the \"logical,\" \"date/time,\" and \"character\" data types are assigned the role of Factor. The \"double\" data type are assigned the role of num and \"integers\" are assigned the role of int. (The \"complex\" data type is assigned the role of \"cplx,\" but don't worry about that now.) These roles correspond to the statistical data types as follows: Factor = nominal, int = ordinal, and num = interval. (We usually transform the ratio data type into an interval data type before doing statistical analysis. This is normally done by taking the logarithm of the ratio data. More on this in later chapters.) We can discover the roles each variable will play within a data frame by using the structure command in R: `str()`. We will explain what \"factors\" are in latter chapters.\n\n## Assignment/Exercise\n\nThis assignment should be done in a group of 3 or 4 students. The groups need to be composed of different people from the previous two homework groups. All should interact with the R programming language. The group can help each other both learn the concepts and figure out how to make R work. Practice with R by trying out different ways of using the commands that are described below.\n\n#### Find Data Types in R\n\nUse the `typeof()` command to verify data types. See if you can guess what the output will look like before you press the enter key.\n\n``` > a <- as.integer(1)\n> typeof(a)\n> a\n\n> b <- as.double(1)\n> typeof(b)\n> b\n\n> d <- as.character(1)\n> typeof(d)\n> d\n\n> e <- as.logical(\"true\")\n> typeof(e)\n> e\n\n> f <- as.complex(-25)\n> typeof(f)\n> f\n\n> g <- as.null(0)\n> typeof(g)\n> g\n\n> h <- as.Date(\"2012-07-04\")\n> typeof(h)\n> class(h)\n> h\n\n> i <-as.POSIXct(\"2012/07/04 10:15:59\")\n> typeof(i)\n> class(i)\n> i\n\n> j <-as.POSIXlt(\"2012/07/04 10:15:59\")\n> typeof(j)\n> class(j)\n> j\n\n> k <- list(\"Get haircut\", \"Buy Groceries\", \"Take shower\")\n> typeof(k)\n> k\n```\n\nIf you don't specifically specify a data type through the as.* commands, R tries to figure out what data type you intended. It does not always guess your mind correctly. Play around with R, assigning some values to some variables and then use the ` typeof() ` command to see the automatic assignments of data types that R made for you. Then see if you can convert a value from one data type to another.\n\n#### Objects, Variables, Values, and Vectors in R\n\nThe R language is based on an object-oriented programming language. Thus, things in R are called objects. So, when we assign a value to the letter \"X,\" in R we would say we have assigned a value to the object \"X.\" Objects in R may have different properties from each other, depending on how they are used. For this exercise, we will concern ourselves with objects that behave like variables. Those types of objects are called vector objects. So, when we talk—in the language of data science—about the variable \"X,\" in R we could call it the vector \"X.\" As you remember, a variable is something that varies. Let's create a character vector in R and assign it three values. We will use the concatenate ` c() ` command in R. Let's also create an integer vector using the same concatenate command.\n\n``` > name <- c(\"Maria\", \"Fred\", \"Sakura\")\n> typeof(name)\n> name\n\n> age <- as.integer(c(24,19,21))\n> typeof(age)\n> age\n```\n\nBoth vectors now have three values each. The character string \"Maria\" is in the first position of the vector \"name,\" \"Fred\" is in the second position, and \"Sakura\" is in the third position. Similarly, the integer 24 is in the first position of the vector \"age,\" 19 is in the second position, and 21 is in the third position. Let's examine each of these individually.\n\n``` > name\n> name\n> name\n> age\n> age\n> age\n```\n\nThe number with in the brackets is called the index or the subscript.\n\n#### Data Sets and Data Frames\n\nIf we had observed the actual names and ages of three people so that `name` corresponded to `age`, we would have a data set that looks like the following.\n\nName Age\nMaria 24\nFred 19\nSakura 21\n\nLet us put our data set into an R data frame object. We need to think of a name for our data frame object. Let's call it \"project.\" After we put our data set into the data frame, we will inspect it using R's \"typeof,\" \"class,\" \"ls,\" and \"structure\" commands, `str()`. Remember, upper and lower cases are meaningful.\n\n``` > project <- data.frame(name, age)\n> typeof(project)\n> class(project)\n> ls(project)\n> str(project)\n```\n\nThe typeof() function told us we had created a list object. The class() function told us it is a special type of list object known as a data.frame. The ls() function tells us what \"key-value\" pairs exist inside our list object. Please don't worry too much about all of that detail right now. What is important is what the str() function tells us.\n\nThe structure command tells us we have three observations and two variables. That is great. It tells us the names of the variables are `\\$name` and `\\$age`. This tells us that when we put a data set into an R data frame list object, we need to reference the variable WITHIN the data frame as follows: `project\\$name` and `project\\$age`. The structure command also tells us that `project\\$name` was assigned a the role of a \"Factor\" variable and that `project\\$age` was assigned the role of \"int.\" These correspond to the \"nominal\" and \"ordinal\" data types that statistitians use. R needs to know the role variables play in order to perform the correct statistical functions on the data. One might argue that the age variable is more like the statistical interval data type than the statistical ordinal data type. We would then have to change the R data type from integer to double. This will change its role to \"number\" within the data frame.\n\nRather than change the data type of `project\\$age`, it is a good practice to create a new variable, so the original is not lost. We will call the new variable `project\\$age.n`, so we can tell that is the transformed `project\\$age` variable.\n\n``` > project\\$age.n <- as.double(project\\$age)\n> str(project)\n```\n\nWe can now see that `project\\$age` and the `project\\$age.n` variables play different roles in the data frame, one as \"int\" and one as \"num.\" Now, confirm that the complete data set has been properly implemented in R by displaying the data frame object.\n\n``` > project\nname age age.n\n1 Maria 24 24\n2 Fred 19 19\n3 Sakura 21 21\n```\n\nNow let's double check the data types.\n\n``` > typeof(project\\$name)\n> typeof(project\\$age)\n> typeof(project\\$age.n)\n```\n\nWhoops! We see some of the quirkiness of R. When we created the variable \"name,\" it had a data type of \"character.\" When we put it into a data frame not only did R assign it the role of a \"Factor\" but it also changed its data type to \"integer.\" What is going on here? This is more than you want to know right now. We will explain it now, but you really don't have to understand it until later.\n\n• Because all statistical computations are done on numbers, R gave each value of the variable \"name\" an arbitrary integer number. It calls these arbitrary numbers levels. It then labeled these levels with the original values, so we would know what is going on. So under the covers, `project\\$name`, has the values: 2 (labeled \"Maria), 1 (labeled \"Fred\") and 3 (labeled Sakura). We can convert `project\\$name` back into the character data type, but we won't be able to perform statistical calculations on it.\n``` > project\\$name.c <- as.character(project\\$name)\n> typeof(project\\$name.c)\n> str(project)\n'data.frame':\t3 obs. of 4 variables:\n\\$ name : Factor w/ 3 levels \"Fred\",\"Maria\",..: 2 1 3\n\\$ age : int 24 19 21\n\\$ age.n : num 24 19 21\n\\$ name.c: chr \"Maria\" \"Fred\" \"Sakura\"\n```\n\nWe can now see that `project\\$name.c` has a data type of character, and has been assigned a data frame role of \"chr.\""
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https://de.mathworks.com/help/sps/ref/smst7c.html | [
"# SM ST7C\n\nDiscrete-time or continuous-time synchronous machine ST7C static excitation system with automatic voltage regulator\n\nSince R2023a\n\n•",
null,
"Libraries:\nSimscape / Electrical / Control / SM Control\n\n## Description\n\nThe SM ST7C block implements a synchronous-machine-type ST7C static excitation system model in conformance with IEEE Std 421.5-2016 .\n\nUse this block to model the control and regulation of the field voltage of a synchronous machine.\n\nSwitch between continuous and discrete implementations of the block by using the Sample time (-1 for inherited) parameter. To configure the integrator for continuous time, set the Sample time (-1 for inherited) parameter to `0`. To configure the integrator for discrete time, set the Sample time (-1 for inherited) parameter to a positive scalar. To inherit the sample time from an upstream block, set the Sample time (-1 for inherited) parameter to `-1`.\n\nThe SM ST7C block comprises three major components:\n\n• The Current Compensator component modifies the measured terminal voltage as a function of the terminal current.\n\n• The Voltage Measurement Transducer component simulates the dynamics of a terminal voltage transducer using a low-pass filter.\n\n• The Excitation Control Elements component compares the voltage transducer output with a terminal voltage reference to produce a voltage error value. The component then passes this value through a voltage regulator to produce the field voltage.\n\nThis diagram shows the structure of the ST7C excitation system model:",
null,
