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https://metanumbers.com/156529	[ "# 156529 (number)\n\n156,529 (one hundred fifty-six thousand five hundred twenty-nine) is an odd six-digits composite number following 156528 and preceding 156530. In scientific notation, it is written as 1.56529 × 105. The sum of its digits is 28. It has a total of 2 prime factors and 4 positive divisors. There are 155,376 positive integers (up to 156529) that are relatively prime to 156529.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Odd\n• Number length 6\n• Sum of Digits 28\n• Digital Root 1\n\n## Name\n\nShort name 156 thousand 529 one hundred fifty-six thousand five hundred twenty-nine\n\n## Notation\n\nScientific notation 1.56529 × 105 156.529 × 103\n\n## Prime Factorization of 156529\n\nPrime Factorization 157 × 997\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 2 Total number of prime factors rad(n) 156529 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 156,529 is 157 × 997. Since it has a total of 2 prime factors, 156,529 is a composite number.\n\n## Divisors of 156529\n\n4 divisors\n\n Even divisors 0 4 4 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 4 Total number of the positive divisors of n σ(n) 157684 Sum of all the positive divisors of n s(n) 1155 Sum of the proper positive divisors of n A(n) 39421 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 395.637 Returns the nth root of the product of n divisors H(n) 3.9707 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 156,529 can be divided by 4 positive divisors (out of which 0 are even, and 4 are odd). The sum of these divisors (counting 156,529) is 157,684, the average is 39,421.\n\n## Other Arithmetic Functions (n = 156529)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 155376 Total number of positive integers not greater than n that are coprime to n λ(n) 12948 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 14354 Total number of primes less than or equal to n r2(n) 16 The number of ways n can be represented as the sum of 2 squares\n\nThere are 155,376 positive integers (less than 156,529) that are coprime with 156,529. And there are approximately 14,354 prime numbers less than or equal to 156,529.\n\n## Divisibility of 156529\n\n m n mod m 2 3 4 5 6 7 8 9 1 1 1 4 1 2 1 1\n\n156,529 is not divisible by any number less than or equal to 9.\n\n## Classification of 156529\n\n• Arithmetic\n• Semiprime\n• Deficient\n\n• Polite\n\n• Square Free\n\n### Other numbers\n\n• LucasCarmichael\n\n## Base conversion (156529)\n\nBase System Value\n2 Binary 100110001101110001\n3 Ternary 21221201101\n4 Quaternary 212031301\n5 Quinary 20002104\n6 Senary 3204401\n8 Octal 461561\n10 Decimal 156529\n12 Duodecimal 76701\n20 Vigesimal jb69\n36 Base36 3cs1\n\n## Basic calculations (n = 156529)\n\n### Multiplication\n\nn×y\n n×2 313058 469587 626116 782645\n\n### Division\n\nn÷y\n n÷2 78264.5 52176.3 39132.2 31305.8\n\n### Exponentiation\n\nny\n n2 24501327841 3835168345623889 600315065972161721281 93966716961556502070393649\n\n### Nth Root\n\ny√n\n 2√n 395.637 53.8929 19.8906 10.9375\n\n## 156529 as geometric shapes\n\n### Circle\n\n Diameter 313058 983501 7.69732e+10\n\n### Sphere\n\n Volume 1.60647e+16 3.07893e+11 983501\n\n### Square\n\nLength = n\n Perimeter 626116 2.45013e+10 221365\n\n### Cube\n\nLength = n\n Surface area 1.47008e+11 3.83517e+15 271116\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 469587 1.06094e+10 135558\n\n### Triangular Pyramid\n\nLength = n\n Surface area 4.24375e+10 4.51979e+14 127805\n\n## Cryptographic Hash Functions\n\nmd5 a54e3548fa3b4b53857a6177b9b73e8a f3b19c951809683d8ebc487a37a36b36a19acfc5 c5eff768b20b3f1b40d256a2ea1ea4b9028bbd9eff4c5f5ea224deb01ed1c848 715fbd82cecadced3358f4fc842a8d90c1804e80b565472fc48964d6a416e3c7baa8a2d41f5c884c954493c70bd08e5ac25a80972cc599c53bb707a9a5d37e71 ebce2005860a7ca1121ccfd6d548ce41c53fa598" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62182534,"math_prob":0.9772478,"size":4667,"snap":"2021-43-2021-49","text_gpt3_token_len":1631,"char_repetition_ratio":0.12030882,"word_repetition_ratio":0.032210834,"special_character_ratio":0.45296764,"punctuation_ratio":0.0747782,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9952584,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-02T18:33:41Z\",\"WARC-Record-ID\":\"<urn:uuid:29ae8402-c529-48a1-8184-dc7e18df8d61>\",\"Content-Length\":\"39999\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c9bb87f1-f983-438e-941a-abefe0c97eaa>\",\"WARC-Concurrent-To\":\"<urn:uuid:3560589c-8b3b-4481-aef6-c8bb0692e307>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/156529\",\"WARC-Payload-Digest\":\"sha1:JHIOHREFP3BS3KR4EOIU4IYVEWTO5OKJ\",\"WARC-Block-Digest\":\"sha1:FWVY2WTUQ7P72SWI7NDNPNOGYJEJG4GX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964362287.26_warc_CC-MAIN-20211202175510-20211202205510-00113.warc.gz\"}"}
https://answers.everydaycalculation.com/compare-fractions/84-8-and-14-63	[ "Solutions by everydaycalculation.com\n\n## Compare 84/8 and 14/63\n\n1st number: 10 4/8, 2nd number: 14/63\n\n84/8 is greater than 14/63\n\n#### Steps for comparing fractions\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 8 and 63 is 504\n\nNext, find the equivalent fraction of both fractional numbers with denominator 504\n2. For the 1st fraction, since 8 × 63 = 504,\n84/8 = 84 × 63/8 × 63 = 5292/504\n3. Likewise, for the 2nd fraction, since 63 × 8 = 504,\n14/63 = 14 × 8/63 × 8 = 112/504\n4. Since the denominators are now the same, the fraction with the bigger numerator is the greater fraction\n5. 5292/504 > 112/504 or 84/8 > 14/63\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:\nAndroid and iPhone/ iPad\n\nand" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8502146,"math_prob":0.99144095,"size":881,"snap":"2022-40-2023-06","text_gpt3_token_len":319,"char_repetition_ratio":0.18928164,"word_repetition_ratio":0.0,"special_character_ratio":0.44721907,"punctuation_ratio":0.07253886,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9910695,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T18:47:50Z\",\"WARC-Record-ID\":\"<urn:uuid:bb044350-8e18-445f-80dc-0acd6f75fb30>\",\"Content-Length\":\"7697\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a87db103-5c62-4897-a538-41372f5d26e9>\",\"WARC-Concurrent-To\":\"<urn:uuid:aacc189c-87d9-4f2c-be5f-d0563820a3a7>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/compare-fractions/84-8-and-14-63\",\"WARC-Payload-Digest\":\"sha1:QMLXX4W3HU5HDXTLJPAKY4JYMGGPQNCI\",\"WARC-Block-Digest\":\"sha1:PKUXYHMKZ5VJQ643TMWPBDYEIQXK32AF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337663.75_warc_CC-MAIN-20221005172112-20221005202112-00206.warc.gz\"}"}
https://biz.libretexts.org/Courses/Kwantlen_Polytechnic_University/BUSI1215_Organizational_Behaviour/09%3A_Leading_People_Within_Organizations/9.8%3A_Conclusion	[ "$$\\newcommand{\\vecs}{\\overset { \\rightharpoonup} {\\mathbf{#1}} }$$ $$\\newcommand{\\vecd}{\\overset{-\\!-\\!\\rightharpoonup}{\\vphantom{a}\\smash {#1}}}$$$$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\\| #1 \\\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\\| #1 \\\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$$$\\newcommand{\\AA}{\\unicode[.8,0]{x212B}}$$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9592598,"math_prob":1.000003,"size":932,"snap":"2022-40-2023-06","text_gpt3_token_len":163,"char_repetition_ratio":0.1487069,"word_repetition_ratio":0.0,"special_character_ratio":0.17167382,"punctuation_ratio":0.1118421,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96362203,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-25T05:27:54Z\",\"WARC-Record-ID\":\"<urn:uuid:303dc392-8ad4-4697-9ea4-3da8e574f939>\",\"Content-Length\":\"98207\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:43399a98-e750-4421-bed9-4e1adce32ec3>\",\"WARC-Concurrent-To\":\"<urn:uuid:cce90b4f-a8db-49c0-a315-a2cf77d6738b>\",\"WARC-IP-Address\":\"18.160.46.83\",\"WARC-Target-URI\":\"https://biz.libretexts.org/Courses/Kwantlen_Polytechnic_University/BUSI1215_Organizational_Behaviour/09%3A_Leading_People_Within_Organizations/9.8%3A_Conclusion\",\"WARC-Payload-Digest\":\"sha1:D6REWLJWXSL7Z4NCDT6ZMF2FMXKZR3LY\",\"WARC-Block-Digest\":\"sha1:MR4R6FZSL574VB2BSZEN4EYXCOHKULTE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334514.38_warc_CC-MAIN-20220925035541-20220925065541-00401.warc.gz\"}"}
https://www.sanfoundry.com/basic-electrical-engineering-questions-answers-generation-alternating-emf/	[ "# Basic Electrical Engineering Questions and Answers – Generation of an Alternating EMF\n\n«\n»\n\nThis set of Basic Electrical Engineering Multiple Choice Questions & Answers (MCQs) focuses on “Generation of an Alternating EMF”.\n\n1. Which, among the following, is the correct expression for alternating emf generated?\na) e=2Blvsin(θ)\nb) e=2B2lvsin(θ)\nc) e=Blvsin(θ)\nd) e=4Blvsin(θ)\n\nExplanation: The correct expression for alternating emf generated is e=Blvsin(θ). Where B stands for magnetic field density, l is the length of each of the parallel sides v is the velocity with which the conductor is moved and θ is the angle between the velocity and the length.\n\n2. What should theta be in order to get maximum emf?\na) 00\nb) 900\nc) 1800\nd) 450\n\nExplanation: The value of θ should be 900 in order to get maximum emf because e = Blvsin(θ) and sin is maximum when θ is 900.\n\n3. Calculate the maximum emf when the velocity is 10m/s, the length is 3m and the magnetic field density is 5T.\na) 150V\nb) 100V\nc) 300V\nd) 0V\n\nExplanation: We know that: emax=Bvl\nSubstituting the values from the given question, we get e=150V.\n\n4. When a coil is rotated in a magnetic field, the emf induced in it?\na) Is maximum\nb) Is minimum\nc) Continuously varies\nd) Remains constant\n\nExplanation: When a coil is rotated in a magnetic field, cross sectional area varies due to which the number of flux lines crossing it varies, which causes the emf to vary continuously.\n\n5. emf is zero if the angle between velocity and length is _____\na) 00\nb) 900\nc) 2700\nd) 450\n\nExplanation: If the angle between velocity and length is zero, sinθ=0\nSo, e=Bvlsinθ = 0.\n\n6. In an A.C. generator, increase in number of turns in the coil _________\na) Increases emf\nb) Decreases emf\nc) Makes the emf zero\nd) Maintains the emf at a constant value\n\nExplanation: In an A.C. generator, the emf increases as the number of turns in the coil increases because the emf is directly proportional to the number of turns.\n\n7. The number of cycles that occur in one second is termed as ___________\na) Waveform\nb) Frequency\nc) Amplitude\nd) Period\n\nExplanation: The number of cycles that occur in one second is known as the frequency. It is the reciprocal of the time period.\n\nSanfoundry Global Education & Learning Series – Basic Electrical Engineering.\n\nTo practice all areas of Basic Electrical Engineering, here is complete set of 1000+ Multiple Choice Questions and Answers.", null, "" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns=%22http://www.w3.org/2000/svg%22%20viewBox=%220%200%20150%20150%22%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8240558,"math_prob":0.9921012,"size":2749,"snap":"2022-27-2022-33","text_gpt3_token_len":772,"char_repetition_ratio":0.11876138,"word_repetition_ratio":0.12083333,"special_character_ratio":0.2648236,"punctuation_ratio":0.12164579,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99786526,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-07T18:45:29Z\",\"WARC-Record-ID\":\"<urn:uuid:72cb0f88-7ec0-42a1-8634-8d2060832f75>\",\"Content-Length\":\"151908\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7d8315ec-145d-4920-a05b-5c21905bc975>\",\"WARC-Concurrent-To\":\"<urn:uuid:c5f09bda-6d8c-451c-b34d-7987f8398f32>\",\"WARC-IP-Address\":\"104.25.131.119\",\"WARC-Target-URI\":\"https://www.sanfoundry.com/basic-electrical-engineering-questions-answers-generation-alternating-emf/\",\"WARC-Payload-Digest\":\"sha1:WW5THWP4S44DTOVRNNYB2AX5ZR7X3SZZ\",\"WARC-Block-Digest\":\"sha1:VC7GO5WDBLUPU7BUGO7NELUO4JEAAL3V\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882570692.22_warc_CC-MAIN-20220807181008-20220807211008-00304.warc.gz\"}"}
http://www.youdao.com/example/blng/eng/material_balance/	[ "go top 返回词典\n• So the simulated annealing algorithm is introduced to solving the material balance equation.\n\n因此本文引入模拟退火算法求解物质平衡方程\n\ndict.cnki.net\n\n• The method is derived by using material balance equation in combination with formation pressure.\n\n方法考虑地层压力采用物质平衡方程得到\n\ndict.cnki.net\n\n• Based on the special geologic and hydrodynamic features, the conventional material balance equation is modified.\n\n根据特殊地质水动力学特征常规物质平衡方程进行了改进。\n\ndict.cnki.net\n\n• Add fuel stream to the heat and material balance.\n\n增设燃料达到热平衡物料平衡\n\nwww.websaru.net\n\n• The general material balance equation was applied widely in the field, but few studies about material balance equation of gas injection flooding has been conducted.\n\n常规物质平衡方程油田已广泛应用然而油田物质平衡方程研究较少\n\ndict.cnki.net\n\n• The specifications should provide a reasonable material balance.\n\n应该提供一个合理物料平衡控制标准。\n\nwww.websaru.net\n\n• Filtered process mathematical model includes material balance equations of solid filter material and liquid sewage respectively.\n\n污水过滤过程数学模型包括固体相和液体污水物料衡算方程\n\ndict.cnki.net\n\n• Using the equilibrium constant method, the thermal and material balance calculation, and has set up the calculation model of straight Claus sulfur recovery process.\n\n应用平衡常数物料热量建立了直流法克劳斯硫磺回收工艺计算模型\n\ndict.cnki.net\n\n• Based on the material balance, energy balance and reaction kinetics, a dynamic model, having the form of a nonlinear distributed parameter has been presented for catalytic cracking process.\n\n基于物料算、热量反应动力学建立催化裂化过程动态模型具有非线性分布参数模型形式\n\ndict.cnki.net\n\n• After the material balance for any a mixing pool, the analytical solutions of concentration on the evaporating surface, the distillate rate and the residue rate obtained.\n\n每个混合进行物料衡算，可以得到蒸发壁面上浓度变化轻、馏分流量含量解析\n\ndict.cnki.net\n\n• Material balance equation of gas reservoir is an important method in gas reservoir engineering.\n\n气藏物质平衡方程序气藏工程中的重要方法\n\nwww.chemyq.com\n\n• Material balance method is most used in reserve calculation of gas reservoirs, and its calculating results are accurate relatively.\n\n物质平衡气藏储量计算中用得最多计算较为准确的一种方法。\n\ndict.cnki.net\n\n• Material balance equation for gas reservoir is an important method in gas reservoir engineering.\n\n气藏物质平衡方程气藏工程中的重要方法\n\ndict.cnki.net\n\n• This paper presents the dynamic evaluation method for calculation of water energy, based on the performances and material balance theory.\n\n将气藏一具体地质开发特征物质平衡理论结合给出水体能量动态评价方法\n\nwww.dictall.com\n\n• The breakthrough curve inside the bed under another operation is predicted by the adsorption rate equation, the adsorption isotherm equation and the material balance equation.\n\n吸附速率方程吸附等温方程物料算相结合预测其他条件下穿透曲线\n\ndict.cnki.net\n\n• When calculating with material balance equation, the accuracy of the calculation result is often affected by the error of PVT parameters.\n\n采用物质平衡方程计算常常由于PVT参数误差影响最终计算结果精度\n\ndict.cnki.net\n\n• Large deviation will appear if material balance equation of the gas reservoir for calculating OIP is used directly in this production scheme.\n\n开采方式下，直接采用气藏物质平衡方程进行地质储量计算出现很大偏差\n\nwww.chemyq.com\n\n物质平衡方式不定态公式等油藏工程理论基础，得出期采收率油藏见时间计算公式\n\nwww.chemyq.com\n\n• Material balance method is one of the methods used commonly in estimating gas reserves in a reservoir.\n\n物质平衡方法计算气藏储量常用方法之一\n\nwww.chemyq.com\n\n• On the basis of the material balance equation, this paper develops a method to predict the formation pressure using cumulative production data for normal pressure confined gas reservoir.\n\n正常压力系统封闭气藏物质平衡方程序基础建立一种利用累积产气量计算气藏地层压力方法\n\ndict.cnki.net\n\n• Traditional well-test models for liquid flow are not consistent with material balance equation.\n\n传统试井模型物质平衡方程都是一致的。\n\ndict.cnki.net\n\n• Utilizing the optimal principle and the material balance equation, with the inheritance arithmetic, this paper has programmed to solve the geological reserves .\n\n利用最优化原理物质平衡方程结合遗传算法编程求解了水驱气藏地质储量\n\ndict.youdao.com\n\n• Analyzing methods based on material balance are currently important tools for performance analysis and reserve calculation of oil and gas reservoirs.\n\n物质平衡原理为基础动态分析方法现今油气藏动态分析储量核实重要工具\n\nwww.cngascn.com\n\n• Methods the effect factors of the steady producing for single gas well production was discussed by using the model of material balance with compensating as well as the parameters of the model.\n\n方法采用具有补给物质平衡模型确定气井单井稳产水平，探讨影响单井稳产水平因素及其模型中的反映。\n\ndict.cnki.net\n\n• The model includes the calculation of material balance, the vapor and liquid equilibrium compositions, the interfacial area of mass transfer, mass transfer rate.\n\n模型包括物料、相平衡计算面积计算速率计算等。\n\ndict.cnki.net\n\n• The continuous suspension crystallization process for xylene separation was optimized by calculation of solid-liquid phase equilibrium, material balance and energy balance.\n\n采用混合二甲苯二组分物系的液相平衡方程及物料热量方程对混合二甲苯连续悬浮结晶工艺进行模拟计算。\n\nwww.dictall.com\n\n• Material balance equation of gas reservoirs is an important tool in gas reservoir engineering. It can be used to determine the original gas-in-place of the gas reservoirs.\n\n气藏物质平衡方程序可以确定气藏原始地质储量，气藏工程中的重要方法\n\ndict.cnki.net\n\n• We can learn from above that the study of material balance method during gas injection flooding is very important.\n\n因此，研究开采过程中的物质平衡方程序具有十分重要的意义。\n\ndanci.911cha.com\n\n• The material balance equation for gas reservoir has been used to determine reserves and OGIP of gas reservoir, judge its driving type and predict its production performance.\n\n气藏物质平衡方程气藏工程重要方法可以确定气藏原始地质储量可采储量，也可以判断气藏驱动类型预测气藏开发动态\n\ndict.cnki.net\n\n• And using material balance with time equation to calculate recoverable reservoir controlled by single well.\n\n采用物质平衡时间方程分析法求取单井控制储量\n\ndict.cnki.net\n\n\\$firstVoiceSent\n- 来自原声例句" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6833623,"math_prob":0.97252864,"size":5166,"snap":"2020-45-2020-50","text_gpt3_token_len":1904,"char_repetition_ratio":0.21619527,"word_repetition_ratio":0.02519685,"special_character_ratio":0.14711575,"punctuation_ratio":0.073305674,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9718071,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-05T03:19:54Z\",\"WARC-Record-ID\":\"<urn:uuid:442d355c-396a-4c7c-b9bd-432d7bdb2373>\",\"Content-Length\":\"133836\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e2ebcf72-4562-44d0-9cb0-8c3d32630e76>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c09a67a-a12f-4975-82a6-eb2eee109d8f>\",\"WARC-IP-Address\":\"103.72.47.245\",\"WARC-Target-URI\":\"http://www.youdao.com/example/blng/eng/material_balance/\",\"WARC-Payload-Digest\":\"sha1:FQMFJX3TLN6WNVAKJ7VZS24JXVIJZSCR\",\"WARC-Block-Digest\":\"sha1:QRCPBEIKUKWZPX4MNOBPFGWGIBMKYP7Y\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141746033.87_warc_CC-MAIN-20201205013617-20201205043617-00084.warc.gz\"}"}
https://www.w3definitions.com/mcq/number-systems-and-codes/	[ "# Number Systems And Codes MCQs\n\n## 3428 is the decimal value for which of the following binary coded decimal (BCD) groupings?\n\n• A. 11010001001000\n• B. 11010000101000\n• C. 011010010000010\n• D. 110100001101010\n\n## A BCD code that represents each digit of a decimal number by a binary number derived by adding 3 to its 4-bit true binary value is _________.\n\n• A. 9's complement code\n• B. excess-3 code\n• C. 8421 code\n• D. gray code\n\n## A binary code that progresses such that only one bit changes between two successive codes is _________.\n\n• A. 9's complement code\n• B. excess-3 code\n• C. 8421 code\n• D. gray code\n\n## A binary number’s value changes most drastically when the ____ is changed.\n\n• A. LSB\n• B. duty cycle\n• C. MSB\n• D. frequency\n\n## Base 10 refers to which number system?\n\n• A. binary coded decimal\n• B. decimal\n• C. octal\n\n• A. 201\n• B. 2001\n• C. 20\n• D. 210\n\n• A. 125\n• B. 12.5\n• C. 90.125\n• D. 9.125\n\n• A. 5B\n• B. 5F\n• C. 5A\n• D. 5C\n\n## Convert the decimal number 151.75 to binary.\n\n• A. 10000111.11\n• B. 11010011.01\n• C. 00111100.00\n• D. 10010111.11\n\n• A. decimal\n• C. binary\n• D. octal\n\n## Sample-and-hold circuits in ADCs are designed to:\n\n• A. sample and hold the output of the binary counter during the conversion process\n• B. stabilize the ADCs threshold voltage during the conversion process\n• C. stabilize the input analog signal during the conversion process\n• D. sample and hold the ADC staircase waveform during the conversion process\n\n• A. excess-3\n• B. gray\n• C. multibit\n• D. minival\n\n## The binary coded decimal (BCD) code is a system that represents each of the 10 decimal digits as a(n) ____________.\n\n• A. 4-bit binary code\n• B. 8-bit binary code\n• C. 16-bit binary code\n• D. ASCII code\n\n• A. 8\n• B. 4\n• C. 1\n• D. 2\n\n• A. 1\n• B. 2\n• C. 3\n• D. 4\n\n• A. 191\n• B. 1911\n• C. 19\n• D. 19111\n\n## What is the difference between binary coding and binary coded decimal?\n\n• A. Binary coding is pure binary.\n• B. BCD is pure binary.\n• C. Binary coding has a decimal format.\n• D. BCD has no decimal format.\n\n• A. 327.375\n• B. 12166\n• C. 1388\n• D. 1476" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7260484,"math_prob":0.9105126,"size":2736,"snap":"2021-31-2021-39","text_gpt3_token_len":770,"char_repetition_ratio":0.1954612,"word_repetition_ratio":0.14401622,"special_character_ratio":0.33369884,"punctuation_ratio":0.15837938,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99689883,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-23T21:28:04Z\",\"WARC-Record-ID\":\"<urn:uuid:8f58b118-b222-48e9-8062-55015719caeb>\",\"Content-Length\":\"82735\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a64bfa25-81fe-4879-bcb8-f635b0f53f31>\",\"WARC-Concurrent-To\":\"<urn:uuid:40c30d49-7918-4306-abfb-603fe4f4bc8a>\",\"WARC-IP-Address\":\"23.111.187.131\",\"WARC-Target-URI\":\"https://www.w3definitions.com/mcq/number-systems-and-codes/\",\"WARC-Payload-Digest\":\"sha1:FDELX4NGFW6PBHGSMCW46ZZOO5NVY5Z3\",\"WARC-Block-Digest\":\"sha1:JQVMWJP4UVFQCMO36HK3PDVNPXK3DGDW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057447.52_warc_CC-MAIN-20210923195546-20210923225546-00665.warc.gz\"}"}
http://textbooks.math.gatech.edu/ila/solution-sets.html	[ "##### Objectives\n1. Understand the relationship between the solution set of and the solution set of\n2. Understand the difference between the solution set and the column span.\n3. Recipes: parametric vector form, write the solution set of a homogeneous system as a span.\n4. Pictures: solution set of a homogeneous system, solution set of an inhomogeneous system, the relationship between the two.\n5. Vocabulary words: homogeneous/inhomogeneous, trivial solution.\n\nIn this section we will study the geometry of the solution set of any matrix equation\n\nThe equation is easier to solve when so we start with this case.\n\n##### Definition\n\nA system of linear equations of the form is called homogeneous.\n\nA system of linear equations of the form for is called inhomogeneous.\n\nA homogeneous system is just a system of linear equations where all constants on the right side of the equals sign are zero.\n\nA homogeneous system always has the solution This is called the trivial solution. Any nonzero solution is called nontrivial.\n\n##### Observation\n\nThe equation has a nontrivial solution there is a free variable has a column without a pivot position.\n\n##### Observation\n\nWhen we row reduce the augmented matrix for a homogeneous system of linear equations, the last column will be zero throughout the row reduction process. We saw this in the last example:\n\nSo it is not really necessary to write augmented matrices when solving homogeneous systems.\n\nWhen the homogeneous equation does have nontrivial solutions, it turns out that the solution set can be conveniently expressed as a span.\n\n##### Parametric Vector Form (homogeneous case)\n\nConsider the following matrix in reduced row echelon form:\n\nThe matrix equation corresponds to the system of equations\n\nWe can write the parametric form as follows:\n\nWe wrote the redundant equations and in order to turn the above system into a vector equation:\n\nThis vector equation is called the parametric vector form of the solution set. Since and are allowed to be anything, this says that the solution set is the set of all linear combinations of and In other words, the solution set is\n\nHere is the general procedure.\n\n##### Recipe: Parametric vector form (homogeneous case)\n\nLet be an matrix. Suppose that the free variables in the homogeneous equation are, for example, and\n\n1. Find the reduced row echelon form of\n2. Write the parametric form of the solution set, including the redundant equations Put equations for all of the in order.\n3. Make a single vector equation from these equations by making the coefficients of and into vectors and respectively.\n\nThe solutions to will then be expressed in the form\n\nfor some vectors in and any scalars This is called the parametric vector form of the solution.\n\nIn this case, the solution set can be written as\n\nWe emphasize the following fact in particular.\n\nThe set of solutions to a homogeneous equation is a span.\n\nSince there were two variables in the above example, the solution set is a subset of Since one of the variables was free, the solution set is a line:\n\nIn order to actually find a nontrivial solution to in the above example, it suffices to substitute any nonzero value for the free variable For instance, taking gives the nontrivial solution Compare to this important note in Section 1.3.\n\nSince there were three variables in the above example, the solution set is a subset of Since two of the variables were free, the solution set is a plane.\n\nThere is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? We will see in example in Section 2.5 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors.\n\nAnother natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans.\n\nThere is a natural relationship between the number of free variables and the “size” of the solution set, as follows.\n\n##### Dimension of the solution set\n\nThe above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc. The number of free variables is called the dimension of the solution set.\n\nWe will develop a rigorous definition of dimension in Section 2.7, but for now the dimension will simply mean the number of free variables. Compare with this important note in Section 2.5.\n\nIntuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. For a line only one parameter is needed, and for a plane two parameters are needed. This is similar to how the location of a building on Peachtree Street—which is like a line—is determined by one number and how a street corner in Manhattan—which is like a plane—is specified by two numbers.\n\n# Subsection2.4.2Inhomogeneous Systems\n\nRecall that a matrix equation is called inhomogeneous when\n\nIn the above example, the solution set was all vectors of the form\n\nwhere is any scalar. The vector is also a solution of take We call a particular solution.\n\nIn the solution set, is allowed to be anything, and so the solution set is obtained as follows: we take all scalar multiples of and then add the particular solution to each of these scalar multiples. Geometrically, this is accomplished by first drawing the span of which is a line through the origin (and, not coincidentally, the solution to ), and we translate, or push, this line along The translated line contains and is parallel to it is a translate of a line.\n\nIn the above example, the solution set was all vectors of the form\n\nwhere and are any scalars. In this case, a particular solution is\n\nIn the previous example and the example before it, the parametric vector form of the solution set of was exactly the same as the parametric vector form of the solution set of (from this example and this example, respectively), plus a particular solution.\n\n##### Key Observation\n\nIf is consistent, the set of solutions to is obtained by taking one particular solution of and adding all solutions of\n\nIn particular, if is consistent, the solution set is a translate of a span.\n\nThe parametric vector form of the solutions of is just the parametric vector form of the solutions of plus a particular solution\n\nIt is not hard to see why the key observation is true. If is a particular solution, then and if is a solution to the homogeneous equation then\n\nso is another solution of On the other hand, if we start with any solution to then is a solution to since\n\nSee the interactive figures in the next subsection for visualizations of the key observation.\n\n##### Dimension of the solution set\n\nAs in this important note, when there is one free variable in a consistent matrix equation, the solution set is a line—this line does not pass through the origin when the system is inhomogeneous—when there are two free variables, the solution set is a plane (again not through the origin when the system is inhomogeneous), etc.\n\nAgain compare with this important note in Section 2.5.\n\n# Subsection2.4.3Solution Sets and Column Spans¶ permalink\n\nTo every matrix we have now associated two completely different geometric objects, both described using spans.\n\n• The solution set: for fixed this is the set of all such that\n\n• This is a span if and it is a translate of a span if (and is consistent).\n• It is a subset of\n• It is computed by solving a system of equations: usually by row reducing and finding the parametric vector form.\n• The span of the columns of : this is the set of all such that is consistent.\n\n• This is always a span.\n• It is a subset of\n• It is not computed by solving a system of equations: row reduction plays no role.\n\nDo not confuse these two geometric constructions! 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https://jov.arvojournals.org/article.aspx?articleid=2279458	[ "", null, "jov", null, "June 2014\nVolume 14, Issue 7\nFree\nMethods \| June 2014\nA formula for human retinal ganglion cell receptive field density as a function of visual field location\nAuthor Affiliations\nJournal of Vision June 2014, Vol.14, 15. doi:https://doi.org/10.1167/14.7.15\n• Views\n• PDF\n• Share\n• Tools\n×\n###### This feature is available to authenticated users only.\n• Get Citation\n\nAndrew B. Watson; A formula for human retinal ganglion cell receptive field density as a function of visual field location. Journal of Vision 2014;14(7):15. doi: https://doi.org/10.1167/14.7.15.\n\n© ARVO (1962-2015); The Authors (2016-present)\n\n×\n• Supplements\nAbstract\nAbstract\nAbstract:\n\nAbstract In the human eye, all visual information must traverse the retinal ganglion cells. The most numerous subclass, the midget retinal ganglion cells, are believed to underlie spatial pattern vision. Thus the density of their receptive fields imposes a fundamental limit on the spatial resolution of human vision. This density varies across the retina, declining rapidly with distance from the fovea. Modeling spatial vision of extended or peripheral targets thus requires a quantitative description of midget cell density throughout the visual field. Through an analysis of published data on human retinal topography of cones and ganglion cells, as well as analysis of prior formulas, we have developed a new formula for midget retinal ganglion cell density as a function of position in the monocular or binocular visual field.\n\nIntroduction\nThe spatial resolution of human photopic vision is limited by optical blur, and beyond that by spatial sampling by retinal neurons. The initial sampling is by the inner segments of the cone photoreceptors, and subsequent resampling of their signals is performed, via various interneurons, by the retinal ganglion cells (RGC). These are the output cells of the human eye and consequently their properties limit the signal that travels to the rest of the brain. One class of these cells, the midget retinal ganglion cells (mRGC) are the most numerous; near the fovea they appear to sample a single cone while in peripheral retina they gather signals from multiple cones (Ahmad, Klug, Herr, Sterling, & Schein, 2003; Dacey, 1993; Dacey & Petersen, 1992; Goodchild, Ghosh, & Martin, 1996; Kolb & Dekorver, 1991; Schein, 1988). In consequence the mRGC likely set an upper bound on the spatial resolution of human vision, especially at low temporal frequencies (Hirsch & Curcio, 1989; Merigan & Eskin, 1986; Merigan & Katz, 1990; Rossi & Roorda, 2010; Thibos, Cheney, & Walsh, 1987).\nFrom a sampling point of view, the critical metric of the mRGC lattice is the local density or spacing of adjacent mRGC receptive fields (mRGCf). Because this spacing varies across the visual field, and because of its fundamental role in modeling human visual spatial processing, it would be valuable to have a formula for mRGCf spacing as a function of location in the visual field.\nAn earlier formula for mRGCf density was developed by Drasdo, Millican, Katholi, and Curcio (2007). While this formula was an important contribution, it was largely based on psychophysical results (acuity vs. eccentricity). Since we would like to use our formula to make psychophysical predictions, we sought to develop a formula based only on anatomical data. Barten (1999) also produced a formula for RGC density along an average meridian, but did not provide a derivation for his result.\nDacey (1993) also provided a figure depicting estimated average midget ganglion cell density as a function of eccentricity. However, his estimates are based on highly variable estimates of dendritic field size and an assumption of unit coverage (the product of density and field area) throughout the visual field. Also, separate estimates for the four meridians are not provided. Nonetheless, we show in the Discussion that our new formula is consistent with his empirical results.\nThe approach that we have taken is to seek a simple analytic formula that approximately satisfies known or probable anatomical constraints.\nCurcio, Sloan, Kalina, and Hendrickson (1990) measured the distribution of cone photoreceptors across the retina in a set of eight human eyes. Consistent with earlier fragmentary reports (Osterberg, 1935), they found that density declined rapidly with eccentricity. They also described substantial meridional asymmetries, and large individual differences in peak density.\nIn a second paper six of those eyes, along with one additional eye, were used to measure the distribution of retinal ganglion cells (Curcio & Allen, 1990). This distribution also varies markedly with eccentricity, but unlike the cone distribution, it does not peak at the fovea. This is because in a central retinal zone the ganglion cell bodies are displaced centrifugally some distance from the inner segments of the cones to which they are connected through the bipolar cells, and thus from their receptive fields. This displacement zone continues up to eccentricities of around 13°–17°, depending on the meridian (Drasdo et al., 2007). The extent of ganglion cell displacement as a function of eccentricity of the cell body has been measured in primate (Schein, 1988; Wassle, Grunert, Rohrenbeck, & Boycott, 1990) and human (Drasdo et al., 2007). In human these displacements are as large as 2.2°. As a result, the peak RGC density occurs some 4°–5° away from the center of the fovea. Thus the local density of the cell bodies does not reflect the local density of the RGC receptive fields (RGCf).\nHowever, the RGC distribution combined with several other constraints does allow a plausible reconstruction of the distribution of RGCf. The constraints that we consider are:\n1.\nAlong a given meridian, the cumulative distribution of RGC and RGCf must agree outside the displacement zone.\n2.\nIn the fovea, it is likely that each cone connects (via bipolars) to exactly two mRGC (Kolb & Dekorver, 1991).\n3.\nNear the fovea midgets constitute most but not all of the ganglion cells. The ratio RGC/mRGC is given as about 1.12 by Drasdo et al. (2007).\n4.\nThe hypothetical distribution of RGCf must be consistent with the measured distribution of RGC outside of the displacement zone.\nMaking use of these constraints, this report derives a new formula for RGC density as a function of eccentricity along the four principal meridians, and more generally of position on the retina or in the visual field in degree coordinates.\nThe derivation relies on transformations from retinal coordinates in mm to degree, based on a model eye (Drasdo & Fowler, 1974) and an assumed offset between optical and visual axes (Charman, 1991), which are described in Appendix 6. We also place in Appendix 5 a number of formulas relating various metrics of points in a hexagonal sampling lattice, such as spacing, density, and row spacing. We also provide (see Appendix 2) with this report a supplementary file of Mathematica functions that implement our formulas (Wolfram Research Inc., interactive demonstration that computes RGCf density and spacing at a selected visual field location (see Appendix 1).\nConventions regarding meridians, locations, and the visual center\nIn retinal anatomy locations are often specified as a distance from a retinal center along one of the four principal meridians. Because the topography is not radially symmetric, or even bilaterally symmetric, measurements differ along the four meridians. The naming of the meridians is a possible source of confusion, since different conventions are typically used for retinal anatomy and visual fields. Specifically, the nasal retina (the part nearest the nose) images the temporal visual field (the part away from the nose), and vice versa, in both eyes. We avoid this confusion by always referring to the visual field locations, even when discussing anatomy. We order the meridians temporal, superior, nasal, inferior, consistent with increasing polar angle in the visual field of the right eye, and assign them indexes of 1–4. When cartesian coordinates are used, positive x-coordinates refer to the temporal visual field of either eye, or the right binocular visual field.\nA further possible confusion is that in the right eye, the temporal visual field is the right visual field, while in the left eye, it is the left visual field. Thus when we compute binocular visual fields, we combine the temporal field of the right eye with the nasal field of the left eye, and vice versa.\nA final possible confusion is the definition of the retinal center. Remarkably, there appears to be no consistent term for this concept. What we want is the “visual center,” defined essentially as the intersection of the visual axis with the retina. It is not the same as the “fovea” which is an area, not a point. It may well correspond to the point of highest cone density, and is operationally defined as the retinal location that images a fixated point, sometimes called the “preferred retinal locus of fixation” or PRLF (Rossi & Roorda, 2010). We will assume that anatomical measurements and visual field locations are referenced to this common visual center.\nNotation\nIn Appendix 3 we provide a more complete review of notation, but here we introduce some general conventions. The symbol r will indicate eccentricity in degree, d will indicate density in deg−2, and s will indicate spacing (of adjacent cells or receptive fields) in degrees. We use subscripts g, m, and c to denote RGC, mRGC, and cones respectively, and gf and mf to denote RGC and mRGC receptive fields. A particular meridian will be indicated by an integer index k\nCone densities\nCurcio et al. (1990) measured cone photoreceptor density across the retina in eight human eyes. Average data are provided in tables of cone densities in cones/mm2 in the four principal meridians at each of 34 eccentricities in mm (Curcio, 2006). Using the conversion formulas described in Appendix 6, we have converted the densities to cones/deg2 as a function of eccentricity in degree as shown in Figure 1. Writing dc(r, k) for the cone density at eccentricity r degree along meridian k, we note that the foveal peak is dc(0, 1) = dc(0) = 14,804.6. This peak density is plotted at the upper left on this log-log plot.\nFigure 1\n\nCone density as a function of eccentricity (Curcio et al., 1990). Foveal density is indicated by the line segment at the upper left. The gap in the temporal meridian corresponds to the blind spot.