"In the diagram:\n\n• VT and IT are the measured terminal voltage and current of the synchronous machine, respectively.\n\n• VC1 is the current-compensated terminal voltage.\n\n• VC is the filtered, current-compensated terminal voltage.\n\n• VREF is the reference terminal voltage.\n\n• VS is the power system stabilizer voltage.\n\n• EFD is the field voltage.\n\n### Current Compensator and Voltage Measurement Transducer\n\nThe block models the current compensator by using this equation:\n\n`${V}_{C1}={V}_{T}+{I}_{T}\\sqrt{{R}_{C}^{2}+{X}_{C}^{2}},$`\n\nwhere:\n\n• RC is the load compensation resistance.\n\n• XC is the load compensation reactance.\n\nThe block implements the voltage measurement transducer as a Low-Pass Filter block with the time constant TR. Refer to the documentation for the Low-Pass Filter block for information about the exact discrete and continuous implementations.\n\n### Excitation Control Elements\n\nThis diagram shows the structure of the excitation control elements:",
null,
"In the diagram:\n\n• The Summation Point Logic subsystem models the summation point input location for the overexcitation limiter (OEL), underexcitation limiter (UEL), and stator current limiter (SCL) voltages. For more information about using limiters with this block, see Field Current Limiters.\n\n• There are two Take-over Logic subsystems. The subsystems model the take-over point input location for the OEL, UEL and SCL voltages. For more information about using limiters with this block, see Field Current Limiters.\n\n• A Lead-Lag block in series introduces a derivative function and transforms the regulator in a PID. This is typically used only with brushless excitation systems.\n\n• KL and KH model a high-bandwidth regulator inner loop that regulates the generator exciting current.\n\n### Field Current Limiters\n\nYou can use different types of field current limiter to modify the output of the voltage regulator under unsafe operating conditions:\n\n• Use an overexcitation limiter to prevent overheating of the field winding due to excessive field current demand.\n\n• Use an underexcitation limiter to boost field excitation when it is too low, which risks desynchronization.\n\n• Use a stator current limiter to prevent overheating of the stator windings due to excessive current.\n\nAttach the output of any of these limiters at one of these points:\n\n• Summation point — Use the limiter as part of the automatic voltage regulator (AVR) feedback loop.\n\n• Take-over point — Override the usual behavior of the AVR.\n\nIf you are using the stator current limiter at the summation point, use the input VSCLsum. If you are using the stator current limiter at the take-over point, use the overexcitation input VSCLoel, and the underexcitation input VSCLuel.\n\n## Ports\n\n### Input\n\nexpand all\n\nVoltage regulator reference set point, in per-unit representation, specified as a scalar.\n\nData Types: `single` | `double`\n\nInput from the power system stabilizer, in per-unit representation, specified as a scalar.\n\nData Types: `single` | `double`\n\nTerminal voltage magnitude, in per-unit representation, specified as a scalar.\n\nData Types: `single` | `double`\n\nTerminal current magnitude, in per-unit representation, specified as a scalar.\n\nData Types: `single` | `double`\n\nInput from the overexcitation limiter, in per-unit representation, specified as a scalar.\n\n#### Dependencies\n\n• To ignore the input from the overexcitation limiter, set Alternate OEL input locations (V_OEL) to `Unused`.\n\n• To use the input from the overexcitation limiter at the summation point, set Alternate OEL input locations (V_OEL) to ```Summation point at voltage reference```.\n\n• To use the input from the overexcitation limiter at the Take-over Logic subsystem, set Alternate OEL input locations (V_OEL) to ```Take-over at voltage error```.\n\n• To use the input from the overexcitation limiter at the Take-over Logic1 subsystem, set Alternate OEL input locations (V_OEL) to ```Take-over at voltage regulator output```.\n\nData Types: `single` | `double`\n\nInput from the underexcitation limiter, in per-unit representation, specified as a scalar.\n\n#### Dependencies\n\n• To ignore the input from the underexcitation limiter, set Alternate UEL input locations (V_UEL) to `Unused`.\n\n• To use the input from the underexcitation limiter at the summation point, set Alternate UEL input locations (V_UEL) to ```Summation point at voltage reference```.\n\n• To use the input from the underexcitation limiter at the Take-over Logic subsystem, set Alternate UEL input locations (V_UEL) to ```Take-over at voltage error```.\n\n• To use the input from the underexcitation limiter at the Take-over Logic1 subsystem, set Alternate UEL input locations (V_UEL) to ```Take-over at voltage regulator output```.\n\nData Types: `single` | `double`\n\nInput from the stator current limiter when using the summation point, in per-unit representation, specified as a scalar.\n\n#### Dependencies\n\n• To ignore the input from the stator current limiter, set Alternate SCL input locations (V_SCL) to `Unused`.\n\n• To use the input from the stator current limiter at the summation point, set Alternate SCL input locations (V_SCL) to ```Summation point at voltage reference```.\n\nData Types: `single` | `double`\n\nInput from the stator current limiter to prevent field overexcitation when using the take-over point, in per-unit representation, specified as a scalar.\n\n#### Dependencies\n\n• To ignore the input from the stator current limiter, set Alternate SCL input locations (V_SCL) to `Unused`.\n\n• To use the input from the stator current limiter at the Take-over Logic subsystem, set Alternate SCL input locations (V_SCL) to ```Take-over at voltage reference```.\n\n• To use the input from the stator current limiter at the Take-over Logic1 subsystem, set Alternate SCL input locations (V_SCL) to ```Take-over at voltage regulator output```.\n\nData Types: `single` | `double`\n\nInput from the stator current limiter to prevent field underexcitation when using the take-over point, in per-unit representation, specified as a scalar.\n\n#### Dependencies\n\n• To ignore the input from the stator current limiter, set Alternate SCL input locations (V_SCL) to `Unused`.\n\n• To use the input from the stator current limiter at the Take-over Logic subsystem, set Alternate SCL input locations (V_SCL) to ```Take-over at voltage error```.\n\n• To use the input from the stator current limiter at the Take-over Logic1 subsystem, set Alternate SCL input locations (V_SCL) to ```Take-over at voltage regulator output```.\n\nData Types: `single` | `double`\n\n### Output\n\nexpand all\n\nPer-unit field voltage to apply to the field circuit of the synchronous machine, returned as a scalar.\n\nData Types: `single` | `double`\n\n## Parameters\n\nexpand all\n\n### General\n\nInitial per-unit voltage to apply to the field circuit of the synchronous machine.\n\nTime between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state. For more information, see What Is Sample Time? and Specify Sample Time.\n\nFor inherited discrete-time operation, set this parameter to `-1`. For discrete-time operation, set this parameter to a positive integer. For continuous-time operation, set this parameter to `0`.\n\nIf this block is in a masked subsystem or a variant subsystem that supports switching between continuous operation and discrete operation, promote this parameter to ensure correct switching between the continuous and discrete implementations of the block. For more information, see Promote Block Parameters on a Mask.\n\n### Pre-Control\n\nResistance used in the current compensation system. Set this parameter and Reactance component of load compensation, X_C (pu) to `0` to disable current compensation.\n\nReactance used in the current compensation system. Set this parameter and Resistive component of load compensation, R_C (pu) to `0` to disable current compensation.\n\nEquivalent time constant for the voltage transducer filtering.\n\n### Control\n\nEquivalent time constant for the voltage filtering.\n\nEquivalent lag time constant of the voltage regulator, in seconds.\n\nMaximum per-unit voltage reference of the voltage regulator.\n\nMinimum per-unit voltage reference of the voltage regulator.\n\nPer-unit gain of the voltage regulator.\n\nEquivalent lag time constant of the voltage regulator. Set this parameter to `0` when the additional lag dynamics are negligible.\n\nEquivalent lead time constant of the voltage regulator. Set this parameter to `0` when the additional lead dynamics are negligible.\n\nEquivalent time constant in the thyristor bridge firing control.\n\nMaximum per-unit output of the regulator.\n\nMinimum per-unit output of the regulator.\n\nOverexcitation limiter input location, specified as one of these options:\n\n• ```Summation point at voltage reference```V_OEL is an input of the Summation Point Logic subsystem.\n\n• `Take-over at voltage reference`V_OEL is an input of the Take-over Logic subsystem.\n\n• ```Take-over at voltage regulator output```V_OEL is an input of the Take-over Logic 1 subsystem.\n\nUnderexcitation limiter input location, specified as one of these options:\n\n• ```Summation point at voltage reference```V_UEL is an input of the Summation Point Logic subsystem.\n\n• `Take-over at voltage reference`V_UEL is an input of the Take-over Logic subsystem.\n\n• ```Take-over at voltage regulator output```V_UEL is an input of the Take-over Logic 1 subsystem.\n\nStator current limiter input location. specified as one of these options:\n\n• `Summation point` — Use the V_SCLsum input port.\n\n• Any of the `Take-over` options — Use the V_SCLoel and V_SCLuel input ports.\n\n### Exciter\n\nMinimum per-unit gain of the exciter.\n\nMaximum per-unit gain of the exciter.\n\nPer-unit feedback gain of the PI regulator.\n\nFeedback time constant of the Pi regulator, in seconds.\n\n IEEE Std 421.5-2016 (Revision of IEEE Std 421.5-2005). \"IEEE Recommended Practice for Excitation System Models for Power System Stability Studies.\" Piscataway, NJ: IEEE, 2016."
] | [
null,
"https://de.mathworks.com/help/sps/ref/sm_st7c_ic.png",
null,
"https://de.mathworks.com/help/sps/ref/sm_st7c_01.png",
null,
"https://de.mathworks.com/help/sps/ref/sm_st7c_excitation_control_elements.png",
null
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https://xianblog.wordpress.com/tag/frechet/ | [
"## A survey of the [60′s] Monte Carlo methods \n\nPosted in Books, R, Statistics, University life with tags , , , , , on May 18, 2011 by xi'an",
null,
"The 24 questions asked by John Halton in the conclusion of his 1970 survey are\n\n1. Can we obtain a theory of convergence for random variables taking values in Fréchet spaces?\n2. Can the study of Monte Carlo estimates in separable Fréchet spaces give a theory of global approximation?\n3. When sampling functions, what constitutes a representative sample of function values?\n4. Can one apply Monte Carlo to pattern recognition?\n5. Relate Monte Carlo theory to the theory of random equations.\n6. What can be said about quasi-Monte Carlo estimates for finite-dimensional and infinite-dimensional integrals?\n7. Obtain expression, asymptotic forms or upper bounds for L² and L discrepancies of quasirandom sequences.\n8. How should one improve quasirandom sequences?\n9. How to interpret the results of statistical tests applied to pseudo- or quasirandom sequences?\n10. Can we develop a meaningful statistical theory of quasi-Monte Carlo estimates?\n11. Can existing Monte Carlo techniques be improved and applied to new classes of problems?\n12. Can the design of Monte Carlo estimators be made more systematic?\n13. How can the idea of sequential Monte Carlo be extended?\n14. Can sampling with signed probabilities be made practical?\n15. What is the best allocation effort in obtaining zeroth- and first-level estimators in algebraic problems?\n16. Examine the Monte Carlo analogues of the various matrix iterative schemes.\n17. Develop the schemes of grid refinement in continuous problems.\n18. Develop new Monte Carlo eigenvectors and eigenvalue techniques.\n19. Develop fast, reliable true canonical random generators.\n20. How is the output of a true random generator to be tested?\n21. Develop fast, efficient methods for generating arbitrary random generators.\n22. Can we really have useful general purpose pseudorandom sequences.\n23. What is the effect of the discreteness of digital computers on Monte Carlo calculations?\n24. Is there a way to estimate the accuracy of Monte Carlo estimates?"
] | [
null,
"https://xianblog.files.wordpress.com/2011/05/img_8168-e1305488888661.jpg",
null
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https://moodle.tbaisd.org/mod/book/view.php?id=51433&chapterid=50228 | [
"Negative Reciprocal\n\nRemember that a whole number to a negative power is equivalent to its reciprocal to the positive power. For instance, y = 5 · 2 ?x is equivalent to",
null,
". Therefore, these types of functions also create exponential decay.\n\nExample Graph the exponential function y = 8 · 3?x.\n\nStep 1. Make a table of values.",
null,
"Step 2. Graph the function by plotting points.",
null,
""
] | [
null,
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null,
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",
null,
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null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.81582296,"math_prob":0.99844295,"size":319,"snap":"2021-21-2021-25","text_gpt3_token_len":76,"char_repetition_ratio":0.104761906,"word_repetition_ratio":0.0,"special_character_ratio":0.23510972,"punctuation_ratio":0.15625,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9934082,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-16T01:28:02Z\",\"WARC-Record-ID\":\"<urn:uuid:da389977-ec11-4684-bacf-6984454716ff>\",\"Content-Length\":\"73042\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2270bd50-ab27-4d0e-a9df-18c7e19fde8c>\",\"WARC-Concurrent-To\":\"<urn:uuid:89c67d1a-51bd-4838-9d27-dd5d4dad2420>\",\"WARC-IP-Address\":\"136.228.36.41\",\"WARC-Target-URI\":\"https://moodle.tbaisd.org/mod/book/view.php?id=51433&chapterid=50228\",\"WARC-Payload-Digest\":\"sha1:D5CHXIWVO2GAS6R36HPCRZ7AOHOQMJLM\",\"WARC-Block-Digest\":\"sha1:U3SZD2GTERPJOQGWSLWFNCPJCS5P3LZD\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487621699.22_warc_CC-MAIN-20210616001810-20210616031810-00083.warc.gz\"}"} |
https://fr.mathworks.com/help/bioinfo/ug/analyzing-array-based-cgh-data-using-bayesian-hidden-markov-modeling.html | [
"# Analyzing Array-Based CGH Data Using Bayesian Hidden Markov Modeling\n\nThis example shows how to use a Bayesian hidden Markov model (HMM) technique to identify copy number alteration in array-based comparative genomic hybridization (CGH) data.\n\n### Introduction\n\nArray-based CGH is a high-throughput technique to measure DNA copy number change across the genome. The DNA fragments or \"clones\" of test and reference samples are hybridized to mapped array fragments. Log2 intensity ratios of test to reference provide useful information about genome-wide profiles in copy number. In an ideal situation, the log2 ratio of normal (copy-neutral) clones is log2(2/2) = 0, single copy losses is log2(1/2) = -1, and single copy gains is log2(3/2) = 0.58. Multiple copy gains or amplifications would have values of log2(4/2), log2(5/2),.... Loss of both copies, or a deletion would correspond to the value of -inf. In real applications, even after accounting for measurement error, the log2 ratios differ considerably from the theoretical values. The ratios typically shrink towards zero. One main factor is contamination of the tumor samples with normal cells. There is also a dependence between the intensity ratios of neighboring clones. These issues necessitate the use of efficient statistical algorithms characterizing the genomic profiles.\n\n### Bayesian HMM\n\nGuha et al., (2006) proposed a Bayesian statistical approach depending on a hidden Markov model (HMM) for analyzing array CGH data. The hidden Markov model accounts for the dependence between neighboring clones. The intensity ratios are generated by hidden copy number states. Bayesian learning is used to identify genome-wide changes in copy number from the data. Posterior inferences are made about the copy number gains and losses.\n\nIn this Bayesian HMM algorithm, there are four states, defined as copy number loss state (1), copy number neutral state (2), single copy gain state (3), and amplification (multiple gain) state (4). A copy number state is associated with each clone. The normalized log2 ratios are assumed to be distributed as",
null,
"The mu is a unknown parameter for each state with this constraint:",
null,
"The priors for mean copy number changes are:",
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"",
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"",
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"",