\nFigure 1\n\nCone density as a function of eccentricity (Curcio et al., 1990). Foveal density is indicated by the line segment at the upper left. The gap in the temporal meridian corresponds to the blind spot.", null, "RGC densities\nCurcio and Allen (1990) measured local density of RGC cell bodies in seven retinas (including six of those used above to measure cone densities). Curcio (2013) has provided tables of average RGC densities in RGC/mm2 in the four principal meridians at each of 35 eccentricities in millimeters. We have again converted these values to densities in RGC/deg2 as a function of eccentricity in degree, and the results are plotted in Figure 2. We omit one point in the inferior meridian with a density below 1. The peak density of about 2,375 RGC/deg2 occurs not at the foveal center but at an eccentricity of about 3.7°. This is because, as noted above, ganglion cell bodies within a displacement zone extending out as far as 17° are displaced centrifugally from their cone inputs. Thus the RGC densities within this zone cannot be used directly as an estimate of the densities of the RGCf.\nFigure 2\n\nRGC density as a function of eccentricity in four meridians (Curcio & Allen, 1990). The gap in the temporal meridian corresponds to the blind spot.\nFigure 2\n\nRGC density as a function of eccentricity in four meridians (Curcio & Allen, 1990). The gap in the temporal meridian corresponds to the blind spot.", null, "The first constraint noted above is that along a given meridian, the cumulative distribution of RGC and RGCf must agree at the limit of the displacement zone. In Figure 3 we show the estimated cumulative number of RGC as a function of eccentricity along each meridian for eccentricities up to 20°. For each meridian, the counts assume a radially symmetric density function, and are produced through linear interpolation of the data shown in Figure 2. To compute cumulative counts as a function of eccentricity, we integrate density at eccentricity r multiplied by 2πr to account for the increasing area. The circular point on each curve marks the cumulative density at the approximate limit of the displacement zone (11 for temporal, 17 for the others). When we construct a candidate function for the density of RGCf, its cumulative value must approximately agree at these points. In other words, the total number of receptive fields must equal the total number of cell bodies within the displacement zone.\nFigure 3\n\nCumulative total number of RGC along four meridians.\nFigure 3\n\nCumulative total number of RGC along four meridians.", null, "Foveal density of RGC receptive fields\nIf each foveal cone drives exactly two midget retinal ganglion cells, as specified in our second constraint, then the foveal density of midget retinal ganglion cell receptive fields must be twice the cone density,", null, "If midget cells constitute a fraction f(r) of all ganglion cells at eccentricity r, then", null, "", null, "As noted below, f(0)−1 has been estimated as 1.12 (Drasdo et al., 2007) or 1.09 (Dacey, 1993). For reasons outlined below, here we adopt the value of 1.12, in which case dgf(0) = 2 × 1.12 × 14,804.6 = 33,163.2 deg−2\nDensity of RGC receptive fields\nWe now attempt to discover a function that will describe RGCf density as a function of eccentricity and satisfy the constraints outlined in the Introduction. For each candidate function, we optimized the several parameters with respect to an error function consisting of the sum of the squared errors between empirical and computed log densities for eccentricities outside the exclusion zone, and the weighted squared log of the ratio of empirical and computed cumulative counts within the exclusion zone (see points in Figure 3). This ensures a reasonable fit to peripheral RGC densities, and to the cumulative counts. We used log densities in the fit to accommodate the very wide range of densities, and to avoid giving the larger densities undue influence in the fit. We explored a wide range of functions, leading to the best-fitting one described below.\nSince the work of Aubert and Foerster (1857) it has been observed that many measures of visual resolution decline in an approximately linear fashion with eccentricity, at least up to the eccentricity of the blind spot (Strasburger, Rentschler, & Juttner, 2011). Because resolution may depend on receptive field spacing, and since density is proportional to the inverse of spacing squared (see Appendix 5), this suggests that density might vary with eccentricity as", null, "where dgf(0) is the density at r = 0, and r2 is the eccentricity at which density is reduced by a factor of four (and spacing is doubled). By itself, this did not provide a good fit, especially at larger eccentricities. However we found that a simple modification, the addition of an exponential, yielded an acceptable fit. The new function is given by", null, "where ak is the weighting of the first term, and re,k is the scale factor of the exponential. The meridian is indicated by the index k. We have fit this expression separately for each meridian and optimized parameters relative to the error function described above. The results are shown in Figure 4. For each meridian, we show the average RGC densities reported by Curcio and Allen (1990), along with the fitted function. The vertical gray line in each figure shows the assumed limit of the displacement zone. Note that only data points outside the displacement zone are used in the fit. The estimated parameters, predicted cell counts, and fitting error are given in Table 1.\nFigure 4\n\nGanglion cell density as a function of eccentricity in four meridians. Data points are from Curcio and Allen (1990). The solid curve is the fit of Equation 4 to the points outside the displacement zone. The dashed gray line indicates the approximate limit of the displacement zone. The density at eccentricity of zero is indicated by the line segment at the upper left.\nFigure 4\n\nGanglion cell density as a function of eccentricity in four meridians. Data points are from Curcio and Allen (1990). The solid curve is the fit of Equation 4 to the points outside the displacement zone. The dashed gray line indicates the approximate limit of the displacement zone. The density at eccentricity of zero is indicated by the line segment at the upper left.", null, "Table 1\n\nParameters and error for fits of Equation 4 in four meridians. Note: Also shown are measured and predicted cumulative counts along each meridian within the displacement zone. The next-to-last column shows the fitting error outside the displacement zone. The last column indicates the assumed limit of the displacement zone.\nTable 1\n\nParameters and error for fits of Equation 4 in four meridians. Note: Also shown are measured and predicted cumulative counts along each meridian within the displacement zone. The next-to-last column shows the fitting error outside the displacement zone. The last column indicates the assumed limit of the displacement zone.\n Meridian k a r2 re Data count (× 1000) Model count (× 1000) Error rz Temporal 1 0.9851 1.058 22.14 485.1 485.7 0.23 11 Superior 2 0.9935 1.035 16.35 526.1 528.9 0.12 17 Nasal 3 0.9729 1.084 7.633 660.9 661.1 0.01 17 Inferior 4 0.996 0.9932 12.13 449.3 452.1 0.93 17\nThe fits are good for three of the four meridians. Both the peripheral densities and the cumulative counts are in close agreement. The agreement is less good for the inferior meridian, largely due to the unusual distribution of the far peripheral densities. The anomalous bump at around 60° and subsequent rapid decline are difficult to fit with simple analytic functions. For comparison, in Figure 5 we show the RGCf density formula in the four meridians, replotted from Figure 4\nFigure 5\n\nRGCf density formula as a function of eccentricity for four meridians, replotted from Figure 4.\nFigure 5\n\nRGCf density formula as a function of eccentricity for four meridians, replotted from Figure 4.", null, "RGC displacement\nOne test of our formula for the density of RGCf is to compare the predicted density of RGC within the displacement zone with the actual densities measured by Curcio and Allen (1990). To generate these predictions we make use of measured displacements from cone inner segment to retinal ganglion cell body in six human retinas along the horizontal meridian (Drasdo et al., 2007). For the separate nasal and temporal meridians, they provided a mathematical formula to describe the average displacement as a function of RGC eccentricity. This can be converted to a function of cone inner segment eccentricity. A difficulty with this function is that it does not provide the correct result at an eccentricity of zero. It should indicate the displacement corresponding to the RGC closest to the fovea, but instead yields a value of 0. Rather than use their formula directly, we have instead constructed a new formula to describe displacement as a function of cone inner segment eccentricity. We constructed this new function to be a reasonable fit to the function of Drasdo but to also yield a displacement of h(0) > 0° at 0° eccentricity. There is some disagreement about the value of h(0). Drasdo stated that the first RGC are located 0.15 to 0.2 mm (0.53°–0.71°), whereas Curcio provides nonzero RGC densities at eccentricities as small as about 0.2° (see Figure 2). We have assumed a value of 0.5°, but a value of 0.3 gives very similar results.\nThe displacement function we used is the probability density function of the generalized Gamma distribution (multiplied by the gain δ), given by", null, "We attach no particular significance to the function or its parameters. It serves only as a device to displace hypothetical RGC cell bodies, as will be discussed below. This new function is shown in Figure 6, along with the corresponding functions provided by Drasdo for the two meridians. The parameters are provided in Table 2. Note that only three of the parameters are independent, the other two are constrained by the value of h(0) and by the peak value, which we set to the maximum of the fitted values.\nFigure 6\n\nModeled displacement functions (solid curve) and points from the formula of Drasdo et al. (2007).\nFigure 6\n\nModeled displacement functions (solid curve) and points from the formula of Drasdo et al. (2007).", null, "Table 2\n\nParameters of the displacement function (Equation 5 and Figure 6).\nTable 2\n\nParameters of the displacement function (Equation 5 and Figure 6).\n Meridian α β(deg) γ δ μ(deg) Temporal 1.8938 2.4598 0.91565 14.904 −0.09386 Nasal 2.4607 1.7463 0.77754 15.111 −0.15933\nNext we generated a population of RGC with eccentricities based on our density formula (Equation 4). Eccentricities were random within annuli of 0.05°. We then displaced each RGC centrifugally according to the displacement function (Equation 5, Figure 6). The density of the displaced cells was then computed. Results are shown in Figure 7. Note that this test can only be conducted on the two horizontal meridians along which displacement was measured. The actual and predicted densities are in reasonable agreement. In one case, the peak densities are too low and in the other, too high, but the height and shape of the distributions are approximated. Note that the displacements and densities were estimated from a different (but overlapping) set of eyes, and thus complete agreement is not expected.\nFigure 7\n\nModeled and measured density of displaced retinal ganglion cells in temporal (left) and nasal (right) meridians.\nFigure 7\n\nModeled and measured density of displaced retinal ganglion cells in temporal (left) and nasal (right) meridians.", null, "Midget RGCf density\nMidget cells make up most but not all of the retinal ganglion cells, and their proportion varies with eccentricity. In general, the density of midget retinal ganglion cell receptive fields is given by", null, "where f(r) is the fraction of retinal ganglion cells that are midgets, as introduced in Equation 2. Two estimates of f(r) have been provided in the literature.\nDacey (1993) estimated mRGC dendritic field diameters at various eccentricities in whole mounts of human retina. Careful examination of dendritic trees at several midperipheral and peripheral locations indicated a coverage (dendritic field area × density) of approximately 1. By assuming that this coverage remained constant at 1 throughout the retina Dacey estimated the density of midget cells as the inverse of dendritic field area. The ratio of that density to Curcio and Allen's (1990) estimates of RGC density (beyond about 3.5°) provided an estimate of the fraction of RGC that are midget cells. These estimates of Dacey are shown as the points in Figure 8. They range from over 95% near the fovea to less than 50% in the far periphery.\nFigure 8\n\nEstimated fraction of RGC that are midget cells as a function of eccentricity.\nFigure 8\n\nEstimated fraction of RGC that are midget cells as a function of eccentricity.", null, "A second estimate of f(r) was provided by Drasdo et al. (2007) as the formula", null, "where f(0) = 1/1.12 = 0.8928 and rm = 41.03°. The formula is shown by the curve in Figure 8. This formula was derived from iterative fit of a more elaborate expression involving both psychophysical measures and anatomical measures (Drasdo et al., 2007). Drasdo and Dacey's measures agree generally that the fraction declines with eccentricity, and roughly agree in foveal and peripheral asymptotes.\nOne problem with Dacey's method is that from about 3.5° to 17° it includes both RGC densities and dendritic field measurements of displaced cells. Hence his estimates at those eccentricities may not reflect the fraction of receptive fields belonging to midget cells. For this reason, and until better estimates can be obtained, we adopt Drasdo's formula to compute the density of mRGCf. Combining Equations 2, 4, 6, and 7, we have", null, "This formula is plotted in Figure 9 for the four meridians.\nFigure 9\n\nmRGCf density formula as a function of eccentricity (Equation 8).\nFigure 9\n\nmRGCf density formula as a function of eccentricity (Equation 8).", null, "mRGCf spacing\nOn the assumption of hexagonal packing (Equation A4), the spacing of adjacent midget receptive fields is given by", null, "Spacing at a binocular horizontal eccentricity can be computed by averaging densities at corresponding eccentricities in temporal and nasal meridians and converting to spacing. We can also compute the “mean” spacing, the average of all four meridians, by averaging densities and converting to spacing. These formulas for spacing are plotted in Figure 10. We show the individual meridians and the “Horizontal” and “Mean” versions.\nFigure 10\n\nFormula for mRGCf spacing.\nFigure 10\n\nFormula for mRGCf spacing.", null, "Note that the midget RGC are composed of approximately equal numbers of “on” and “off” center cells (we neglect reports of asymmetry between the two types). Typically we are concerned with the spacing within one class, in which case density is halved and the spacings should be multiplied by √2. In Figure 11 we show the formula scaled in this way for eccentricities between 0° and 10°, and we express the spacings in arcmin.\nFigure 11\n\nFormula for on or off mRGCf spacing for eccentricities 0°–10°.\nFigure 11\n\nFormula for on or off mRGCf spacing for eccentricities 0°–10°.", null, "Averaged across meridians, the computed spacing of mRGCf is very nearly linear (R2 = 0.9997). This allows us to write the following simple approximation for average spacing in either on- or off-center mosaic, where both s and r are expressed in degrees.", null, "This convenient result is the due to the form of Equation 4, and the fact that the second exponential term does not have much effect for eccentricities less than 30°.\nExtension to arbitrary retinal locations\nTo this point we have developed formulas describing spacing as a function of eccentricity along each of the four principal meridians. We would like to extend these formulas to describe spacing at an arbitrary point {x, y} in the retina. To do this we make the assumption that within any one quadrant of the retina the iso-spacing contours are ellipses. This is consistent with the idea that spacing changes smoothly with the angle of a ray extending from the visual center. An example is shown in Figure 12. The eccentricities at the intersections of the ellipse with the two enclosing meridians are rx and ry\nFigure 12\n\nA hypothetical iso-spacing curve in one quadrant, including a point {x, y} and points on the two enclosing meridians.\nFigure 12\n\nA hypothetical iso-spacing curve in one quadrant, including a point {x, y} and points on the two enclosing meridians.", null, "Under the ellipse assumption, we can write", null, "And because they are on an iso-spacing curve,", null, "Given numerical values for x and y, we can solve Equations 11 and 12 together to find numerical solutions for rx and ry, and then we can compute", null, "To avoid solving a system of equations, we have found that the following approximation works well. Let rxy be the radial eccentricity of the point {x, y},", null, "Then we compute", null, "", null, "This approximation is always within 1.7% of the value obtained by Equations 10 through 12. Equation 15 is easily generalized to work for arbitrary retinal quadrants. Since the sign of the horizontal coordinate is arbitrary, we define positive x values to mean the temporal visual field. This corresponds to the right visual field for the right eye, and the left visual field for the left eye.\nExtension to arbitrary binocular visual field locations\nEquation 15 computes mRGC spacing at locations specified in visual field coordinates in one eye. In psychophysical modeling of natural vision, it is useful to compute spacing at locations specified in the binocular visual field. To do this we compute the spacing at corresponding visual field locations in the two eyes, convert them to densities (Equation 2), compute their mean, and convert back to spacing (Equation A4). After simplifying, we obtain the following result for binocular spacing sB:", null, "With this function we can compute a plot of the Nyquist frequency over the binocular visual field, as shown in Figure 13. In this calculation we have divided the value by √2, based on the assumption of overlapping lattices of on- and off-center cells. The peak value is 65.4 cycles/deg. Because there is some ambiguity about the best way to combine the densities of the two eyes, in our supplementary materials we also provide functions that compute the maximum density, or the total density of the two eyes.\nFigure 13\n\nNyquist frequency over the binocular visual field based on the density formula and assuming separate and equal on- and off-center populations.\nFigure 13\n\nNyquist frequency over the binocular visual field based on the density formula and assuming separate and equal on- and off-center populations.", null, "Discussion\nRatio of midget RGC and cones\nIn Figure 14 we plot the ratio between mRGCf densities computed from Equation 8 and cone densities reported by Curcio et al. (1990). Although we did not impose this as a constraint in estimating a function for the density of ganglion cells, the ratio remains close to 2 for the central several degrees. This is roughly consistent with Dacey's report that up to about 4° mRGC dendritic fields remain at a minimum size appropriate to connection with a single bipolar, and thereby to a single cone. Beyond about 6°, he found that dendritic fields enlarge and begin to show clusters suggesting input from multiple cones, and consistent with the decline in the ratio in Figure 14. Our result is also consistent with Schein's estimate that the ratio remains constant out to about 2.5° in primates (Schein, 1988).\nFigure 14\n\nRatio of mRGCf to cones as a function of eccentricity based on the Watson formula for mRGCf density.\nFigure 14\n\nRatio of mRGCf to cones as a function of eccentricity based on the Watson formula for mRGCf density.", null, "Comparison with Drasdo et al. (2007)\nIn Figure 15 we compare mRGCF spacing computed from our formula to densities computed from the formula of Drasdo et al. (2007), converted to spacings by Equation A4. While there is considerable agreement, there are some significant discrepancies. In particular the curves for the superior and inferior meridians are nearly interchanged in the two formulas. Our formula is consistent with Curcio and Allen (1990) who clearly show higher density (smaller spacing) in superior versus inferior meridians beyond about 6° (see Figure 2), while Drasdo's formula shows the opposite, for unknown reasons. We also note that the Drasdo formula is defined only up to 30°, while ours extends at least as far as 90°.\nFigure 15\n\nComparison of formulas of Watson and Drasdo et al. (2007) (dashed).\nFigure 15\n\nComparison of formulas of Watson and Drasdo et al. (2007) (dashed).", null, "Comparison with dendritic field diameter\nDacey measured dendritic field diameters of midget ganglion cells in human retina (Dacey, 1993; Dacey & Petersen, 1992). In Figure 16 we have reproduced his Figure 4B, that shows diameter as a function of eccentricity. The filled points are for the temporal quadrant, open points are for the other quadrants. Near the fovea, where mRGC connect to single cone, we do not expect any relationship between field diameter and cell spacing. However in the periphery, Dacey found that within either the on- or off-center lattice, spacing and diameter were about equal. This allows us to compare his measures of diameter to our spacing formula for equivalent eccentricities and quadrants. Formula values for the temporal meridian and the mean of the other three meridians are shown by the colored curves in Figure 16. The computed spacing is for either the on- or off-center lattice. The agreement is reasonable, especially considering the sizable scatter in measurements of diameter.\nFigure 16\n\nDendritic field diameter as a function of eccentricity for human mRGC as measured by Dacey (1993). Filled points are for the temporal meridian; open points are for other quadrants. Gray curve is Dacey's estimate of the mean. Red and blue curves show spacing calculated from our formula for the temporal meridian or the mean of the other meridians.\nFigure 16\n\nDendritic field diameter as a function of eccentricity for human mRGC as measured by Dacey (1993). Filled points are for the temporal meridian; open points are for other quadrants. Gray curve is Dacey's estimate of the mean. Red and blue curves show spacing calculated from our formula for the temporal meridian or the mean of the other meridians.", null, "Comparison with acuity\nRossi and Roorda (2010) provide estimates of letter acuity, expressed as minimum angle of resolution (MAR) for five observers viewing targets under adaptive optics conditions. The targets were at nasal visual field locations between 0° and 2.5°. In Figure 17 we compare their results with the computed row spacing, assuming separate on- and off-cell lattices. We use separate lattices on the assumption that, at least near the fovea, both an on and an off midget cell are required to signal the signed value of local contrast. In addition, where each cone drives one on and one off midget, we know the two midgets have the same receptive field location. Specifically, the function returned by Equation 9 is multiplied by 60 √2 √3 / 2 = 30 √6 to reflect the halved density, conversion to row spacing, and conversion to arcmin. The agreement is excellent. This is not surprising, since Rossi and Roorda previously showed good agreement with the formula of Drasdo et al. (2007), to which ours is similar for small eccentricities.\nFigure 17\n\nHuman letter acuity (points) along the nasal meridian of five observers from Rossi and Roorda (2010) and row spacing from our formula for the same meridian (line).\nFigure 17\n\nHuman letter acuity (points) along the nasal meridian of five observers from Rossi and Roorda (2010) and row spacing from our formula for the same meridian (line).", null, "Estimates of peripheral acuity are complicated by the possibility of aliasing. Anderson, Mullen, and Hess (1991) attempted to bypass this problem by using direction discrimination of drifting gratings. Their results are plotted in Figure 18, along with calculations of Nyquist frequency of the on- or off-center mRGCf lattice from Equations 8 and A3. The agreement is reasonable. One caveat regarding the comparison at r = 0 is that these data were collected with Gabor targets that extended (at half height) well over 0.5°, so that performance may reflect the averaging spacing over that area. The precise relationship between mRGCf spacing and acuity is beyond the scope of this paper (Anderson & Thibos, 1999), here we only point to the general agreement in both the shape and absolute level of the calculations.\nFigure 18\n\nHuman grating acuity (points) from Anderson et al. (1991) and calculated Nyquist frequency of midget RGC.\nFigure 18\n\nHuman grating acuity (points) from Anderson et al. (1991) and calculated Nyquist frequency of midget RGC.", null, "Comparison with Sjöstrand\nIn a series of papers Sjöstrand, Popovic, and colleagues measured human RGC densities at eccentricities from about 2° to 34° eccentricity along the vertical meridian in sectioned human retinas (Popovic & Sjöstrand, 2001, 2005; Sjöstrand, Olsson, Popovic, & Conradi, 1999; Sjostrand, Popovic, Conradi, & Marshall, 1999). From these densities, using their own estimates of displacement, the inferred RGC spacing at various eccentricities. Their formula for conversion from density to spacing actually yields the row spacing (Equation A1) not the spacing between cells (Equation A4), which is 2 / √3 larger. Even taking this into account, their values are about a factor 0.75 smaller than those computed from our formula for the mean of superior and inferior meridians. However their values are also discrepant with Drasdo's formula (Figure 6) and with spacing estimated from Dacey's estimates of mRGC field diameter (Figure 16). Some part of this discrepancy may arise from their formula for displacement, which though similar in form is only half the magnitude of ours or that of Drasdo, who has also commented on this discrepancy (Drasdo et al., 2007).\nPopovic and Sjöstrand (2005) measured acuity of three observers at eccentricities between 5.8° and 26.4° in both eyes, one of which was subsequently enucleated. Ganglion cell densities and spacings were measured along the vertical meridian. Acuity (MAR) was measured using high-pass resolution perimetry. They found MAR was approximately proportional to RGC spacing over the full range of eccentricities. The constant of proportionality was rather large (4.24), especially compared to the value of 1 we have used in Figures 17 and 18. They note that correcting for the low contrast of their target (0.25), and considering only spacing in one class (on or off) of midget cells as we have done, would lower the constant to 1.43. The remaining discrepancy may be due to their low estimates of spacing, as noted above. In general, their results support the notion that psychophysical resolution is governed by the mRGC spacing.\nMidget fraction\nPerhaps the least secure element of our formula is the midget fraction, the function describing the fraction of all ganglion cells that are midget as a function of eccentricity. As noted above (Figure 8) there are discrepancies between available estimates. We have adopted the formula of Drasdo, but it is unclear whether that is accurate, especially at very large eccentricities, where it continues to descend to values as low as 0.25. In the periphery, where his estimates are arguably most accurate, Dacey's estimates appear the level off at about 0.5. Until more definitive estimates are available, we will have to acknowledge the speculative nature of this element.\nVariability\nI have based my formula on average densities of cones and retinal ganglion cells (Curcio & Allen, 1990; Curcio et al., 1990). Curcio et al. (1990) note a very large variation in peak cone density in their set of nine eyes, ranging from 98,200 to 324,100 mm−2 (7,385 to 24,372 deg−2) (coefficient of variation ∼0.46). When two anomalous eyes are excluded, the lower bound only increases to 166,000 mm−2 (12,483 deg−2). This variation largely disappeared at eccentricities beyond 1°, so that the total number of cones within a radius of about 3.6° (or over the entire retina) was nearly constant (coefficient of variation ∼0.1). However, more recent density estimates from in vivo measurement show a fairly consistent coefficient of variation (∼0.2) regardless of eccentricity (Song, Chui, Zhong, Elsner, & Burns, 2011). These latter authors have also shown an up to 25% decrement in density with age, primarily at eccentricities less than 1.6°. Curcio's data, and our formula, are consistent with the data for their younger group of observers.\nGanglion cell densities also show sizable individual differences (Curcio & Allen, 1990). However, perhaps in contrast to cones, the total number varies considerably, from 0.71 to 1.54 million cells over a set of six eyes. This variation seems to be consistent across the retina. Variation at the fovea cannot be directly determined because of the displacement of the RGC.\nBeyond these variations between individuals and with respect to age, there may be additional sources of variability and measurement error. Thus while a formula for the average may be useful, it is important to note that individuals may differ considerably from these computed values.\nConclusions\nWe have derived a mathematical formula for the density of receptive fields of human retinal ganglion cells as a function of position in the monocular or binocular visual field. Densities can also be computed for the receptive fields of the midget subclass of ganglion cells. Both spacing and position are expressed in degrees. The formula has several advantages over existing formulas, which are based on psychophysics, limited to small eccentricities, confined to specific meridians, or are inaccurate in the foveal region. Since the midget retinal ganglion cells provide the primary limit on human visual spatial resolution across the visual field, this formula may be useful in the modeling of human spatial vision.\nSupplementary Materials\nAcknowledgments\nI thank Albert Ahumada, Jeffrey Mulligan, Dennis Dacey, Heinz Wässle, Joy Hirsch, Neville Drasdo, Tony Movshon, and Denis Pelli and two anonymous referees for comments on earlier versions of the manuscript. I thank Christine Curcio for providing the cone density and retinal ganglion cell data, and Ethan Rossi for providing the acuity data in Figure 17. This work supported by the NASA Space Human Factors Research Project WBS 466199.\nCommercial relationships: none.\nCorresponding author: Andrew B. Watson.\nEmail: [email protected].\nAddress: NASA Ames Research Center, Moffett Field, CA, USA.\nReferences\nAhmad K. M. Klug K. Herr S. Sterling P. Schein S. (2003). Cell density ratios in a foveal patch in macaque retina. Visual Neuroscience, 20 (2), 189– 209.\nAnderson R. S. Thibos L. N. (1999). Relationship between acuity for gratings and for tumbling-E letters in peripheral vision. Journal of the Optical Society of America A, 16 (10), 2321– 2333, http://josaa.osa.org/abstract.cfm?URI=josaa-16-10-2321.\nAnderson S. J. Mullen K. T. Hess R. F. (1991). Human peripheral spatial resolution for achromatic and chromatic stimuli: Limits imposed by optical and retinal factors. Journal of Physiology, 442, 47– 64, http://www.ncbi.nlm.nih.gov/pubmed/1798037.\nAubert H. R. Foerster C. F. R. (1857). Beiträge zur Kenntniss des indirecten Sehens. (I). Untersuchungen über den Raumsinn der Retina [Translation: Contributions to the knowledge of the indirect vision: I. Studies on the sense of space of the retina]. Archiv für Ophthalmologie, 3, 1– 37.\nBarten P. G. J. (1999). Contrast sensitivity of the human eye and its effects on image quality. Bellingham, WA: SPIE Optical Engineering Press.\nCharman W. N. (1991). Optics of the human eye. In Cronly Dillon J. Ed.), Visual optics and instrumentation ( pp. 1– 26). Boca Raton: CRC Press.\nCurcio C. (2013). Curcio_JCompNeurol1990_GCtopo_F6.xls.\nCurcio C. A. Allen K. A. (1990). Topography of ganglion cells in human retina. Journal of Comparative Neurology, 300 (1), 5– 25.\nCurcio C. A. Sloan K. R. Kalina R. E. Hendrickson A. E. (1990). Human photoreceptor topography. Journal of Comparative Neurology, 292 (4), 497– 523, http://www.ncbi.nlm.nih.gov/pubmed/2324310.\nDacey D. M. (1993). The mosaic of midget ganglion cells in the human retina. Journal of Neuroscience, 13 (12), 5334– 5355, http://www.ncbi.nlm.nih.gov/pubmed/8254378.\nDacey D. M. Petersen M. R. (1992). Dendritic field size and morphology of midget and parasol ganglion cells of the human retina. Proceedings of the National Academy of Sciences, USA, 89 (20), 9666– 9670, http://www.ncbi.nlm.nih.gov/pubmed/1409680.\nDrasdo N. Fowler C. W. (1974). Non-linear projection of the retinal image in a wide-angle schematic eye. British Journal of Ophthalmology, 58 (8), 709– 714, http://www.ncbi.nlm.nih.gov/pubmed/4433482.\nDrasdo N. Millican C. L. Katholi C. R. Curcio C. A. (2007). The length of Henle fibers in the human retina and a model of ganglion receptive field density in the visual field. Vision Research, 47 (22), 2901– 2911, http://www.ncbi.nlm.nih.gov/pubmed/17320143.\nEmsley H. H. (1952). Visual optics (5th ed.). London: Hatton Press.\nGoodchild A. K. Ghosh K. K. Martin P. R. (1996). Comparison of photoreceptor spatial density and ganglion cell morphology in the retina of human, macaque monkey, cat, and the marmoset Callithrix jacchus. Journal of Comparative Neurology, 366 (1), 55– 75, http://www.ncbi.nlm.nih.gov/pubmed/8866846.\nHirsch J. Curcio C. A. (1989). The spatial resolution capacity of human foveal retina. Vision Research, 29 (9), 1095– 1101.\nKolb H. Dekorver L. (1991). Midget ganglion cells of the parafovea of the human retina: A study by electron microscopy and serial section reconstructions. Journal of Comparative Neurology, 303 (4), 617– 636, http://www.ncbi.nlm.nih.gov/pubmed/1707423.\nMerigan W. H. Eskin T. A. (1986). Spatio-temporal vision of macaques with severe loss of P-beta retinal ganglion cells. Vision Research, 26, 1751– 1761.\nMerigan W. H. Katz L. M. (1990). Spatial resolution across the macaque retina. Vision Research, 30 (7), 985– 991.\nOsterberg G. A. (1935). Topography of the layer of rods and cones in the human retina. Acta Ophthalmologica, 6 (Suppl. VI), 1– 97.\nPopovic Z. Sjöstrand J. (2005). The relation between resolution measurements and numbers of retinal ganglion cells in the same human subjects. Vision Research, 45 (17), 2331– 2338.\nPopovic Z. Sjöstrand J. (2001). Resolution, separation of retinal ganglion cells, and cortical magnification in humans. Vision Research, 41 (10), 1313– 1319.\nRossi E. A. Roorda A. (2010). The relationship between visual resolution and cone spacing in the human fovea. Nature Neuroscience, 13 (2), 156– 157, http://dx.doi.org/10.1038/nn.2465.\nSchein S. J. (1988). Anatomy of macaque fovea and spatial densities of neurons in foveal representation. Journal of Comparative Neurology, 269 (4), 479– 505, http://www.ncbi.nlm.nih.gov/pubmed/3372725.\nSjöstrand J. Olsson V. Popovic Z. Conradi N. (1999). Quantitative estimations of foveal and extra-foveal retinal circuitry in humans. Vision Research, 39 (18), 2987– 2998.\nSjostrand J. Popovic Z. Conradi N. Marshall J. (1999). Morphometric study of the displacement of retinal ganglion cells subserving cones within the human fovea. Graefes Archives for Clinical & Experimental Ophthalmology, 237 (12), 1014– 1023.\nSong H. Chui T. Y. P. Zhong Z. Elsner A. E. Burns S. A. (2011). Variation of cone photoreceptor packing density with retinal eccentricity and age. Investigative Ophthalmology & Visual Science, 52 (10), 7376– 7384, http://www.iovs.org/content/52/10/7376. [Pubmed] [Article]\nStrasburger H. Rentschler I. Juttner M. (2011). Peripheral vision and pattern recognition: A review. Journal of Vision, 11( 5): 13, 1-82, http://www.journalofvision.org/content/11/5/13, doi:10.1167/11.5.13. [Pubmed] [Article]\nThibos L. Cheney F. Walsh D. (1987). Retinal limits to the detection and resolution of gratings. Journal of the Optical Society of America A, 4 (8), 1524– 1529.\nWassle H. Grunert U. Rohrenbeck J. Boycott B. B. (1990). Retinal ganglion cell density and cortical magnification factor in the primate. Vision Research, 30 (11), 1897– 1911, http://www.ncbi.nlm.nih.gov/pubmed/2288097.\nWolfram Research Inc. (2013). Mathematica (Version 9.0). Champaign, IL: Author.\nAppendix 1: Demonstration\nAs a supplement to this paper we provide an interactive calculator that returns density and spacing at arbitrary visual field locations. The demonstration requires the use of the Wolfram CDF player, available at https://www.wolfram.com/cdf-player/. An illustration of the calculator is shown in Figure 19\nFigure 19\n\nDemonstration of Retinal Topography Calculator.\nFigure 19\n\nDemonstration of Retinal Topography Calculator.", null, "Appendix 2: Mathematica tables and functions\nAs a supplement we provide a Mathematica Notebook that contains a number of tables and functions derived and used in this report. The Notebook is a text file, but is most readable using the free Wolfram CDF Player available at https://www.wolfram.com/cdf-player/\nAppendix 3: Notation\nThe following is a table of notation used in this report.\nTable A1\n\nNotation.\nTable A1\n\nNotation.\n Symbol Definition Unit r Eccentricity in deg relative to the visual axis deg k Meridian index d(r, k) Density of cells or receptive fields at eccentricity r along meridian k deg−2 s(r, k) Spacing between receptive fields at eccentricity r along meridian k deg c, g, m, gf, mf Subscripts to indicate cones, RGC, mRGC, RGCf, and mRGCf h(r) Displacement of RGC from RGCf at eccentricity r deg f(r) Fraction of RGC that are midget, as a function of eccentricity dimensionless Δk Offset between optic and visual axis along the specified meridian mm r′ Eccentricity in deg relative to the optic axis deg rmm Eccentricity in mm relative to the visual axis mm r′mm Eccentricity in mm relative to the optic axis mm N Nyquist frequency of hexagonal lattice cycles/deg S Point spacing of hexagonal lattice deg R Row spacing of hexagonal lattice deg D Point density of hexagonal lattice deg−2\nAppendix 4: Formula parameters and useful numbers\nTable A2\n\nFormula parameters and useful numbers.\nTable A2\n\nFormula parameters and useful numbers.\n Item Value Unit Peak cone density 14,804.6 deg−2 Peak RGCf density 33,162.3 deg−2 Peak mRGCf density 29,609.2 deg−2 Minimum on-center mRGCf (or cone) spacing 0.5299 arcmin Peak on-center mRGCf (or cone) Nyquist 65.37 cycles/deg f(0), midget fraction at zero eccentricity 1/1.12 = 0.8928 rm, scale factor for decline in midget fraction with eccentricity 41.03 deg\nAppendix 5: Density, spacing, row spacing, and Nyquist frequency\nAt least in the fovea, both photoreceptors and midget ganglion cell receptive fields form an approximately hexagonal lattice. Because we will make use of these formulas in the text, we review here the relationships among various metrics of a hexagonal lattice of points: S(deg) = the spacing between adjacent points, R(deg) = the spacing between rows of points, D(deg−2) = the density of points, and N(c/deg) = the Nyquist frequency of the lattice (the highest frequency that can be supported by a particular row spacing). These formulas will allow us to convert among these metrics. Then", null, "", null, "", null, "", null, "Appendix 6: Conversion formulas for retinal dimensions\nConversion of eccentricities in millimeters to degrees\nEccentricity is defined as distance from a visual center. Anatomical measurements of the retina often express eccentricity in millimeters. We would like to convert these measurements to degrees of visual angle (deg) relative to the visual axis. Drasdo and Fowler (1974) used a model eye to compute conversions from retinal distances in millimeters to degrees. Their presentation does not, however, provide analytical expressions of the relevant quantities, so we must derive them from the figures. In their figure 2 they show a “curve showing computed relationship between retinal arc lengths and visual angles from the optic axis.” We have extracted the contour and fit it with a third order polynomial,", null, "In this equation, rmm refers to distance in mm, while r′ is the comparable measurement in degree. The prime marking indicates a measurement relative to the optic axis.\nWe use this equation to translate optical eccentricities in degrees to millimeters. It is plotted in Figure A1a, along with a linear function with slope 0.268. The linear approximation is acceptable up to angles of 40°. We also fit to the transpose of the contour data, to obtain an inverse function,", null, "Figure A1\n\nRelation between retinal distance from the optic axis in millimeters and degrees. The linear approximations shown as red dashed lines have slopes of 0.268 and 3.731, respectively.