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"Guha et al., (2006) also described an Metropolis-within-Gibbs algorithm to generate posterior samples. The MCMC algorithm is independently run for each chromosome to generate an MCMC sample for the chromosome parameters. The starting values of the parameters are generated from the priors. The generated copy number states represent draws from the marginal posterior of interest, For each MCMC draw, the generated states are inspected and classified as focal ablations, transition points, amplifications, outliers and whole chromosomal changes.\n\nIn this example, you will apply the Bayesian HMM algorithm to analyze the array CGH profiles of some pancreatic cancer samples .\n\nThe data in this example is the array CGH profiles of 24 pancreatic adenocarcinoma cell lines and 13 primary tumor specimens from Alguirre et al.,(2004). Labeled DNA fragments were hybridized to Agilent® human cDNA microarrays containing 14,160 cDNA clones. About 9,420 clones have unique map positions with a median interval between mapped elements of 100.1 kb. More details of the data and experiment can be found in . For convenience, the data has already been normalized and the log2 based intensity ratios are provided by the MAT file pancrea_oligocgh.mat.\n\nYou will apply the Bayesian HMM algorithm to analyze chromosome 12 of sample 6 of the pancreatic adenocarcinoma data, and compare the results with the segments found by the circular binary segmentation (CBS) algorithm of Olshen et al.,(2004).\n\nLoad the MAT file containing the log2 intensity ratios and mapped genomic positions of the 37 samples.\n\nload pancrea_oligocgh pancrea_data \npancrea_data = struct with fields: Sample: {37x1 cell} Chromosome: [13446x1 int8] GenomicPosition: [13446x1 int32] Log2Ratio: [13446x37 double] Log2RatioMed: [13446x37 double] Log2RatioSeg: [13446x37 double] CloneIDs: [13446x1 int32] \n\nSpecify the chromosome number and sample to analyze.\n\nsampleIndex = 6; chromID = 12; sample = pancrea_data.Sample{sampleIndex} \nsample = 'PA.C.Dan.G' \n\nLoad and plot the log2 ratio data of chromosome 12 from sample PA.C.Dan.G.\n\nidx = pancrea_data.Chromosome == chromID; X = double(pancrea_data.GenomicPosition(idx)); Y = pancrea_data.Log2Ratio(idx, sampleIndex); % Remove NaN data points idx = ~isnan(Y); X = X(idx); Y = Y(idx); % Plot the data figure; plot(X, Y, '.', 'color', [0.6 0.6 1]) ylims = [-1.5, 3.5]; ylim(gca, ylims) title(sprintf('%s - Chromosome %d', sample, chromID)) xlabel('Genomic Position'); ylabel('Log2(Ratio)')",
null,
"Number of clones on chromosome 12 to be analyzed\n\nN = numel(Y) \nN = 437 \n\n### Performing Circular Binary Segmentation\n\nYou can start the analysis by performing chromosomal segmentation using the CBS algorithm , which is implemented in the cghcbs function. The process will take several seconds. You can view the plot of the segment means over the original data by specifying the SHOWPLOT parameter. Note: You can type doc cghcbs for more details on this function.\n\nPS = cghcbs(pancrea_data, 'SampleInd', sampleIndex, ... 'Chromosome', chromID, 'ShowPlot', chromID); ylim(gca, ylims) \nAnalyzing: PA.C.Dan.G. Current chromosome 12",
null,
"As shown in the figure, the CBS procedure declared the set of high intensity ratios as two separate segments. The CBS procedure also found a region with copy number losses.\n\n### Initializing Parameters\n\nThe Bayesian HMM approach uses a Metropolis-within-Gibbs algorithm to generate posterior samples of the parameters . The model parameters are grouped into four blocks. The algorithm iteratively generates each of the four blocks conditional on the remaining blocks and the data.\n\nTo analyze the data with the Bayesian HMM algorithm, you need to initialize the parameters. More details on prior parameters can be found in references and .\n\nInitialize the state of the random number generator to ensure that the figures generated by these command match the figures in the HTML version of this example.\n\nrng('default'); \n\nDefine the number of states\n\nNS = 4; \n\nDefine the number of MCMC iterations\n\nNMC = 100; \n\nDetermine the hyperparameters of the prior distributions for the four states.\n\nmus_hyper = [-1, 0, 0.58, 1]; taus_hyper = [1, 1, 1, 2]; \n\nSet the parameter epsilon which determines the constrains of the means.\n\neps = 0.1; \n\nSet the bounds of the prior means of each state.\n\nmu_low_bounds = [-Inf, -eps, eps, 0.58]; mu_up_bounds = [-eps, eps, 0.58, Inf]; \n\nGuha et al., (2006) assumes the inverse of the prior error variances (sigma^2) as gamma distributions with lower bounds of 0.41 for states 1, 2 and 3. Set the scale parameters for the gamma distributions for each state.\n\nsg_alpha = [1 1 1 1]; sg_beta = [1, 1, 1, 1]; sg_bounds = [0.41 0.41 0.41 1]; \n\nDefine a variable states to store the copy number state sequences of the clones for each MCMC iteration.\n\nstates = zeros(N, NMC); \n\nDefine a variable st_counts to hold the state transition counts for each copy number state.\n\nst_counts = zeros(NS, NS); \n\n### Determining the Prior Distributions\n\nThe MCMC iteration starts at\n\niloop = 1; \n\nDetermine sigmas for the four states by sampling from gamma distribution with prior scale parameter alpha and beta.\n\nsigmas = zeros(NS, NMC); for i = 1:NS sigmas(i, iloop) = acghhmmsample('gamma', sg_alpha(i), sg_beta(i), sg_bounds(i)); end \n\nDetermine means for the four states by sampling from truncated normal distribution between the lower and upper bounds of the means. Note: The fourth state lower bound will be determined by the third state.\n\nmus = zeros(NS, NMC); for i = 1:NS if i == 4 mu_low_bounds(4) = mus(3,iloop) + 3*sigmas(3,iloop); end mus(i, iloop) = acghhmmsample('normal', mus_hyper(i), taus_hyper(i),... mu_low_bounds(i), mu_up_bounds(i)); end \n\nAssume independent Dirichlet priors for the rows of the stochastic 4x4 transition probability matrix . Generate the stochastic prior transition matrix A from the Dirichlet distributions.\n\na = ones(NS, NS); A = acghhmmsample('dirichlet', a, NS); \n\nThe transition matrix has a unique stationary distribution. The stationary distribution PI is an eigenvector of the transition matrix associated with the eigenvalue 1.\n\nPI =@(x, n) (ones(1,n)/(eye(n) -x + ones(n)))'; \n\nGenerate the prior stationary distribution PI.\n\nPi = PI(A, NS); \n\nGenerate the initial emission matrix B\n\nB = zeros(NS, N); for i = 1:NS B(i,:) = normpdf(Y, mus(i,iloop), sigmas(i,iloop)); end \n\nDecode initial hidden states of the clones using a stochastic forward-backward algorithm .\n\nstates(:, iloop) = acghhmmfb(Pi, A, B); \n\n### Generating Posterior Samples\n\nFor each MCMC iteration, the four blocks of parameters are generated as follows : Update block B1 using a Metropolis-Hastings step to generate the transition matrix, update block B2 the copy number states using a stochastic forward propagate backward sampling algorithm, update block B3 by computing the mus, and update block B4 to generate sigmas.