\nFigure A1\n\nRelation between retinal distance from the optic axis in millimeters and degrees. The linear approximations shown as red dashed lines have slopes of 0.268 and 3.731, respectively.", null, "We use this function, shown in Figure A1b, to convert optical eccentricities in millimeters to degrees. Dacey (1993) used a second order polynomial here, but its fit is poor at very small (0 mm) or large (22 mm) eccentricities. The linear approximation shown in the figure has a slope of 1/0.268 = 3.731.\nConversion of mm2 to deg2\nThe preceding formulas convert eccentricities (distances) from millimeters to degrees. We also need to convert local areas from mm2 to deg2. In their figure 5, Drasdo and Fowler (1974) show “variation of retinal area per solid degree with peripheral angle from the optic axis.” We have fit this with a polynomial", null, "where a is the ratio of areas mm2/deg2. We have used this to convert cell densities in mm−2 to deg−2. The function is illustrated in Figure A2.\nFigure A2\n\nArea ratio as a function of angular distance from the optic axis.\nFigure A2\n\nArea ratio as a function of angular distance from the optic axis.", null, "Visual versus optical axis\nDrasdo and Fowler's (1974) angular measurements are relative to the optic axis. Measurements of cone densities and psychophysical predictions are usually expressed relative to the visual axis. According to Charman (1991), quoting Emsley (1952), the optical axis intersects the retina 1.5 mm nasal and 0.5 mm superior to the visual axis. In the visual field, the optical center is thus 1.5 mm temporal of the visual center. To convert eccentricities from millimeters relative to the visual axis to degrees relative to the visual axis, we use the following approximation", null, "where m is the index of the meridian, and the corresponding offsets Δ from optic to visual axis in mm are given by", null, "We remind that primed eccentricities are relative to the optic axis. The relation between visual millimeters and visual degrees for the four meridians is shown in Figure A3. Note that without the correction of visual-optical offset, all curves would be superimposed and look like Figure A1b above.\nFigure A3\n\nRelation between distance from the visual axis in millimeters and degrees, with correction for offset between visual and optic axes.\nFigure A3\n\nRelation between distance from the visual axis in millimeters and degrees, with correction for offset between visual and optic axes.", null, "In Figure A4 we show the relation between eccentricities in degrees computed with and without the correction for the offset. Each curve is a parametric plot, where the millimeter argument is relative to either optical or visual axis. The correction is only significant above 40°.\nFigure A4\n\nEccentricities in degree computed with and without correction for offset between visual and optic axes.\nFigure A4\n\nEccentricities in degree computed with and without correction for offset between visual and optic axes.", null, "Figure 1\n\nCone density as a function of eccentricity (Curcio et al., 1990). Foveal density is indicated by the line segment at the upper left. The gap in the temporal meridian corresponds to the blind spot.\nFigure 1\n\nCone density as a function of eccentricity (Curcio et al., 1990). Foveal density is indicated by the line segment at the upper left. The gap in the temporal meridian corresponds to the blind spot.", null, "Figure 2\n\nRGC density as a function of eccentricity in four meridians (Curcio & Allen, 1990). The gap in the temporal meridian corresponds to the blind spot.\nFigure 2\n\nRGC density as a function of eccentricity in four meridians (Curcio & Allen, 1990). The gap in the temporal meridian corresponds to the blind spot.", null, "Figure 3\n\nCumulative total number of RGC along four meridians.\nFigure 3\n\nCumulative total number of RGC along four meridians.", null, "Figure 4\n\nGanglion cell density as a function of eccentricity in four meridians. Data points are from Curcio and Allen (1990). The solid curve is the fit of Equation 4 to the points outside the displacement zone. The dashed gray line indicates the approximate limit of the displacement zone. The density at eccentricity of zero is indicated by the line segment at the upper left.\nFigure 4\n\nGanglion cell density as a function of eccentricity in four meridians. Data points are from Curcio and Allen (1990). The solid curve is the fit of Equation 4 to the points outside the displacement zone. The dashed gray line indicates the approximate limit of the displacement zone. The density at eccentricity of zero is indicated by the line segment at the upper left.", null, "Figure 5\n\nRGCf density formula as a function of eccentricity for four meridians, replotted from Figure 4.\nFigure 5\n\nRGCf density formula as a function of eccentricity for four meridians, replotted from Figure 4.", null, "Figure 6\n\nModeled displacement functions (solid curve) and points from the formula of Drasdo et al. (2007).\nFigure 6\n\nModeled displacement functions (solid curve) and points from the formula of Drasdo et al. (2007).", null, "Figure 7\n\nModeled and measured density of displaced retinal ganglion cells in temporal (left) and nasal (right) meridians.\nFigure 7\n\nModeled and measured density of displaced retinal ganglion cells in temporal (left) and nasal (right) meridians.", null, "Figure 8\n\nEstimated fraction of RGC that are midget cells as a function of eccentricity.\nFigure 8\n\nEstimated fraction of RGC that are midget cells as a function of eccentricity.", null, "Figure 9\n\nmRGCf density formula as a function of eccentricity (Equation 8).\nFigure 9\n\nmRGCf density formula as a function of eccentricity (Equation 8).", null, "Figure 10\n\nFormula for mRGCf spacing.\nFigure 10\n\nFormula for mRGCf spacing.", null, "Figure 11\n\nFormula for on or off mRGCf spacing for eccentricities 0°–10°.\nFigure 11\n\nFormula for on or off mRGCf spacing for eccentricities 0°–10°.", null, "Figure 12\n\nA hypothetical iso-spacing curve in one quadrant, including a point {x, y} and points on the two enclosing meridians.\nFigure 12\n\nA hypothetical iso-spacing curve in one quadrant, including a point {x, y} and points on the two enclosing meridians.", null, "Figure 13\n\nNyquist frequency over the binocular visual field based on the density formula and assuming separate and equal on- and off-center populations.\nFigure 13\n\nNyquist frequency over the binocular visual field based on the density formula and assuming separate and equal on- and off-center populations.", null, "Figure 14\n\nRatio of mRGCf to cones as a function of eccentricity based on the Watson formula for mRGCf density.\nFigure 14\n\nRatio of mRGCf to cones as a function of eccentricity based on the Watson formula for mRGCf density.", null, "Figure 15\n\nComparison of formulas of Watson and Drasdo et al. (2007) (dashed).\nFigure 15\n\nComparison of formulas of Watson and Drasdo et al. (2007) (dashed).", null, "Figure 16\n\nDendritic field diameter as a function of eccentricity for human mRGC as measured by Dacey (1993). Filled points are for the temporal meridian; open points are for other quadrants. Gray curve is Dacey's estimate of the mean. Red and blue curves show spacing calculated from our formula for the temporal meridian or the mean of the other meridians.\nFigure 16\n\nDendritic field diameter as a function of eccentricity for human mRGC as measured by Dacey (1993). Filled points are for the temporal meridian; open points are for other quadrants. Gray curve is Dacey's estimate of the mean. Red and blue curves show spacing calculated from our formula for the temporal meridian or the mean of the other meridians.", null, "Figure 17\n\nHuman letter acuity (points) along the nasal meridian of five observers from Rossi and Roorda (2010) and row spacing from our formula for the same meridian (line).\nFigure 17\n\nHuman letter acuity (points) along the nasal meridian of five observers from Rossi and Roorda (2010) and row spacing from our formula for the same meridian (line).", null, "Figure 18\n\nHuman grating acuity (points) from Anderson et al. (1991) and calculated Nyquist frequency of midget RGC.\nFigure 18\n\nHuman grating acuity (points) from Anderson et al. (1991) and calculated Nyquist frequency of midget RGC.", null, "Figure 19\n\nDemonstration of Retinal Topography Calculator.\nFigure 19\n\nDemonstration of Retinal Topography Calculator.", null, "Figure A1\n\nRelation between retinal distance from the optic axis in millimeters and degrees. The linear approximations shown as red dashed lines have slopes of 0.268 and 3.731, respectively.\nFigure A1\n\nRelation between retinal distance from the optic axis in millimeters and degrees. The linear approximations shown as red dashed lines have slopes of 0.268 and 3.731, respectively.", null, "Figure A2\n\nArea ratio as a function of angular distance from the optic axis.\nFigure A2\n\nArea ratio as a function of angular distance from the optic axis.", null, "Figure A3\n\nRelation between distance from the visual axis in millimeters and degrees, with correction for offset between visual and optic axes.\nFigure A3\n\nRelation between distance from the visual axis in millimeters and degrees, with correction for offset between visual and optic axes.", null, "Figure A4\n\nEccentricities in degree computed with and without correction for offset between visual and optic axes.\nFigure A4\n\nEccentricities in degree computed with and without correction for offset between visual and optic axes.", null, "Table 1\n\nParameters and error for fits of Equation 4 in four meridians. Note: Also shown are measured and predicted cumulative counts along each meridian within the displacement zone. The next-to-last column shows the fitting error outside the displacement zone. The last column indicates the assumed limit of the displacement zone.\nTable 1\n\nParameters and error for fits of Equation 4 in four meridians. Note: Also shown are measured and predicted cumulative counts along each meridian within the displacement zone. The next-to-last column shows the fitting error outside the displacement zone. The last column indicates the assumed limit of the displacement zone.\n Meridian k a r2 re Data count (× 1000) Model count (× 1000) Error rz Temporal 1 0.9851 1.058 22.14 485.1 485.7 0.23 11 Superior 2 0.9935 1.035 16.35 526.1 528.9 0.12 17 Nasal 3 0.9729 1.084 7.633 660.9 661.1 0.01 17 Inferior 4 0.996 0.9932 12.13 449.3 452.1 0.93 17\nTable 2\n\nParameters of the displacement function (Equation 5 and Figure 6).\nTable 2\n\nParameters of the displacement function (Equation 5 and Figure 6).\n Meridian α β(deg) γ δ μ(deg) Temporal 1.8938 2.4598 0.91565 14.904 −0.09386 Nasal 2.4607 1.7463 0.77754 15.111 −0.15933\nTable A1\n\nNotation.\nTable A1\n\nNotation.\n Symbol Definition Unit r Eccentricity in deg relative to the visual axis deg k Meridian index d(r, k) Density of cells or receptive fields at eccentricity r along meridian k deg−2 s(r, k) Spacing between receptive fields at eccentricity r along meridian k deg c, g, m, gf, mf Subscripts to indicate cones, RGC, mRGC, RGCf, and mRGCf h(r) Displacement of RGC from RGCf at eccentricity r deg f(r) Fraction of RGC that are midget, as a function of eccentricity dimensionless Δk Offset between optic and visual axis along the specified meridian mm r′ Eccentricity in deg relative to the optic axis deg rmm Eccentricity in mm relative to the visual axis mm r′mm Eccentricity in mm relative to the optic axis mm N Nyquist frequency of hexagonal lattice cycles/deg S Point spacing of hexagonal lattice deg R Row spacing of hexagonal lattice deg D Point density of hexagonal lattice deg−2\nTable A2\n\nFormula parameters and useful numbers.\nTable A2\n\nFormula parameters and useful numbers.\n Item Value Unit Peak cone density 14,804.6 deg−2 Peak RGCf density 33,162.3 deg−2 Peak mRGCf density 29,609.2 deg−2 Minimum on-center mRGCf (or cone) spacing 0.5299 arcmin Peak on-center mRGCf (or cone) Nyquist 65.37 cycles/deg f(0), midget fraction at zero eccentricity 1/1.12 = 0.8928 rm, scale factor for decline in midget fraction with eccentricity 41.03 deg\nSupplement 1\nSupplement 2\n×\n\nThis PDF is available to Subscribers Only" ]	[ null, "https://jov.arvojournals.org/UI/app/images/arvo_journals_logo-white.png", null, "https://jov.arvojournals.org/Images/fallbackCovers/170.jpg", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://jov.arvojournals.org/Images/grey.gif", null, "https://arvo.silverchair-cdn.com/arvo/content_public/journal/jov/933548/m_i1534-7362-14-7-15-e01.png", null, "https://arvo.silverchair-cdn.com/arvo/content_public/journal/jov/933548/m_i1534-7362-14-7-15-e02.png", null, "https://arvo.silverchair-cdn.com/arvo/content_public/journal/jov/933548/m_i1534-7362-14-7-15-e03.png", null, "https://arvo.silverchair-cdn.com/arvo/content_public/journal/jov/933548/m_i1534-7362-14-7-15-e04.png", null, "https://arvo.silverchair-cdn.com/arvo/content_public/journal/jov/933548/m_i1534-7362-14-7-15-e05.png", null, 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https://electronics.stackexchange.com/questions/301239/mosfet-conventional-current-flow-direction-in-the-circuit/301252	[ "# Mosfet Conventional current flow direction in the circuit\n\nI am working on a project to control heavy loads with Arduino up to 10 Amperes. I found the circuit which is made using p-channel Mosfet and a p-type transistor. I am confused in the flow of current through the circuit. I uploaded the diagram please see if the conventional current flow is right in the diagram? and what about the current through the red box (Gate of Mosfet) what will be IL=?. If the input current is up to 10 Amperes does it effects my arduino digital pin? Also if you have any recommendations regarding the circuit please share them.", null, "• To the OP, Olin was rather harsh. I guess I would just say that the black arrows do not help us. We all know which way the current is flowing. And we only care about conventional current. Nobody cares if electrons are flowing the opposite way. It is better not to even mention it. – mkeith Apr 23 '17 at 19:14\n\nMOSFET gates are very high impedance, so no current (or almost no current) flows into them in steady-state conditions.\n\nDuring switching on/off there is indeed current flowing to/from the gate as it charges/discharges and reaches its required Vgs level. But this is only a transient condition. If your load is switched only from time to time, its steady-state condition is no current flowing to/from the MOSFET gate.\n\n1. If you plan to control inductive loads such as motors, use a flyback diode across the load terminals to avoid destroying the P-MOSFET due to inductive voltage spikes at its drain when the load is turned off.\n\n2. Decouple the +12V supply rail with a big capacitor to avoid destroying the P-MOSFET due to inductive voltage spikes at its source when the load is turned off.\n\n3. Due to the high currents involved, consider using an optocoupler instead of a BJT, to fully isolate the 12V circuit from the Arduino.\n\n4. Consider using a logic-level N-MOSFET instead of a BJT for T1. If you decide to keep the BJT, then add a base resistor lo limit the current into the base. Also, add a pull-down resistor at the base to ensure the BJT is cut-off when the Arduino pin is at high impedance (something that can happen when the Arduino is off or when it is starting up, before the pin is configured as OUTPUT).\n\n• Agree with item 4. I would keep the BJT, because I like BJT's, but use a resistor in series with the base, and a pulldown on the base. The value of the series resistor can be around 10x the value of R1. Just as a general guide. – mkeith Apr 23 '17 at 19:11\n\nYour diagram is right (but difficult to read due the blocky arrows). The current through the red element is substantial only when the load current is switched on or off. The current \"?\" exists during the state transition, because the mosfet is controlled by voltage. The current is needed because there exixts a substantial internal capacitance between the gate and the drain & source. Ic charges that capacitance when the loar current is turned to ON. The capacitange gets discharged through R1 when the load current is turned to OFF.\n\nT1 is not P-type, but NPN\n\nThe red element can be a wire. Often a small resistor is used to damp unwanted radio frequency oscillations that are common in fast pulse circuits with no precautions.\n\nIf this circuit is properly realized, I1 is only a few milliamperes, the major part of the Is goes to the load.\n\nThose arrows make everything hard to decipher, but they look correct.\n\nThe gate of a MOSFET behaves pretty much like a capacitor. So a current will flow into or out of the gate only when you are switching. (The amount of gate charge should be found in the datasheet.)\n\nThe source/drain channel of a MOSFET behaves pretty much like a resistor. (The resistance (RDS(on)) should be found in the datasheet.) In a MOSFET rated for large currents, this resistance typically is very low (milliohms), so it's usually ignored. In other words, you can assume that the load behaves as if it were connected directly to +12V.\n\nIf the load is inductive (e.g., motor, relay, transformer), it can generate large voltage spikes when switched off, and you need to add a snubber to protect the rest of your circuit.\n\nSince the voltages throughout the circuit weren't labeled or discussed, perhaps your question stems from a common misunderstanding. This is the misconception that circuitry is based on electric current ...and that to understand circuits, we sketch in all the currents.\n\nActually, engineers and scientists view most circuits as voltage-controlled systems. Everything is powered by constant-voltage supplies, and the signalling is voltage-based. To understand a circuit, we sketch in all the voltages. Then, using Ohm's law, we can determine the currents if necessary (or even ignore them entirely, and instead concentrate on input/output voltages and load wattage.)\n\nFor a nice animated view of voltages (and currents) inside circuits, try the little simulator at Falstad's site. (a java applet)\n\nFor example, the current in the gate-wire of the PMOS remains zero, whether the transistor is on or off. MOS transistors are voltage-driven devices, and their gate-current is usually irrelevant.\n\nTo analyze this circuit, notice that transistor T1 and resistor R1 form a voltage-divider between 12V and 0V. When T1 is on, T1 forms a short circuit to ground, and it pulls down the PMOS gate to zero volts. When T1 is off, it acts like an open circuit, and then R1 pulls the PMOS gate up to 12V.\n\nIn other words, T1 and R1 have converted the Arduino's small output voltage into a 12V signal. This 12V signal then drives the PMOS transistor gate.\n\nThe PMOS transistor is wired as an inverter: when the voltage on the PMOS gate is zero, that transistor turns fully on, and when the voltage is 12V, it turns off. (Yes, it should handle 10amps just fine. If its on-resistance is low enough, it might not even need any heat-sink.)\n\nAlso, note that you'll need a resistor in series with the base of transistor T1. The transistor's input acts as a diode to ground, and this diode would short out your Arduino's output pin. (LEDs need a current-limiting resistor, and so does this transistor's base lead.) The added resistor should be around 10x larger than the value of R1 (so if R1 is 10K, then add a 100K resistor to the connection between the Arduino and T1.)" ]	[ null, "https://i.stack.imgur.com/opHX0.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93555415,"math_prob":0.9138104,"size":6848,"snap":"2020-34-2020-40","text_gpt3_token_len":1588,"char_repetition_ratio":0.1354471,"word_repetition_ratio":0.2006689,"special_character_ratio":0.22108644,"punctuation_ratio":0.1003663,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9719087,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-14T03:47:30Z\",\"WARC-Record-ID\":\"<urn:uuid:03a6c57b-9687-4c99-abae-869f7aae97c1>\",\"Content-Length\":\"175152\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6c76dac5-8337-40aa-a1d5-f3640f49fb8e>\",\"WARC-Concurrent-To\":\"<urn:uuid:f99d6f03-f5fc-49e6-9584-9e11e6876c90>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://electronics.stackexchange.com/questions/301239/mosfet-conventional-current-flow-direction-in-the-circuit/301252\",\"WARC-Payload-Digest\":\"sha1:PJHL2WW4VB4ZYAV65FWT3SGBEELHKKBG\",\"WARC-Block-Digest\":\"sha1:QKXTZYW6PDWVVG2GU2BGURKQIPJNCVKF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439739134.49_warc_CC-MAIN-20200814011517-20200814041517-00077.warc.gz\"}"}
https://zbmath.org/?q=an%3A1063.37068	[ "zbMATH — the first resource for mathematics\n\nA new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling. (English) Zbl 1063.37068\nSummary: A type of new interesting loop algebra $$\\widetilde G_M$$, $$M = 1,2,\\dots$$, with a simple commutation operation just like that in the loop algebra $$\\widetilde A_1$$ is constructed. With the help of the loop algebra $$\\widetilde G_M$$, a new multicomponent integrable system, M-AKNS-KN hierarchy, is worked out. As reduction cases, the M-AKNS hierarchy and M-KN hierarchy are engendered, respectively. In addition, the system 1-AKNS-KN, which is a reduced case of the M-AKNS-KN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the M-AKNS-KN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (M-AKNS) and the multicomponent KN hierarchy(M-KN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the M-AKNS-KN hierarchy.\n\nMSC:\n 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures\nFull Text:\nReferences:\n M. J. Ablowitz and P. A. Clarkson,Solitons, Nonlinear Evolution Equations and Inverse Scattering(Cambridge University Press, Cambridge, 1991). · Zbl 0762.35001 A. C. Newell,Soliton in Mathematics and Physics(SIAM, Philadelphia, 1985). Wadati, J. Phys. Soc. Jpn. 32 pp 1447– (1972) Wadati, J. Phys. Soc. Jpn. 34 pp 1289– (1973) Wadati, Prog. Theor. Phys. 53 pp 417– (1975) Wadati, J. Phys. Soc. Jpn. 47 pp 1698– (1979) Shimizu, Prog. Theor. Phys. 63 pp 808– (1980) Tu, J. Math. Phys. 30 pp 330– (1989) Guo, Acta Math. Phys. Sin. 19 pp 507– (1999) Tsuchida, J. Phys. Soc. Jpn. 69 pp 2241– (1999) Tsuchida, Phys. Lett. A 257 pp 53– (1999) Guo, Acta Math. Appl. Sin. 23 pp 181– (2000) Guo, J. Syst. Sci. Math. Sci. 22 pp 36– (2003) Ma, Chin. Ann. Math., Ser. A 13 pp 115– (1992) Ma, Chin. Ann. Math., Ser. B 23 pp 373– (2002) F. Guo and Y. Zhang, Chaos, Solitons Fractals, 2003 (accepted for publication). Guo, Acta Phys. Sin. 51 pp 951– (2002) Ma, Chaos, Solitons Fractals 7 pp 1227– (1996)\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7025288,"math_prob":0.7874848,"size":2915,"snap":"2022-05-2022-21","text_gpt3_token_len":947,"char_repetition_ratio":0.124012366,"word_repetition_ratio":0.024229076,"special_character_ratio":0.33962265,"punctuation_ratio":0.22364217,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9674566,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-28T10:57:31Z\",\"WARC-Record-ID\":\"<urn:uuid:158a5239-dd76-46ee-ae3a-0d1aef8c3c26>\",\"Content-Length\":\"51323\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f019175d-8c36-4fcb-9160-16e74f0849fd>\",\"WARC-Concurrent-To\":\"<urn:uuid:066146a0-2a4f-45c6-86a4-a41b250fd319>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/?q=an%3A1063.37068\",\"WARC-Payload-Digest\":\"sha1:XHJL6A5EMA5LXTMWK2TNY5ITOBCITT2M\",\"WARC-Block-Digest\":\"sha1:OSQDEPCXTIOR25PMDY2PDBEULUCOHB46\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320305494.6_warc_CC-MAIN-20220128104113-20220128134113-00048.warc.gz\"}"}
https://crypto.stackexchange.com/questions/43748/how-would-you-explain-this-very-simple-add-modulo-asymetric-encryption	[ "# How would you explain this very simple add/modulo asymetric encryption?\n\nI am working on a presentation aimed at developers without a strong mathematical background (like me). I would like to explain how asymetric encryption works with an non-fully-secure but working example.\nI found one on Stackoverflow based on addition and modulo :\n\n• given a range from 1 to M\n• given a and b where a+b = M\n• a is the public key, b is the private key\n• x is the clear data (just a number)\n• y is the encrypted data\n• y = (x + a) mod M\n• x = (y + b) mod M\n\nExample :\n\nM=10, a=3, b=7, x=4\ny = (4+3) mod 10 = 7\n(7 + 7) mod 10 = 4\n\n\nVilain knows the algorithm and a, y, M.\nWe suppose Vilain cannot guess b from a.\nReverse modulo has multiple results :\n\n4 mod 3 = 1\n7 mod 3 = 1\n13 mod 3 = 1\n\n\nMeaning Vilain cannot easyly find\n\nx+a\n\n\nfrom\n\n(x+a) mod M\n\n\nEven this is really simple, and I \"feel\" the modulo \"wrap-around\" I could not find anyway to explain WHY this works :\n\ny = (x + a) mod M\nx = (y + b) mod M\n\n\nDoes anyone could try to explain it to me ?\n\n• Note: in crypto, we assume the adversary (vilain) is smart. Knowing $a$, $M$, and with given that $a+b=M$, the adversary can find $b$, thus the system is insecure. Hint at why $x=(y+b)\\bmod M$ holds: in the expression $(y+b)\\bmod M$, replace $y$ according to the given $y=(x+a)\\bmod M$, then use the given $a+b=M$, and the properties of the $\\bmod$ operator (which stands for rest of the Euclidean division by); the property you want is that $(((u\\bmod M)+(v\\bmod M))\\bmod M)=((u+v)\\bmod M)$.\n– fgrieu\nFeb 9 '17 at 16:16\n• If you want one with a less obvious backdoor than $b = M-a$, you might want to try the same with modular multiplication; if $M$ is a large number, and $a \\cdot b = 1 \\bmod M$, then $y = (x \\cdot a) \\bmod M$ is \"encryption\", and $x = (y \\cdot b) \\bmod M$ is \"decryption\". One way to select such a 'key pair' is by selecting $a, b$ and setting $M = a \\cdot b - 1$. Of course, this doesn't really work, it's actually fairly easy to find $b$ given $a, M$ (which you need to highlight, lest anyone thing this is actually usable), however it isn't as immediately obvious as $b = M-a$... Feb 9 '17 at 16:31\n• Thanks both of you. I know how much insecure is the modular addition. But my goal is to explain the basic principles, not ensure a \"real\" crypto algorithm. @poncho I am still struggling, could you give the whole process ? Feb 10 '17 at 8:30\n• Whole process: key generation might be 'select $a, b$ at random, compute $M = ab - 1$, publish $M, a$ as your public key, keep $M, b$ as your private key; to encrypt the message $x$, you'd compute $y = ax \\bmod M$; to decrypt the message $y$, you'd compute $x = by \\bmod M$. This works because the result of an encryption followed by decryption is $b(ax \\bmod M) \\bmod M = abx \\bmod M = (M+1)x \\bmod M = x \\bmod M$ (which is $x$ if the original plaintext are $0 \\le x < M$) Feb 10 '17 at 13:47\n• I am struggling with the modular addition (sorry !). Given y = (x+a)%M and x = (y+b)%M, if I replace y I have : x = ((x+a)%M+b)%M and then what ? Feb 10 '17 at 14:06\n\nIf you were to plot the numbers 0-M on a straight horizontal line in increasing order from left to right, your message would become a point on that line. When you apply the key, it moves the message over $a$ spaces to the right. The modulo operator means that if your message goes too far off the right side of the line, then it wraps back around to the left side of the line.\n\nIf you were to apply the key $a$ again, your ciphertext message would move over another $a$ spaces to the right, which, assuming an appropriate modulus, implies that the result will not line up with the original plaintext message.\n\nWith this in mind, if you were move the ciphertext message instead $b$ spaces forward, you would end up back at the original message.\n\nIf you wanted to use $a$ as a symmetric key, you would instead perform modular subtraction on the ciphertext using $a$. Basically, symmetric decryption \"reverses\" the transformation, while in the case of asymmetric encryption, the ciphertext is decrypted by going forward, as opposed to going backwards. This is what enables us to build public key encryption from non-invertible functions - we do not invert the transformation, we design it such that \"going forward\" (in a particular way) happens to \"line up\" and result in the original message.\n\nOther operations can be used to create this asymmetric effect, usually employing a modulus. As mentioned by @poncho in the comments, multiplication is often used as a way to demonstrate \"asymmetric crypto\", but it should be noted neither addition nor multiplication is \"strong enough\", typically: modular exponentiation is what is used in practice to achieve the effect.\n\n-----0-----1-----2-----3-----4-----5-----6-----7-----8-----9-----10-----\n\n\nSetting $a$ at 3:\n\n-----0-----1-----2-----a-----4-----5-----6-----7-----8-----9-----10-----\n\n\nAfter we add $x$ (which is 4) to $a$ (which is 3); Since $a$ has been encrypted, we shall refer to the ciphertext as $b$ (as you did), which now has the value of $7$; The addition moves the value to the right on the number line:\n\n +x\n+-----------------------+\n\| v\n-----0-----1-----2-----a-----4-----5-----6-----b-----8-----9-----10-----\n\n\nIf we were to add $x$ to our ciphertext again, we would end up at $1$ ($11 \\bmod 10 = 1$)\n\n +x\n... -------+ +------------------ ...\nv \|\n-----0-----1-----2-----3-----4-----5-----6-----b-----8-----9-----10-----\n\n\nThis obviously leads to incorrect decryption. To decrypt correctly, we can add the corresponding private key $y=6$ ($x + y = 10$) to make the ciphertext \"go forward\" to the correct plaintext, remembering to wrap around when we get to the end of the number line:\n\n +y\n... -------------------+ +------------------ ...\nv \|\n-----0-----1-----2-----3-----4-----5-----6-----b-----8-----9-----10-----\n\n\nRemark: here $10$ and $0$ are the same number (because $\\mod 10$) !\n\n• This is a very good explanation, thank you very much ! I think I will use your process for my presentation. I am also looking for something \"more mathematical\", meaning : given y = (a+x) mod M, what is the process to find back x with the help of b (what is the process to go from y = (a+x) mod m to x = (y+b) mod m) Feb 10 '17 at 8:36" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87689495,"math_prob":0.9918152,"size":2905,"snap":"2021-43-2021-49","text_gpt3_token_len":699,"char_repetition_ratio":0.1461565,"word_repetition_ratio":0.004494382,"special_character_ratio":0.3624785,"punctuation_ratio":0.105166055,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99940956,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-29T12:05:15Z\",\"WARC-Record-ID\":\"<urn:uuid:33a1aa3f-f621-4b00-a8ea-6d68cb8c6ff4>\",\"Content-Length\":\"148365\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b0bfd8b9-f7b7-4b0d-ab64-43ec887ab002>\",\"WARC-Concurrent-To\":\"<urn:uuid:3aaa8c52-2e51-43aa-9203-bb7f4844be71>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://crypto.stackexchange.com/questions/43748/how-would-you-explain-this-very-simple-add-modulo-asymetric-encryption\",\"WARC-Payload-Digest\":\"sha1:VYEHOXT6ENODM45LXZPBGHD7CHINBKHS\",\"WARC-Block-Digest\":\"sha1:4XNLYMGAGMTCSHBB6FGI45FATST36EXF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358705.61_warc_CC-MAIN-20211129104236-20211129134236-00429.warc.gz\"}"}
https://pyklip.readthedocs.io/en/latest/fm_spect.html	[ "# Spectrum Extraction using extractSpec FM¶\n\nThis document describes with an example how to use KLIP-FM to extract a spectrum, described in Pueyo et al. (2016) to account the effect of the companion signal in the reference library when measuring its spectrum.\n\n## Set up:¶\n\nHere we will just read in the dataset and grab the instrumental PSF. The example code here shows how it is done with GPI, but you will want to refer to the Instrument Tutorials for the instrument you are working. As the code notes, it is important what the units of your instrumental PSF is in, as the code will return the spectrum relative to the input PSF model.\n\nimport glob\nimport numpy as np\nimport pyklip.instruments.GPI as GPI\nimport pyklip.fmlib.extractSpec as es\nimport pyklip.fm as fm\nimport pyklip.fakes as fakes\nimport matplotlib.pyplot as plt\n\nfiles = glob.glob(\"\\path\\to\\dataset\\.fits\")\ndataset = GPI.GPIData(files, highpass=True)\n# Need to specify a model PSF (either via this method, or any other way)\nmodel_psfs = dataset.generate_psf_cube(20)\n# in this case model_psfs has shape (N_lambda, 20, 20)\n# The units of your model PSF are important, the return spectrum will be\n# relative to the input PSF model, see next example\n\n###### Useful values based on dataset ######\nN_frames = len(dataset.input)\nN_cubes = np.size(np.unique(dataset.filenums))\nnl = N_frames // N_cubes\n\n\n### Calibrating stellar flux for GPI example:¶\n\nConverting to contrast units for GPI data is done using the flux of the satellite spots. The GPI dataset object has attribute spot_flux that represent the average peak flux of the four spots. The normalization factor is computed by dividing the spot flux spectrum by the ratio between the stellar flux and the spot flux (stored in spot_ratio) and adjusting for the ratio between the peak and the sum of the spot PSF.\n\nFor any instrument you can scale your model PSF by its respective calibration factors if the model PSF is not already scaled to be the flux of the star. Alternatively, you can choose to skip this step and calibrate your spectrum into astrophysical units as the very end.\n\nGPI Example:\n\n# First set up a PSF model and sums -- this is necessary for GPI because\n# dataset.spot_flux contains peak values of the satellite spots and we\n# have to correct for the full aperture.\nPSF_cube = dataset.psfs\nmodel_psf_sum = np.nansum(PSF_cube, axis=(1,2))\nmodel_psf_peak = np.nanmax(PSF_cube, axis=(1,2))\n# Now divide the sum by the peak for each wavelength slice\naper_over_peak_ratio = model_psf_sum/model_psf_peak\n\n# star-to-spot calibration factor\nband = dataset.prihdrs['APODIZER'].split('_')\nspot_to_star_ratio = dataset.spot_ratio[band]\n\nspot_peak_spectrum = \\\nnp.median(dataset.spot_flux.reshape(len(dataset.spot_flux)//nl, nl), axis=0)\ncalibfactor = aper_over_peak_ratiospot_peak_spectrum / spot_to_star_ratio\n\n# calibrated_PSF_model is the stellar flux in counts for each wavelength\ncalibrated_PSF_model = PSF_cubecalibfactor\n\n\nThis is your model_psf for generating the forward model and will return the spectrum in contrast units relative to the star.\n\n## Computing the forward model and recovering the spectrum with invert_spect_fmodel¶\n\nWe will use the ExtractSpec class to forward model the PSF of the planet and the invert_spect_fm function in pyklip.fmlib.extractSpec to recover the spectrum. invert_spect_fm returns a spectrum in units relative to the input PSF.\n\nThese are the numbers you change:\n\n###### parameters you specify ######\npars = (45, 222) # replace with known separation and pa of companion\nplanet_sep, planet_pa = pars\nnumbasis = [50,] # \"k_klip\", this can be a list of any size.\n# a forward model will be computed for each element.\nnum_k_klip = len(numbasis) # how many k_klips running\nmaxnumbasis = 100 # Max components to be calculated\nmovement = 2.0 # aggressiveness for choosing reference library\nstamp_size = 10.0 # how big of a stamp around the companion in pixels\n# stamp will be stamp_size2 pixels\nnumthreads=4 # number of threads, machine specific\nspectra_template = None # a template spectrum, if you want\n\n\nGenerating the forward model with pyKLIP:\n\n###### The forward model class ######\nfm_class = ExtractSpec(dataset.input.shape,\nnumbasis,\nplanet_sep,\nplanet_pa,\ncalibrated_PSF_model,\nnp.unique(dataset.wvs),\nstamp_size = stamp_size)\n\n###### Now run KLIP! ######\nfm.klip_dataset(dataset, fm_class,\nfileprefix=\"fmspect\",\nannuli=[[planet_sep-stamp_size,planet_sep+stamp_size]],\nsubsections=[[(planet_pa-stamp_size)/180.np.pi,\\\n(planet_pa+stamp_size)/180.np.pi]],\nmovement=movement,\nnumbasis = numbasis,\nmaxnumbasis=maxnumbasis,\nspectrum=spectra_template,\nsave_klipped=True, highpass=True,\noutputdir=\"\\path\\to\\output\")\n\n# Forward model is stored in dataset.fmout, this is how it is organized:\n# the klipped psf\nklipped = dataset.fmout[:,:,-1,:]\n# The rest is the forward model, dimensions:\n# [num_k_klip, N_frames, N_frames, stamp_sizestamp_size]\n# If numbasis is a list, the first dimension will be the size of that list,\n# a forward model calculated at each value of numbasis.\n\n\nNow you can recover the spectrum:\n\n# If you want to scale your spectrum by a calibration factor:\nunits = \"scaled\"\nscaling_factor = my_calibration_factor\n#e.g., for GPI this could be the star-to-spot ratio\n# otherwise, the defaults are:\nunits = \"natural\" # (default) returned relative to input PSF model\nscale_factor=1.0 # (default) not used if units not set to \"scaled\"\n\nexspect, fm_matrix = es.invert_spect_fmodel(dataset.fmout, dataset, units=units,\nscaling_factor=scaling_factor,\nmethod=\"leastsq\")\n# method indicates which matrix inversion method to use, they all tend\n# to yield similar results when things are well-behaved. Here are the options:\n# \"JB\" matrix inversion adds up over all exposures, then inverts\n# \"leastsq\" uses a leastsq solver.\n# \"LP\" inversion adds over frames and one wavelength axis, then inverts\n# (LP is not generally recommended)\n\n\nThe units of the spectrum, FM matrix, and klipped data are all in raw data units in this example. Calibration of instrument and atmospheric transmmission and stellar spectrum can be done via the input PSF model and optionally applying the scaling factor to invert_spect_fmodel. It can also be done after extracting the spectrum.\n\n## Simulating + recovering a simulated source¶\n\nExample:\n\n# PSF model template for each cube observation, copies of the PSF model:\ninputpsfs = np.tile(calibrated_PSF_model, (N_cubes, 1, 1))\nbulk_contrast = 1e-2\nfake_psf = inputpsfsbulk_contrast\nfake_flux = bulk_contrastnp.ones(dataset.wvs.shape)\n#for ll in range(N_cubes):\n# fake_flux[llnl:(ll+1)nl] = exspect[0, :]\npa = planet_pa+180\n\ntmp_dataset = GPI.GPIData(files, highpass=False)\nfakes.inject_planet(tmp_dataset.input, tmp_dataset.centers, fake_psf,\\\ntmp_dataset.wcs, planet_sep, pa)\n\nfm_class = es.ExtractSpec(tmp_dataset.input.shape,\nnumbasis,\nplanet_sep,\npa,\ncalibrated_PSF_model,\nnp.unique(dataset.wvs),\nstamp_size = stamp_size)\n\nfm.klip_dataset(tmp_dataset, fm_class,\nfileprefix=\"fakespect\",\nannuli=[[planet_sep-stamp_size,planet_sep+stamp_size]],\nsubsections=[[(pa-stamp_size)/180.np.pi,\\\n(pa+stamp_size)/180.np.pi]],\nmovement=movement,\nnumbasis = numbasis,\nmaxnumbasis=maxnumbasis,\nspectrum=spectra_template,\nsave_klipped=True, highpass=True,\noutputdir=\"demo_output/\")\n\nfake_spect, fakefm = es.invert_spect_fmodel(tmp_dataset.fmout, tmp_dataset,\nmethod=\"leastsq\", units=\"scaled\", scaling_factor=2.0)\n\n\n## Comparing the klipped data to the FM¶\n\nYou may want to look at how well your forward model represents the klipped data, measure residual error, etc. All the information you need is in the output of invert_spect_fmodel: the spectrum and FM matrix.\n\nRecall the klipped data is in fmout:\n\nklipped_data = tmp_dataset.fmout[:,:,-1, :]\nklipped_coadd = np.zeros((num_k_klip, nl, stamp_sizestamp_size))\nfor ll in range(N_cubes):\nklipped_coadd = klipped_coadd + klipped_data[0, llnl:(ll+1)nl, :]\n# turn it back into a 2D arrat at each wavelength, k_klip\nklipped_coadd.shape = [nl, int(stamp_size), int(stamp_size)]\n# summed over each wavelength channel, but you can view them individually\nplt.colorbar()\n\n\nPlot the forward model by taking the dot product with the extracted spectrum:\n\nk=0 # choose which numbasis\nfm_image_k = np.dot(fakefm[k,:,:], fake_spect[k].transpose())\n# reshape the image back to 2D\nfm_image_k = fm_image_k.reshape(nl, stamp_size, stamp_size)\n# summed over each wavelength channel\nplt.imshow(fm_image_k.sum(axis=0), interpolation=\"nearest\")\nplt.colorbar()\n\n\n## Calculating Errobars¶\n\nOne may want to calculate errorbars by injecting signals at an annulus of same separation as the real signal and measuring the spread of the recovered spectra (loop through the procedure above):\n\ndef recover_fake(files, position, fake_flux):\n# We will need to create a new dataset each time.\n\n# PSF model template for each cube observation, copies of the PSF model:\ninputpsfs = np.tile(calibrated_PSF_model, (N_cubes, 1, 1))\nbulk_contrast = 1e-2\nfake_psf = inputpsfsfake_flux[0,None,None]\npa = planet_pa+180\n\ntmp_dataset = GPI.GPIData(files, highpass=False)\nfakes.inject_planet(tmp_dataset.input, tmp_dataset.centers, fake_psf,\\\ntmp_dataset.wcs, planet_sep, pa)\n\nfm_class = es.ExtractSpec(tmp_dataset.input.shape,\nnumbasis,\nplanet_sep,\npa,\ncalibrated_PSF_model,\nnp.unique(dataset.wvs),\nstamp_size = stamp_size)\n\nfm.klip_dataset(tmp_dataset, fm_class,\nfileprefix=\"fakespect\",\nannuli=[[planet_sep-stamp_size,planet_sep+stamp_size]],\nsubsections=[[(pa-stamp_size)/180.np.pi,\\\n(pa+stamp_size)/180.np.pi]],\nmovement=movement,\nnumbasis = numbasis,\nmaxnumbasis=maxnumbasis,\nspectrum=spectra_template,\nsave_klipped=True, highpass=True,\noutputdir=\"demo_output/\")\nfake_spect, fakefm = es.invert_spect_fmodel(tmp_dataset.fmout,\ntmp_dataset, method=\"leastsq\",\nunits=\"scaled\", scaling_factor=2.0)\ndel tmp_dataset\nreturn fake_spect\n\n# This could take a long time to run\n# Define a set of PAs to put in fake sources\nnpas = 11\npas = (np.linspace(planet_pa, planet_pa+360, num=npas+2)%360)[1:-1]\n\n# For numbasis \"k\"\n# repeat the spectrum over each cube in the dataset\ninput_spect = np.tile(exspect[k,:], N_cubes)[0,:]\nfake_spectra = np.zeros((npas, nl))\nfor p, pa in enumerate(pas):\nfake_spectra[p,:] = recover_fake(files, (planet_sep, pa), input_spect)\n\n\nOther details, like the forward model or klipped data for the injected signal could be useful.\n\nIf the real companion signal is too bright, the forward model may fail to capture all the flux It could be helpful to look at whether the recovered spectra for the simulated signal are evenly distributed around the simulated spectrum or if they are systematically lower flux:\n\noffset[ii] = estim_spec[ii] - np.median(fake_spectra, axis=0)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6664645,"math_prob":0.9671161,"size":10747,"snap":"2022-40-2023-06","text_gpt3_token_len":2749,"char_repetition_ratio":0.14372149,"word_repetition_ratio":0.09913793,"special_character_ratio":0.2446264,"punctuation_ratio":0.18109821,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9924466,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-06T11:29:44Z\",\"WARC-Record-ID\":\"<urn:uuid:3db4eef9-09db-474a-90d7-ede4c4924e83>\",\"Content-Length\":\"52758\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4b199ed9-ddc0-4846-a983-e6b320f3b43a>\",\"WARC-Concurrent-To\":\"<urn:uuid:5c1a0284-9400-40e4-bff3-2a738d2c619f>\",\"WARC-IP-Address\":\"104.17.32.82\",\"WARC-Target-URI\":\"https://pyklip.readthedocs.io/en/latest/fm_spect.html\",\"WARC-Payload-Digest\":\"sha1:OXOXJRHPSTW2CWHSA2TI7I5FYTZUR3FE\",\"WARC-Block-Digest\":\"sha1:N2YVBGZHTZ5ONJ63XAALSSAXDXZA7DR7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337803.86_warc_CC-MAIN-20221006092601-20221006122601-00562.warc.gz\"}"}
https://www.jpost.com/israel/would-be-female-suicide-bomber-arrested-in-nablus	[ "(function (a, d, o, r, i, c, u, p, w, m) { m = d.getElementsByTagName(o), a[c] = a[c] \|\| {}, a[c].trigger = a[c].trigger \|\| function () { (a[c].trigger.arg = a[c].trigger.arg \|\| []).push(arguments)}, a[c].on = a[c].on \|\| function () {(a[c].on.arg = a[c].on.arg \|\| []).push(arguments)}, a[c].off = a[c].off \|\| function () {(a[c].off.arg = a[c].off.arg \|\| []).push(arguments) }, w = d.createElement(o), w.id = i, w.src = r, w.async = 1, w.setAttribute(p, u), m.parentNode.insertBefore(w, m), w = null} )(window, document, \"script\", \"https://95662602.adoric-om.com/adoric.js\", \"Adoric_Script\", \"adoric\",\"9cc40a7455aa779b8031bd738f77ccf1\", \"data-key\");\nvar domain=window.location.hostname; var params_totm = \"\"; (new URLSearchParams(window.location.search)).forEach(function(value, key) {if (key.startsWith('totm')) { params_totm = params_totm +\"&\"+key.replace('totm','')+\"=\"+value}}); var rand=Math.floor(10*Math.random()); var script=document.createElement(\"script\"); script.src=`https://stag-core.tfla.xyz/pre_onetag?pub_id=34&domain=\\${domain}&rand=\\${rand}&min_ugl=0\\${params_totm}`; document.head.append(script);" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9625428,"math_prob":0.96919554,"size":465,"snap":"2023-40-2023-50","text_gpt3_token_len":96,"char_repetition_ratio":0.0867679,"word_repetition_ratio":0.0,"special_character_ratio":0.18924731,"punctuation_ratio":0.047058824,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9789482,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-03T14:20:35Z\",\"WARC-Record-ID\":\"<urn:uuid:437e26e7-571d-45ee-ab04-de03b38679ee>\",\"Content-Length\":\"78290\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:288c6d17-63a5-4fa2-b9ac-8d011a001af8>\",\"WARC-Concurrent-To\":\"<urn:uuid:aa22dba9-a411-41f0-b48e-5a500acbfdba>\",\"WARC-IP-Address\":\"159.60.130.79\",\"WARC-Target-URI\":\"https://www.jpost.com/israel/would-be-female-suicide-bomber-arrested-in-nablus\",\"WARC-Payload-Digest\":\"sha1:57JKNZ7OYHM7KNXVL4XWHDAUVUZHLHIM\",\"WARC-Block-Digest\":\"sha1:H5O56RGT7JA3KCAKAZYWNZEGD3LQJUTG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511106.1_warc_CC-MAIN-20231003124522-20231003154522-00000.warc.gz\"}"}
https://staging.physicsclassroom.com/class/vectors/Lesson-2/Initial-Velocity-Components	[ "Vectors - Motion and Forces in Two Dimensions - Lesson 2 - Projectile Motion\n\n# Initial Velocity Components\n\nIt has already been stated and thoroughly discussed that the horizontal and vertical motions of a projectile are independent of each other. The horizontal velocity of a projectile does not affect how far (or how fast) a projectile falls vertically. Perpendicular components of motion are independent of each other. Thus, an analysis of the motion of a projectile demands that the two components of motion are analyzed independent of each other, being careful not to mix horizontal motion information with vertical motion information. That is, if analyzing the motion to determine the vertical displacement, one would use kinematic equations with vertical motion parameters (initial vertical velocity, final vertical velocity, vertical acceleration) and not horizontal motion parameters (initial horizontal velocity, final horizontal velocity, horizontal acceleration). It is for this reason that one of the initial steps of a projectile motion problem is to determine the components of the initial velocity.\n\n### Determining the Components of a Velocity Vector\n\nEarlier in this unit, the method of vector resolution was discussed. Vector resolution is the method of taking a single vector at an angle and separating it into two perpendicular parts. The two parts of a vector are known as components and describe the influence of that vector in a single direction. If a projectile is launched at an angle to the horizontal, then the initial velocity of the projectile has both a horizontal and a vertical component. The horizontal velocity component (vx) describes the influence of the velocity in displacing the projectile horizontally. The vertical velocity component (vy) describes the influence of the velocity in displacing the projectile vertically. Thus, the analysis of projectile motion problems begins by using the trigonometric methods discussed earlier to determine the horizontal and vertical components of the initial velocity.", null, "Consider a projectile launched with an initial velocity of 50 m/s at an angle of 60 degrees above the horizontal. Such a projectile begins its motion with a horizontal velocity of 25 m/s and a vertical velocity of 43 m/s. These are known as the horizontal and vertical components of the initial velocity. These numerical values were determined by constructing a sketch of the velocity vector with the given direction and then using trigonometric functions to determine the sides of the velocity triangle. The sketch is shown at the right and the use of trigonometric functions to determine the magnitudes is shown below. (If necessary, review this method on an earlier page in this unit.)", null, "", null, "All vector resolution problems can be solved in a similar manner. As a test of your understanding, utilize trigonometric functions to determine the horizontal and vertical components of the following initial velocity values. When finished, click the button to check your answers.\n\nPractice A: A water balloon is launched with a speed of 40 m/s at an angle of 60 degrees to the horizontal.\n\nPractice B: A motorcycle stunt person traveling 70 mi/hr jumps off a ramp at an angle of 35 degrees to the horizontal.\n\nPractice C: A springboard diver jumps with a velocity of 10 m/s at an angle of 80 degrees to the horizontal.\n\n### Try Some More!\n\nNeed more practice? Use the Velocity Components for a Projectile widget below to try some additional problems. Enter any velocity magnitude and angle with the horizontal. Use your calculator to determine the values of vx and vy. Then click the Submit button to check your answers.\n\nAs mentioned above, the point of resolving an initial velocity vector into its two components is to use the values of these two components to analyze a projectile's motion and determine such parameters as the horizontal displacement, the vertical displacement, the final vertical velocity, the time to reach the peak of the trajectory, the time to fall to the ground, etc. This process is demonstrated on the remainder of this page. We will begin with the determination of the time.\n\n### Determination of the Time of Flight", null, "The time for a projectile to rise vertically to its peak (as well as the time to fall from the peak) is dependent upon vertical motion parameters. The process of rising vertically to the peak of a trajectory is a vertical motion and is thus dependent upon the initial vertical velocity and the vertical acceleration (g = 9.8 m/s/s, down). The process of determining the time to rise to the peak is an easy process - provided that you have a solid grasp of the concept of acceleration. When first introduced, it was said that acceleration is the rate at which the velocity of an object changes. An acceleration value indicates the amount of velocity change in a given interval of time. To say that a projectile has a vertical acceleration of -9.8 m/s/s is to say that the vertical velocity changes by 9.8 m/s (in the - or downward direction) each second. For example, if a projectile is moving upwards with a velocity of 39.2 m/s at 0 seconds, then its velocity will be 29.4 m/s after 1 second, 19.6 m/s after 2 seconds, 9.8 m/s after 3 seconds, and 0 m/s after 4 seconds. For such a projectile with an initial vertical velocity of 39.2 m/s, it would take 4 seconds for it to reach the peak where its vertical velocity is 0 m/s. With this notion in mind, it is evident that the time for a projectile to rise to its peak is a matter of dividing the vertical component of the initial velocity (viy) by the acceleration of gravity.", null, "Once the time to rise to the peak of the trajectory is known, the total time of flight can be determined. For a projectile that lands at the same height which it started, the total time of flight is twice the time to rise to the peak. Recall from the last section of Lesson 2 that the trajectory of a projectile is symmetrical about the peak. That is, if it takes 4 seconds to rise to the peak, then it will take 4 seconds to fall from the peak; the total time of flight is 8 seconds. The time of flight of a projectile is twice the time to rise to the peak.", null, "### Determination of Horizontal Displacement\n\nThe horizontal displacement of a projectile is dependent upon the horizontal component of the initial velocity. As discussed in the previous part of this lesson, the horizontal displacement of a projectile can be determined using the equation\n\nx = vix • t\n\nIf a projectile has a time of flight of 8 seconds and a horizontal velocity of 20 m/s, then the horizontal displacement is 160 meters (20 m/s • 8 s). If a projectile has a time of flight of 8 seconds and a horizontal velocity of 34 m/s, then the projectile has a horizontal displacement of 272 meters (34 m/s • 8 s). The horizontal displacement is dependent upon the only horizontal parameter that exists for projectiles - the horizontal velocity (vix).\n\n### Determination of the Peak Height\n\nA non-horizontally launched projectile with an initial vertical velocity of 39.2 m/s will reach its peak in 4 seconds. The process of rising to the peak is a vertical motion and is again dependent upon vertical motion parameters (the initial vertical velocity and the vertical acceleration). The height of the projectile at this peak position can be determined using the equation\n\ny = viy • t + 0.5 • g • t2\n\nwhere viy is the initial vertical velocity in m/s, g is the acceleration of gravity (-9.8 m/s/s) and t is the time in seconds it takes to reach the peak. This equation can be successfully used to determine the vertical displacement of the projectile through the first half of its trajectory (i.e., peak height) provided that the algebra is properly performed and the proper values are substituted for the given variables. Special attention should be given to the facts that the t in the equation is the time up to the peak and the g has a negative value of -9.8 m/s/s.\n\n### We Would Like to Suggest ...", null, "Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of our Projectile Motion Simulator. You can find it in the Physics Interactives section of our website. The simulator allows one to explore projectile motion concepts in an interactive manner. Change a height, change an angle, change a speed, and launch the projectile.\n\nAnswer the following questions and click the button to see the answers.\n\n1. Aaron Agin is resolving velocity vectors into horizontal and vertical components. For each case, evaluate whether Aaron's diagrams are correct or incorrect. If incorrect, explain the problem or make the correction.", null, "2. Use trigonometric functions to resolve the following velocity vectors into horizontal and vertical components. Then utilize kinematic equations to calculate the other motion parameters. Be careful with the equations; be guided by the principle that \"perpendicular components of motion are independent of each other.\"", null, "See Answer", null, "A: vix = 9.5 m/s • cos(40 degrees) = 7.28 m/s B : viy = 9.5 m/s • sin(40 degrees) = 6.11 m/s C: tup = (6.11 m/s) / (9.8 m/s/s) = 0.623 s D: ttotal = 2 • (0.623 s) = 1.25 s E: x = 7.28 m/s • 1.25 s = 9.07 m F: y = (6.11 m/s) • (0.623 s) + 0.5*(-9.8 m/s/s) • (0.623 s )2 = 1.90 m See Answer", null, "G: vix = 25 m/s • cos (60 degrees) = 12.5 m/s H: viy = 25 m/s • sin (60 degrees) = 21.7 m/s I: tup = (21.7 m/s) / (9.8 m/s/s) = 2.21 s J: ttotal = 2 • 2.21 s = 4.42 s K: x = 12.5 m/s • 4.42 s = 55.2 m L: y = 21.7 m/s • 2.21 s + 0.5 • (-9.8 m/s/s) • (2.21 s )2 = 23.9 m See Answer", null, "M: vix = 30 m/s • cos (30 degrees) = 26.0 m/s N: viy = 30 m/s • sin (30 degrees) = 15.0 m/s O: tup = (15.0 m/s) / (9.8 m/s/s) = 1.53 s P: ttotal = 2 • 1.53 s = 3.06 s Q: x = 26.0 m/s • 3.06 s = 79.5 m R: y = 15.0 m/s • 1.53 s + 0.5 • (-9.8 m/s/s) • (1.53 s )2 = 11.5 m\n\n3. Utilize kinematic equations and projectile motion concepts to fill in the blanks in the following tables.", null, "Next Section:" ]	[ null, "http://www.physicsclassroom.com/Class/vectors/u3l2d2.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d3.gif", null, "http://www.physicsclassroom.com/Class/images/qwikquiz.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d4.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d5.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d7.gif", null, "https://staging.physicsclassroom.com/PhysicsClassroom/media/Images/archive/Class/images/Interact.png", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d6.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d8.gif", null, "http://www.physicsclassroom.com/Class/images/cyuheader.gif", null, "http://www.physicsclassroom.com/Class/images/cyuheader.gif", null, "http://www.physicsclassroom.com/Class/images/cyuheader.gif", null, "http://www.physicsclassroom.com/Class/vectors/u3l2d9.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.888141,"math_prob":0.97722274,"size":9091,"snap":"2019-51-2020-05","text_gpt3_token_len":1836,"char_repetition_ratio":0.2030373,"word_repetition_ratio":0.08420365,"special_character_ratio":0.20272797,"punctuation_ratio":0.08215962,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998529,"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],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-23T23:52:36Z\",\"WARC-Record-ID\":\"<urn:uuid:707d9a32-7b5a-4dfd-bec9-d13da250f9c1>\",\"Content-Length\":\"105220\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:56c7d00b-14f6-4eee-a423-922f7609316e>\",\"WARC-Concurrent-To\":\"<urn:uuid:865784b1-a282-4673-ba02-e48250ffa001>\",\"WARC-IP-Address\":\"64.25.122.121\",\"WARC-Target-URI\":\"https://staging.physicsclassroom.com/class/vectors/Lesson-2/Initial-Velocity-Components\",\"WARC-Payload-Digest\":\"sha1:TOORASRGXV2R5VQIM3K6S25Q64R62RLD\",\"WARC-Block-Digest\":\"sha1:VFRHOQEBE64XGVYEHIVPDLJUK6I37IUU\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250614086.44_warc_CC-MAIN-20200123221108-20200124010108-00094.warc.gz\"}"}
https://mesak.tw/code/javascript/11932/js-table-convert-to-object-part1	[ "# [JS] Table 轉 Object 練習 part1\n\n``````<table border=1>\n<tr>\n<th>Company</th>\n<th>Contact</th>\n<th>Country</th>\n</tr>\n<tbody>\n<tr>\n<td>Alfreds Futterkiste</td>\n<td>Maria Anders</td>\n<td>Germany</td>\n</tr>\n<tr>\n<td>Centro comercial Moctezuma</td>\n<td>Francisco Chang</td>\n<td>Mexico</td>\n</tr>\n</tbody>\n</table>``````\n\n``````let heads = [];\n})``````\n\n``````let tableObject = []\n\\$('tbody tr').each( function(i,trNode) {\nlet data = {}\n\\$(trNode).find('td').each( function(index, tdNode) {\n})\ntableObject.push(data)\n})\nconsole.log( tableObject );``````\n\n``````let headData = Array.from(document.querySelectorAll('thead > tr > th')).map(n=>n.innerText)\n\nlet tableObject = []\nfor(let trNode of document.querySelectorAll('tbody > tr') ){\nlet data = {};\nArray.from(trNode.querySelectorAll('td')).forEach((n , index)=>{\n})\ntableObject.push(data)\n}\nconsole.log( 'tableObject ',tableObject )\n``````\n\n0 comment" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8552624,"math_prob":0.485695,"size":2024,"snap":"2022-40-2023-06","text_gpt3_token_len":1322,"char_repetition_ratio":0.10891089,"word_repetition_ratio":0.0,"special_character_ratio":0.23666008,"punctuation_ratio":0.113793105,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9843928,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-30T11:45:25Z\",\"WARC-Record-ID\":\"<urn:uuid:dc60d632-e879-46e5-86da-713a1cb2ba46>\",\"Content-Length\":\"112068\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ee01038-85b3-46ea-ba65-a11401d33c9a>\",\"WARC-Concurrent-To\":\"<urn:uuid:179a942d-82fa-4565-8ae6-d233fa9808da>\",\"WARC-IP-Address\":\"104.21.0.229\",\"WARC-Target-URI\":\"https://mesak.tw/code/javascript/11932/js-table-convert-to-object-part1\",\"WARC-Payload-Digest\":\"sha1:L3QKMSKK2NCPTJIRTJ7IADAXEAF6MEBT\",\"WARC-Block-Digest\":\"sha1:57O5GVHHO7BYTK7YMW5XVVC4BTHSP266\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499816.79_warc_CC-MAIN-20230130101912-20230130131912-00671.warc.gz\"}"}
https://crypto.stackexchange.com/questions/8564/hmac-and-assumptions-on-the-cryptographic-hash	[ "# HMAC and assumptions on the cryptographic hash\n\nAccording to Wikipedia, a cryptographic hash function has the following properties:\n\n1. Pre-image resistance: Given $h$, it's difficult to find any message $m$ such that $h = H(m)$.\n2. Second pre-image resistance: Given $m_1$, it's difficult to find another $m_2$ such that $m_1 \\ne m_2$ and $H(m_1) = H(m_2)$\n3. Collision Resistance: It's difficult to find two distinct messages $m_1$ and $m_2$ such that $H(m_1) = H(m_2)$\n\nAssuming $H$ is a hash function, the following function $H'$ should — to my understanding — also be a hash function:\n\n$$H'(m) = m_0 \|\| H(m_1\|\|m_2\|\|\\dots\|\|m_k)$$\n\nwhere $m_i$ is the $i$th byte of $m$.\n\n$H'$ leaks the first byte of $m$, but even just leaking one byte, you still can't find a pre-image or any collisions.\n\nWhen looking at HMAC, Wikipedia says that it takes a hash function (without maker further assumptions on the hash function). Taking my hash function $H'$ (just let it leak $\\operatorname{len}(key)$ bytes, but not the long message) for HMAC, HMAC would be insecure, since everybody now sees the key.\n\nSo maybe my $H'$ is not a cryptographic hash function after all — in which case my question is: Why not?\n\nOr $H'$ is a cryptographic hash function, but building an HMAC requires additional assumptions. What additional assumptions must be fulfilled by the hash function to be secure in HMAC? Do the three properties above imply some other properties I don't see?\n\nAlso PBKDF2 takes a PRF, where HMAC-SHA256 should be secure, but HMAC just given a hash function which has the three properties, won't be a PRF to my understanding. Again the same questions: Are there more assumptions? Even more than on HMAC?\n\n## migrated from security.stackexchange.comJun 4 '13 at 12:53\n\nThis question came from our site for information security professionals.\n\n• > In order for a hash function with n bits of output to be collision resistant, it must take at least $2^n$ work and storage to find a collision Don't you just need $2^{\\frac{n}{2}}$ due the birthday attack to find a collision? – So Morr Jun 4 '13 at 15:02\n• In order for a hash function with $n$ bits of output to be collision resistant, it must take at least $2^{n/2}$ work and storage to find a collision. Presuming your hash function $H$ is collision resistant, it is not obvious $H′$ also is collision resistant. Finding two different messages that both share the same prefix, and collide in the last $n−8$ bits, requires less work and less storage than finding a collision in all $n$ bits of $H$. – Henrick Hellström Jun 4 '13 at 15:44\n\nBefore we jump into this question, you first need to know a bit about the internals of hash functions with the Merkle-Dåmgard construction. Here's a pretty picture from Wikipedia:", null, "In this diagram, you see the compression function $f$ being fed the message blocks along with the output of the state of the previous compression block (or the IV). The final output is the result of the last compression function. (You can ignore the finalization step for our purposes.)\n\nNow, let's focus on the original NMAC/HMAC paper. In it, the authors state:\n\nIn the rest of this paper we will concentrate on iterated hash functions, except if stated otherwise.\n\nThis should be the first clue that your scheme is not one that will work with NMAC/HMAC: it's not iterated! Not all of it, at least. The fact that the first $n$ bytes are concatenated (leaked, what have you) means that your hash function's output is no longer solely the result from the compression function evaluated on the last block. This changes the construction of the scheme drastically.\n\nIn regular circumstances, the (almost implicit) assumption that the underlying hash function is iterated is not an unreasonable one at all: all of the popular hash functions of today are. (SHA3/Keccak is a bit of a special case. It's not clear one even needs the HMAC construction for it. But that's a topic for another question.)\n\nFor example, what do you do with the IV (initialization vector, or as Bellare et al call it, \"initial variable\")? Do you simply pass it along to the $H$ in $H'$? If so, then your scheme doesn't actually leak the key with NMAC, although it does leak $k \\oplus \\mathtt{opad}$ in HMAC. In case you're unfamiliar with NMAC, the basic idea of the scheme is replace the IVs of the regular hash functions with the keys $k = (k_1, k_2)$. In the case of HMAC, the \"new IVs\" are (where $f$ is the compression function for the hash in question) $k_1 = f(k \\oplus \\mathtt{opad})$ and $k_2 = f(k \\oplus \\mathtt{ipad})$. But note that this, too, carries with it the implicit assumption that the starting state of the next $f$ in the chain is the previous evaluation of $f$.\n\nTrying to discuss $H'$ in the context of HMAC is difficult, though. The primary issue is that your $H'$ doesn't have a clearly available compression function. Sure, $H$ (probably) has a compression function, but for $H'$, things are much less certain. Even if you attempted to define a compression function for $H'$, in order to be an iterated hash function, it would somehow have to leak the first $n$ bytes of the original message while simultaneously evaluating $H$ for the rest of the message.\n\nHere is the problem, though: even if you created such a compression function, it would be insecure, naturally. Namely, it no longer would act as a pseudorandom function (PRF) (or for Wikipedia's less thorough, but perhaps easier, explanation, see here). In this relatively new paper, Bellare proves that HMAC is a PRF itself if the compression function of the underlying hash is a PRF. If $H'$ had a compression function, then it definitely would not be a PRF, since it (quite literally) leaks part of the input.\n\nThe prerequisite that the compression function is a PRF is quite a weak requirement, too. Given that $H'$ (even if you did somehow come up with a compression function for it) simply fails this requirement, the security proofs for HMAC do not cover it. Further, pretty much all of the security proofs given in favor of HMAC assume that the attacker doesn't know the key. Those proofs are possibly invalid if this assumption is invalid.\n\nSo, to answer your question directly, HMAC requires an iterated hash scheme whose compression function is, as best as we can tell, a PRF. A generic cryptographic hash function, at least using the definition Wikipedia gives, is not strong enough to guarantee a solid MAC. But the PRF requirement is relatively weak, as even MD5 (which is completely broken as far as collisions go) still appears to satisfy it.\n\nInspired by Henrick Hellström's comment, I think you need to dig a bit deeper into what “difficult” means.\n\nPre-image resistance means that given $h$, it's difficult to find $m$ such that $h = H(m)$. Intuitively speaking, your only chance is to have started with an $h$ that is in the relatively small set of already-computed hashes.\n\nNow suppose you have $h'$ and are looking for $m'$ such that $h' = H'(m)$. You have a better chance of finding $m'$ than you had of finding $m$, because if you had previously computed $H'(a\|\|m'') = a\|\|H(m'')$ such that $h' = b\|\|H(m'')$, you can take $m' = b\|\|m''$. With an alphabet of size $n$, $H'$ gives you $n$ times better chances to find a pre-image.\n\nThis is perhaps more visible if you think of $H'$ consisting of, say, the first 128 bits of the message followed by a 128-bit hash of the rest. Then $H'$ hashes are 256 bits long but the preimage resistance of $H'$ is no more than that of a 128-bit hash, only half as strong as expected.\n\nA natural question at this point is, what if you define $H''(m) = m_0 \|\| H(m)$? (That is, calculate the same hash, but leak a fixed-size prefix of the message.) You do not get a better chance of finding a pre-computed pre-image. However, the amount of work required to find a pre-image by brute-force is clearly less, because you can concentrate your efforts on messages with the prefix $m_0$. This is actually closer to the mathematical definition of difficulty (computational complexity of finding a pre-image) than the informal explanation above.\n\nI'm not fully satisfied with this explanation. What about $H^\\circ(m) = \\mathbf{0} \|\| H(m)$ where $\\mathbf{0}$ is some constant string? This is obviously as good a hash as $H$ since it doesn't leak anything. Yet the amount of work required to find a pre-image is only as good as $H$, despite the longer hash, which is no different from the complaint against $H'$ above.\n\nIn any case, as far as I know, the Wikipedia article is a simplification: there is no general result that any hash function can be used to build a HMAC in this way. The security proofs of HMAC only apply to hash functions of a certain form, which includes all Merkle-Damgård hash functions, but not the oddball variants considered in this thread.\n\n• The last part of this answer is the crucial bit: HMAC's security proof relies iterated hash functions, specifically MD constructions. The construction in the question fails this criterion. – Reid Jun 4 '13 at 17:07\n• For modern criteria for HMAC security, I recommend New Proofs for NMAC and HMAC: Security without Collision-Resistance, although it is hard reading and some of it goes over my head. – fgrieu Jun 4 '13 at 17:19\n• @Reid I don't find this fully satisfactory: sure, the proof doesn't apply, but what causes the result not to hold? – Gilles 'SO- stop being evil' Jun 4 '13 at 18:20\n• @Gilles: The structure of the above construction is entirely different from usual hash schemes. How would you define the compression function for $H'$? And note that in order for NMAC's security proof to be relevant, the scheme needs to be iterated, so you somehow have to define the compression function in such a way that the final state will include the first $n$ bytes of the message. Of course, the compression function needs to be a PRF, but leaking even the first byte of the message sounds like a death knell for that idea. – Reid Jun 4 '13 at 19:55\n• @Reid Please write an answer that expands on this! I'd be very interested. – Gilles 'SO- stop being evil' Jun 4 '13 at 20:01" ]	[ null, "https://i.stack.imgur.com/KCXnf.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9342405,"math_prob":0.977967,"size":3984,"snap":"2019-43-2019-47","text_gpt3_token_len":920,"char_repetition_ratio":0.15728644,"word_repetition_ratio":0.0,"special_character_ratio":0.22389558,"punctuation_ratio":0.10824742,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969012,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-17T12:37:50Z\",\"WARC-Record-ID\":\"<urn:uuid:666b10dc-e990-4a4d-897d-5dece7f221a4>\",\"Content-Length\":\"158029\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a74dfdcf-6e1c-4b47-beb4-8a12b88fb22f>\",\"WARC-Concurrent-To\":\"<urn:uuid:ebf66f68-bcf9-40c0-b5a1-8fd62b285cbf>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://crypto.stackexchange.com/questions/8564/hmac-and-assumptions-on-the-cryptographic-hash\",\"WARC-Payload-Digest\":\"sha1:A5H4LDPY4VRLCTOVYXX76FU3YPZOSVOO\",\"WARC-Block-Digest\":\"sha1:V6YHCCNLIWKZOOXLFGWPE52ES3M73FHC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496668954.85_warc_CC-MAIN-20191117115233-20191117143233-00401.warc.gz\"}"}
https://crypto.stackexchange.com/questions/64783/inversion-in-gf28-aes	[ "# Inversion in $GF(2^8)$ AES [duplicate]\n\nI was wondering how you would calculate the S-box in AES. I found that you have to calculate the inverse of the polynomials in $$GF(2^8)$$. I found out that to calculate the inverse, you have to use the Extended Euclidean Algorithm. What I can't figure out is how do you apply this to a polynomial?\n\n• This question has a detailed answer here: https://crypto.stackexchange.com/questions/34212/how-to-use-the-extended-euclidean-algorithm-to-invert-a-finite-field-element – kodlu Dec 11 '18 at 19:20\n• Your question is a duplicate of the one in the comment above. – kodlu Dec 11 '18 at 19:21\n• \"... you have to use the Extended Euclidean Algorithm\"; nope, there are a bunch of other ways to compute inverses. For one, you can take advantage that, in GF(256), $x^{-1} = x^{254}$. For another, to find the inverse of $x$, we just search for a $y$ with $x \\times y = 1$; this takes no more than 64k multiplications to do the full search for all possible inputs, which is trivial as a one time effort... – poncho Dec 11 '18 at 19:53\n\nOne neat trick for the AES Sbox is to use logarithm. In that specific field, the value 3 (0x03) is a multiplicative generator for the non-zero elements. Thus, if you have a \"multiplication by 0x03\" function, you can compute a mapping $$n \\mapsto \\mathrm{\\texttt{03}}^n$$ for all $$0 \\le n \\le 254$$; this would be an \"exponentiation table\". From that table, you can make the inverse mapping as another table, which would be a \"logarithm tables\". By going to logarithms, you transform multiplications into additions modulo 255, and divisions into subtractions modulo 255. In particular, inversion becomes negation.\n\nIn Java code, this may look like this:\n\n /\n Make tmp[n] = pow(0x03, n) in the field.\n/\nint[] tmp = new int;\nfor (int i = 0, x = 1; i < 255; i ++) {\ntmp[i] = x;\n\n/\n* Multiply x by 0x03 in the field.\n/\nint x2 = x << 1;\nx2 ^= -(x2 >> 8) & 0x11B;\nx ^= x2;\n}\n\n/\n* Make Sbox[] as the inverse table. We use the fact that\n* 1/pow(0x03,n) = pow(0x03,255-n) in the field (since 0x03 is\n* a generator of the multiplicative subgroup of size 255).\n/\nSbox = new int;\nSbox = 0; Sbox = 1;\nfor (int i = 1; i < 255; i ++) {\nSbox[tmp[i]] = tmp[255 - i];\n}\n\n/\n* At that point, SBox[i] contains the inverse of i.\n* Now we apply the affine transform in GF(2).\n*/\nfor (int i = 0; i < 256; i ++) {\nint x = Sbox[i];\nx \|= x << 8;\nx ^= (x >> 4) ^ (x >> 5) ^ (x >> 6) ^ (x >> 7);\nSbox[i] = (x ^ 0x63) & 0xFF;\n}\n\n\nThis kind of code can be convenient to recompute the S-box and related tables during an initialization process (assuming that you prefer the extra startup cost over the size cost of embedding the tables in the code).\n\n(Of course, table-based AES implementations are not constant-time, you should not do that in practice, etc.)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8531885,"math_prob":0.9928688,"size":2842,"snap":"2020-34-2020-40","text_gpt3_token_len":860,"char_repetition_ratio":0.10394644,"word_repetition_ratio":0.03816794,"special_character_ratio":0.34517944,"punctuation_ratio":0.13804714,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994321,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-07T03:48:59Z\",\"WARC-Record-ID\":\"<urn:uuid:df15a480-0d59-47ea-8ecc-2a3a3d07e7ca>\",\"Content-Length\":\"135601\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:befb3b3d-13ab-441f-a340-63911b5c8da8>\",\"WARC-Concurrent-To\":\"<urn:uuid:37f2cda0-0a27-4002-a749-29e2688a11fb>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://crypto.stackexchange.com/questions/64783/inversion-in-gf28-aes\",\"WARC-Payload-Digest\":\"sha1:YHAKKZO4JEUHSIDQI3GA7QYCVORURORA\",\"WARC-Block-Digest\":\"sha1:6AJBWZWDEH55AJIZRCJWAWDFIXY4GNHW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737152.0_warc_CC-MAIN-20200807025719-20200807055719-00202.warc.gz\"}"}
https://patents.google.com/patent/US7788051B2/en	[ "# US7788051B2 - Method and apparatus for parallel loadflow computation for electrical power system - Google Patents\n\nMethod and apparatus for parallel loadflow computation for electrical power system Download PDF\n\n## Info\n\nPublication number\nUS7788051B2\nUS7788051B2 US10/594,715 US59471505A US7788051B2 US 7788051 B2 US7788051 B2 US 7788051B2 US 59471505 A US59471505 A US 59471505A US 7788051 B2 US7788051 B2 US 7788051B2\nAuthority\nUS\nUnited States\nPrior art keywords\nnetwork\nnode\npower\nnodes\npq\nPrior art date\nLegal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)\nActive\nApplication number\nUS10/594,715\nOther versions\nUS20070203658A1 (en\nInventor\nSureshchandra B. Patel\nOriginal Assignee\nPatel Sureshchandra B\nPriority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)\nFiling date\nPublication date\nPriority to CA2479603 priority Critical\nPriority to CA002479603A priority patent/CA2479603A1/en\nApplication filed by Patel Sureshchandra B filed Critical Patel Sureshchandra B\nPriority to PCT/CA2005/001537 priority patent/WO2006037231A1/en\nPublication of US20070203658A1 publication Critical patent/US20070203658A1/en\nApplication granted granted Critical\nPublication of US7788051B2 publication Critical patent/US7788051B2/en\nApplication status is Active legal-status Critical\nAnticipated expiration legal-status Critical\n\n## Images\n\n•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "## Classifications\n\n• GPHYSICS\n• G06COMPUTING; CALCULATING; COUNTING\n• G06QDATA PROCESSING SYSTEMS OR METHODS, SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL, SUPERVISORY OR FORECASTING PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL, SUPERVISORY OR FORECASTING PURPOSES, NOT OTHERWISE PROVIDED FOR\n• G06Q50/00Systems or methods specially adapted for specific business sectors, e.g. utilities or tourism\n• G06Q50/06Electricity, gas or water supply\n• HELECTRICITY\n• H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER\n• H02JCIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY\n• H02J3/00Circuit arrangements for ac mains or ac distribution networks\n• H02J2003/007Simulating, e. g. planning, reliability check, modeling\n• HELECTRICITY\n• H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER\n• H02JCIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY\n• H02J3/00Circuit arrangements for ac mains or ac distribution networks\n• H02J3/04Circuit arrangements for ac mains or ac distribution networks for connecting networks of the same frequency but supplied from different sources\n• H02J3/06Controlling transfer of power between connected networks; Controlling sharing of load between connected networks\n• YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS\n• Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE\n• Y02EREDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION\n• Y02E40/00Technologies for an efficient electrical power generation, transmission or distribution\n• Y02E40/70Systems integrating technologies related to power network operation and communication or information technologies for improving the carbon footprint of electrical power generation, transmission or distribution, i.e. smart grids as climate change mitigation technology in the energy generation sector\n• Y02E40/76Computing methods or systems for efficient or low carbon management or operation of electric power systems\n• YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS\n• Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE\n• Y02EREDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION\n• Y02E60/00Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation\n• Y02E60/70Systems integrating technologies related to power network operation and communication or information technologies mediating in the improvement of the carbon footprint of electrical power generation, transmission or distribution, i.e. smart grids as enabling technology in the energy generation sector\n• Y02E60/76Computer aided design [CAD]; Simulation; Modelling\n• YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS\n• Y04INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS\n• Y04SSYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS\n• Y04S10/00Systems supporting electrical power generation, transmission or distribution\n• Y04S10/50Systems or methods supporting the power network operation or management, involving a certain degree of interaction with the load-side end user applications\n• Y04S10/54Management of operational aspects\n• Y04S10/545Computing methods or systems for efficient or low carbon management or operation of electric power systems\n• YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS\n• Y04INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS\n• Y04SSYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS\n• Y04S40/00Systems for electrical power generation, transmission, distribution or end-user application management characterised by the use of communication or information technologies, or communication or information technology specific aspects supporting them\n• Y04S40/20Information technology specific aspects\n• Y04S40/22Computer aided design [CAD]; Simulation; Modelling\n\n## Abstract\n\nGauss-Seidel-Patel Loadflow (GSPL) loadflow calculation method is invented involving self-iteration over a node within global iteration over (n−1) nodes in n-node power network. Also invented is a network decomposition technique referred to as Suresh's diakoptics that determines a sub-network for each node involving directly connected nodes referred to as level-1 nodes and their directly connected nodes referred to as level-2 nodes, wherein the level of outward connectivity for local solution of a sub-network around a given node is to be determined experimentally. Sub-networks are solved in parallel, and local solution of sub networks are related into network wide global solution using an invented technique. These led to the invention of the best possible parallel computer—a server-processor parallel-processors architecture wherein each of the parallel processors communicate only with the server processor, commonly shared memory locations, and each processor's private memory locations but not among themselves.\n\n## Description\n\nTECHNICAL FIELD\n\nThe present invention relates to methods of loadflow computation in power flow control and voltage control in an electrical power system. It also relates to the parallel computer architecture and distributed computing architecture.