\n\nfor iloop = 2:NMC % Compute the number of transitions from state i to state j for i =1:NS for j = 1:NS st_counts(i, j) = sum((states(1:N-1, iloop-1) == i) .* (states(2:N, iloop-1) == j)); end end % Updating block B1 % Generate the transition matrix from the Dirichlet distributions C = acghhmmsample('dirichlet', st_counts + 1, NS); % Compute the state probabilities under stationary distribution of a % given transition matrix C. PiC = PI(C, NS); % Compute the accepting probability using a Metropolis-Hastings step beta = min([1, exp(log(PiC(states(1, iloop-1))) - log(Pi(states(1, iloop-1))))]); if rand < beta A = C; Pi = PiC; end % Updating block B2 % Generate copy number states using Forward propagate, backward sampling . states(:, iloop) = acghhmmfb(Pi, A, B); % Updating blocks B3 and B4 for i = 1:NS idx_s = states(:, iloop) == i; num_states = sum(idx_s); % If state i is not observed, then draw from its prior parameters if num_states == 0 mus(i, iloop) = acghhmmsample('normal', mus_hyper(i),... taus_hyper(i), mu_low_bounds(i), mu_up_bounds(i)); sigmas(i, iloop)= acghhmmsample('gamma', sg_alpha(i),... sg_beta(i), sg_bounds(i)); else Y_avg = mean(Y(idx_s)); theta_prec = 1/taus_hyper(i)^2 + num_states/sigmas(i,iloop-1)^2; weight_means = (mus_hyper(i)/(taus_hyper(i)^2) +... Y_avg * num_states/(sigmas(i, iloop-1)^2))/theta_prec; % Compute mus - B3 mus(i, iloop) = acghhmmsample('normal', weight_means, ... 1/sqrt(theta_prec), mu_low_bounds(i), mu_up_bounds(i)); % Compute sigmas - B4 Y_v = sum((Y(idx_s) - mus(i, iloop)).^2); sigmas(i, iloop) = acghhmmsample('gamma', sg_alpha(i)+num_states/2,... sg_beta(i)+Y_v/2, sg_bounds(i)); end % Update the emission matrix with new mus and sigmas. B(i,:) = normpdf(Y, mus(i,iloop),sigmas(i,iloop)); end end \n\nPlot the posterior mean mu distributions of the four states.\n\nfigure; for j = 1:NS subplot(2,2,j) ksdensity(mus(j,:)); title(sprintf('State %d', j)) end sgtitle('Distribution of Mu of States'); hold off;",
null,
"Plot the posterior sigma distributions of the four states.\n\nfigure; for j = 1:NS subplot(2,2,j) ksdensity(sigmas(j,:)); title(sprintf('State %d', j)) end sgtitle('Distribution of Sigma of States'); hold off;",
null,
"### Posterior Inference\n\nDraw a state label for each clone from the MCMC sampling and compute the posterior probabilities of each state.\n\nclone_states = zeros(1, N); state_prob = zeros(NS, N); state_count = zeros(NS, N); for i = 1:N % for each clone state = states(i, :); for j=1:NS state_count(j, i) = sum(state == j); end selState = find(state_count(:,i) == max(state_count(:,i))); if length(selState) > 1 if i ~= 1 clone_states(i) = clone_states(i-1); else clone_states(i) = min(selState); end else clone_states(i) = selState; end state_prob(:, i) = state_count(:,i)/NMC; end clone_states = clone_states'; \n\nPlot the state label for each clone on chromosome 12 of sample PA.C.Dan.G.\n\nfigure; leg = zeros(1,4); for i = 1:N if clone_states(i) == 1 leg(1) = plot(i,Y(i),'v', 'MarkerFaceColor', [1 0.2 0.2],... 'MarkerEdgeColor', 'none'); elseif clone_states(i) == 2 leg(2) = plot(i,Y(i),'o', 'Color', [0.4 0.4 0.4]); elseif clone_states(i) == 3 leg(3) = plot(i,Y(i),'^', 'MarkerFaceColor', [0.2 1 0.2],... 'MarkerEdgeColor', 'none'); elseif clone_states(i) == 4 leg(4) = plot(i, Y(i), '^', 'MarkerFaceColor', [0.2 0.2 1],... 'MarkerEdgeColor', 'none'); end hold on; end ylim(gca, ylims) legend(leg, 'State 1', 'State 2','State 3','State 4') xlabel('Index') ylabel('Log2(ratio)') title('State Label') hold off",
null,
"### Classifying Array CGH Profiles\n\nFor each MCMC draw, the generated states can be classified as focal aberrations, transition points, amplifications, outliers and whole chromosomal changes . In this example, you will find the high-level amplifications, transition points and outliers on chromosome 12 of sample PA.C.Dan.G.\n\nA clone with state = 4 is considered a high-level amplification . Find high-level amplifications.\n\nhigh_lvl_amp_idx = find(clone_states == 4); \n\nA transition point is associated with large-scale regions of gains and losses and is declared when the width of the altered region exceeds 5 mega base pair . Find transition points.\n\nregion_lim = 5e6; focalabr_idx=[1;find(diff(clone_states)~=0);N]; istranspoint = false(length(focalabr_idx), 1); for i = 1:length(focalabr_idx)-1 region_x = X(focalabr_idx(i+1)) - X(focalabr_idx(i)); istranspoint(i+1) = region_x > region_lim; end trans_idx = focalabr_idx(istranspoint); % Remove adjacent trans_idx that have the same states. hasadjacentstate = [diff(clone_states(trans_idx))==0; true]; trans_idx = trans_idx(~hasadjacentstate) focalabr_idx = focalabr_idx(~istranspoint); focalabr_idx = focalabr_idx(2:end-1); \ntrans_idx = 107 135 323 \n\nAn outlier for gains is a focal aberration satisfying its z-score greater than 2, while an outlier for losses has a z-score less than -2 .\n\nFind outliers for losses\n\noutlier_loss_idx = focalabr_idx(clone_states(focalabr_idx) == 1) if ~isempty(outlier_loss_idx) [F,Xi] = ksdensity(mus(1,:)); [dummy, idx] = max(F); mu_1 = Xi(idx); [F,Xi] = ksdensity(sigmas(1,:)); [dummy, idx] = max(F); sigma_1 = Xi(idx); outlier_loss_idx = outlier_loss_idx((Y(outlier_loss_idx) - mu_1)/sigma_1 < -2) end \noutlier_loss_idx = 0x1 empty double column vector \n\nFind outliers for gains\n\noutlier_gain_idx = focalabr_idx(clone_states(focalabr_idx) == 3); if ~isempty(outlier_gain_idx) [F,Xi] = ksdensity(mus(3,:)); [dummy, idx] = max(F); mu_1 = Xi(idx); [F,Xi] = ksdensity(sigmas(3,:)); [dummy, idx] = max(F); sigma_1 = Xi(idx); outlier_gain_idx = outlier_gain_idx((Y( outlier_gain_idx) - mu_1)/sigma_1 > 2) end \noutlier_gain_idx = 0x1 empty double column vector \n\nAdd the classified labels to the intensity ratio plot of chromosome 12 of sample PA.C.Dan.G. Plot the segment means from the CBS procedure for comparison.\n\nfigure; hl1 = plot(X, Y, '.', 'color', [0.4 0.4 0.4]); hold on; if ~isempty(high_lvl_amp_idx) hl2 = line(X(high_lvl_amp_idx), Y(high_lvl_amp_idx),... 'LineStyle', 'none',... 'Marker', '^',... 'MarkerFaceColor', [0.2 0.2 1],... 'MarkerEdgeColor', 'none'); end if ~isempty(trans_idx) for i = 1:numel(trans_idx) hl3 = line(ones(1,2)*X(trans_idx(i)), [-3.5, 3.5],... 'LineStyle', '--',... 'Color', [1 0.6 0.2]); end end if ~isempty(outlier_gain_idx) line(X(outlier_gain_idx), Y(outlier_gain_idx),... 'LineStyle', 'none',... 'Marker', 'v',... 'MarkerFaceColor', [1 0 0],... 'MarkerEdgeColor', 'none'); end if ~isempty(outlier_loss_idx) hl4 = line(X(outlier_loss_idx), Y(outlier_loss_idx),... 'LineStyle', 'none',... 'Marker', 'v',... 'MarkerFaceColor', [1 0 0],... 'MarkerEdgeColor', 'none'); end % Plot segment means from the CBS procedure. for i = 1:numel(PS.SegmentData.Start) hl5 = line([PS.SegmentData.Start(i) PS.SegmentData.End(i)],... [PS.SegmentData.Mean(i) PS.SegmentData.Mean(i)],... 'Color', [1 0 0],... 'LineWidth', 1.5); end ylim(gca, ylims) ylabel('Log2(Ratio)') title(sprintf('%s - Chromosome %d', sample, chromID)) % Adding chromosome 12 ideogram and legends to the plot. chromosomeplot('hs_cytoBand.txt', chromID, 'addtoplot', gca) legend([hl1, hl2, hl3,hl5], 'IntensityRatio', 'Amplification',... 'TransitionPoint', 'CBS SegmentMean')",