\n\nBACKGROUND OF THE INVENTION\n\nThe present invention relates to power-flow/voltage control in utility/industrial power networks of the types including many power plants/generators interconnected through transmission/distribution lines to other loads and motors. Each of these components of the power network is protected against unhealthy or alternatively faulty, over/under voltage, and/or over loaded damaging operating conditions. Such a protection is automatic and operates without the consent of power network operator, and takes an unhealthy component out of service by disconnecting it from the network. The time domain of operation of the protection is of the order of milliseconds.\n\nThe purpose of a utility/industrial power network is to meet the electricity demands of its various consumers 24-hours a day, 7-days a week while maintaining the quality of electricity supply. The quality of electricity supply means the consumer demands be met at specified voltage and frequency levels without over loaded, under/over voltage operation of any of the power network components. The operation of a power network is different at different times due to changing consumer demands and development of any faulty/contingency situation. In other words healthy operating power network is constantly subjected to small and large disturbances. These disturbances could be consumer/operator initiated, or initiated by overload and under/over voltage alleviating functions collectively referred to as security control functions and various optimization functions such as economic operation and minimization of losses, or caused by a fault/contingency incident.\n\nFor example, a power network is operating healthy and meeting quality electricity needs of its consumers. A fault occurs on a line or a transformer or a generator which faulty component gets isolated from the rest of the healthy network by virtue of the automatic operation of its protection. Such a disturbance would cause a change in the pattern of power flows in the network, which can cause over loading of one or more of the other components and/or over/under voltage at one or more nodes in the rest of the network. This in turn can isolate one or more other components out of service by virtue of the operation of associated protection, which disturbance can trigger chain reaction disintegrating the power network.\n\nTherefore, the most basic and integral part of all other functions including optimizations in power network operation and control is security control. Security control means controlling power flows so that no component of the network is over loaded and controlling voltages such that there is no over voltage or under voltage at any of the nodes in the network following a disturbance small or large. As is well known, controlling electric power flows include both controlling real power flows which is given in MWs, and controlling reactive power flows which is given in MVARs. Security control functions or alternatively overloads alleviation and over/under voltage alleviation functions can be realized through one or combination of more controls in the network. These involve control of power flow over tie line connecting other utility network, turbine steam/water/gas input control to control real power generated by each generator, load shedding function curtails load demands of consumers, excitation controls reactive power generated by individual generator which essentially controls generator terminal voltage, transformer taps control connected node voltage, switching in/out in capacitor/reactor banks controls reactive power at the connected node.\n\nControl of an electrical power system involving power-flow control and voltage control commonly is performed according to a process shown in FIG. 4. The various steps entail the following.\n\n• Step-10: Obtain on-line/simulated readings of open/close status of all switches and circuit breakers, and read data of maximum and minimum reactive power generation capability limits of PV-node generators, maximum and minimum tap positions limits of tap changing transformers, and maximum power carrying capability limits of transmission lines, transformers in the power network, or alternatively read data of operating limits of power network components.\n• Step-20: Obtain on-line readings of real and reactive power assignments/schedules/specifications/settings at PQ-nodes, real power and voltage magnitude assignments/schedules/specifications/settings at PV-nodes and transformer turns ratios. These assigned/set values are controllable and are called controlled variables/parameters.\n• Step-30: Resulting voltage magnitudes and angles at power network nodes, power flows through various power network components, reactive power generations by generators and turns ratios of transformers in the power network are determined by performance of loadflow computation, for the assigned/set/given/known values from step-20 or previous process cycle step-60, of controlled variables/parameters.\n• Step-40: The results of Loadflow computation of step-30 are evaluated for any over loaded power network components like transmission lines and transformers, and over/under voltages at different nodes in the power system\n• Step-50: If the system state is acceptable implying no over loaded transmission lines and transformers and no over/under voltages, the process branches to step-70, and if otherwise, then to step-60\n• Step-60: Changes the controlled variables/parameters set in step-20 or as later set by the previous process cycle step-60 and returns to step-30\n• Step-70: Actually implements the corrected controlled variables/parameters to obtain secure/correct/acceptable operation of power system\n\nOverload and under/over voltage alleviation functions produce changes in controlled variables/parameters in step-60 of FIG. 5. In other words controlled variables/parameters are assigned or changed to the new values in step-60. This correction in controlled variables/parameters could be even optimized in case of simulation of all possible imaginable disturbances including outage of a line and loss of generation for corrective action stored and made readily available for acting upon in case the simulated disturbance actually occurs in the power network. In fact simulation of all possible imaginable disturbances is the modern practice because corrective actions need be taken before the operation of individual protection of the power network components.\n\nIt is obvious that loadflow computation consequently is performed many times in real-time operation and control environment and, therefore, efficient and high-speed loadflow computation is necessary to provide corrective control in the changing power system conditions including an outage or failure of any of the power network components. Moreover, the loadflow computation must be highly reliable to yield converged solution under a wide range of system operating conditions and network parameters. Failure to yield converged loadflow solution creates blind spot as to what exactly could be happening in the network leading to potentially damaging operational and control decisions/actions in capital-intensive power utilities.\n\nThe power system control process shown in FIG. 5 is very general and elaborate. It includes control of power-flows through network components and voltage control at network nodes. However, the control of voltage magnitude at connected nodes within reactive power generation capabilities of electrical machines including generators, synchronous motors, and capacitor/inductor banks, and within operating ranges of transformer taps is normally integral part of loadflow computation as described in “LTC Transformers and MVAR violations in the Fast Decoupled Load Flow, IEEE Trans., PAS-101, No. 9, PP. 3328-3332, September 1982.” If under/over voltage still exists in the results of loadflow computation, other control actions, manual or automatic, may be taken in step-60 in the above and in FIG. 5. For example, under voltage can be alleviated by shedding some of the load connected.\n\nThe prior art and present invention are described using the following symbols and terms:\n\nYpq=Gpq'jBpq: (p-q) th element of nodal admittance matrix without shunts\n\nYpp=Gpp+jBpp: p-th diagonal element of nodal admittance matrix without shunts\n\nyp=gp+jbp: total shunt admittance at any node-p\n\nVp=ep+jfp=Vpp: complex voltage of any node-p\n\nΔθp, ΔVp: voltage angle, magnitude corrections\n\nΔep, Δfp: real, imaginary components of voltage corrections\n\nPp+jQP: net nodal injected power calculated\n\nΔPP+jΔQp: nodal power residue or mismatch\n\nRPP+jRQp: modified nodal power residue or mismatch\n\nΦp: rotation or transformation angle\n\n[RP]: vector of modified real power residue or mismatch\n\n[RQ]: vector of modified reactive power residue or mismatch\n\n[Yθ]: gain matrix of the P-θ loadflow sub-problem defined by eqn. (1)\n\n[YV]: gain matrix of the Q-V loadflow sub-problem defined by eqn. (2)\n\nm: number of PQ-nodes\n\nk: number of PV-nodes\n\nn=m+k+1: total number of nodes\n\nq>p: q is the node adjacent to node-p excluding the case of q=p\n\n[ ]: indicates enclosed variable symbol to be a vector or a matrix\n\nPQ-node: load-node, where, Real-Power-P and Reactive-Power-Q are specified\n\nPV-node: generator-node, where, Real-Power-P and Voltage-Magnitude-V are specified Bold lettered symbols represent complex quantities in description.\n\n• Loadflow Computation: Each node in a power network is associated with four electrical quantities, which are voltage magnitude, voltage angle, real power, and reactive power. The loadflow computation involves calculation/determination of two unknown electrical quantities for other two given/specified/scheduled/set/known electrical quantities for each node. In other words the loadflow computation involves determination of unknown quantities in dependence on the given/specified/scheduled/set/known electrical quantities.\n• Loadflow Model: a set of equations describing the physical power network and its operation for the purpose of loadflow computation. The term ‘loadflow model’ can be alternatively referred to as ‘model of the power network for loadflow computation’. The process of writing Mathematical equations that describe physical power network and its operation is called Mathematical Modeling. If the equations do not describe/represent the power network and its operation accurately the model is inaccurate, and the iterative loadflow computation method could be slow and unreliable in yielding converged loadflow computation. There could be variety of Loadflow Models depending on organization of set of equations describing the physical power network and its operation, including Decoupled Loadflow Models, Super Decoupled Loadflow (SDL) Models, Fast Super Decoupled Loadflow (FSDL) Model, and Novel Fast Super Decoupled Loadflow (NFSDL) Model.\n• Loadflow Method: sequence of steps used to solve a set of equations describing the physical power network and its operation for the purpose of loadflow computation is called Loadflow Method, which term can alternatively be referred to as ‘loadflow computation method’ or ‘method of loadflow computation’. One word for a set of equations describing the physical power network and its operation is: Model. In other words, sequence of steps used to solve a Loadflow Model is a Loadflow Method. The loadflow method involves definition/formation of a loadflow model and its solution. There could be variety of Loadflow Methods depending on a loadflow model and iterative scheme used to solve the model including Decoupled Loadflow Methods, Super Decoupled Loadflow (SDL) Methods, Fast Super Decoupled Loadflow (FSDL) Method, and Novel Fast Super Decoupled Loadflow (NFSDL) Method. All decoupled loadflow methods described in this application use either successive (1θ, 1V) iteration scheme or simultaneous (1V, 1θ), defined in the following.\n\nPrior art methods of loadflow computation of the kind carried out as step-30 in FIG. 5, are well known Gauss-Seidel Loadflow (GSL) and Super Super Decoupled Loadflow (SSDL) methods. The GSL method is well known to be not able to converge to high accuracy solution because of its iteration scheme that lacks self iterations, which realization led to the invention of Gauss-Seidel-Patel Loadflow (GSPL) method. The prior art methods will now the described.\n\nGauss-Seidel Loadflow: GSL\n\nThe complex power injected into the node-p of a power network is given by the following relation,\n\n$P p - j ⁢ ⁢ Q p = V p * ⁢ ∑ q = 1 n ⁢ Y pq ⁢ V q = V p * ⁢ Y pp ⁢ V p + V p * ⁢ ∑ q > p ⁢ Y pq ⁢ V q ⁢ ⁢ Where , ( 1 ) P p = Re ⁢ ⁢ { V p * ⁢ ∑ q = 1 n ⁢ Y pq ⁢ V q } ( 2 ) Q p = - Im ⁢ ⁢ { V p * ⁢ ∑ q = 1 n ⁢ Y pq ⁢ V q } ( 3 )$\nWhere, words “Re” means “real part of” and words “Im” means “imaginary part of”.\n\nThe Gauss-Seidel (GS) method is used to solve a set of simultaneous algebraic equations iteratively. The GSL-method calculates complex node voltage from any node-p relation (1) as given in relation (4).\n\n$V p = [ { ( PSH p - j ⁢ ⁢ QSH p ) / V p * } - ∑ q > p ⁢ Y pq ⁢ V q ] / Y pp ( 4 )$\nIteration Process\n\nIterations start with the experienced/reasonable/logical guess for the solution. The reference node also referred to as the slack-node voltage being specified, starting voltage guess is made for the remaining (n−1)-nodes in n-node network. Node voltage value is immediately updated with its newly calculated value in the iteration process in which one node voltage is calculated at a time using latest updated other node voltage values. A node voltage value calculation at a time process is iterated over (n−1)-nodes in an n-node network, the reference/slack node voltage being specified not required to be calculated. Now, for the iteration-(r+1), the complex voltage calculation at node-p equation (4) and reactive power calculation at node-p equation (3), becomes\n\n$V p ( r + 1 ) = [ { ( PSH p - j ⁢ ⁢ QSH p ) / ( V p * ) r } - ∑ q = 1 p - 1 ⁢ Y pq ⁢ V q ( r + 1 ) - ∑ q = p + 1 n ⁢ Y pq ⁢ V q r ] / Y pp ( 5 ) Q p ( r + 1 ) = - Im ⁢ { ( V p * ) r ⁢ ∑ q = 1 p - 1 ⁢ Y pq ⁢ V q ( r + 1 ) - ( V q * ) r ⁢ ∑ q = p n ⁢ Y pq ⁢ V q r } ( 6 )$\nConvergence\n\nThe iteration process is carried out until changes in the real and imaginary parts of the set of (n−1)-node voltages calculated in two consecutive iterations are all less than the specified tolerance—ε, as shown in relations (7) and (8). The lower the value of the specified tolerance for convergence check, the greater the solution accuracy.\ne P (r+1) \|=\|e P (r+1) −e P r \|<ε (7)\nf P (r+1) \|=\|f P (r+1) −f P r \|<ε (8)\nAccelerated Convergence\n\nThe GS-method being inherently slow to converge, it is characterized by the use of an acceleration factor applied to the difference in calculated node voltage between two consecutive iterations to speed-up the iterative solution process. The accelerated value of node-p voltage at iteration-(r+1) is given by\nV p (r+1)(accelerated)=V p r+β(V p (r+1) −V P r) (9)\nWhere, β is the real number called acceleration factor, the value of which for the best possible convergence for any given network can be determined by trial solutions. The GS-method is very sensitive to the choice of β, causing very slow convergence and even divergence for the wrong choice.\nScheduled or Specified Voltage at a PV-Node\n\nOf the four variables, real power PSHp and voltage magnitude VSHp are specified at a PV-node. If the reactive power calculated using VSHp at the PV-node is within the upper and lower generation capability limits of a PV-node generator, it is capable of holding the specified voltage at its terminal. Therefore the complex voltage calculated by relation (5) by using actually calculated reactive power Qp in place of QSHp is adjusted to specified voltage magnitude by relation (10).\nV p (r+1)=(VSH p V p (r+1))/\|V p (r+1)\| (10)\nCalculation Steps of Gauss-Seidel Loadflow (GSL) Method\n\nThe steps of loadflow computation method, GSL method are shown in the flowchart of FIG. 1 a. Referring to the flowchart of FIG. 1 a, different steps are elaborated in steps marked with similar numbers in the following. The words “Read system data” in Step-1 correspond to step-10 and step-20 in FIG. 5, and step-14, step-20, step-32, step-44, step-50 in FIG. 6. All other steps in the following correspond to step-30 in FIG. 5, and step-60, step-62, and step-64 in FIG. 6.\n\n• 1. Read system data and assign an initial approximate solution. If better solution estimate is not available, set specified voltage magnitude at PV-nodes, 1.0 p.u. voltage magnitude at PQ-nodes, and all the node angles equal to that of the slack-node angle, which is referred to as the flat-start.\n• 2. Form nodal admittance matrix, and Initialize iteration count r=1\n• 3. Scan all the node of a network, except the slack-node whose voltage having been specified need not be calculated. Initialize node count p=1, and initialize maximum change in real and imaginary parts of node voltage variables DEMX=0.0 and DFMX=0.0\n• 4. Test for the type of a node at a time. For the slack-node go to step-12, for a PQ-node go to the step-9, and for a PV-node follow the next step.\n• 5. Compute Qp (r+1) for use as an imaginary part in determining complex schedule power at a PV-node from relation (6) after adjusting its complex voltage for specified value by relation (10)\n• 6. If Qp (r+1) is greater than the upper reactive power generation capability limit of the PV-node generator, set QSHp=the upper limit Qp max for use in relation (5), and go to step-9. If not, follow the next step.\n• 7. If Qp (r+1) is less than the lower reactive power generation capability limit of the PV-node generator, set QSHp=the lower limit Qp min for use in relation (5), and go to step-9. If not, follow the next step.\n• 8. Compute Vp (r+1) from relation (5) using QSHp=Qp (r+1), and adjust its value for specified voltage at the PV-node by relation (10), and go to step-10\n• 9. Compute Vp (r+1) from relation (5)\n• 10. Compute changes in the real and imaginary parts of the node-p voltage by using relations (7) and (8), and replace current value of DEMX and DFMX respectively in case any of them is larger.\n• 11. Calculate accelerated value of Vp (r+1) by using relation (9), and update voltage by Vp r=Vp (r+1) for immediate use in the next node voltage calculation.\n• 12. Check for if the total numbers of nodes-n are scanned. That is if p<n, increment p=p+1, and go to step-4. Otherwise follow the next step.\n• 13. If DEMX and DFMX both are not less than the convergence tolerance (ε) specified for the purpose of the accuracy of the solution, advance iteration count r=r+1 and go to step-3, otherwise follow the next step\n• 14. From calculated and known values of complex voltage at different power network nodes, and tap position of tap changing transformers, calculate power flows through power network components, and reactive power generation at PV-nodes.\nDecoupled Loadflow\n\nIn a class of decoupled loadflow methods, each decoupled method comprises a system of equations (11) and (12) differing in the definition of elements of [RP], [RQ], and [Yθ] and [YV]. It is a system of equations for the separate calculation of voltage angle and voltage magnitude corrections.\n[RP]=[Yθ][Δθ] (11)\n[RQ]=[YV][ΔV] (12)\nSuccessive (1θ, 1V) Iteration Scheme\n\nIn this scheme (11) and (12) are solved alternately with intermediate updating. Each iteration involves one calculation of [RP] and [Δθ] to update [θ] and then one calculation of [RQ] and [ΔV] to update [V]. The sequence of relations (13) to (16) depicts the scheme.\n[Δθ]=[Yθ]−1[RP] (13)\n[θ]=[θ]+[Δθ] (14)\n[ΔV]=[YV]−1[RQ] (15)\n[V]=[V]+[ΔV] (16)\n\nThe scheme involves solution of system of equations (11) and (12) in an iterative manner depicted in the sequence of relations (13) to (16). This scheme requires mismatch calculation for each half-iteration; because [RP] and [RQ] are calculated always using the most recent voltage values and it is block Gauss-Seidel approach. The scheme is block successive, which imparts increased stability to the solution process. This in turn improves convergence and increases the reliability of obtaining solution.\n\nSuper Super Decoupled Loadflow: SSDL\n\nThis method is not very sensitive to the restriction applied to nodal transformation angles; SSDL restricts transformation angles to the maximum of −48 degrees determined experimentally for the best possible convergence from non linearity considerations, which is depicted by relations (19) and (20). However, it gives closely similar performance over wide range of restriction applied to the transformation angles say, from −36 to −90 degrees.\nRP p=(ΔP p Cos Φp +ΔQ p Sin Φp)/V p 2 —for PQ-nodes (17)\nRQ p=(ΔQ p Cos Φp −ΔP p Sin Φp)/V p —for PQ-nodes (18)\nCos Φp=Absolute(B pp /v(G pp 2 +B pp 2))≧Cos(−48°) (19)\nSin Φp=−Absolute(G pp /v(G pp 2 +B pp 2))≧Sin(−48°) (20)\nRP p =ΔP p/(K p V p 2) —for PV-nodes (21)\nK p=Absolute(B pp /Yθ pp) (22)\n\n$Y ⁢ ⁢ θ pq = [ - ⁢ Y pq - ⁢ for ⁢ ⁢ branch ⁢ ⁢ ⁢ r / x ⁢ ⁢ ratio ≤ 3.0 - ⁢ ( B pq + 0.9 ⁢ ( Y pq - B pq ) ) - ⁢ for ⁢ ⁢ branch ⁢ ⁢ r / x ⁢ ⁢ ⁢ ratio > 3.0 - ⁢ B pq - ⁢ for ⁢ ⁢ branches ⁢ ⁢ connected ⁢ ⁢ between ⁢ ⁢ two PV ⁢ - ⁢ nodes ⁢ ⁢ or ⁢ ⁢ ⁢ a ⁢ ⁢ PV ⁢ - ⁢ node ⁢ ⁢ and ⁢ ⁢ the ⁢ ⁢ slack ⁢ - ⁢ node ⁢ ( 23 ) YV pq = [ - ⁢ Y pq - ⁢ for ⁢ ⁢ branch ⁢ ⁢ ⁢ r / x ⁢ ⁢ ratio ≤ 3.0 - ⁢ ( B pq + 0.9 ⁢ ( Y pq - B pq ) ) - ⁢ for ⁢ ⁢ branch ⁢ ⁢ r / x ⁢ ⁢ ⁢ ratio > 3.0 ( 24 ) Y ⁢ ⁢ θ pp = ∑ q > p ⁢ - Y ⁢ ⁢ θ pq ⁢ ⁢ and ⁢ ⁢ YV pp = b p ′ + ∑ q > p ⁢ - YV pq ( 25 ) b p ′ = ( QSH p ⁢ Cos ⁢ ⁢ Φ p - PSH p ⁢ Sin ⁢ ⁢ Φ p / V s 2 ) - b p ⁢ Cos ⁢ ⁢ Φ p ⁢ ⁢ or b p ′ = 2 ⁢ ( QSH p ⁢ Cos ⁢ ⁢ Φ p - PSH p ⁢ Sin ⁢ ⁢ Φ p ) / V s 2 ( 26 )$\nb p′=(QSH p Cos Φp −PSH p Sin Φp /V s 2)−b p Cos Φp or\nb p′=2(QSH p Cos Φp −PSH p Sin Φp)/Vs 2 (26)\nwhere, Kp is defined in relation (22), which is initially restricted to the minimum value of 0.75 determined experimentally; however its restriction is lowered to the minimum value of 0.6 when its average over all less than 1.0 values at PV nodes is less than 0.6. Restrictions to the factor Kp as stated in the above is system independent. However it can be tuned for the best possible convergence for any given system. In case of systems of only PQ-nodes and without any PV-nodes, equations (23) and (24) simply be taken as Yθpq=YVpq=−Ypq.\n\nBranch admittance magnitude in (23) and (24) is of the same algebraic sign as its susceptance. Elements of the two gain matrices differ in that diagonal elements of [YV] additionally contain the b′ values given by relations (25) and (26) and in respect of elements corresponding to branches connected between two PV-nodes or a PV-node and the slack-node. Relations (19) and (20) with inequality sign implies that transformation angles are restricted to maximum of −48 degrees for SSDL. The method consists of relations (11) to (26). In two simple variations of the SSDL method, one is to make YVpq=Yθpq and the other is to make Yθpq=YVpq.\n\nCalculation Steps of Super Super Decoupled Loadflow (SSDL) Method\n\nThe steps of loadflow computation method, SSDL method are shown in the flowchart of FIG. 1 b. Referring to the flowchart of FIG. 1 b, different steps are elaborated in steps marked with similar letters in the following. The words “Read system data” in Step-1 correspond to step-10 and step-20 in FIG. 5, and step-14, step-20, step-32, step-44, step-50 in FIG. 6. All other steps in the following correspond to step-30 in FIG. 5, and step-60, step-62, and step-64 in FIG. 6.\n\n• a. Read system data and assign an initial approximate solution. If better solution estimate is not available, set voltage magnitude and angle of all nodes equal to those of the slack-node. This is referred to as the slack-start.\n• b. Form nodal admittance matrix, and Initialize iteration count ITRP=ITRQ=r=0\n• c. Compute Cosine and Sine of nodal rotation angles using relations (19) and (20), and store them. If they, respectively, are less than the Cosine and Sine of −48 degrees, equate them, respectively, to those of −48 degrees.\n• d. Form (m+k)×(m+k) size matrices [Yθ] and [YV] of (11) and (12) respectively each in a compact storage exploiting sparsity. The matrices are formed using relations (23), (24), (25), and (26). In [YV] matrix, replace diagonal elements corresponding to PV-nodes by very large value (say, 10.0*10). In case [YV] is of dimension (m×m), this is not required to be performed. Factorize [Yθ] and [YV] using the same ordering of nodes regardless of node-types and store them using the same indexing and addressing information. In case [YV] is of dimension (m×m), it is factorized using different ordering than that of [Yθ].\n• e. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at only PQ-nodes. If all are less than the tolerance (ε), proceed to step-n. Otherwise follow the next step.\n• f. Compute the vector of modified residues [RP] as in (17) for PQ-nodes, and using (21) and (22) for PV-nodes.\n• g. Solve (13) for [Δθ] and update voltage angles using, [θ]=[θ]+[Δθ].\n• h. Set voltage magnitudes of PV-nodes equal to the specified values, and Increment the iteration count ITRP=ITRP+1 and r=(ITRP+ITRQ)/2.\n• i. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at PQ-nodes only. If all are less than the tolerance (ε), proceed to step-n. Otherwise follow the next step.\n• j. Compute the vector of modified residues [RQ] as in (18) for only PQ-nodes.\n• k. Solve (15) for [ΔV] and update PQ-node magnitudes using [V]=[V]+[ΔV]. While solving equation (15), skip all the rows and columns corresponding to PV-nodes.\n• l. Calculate reactive power generation at PV-nodes and tap positions of tap changing transformers. If the maximum and minimum reactive power generation capability and transformer tap position limits are violated, implement the violated physical limits and adjust the loadflow solution.\n• m. Increment the iteration count ITRQ=ITRQ+1 and r=(ITRP+ITRQ)/2, and Proceed to step-e.\n• n. From calculated and known values of voltage magnitude and voltage angle at different power network nodes, and tap position of tap changing transformers, calculate power flows through power network components, and reactive power generation at PV-nodes.\nSUMMARY OF THE INVENTION\n\nIt is a primary object of the present invention to improve solution accuracy, convergence and efficiency of the prior art GSL and SSDL computations method under wide range of system operating conditions and network parameters for use in power flow control and voltage control in the power system.\n\nThe above and other objects are achieved, according to the present invention, with Gauss-Seidel-Patel loadflow (GSPL), the prior art Super Super Decoupled Loadflow (SSDL) and their parallel versions PGSPL, PSSDL loadflow computation methods for Electrical Power. System. In context of voltage control, the inventive system of parallel loadflow computation for Electrical Power system consisting of plurality of electromechanical rotating machines, transformers and electrical loads connected in a network, each machine having a reactive power characteristic and an excitation element which is controllable for adjusting the reactive power generated or absorbed by the machine, and some of the transformers each having a tap changing element, which is controllable for adjusting turns ratio or alternatively terminal voltage of the transformer, said system comprising:\n\n• means for defining and solving loadflow model of the power network characterized by inventive PGSPL or PSSDL model for providing an indication of the quantity of reactive power to be supplied by each generator including the reference/slack node generator, and for providing an indication of turns ratio of each tap-changing transformer in dependence on the obtained-online or given/specified/set/known controlled network variables/parameters, and physical limits of operation of the network components,\n• means for machine control connected to the said means for defining and solving loadflow model and to the excitation elements of the rotating machines for controlling the operation of the excitation elements of machines to produce or absorb the amount of reactive power indicated by said means for defining and solving loadflow model in dependence on the set of obtained-online or given/specified/set controlled network variables/parameters, and physical limits of excitation elements,\n• means for transformer tap position control connected to said means for defining and solving loadflow model and to the tap changing elements of the controllable transformers for controlling the operation of the tap changing elements to adjust the turns ratios of transformers indicated by the said means for defining and solving loadflow model in dependence on the set of obtained-online or given/specified/set controlled network variables/parameters, and operating limits of the tap-changing elements.\n\nThe method and system of voltage control according to the preferred embodiment of the present invention provide voltage control for the nodes connected to PV-node generators and tap changing transformers for a network in which real power assignments have already been fixed. The said voltage control is realized by controlling reactive power generation and transformer tap positions.\n\nThe inventive system of parallel loadflow computation can be used to solve a model of the Electrical Power System for voltage control. For this purpose real and reactive power assignments or settings at PQ-nodes, real power and voltage magnitude assignments or settings at PV-nodes and transformer turns ratios, open/close status of all circuit breaker, the reactive capability characteristic or curve for each machine, maximum and minimum tap positions limits of tap changing transformers, operating limits of all other network components, and the impedance or admittance of all lines are supplied. GSPL or SSDL loadflow equations are solved by a parallel iterative process until convergence. During this solution the quantities which can vary are the real and reactive power at the reference/slack node, the reactive power set points for each PV-node generator, the transformer transformation ratios, and voltages on all PQ-nodes nodes, all being held within the specified ranges. When the iterative process converges to a solution, indications of reactive power generation at PV-nodes and transformer turns-ratios or tap-settings are provided. Based on the known reactive power capability characteristics of each PV-node generator, the determined reactive power values are used to adjust the excitation current to each generator to establish the reactive power set points. The transformer taps are set in accordance with the turns ratio indication provided by the system of loadflow computation.\n\nFor voltage control, system of parallel GSPL or SSDL computation can be employed either on-line or off-line. In off-line operation, the user can simulate and experiment with various sets of operating conditions and determine reactive power generation and transformer tap settings requirements. An invented parallel computer System can implement any of the parallel loadflow computation methods. For on-line operation, the loadflow computation system is provided with data identifying the current real and reactive power assignments and transformer transformation ratios, the present status of all switches and circuit breakers in the network and machine characteristic curves in steps-10 and -20 in FIG. 5, and steps 12, 20, 32, 44, and 50 in FIG. 6 described below. Based on this information, a model of the system provide the values for the corresponding node voltages, reactive power set points for each machine and the transformation ratio and tap changer position for each transformer.\n\nInventions include Gauss-Seidel-Patel Loadflow (GSPL) method for the solution of complex simultaneous algebraic power injection equations or any set, of complex simultaneous algebraic equations arising in any other subject areas. Further inventions are a technique of decomposing a network into sub-networks for the solution of sub-networks in parallel referred to as Suresh's diakoptics, a technique of relating solutions of sub-networks into network wide solution, and a best possible parallel computer architecture ever invented to carry out solution of sub-networks in parallel.\n\nBRIEF DESCRIPTION OF DRAWINGS\n\nFIG. 1 is a flow-charts of the prior art GSL and SSDL methods\n\nFIG. 2 is a one-line diagram of IEEE 14-node network and its decomposition into level-1 connectivity sub-networks\n\nFIG. 3 is a flow-charts embodiment of the invented GSPL, PGSPL methods\n\nFIG. 4 is a block diagram of invented parallel computer architecture/organization\n\nFIG. 5 prior art is a flow-chart of the overall controlling method for an electrical power system involving loadflow computation as a step which can be executed using one of the invented loadflow computation method of FIG. 3.\n\nFIG. 6 prior art is a flow-chart of the simple special case of voltage control system in overall controlling system of FIG. 5 for an electrical power system\n\nFIG. 7 prior art is a one-line diagram of an exemplary 6-node power network having a reference/slack/swing node, two PV-nodes, and three PQ-nodes\n\nDESCRIPTION OF A PREFERRED EMBODIMENT\n\nA loadflow computation is involved as a step in power flow control and/or voltage control in accordance with FIG. 5 or FIG. 6. A preferred embodiment of the present invention is described with reference to FIG. 7 as directed to achieving voltage control.\n\nFIG. 7 is a simplified one-line diagram of an exemplary utility power network to which the present invention may be applied. The fundamentals of one-line diagrams are described in section 6.11 of the text ELEMENTS OF POWER SYSTEM ANALYSIS, forth edition, by William D. Stevenson, Jr., McGrow-Hill Company, 1982. In FIG. 7, each thick vertical line is a network node. The nodes are interconnected in a desired manner by transmission lines and transformers each having its impedance, which appears in the loadflow models. Two transformers in FIG. 7 are equipped with tap changers to control their turns ratios in order to control terminal voltage of node-1 and node-2 where large loads are connected.\n\nNode-6 is a reference node alternatively referred to as the slack or swing-node, representing the biggest power plant in a power network. Nodes-4 and -5 are PV-nodes where generators are connected, and nodes-1, -2, and -3 are PQ-nodes where loads are connected. It should be noted that the nodes-4, -5, and -6 each represents a power plant that contains many generators in parallel operation. The single generator symbol at each of the nodes-4, -5, and -6 is equivalent of all generators in each plant. The power network further includes controllable circuit breakers located at each end of the transmission lines and transformers, and depicted by cross markings in one-line diagram of FIG. 7. The circuit breakers can be operated or in other words opened or closed manually by the power system operator or relevant circuit breakers operate automatically consequent of unhealthy or faulty operating conditions. The operation of one or more circuit breakers modify the configuration of the network. The arrows extending certain nodes represent loads.\n\nA goal of the present invention is to provide a reliable and computationally efficient loadflow computation that appears as a step in power flow control and/or voltage control systems of FIG. 5 and FIG. 6. However, the preferred embodiment of loadflow computation as a step in control of terminal node voltages of PV-node generators and tap-changing transformers is illustrated in the flow diagram of FIG. 6 in which present invention resides in function steps 60 and 62.\n\nShort description of other possible embodiment of the present invention is also provided herein. The present invention relates to control of utility/industrial power networks of the types including plurality of power plants/generators and one or more motors/loads, and connected to other external utility. In the utility/industrial systems of this type, it is the usual practice to adjust the real and reactive power produced by each generator and each of the other sources including synchronous condensers and capacitor/inductor banks, in order to optimize the real and reactive power generation assignments of the system. Healthy or secure operation of the network can be shifted to optimized operation through corrective control produced by optimization functions without violation of security constraints. This is referred to as security constrained optimization of operation. Such an optimization is described in the U.S. Pat. No. 5,081,591 dated Jan. 13, 1992: “Optimizing Reactive Power Distribution in an Industrial Power Network”, where the present invention can be embodied by replacing the step nos. 56 and 66 each by a step of constant gain matrices [Yθ] and [YV], and replacing steps of “Exercise Newton-Raphson Algorithm” by steps of “Exercise parallel GSPL or SSDL Computation” in places of steps 58 and 68. This is just to indicate the possible embodiment of the present invention in optimization functions like in many others including state estimation function. However, invention is being claimed through a simplified embodiment without optimization function as in FIG. 6 in this application. The inventive steps-60 and -62 in FIG. 6 are different than those corresponding steps-56, and -58, which constitute a well known Newton-Raphson loadflow method, and were not inventive even in U.S. Pat. No. 5,081,591.\n\nIn FIG. 6, function step 10 provides stored impedance values of each network component in the system. This data is modified in a function step 12, which contains stored information about the open or close status of each circuit breaker. For each breaker that is open, the function step 12 assigns very high impedance to the associated line or transformer. The resulting data is than employed in a function step 14 to establish an admittance matrix for the power network. The data provided by function step 10 can be input by the computer operator from calculations based on measured values of impedance of each line and transformer, or on the basis of impedance measurements after the power network has been assembled.\n\nEach of the transformers T1 and T2 in FIG. 7 is a tap changing transformer having a plurality of tap positions each representing a given transformation ratio. An indication of initially assigned transformation ratio for each transformer is provided by function step 20.\n\nThe indications provided by function steps 14, and 20 are supplied to a function step 60 in which constant gain matrices [Yθ] and [YV] of any of the invented super decoupled loadflow models are constructed, factorized and stored. The gain matrices [Yθ] and [YV] are conventional tools employed for solving Super Decoupled Loadflow model defined by equations (1) and (2) for a power system.\n\nIndications of initial reactive power, or Q on each node, based on initial calculations or measurements, are provided by a function step 30 and these indications are used in function step 32, to assign a Q level to each generator and motor. Initially, the Q assigned to each machine can be the same as the indicated Q value for the node to which that machine is connected.\n\nAn indication of measured real power, P, on each node is supplied by function step 40. Indications of assigned/specified/scheduled/set generating plant loads that are constituted by known program are provided by function step 42, which assigns the real power, P, load for each generating plant on the basis of the total P which must be generated within the power system. The value of P assigned to each power plant represents an economic optimum, and these values represent fixed constraints on the variations, which can be made by the system according to the present invention. The indications provided by function steps 40 and 42 are supplied to function step 44 which adjusts the P distribution on the various plant nodes accordingly. Function step 50 assigns initial approximate or guess solution to begin iterative method of loadflow computation, and reads data file of operating limits on power network components, such as maximum and minimum reactive power generation capability limits of PV-nodes generators.\n\nThe indications provided by function steps 32, 44, 50 and 60 are supplied to function step 62 where inventive Fast Super Decoupled Loadflow computation or Novel Fast Super Decoupled Loadflow computation is carried out, the results of which appear in function step 64. The loadflow computation yields voltage magnitudes and voltage angles at PQ-nodes, real and reactive power generation by the reference/slack/swing node generator, voltage angles and reactive power generation indications at PV-nodes, and transformer turns ratio or tap position indications for tap changing transformers. The system stores in step 62 a representation of the reactive capability characteristic of each PV-node generator and these characteristics act as constraints on the reactive power that can be calculated for each PV-node generator for indication in step 64. The indications provided in step 64 actuate machine excitation control and transformer tap position control. All the loadflow computation methods using SSDL models can be used to effect efficient and reliable voltage control in power systems as in the process flow diagram of FIG. 6.\n\nInventions include Gauss-Seidel-Patel Loadflow (GSPL) method for the solution of complex simultaneous algebraic power injection equations or any set of complex simultaneous algebraic equations arising in any other subject areas. Further inventions are a technique of decomposing a network into sub-networks for the solution of sub-networks in parallel referred to as Suresh's diakoptics, a technique of relating solutions of sub-networks into network wide solution, and a best possible parallel computer architecture ever invented to carry out solution of sub-networks in parallel.\n\nGauss-Seidel-Patel Loadflow: GSPL\n\nGauss-Seidel numerical method is well-known to be not able to converge to the high accuracy solution, which problem has been resolved for the first-time in the proposed Gauss-Seidel-Patel (GSP) numerical method.\n\nThe GSP method introduces the concept of self-iteration of each calculated variable until convergence before proceeding to calculate the next. This is achieved by replacing relation (5) by relation (27) stated in the following where self-iteration-(sr+1) over a node variable itself within the global iteration-(r+1) over (n−1) nodes in the n-node network is depicted. During the self-iteration process only vp changes without affecting any of the terms involving vq. At the start of the self-iteration Vp sr=Vp r and at the convergence of the self-iteration Vp (r+1)=Vp (sr+1).\n\n$( V p ( sr + 1 ) ) ( r + 1 ) = [ { ( PSH p - j ⁢ ⁢ QSH p ) / ( ( V p ) sr ) r } - ∑ q = 1 p - 1 ⁢ Y pq ⁢ V q ( r + 1 ) - ∑ q = p + 1 n ⁢ Y pq ⁢ V q r ] / Y pp ( 27 )$\nSelf-Convergence\n\nThe self-iteration process is carried out until changes in the real and imaginary parts of the node-p voltage calculated in two consecutive self-iterations are less than the specified tolerance. It has been possible to establish a relationship between the tolerance specification for self-convergence and the tolerance specification for global-convergence. It is found sufficient for the self-convergence tolerance specification to be ten times the global-convergence tolerance specification.\n\|Δep (sr+1)\|=\|ep (sr+1)−ep sr\|<10ε (28)\n\|Δfp (sr+1)\|=\|fp (sr+1)−fp sr\|<10ε (29)\n\nFor the global-convergence tolerance specification of 0.000001, it has been found sufficient to have the self-convergence tolerance specification of 0.00001 in order to have the maximum real and reactive power mismatches of 0.0001 in the converged solution. However, for small networks under not difficult to solve conditions they respectively could be 0.00001 and 0.0001 or 0.000001 and 0.0001, and for large networks under difficult to solve conditions they sometimes need to be respectively 0.0000001 and 0.000001.\n\nNetwork Decomposition Technique: Suresh's Diakoptics\n\nA network decomposition technique referred to as Suresh's diakoptics involves determining a sub-network for each node involving directly connected nodes referred to as level-1 nodes and their directly connected nodes referred to as level-2 nodes and so on. The level of outward connectivity for local solution of a sub-network around a given node is to be determined experimentally. This is particularly true for gain matrix based methods such as Newton-Raphson (NR), SSDL methods. Sub-networks can be solved by any of the known methods including Gauss-Seidel-Patel Loadflow (GSPL) method.\n\nIn the case of GSPL-method only one level of outward connectivity around each node is found to be sufficient for the formation of sub-networks equal to the number of nodes. Level-1 connectivity sub-networks for IEEE 14-node network is shown in FIG. 2 b. The local solution of equations of each sub-network could be iterated for experimentally determined two or more iteration. However, maximum of two iterations are fond to be sufficient. In case of GSPL-method, processing load on processors can be attempted equalization by assigning two or more smaller sub-networks to the single processor for solving separately in sequence.\n\nSometimes it is possible that a sub-network around any given node could be a part of the sub-network around another node making it redundant needing local solution of less than (m+k) sub-networks in case of gain matrix based methods like SSDL. Level-1 connectivity sub-networks for IEEE 14-node for parallel solution by say, SSDL-method is shown in FIG. 2 b. The local solution iteration over a sub-network is not required for gain matrix based methods like SSDL. FIG. 2 c shows the grouping of the non-redundant sub-networks in FIG. 2 b in an attempt to equalize size of the sub-networks reducing the number of processors without increasing time for the solution of the whole network.\n\nIt should be noted that no two decomposed network parts contain the same set of nodes, or the same set of lines connected to nodes, though some same nodes could appear in two or more sub-networks.\n\nDecomposing network in level-1 connectivity sub-networks provides for maximum possible parallelism, and hopefully fastest possible solution. However, optimum outward level of connectivity around a node sub-network can be determined experimentally for the solution of large networks by a gain matrix based method like SSDL.\n\nRelating Sub-Network Solutions to Get the Network-Wide Global Solution\n\nSuresh's decomposition subroutine run by server-processor decomposes the network into sub-networks and a separate processor is assigned to solve each sub-network simultaneously in parallel. A node-p of the network could be contained in two or more sub-networks. Say a node-p is contained in or part of ‘q’ sub-networks. If GSPL-method is used, voltage calculation for a node-p is performed by each of the ‘q’ sub-networks. Add ‘q’ voltages calculated for a node-p by ‘q’ number of sub-networks and divide by ‘q’ to take an average as given by relation (30).\nV p (r+1)=(Vp1 (r+1) +V p2 (r+1) +V p3 (r+1) + . . . +V pq (r+1))/q (30)\n\nIf a gain matrix based method like SSDL is used, voltage angle correction and voltage magnitude correction calculation for a node-p is performed by each of the ‘q’ sub-networks in which node-p is contained. Add ‘q’ voltage angle corrections and ‘q’ voltage magnitude corrections calculated for the node-p by ‘q’ sub-networks and divide by number ‘q’ to take average as given by relations (31) and (32).\nΔθp (r+1)=(Δθp1 (r+1)+Δθp2 (r+1)+Δθp3 (r+1)+ . . . +Δθpq (r+1))/q (31)\nΔV p (r+1)=(ΔV p1 (r+1) +ΔV p2 (r+1) +ΔV p3 (r+1) + . . . +ΔV pq (r+1))/q (32)\n\nSometimes, gain matrix based methods can be organized to directly calculate real and imaginary components of complex node voltages or GSPL-method can be decoupled into calculating real (ep) and imaginary (fp) components of complex voltage calculation for a node-p, which is contributed to by each of the ‘q’ sub-networks in which node-p is contained. Add ‘q’ real parts of voltages calculated for a node-p by ‘q’ sub-networks and divide by number ‘q’. Similarly, add ‘q’ imaginary parts of voltages calculated for the same node-p by ‘q’ sub-networks and divide by number ‘q’ to take an average as given by relation (33) and (34).\ne p (r+1)=(e p1 (r+1) +e p2 (r+1) +e p3 (r+1) + . . . +e pq (r+1))/q (33)\nf p (r+1)=(f p1 (r+1) +f p2 (r+1) +f p3 (r+1) + . . . +f pq (r+1))/q (34)\n\nThe relations (30) to (34), can also alternatively be written as relations (35) to (39).\nV p (r+1)=√{square root over ((Re((V p1 (r+1))2)+Re((V p2 (r+1))2)+ . . . +Re((V pq (r+1))2)2)/q)}{square root over ((Re((V p1 (r+1))2)+Re((V p2 (r+1))2)+ . . . +Re((V pq (r+1))2)2)/q)}{square root over ((Re((V p1 (r+1))2)+Re((V p2 (r+1))2)+ . . . +Re((V pq (r+1))2)2)/q)}+j√{square root over ((Im((V p1 (r+1))2)+Im((V p2 (r+1))2)+ . . . +Im((Vpq (r+1))2))/q)}{square root over ((Im((V p1 (r+1))2)+Im((V p2 (r+1))2)+ . . . +Im((Vpq (r+1))2))/q)}{square root over ((Im((V p1 (r+1))2)+Im((V p2 (r+1))2)+ . . . +Im((Vpq (r+1))2))/q)} (35)\nΔθp (r+1)=√{square root over ((Δθp1 (r+1))2+(Δθp2 (r+1))2+ . . . +(Δθpq (r+1))2)/q)}{square root over ((Δθp1 (r+1))2+(Δθp2 (r+1))2+ . . . +(Δθpq (r+1))2)/q)}{square root over ((Δθp1 (r+1))2+(Δθp2 (r+1))2+ . . . +(Δθpq (r+1))2)/q)} (36)\nΔV p (r+1)=√{square root over ((ΔV p1 (r+1))2+(ΔV p2 (r+1))2+ . . . +(ΔV pq (r+1))2)/q)}{square root over ((ΔV p1 (r+1))2+(ΔV p2 (r+1))2+ . . . +(ΔV pq (r+1))2)/q)}{square root over ((ΔV p1 (r+1))2+(ΔV p2 (r+1))2+ . . . +(ΔV pq (r+1))2)/q)} (37)\ne p (r+1)=√{square root over (((e p1 (r+1))2+(e p2 (r+1))2+ . . . +(e pq (r+1))2)/q)}{square root over (((e p1 (r+1))2+(e p2 (r+1))2+ . . . +(e pq (r+1))2)/q)}{square root over (((e p1 (r+1))2+(e p2 (r+1))2+ . . . +(e pq (r+1))2)/q)} (38)\nf p (r+1)=√{square root over (((f p1 (r+1))2+(f p2 (r+1))2+ . . . +(f pq (r+1))2)/q)}{square root over (((f p1 (r+1))2+(f p2 (r+1))2+ . . . +(f pq (r+1))2)/q)}{square root over (((f p1 (r+1))2+(f p2 (r+1))2+ . . . +(f pq (r+1))2)/q)} (39)\n\nMathematically, square of any positive or negative number is positive. Therefore, in the above relations if the original not-squared value of any number is negative, the same algebraic sign is attached after squaring that number. Again if the mean of squared values turns out to be a negative number, negative sign is attached after taking the square root of the unsigned number.\n\nParallel Computer Architecture\n\nThe Suresh's diakoptics along with the technique of relating sub-network solution estimate to get the global solution estimate does not require any communication between processors assigned to solve each sub-network. All processors access the commonly shared memory through possibly separate port for each processor in a multi-port memory organization to speed-up the access. Each processor access the commonly shared memory to write results of local solution of sub-network assigned to contribute to the generation of network-wide or global solution estimate. The generation of global solution estimate marks the end of iteration. The iteration begins by processors accessing latest global solution estimate for local solution of sub-networks assigned. That means only beginning of the solution of sub-networks by assigned processors need to be synchronized in each iteration, which synchronization can be affected by server-processor.\n\nThere are two possible approaches of achieving parallel processing for a problem. The first is to design and develop a solution technique for the best possible parallel processing of a problem and then design parallel computer organization/architecture to achieve it. The second is to design and develop parallel processing of solution technique that can best be carried out on any of the existing available parallel computer. The inventions of this application follow the first approach. The trick is in breaking the large problem into small pieces of sub-problems, and solving sub-problems each on a separate processor simultaneously in parallel, and then relating solution of sub-problems into obtaining global solution of the original whole problem. That exactly is achieved by the inventions of Suresh's diakoptics of breaking the large network into small pieces of sub-networks, and solving sub-networks each on a separate processor simultaneously in parallel, and then by the technique of relating solutions of sub-networks into obtaining network wide global solution of the original whole network.\n\nInvented technique of parallel loadflow computation can best be carried out on invented parallel computer architecture of FIG. 4. The main inventive feature of the architecture of FIG. 4 is that processors are not required to communicate with each other and provision of private local main memory for each processor for local solution of sub-problem for contribution to the generation of network wide global solution in commonly shared main memory. Other applications can be developed that can best be carried out using the parallel computer architecture of FIG. 4.\n\nFIG. 4 is the generalized and simplified block diagram of a multiprocessor computer system comprising few to thousands of processors meaning the value of number ‘n’ in ‘processor-n’ could be small to in the range of thousands. The invention of the server processor-array processor architecture of the computer of FIG. 4 comprises a multiprocessor system with processors and input/output (I/O) adapter coupled, by a common bus arrangement, to the commonly shared main memory. One of the processors is the main/server processor coupled to the I/O adapter, which is only one for the system and coupled to the I/O devices. The I/O adapter and I/O devices are not explicitly shown but are considered to have been included in the block marked I/O unit. Similarly, dedicated cache memories if required for each processor and I/O adapter are considered to have been included in each processor block and the block of I/O unit. Each processor is also provided with its private local main memory for local processing, and it is not visible or accessible to any other processor or I/O unit. The FIG. 4 also explicitly depicts that no communication of any short is required between processors except that each processor communicates only with the main/server processor for control and coordination purposes. All connecting lines with arrows at each end indicates two way asynchronous communication.\n\nDistributed Computing Architecture\n\nThe parallel computer architecture depicted in FIG. 4 land itself into distributed computing architecture. This is achieved when each processor and associated memory forming a self-contained computer in itself is physically located at each network node or a substation, and communicates over communication lines with commonly shared memory and server processor both located at central station or load dispatch center in a power network. It is possible to have an input/output unit with a computer at each network node or substation, which can be used to read local sub-network data in parallel and communicate over communication line to commonly shared memory for the formation and storage of network wide global data at the central load dispatch center in the power network.\n\nCONCLUSION\n\nThe inventions of Suresh's diakoptics, technique of relating local solution of sub-networks into network-wide global solution, and parallel computer architecture depicted in FIG. 4 afford an opportunity for the maximum possible parallelism with minimum possible communication and synchronization requirements. Also parallel computer architecture and parallel computer program are scalable, which is not possible with most of the parallel computers built so far. Moreover, these inventions provide bridging and unifying model for parallel computation.\n\nCalculation Steps for Parallel Gauss-Seidel-Patel Loadflow Method\n\nThe steps of parallel Gauss-Seidel-Patel loadflow (PGSPL) computation method, using invented parallel computer of FIG. 4 are shown in the flowchart of FIG. 3 b. Referring to the flowchart of FIG. 3 b, different steps are elaborated in steps marked with similar numbers in the following. The words “Read system data” in Step-1 correspond to step-10 and step-20 in FIG. 5, and step-14, step-20, step-32, step-44, step-50 in FIG. 6. All other steps in the following correspond to step-30 in FIG. 5, and step-60, step-62, and step-64 in FIG. 6.\n\n• 21. Read system data and assign an initial approximate solution. If better solution estimate is not available, set specified voltage magnitude at PV-nodes, 1.0 p.u. voltage magnitude at PQ-nodes, and all the node angles equal to that of the slack-node, which is referred to as the flat-start. The solution guess is stored in complex voltage vector say, V (I) where “I” takes values from 1 to n, the number of nodes in the whole network.\n• 22. All processors simultaneously access network-wide global data stored in commonly shared memory, which can be under the control of server-processor, to form and locally store required admittance matrix for each sub-network.\n• 23. Initialize complex voltage vector, say VV (I)=CMPLEX (0.0, 0.0) that receives solution contributions from sub-networks.\n• 24. All processors simultaneously access network-wide global latest solution estimate vector V (I) available in the commonly shared memory to read into the local processor memory the required elements of the vector V (I), and perform 2-iterations of the GSPL-method in parallel for each sub-network to calculate node-voltages.\n• 25. As soon as 2-iterations are performed for a sub-network, its new local solution estimate is contributed to the vector VV (I) in commonly shared memory under the control of server processor without any need for the synchronization. It is possible that small sub-network finished 2-iterations and already contributed to the vector VV (I) while 2-iterations are still being performed for the larger sub-network.\n• 26. Contribution from a sub-network to the vector VV (I) means, the complex voltage estimate calculated for the nodes of the sub-network are added to the corresponding elements of the vector VV (I). After all sub-networks finished 2-iterations and contributed to the vector VV (I), its each element is divided by the number of contributions from all sub-networks to each element or divided by number of nodes directly connected to the node represented by the vector element, leading to the transformation of vector VV (I) into the new network-wide global solution estimate. This operation is performed as indicated in relation (30) or (35). This step requires synchronization in that the division operation on each element of the vector VV(I) can be performed only after all sub-networks are solved and have made their contribution to the vector VV(I).\n• 27. Find the maximum difference in the real and imaginary parts of [VV(I)−V(I)]\n• 28. Calculate accelerated value of VV(I) by relation (9) as VV(I)=V(I)+β[VV(I)−V(I)] and perform V(I)=VV(I)\n• 29. If the maximum difference calculated in step-27 is not less than certain solution accuracy tolerance specified as stopping criteria for the iteration process, increment iteration count and go to step-23, or else follow the next step.\n• 30. From calculated values of complex voltage at different power network nodes, and tap position of tap changing transformers, calculate power flows through power network components, and reactive power generation at PV-nodes.\n\nIt can be seen that steps-22, -24, and -25 are performed in parallel. While other steps are performed by the server-processor. However, with the refined programming, it is possible to delegate some of the server-processor tasks to the parallel-processors. For example, any assignment functions of step-21 and step-22 can be performed in parallel. Even reading of system data can be performed in parallel particularly in distributed computing environment where each sub-network data can be read in parallel by substation computers connected to operate in parallel.\n\nCalculation Steps for Parallel Super Super Decoupled Loadflow Method\n\nThe steps of Parallel Super Super Decoupled Loadflow (PSSDL) computation method using invented parallel computer of FIG. 4 are given in the following without giving its flowchart.\n\n• 41. Read system data and assign an initial approximate solution. If better solution estimate is not available, set all node voltage magnitudes and all node angles equal to those of the slack-node, which is referred to as the slack-start. The solution guess is stored in voltage magnitude and angle vectors say, VM (I) and VA(I) where “I” takes values from 1 to n, the number of nodes in the whole network.\n• 42. All processors simultaneously access network-wide global data stored in commonly shared memory, which can be under the control of server-processor to form and locally store required admittance matrix for each sub-network. Form gain matrices of SSDL-method for each sub-network, factorize and store them locally in the memory associated with each processor.\n• 43. Initialize vectors, say DVM (I)=0.0, and DVA(I)=0.0 that receives respectively voltage magnitude corrections and voltage angle corrections contributions from sub-networks.\n• 44. Calculate real and reactive power mismatches for all the nodes in parallel, find real power maximum mismatch and reactive power maximum mismatch by the server-computer. If both the maximum values are less then convergence tolerance specified, go to step-49. Otherwise, follow the next step.\n• 45. All processors simultaneously access network-wide global latest solution estimate VM(I) and VA(I) available in the commonly shared memory to read into the local processor memory the required elements of the vectors VM(I) and VA(I), and perform 1-iteration of SSDL-method in parallel for each sub-network to calculate node-voltage-magnitudes and node-voltage-angles.\n• 46. As soon as 1-iteration is performed for a sub-network, its new local solution corrections estimate are contributed to the vectors DVM (I) and DVA(I) in commonly shared memory under the control of server processor without any need for the synchronization. It is possible that small sub-network finished 1-iteration and already contributed to the vectors DVM (I) and DVA(I) while 1-iteration is still being performed for the larger sub-network.\n• 47. Contribution from a sub-network to the vectors DVM (I) and DVA(I) means, the complex voltage estimate calculated for the nodes of the sub-network are added to the corresponding elements of the vectors DVM (I) and DVA(I). After all sub-networks finished 1-iteration and contributed to the vectors DVM (I) and DVA(I), its each element is divided by the number of contributions from all sub-networks to each element or divided by number of nodes directly connected to the node represented by the vector element, leading to the transformation of vectors DVM (I) and DVA(I) into the new network-wide global solution correction estimates. This operation is performed as indicated in relation (31) and (32) or (38) and (39). This step requires synchronization in that the division operation on each element of the vectors DVM (I) and DVA(I) can be performed only after all sub-networks are solved and made their contribution to the vectors DVM (I) and DVA(I).\n• 48. Update solution estimates VM(I) and VA(I), and proceed to step-43\n• 49. From calculated values of complex voltage at different power network nodes, and tap position of tap changing transformers, calculate power flows through power network components, and reactive power generation at PV-nodes.\n\nIt can be seen that steps-42, -44, and -45 are performed in parallel. While other steps ate performed by the server-processor. However, with the refined programming, it is possible to delegate some of the server-processor tasks to the parallel-processors. For example, any assignment functions such as in step-43 can be performed in parallel. Even reading of system data can be performed in parallel particularly in distributed computing environment where each sub-network data can be read in parallel by substation computers connected to operate in parallel.\n\nGeneral Statements\n\nThe system stores a representation of the reactive capability characteristic of each machine and these characteristics act as constraints on the reactive power, which can be calculated for each machine.\n\nWhile the description above refers to particular embodiments of the present invention, it will be understood that many modifications may be made without departing from the spirit thereof. The accompanying claims are intended to cover such modifications as would fall within the true scope and spirit of the present invention.\n\nThe presently disclosed embodiments are therefore to be considered in all respect as illustrative and not restrictive, the scope of the invention being indicated by the appended claims in addition to the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.\n\nREFERENCES\n\nForeign Patent Document\n\n• 1. U.S. Pat. No. 4,868,410 dated Sep. 19, 1989: “System of Load Flow Calculation for Electric Power System”\n• 2. U.S. Pat. No. 5,081,591 dated Jan. 14, 1992: “Optimizing Reactive Power Distribution in an Industrial Power Network”\nPublished Pending Patent Applications\n• 3. Canadian Patent Application Number: CA2107388 dated 9 Nov., 1993: “System of Fast Super Decoupled Loadflow Calcutation for Electrical Power System”\n• 4. International Patent Application Number: PCT/CA/2003/001312 dated 29 Aug., 2003: “System of Super Super Decoupled Loadflow Computation for Electrical Power System”\nOther Publications\n• 5. Stagg G. W. and El-Abiad A. H., “Computer methods in Power System Analysis”, McGrow-Hill, New York, 1968\n• 6. S. B. Patel, “Fast Super Decoupled Loadflow”, IEE proceedings Part-C, Vol. 139, No. 1, pp. 13-20, January 1992\n• 7. Shin-Der Chen, Jiann-Fuh Chen, “Fast loadflow using multiprocessors”, Electrical Power & Energy Systems, 22 (2000) 231-236\n\n## Claims (4)\n\n1. A method of forming/defining and solving a model of a power network to affect control of voltages and power flows in a power system, comprising the steps of:\nobtaining on-line/simulated data of open/close status of switches and circuit breakers in the power network, and reading data of operating limits of components of the power network including PV-node, a generator-node where Real-Power-P and Voltage-Magnitude-V are given/assigned/specified/set, maximum and minimum reactive power generation capability limits of generators, and transformers tap position limits,\nobtaining on-line readings of given/assigned/specified/set Real-Power-P and Reactive-Power-Q at PQ-nodes, Real-Power-P and voltage-magnitude-V at PV-nodes, voltage magnitude and angle at a reference/slack node, and transformer turns ratios, wherein said on-line readings are the controlled variables/parameters,\ninitiating loadflow computation with initial approximate/guess solution of the same voltage magnitude and angle as those of the reference/slack node for all the PQ-nodes and the PV-nodes, said initial approximate/guess solution is referred to as a slack-start,\nperforming loadflow computation to calculate complex voltages or their real and imaginary components or voltage magnitude corrections and voltage angle corrections at nodes of the power network providing for calculation of power flow through different components of the power network, and to calculate reactive power generation and transformer tap-position indications,\ndecomposing the power network for performing said loadflow computation in parallel by a method referred to as Suresh's diakoptics that involves determining a sub-network for each node involving directly connected nodes referred to as level-1 nodes and directly connected nodes to level-1 nodes referred to as level-2 nodes, and a level of outward connectivity for local solution of a sub-power-network around a given node is determined experimentally,\ninitializing, at the beginning of each new iteration, a vector of dimension equal to the number of nodes in the power network with each element value zero,\nsolving all sub-networks in parallel using available solution estimate at the start of the iteration,\nadding newly calculated solution estimates or corrections to the available solution estimate for a node resulting from different sub-networks, ‘q’ number of sub-networks, in which a node is contained, in a corresponding vector element that gets initialized zero at the beginning of each new iteration,\ncounting the number of additions and calculating new solution estimate or corrections to the available solution estimate by taking the average or root mean square value using any relevant relations (30) to (39) in the following depending on the loadflow computation method used, and\nstoring the new solution estimate at the end of the current iteration as initial available estimate for the next iteration,\nwherein said Suresh's diakoptics method uses the following relations,\n$V p ( r + 1 ) = ( V p ⁢ ⁢ 1 ( r + 1 ) + V p ⁢ ⁢ 2 ( r + 1 ) + V p ⁢ ⁢ 3 ( r + 1 ) + … + V pq ( r + 1 ) ) / q ( 30 ) Δθ p ( r + 1 ) = ( Δθ p ⁢ ⁢ 1 ( r + 1 ) + Δθ p ⁢ ⁢ 2 ( r + 1 ) + Δθ p ⁢ ⁢ 3 ( r + 1 ) + … + Δθ pq ( r + 1 ) ) / q ( 31 ) Δ ⁢ V p ( r + 1 ) = ( Δ ⁢ V p ⁢ ⁢ 1 ( r + 1 ) + Δ ⁢ V p ⁢ ⁢ 2 ( r + 1 ) + Δ ⁢ V p ⁢ ⁢ 3 ( r + 1 ) + … + Δ ⁢ V pq ( r + 1 ) ) / q ( 32 ) e p ( r + 1 ) = ( e p ⁢ ⁢ 1 ( r + 1 ) + e p ⁢ ⁢ 2 ( r + 1 ) + e p ⁢ ⁢ 3 ( r + 1 ) + … + e pq ( r + 1 ) / q ( 33 ) f p ( r + 1 ) = ( f p ⁢ ⁢ 1 ( r + 1 ) + f p ⁢ ⁢ 2 ( r + 1 ) + f p ⁢ ⁢ 3 ( r + 1 ) + … + f pq ( r + 1 ) ) / q ( 34 )$\nwherein relations (30) to (34), can also alternatively be written as relations (35) to (39) as below,\n$V p ( r + 1 ) = ( Re ⁡ ( ( V p ⁢ ⁢ 1 ( r + 1 ) 2 ) 2 ) + Re ⁡ ( ( V p ⁢ ⁢ 2 ( r + 1 ) ) 2 ) + … + Re ⁡ ( ( V pq ( r + 1 ) ) 2 ) / q + j ⁢ ( Im ( ( V p ⁢ ⁢ 1 ( r + 1 ) ) 2 ) + Im ( ( V p ⁢ ⁢ 2 ( r + 1 ) ) 2 ) + … + Im ( ( V pq ( r + 1 ) ) 2 ) ) / q ( 35 ) Δθ p ( r + 1 ) = ( Δθ p ⁢ ⁢ 1 ( r + 1 ) ) 2 + ( Δθ p ⁢ ⁢ 2 ( r + 1 ) ) 2 + … + ( Δθ pq ( r + 1 ) ) 2 ) / q ( 36 ) Δ ⁢ V p ( r + 1 ) = ( ΔV p ⁢ ⁢ 1 ( r + 1 ) ) 2 + ( ΔV p ⁢ ⁢ 2 ( r + 1 ) ) 2 + … + ( ΔV pq ( r + 1 ) ) 2 ) / q ( 37 ) e p ( r + 1 ) = ( ( e p ⁢ ⁢ 1 ( r + 1 ) ) 2 + ( e p ⁢ ⁢ 2 ( r + 1 ) ) 2 + … + ( e pq ( r + 1 ) ) 2 ) / q ( 38 ) f p ( r + 1 ) = ( ( f p ⁢ ⁢ 1 ( r + 1 ) ) 2 + ( f p ⁢ ⁢ 2 ( r + 1 ) ) 2 + … + ( f pq ( r + 1 ) ) 2 ) / q ( 39 )$\nwherein, square of any positive or negative number being positive, if the original not-squared value of any number is negative, the same algebraic sign is attached after squaring that number, and if the mean of squared values turns out to be a negative number, negative sign is attached after taking the square root of the unsigned number, Vp, θp are voltage magnitude and voltage angle at node-p, ep and fp are the real and imaginary parts of the complex voltage Vp of node-p, symbol Δ before any of defined electrical quantities defines the change in the value of electrical quantity, and superscript ‘r’ indicates the iteration count,\nevaluating loadflow computation for any over loaded components of the power network and for under/over voltage at any of the nodes of the power network,\ncorrecting one or more controlled parameters and repeating the performing loadflow computation by decomposing, initializing, solving, adding, counting, storing, evaluating, and correcting steps until evaluating step finds no over loaded components and no under/over voltages in the power network, and\naffecting a change in power flow through components of the power network and voltage magnitudes and angles at the nodes of the power network by actually implementing the finally obtained values of controlled variables/parameters after evaluating step finds a good power system or alternatively the power network without any overloaded components and under/over voltages, which finally obtained controlled variables/parameters however are stored for acting upon fast in case a simulated event actually occurs.\n2. A method as defined in claim 1 wherein the loadflow computation method referred to as Gauss-Seidel-Patel Loadflow (GSPL) computation method is characterized in using self-iteration denoted by ‘sr’ within a network-wide/sub-network-wide global iteration depicted by ‘r’ in the GSPL model defined by equation (27) given in the following,\n$( V p ( sr + 1 ) ) ( r + 1 ) = [ { ( PSH p - j ⁢ ⁢ QSH p ) / ( ⁢ ( ⁢ V p * ) sr ) r } - ∑ q = 1 p - 1 ⁢ Y pq ⁢ V q ( r + 1 ) - ∑ q = p + 1 n ⁢ Y pq ⁢ V q r ] / Y pp ( 27 )$\nwherein, PSHp and QSHp are scheduled/specified/known/set real and reactive power, Vp is the complex node-p voltage, and Ypq and Ypp are off-diagonal and diagonal complex elements of the network admittance matrix.\n3. A method as defined in claim 1 wherein a parallel loadflow computation is performed using a parallel computer: a server processor-array processors architecture, wherein each of the array processors send communication to and receive communication from only the server processor, commonly shared memory locations, and each processor's private memory locations, but not among themselves.\n4. A multiprocessor computing apparatus for performing the said parallel loadflow computation as defined in claim 1 comprising in combination:\na plurality of processing units adapted to receive and process data, instructions and control signals, and connected to common system bus in parallel asynchronous fashion;\na plurality of local private main memory means for storing the data, instructions and control signals, each said main memory means being directly and asynchronously connected to each said processing unit;\ncommon shared memory coupled directly to said common system bus for sending/receiving the data, instructions and control signals asynchronously to/from each said processing unit, without providing inter-processor communications;\nI/O adapter/control unit coupled directly and asynchronously to a main/server processor, which is one of the said plurality of processing units;\nwherein said I/O adapter/control unit coupled directly and asynchronously to each of said plurality of processing units physically located at far distances in case of said multiprocessor computing apparatus organized for distributed processing.\nUS10/594,715 2004-10-01 2005-09-30 Method and apparatus for parallel loadflow computation for electrical power system Active US7788051B2 (en)\n\n## Priority Applications (3)\n\nApplication Number Priority Date Filing Date Title\nCA2479603 2004-10-01\nCA002479603A CA2479603A1 (en) 2004-10-01 2004-10-01 Sequential and parallel loadflow computation for electrical power system\nPCT/CA2005/001537 WO2006037231A1 (en) 2004-10-01 2005-09-30 System and method of parallel loadflow computation for electrical power system\n\n## Publications (2)\n\nPublication Number Publication Date\nUS20070203658A1 US20070203658A1 (en) 2007-08-30\nUS7788051B2 true US7788051B2 (en) 2010-08-31\n\n# Family\n\n## Family Applications (1)\n\nApplication Number Title Priority Date Filing Date\nUS10/594,715 Active US7788051B2 (en) 2004-10-01 2005-09-30 Method and apparatus for parallel loadflow computation for electrical power system\n\n## Country Status (3)\n\nCountry Link\nUS (1) US7788051B2 (en)\nCA (1) CA2479603A1 (en)\nWO (1) WO2006037231A1 (en)\n\n## Cited By (8)\n\n* Cited by examiner, † Cited by third party\nPublication number Priority date Publication date Assignee Title\nUS20100217550A1 (en) * 2009-02-26 2010-08-26 Jason Crabtree System and method for electric grid utilization and optimization\nUS20120078436A1 (en) * 2010-09-27 2012-03-29 Patel Sureshchandra B Method of Artificial Nueral Network Loadflow computation for electrical power system\nUS20140228976A1 (en) * 2013-02-12 2014-08-14 Nagaraja K. 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Configuration management system & method\nWO2004023622A2 (en) * 2002-09-03 2004-03-18 Sureshchandra Patel System of super super decoupled loadflow computation for electrical power system\nUS20080281474A1 (en) * 2002-09-03 2008-11-13 Patel Sureshchandra B System of Super Super Decoupled Loadflow Computation for Electrical Power System\nUS20060111860A1 (en) * 2002-11-06 2006-05-25 Aplicaciones En Informatica Avanzada, S.A. System and method for monitoring and managing electrical power transmission and distribution networks\nWO2008025162A1 (en) * 2007-08-27 2008-03-06 Sureshchandra Patel System and method of loadflow calculation for electrical power system\n\n## Non-Patent Citations (5)\n\n* Cited by examiner, † Cited by third party\nTitle\nAllan, R.N. et al. \"LTC Transformers and MVAR Violations in the Fast Decoupled Load Flow,\" IEEE Transactions on Power Apparatus and Systems, vol. PAS-101, Issue 9, Sep. 1982, pp. 3328-3332. \nPatel S.B., \"Super Super Decoupled Loadflow\", proceedings of the IEEE Toronto International Conference on Science and Technology for Humanity (TIC-STH 2009), Sep. 2009, pp. 652-659. \nPatel S.B., \"Transformation Based Fast Decoupled Loadflow,\" IEEE Region 10 International Conference on EC3-Energy, Computer, Communication and Control Systems, vol. 1, Aug. 28-30, 1991, pp. 183-187. \nPatel, S.B., \"Fast super decoupled loadflow,\" IEEE Proceedings on Generation, Transmission, and Distribution, vol. 139, Issue 1, Jan. 1992, pp. 13-20. \nVan Amerongen, R.A.M., \"A general-purpose version of the fast decoupled load flow,\" IEEE Transactions on Power Systems, vol. 4, Issue 2, May 1989, pp. 760-770. \n\n## Cited By (10)\n\n Cited by examiner, † Cited by third party\nPublication number Priority date Publication date Assignee Title\nUS8897923B2 (en) 2007-12-19 2014-11-25 Aclara Technologies Llc Achieving energy demand response using price signals and a load control transponder\nUS20100217550A1 (en) * 2009-02-26 2010-08-26 Jason Crabtree System and method for electric grid utilization and optimization\nUS20120078436A1 (en) * 2010-09-27 2012-03-29 Patel Sureshchandra B Method of Artificial Nueral Network Loadflow computation for electrical power system\nUS8756047B2 (en) * 2010-09-27 2014-06-17 Sureshchandra B Patel Method of artificial nueral network loadflow computation for electrical power system\nUS20140228976A1 (en) * 2013-02-12 2014-08-14 Nagaraja K. S. Method for user management and a power plant control system thereof for a power plant system\nUS9891827B2 (en) 2013-03-11 2018-02-13 Sureshchandra B. Patel Multiprocessor computing apparatus with wireless interconnect and non-volatile random access memory\nUS10365310B2 (en) * 2014-06-12 2019-07-30 National Institute Of Advanced Industrial Science And Technology Impedance estimation device and estimation method for power distribution line\nUS10197606B2 (en) 2015-07-02 2019-02-05 Aplicaciones En Informática Avanzada, S.A System and method for obtaining the powerflow in DC grids with constant power loads and devices with algebraic nonlinearities\nCN105205244A (en) * 2015-09-14 2015-12-30 国家电网公司 Closed loop operation simulation system based on electromechanical-electromagnetic hybrid simulation technology\nCN105205244B (en) * 2015-09-14 2018-10-30 国家电网公司 Electromagnetic Simulation Technology rings operating systems Simulation - Based Electromechanical\n\n## Also Published As\n\nPublication number Publication date\nWO2006037231A8 (en) 2006-10-05\nCA2479603A1 (en) 2006-04-01\nWO2006037231A1 (en) 2006-04-13\nUS20070203658A1 (en) 2007-08-30\n\n## Similar Documents\n\nPublication Publication Date Title\nDa Silva et al. 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TCSC allocation based on line flow based equations via mixed-integer programming\n\n## Legal Events\n\nDate Code Title Description\nSTCF Information on status: patent grant\n\nFree format text: PATENTED CASE\n\nFPAY Fee payment\n\nYear of fee payment: 4\n\nMAFP Maintenance fee payment\n\nFree format text: PAYMENT OF MAINTENANCE FEE, 8TH YEAR, MICRO ENTITY (ORIGINAL EVENT CODE: M3552)\n\nYear of fee payment: 8" ]	[ null, "https://patentimages.storage.googleapis.com/92/21/b9/4787fd0fa9697b/US07788051-20100831-D00000.png", null, "https://patentimages.storage.googleapis.com/58/7d/b7/582691c0e0dfe2/US07788051-20100831-D00001.png", null, "https://patentimages.storage.googleapis.com/08/39/bb/fb441d1ea871e9/US07788051-20100831-D00002.png", null, "https://patentimages.storage.googleapis.com/cc/e3/af/7f705c58112ab4/US07788051-20100831-D00003.png", null, "https://patentimages.storage.googleapis.com/cc/b7/2b/a7bf70bdfee20b/US07788051-20100831-D00004.png", null, "https://patentimages.storage.googleapis.com/b4/d0/22/82f5dc9501e648/US07788051-20100831-D00005.png", null, "https://patentimages.storage.googleapis.com/a0/d4/34/4edcf95ad0c20a/US07788051-20100831-D00006.png", null, "https://patentimages.storage.googleapis.com/b1/6b/d0/6fd8060da31199/US07788051-20100831-D00007.png", null, "https://patentimages.storage.googleapis.com/3f/e5/b4/4546d72ee0cda8/US07788051-20100831-D00008.png", null, "https://patentimages.storage.googleapis.com/f8/a6/5c/cbd084153a94d3/US07788051-20100831-D00009.png", null, "https://patentimages.storage.googleapis.com/eb/c6/3f/95607123822fa9/US07788051-20100831-D00010.png", null, "https://patentimages.storage.googleapis.com/13/85/fc/6091b4c8b5ab5b/US07788051-20100831-D00011.png", null, "https://patentimages.storage.googleapis.com/a1/e6/6b/ace10adedb874e/US07788051-20100831-D00012.png", null, "https://patentimages.storage.googleapis.com/07/ba/c7/4d2da669a90d65/US07788051-20100831-D00013.png", null, "https://patentimages.storage.googleapis.com/57/70/24/9a852fad2ad16e/US07788051-20100831-D00014.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82650137,"math_prob":0.89247894,"size":77748,"snap":"2019-35-2019-39","text_gpt3_token_len":18962,"char_repetition_ratio":0.1689648,"word_repetition_ratio":0.25801325,"special_character_ratio":0.24948552,"punctuation_ratio":0.07639095,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9610973,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-25T02:41:04Z\",\"WARC-Record-ID\":\"<urn:uuid:05242c35-b697-4298-b71c-b9656358ebd6>\",\"Content-Length\":\"313620\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:47f15db0-5cf1-450c-8524-a9a47a8bf516>\",\"WARC-Concurrent-To\":\"<urn:uuid:6522c112-0452-4860-92af-c7b490634944>\",\"WARC-IP-Address\":\"172.217.12.238\",\"WARC-Target-URI\":\"https://patents.google.com/patent/US7788051B2/en\",\"WARC-Payload-Digest\":\"sha1:5O536QKPQGMBFEUWW65R3PO5B4M5DV6D\",\"WARC-Block-Digest\":\"sha1:AUNIKNT44AEWZUXDINMSQRTV7FX3Z73S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027322170.99_warc_CC-MAIN-20190825021120-20190825043120-00129.warc.gz\"}"}