null,
"The Bayesian HMM algorithm found 3 transition points indicated by the broken vertical lines in the plot. The Bayesian HMM algorithm identified two high-level amplified regions marked by blue up-triangles in the plot. The two high-level amplified regions correspond to the two minimal common regions (MCRs) on chromosome 12, associated with copy number gains as explained by Aguirre et al.,(2004). The Bayesian HMM declared the first set of high intensity rations as a single region of high-level amplification. In comparison, the CBS procedure failed to detect the second MCR and segmented the first MCR into two regions. No outlier was detected in this example.\n\n### References\n\n Guha, S., Li, Y. and Neuberg, D., \"Bayesian hidden Markov modeling of array CGH data\", Journal of the American Statistical Association, 103(482):485-497, 2008.\n\n Aguirre, A.J., et al., \"High-resolution characterization of the pancreatic adenocarcinoma genome\", PNAS, 101(24):9067-72, 2004.\n\n Olshen, A.B., et al., \"Circular binary segmentation for the analysis of array-based DNA copy number data\", Biostatistics, 5(4):557-7, 2004.\n\n Shah, S.P., et al., \"Integrating copy number polymorphisms into array CGH analysis using a robust HMM\", Bioinformatics, 22(14):e431-e439, 2006\n\n## Support",
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https://stat.ethz.ch/pipermail/r-help/2009-March/417552.html | [
"# [R] non-positive definite matrix remedies?\n\nAlbyn Jones jones at reed.edu\nThu Mar 12 00:54:03 CET 2009\n\n```That's an interesting problem.\n\nMy first thought was to choose the closest positive definite matrix to\nthe given matrix, say in the least squares sense. However, the\n2x2 diagonal matrix with diagonal (1,0) makes it clear that there\nisn't a closest pd symmetric matrix.\n\nPerhaps multiple imputation would work: impute a complete data\nmatrix X, compute polycor(X), and repeat. Will the average of these\npositive definite matrices be positive definite???? I think it would if\nyou were computing pearson correlations, but I am not sure about the\npolychoric case.\n\nalbyn\n\nOn Wed, Mar 11, 2009 at 04:20:27PM -0600, Matthew Keller wrote:\n> Hi all,\n>\n> For computational reasons, I need to estimate an 18x18 polychoric\n> correlation matrix two variables at a time (rather than trying to\n> estimate them all simultaneously using ML). The resulting polychoric\n> correlation matrix I am getting is non-positive definite, which is\n> problematic because I'm using this matrix later on as if it were a\n> legitimately estimated correlation matrix (in order to fit an SEM\n> model). I could add to the diagonal I suppose until it becomes\n> positive definite. Does anyone have any other ideas on how to deal\n> with this problem, and what the strengths and weaknesses of different\n> approaches are?\n>\n>\n> Matt\n>\n>\n> --\n> Matthew C Keller\n> Asst. Professor of Psychology\n> University of Colorado at Boulder\n> www.matthewckeller.com\n>\n> ______________________________________________\n> R-help at r-project.org mailing list\n> https://stat.ethz.ch/mailman/listinfo/r-help"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.88248205,"math_prob":0.7338095,"size":1864,"snap":"2021-31-2021-39","text_gpt3_token_len":444,"char_repetition_ratio":0.13333334,"word_repetition_ratio":0.0,"special_character_ratio":0.26233906,"punctuation_ratio":0.14534883,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9692237,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T03:52:19Z\",\"WARC-Record-ID\":\"<urn:uuid:abfd35f9-5f73-480c-a1e7-b05fae0150a1>\",\"Content-Length\":\"4949\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eb6a656e-7fd0-4024-b019-f453dc6d39d6>\",\"WARC-Concurrent-To\":\"<urn:uuid:88406a51-1937-4d92-8172-d6910a12e0d5>\",\"WARC-IP-Address\":\"129.132.119.195\",\"WARC-Target-URI\":\"https://stat.ethz.ch/pipermail/r-help/2009-March/417552.html\",\"WARC-Payload-Digest\":\"sha1:72DXU5LQND66WEN3N2KDBZZWQZU77CB2\",\"WARC-Block-Digest\":\"sha1:K2GY7QUCQMU6AVIFGQ524NW74IRNAK43\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152168.38_warc_CC-MAIN-20210727010203-20210727040203-00519.warc.gz\"}"} |
https://geoenergymath.com/2014/07/ | [
"# Correlation of time series\n\n[mathjax]The Southern Oscillation embedded with the ENSO behavior is what is called a dipole , or in other vernacular, a standing wave. Whenever the atmospheric pressure at Tahiti is high, the pressure at Darwin is low, and vice-versa. Of course the standing wave is not perfect and far from being a classic sine wave.\n\nTo characterize the quality of the dipole, we can use a measure such as a correlation coefficient applied to the two time series. Flipping the sign of Tahiti and applying a correlation coefficient to SOI, we get Figure 1 below:",
null,
"Fig 1 : Anti-correlation between Tahiti and Darwin. The sign of Tahiti is reversed to see better the correlation. The correlation coefficient is calculated to be 0.55 or 55/100.\n\nNote that this correlation coefficient is “only” 0.55 when comparing the two time-series, yet the two sets of data are clearly aligned. What this tells us is that other factors, such as noise in the measurements, can easily drop correlated waveforms well below unity.\n\nThis is what we have to keep in mind when evaluating correlations of data with models as we can see in the following examples.\n\n# Sloshing Animation\n\nThe models of ENSO for SOI and proxy records apply sloshing dynamics to describe the quasi-periodic behavior. see J. B. Frandsen, “Sloshing motions in excited tanks,” Journal of Computational Physics, vol. 196, no. 1, pp. 53–87, 2004.\n\nThe following GIF animations are supplementary material from S. S. Kolukula and P. Chellapandi, “Finite Element Simulation of Dynamic Stability of plane free-surface of a liquid under vertical excitation.”\n\nDetuning Effect.gif shows the animation of sloshing fluid for the fourth test case, with frequency ratio Ω3 = 0.5 and forcing amplitudeV = 0.2: test case 4 as shown in Figure 4. This case corresponds to instability in the second sloshing mode lying in the first instability region. Figure 8(b) shows the free-surface elevation and Figure 9 shows the moving mesh generated in this case.",
null,
"Dynamic Instability.gif shows the animation of sloshing fluid for the second test case which lies in the unstable region, with frequency ratio Ω1 = 0.5 and forcing amplitude kV =0.3: test case 2 as shown in Figure 4. Figure 6 shows the free-surface elevation and Figure 7 shows the moving mesh generated in this case."
] | [
null,
"https://i0.wp.com/imageshack.com/a/img539/1412/48fe30.gif",
null,
"https://i0.wp.com/imageshack.com/a/img823/6842/6fd5.gif",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.8981518,"math_prob":0.95029724,"size":2291,"snap":"2023-14-2023-23","text_gpt3_token_len":536,"char_repetition_ratio":0.11980761,"word_repetition_ratio":0.05835544,"special_character_ratio":0.22086425,"punctuation_ratio":0.124168515,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9772026,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,5,null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-05T00:40:04Z\",\"WARC-Record-ID\":\"<urn:uuid:ca22001d-2708-490b-b733-314dcd62b1e3>\",\"Content-Length\":\"88357\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:caa741cb-5328-4429-b751-36d5dbf84a79>\",\"WARC-Concurrent-To\":\"<urn:uuid:2360aadf-eafb-4450-8bf7-5e83454b21a3>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://geoenergymath.com/2014/07/\",\"WARC-Payload-Digest\":\"sha1:4F7QCJQPP66ITUFWP2HLLYTJLBDFPLP4\",\"WARC-Block-Digest\":\"sha1:O3CNYG7DBXEABJMX4B33DIGF27ZRYJ2P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224650409.64_warc_CC-MAIN-20230604225057-20230605015057-00261.warc.gz\"}"} |
https://www.cs.auckland.ac.nz/software/AlgAnim/search_trees.html | [
"Data Structures and Algorithms 8 Searching Revisited\n\nBefore we examine some more searching techniques, we need to consider some operations on trees - in particular means of traversing trees.\n\n#### Tree operations\n\nA binary tree can be traversed in a number of ways:\n pre-order Visit the root Traverse the left sub-tree, Traverse the right sub-tree in-order Traverse the left sub-tree, Visit the root Traverse the right sub-tree post-order Traverse the left sub-tree, Traverse the right sub-tree Visit the root\n\nIf we traverse the standard ordered binary tree in-order, then we will visit all the nodes in sorted order.\n\n#### Parse trees\n\n If we represent the expression: A*(((B+C)*(D*E))+F) as a tree:",
null,
"then traversing it post-order will produce:\nA B C + D E * * F + *\nwhich is the familiar reverse-polish notation used by a compiler for evaluating the expression.\n\n#### Search Trees\n\nWe've seen how to use a heap to maintain a balanced tree for a priority queue. What about a tree used to store information for retrieval (but not removal)? We want to be able to find any item quickly in such a tree based on the value of its key. The search routine on a binary tree:\n```tree_search(tree T, Key key) {\nif (T == NULL) return NULL;\nif (key == T->root) return T->root;\nelse\nif (key < T->root) return tree_search( T->left, key );\nelse return tree_search( T->right, key );\n}\n```\nis simple and provides us with a O(log n) searching routine as long as we can keep the tree balanced. However, if we simply add items to a tree, producing an unbalanced tree is easy!\n This is what happens if we add the letters A B C D E F in that order to a tree: Not exactly well balanced!",
null,
"### Key terms\n\nPre-order tree traversal\nTraversing a tree in the order: root | left | right\nIn-order tree traversal\nTraversing a tree in the order: left | root | right\nPost-order tree traversal\nTraversing a tree in the order: left | right | root\n\n©",
null,
", 1998"
] | [
null,
"https://www.cs.auckland.ac.nz/software/AlgAnim/fig/parse_tree.gif",
null,
"https://www.cs.auckland.ac.nz/software/AlgAnim/fig/unbal_tree.gif",
null,
"https://www.cs.auckland.ac.nz/software/AlgAnim/john_email.gif",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7771392,"math_prob":0.89943033,"size":935,"snap":"2020-34-2020-40","text_gpt3_token_len":225,"char_repetition_ratio":0.116004296,"word_repetition_ratio":0.0,"special_character_ratio":0.2513369,"punctuation_ratio":0.10382514,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9647573,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,2,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-27T23:15:50Z\",\"WARC-Record-ID\":\"<urn:uuid:d5937a5e-72df-420e-b1f6-8d149f2ebef5>\",\"Content-Length\":\"4328\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3bca03ac-a6db-45cf-b3ba-2c2669ddccab>\",\"WARC-Concurrent-To\":\"<urn:uuid:d57cee54-dd11-4e4d-81e7-f2d44f52d3ab>\",\"WARC-IP-Address\":\"130.216.158.22\",\"WARC-Target-URI\":\"https://www.cs.auckland.ac.nz/software/AlgAnim/search_trees.html\",\"WARC-Payload-Digest\":\"sha1:UIAYZDBS5Y4QSNUFZGL4SRLEWMQDH2UY\",\"WARC-Block-Digest\":\"sha1:ZEGY54ZAZW3GISIKK2UC3U4P6IENAVVW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600401582033.88_warc_CC-MAIN-20200927215009-20200928005009-00184.warc.gz\"}"} |
https://findfilo.com/math-question-answers/choose-the-best-possible-option-displaystyle-x-3-534p | [
"",
null,
"Choose the best possible option.\\displaystyle { x }^{ 3 }-5x+2{ x | Filo",
null,
"",
null,
"Class 10\n\nMath\n\nAll topics\n\nQuadratic Equations",
null,
"537\n\nChoose the best possible option.\nis quadratic equation.\n\n1. Yes\n2. No\n3. Can't predict\n4. None\nCorrect Answer: Option(b)\nSolution: The degree of equation is 3.\nis not quadratic.\nIt is cubic equation.",
null,
"537",
null,
"Connecting you to a tutor in 60 seconds.\n\nGet answers to your doubts.\n\nSimilar Topics\nintroduction to trigonometry\nfunctions\nsome applications of trigonometry\nquadratic equations\nsurface areas and volumes"
] | [
null,
"https://www.facebook.com/tr",
null,
"https://findfilo.com/images/logo.svg",
null,
"https://findfilo.com/images/icons/navbar.png",
null,
"https://findfilo.com/images/icons/view.svg",
null,
"https://findfilo.com/images/icons/view.svg",
null,
"https://findfilo.com/images/common/mobile-widget.png",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.7937773,"math_prob":0.9834267,"size":268,"snap":"2021-21-2021-25","text_gpt3_token_len":91,"char_repetition_ratio":0.15530303,"word_repetition_ratio":0.0,"special_character_ratio":0.2947761,"punctuation_ratio":0.12727273,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99733675,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-23T11:01:24Z\",\"WARC-Record-ID\":\"<urn:uuid:7a387746-83a5-491a-b3e9-85df3a4b69d2>\",\"Content-Length\":\"72261\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6117ffb-b73e-4cf7-8b76-c6f22b878bf4>\",\"WARC-Concurrent-To\":\"<urn:uuid:b2e8526b-0612-4cb1-ba59-04ec451599b3>\",\"WARC-IP-Address\":\"34.117.94.82\",\"WARC-Target-URI\":\"https://findfilo.com/math-question-answers/choose-the-best-possible-option-displaystyle-x-3-534p\",\"WARC-Payload-Digest\":\"sha1:R4NJDFW4A2RYFBXQBLQOG63K5MM5KK7V\",\"WARC-Block-Digest\":\"sha1:ZLMGFYAEXVKK3TJVFFSB4CEBNSGYDDXK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488538041.86_warc_CC-MAIN-20210623103524-20210623133524-00235.warc.gz\"}"} |
https://aakashdigitalsrv1.meritnation.com/ask-answer/question/a-small-note-on-aryabhatta-s-contribution-to-maths-plzzz-i-w/the-triangle-and-its-properties/6776732 | [
"# A small note on aryabhatta's contribution to maths. plzzz i want it fast. First answer will get two thumbs up!!\n\nARYABHATTA CONTRIBUTIONS TO MATHEMATICSNUMBER NOTATIONØNumerical values:he made a notation system in which digits are denoted with the help of alphabet numerals e.g., 1 = ka, 2 = Kha, etc.ØNotation system:He invented a notation system consisting of alphabet numerals Digits were denoted by alphabet numeralsØPlace-value:Aryabhatta was familiar with the place-value system.ALGEBRAØInteger solutions:Aryabhatta was the first one to explore integer solutions to the equations of the form by =ax+c and by =ax-c, where a,b,c are integers. He used kuttuka method to solve problems.ØIndeterminate equations:He gave general solutions to linear indeterminate equations ax+by+c= 0 by the method of continued fraction.GEOMETRYØDiscover thePValue :The credit for discovering the exact valuesPmay be ascribed to the celebrated mathematician Aryabhatta.Rule: Add 4 to 100, multiply by 8, add 62000. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.This givesP=62832/20000=3.1416. Which is an accurate value ofP. Aryabhatta discovered this value independently and also realized thatPis an irrational numberØPythagorean Theorem:The Pythagorean theorem is stated as follows in his work the square of the Bhuja (base) plus the square of the koti (perpendicular) is the square of the Karna(Buja and koti are the sides of a right-angled triangle. The Karna is the hypotenuse)ØCircle Theorem:He has postulated a theorem relating to circle as follows In a circle the product of two Saras is the square of the half chord of the two arcs i.e. a*b=c2where c is half the chord and the saras or arrows are the segments of a diameter which bisect any chord.ØFormula:Aryabhatta gives formulae for the areas of a triangle, square, rectangle, rhombus, circle etc.\n\n• 6\nARYABHATTA CONTRIBUTIONS TO MATHEMATICSNUMBER NOTATIONØNumerical values:he made a notation system in which digits are denoted with the help of alphabet numerals e.g., 1 = ka, 2 = Kha, etc.ØNotation system:He invented a notation system consisting of alphabet numerals Digits were denoted by alphabet numeralsØPlace-value:Aryabhatta was familiar with the place-value system.ALGEBRAØInteger solutions:Aryabhatta was the first one to explore integer solutions to the equations of the form by =ax+c and by =ax-c, where a,b,c are integers. He used kuttuka method to solve problems.ØIndeterminate equations:He gave general solutions to linear indeterminate equations ax+by+c= 0 by the method of continued fraction.GEOMETRYØDiscover thePValue :The credit for discovering the exact valuesPmay be ascribed to the celebrated mathematician Aryabhatta.Rule: Add 4 to 100, multiply by 8, add 62000. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.This givesP=62832/20000=3.1416. Which is an accurate value ofP. Aryabhatta discovered this value independently and also realized thatPis an irrational numberØPythagorean Theorem:The Pythagorean theorem is stated as follows in his work the square of the Bhuja (base) plus the square of the koti (perpendicular) is the square of the Karna(Buja and koti are the sides of a right-angled triangle. The Karna is the hypotenuse)ØCircle Theorem:He has postulated a theorem relating to circle as follows In a circle the product of two Saras is the square of the half chord of the two arcs i.e. a*b=c2where c is half the chord and the saras or arrows are the segments of a diameter which bisect any chord.ØFormula:Aryabhatta gives formulae for the areas of a triangle, square, rectangle, rhombus, circle etc.\n• 1\nWhat are you looking for?"