https://zh.m.wikipedia.org/wiki/%E9%85%8D%E4%BD%8D	[ "# 配位键\n\n（重定向自配位\n\n## 相关概念\n\n### 形成条件\n\n• 一是中心原子或离子，它必须有能接受电子对的空轨域；\n• 二是配位体，组成y配位体的原子必须能提供配对的孤对电子(L.P)。\n\n${\\ce {NH3(g)}}$ $+$ ${\\ce {BF3(g) ->NH3BF3(s)}}$\n\n## 常见配位键化合物\n\n• 一氧化碳${\\ce {CO}}$ ，其中碳氧間的三對共用電子對有一配位鍵，兩個正常共價鍵。\n• 铵根${\\ce {NH4+}}$ ，其中N原子与左下右的${\\ce {H}}$ 原子以极性键结合，与上方的${\\ce {H}}$ 以配位键结合，由${\\ce {N}}$ 原子提供一对價电子。\n• 水合氫離子${\\ce {H3O+}}$" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.98866093,"math_prob":0.9998661,"size":421,"snap":"2020-34-2020-40","text_gpt3_token_len":451,"char_repetition_ratio":0.09832134,"word_repetition_ratio":0.0,"special_character_ratio":0.16152018,"punctuation_ratio":0.023809524,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96885806,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-10T04:06:59Z\",\"WARC-Record-ID\":\"<urn:uuid:04b9cb04-69da-465d-9fcc-5a96026b3f33>\",\"Content-Length\":\"45729\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b2f45f60-3dfa-4922-a684-d0bdf482edfd>\",\"WARC-Concurrent-To\":\"<urn:uuid:42c6d51e-99e5-4eff-b6e3-2726eba68644>\",\"WARC-IP-Address\":\"208.80.154.224\",\"WARC-Target-URI\":\"https://zh.m.wikipedia.org/wiki/%E9%85%8D%E4%BD%8D\",\"WARC-Payload-Digest\":\"sha1:4N3MOJSRCRCOOXREJNVGWYCXBRNTVDHT\",\"WARC-Block-Digest\":\"sha1:LZCYCSPJ32J6RZPFNXQROHGYCTVAL36H\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738603.37_warc_CC-MAIN-20200810012015-20200810042015-00210.warc.gz\"}"}
https://japanese.stackexchange.com/questions/19619/why-in-this-example-sentence-theres-no-difference-between-%E3%81%94%E3%81%A8%E3%81%AB-and-%E3%81%8A%E3%81%8D%E3%81%AB	[ "# Why in this example sentence there's no difference between ごとに and おきに?\n\nI am using the \"A Dictionary of Basic Japanese Grammar\" to help me with Japanese studies. As explained from page 128 on I understood the difference between ごとに and おきに.\n\nFor example:\n\n• ２日おきに = every third day\n• ２日ごとに = every second day\n\nThe Answer what is the difference between ごとに and おきに? also backs this understanding.\n\nNow in the mentioned grammar book there is one last example where both forms do yield the same meaning.\n\nQuoting:\n\nWhen a time expression precedes oki no or goto ni, there is no difference in meaning, if an event takes place at one point in time:\n[電車]{でんしゃ}は[五分]{ごふん}おきに／ごとに[出]{で}る = The train leaves every five minutes.\n\nI don't get the difference between this \"time expression\" and the other ones like 二日. Can somebody give me a hint here?\n\n• I don't think that anything special with Japanese. Time as measured in minutes/seconds/... is continous. If measured as an amount of days, it's discrete. Let A and B be two points in time, then `there are days between A (eg Monday) and B (eg Thursday)` and `there are 5 minutes between A (eg 1:00am) and B (eg 1:05am)` use the same expression in English as well. Nov 21, 2014 at 16:36\n• @blutorange You can leave that as an answer, if you like.\n– user1478\nNov 21, 2014 at 17:37\n• I must have read a paper on this topic years ago. In short, the difference pretty much depends on how you perceive the noun. If you view it as a mass noun, then ごとに and おきに are the same. If you view it as a countable noun, the case becomes more complicated, depending on whether two types of things or a single type is involved. For example, オリンピック and 年 are two types of things, so there are 4年 between two オリンピック. But we often think days are time slots, which can be assigned to something. Therefore 2日おきに is more like 1日 between 2日, which only involves only a single type of noun. Nov 21, 2014 at 21:47\n• An even more impressive example than the Olympic Games is imo 4年おきに閏年を入れる. Nov 21, 2014 at 22:59\n\n`毎【ごと】に` means \"every\", so `2日ごとに` is \"every second day\".\n\nOn the other hand, `X置【お】きに` literally means \"leaving (an amount of time/space/...) X (between each occurence)\". It comes from the verb `置く`, \"to put\", \"to place\", \"to leave (sth. somewhere)\".\n\nHere is an article from NHK's 身近なことばの疑問にお答えします about ごとに and おきに.\n\nSo how come `おきに` sometimes means the same as `ごとに`, and sometimes not? Let's think about English for a moment, the same phenomenon happens in English as well.\n\nLet A and B be two events separated by a certain amount of time. How much time is there between A and B? You might be tempted to answer `B minus A`, but there are two different answers, depending on how we count time:\n\n(a) There are 5 minutes between A = 1:05 am and B = 1:10 am. B - A = 5 minutes. This is 5分おきに - leaving 5 minutes between A and B.\n\n(b) There are 2 days between A = Monday and B = Thursday. B - A = 3 days, not(!) two. This is 2日おきに - leaving two days between A and B.\n\nThe difference between these two cases is that in (a), time as measured in hours, minutes, seconds etc. is considered continous (uncountable) - there's 5 minutes, 5.3 minutes, 5.000321 minutes and so on. In scenario (b), time counted as weekdays or as a number of days is considered discrete (countable) - there's Monday and Tuesday, but nothing in between, we don't talk about `Monday and a half`.\n\nTo illustrate this point:\n\n``````1 2 3 4 5\n``````\n\nHow many numbers are there between 1 and 5? Three, namely 2, 3, and 4.\n\n``````The scale of a measuring cup (for water)\n\n\|---+---+---+---+---\|\n0 1 2 3 4 5 (deciliter)\n``````\n\nHow many milliliters are there between 1 dl and 5 dl? The answer is 400 ml = 4 dl, not 3 dl.\n\nConclusion:\n\nBoth `おきに` and `ごとに` have got only one basic meaning. Depending on the noun they apply to, both can refer to the same (temporal) interval.\n\n• 2日ごとに every 2 (week) days\n• 2分ごとに every 2 minutes\n• 2日おきに 2 (week) days between each occurence\n• 2分おきに 2 minutes between each occurence\n\nA day consists of 24 hours. Note the difference between `48 hours between two events` and `2 days between two events`. And would you say that there are 2880 minutes between Monday and Thursday?\n\nIf you were looking at a clock face with each minute marked individually, `2分おきに` might mean something different.\n\nNote that this is not limited to temporal intervals. `一行おきに書く` \"write on every other line\", `5メートルおきに杭【くい】を立てる` \"place stakes with a space of five meters in between\".\n\nTo put it another way, you take the open interval (A,B). In the case of a continous variable, everything just a split second after 1:00 am is part of that interval, and thus you get 5 minutes between 1:00 am and 1:05 am. But in the case of a discrete variable, you get less. Consider (Monday,Thursday), Monday midnight + 1hour is not part of the interval because you're counting only in days, but not hours. There's only Monday and Tuesday, but nothing in between. So you get only two days inside the interval (Monday,Thursday), even though Thursday is 3 days after Monday.\n\n• For the record, my (non-linguistic) explanation for the case Yang Muye mentioned would be the different length of the two types of things involved, `Olympic Games` and `years`. The Olympic Games take place over the course of a few weeks, much shorter than a year, so instead of year-time-slots, we think about the continous time (mass noun) between two Olympic Games. Which happens to be approx. 4.0 (not 4) years, or ~208 weeks. Thus you can say 「オリンピックは４年おきに開催される」. Nov 21, 2014 at 22:21\n• I forget to mention that when `X` is viewed as slots, Xごとに is actually `every X on average`. Nov 21, 2014 at 22:48" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95477784,"math_prob":0.93652374,"size":3118,"snap":"2023-40-2023-50","text_gpt3_token_len":924,"char_repetition_ratio":0.13712268,"word_repetition_ratio":0.013961606,"special_character_ratio":0.27774215,"punctuation_ratio":0.13323353,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9636991,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T20:43:59Z\",\"WARC-Record-ID\":\"<urn:uuid:26091a4e-a495-4f54-b7c0-2fcbf7fe7ed9>\",\"Content-Length\":\"157268\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dcd61298-291d-40f6-9c0e-f670975c1d2a>\",\"WARC-Concurrent-To\":\"<urn:uuid:327d3789-bc12-494f-94c6-2aeeb12487a8>\",\"WARC-IP-Address\":\"172.64.144.30\",\"WARC-Target-URI\":\"https://japanese.stackexchange.com/questions/19619/why-in-this-example-sentence-theres-no-difference-between-%E3%81%94%E3%81%A8%E3%81%AB-and-%E3%81%8A%E3%81%8D%E3%81%AB\",\"WARC-Payload-Digest\":\"sha1:FTFQM4Y5TK3HWIL2RM4QU7JVFB2FTUKG\",\"WARC-Block-Digest\":\"sha1:FDSWM7EDRY3256RWBHS67IRDH3FQ3XPA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100769.54_warc_CC-MAIN-20231208180539-20231208210539-00383.warc.gz\"}"}
https://plainmath.net/16640/solve-the-equation-%E2%88%927-%E2%88%923n-plus-3-equal-%E2%88%9221-plus-21n	[ "Question", null, "# Solve the equation −7(−3n+3)=−21+21n\n\nEquations and inequalities\nANSWERED", null, "Solve the equation −7(−3n+3)=−21+21n", null, "From the given equation we have $$\\displaystyle-{7}{\\left(-{3}{n}+{3}\\right)}=-{21}+{21}{n}\\to{21}{n}-{21}=-{21}+{21}{n}\\to{21}{n}={21}{n}$$" ]	[ null, "https://plainmath.net/qa-theme/BTMath/images/search.png", null, "https://plainmath.net/", null, "https://plainmath.net/", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7519537,"math_prob":1.000009,"size":445,"snap":"2021-31-2021-39","text_gpt3_token_len":182,"char_repetition_ratio":0.17687075,"word_repetition_ratio":0.6666667,"special_character_ratio":0.48539326,"punctuation_ratio":0.021276595,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99986553,"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\":\"2021-09-21T02:42:24Z\",\"WARC-Record-ID\":\"<urn:uuid:63dbddc9-0358-4a19-a0b9-496952bca5c0>\",\"Content-Length\":\"32654\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4cdb5b98-d071-4da8-8c57-ea98271da03e>\",\"WARC-Concurrent-To\":\"<urn:uuid:a57d87ad-10e9-4fba-b96f-f20437e4508f>\",\"WARC-IP-Address\":\"172.67.217.47\",\"WARC-Target-URI\":\"https://plainmath.net/16640/solve-the-equation-%E2%88%927-%E2%88%923n-plus-3-equal-%E2%88%9221-plus-21n\",\"WARC-Payload-Digest\":\"sha1:42KIAH4JBNCC2HEOY3YTX7QVIBCVNHM6\",\"WARC-Block-Digest\":\"sha1:J3VWMRCSG72OYTYTW4LGUUTPN6O4MB3R\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057131.88_warc_CC-MAIN-20210921011047-20210921041047-00708.warc.gz\"}"}
https://www.prodekdecoracion.com.mx/category/bad-credit-installment-loans-indiana-2/	[ "### Exactly what are the needs to borrow a SELF Loan?\n\nExactly what are the needs to borrow a SELF Loan? Exactly what are the needs to borrow a SELF Loan? What’s the interest rate that is current? Just how much am I able to borrow? So how [...]" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.920122,"math_prob":0.5377447,"size":263,"snap":"2021-21-2021-25","text_gpt3_token_len":65,"char_repetition_ratio":0.16602316,"word_repetition_ratio":0.33333334,"special_character_ratio":0.26615968,"punctuation_ratio":0.14516129,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9669525,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-19T11:53:08Z\",\"WARC-Record-ID\":\"<urn:uuid:1d2e5a31-69e6-497b-b837-3d1e9b38924e>\",\"Content-Length\":\"51494\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:707051c6-abf9-44ff-9b56-7a502c5f213b>\",\"WARC-Concurrent-To\":\"<urn:uuid:8ecc85ee-78fe-4df7-b3a8-302f7e7545a2>\",\"WARC-IP-Address\":\"75.119.211.233\",\"WARC-Target-URI\":\"https://www.prodekdecoracion.com.mx/category/bad-credit-installment-loans-indiana-2/\",\"WARC-Payload-Digest\":\"sha1:OFP3P3QD2G43WMATT4K23GTMIZS7GED5\",\"WARC-Block-Digest\":\"sha1:OCVBZJDBCX3QEW4BME2NVRY3UNYXTFU3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487648194.49_warc_CC-MAIN-20210619111846-20210619141846-00199.warc.gz\"}"}
https://calculator.academy/phasor-calculator/	[ "# Phasor Calculator\n\nEnter the real and imaginary numbers of a rectangular form into the phasor calculator. The calculator will return the phasor value of that equation.\n\n## Phasor Formula\n\nThe following formula is used to convert a rectangular form to a phasor form.\n\nP = arctan (y/x)\n\n• Where P is the phasor angle\n• y is the imaginary part of the rectangular form x + jy\n• x is the real part of the rectangular form\n\nHow to calculate a phasor angle\n\n1. First, determine the rectangular form\n\nThis will be an equation of the form x + jy, where x is the real part and y is the imaginary part.\n\n2. Next, separate x and y from the equation in step 1\n\nTake out the real and imaginary portions.\n\n3. Calculate the phasor\n\nEnter the x and y components into the formula above.\n\n## FAQ\n\nWhat is rectangular form?\n\nA rectangular form is a way to represent a vector in terms of a real and imaginary part.\n\nWhat is a phasor?\n\nA phasor is an angle that represents the angle between the x-axis and a vector." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81890005,"math_prob":0.9960997,"size":1524,"snap":"2020-45-2020-50","text_gpt3_token_len":366,"char_repetition_ratio":0.23421052,"word_repetition_ratio":0.015936255,"special_character_ratio":0.19947506,"punctuation_ratio":0.049808428,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99981827,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-27T09:14:40Z\",\"WARC-Record-ID\":\"<urn:uuid:414bdf67-a529-4b8e-b7d5-f5d8a20e6987>\",\"Content-Length\":\"93816\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8fcdb3d4-174c-42aa-a992-19ff3aa314c6>\",\"WARC-Concurrent-To\":\"<urn:uuid:0eb3bc80-6e0e-4bb9-861f-e75189eec738>\",\"WARC-IP-Address\":\"18.232.245.187\",\"WARC-Target-URI\":\"https://calculator.academy/phasor-calculator/\",\"WARC-Payload-Digest\":\"sha1:TKLGXWHHWIKNGMSXAKTHK75WPVJUDLRY\",\"WARC-Block-Digest\":\"sha1:K5SL7HIWAEYB5ELF4DLCCG5GKRQRG5H3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141191511.46_warc_CC-MAIN-20201127073750-20201127103750-00556.warc.gz\"}"}
https://www.meanings4all.com/2021/05/1-million-means-or-1-million-meaning.html	[ "# 1 Million Means \| 1 Million Meaning \| 1 Million in Rupees, Lakhs and Crores\n\n## What does 1 Million Means?\n\nMillion is a number equal to 1 followed by 6 zeros\n\n(or)\n\n1 Million is the product of thousand and thousand\n\nie 1,000 * 1,000 = 1,000,000.\n\nThe word “million” is derived from the early Italian million (milione in modern Italian), from mille, “thousand”, plus the augmentative suffix -one. It is abbreviated as m.\n\n1 Million in figures = 1,000,000\n\n1 Million in numbers = 1,000,000\n\nTotal zeros in million = 6 ie 2 sets of 3 zeros\n\n1 Million in scientific notation = 1 * 106 = 106\n\n1 Million in roman numerals = M\n\n## International and Indian Numbering System\n\n International Indian Figures Unit Unit 1 Ten Ten 10 Hundred Hundred 100 Thousand Thousand 1,000 Ten Thousand Ten Thousand 10,000 Hundred Thousand Lakh 100,000 Million Ten Lakh 1,000,000 Ten Million Crore 10,000,000 Hundred Million Ten Crore 100,000,000 Billion Arab 1,000,000,000 Ten Billion Ten Arab 10,000,000,000 Hundred Billion Kharab 100,000,000,000 Trillion Ten Kharab 1000000000000\n\n## 1 Million in Lakhs\n\nFrom the above table, 1 Million in International Numbering System is equal to 10 Lakhs in Indian Numbering System\n\ni.e 1 Million = 10 Lakhs\n\n Millions Lakhs 1 Million 10 Lakhs 2 Million 20 Lakhs 5 Million 50 Lakhs 10 Million 100 Lakhs 25 Million 250 Lakhs 50 Million 500 Lakhs 100 Million 1000 Lakhs 250 Million 2500 Lakhs 500 Million 5000 Lakhs 999 Million 9990 Lakhs\n\n## 1 Million in Crores\n\nFrom the above table, 10 Millions in International Numbering System is equal to 1 Crore in Indian Numbering System\n\ni.e 10 Millions = 1 Crore (or) 1 Million = 10 Lakhs = 0.1 Crores\n\n Millions Lakhs Crores 1 Million 10 Lakhs 0.1 Crores 2 Million 20 Lakhs 0.2 Crores 5 Million 50 Lakhs 0.5 Crores 10 Million 100 Lakhs 1 Crores 25 Million 250 Lakhs 2.5 Crores 50 Million 500 Lakhs 5 Crores 100 Million 1000 Lakhs 10 Crores 250 Million 2500 Lakhs 25 Crores 500 Million 5000 Lakhs 50 Crores 999 Million 9990 Lakhs 99.9 Crores\n\n## 1 Million in Rupees \| 1 Million Dollars in Rupees\n\nFor money conversion, we need to do following 2 steps\n\nStep 1: As we already know that, 1 Million in International Numbering System is equal to 10 Lakhs in Indian Numbering System (1M = 10L). So, convert given millions into lakhs\n\nStep 2: Multiply Step 1 - Lakhs with current dollar rate.\n\nLets understand this with examples\n\na) 1 Million in Rupees \| 1 Million Dollars in Rupees\n\nStep 1: Converting million to lakhs ie 1 Million = 10 Lakhs\n\nStep 2: On 1st-Jan-2021 1 USD = 73.092 INR\n\nSo, 1 Million USD = 10 Lakhs * 73.092 = 730.92 Lakhs INR = 7.3092 Crores = 7,30,92,000 INR\n\nHence, 1 Million Dollars = 7.3092 Crores Rupees\n\nb) 10 Million in Rupees \| 10 Million Dollars in Rupees\n\nStep 1: Converting million to lakhs ie 10 Million = 100 Lakhs\n\nStep 2: On 1st-Jan-2021 1 USD = 73.092 INR\n\nSo, 10 Million USD = 100 Lakhs * 73.092 = 7309.2 Lakhs INR = 73.092 Crores = 73,09,20,000 INR\n\nHence, 10 Million Dollars = 73.092 Crores Rupees\n\n## 1 Million is equal to\n\n1 Million is equal to 10 Lakhs\n\n1 Million is equal to 0.1 Crores\n\n1 Million is equal to 0.001 Billions\n\n1 Million is equal to 0.001 Arab\n\n1 Million is equal to 0.00001 Kharab\n\n1 Million is equal to 0.000001 Trillion\n\n1 Million is equal to 1,000 Thousands\n\n1 Million is equal to 10,000 Hundreds\n\n## The Word \"Million\" in Example Sentences\n\n1. It was the million dollar question.\n\n2. Land on Mars, a round-trip ticket - half a million dollars. It can be done.\n\n3. There are hundreds of millions of videos on YouTube.\n\n4. Helium is present in the atmosphere, of which it constitutes four parts in a million.\n\n5. Compared with 90.5 million sq.\n\n6. They sell 100 million gallons of crude oil annually.\n\n7. I have five million dollars.\n\n8. If you won a 1 million dollars, what would you do?\n\n9. There are more than millions of apps on playstore.\n\n10. At least a hundred million websites are out there.\n\n### People also search for 1 Million Means like\n\n1million mean, one million means, one in a million meaning, define million, i million means, million definition, 100 million meaning, 5 million meaning, million meaning in english, 1000000 means, meaning of 1 million, multi million meaning, l million means, 1 million meaning in english, one in million meaning, a million means, one in a million definition, one of a million meaning, meaning of one million" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8115806,"math_prob":0.95104617,"size":4420,"snap":"2023-40-2023-50","text_gpt3_token_len":1466,"char_repetition_ratio":0.2574728,"word_repetition_ratio":0.11082474,"special_character_ratio":0.34977376,"punctuation_ratio":0.13440861,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9596054,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-22T17:12:58Z\",\"WARC-Record-ID\":\"<urn:uuid:853f2a92-2784-4109-bffb-f0efdc721a4a>\",\"Content-Length\":\"60708\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4ae8f58f-4d0b-46f6-a0ef-c2afccc876a5>\",\"WARC-Concurrent-To\":\"<urn:uuid:3ffa444d-989f-4ad0-be88-1c5cc187883d>\",\"WARC-IP-Address\":\"172.253.115.121\",\"WARC-Target-URI\":\"https://www.meanings4all.com/2021/05/1-million-means-or-1-million-meaning.html\",\"WARC-Payload-Digest\":\"sha1:WU3IAL3MZKVY3R37JFIV57RVKBBA6ZBJ\",\"WARC-Block-Digest\":\"sha1:6F4S7KYQPVISARZH2ZIYDDEHRF4L7FYE\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506421.14_warc_CC-MAIN-20230922170343-20230922200343-00513.warc.gz\"}"}
https://www.colorhexa.com/dcd8fb	[ "# #dcd8fb Color Information\n\nIn a RGB color space, hex #dcd8fb is composed of 86.3% red, 84.7% green and 98.4% blue. Whereas in a CMYK color space, it is composed of 12.4% cyan, 13.9% magenta, 0% yellow and 1.6% black. It has a hue angle of 246.9 degrees, a saturation of 81.4% and a lightness of 91.6%. #dcd8fb color hex could be obtained by blending #ffffff with #b9b1f7. Closest websafe color is: #ccccff.\n\n• R 86\n• G 85\n• B 98\nRGB color chart\n• C 12\n• M 14\n• Y 0\n• K 2\nCMYK color chart\n\n#dcd8fb color description : Light grayish blue.\n\n# #dcd8fb Color Conversion\n\nThe hexadecimal color #dcd8fb has RGB values of R:220, G:216, B:251 and CMYK values of C:0.12, M:0.14, Y:0, K:0.02. Its decimal value is 14473467.\n\nHex triplet RGB Decimal dcd8fb `#dcd8fb` 220, 216, 251 `rgb(220,216,251)` 86.3, 84.7, 98.4 `rgb(86.3%,84.7%,98.4%)` 12, 14, 0, 2 246.9°, 81.4, 91.6 `hsl(246.9,81.4%,91.6%)` 246.9°, 13.9, 98.4 ccccff `#ccccff`\nCIE-LAB 87.627, 8.027, -16.55 71.48, 71.292, 101.256 0.293, 0.292, 71.292 87.627, 18.394, 295.875 87.627, 0.092, -27.555 84.435, 3.353, -11.998 11011100, 11011000, 11111011\n\n# Color Schemes with #dcd8fb\n\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #f7fbd8\n``#f7fbd8` `rgb(247,251,216)``\nComplementary Color\n• #d8e6fb\n``#d8e6fb` `rgb(216,230,251)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #eed8fb\n``#eed8fb` `rgb(238,216,251)``\nAnalogous Color\n• #e6fbd8\n``#e6fbd8` `rgb(230,251,216)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #fbeed8\n``#fbeed8` `rgb(251,238,216)``\nSplit Complementary Color\n• #d8fbdc\n``#d8fbdc` `rgb(216,251,220)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #fbdcd8\n``#fbdcd8` `rgb(251,220,216)``\n• #d8f7fb\n``#d8f7fb` `rgb(216,247,251)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #fbdcd8\n``#fbdcd8` `rgb(251,220,216)``\n• #f7fbd8\n``#f7fbd8` `rgb(247,251,216)``\n• #9e93f4\n``#9e93f4` `rgb(158,147,244)``\n• #b2aaf6\n``#b2aaf6` `rgb(178,170,246)``\n• #c7c1f9\n``#c7c1f9` `rgb(199,193,249)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #f1effd\n``#f1effd` `rgb(241,239,253)``\n• #ffffff\n``#ffffff` `rgb(255,255,255)``\n• #ffffff\n``#ffffff` `rgb(255,255,255)``\nMonochromatic Color\n\n# Alternatives to #dcd8fb\n\nBelow, you can see some colors close to #dcd8fb. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #d8ddfb\n``#d8ddfb` `rgb(216,221,251)``\n• #d8dafb\n``#d8dafb` `rgb(216,218,251)``\n• #d9d8fb\n``#d9d8fb` `rgb(217,216,251)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #dfd8fb\n``#dfd8fb` `rgb(223,216,251)``\n• #e2d8fb\n``#e2d8fb` `rgb(226,216,251)``\n• #e5d8fb\n``#e5d8fb` `rgb(229,216,251)``\nSimilar Colors\n\n# #dcd8fb Preview\n\nThis text has a font color of #dcd8fb.\n\n``<span style=\"color:#dcd8fb;\">Text here</span>``\n#dcd8fb background color\n\nThis paragraph has a background color of #dcd8fb.\n\n``<p style=\"background-color:#dcd8fb;\">Content here</p>``\n#dcd8fb border color\n\nThis element has a border color of #dcd8fb.\n\n``<div style=\"border:1px solid #dcd8fb;\">Content here</div>``\nCSS codes\n``.text {color:#dcd8fb;}``\n``.background {background-color:#dcd8fb;}``\n``.border {border:1px solid #dcd8fb;}``\n\n# Shades and Tints of #dcd8fb\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, #03010e is the darkest color, while #fcfcff is the lightest one.\n\n• #03010e\n``#03010e` `rgb(3,1,14)``\n• #070320\n``#070320` `rgb(7,3,32)``\n• #0a0532\n``#0a0532` `rgb(10,5,50)``\n• #0e0744\n``#0e0744` `rgb(14,7,68)``\n• #120956\n``#120956` `rgb(18,9,86)``\n• #150b67\n``#150b67` `rgb(21,11,103)``\n• #190c79\n``#190c79` `rgb(25,12,121)``\n• #1c0e8b\n``#1c0e8b` `rgb(28,14,139)``\n• #20109d\n``#20109d` `rgb(32,16,157)``\n• #2412ae\n``#2412ae` `rgb(36,18,174)``\n• #2714c0\n``#2714c0` `rgb(39,20,192)``\n• #2b16d2\n``#2b16d2` `rgb(43,22,210)``\n• #2f17e4\n``#2f17e4` `rgb(47,23,228)``\n• #3c26e9\n``#3c26e9` `rgb(60,38,233)``\n• #4c38eb\n``#4c38eb` `rgb(76,56,235)``\n• #5c4aec\n``#5c4aec` `rgb(92,74,236)``\n• #6c5bee\n``#6c5bee` `rgb(108,91,238)``\n• #7c6df0\n``#7c6df0` `rgb(124,109,240)``\n• #8c7ff2\n``#8c7ff2` `rgb(140,127,242)``\n• #9c91f4\n``#9c91f4` `rgb(156,145,244)``\n• #aca3f6\n``#aca3f6` `rgb(172,163,246)``\n• #bcb4f7\n``#bcb4f7` `rgb(188,180,247)``\n• #ccc6f9\n``#ccc6f9` `rgb(204,198,249)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #eceafd\n``#eceafd` `rgb(236,234,253)``\n• #fcfcff\n``#fcfcff` `rgb(252,252,255)``\nTint Color Variation\n\n# Tones of #dcd8fb\n\nA tone is produced by adding gray to any pure hue. In this case, #e9e9ea is the less saturated color, while #d9d5fe is the most saturated one.\n\n• #e9e9ea\n``#e9e9ea` `rgb(233,233,234)``\n• #e7e7ec\n``#e7e7ec` `rgb(231,231,236)``\n• #e6e5ee\n``#e6e5ee` `rgb(230,229,238)``\n• #e5e4ef\n``#e5e4ef` `rgb(229,228,239)``\n• #e4e2f1\n``#e4e2f1` `rgb(228,226,241)``\n• #e2e0f3\n``#e2e0f3` `rgb(226,224,243)``\n• #e1dff4\n``#e1dff4` `rgb(225,223,244)``\n• #e0ddf6\n``#e0ddf6` `rgb(224,221,246)``\n• #dfdbf8\n``#dfdbf8` `rgb(223,219,248)``\n• #dddaf9\n``#dddaf9` `rgb(221,218,249)``\n• #dcd8fb\n``#dcd8fb` `rgb(220,216,251)``\n• #dbd6fd\n``#dbd6fd` `rgb(219,214,253)``\n• #d9d5fe\n``#d9d5fe` `rgb(217,213,254)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #dcd8fb is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51314586,"math_prob":0.69575715,"size":3721,"snap":"2020-34-2020-40","text_gpt3_token_len":1712,"char_repetition_ratio":0.12536992,"word_repetition_ratio":0.011090573,"special_character_ratio":0.51410913,"punctuation_ratio":0.23783186,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9637083,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-23T13:22:05Z\",\"WARC-Record-ID\":\"<urn:uuid:673113b9-a29e-4253-a75a-8433b791c6e5>\",\"Content-Length\":\"36355\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:43b4a0f4-980d-4226-b1d3-96c9556a3448>\",\"WARC-Concurrent-To\":\"<urn:uuid:c31f3218-931a-4c61-b108-c2781d4a9bd8>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/dcd8fb\",\"WARC-Payload-Digest\":\"sha1:NKEZZRVVUJC3WFSLH7GDYYKLAEA4NVQ2\",\"WARC-Block-Digest\":\"sha1:45Z5KHAVLXDZYHSIJKSJKZSSIN47TK2G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400210996.32_warc_CC-MAIN-20200923113029-20200923143029-00489.warc.gz\"}"}
http://newtonphysics.on.ca/einstein/chapter1.html	[ "", null, "Einstein's Theory of Relativity\nVersus Classical Mechanics\nPaul Marmet\n\"It follows from the theory of relativity that mass and energy are both different manifestations of the same thing -a somewhat unfamiliar conception for the average man. . . . the mass and energy in fact were equivalent.\" Albert Einstein\n\"The Quotable Einstein\", Princeton University Press, Princeton New Jersey (1996), also in the Einstein film produced by Nova Television, 1979\n-------------------------------------------------------\nWe must note that the equivalence of mass and energy is different from the principle of mass-energy conservation, which is not applied in Einstein's relativity (see Straumann)\n( Last checked 2017/01/15 - The estate of Paul Marmet )\n\nWhere to get a Hard Copy of this Book\n\nEine deutsche Übersetzung dieses Artikels finden Sie hier.\nChapter One\nThe Physical Reality of Length Contraction.\n1.1 - Introduction.\nIn this first chapter, we will show that it is possible to establish links between quantum mechanics and mass-energy conservation. These links will help us calculate the interatomic distances in molecules and in crystals as a function of their gravitational potential. We will show that the natural interatomic distance calculated using quantum mechanics leads to the length contraction (or dilation) predicted by relativity. This result will be obtained here without using the hypothesis of the constancy of the velocity of light. It will appear instead as a consequence of quantum mechanics when mass-energy conservation is taken into account.\nSince length contraction appears as a consequence of quantum mechanical calculations, the physical reality of those predictions can be verified experimentally. We will show that the results of the most precise quantum mechanical experiments prove that the change of length is real. Two different experiments which have been found to give sufficient accuracy to verify this change of length will be described in detail. We will show that the dimensions of matter really change naturally depending on its location in a gravitational potential.\n\n1.2 - Mass-Energy Conservation at Macroscopic Scale.\nThe most reliable principle in physics seems to be the principle of mass-energy conservation: mass can be transformed into energy and vice versa. Without this principle, one would be able to create mass or energy from nothing. We do not believe that absolute creation from nothing is possible. Surprizingly, most scientists do not know that Einstein's general relativity is not compatible with the principle of mass-energy conservation [Ref]\nIn order to understand the fundamental implications related to mass-energy conservation, let us consider the following example. Suppose momentarily that the Earth is not moving around the Sun, but has been pushed away with a powerful rocket and has reached interstellar space at location P (see figure 1.1). It now has a negligible residual velocity with respect to the Sun and except for the fact that the Sun has faded away, everything appears the same. The Earth is still made of about 1050 atoms, its center contains iron, is surrounded by oceans, deserts, cities and the atmosphere is the same. The planet is still populated by about the same five billion people.", null, "Figure 1.1\n\nLet us assume that after a while, the planet starts falling slowly from P toward the Sun. Due to the solar attraction, the Earth accelerates until it reaches the distance of 150 million kilometers (from the Sun) corresponding to its normal orbit. At that moment, one can calculate that the Earth has reached a velocity of 42 km/s. This velocity is too large for the Earth to be in a stable orbit around the Sun as it is normally. It must be reduced to 30 km/s, the velocity for a stable orbit. The Earth must be slowed down.\nIt is decided that the velocity of the Earth can be reduced with the help of a strong rope attached to a group of stars at the center of our galaxy. The force produced by the rope will generate energy at the center of the galaxy while the Earth is slowed down to the desired velocity for a stable orbit around the Sun.\nKnowing that the Earth has a mass of 5.97×1024 kg, it is easy to calculate the amount of work transferred to the center of the galaxy. It corresponds to slowing down the Earth from 42 km/s to 30 km/s. This represents an amount of work equal to 2.6×1033 joules. Therefore the Earth must get rid of 2.6×1033 joules to go back to its normal orbit and the center of the galaxy must absorb that same amount of energy. The rope used to slow down the Earth could then run a generator located at the center of the galaxy to produce 2.6×1033 joules of energy.\nHowever, due to the principle of mass-energy conservation, the energy carried out to the center of the galaxy to slow down the Earth can be transformed into mass. Using the relation E = mc2, we find that the mass corresponding to 2.6", null, "1033 joules of energy is equal to 2.9×1016 kg. This means that 29 billions of millions of kilograms of mass have been transferred from the Earth to the center of the galaxy through the rope. This mass-energy is a very small fraction of the Earth’s mass but it must be coming from the Earth and received at the center of the galaxy.\nAfter the re-establishment of the Earth’s orbit at one astronomical unit from the Sun, the inhabitants of the Earth find nothing changed. Other than the neighboring Sun, no difference can be noticed compared with when the Earth, still made of its initial 1050 atoms, was away from the Sun. The question is: How can the Earth not lose one single atom or molecule while 29 billions of millions of kilograms of mass have been lost and received at the center of the galaxy? There is only one logical answer. Since each atom on Earth was submitted to the force of the rope, each atom has lost mass in a proportion of approximately one part per one hundred million.\nNote that this situation is equivalent to the formation of a hydrogen atom. When a proton and an electron come together to form a hydrogen atom, energy is released in the form of light. This light corresponds to the work transferred to the center of the galaxy in our problem.\n\n1.3 - Mass-Energy Conservation at a Microscopic Scale.\nThe experiment described above takes place at a macroscopic scale. Each individual atom loses mass because a force interacts on all atoms when the Earth decelerates in the Sun's gravitational potential. It is normally assumed that atoms have a constant mass. For example we learn that the mass of the hydrogen atom is mo = 1.6727406×10-27 kg. Can we have hydrogen atoms with less or more mass? From the thought experiment of section 1.2, we see that the principle of mass-energy conservation requires a transformation of mass into energy on each atom forming the Earth, since each of them has contributed to generate energy transmitted to the center of the galaxy.\nLet us study the following experiment. We first consider that an individual hydrogen atom is placed on a table on the first floor of a house in the gravitational field of the Earth, as shown on figure 1.2. The hydrogen atom is then attached to a fine (weightless) thread so that the atom can be lowered down slowly to the basement of the house, while the experimenter remains on the first floor. When the atom is lowered down, its weight produces a force F in the thread. That force is measured by the experimenter on the first floor. It is given by:\n\n F = mog 1.1", null, "Figure 1.2\n\nThe slow descent of the atom attached to the thread is stopped every time a measurement is made, which means that the kinetic energy is zero at the moment of the measurement. When the atom has traveled a vertical distance Dh, the observer on the first floor observes that the energy DE produced by the atom and transmitted through the thread to the first floor is:\n\n DE = FDh 1.2\nThe work extracted from the descent of the atom is positive when the final position of the atom is under the first floor (Dh is positive). Then, according to the principle of mass-energy conservation, the energy produced at the first floor by the descent of the atom in the basement can be transformed into mass according to the relationship (see reference):\n E = mc2 1.3\nThe important point that must be retained about equation 1.3 is that the energy E is proportional to the mass, independently of the fact that it just happens that the numerical value of the constant of proportionality is equal to the square of the velocity of light. From equations 1.1, 1.2 and 1.3, the amount of mass Dmf generated at the first floor by the descent is:", null, "1.4\nThis amount of mass (or energy) carried by the thread is generated by the weight of the atom which slowly moves down to the basement. When the hydrogen atom lies on the table, its mass is mo. However, during its descent, it produces work (corresponding to the mass Dmf generated at the first floor). The initial mass mo of the particle is now transferred into the mass-energy Dmf generated at the first floor by the falling particle, plus the remaining mass mb of the particle now in the basement. Using equation 1.4, we find:", null, "1.5\nAccording to the principle of mass-energy conservation, the mass of the hydrogen atom in the basement is now different from its initial mass mo on the first floor. It is slightly smaller than mo and is now equal to mb. Any variation of g with height is negligible and can be taken (with g) into account in equations 1.4 and 1.5.\nOf course, the relative change of mass Dmf/mo is extremely small. (It was equally small in the case of the Earth falling back to its normal orbit, as seen above in section 1.2.) The change of mass given by equation 1.5 is so small that it cannot be verified using a weighing scale. However, this reduction of mass must exist, otherwise, mass-energy would be created from nothing. We will see below that this change of mass has actually been measured.\nIt was quite arbitrary for us to assume that the initial mass of hydrogen on the first floor is mo. Physical tables do not mention all the experimental conditions in which an atom is measured. Furthermore, the accuracy of this value is quite insufficient now to detect Dmf (equation 1.5). A change of altitude of one meter near the Earth’s surface gives a relative change of mass of the order of 10-16. Masses are not known with such an accuracy.\nAt this point, we must recall that in the above reasoning, we have made a choice between the principle of mass-energy conservation and the concept of absolute identical mass in all frames. It is illogical to accept both principles simultaneously since they are not compatible. We have chosen to rely on the principle of mass-energy conservation which is equivalent to not believing in \"absolute creation from nothing\" as defined in section 1.2. We must realize that without mass-energy conservation not much of physics remains. Physics becomes magic.\n\n1.4 - Mass Loss of the Electron.\nThere is a way to measure experimentally the mass difference between a hydrogen atom in the basement and one on the first floor. In equation 1.5, we see that a mass Dmf appears and increases when the atom moves down in the gravitational field. Due to mass-energy conservation, the mass mb of the atom moving down decreases by the same amount, that is:\n\n Dmb=Dmf 1.6\nSince the hydrogen atom has lost a part of its mass due to the change of gravitational potential energy, we must expect (according to equation 1.5) that the electron as well as the proton in the atom have individually lost the same relative mass. Let us calculate the relative change of mass of the electron (Dme/me) and of the proton inside the hydrogen atom due to its change of height. From equations 1.5 and 1.6, we have:", null, "1.7\nwhere\n Dme=Dmb 1.8\nWhen Dh is a few meters, equation 1.7 gives a relative change of mass of the order of 10-16. Consequently, the first order term gives an excellent approximation. Let us use:", null, "1.9\nThe electron mass me (as well as the proton mass) is not constant and decreases continuously when the atom is moving down. Equation 1.7 shows that independently of the mass of the particle, the relative change of mass is the same. This means that for the same change of altitude, the relative change of mass of the electron is the same as for the proton.\nDue to the principle of mass-energy conservation, we must conclude that a hydrogen atom at rest has a less massive electron and a less massive proton at a lower altitude than at a higher altitude. The mass of an electron and of a proton can be tested very accurately in atomic physics. Quantum physics shows us how to calculate the exact structure of the hydrogen atom as a function of the electron and proton mass. From that, one can calculate the Bohr radius of an atom having a different mass. Fortunately, the Bohr radius can also be measured with extreme accuracy experimentally.\n\n1.5 - Change of the Radius of the Electron Orbit.\nIt is shown in textbooks how quantum physics predicts the radius of the orbit of the electron in hydrogen for a given electronic state. This is given by the well known Bohr equation:", null, "1.1\nwhere rn is the radius of the Bohr orbit of the electron with principal quantum number n, me is the mass of the electron (actually, me is the reduced mass, but it is approximately the same as the electron mass), h is the Planck constant (= 2p", null, "), k is the Coulomb constant (1/4peo), e is the electronic charge and Z is the number of charges in the nucleus (Z = 1 corresponds to atomic hydrogen). Furthermore when we choose n = 1 and Z = 1, rn becomes ao, which is called the Bohr radius. The Bohr radius is 5.291772×10-11 m at the Earth's surface (for the case of R¥ for which the nucleus is very massive). Equation 1.10 illustrates a simple principle. It illustrates the fact that the circumference of the electron orbit is exactly equal to (or any multiple of) the de Broglie wavelength of the electron orbiting the nucleus.