] | [
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.91308576,"math_prob":0.9887011,"size":3789,"snap":"2022-27-2022-33","text_gpt3_token_len":951,"char_repetition_ratio":0.11519155,"word_repetition_ratio":0.95714283,"special_character_ratio":0.1989971,"punctuation_ratio":0.12044818,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995531,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-18T20:48:58Z\",\"WARC-Record-ID\":\"<urn:uuid:396a85f9-3145-4f89-8e4c-04f9e680c517>\",\"Content-Length\":\"33672\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4c3f61ca-962c-4084-bb3d-0f3d9460b562>\",\"WARC-Concurrent-To\":\"<urn:uuid:27c9281e-d50e-48df-936c-d2f508d62009>\",\"WARC-IP-Address\":\"18.67.76.101\",\"WARC-Target-URI\":\"https://aakashdigitalsrv1.meritnation.com/ask-answer/question/a-small-note-on-aryabhatta-s-contribution-to-maths-plzzz-i-w/the-triangle-and-its-properties/6776732\",\"WARC-Payload-Digest\":\"sha1:IARAS4NWOTUBSIW75ARPNKOKT62YIFRH\",\"WARC-Block-Digest\":\"sha1:23WUWJEIKVVD3Y6VGGD3T3BC3UCVC4QO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573399.40_warc_CC-MAIN-20220818185216-20220818215216-00748.warc.gz\"}"} |
https://dumpz.org/bFcFFZGcGq57 | [
"# Алгоритм шифрования RSA\n\n ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74``` ```def gcd (a, b): \"Compute GCD of two numbers\" if b == 0: return a else: return gcd(b, a % b) def multiplicative_inverse(a, b): \"\"\" Find multiplicative inverse of a modulo b (a > b) using Extended Euclidean Algorithm \"\"\" origA = a X = 0 prevX = 1 Y = 1 prevY = 0 while b != 0: temp = b quotient = a/b b = a % b a = temp temp = X a = prevX - quotient * X prevX = temp temp = Y Y = prevY - quotient * Y prevY = temp return origA + prevY def generateRSAKeys(p, q): \"Generate RSA Public and Private Keys from prime numbers p & q\" n = p * q m = (p - 1) * (q - 1) # Generate a number e so that gcd(n, e) = 1, start with e = 3 e = 3 while 1: if gcd(m, e) == 1: break else: e = e + 2 # start with a number d = m/e will be atleast 1 d = multiplicative_inverse(m, e) # Return a tuple of public and private keys return ((n,e), (n,d)) if __name__ == \"__main__\": print \"RSA Encryption algorithm....\" p = long(raw_input(\"Enter the value of p (prime number):\")) q = long(raw_input(\"Enter the value of q (prime number):\")) print \"Generating public and private keys....\" (publickey, privatekey) = generateRSAKeys(p, q) print \"Public Key (n, e) =\", publickey print \"Private Key (n, d) =\", privatekey n, e = publickey n, d = privatekey input_num = long(raw_input(\"Enter a number to be encrypted:\")) encrypted_num = (input_num ** e) % n print \"Encrypted number using public key =\", encrypted_num decrypted_num = encrypted_num ** d % n print \"Decrypted (Original) number using private key =\", decrypted_num ```"
] | [
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] | {"ft_lang_label":"__label__en","ft_lang_prob":0.5585012,"math_prob":0.9985113,"size":1483,"snap":"2022-05-2022-21","text_gpt3_token_len":471,"char_repetition_ratio":0.1311697,"word_repetition_ratio":0.006968641,"special_character_ratio":0.3499663,"punctuation_ratio":0.14878893,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9993722,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-23T17:13:15Z\",\"WARC-Record-ID\":\"<urn:uuid:f6f8432b-4bf7-40d4-943a-ff15ea178611>\",\"Content-Length\":\"18463\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7deca538-3ef5-42e4-b938-cc6717935029>\",\"WARC-Concurrent-To\":\"<urn:uuid:1f8121ca-ddf0-4e77-9c75-e149b0a06a3d>\",\"WARC-IP-Address\":\"89.248.172.2\",\"WARC-Target-URI\":\"https://dumpz.org/bFcFFZGcGq57\",\"WARC-Payload-Digest\":\"sha1:XJLDCWSZVCB5OWZ6EHSDT6CVNU4HATHP\",\"WARC-Block-Digest\":\"sha1:DCTS4ZMCJBU5QLY26UTWJXWTPFEHTLI3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662560022.71_warc_CC-MAIN-20220523163515-20220523193515-00326.warc.gz\"}"} |
https://homework.cpm.org/category/CON_FOUND/textbook/caac/chapter/4/lesson/4.2.4/problem/4-104 | [
"",
null,
"",
null,
"### Home > CAAC > Chapter 4 > Lesson 4.2.4 > Problem4-104\n\n4-104.",
null,
"",
null,
"The y-intercept occurs when x = 0.\n\ny-intercept = (0, 3)\n\nIf y = mx + b, b = the y-intercept.\n\nWhat is b in this equation?\n\n$-\\frac{5}{12}$\n\ny intercept =\n\n$\\left(0,-\\frac{5}{12}\\right)$\n\nUse the hints from parts (a) and (b)."
] | [
null,
"https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png",
null,
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null,
"https://s3-us-west-2.amazonaws.com/c3po-media-dev/files/ab9b0fb0-259d-11e9-ab6b-d7741872f579/ac4-104im1_original.png",
null,
"https://s3-us-west-2.amazonaws.com/c3po-media-dev/files/7d08e960-259d-11e9-ab6b-d7741872f579/ac4-104im2_original.png",
null
] | {"ft_lang_label":"__label__en","ft_lang_prob":0.84044504,"math_prob":0.9995016,"size":405,"snap":"2021-21-2021-25","text_gpt3_token_len":141,"char_repetition_ratio":0.19950125,"word_repetition_ratio":0.8,"special_character_ratio":0.38765433,"punctuation_ratio":0.15306123,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99796146,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-14T20:17:51Z\",\"WARC-Record-ID\":\"<urn:uuid:5b9b3566-417a-43ee-9813-3b68e38e90ce>\",\"Content-Length\":\"39525\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f646ea8d-26ff-417d-84e2-6b1ee72785ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:22ab4519-16f4-414c-aaa6-4ff49d560ca2>\",\"WARC-IP-Address\":\"104.26.7.16\",\"WARC-Target-URI\":\"https://homework.cpm.org/category/CON_FOUND/textbook/caac/chapter/4/lesson/4.2.4/problem/4-104\",\"WARC-Payload-Digest\":\"sha1:ARODHSRU4RHOLIWND36SR6MTVPDSDUS2\",\"WARC-Block-Digest\":\"sha1:EW3KBTMXV47IBTLMMHF3B2DNMUVU2ZPZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991207.44_warc_CC-MAIN-20210514183414-20210514213414-00459.warc.gz\"}"} |
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