\nSince, as we have seen above, the electron mass me changes with its position in a gravitational potential, let us calculate (using Bohr's equation) the change of radius rn caused by that change of electron mass. This is given by the partial derivative of rn with respect to me. From equation 1.10 we find:", null, "1.11\nEquation 1.11 shows that any relative decrease of electron mass is equal to the same relative increase of the radius of the electron orbit. According to the principle of mass-energy conservation, the electron mass decreases when brought to a lower gravitational potential. Consequently, quantum physics (Bohr's equation) shows that the radius of the electron orbit in hydrogen must increase when the atom is at a lower altitude. Using equation 1.10, quantum physics gives us the possibility to predict the size of the electron orbit rn in an atom for different values of electron mass. Let us study the change of size of the electron orbit as a function of the altitude where the particle is located in a gravitational field.\n\n1.6 - Change of Energy of Electronic States.\nSince it has been observed and accepted that the laws of quantum physics are invariant in any frame of reference, let us calculate the energy states of atoms having an electron (and a proton) with a different mass. The consequences of the change of proton mass are easily calculated since the energy levels depend only on the reduced mass of the electron-proton system. In the Bohr equation, we take me as the reduced mass. This does not produce any relevant difference in the problem here.\nThe binding energy between the electron and the proton is a function of the electrostatic potential between the nucleus and the electron. Quantum physics teaches that the energy En of the nth state as a function of the electron mass is:", null, "1.12\nFrom equation 1.12, we can find the relationship between the change of electron mass and the change of energy:", null, "1.13\nThe Bohr radius ao is the average radius of the electron orbit for n = 1. According to quantum physics the energy of state n is:", null, "1.14\nwhere ao is a function of the electron mass me, given by:", null, "1.15\nWe know that the energy of electronic states of atoms can be measured very accurately in spectroscopy from the light emitted during the transition between any two states En and En'. Extremely accurate results can also be obtained in some nuclear reactions with the help of Mössbauer spectroscopy.\nThe frequency nn of the radiation emitted as a function of the energy En of level n is given by:\n En = hnn 1.16\nBy differentiation of equation 1.16, we find:", null, "1.17\nDifferentiation of equation 1.14 gives:", null, "1.18\nCombining equations 1.11, 1.13, 1.17 and 1.18, we get:", null, "1.19\nSince these quantities are extremely small but finite, we can write:", null, "1.2\nFrom equation 1.7, we have:", null, "1.21\nEquations 1.20 and 1.21 give:", null, "1.22\nEquation 1.22 shows that the relative change of size of the Bohr radius Dao/ao is equal to -gDh/c2.\nThis shows that following the laws of quantum physics, a change of electron mass due to a change of gravitational potential (which results necessarily from the principle of mass-energy conservation) produces a physical change of the Bohr radius.\nWe must notice here that using the relativistic correction given by Dirac's mathematics is irrelevant and does not solve this problem. Relativistic quantum mechanics introduces a relativistic correction due to the electron velocity with respect to the center of mass of the atom. The change in electron mass brought by the relativistic correction implied in this chapter is due to the gravitational potential originating from outside the proton-electron system. It is not due to any internal velocity within the atom. The use of the relativistic Dirac equation is not related to calculating how the Bohr radius changes between its value in the initial gravitational potential and its value in the final gravitational potential.\n\n1.7 - Experimental Measurements of Length Dilation in a Gravitational Potential.\nA measurement proving that there is a change of the Bohr radius due to the change of gravitational potential has already been made. The difference of energy for an atom corresponding to its change of size is observed as a red shift of its spectroscopic lines. The change of mass can be applied quite generally to any particle or subatomic particle in physics placed in a gravitational potential. It can also be applied to astronomical bodies like planets and galaxies since it relies on the principle of mass-energy conservation which is always valid.\n\n1.7.1 - Pound and Rebka's Experiment.\nA spectroscopic measurement of the highest precision has been reported by Pound and Rebka in 1964 with an improved result by Pound and Snider in 1965. Since we have seen that the change of ao corresponds to a change of energy of spectroscopic levels, let us examine Pound and Rebka's experiment. They used Mössbauer spectroscopy to measure the red shift of 14.4 keV gamma rays from Fe57. The emitter and the absorber were placed at rest at the bottom and top of a tower of 22.5 meters at Harvard University.\nThe consequence of the gravitational potential on the particles is such that their mass is lower at the bottom than at the top of the tower. Therefore an electron in an atom located at the base of the tower has a larger Bohr radius than an electron located 22.5 meters above, as given by equation 1.22. The same equation also shows that electrons orbiting with a larger radius have less energy and emit photons with longer wavelengths.\nPound and Rebka reported that the measured red shift agrees within one percent with the equation:", null, "1.23\nNot only is the change of energy predicted by relativity and verified experimentally by Pound and Rebka (equation 1.23) numerically compatible with the change of energy predicted by the conservation of mass-energy, but the predicted relativistic equation is mathematically identical to the one predicting the increase of Bohr's radius (equation 1.22). Since the red shift measured corresponds exactly to the change of the Bohr radius existing between the source and the detector, we see that it cannot be attributed to an absolute increase of energy of the photon during its trip in the gravitational field.\nThis result is exactly the one that proves that matter at the base of the tower is dilated with respect to matter at the top. It is clear that the Bohr radius has actually changed as expected which means that the physical length has really changed. Therefore, this phenomenon is not space dilation. The real physical dilation of matter is observed because electrons (as well as all particles) have a lower mass at the bottom of the tower which gives them a longer de Broglie wavelength. Space dilation is not compatible with a rational interpretation of modern physics. A rational interpretation has already been presented .\nThe equilibrium distance between particles is now increased because the Bohr radius has increased. When atoms are brought to a different gravitational potential, the electron and proton must reach a new distance equilibrium as required by quantum physics in equation 1.12. Quantum physics and the principle of mass-energy conservation lead to a real physical contraction or dilation. This solution solves the mysterious description of space contraction in relativity without involving any new hypothesis or new logic. Length contraction or dilation is real and is demonstrated here as the result of actual experiments. Let us also note that this length dilation is done without producing any internal mechanical stress in solid material. Finally, if the source were above the detector, we would observe a blue shift proving that the Bohr radius in matter above the detector has decreased with respect to the Bohr radius in matter at lower altitude. One can conclude that Pound and Rebka's experiment has shown that matter is contracted or dilated when it is moved to a different gravitational potential.\n\n1.7.2 - The Solar Red Shift.\nOther experiments also show the reality of length contraction or dilation. For example, the atoms at the surface of the Sun have been measured to show exactly the gravitational dilation due to the decrease of mass of the electrons in the solar gravitational potential. The gravitational potential at the Sun's surface is well known. As shown above, it is a change of electron mass in the hydrogen atom due to the gravitational potential that produces a change of the Bohr radius. It is that change of Bohr's radius that produces a change of energy between different atomic states. Brault has reported such a change of energy between atomic states. It corresponds exactly to the change of Bohr's radius caused by the gravitational potential. The atoms on the Sun emit light at a different frequency because the electrons are lighter on the solar surface than on Earth, exactly as required by the principle of mass-energy conservation. The change of electron mass on the Sun produces displaced spectral lines toward longer wavelengths as given by equation 1.22 (see other reference). Since quantum physics is valid on the solar surface, we can understand that the electrons have less mass due to the solar gravitational potential. This leads to an increase of the Bohr radius for the atoms located on the solar surface which leads to atomic transitions having less energy, as observed experimentally.\nThe Mössbauer experiment as well as the solar red shift experiment prove that atoms are really dilated physically. This means that the physical length of objects actually changes. We also find that not only do protons and electrons lose mass in a gravitational potential but so do nuclear particles in the nucleus of Fe57, as observed in the Mössbauer experiment of Pound and Rebka.\n\n1.8 - The Crucial Influence of the Electron Mass on the Fundamental Laws of Relativity.\nMacroscopic matter is formed by an arrangement of atoms. In molecular physics, we learn that quantum physics predicts that interatomic distances are proportional to the Bohr radius. Those distances are calculated as a function of the Bohr radius. According to quantum physics, a smaller Bohr radius will lead to a smaller interatomic distance between atoms in molecular hydrogen. The interatomic distance in molecules is known to be a function of the Bohr radius just as the interatomic distance in a crystalline structure is proportional to the Bohr radius. This means that since the Bohr radius changes with the intensity of the gravitational potential, the size of molecules and crystals also changes in the same proportion. This is true even in the case of large organic molecules. Therefore the size of all biological matter is proportional to the Bohr radius. This point is explained in more details in appendix I.\nBecause the size of macroscopic matter changes with the gravitational potential, the original length of the standard meter transferred to a location having a different gravitational potential will also change. To be more specific, mass-energy conservation requires that the standard meter made of platinum-iridium alloy becomes shorter if we move it to the top of a mountain. Furthermore, due to the increase of electron mass, an atomic clock will increase its frequency by the same ratio when it is moved to the top of the same mountain. However, since the velocity of light (or any other velocity) is the ratio between these two units, it will not change at the top of the mountain with respect to any frame of reference. This point will be discussed later. Because the relative changes of length and clock rate are equal, they will be undetectable when simply using proper values within a frame of reference. All matter, including human bodies, composed of atoms and molecules will change in the same proportion since the intermolecular distance depends on the Bohr radius and consequently on the electron mass which is reduced when located in a gravitational potential.\nIt is important to notice that length dilation or contraction is predicted and explained here without using the relativistic Lorentz equations nor the constancy of the velocity of light. Consequently, we must consider now that we have demonstrated experimentally (using Pound and Rebka's results) the physical change of length of an object in a gravitational potential. More demonstrations will be given in the following chapters.\nThe experiments reported here showing length dilation use atoms that are at rest. They are solely related to the potential energy. We will see that the problems of kinetic energy and velocities require new considerations in the next chapters.\n\n1.9 - References.\n\n C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company San Francisco. page 1056. See also: Pound R. V. and G. A. Rebka, Apparent Weight of Photons, Phys. Rev. Lett., 4, 337 1964. See also: Pound R. V. and Snider, J.L. Effect of gravity on Nuclear Resonance, Phys. Rev. B, 140, 788-803, 1965. This has been measured in a rocket experiment by Vessot and Levine (1976) with an accuracy of 2 x 10-4.\n\n J. W. Brault, The Gravitational Redshift in the Solar Spectrum, Doctoral dissertation, Princeton University, 1962. Also Gravitational Redshift in Solar Lines, Bull. Amer. Phys. Soc., 8, 28, 1963.\n\n P. Marmet, Absurdities in Modern Physics: A Solution, ISBN 0-921272-15-4 Les Éditions du Nordir, c/o R. Yergeau, 165 Waller, Ottawa, Ontario K1N 6N5, 144p. 1993.\n\n1.10 - Symbols and Variables.\n\nDE energy produced by the atom and transmitted to the first floor\nDh distance travelled by the atom\nDmb amount of mass lost by the atom\nDme amount of mass lost by the electron\nDmf amount of mass generated on the first floor\nEn energy of the hydrogen atom in state n\nF weight of the atom\nmo mass of the atom on the table\nnn frequency of the radiation emitted corresponding to En\nrn radius of the orbit of the electron in hydrogen in state n\nZ number of charges in the nucleus\n\n<><><><><><><><><><><><>\nPreface Contents Chapter 2" ]	[ null, "http://newtonphysics.on.ca/einstein/orbit.gif", null, "http://newtonphysics.on.ca/einstein/image1534.gif", null, "http://newtonphysics.on.ca/einstein/image1535.gif", null, "http://newtonphysics.on.ca/einstein/image1536.gif", null, "http://newtonphysics.on.ca/einstein/image1537.gif", null, "http://newtonphysics.on.ca/einstein/image1538.gif", null, "http://newtonphysics.on.ca/einstein/image1539.gif", null, "http://newtonphysics.on.ca/einstein/image1540.gif", null, "http://newtonphysics.on.ca/einstein/image1541.gif", null, "http://newtonphysics.on.ca/einstein/image1542.gif", null, "http://newtonphysics.on.ca/einstein/image1543.gif", null, "http://newtonphysics.on.ca/einstein/image1544.gif", null, "http://newtonphysics.on.ca/einstein/image1545.gif", null, "http://newtonphysics.on.ca/einstein/image1546.gif", null, "http://newtonphysics.on.ca/einstein/image1547.gif", null, "http://newtonphysics.on.ca/einstein/image1548.gif", null, "http://newtonphysics.on.ca/einstein/image1549.gif", null, "http://newtonphysics.on.ca/einstein/image1550.gif", null, "http://newtonphysics.on.ca/einstein/image1551.gif", null, "http://newtonphysics.on.ca/einstein/image1552.gif", null, "http://newtonphysics.on.ca/einstein/image1553.gif", null, "http://newtonphysics.on.ca/einstein/image1554.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88905907,"math_prob":0.96861094,"size":27894,"snap":"2023-14-2023-23","text_gpt3_token_len":6667,"char_repetition_ratio":0.16392973,"word_repetition_ratio":0.051600672,"special_character_ratio":0.21302073,"punctuation_ratio":0.09321876,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99116105,"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],"im_url_duplicate_count":[null,null,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-22T10:15:16Z\",\"WARC-Record-ID\":\"<urn:uuid:4f75a3f6-b69e-4616-8504-4a5f62fa3916>\",\"Content-Length\":\"55978\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1cf7a4ad-cb18-49fb-9fd5-5fb7cee854a8>\",\"WARC-Concurrent-To\":\"<urn:uuid:0503fda3-07d0-4f7d-ad0f-5590e8e82e5b>\",\"WARC-IP-Address\":\"204.44.202.18\",\"WARC-Target-URI\":\"http://newtonphysics.on.ca/einstein/chapter1.html\",\"WARC-Payload-Digest\":\"sha1:NJXWJ4WEFARW4L6MFOTECYSH7OLWXKJC\",\"WARC-Block-Digest\":\"sha1:EK3WC3YB4VIILNIO7QA3NKNC7YS3EX67\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296943809.22_warc_CC-MAIN-20230322082826-20230322112826-00626.warc.gz\"}"}
https://stacks.math.columbia.edu/tag/030K	[ "Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one.\n\nLemma 9.14.6. Let $E/F$ be an algebraic field extension. There exists a unique subextension $E/E_{sep}/F$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is purely inseparable.\n\nProof. If the characteristic is zero we set $E_{sep} = E$. Assume the characteristic is $p > 0$. Let $E_{sep}$ be the set of elements of $E$ which are separable over $F$. This is a subextension by Lemma 9.12.13 and of course $E_{sep}$ is separable over $F$. Given an $\\alpha$ in $E$ there exists a $p$-power $q$ such that $\\alpha ^ q$ is separable over $F$. Namely, $q$ is that power of $p$ such that the minimal polynomial of $\\alpha$ is of the form $P(x^ q)$ with $P$ separable algebraic, see Lemma 9.12.1. Hence $E/E_{sep}$ is purely inseparable. Uniqueness is clear. $\\square$\n\nComment #826 by on\n\nSuggested slogan: Algebraic field extensions break down uniquely into a separable and purely inseparable part.\n\nComment #5488 by Théo de Oliveira Santos on\n\nVery small typo: \"Assume the characteristic if $p>0$.\"\n\nThere are also:\n\n• 3 comment(s) on Section 9.14: Purely inseparable extensions\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":0.8107429,"math_prob":0.99721265,"size":1940,"snap":"2023-14-2023-23","text_gpt3_token_len":554,"char_repetition_ratio":0.13997933,"word_repetition_ratio":0.6060606,"special_character_ratio":0.29536083,"punctuation_ratio":0.10552764,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997619,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-09T23:59:33Z\",\"WARC-Record-ID\":\"<urn:uuid:b64934db-96c1-4208-a82d-8eb76c3b2fdf>\",\"Content-Length\":\"16475\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cbffef70-2791-409b-811a-a18699f8756b>\",\"WARC-Concurrent-To\":\"<urn:uuid:6a3d08f4-308f-4da5-80b4-e64a4173482f>\",\"WARC-IP-Address\":\"128.59.222.85\",\"WARC-Target-URI\":\"https://stacks.math.columbia.edu/tag/030K\",\"WARC-Payload-Digest\":\"sha1:2DCJAPT36OZNRQF7FXCPKGHP4JARP56E\",\"WARC-Block-Digest\":\"sha1:5A44XAKKVL33P4JDBM5MWFDE3TR4BF5M\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224656869.87_warc_CC-MAIN-20230609233952-20230610023952-00197.warc.gz\"}"}
https://programs.team/leetcode-find-the-median-of-two-positively-ordered-arrays.html	[ "# LeetCode - find the median of two positively ordered arrays\n\n## Topic information\n\nSource address: Find the median of two positive arrays\n\nGive two positively ordered (from small to large) arrays nums1 and nums2 of sizes m and n, respectively. Please find out and return the median of these two positive arrays.\n\nThe time complexity of the algorithm should be \\ (O(log (m+n)) \\).\n\n## Prompt information\n\n### Example 1\n\nInput: nums1 = [1,3], nums2 = \nOutput: 2.00000\nExplanation: merge arrays = [1,2,3] ，Median 2\n\n\n### Example 2\n\nInput: nums1 = [1,2], nums2 = [3,4]\nOutput: 2.50000\nExplanation: merge arrays = [1,2,3,4] ，median (2 + 3) / 2 = 2.5\n\n\n### Tips\n\n• nums1.length == m\n• nums2.length == n\n• 0 <= m <= 1000\n• 0 <= n <= 1000\n• 1 <= m + n <= 2000\n• -10^6 <= nums1[i], nums2[i] <= 10^6\n\n## Implementation logic\n\n### Merging method\n\nThe first way to solve the problem is to combine two ordered arrays into an ordered array, and then calculate the median of the result set. This method is a relatively direct idea.\n\nHowever, merging two arrays is the most time-consuming operation of this method, and it reaches the time complexity of \\ (O(m+n) \\), so it does not meet the requirement that the time complexity of the algorithm in the topic reaches \\ (O(log (m+n)) \\).\n\nMoreover, the space complexity of this method has reached the level of \\ (O(m+n) \\), and its space occupation is not optimal.\n\npackage cn.fatedeity.algorithm.leetcode;\n\npublic class MedianOfTwoSortedArrays {\npublic double answer(int[] nums1, int[] nums2) {\nint i = 0, j = 0;\nint[] numbers = new int[nums1.length + nums2.length];\nint index = 0;\nwhile (i < nums1.length \|\| j < nums2.length) {\nif (i == nums1.length) {\nfor (int k = j; k < nums2.length; k++) {\nnumbers[index++] = nums2[k];\n}\nbreak;\n} else if (j == nums2.length) {\nfor (int k = i; k < nums1.length; k++) {\nnumbers[index++] = nums1[k];\n}\nbreak;\n}\n\nif (nums1[i] < nums2[j]) {\nnumbers[index++] = nums1[i++];\n} else if (nums2[j] < nums1[i]) {\nnumbers[index++] = nums2[j++];\n} else {\nnumbers[index++] = nums1[i++];\nnumbers[index++] = nums2[j++];\n}\n}\nif ((numbers.length & 1) == 0) {\n// Array length is even\nint mid = numbers.length >> 1;\nreturn (numbers[mid - 1] + numbers[mid]) / 2f;\n} else {\nreturn numbers[numbers.length >> 1];\n}\n}\n}\n\n\n### Double pointer\n\nFacing two arrays, you can also use double pointers, as long as you find the position of the median. Since the length of the two arrays is known, the sum of the subscripts of the two arrays corresponding to the median is also known.\n\nThe basic idea is to filter the smaller values in the two arrays one by one through double pointers until they reach the subscript position of the median, and then the median of the two arrays can be obtained.\n\nThe whole method is better than the merging method, and the spatial complexity is directly reduced to the degree of \\ (O(1) \\). In terms of running efficiency, it only cycles \\ (\\frac{m+n}2\\) times, but the time complexity is still \\ (O(m+n) \\, which does not meet the requirements of \\ (O(log (m+n)) \\) in the title.\n\npackage cn.fatedeity.algorithm.leetcode;\n\npublic class MedianOfTwoSortedArrays {\npublic double answer(int[] nums1, int[] nums2) {\nint m = nums1.length;\nint n = nums2.length;\nint len = m + n;\nint mStart = 0, nStart = 0;\nint left = 0, right = 0;\nfor (int i = 0; i <= len >> 1; i++) {\nleft = right;\nif (mStart < m && (nStart >= n \|\| nums1[mStart] < nums2[nStart])) {\nright = nums1[mStart++];\n} else {\nright = nums2[nStart++];\n}\n}\nif ((len & 1) == 0) {\nreturn (left + right) / 2f;\n} else {\nreturn right;\n}\n}\n}\n\n\n### Binary search\n\nGenerally, most of the problems requiring a time complexity of \\ (O(log (n)) \\) can be solved using the idea of dichotomy.\n\nThis topic can also use dichotomy to make the time complexity meet the requirements of \\ (O(log (m+n)) \\), but it is indeed a dichotomy search with a little anti thinking logic.\n\nThe main idea of solving the problem is to determine the selected value of \\ (\\frac{1}{2}\\) through the combined length, compare the \\ (\\frac{1}{2}\\) value in the two arrays, filter the smaller part directly until \\ (\\frac{m+n}{2}\\) value is filtered, and the remaining first index of the two arrays is used to calculate the median.\n\npackage cn.fatedeity.algorithm.leetcode;\n\npublic class MedianOfTwoSortedArrays {\npublic double answer(int[] nums1, int[] nums2) {\nint len1 = nums1.length;\nint len2 = nums2.length;\nint total = len1 + len2;\nint left = (total + 1) >> 1;\nint right = (total + 2) >> 1;\nif (left == right) {\nreturn this.findK(nums1, 0, nums2, 0, left);\n} else {\nreturn (this.findK(nums1, 0, nums2, 0, left) + this.findK(nums1, 0, nums2, 0, right)) / 2.0;\n}\n}\n\nprivate int findK(int[] nums1, int i, int[] nums2, int j, int k) {\nif (i >= nums1.length) {\n// num1 has been filtered out\nreturn nums2[j + k - 1];\n} else if (j >= nums2.length) {\n// num2 has been filtered\nreturn nums1[i + k - 1];\n}\nif (k == 1) {\n// First numeric comparison in array\nreturn Math.min(nums1[i], nums2[j]);\n}\n\n// Determine the two values to compare each time\nint mid1 = (i + (k >> 1) - 1) < nums1.length ? nums1[i + (k >> 1) - 1] : Integer.MAX_VALUE;\nint mid2 = (j + (k >> 1) - 1) < nums2.length ? nums2[j + (k >> 1) - 1] : Integer.MAX_VALUE;\n\nif (mid1 < mid2) {\nreturn findK(nums1, i + (k >> 1), nums2, j, k - (k >> 1));\n} else {\nreturn findK(nums1, i, nums2, j + (k >> 1), k - (k >> 1));\n}\n}\n}\n\n\nPosted by trg on Sat, 13 Aug 2022 22:14:53 +0530" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62771636,"math_prob":0.99961275,"size":5232,"snap":"2022-40-2023-06","text_gpt3_token_len":1584,"char_repetition_ratio":0.14211936,"word_repetition_ratio":0.07061267,"special_character_ratio":0.35932723,"punctuation_ratio":0.16698474,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995173,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T22:46:42Z\",\"WARC-Record-ID\":\"<urn:uuid:82a24977-cc31-4954-a78f-52b408969f91>\",\"Content-Length\":\"11575\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:484d5d1b-c2de-4d4f-ab25-156015dd9b21>\",\"WARC-Concurrent-To\":\"<urn:uuid:07ec57b2-769e-4d2f-bfb4-87fccbc1bfe5>\",\"WARC-IP-Address\":\"178.238.237.47\",\"WARC-Target-URI\":\"https://programs.team/leetcode-find-the-median-of-two-positively-ordered-arrays.html\",\"WARC-Payload-Digest\":\"sha1:OGAG65HCF7BCW7NA6UNG7WZJOZLIDURC\",\"WARC-Block-Digest\":\"sha1:TWFNIBBO2R42WO6ATRVINARTSEFZU7O3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500154.33_warc_CC-MAIN-20230204205328-20230204235328-00311.warc.gz\"}"}
https://dmu.ghpc.au.dk/lmt/wiki/index.php?title=Supported_features	[ "# Supported features\n\n### Raw input data support\n\n• Only numeric input is supported. That is, data must be either integer or real numbers, but not characters. The only exemptions are genomic data and files containing boolean data.\n• 64 bit integers are used to store integer data. That is, integer data may range from -9.223372e+18 to +9.223372e+18.\n• All input data are renumbered automatically if required by the job and respective cross-reference files will be provided if necessary.\n• Files containing human readable array data must be in comma-separated-value(csv) format.\n• Files containing human readable vector data must contain a single column vector.\n• Pedigree files must be complete. That is, all individuals occurring as parents (2. and 3. column) must occur as individuals (1. column). All individual ids which occur in the data file must occur in the pedigree.\n• Genomic marker data must be imputed to common density across all genotypes and must not contain missing marker.\n\n### Supported operations\n\nCurrently lmt support the following operations on linear mixed models:\n\n• Solving for BLUP and BLUE solutions conditional on supplied variances for random and fixed factor, respectively;\n• Gibbs sampling of variance components in single pass and blocked mode;\n• AI-REML estimation of variance components using the coefficient matrix of the mixed model equation system (AI-REML-C)\n• MC-EM-REML estimation of variance components\n• Sampling (block)diagonal elements of the inverse of the mixed model coefficient matrix\n• Solving for (block)diagonal elements of the inverse of the mixed model coefficient matrix\n\n### Supported factors and variables\n\nlmt supports\n\n• fixed factors\n• random factors\n• classification variables\n• continuous co-variables, which can be nested. For continuous co-variables lmt support user-defined polynomials(e.g. sin(x) or x^(0.5) ) and hard coded Legendre polynomials up to order 6.\n• genetic group co-variables\n\nAll classification and co-variables can be associated to a fixed or random factor.\n\n### Supported variance structures\n\nFor random factor lmt supports variance structures of\n\n• structure $$\\Gamma\\otimes\\Sigma$$, where $$\\Sigma$$ is an dense symmetric positive definite matrix, and\n• $$\\Theta_L(\\Gamma\\otimes I_{\\Sigma})\\Theta_L^{'}$$, where $$\\Theta$$ is symmetric positive definite block-diagonal matrix of $$n$$ symmetric positive definite martices $$\\Sigma_i, i=1,..,n$$, $$\\Theta_L$$ is the lower Cholesky factor of $$\\Theta$$ and $$I_{\\Sigma}$$ is an identity matrix of dimension $$\\Sigma_i$$.\n\nWhen solving linear mixed models $$\\Sigma$$ and $$\\Gamma$$ are user determined constants, whereas when estimating variances $$\\Gamma$$ is a user determined constant and $$\\Sigma$$ is a function of the data.\n\nSupported type for $$\\Gamma$$ are\n\n• an identity matrix\n• an arbitrary positive definite diagonal matrix\n• a pedigree-based numerator relationship matrix $$A$$ which may contain meta-founders\n• a pedigree- and genotype-based relationship matrix $$H$$ which may contain meta-founders\n• genetic groups absorbed into $$A$$ or $$H$$\n• a user-defined(u.d.) symmetric, positive definite matrix of which inverse is supplied\n• as a sparse upper-triangular matrix stored in csr format\n• as a dense matrix\n• a co-variance matrix of a selected auto-regressive process\n\nFurther lmt supports special variance structures which are not covered by the above description\n\n• SNP-BLUP co-variance structure with the option to model marker co-variances as $$\\Theta_L(\\Gamma\\otimes I_{\\Sigma})\\Theta_L^{'}$$.\n\n### Supported linear mixed model solvers\n\nlmt supports\n\n• a direct solver requiring to explicitly build the linear mixed model equations left-hand-side coefficient matrix($$C$$)\n• a pre-conditioned gradient solver which does not require $$C$$\n\n### Supported features related to genomic data\n\n• direct use of genomic marker data\n• building of genomic relationship matrices($$G$$) from supplied genomic data\n• uploading of a u.d. $$G$$\n• adjustment of $$G$$ to $$A_{gg}$$ in ssGBLUP and ssSNPBLUP\n• solving ssGBLUP models\n• Variance component estimation for ssGBLUP models\n• solving ssGTBLUP models\n• solving ssSNPBLUP models\n• calculation of true H matrix diagonal elements for ssGBLUP models\n• all Single-Step models can be run from \"bottom-up\", that is the user supplies the genotypes and all necessary ingredients(e.g. $$G$$) are built on the fly.\n\n### Supported pedigree types\n\n• ordinary pedigrees\n• probabilistic pedigrees with an unlimited number of parent pairs per individual\n• genetic group pedigrees\n• meta-founder pedigrees\n• ignoring of inbreeding\n• iterative derivation of inbreeding coefficients\n\n### Supported features related to meta-founders and genetic groups\n\n• meta-founders can be modeled for all $$\\Gamma$$ which contain $$A$$(.e.g. $$A$$, $$H$$ for ssGBLUP, ssGTBLUP and ssSNPBLUP)\n• genetic groups can be modeled as an extra factor or can be absorbed into all $$\\Gamma$$ which contain $$A$$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80049056,"math_prob":0.9938445,"size":5670,"snap":"2023-40-2023-50","text_gpt3_token_len":1349,"char_repetition_ratio":0.10889517,"word_repetition_ratio":0.019277109,"special_character_ratio":0.23544973,"punctuation_ratio":0.09263158,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9993117,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-29T22:03:32Z\",\"WARC-Record-ID\":\"<urn:uuid:f946a73a-edd5-4e9e-8745-c9a8f8c205b7>\",\"Content-Length\":\"26568\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d3187779-a538-475b-963b-ff2caf6a1a26>\",\"WARC-Concurrent-To\":\"<urn:uuid:6c1745d4-7110-4aa6-998b-bf7044564c4d>\",\"WARC-IP-Address\":\"130.225.18.162\",\"WARC-Target-URI\":\"https://dmu.ghpc.au.dk/lmt/wiki/index.php?title=Supported_features\",\"WARC-Payload-Digest\":\"sha1:OAJ3WHA26AHOTHF3AUIDLUIWJND5AJTC\",\"WARC-Block-Digest\":\"sha1:3LNDQKPO67ZA7R2KVREKWM4UUBMXDREN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100146.5_warc_CC-MAIN-20231129204528-20231129234528-00511.warc.gz\"}"}
https://optuna.readthedocs.io/en/stable/_modules/optuna/visualization/_edf.html	[ "# Source code for optuna.visualization._edf\n\n```import itertools\nfrom typing import Callable\nfrom typing import cast\nfrom typing import List\nfrom typing import Optional\nfrom typing import Sequence\nfrom typing import Union\n\nimport numpy as np\n\nfrom optuna.logging import get_logger\nfrom optuna.study import Study\nfrom optuna.trial import FrozenTrial\nfrom optuna.trial import TrialState\nfrom optuna.visualization._plotly_imports import _imports\nfrom optuna.visualization._utils import _check_plot_args\n\nif _imports.is_successful():\nfrom optuna.visualization._plotly_imports import go\n\n_logger = get_logger(__name__)\n\n[docs]def plot_edf(\nstudy: Union[Study, Sequence[Study]],\n,\ntarget: Optional[Callable[[FrozenTrial], float]] = None,\ntarget_name: str = \"Objective Value\",\n) -> \"go.Figure\":\n\"\"\"Plot the objective value EDF (empirical distribution function) of a study.\n\nNote that only the complete trials are considered when plotting the EDF.\n\n.. note::\n\nEDF is useful to analyze and improve search spaces.\nFor instance, you can see a practical use case of EDF in the paper\n`Designing Network Design Spaces <https://arxiv.org/abs/2003.13678>`_.\n\n.. note::\n\nThe plotted EDF assumes that the value of the objective function is in\naccordance with the uniform distribution over the objective space.\n\nExample:\n\nThe following code snippet shows how to plot EDF.\n\n.. plotly::\n\nimport math\n\nimport optuna\n\ndef ackley(x, y):\na = 20 math.exp(-0.2 * math.sqrt(0.5 * (x 2 + y 2)))\nb = math.exp(0.5 * (math.cos(2 * math.pi * x) + math.cos(2 * math.pi * y)))\nreturn -a - b + math.e + 20\n\ndef objective(trial, low, high):\nx = trial.suggest_float(\"x\", low, high)\ny = trial.suggest_float(\"y\", low, high)\nreturn ackley(x, y)\n\nsampler = optuna.samplers.RandomSampler(seed=10)\n\n# Widest search space.\nstudy0 = optuna.create_study(study_name=\"x=[0,5), y=[0,5)\", sampler=sampler)\nstudy0.optimize(lambda t: objective(t, 0, 5), n_trials=500)\n\n# Narrower search space.\nstudy1 = optuna.create_study(study_name=\"x=[0,4), y=[0,4)\", sampler=sampler)\nstudy1.optimize(lambda t: objective(t, 0, 4), n_trials=500)\n\n# Narrowest search space but it doesn't include the global optimum point.\nstudy2 = optuna.create_study(study_name=\"x=[1,3), y=[1,3)\", sampler=sampler)\nstudy2.optimize(lambda t: objective(t, 1, 3), n_trials=500)\n\nfig = optuna.visualization.plot_edf([study0, study1, study2])\nfig.show()\n\nArgs:\nstudy:\nA target :class:`~optuna.study.Study` object.\nYou can pass multiple studies if you want to compare those EDFs.\ntarget:\nA function to specify the value to display. If it is :obj:`None` and ``study`` is being\nused for single-objective optimization, the objective values are plotted.\n\n.. note::\nSpecify this argument if ``study`` is being used for multi-objective optimization.\ntarget_name:\nTarget's name to display on the axis label.\n\nReturns:\nA :class:`plotly.graph_objs.Figure` object.\n\nRaises:\n:exc:`ValueError`:\nIf ``target`` is :obj:`None` and ``study`` is being used for multi-objective\noptimization.\n\"\"\"\n\n_imports.check()\n\nif isinstance(study, Study):\nstudies = [study]\nelse:\nstudies = list(study)\n\n_check_plot_args(studies, target, target_name)\n\nreturn _get_edf_plot(studies, target, target_name)\n\ndef _get_edf_plot(\nstudies: List[Study],\ntarget: Optional[Callable[[FrozenTrial], float]] = None,\ntarget_name: str = \"Objective Value\",\n) -> \"go.Figure\":\nlayout = go.Layout(\ntitle=\"Empirical Distribution Function Plot\",\nxaxis={\"title\": target_name},\nyaxis={\"title\": \"Cumulative Probability\"},\n)\n\nif len(studies) == 0:\n_logger.warning(\"There are no studies.\")\nreturn go.Figure(data=[], layout=layout)\n\nall_trials = list(\nitertools.chain.from_iterable(\n(\ntrial\nfor trial in study.get_trials(deepcopy=False)\nif trial.state == TrialState.COMPLETE\n)\nfor study in studies\n)\n)\n\nif len(all_trials) == 0:\n_logger.warning(\"There are no complete trials.\")\nreturn go.Figure(data=[], layout=layout)\n\nif target is None:\n\ndef _target(t: FrozenTrial) -> float:\nreturn cast(float, t.value)\n\ntarget = _target\n\nmin_x_value = min(target(trial) for trial in all_trials)\nmax_x_value = max(target(trial) for trial in all_trials)\nx_values = np.linspace(min_x_value, max_x_value, 100)\n\ntraces = []\nfor study in studies:\nvalues = np.asarray(\n[\ntarget(trial)\nfor trial in study.get_trials(deepcopy=False)\nif trial.state == TrialState.COMPLETE\n]\n)\n\ny_values = np.sum(values[:, np.newaxis] <= x_values, axis=0) / values.size\n\ntraces.append(go.Scatter(x=x_values, y=y_values, name=study.study_name, mode=\"lines\"))\n\nfigure = go.Figure(data=traces, layout=layout)\nfigure.update_yaxes(range=[0, 1])\n\nreturn figure\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5283001,"math_prob":0.9193627,"size":4480,"snap":"2021-43-2021-49","text_gpt3_token_len":1200,"char_repetition_ratio":0.13516532,"word_repetition_ratio":0.075229354,"special_character_ratio":0.27924109,"punctuation_ratio":0.24390244,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99900216,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-19T18:48:03Z\",\"WARC-Record-ID\":\"<urn:uuid:159c6d08-31a2-4dec-93e6-861f76e8b61f>\",\"Content-Length\":\"29154\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0755a74f-5793-46c3-bb65-fcb3dc3e6f35>\",\"WARC-Concurrent-To\":\"<urn:uuid:50605bb1-f072-4ba7-97a6-ea68b52f4142>\",\"WARC-IP-Address\":\"104.17.32.82\",\"WARC-Target-URI\":\"https://optuna.readthedocs.io/en/stable/_modules/optuna/visualization/_edf.html\",\"WARC-Payload-Digest\":\"sha1:3DDVLQG6GH5ASNBXQFKNPL7QCUEUWAUB\",\"WARC-Block-Digest\":\"sha1:EHXANN7F4S6UIJ4YV6M62YG2R6DNOQKN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585280.84_warc_CC-MAIN-20211019171139-20211019201139-00701.warc.gz\"}"}
http://zuowen6.info/node/3215	[ "## 单纯的快乐幸福的微笑\n\n• huān\n• zhè\n• huà\n•\n• 我喜欢这句话。\n•\n•\n• shì\n• men\n• de\n• xiǎo\n• xué\n• bān\n• zhǔ\n• rèn\n• zài\n• qián\n• sòng\n• 　　它是我们的小学班主任在毕业前夕送\n• gěi\n• jiā\n• de\n• huà\n•\n• shēn\n•\n• zhī\n• dào\n• hái\n• yǒu\n• 给大家的话，寓意深刻。我不知道还有几\n• rén\n• zhè\n• huà\n•\n• zhì\n• shǎo\n•\n• 个人记得这句话，可至少我记得。\n•\n•\n• qián\n• jiā\n• zài\n• de\n• shí\n• hòu\n•\n• huān\n• xiào\n• shēng\n• zǒng\n• shì\n• 　　以前大家在一起的时候，欢笑声总是\n• bàn\n• zhe\n• lèi\n• shuǐ\n• fēi\n•\n• xìng\n• de\n• shì\n• qíng\n• bēi\n• shāng\n• de\n• shì\n• qíng\n• 伴着泪水齐飞，幸福的事情与悲伤的事情\n• jiāo\n• zhī\n• zài\n•\n• zhōu\n•\n• jiā\n• ǒu\n• ěr\n• huì\n• chū\n• wán\n• 交织在一起。周末，大家偶尔会出去玩一\n• wán\n•\n• zǒu\n• zǒu\n•\n• kuài\n• de\n• jiā\n• fèn\n• xiǎng\n•\n• ér\n• bēi\n• 玩，走一走，把快乐的与大家分享，而悲\n• shāng\n• de\n•\n• fèn\n• dān\n•\n• tiān\n•\n• xiǎo\n• yún\n• shuō\n• yào\n• bīn\n• 伤的，一起分担。那天，小云说要去滨湖\n• gōng\n• yuán\n• zǒu\n• zǒu\n•\n• shuō\n• hǎo\n•\n• jiān\n• shàng\n• de\n• dān\n• tài\n• zhòng\n•\n• 公园走一走，我说好，肩上的担子太重，\n• yǒu\n• xiē\n• bèi\n• le\n•\n• bīn\n• gōng\n• yuán\n• kàn\n• kàn\n•\n• céng\n• jīng\n• 我有些疲惫了，去滨湖公园看看，把曾经\n• de\n• fèn\n• chún\n• zhēn\n• zhǎo\n• huí\n• lái\n•\n• hái\n• yǒu\n•\n• shì\n• shì\n• mǒu\n• 的那份纯真找回来，还有，我是不是把某\n• xiē\n• luò\n• zài\n• le\n•\n• xiǎng\n• zhǎo\n• huí\n• lái\n•\n• 些记忆遗落在那里了，我想找回来。\n•\n•\n• jiā\n• hào\n• hào\n• dàng\n• dàng\n• lái\n• dào\n• bīn\n• gōng\n• yuán\n•\n• kàn\n• zhe\n• 　　大家浩浩荡荡地来到滨湖公园，看着\n• zhī\n• shǔ\n• xiǎo\n• hái\n• de\n• xuán\n• zhuǎn\n•\n• kàn\n• zhe\n• zhī\n• shǔ\n• xiǎo\n• 只属于小孩子的旋转木马，看着只属于小\n• hái\n• de\n• sài\n• chē\n•\n• kàn\n• zhe\n• zhī\n• shǔ\n• xiǎo\n• hái\n• de\n• bèng\n•\n• 孩子的赛车，看着只属于小孩子的蹦极…\n•\n• men\n• yǒu\n• xiē\n• máng\n•\n•\n• zhēn\n• de\n• zài\n• shǔ\n• zhè\n• le\n• …我们有些迷茫——真的不再属于这里了\n• ma\n•\n• zhēn\n• de\n• zhēn\n• de\n• zài\n• shì\n• xiǎo\n• hái\n• le\n• ma\n•\n• fèn\n• dān\n• 吗？真的真的不再是小孩子了吗？那份单\n• chún\n• de\n• kuài\n•\n• zài\n• zhǎo\n• huí\n• lái\n• le\n• ma\n•\n•\n• 纯的快乐，也再找不回来了吗……\n•\n•\n• yòu\n• shì\n• shuí\n•\n• ?\n• shā\n• le\n• men\n• de\n• kuài\n•\n• 　　那又是谁，抹杀了我们的快乐。记忆\n• zhōng\n•\n• cán\n• de\n• shēng\n• yīn\n• zài\n• ěr\n• biān\n• jiǔ\n• jiǔ\n• huí\n• dàng\n•\n• 中，残酷的声音在耳边久久回荡。\n•\n•\n•\n• zhè\n• me\n• le\n• hái\n• lái\n• gōng\n• yuán\n• wán\n•\n• hài\n• hài\n• xiū\n• ya\n• 　　“这么大了还来公园玩，害不害羞呀\n•\n•\n•\n• ……”\n•\n•\n• zhī\n• shì\n• xiǎng\n• yào\n• fèn\n• dān\n• chún\n• de\n• kuài\n•\n• zhī\n• shì\n• 　　我只是想要一份单纯的快乐，我只是\n• xiǎng\n• yōng\n• yǒu\n• xìng\n• de\n• wēi\n• xiào\n•\n• zhè\n• yàng\n• yǒu\n• cuò\n• me\n•\n• 想拥有一个幸福的微笑。这样有错么？\n•\n•\n• suǒ\n•\n• jiā\n• pāo\n• kāi\n• le\n• qiē\n• fán\n• nǎo\n•\n• míng\n• de\n• 　　所以，大家抛开了一切烦恼，莫名的\n• wǎng\n• hǎo\n•\n• de\n• fǎn\n• duì\n•\n• fǎn\n• 迷惘也好、父母的反对也罢，义无反顾地\n• chōng\n• shàng\n• le\n•\n•\n• zhī\n•\n•\n• céng\n• jīng\n• bèi\n• wàng\n• de\n• màn\n• màn\n• 冲上了“happy之旅”。曾经被遗忘的记忆慢慢\n• chū\n• shuǐ\n• miàn\n•\n• kāi\n• xīn\n• de\n• xiào\n• shēng\n•\n• shī\n• tòu\n• de\n•\n• 浮出水面。开心的大笑声，湿透的衣物，\n• jǐn\n• tiē\n• zhe\n• é\n• tóu\n• de\n• hēi\n• duàn\n• bān\n• guāng\n• huá\n•\n• zhǒng\n• 紧贴着额头的黑发如缎子一般光滑，一种\n• jiào\n•\n• kuài\n•\n• de\n• dōng\n• cóng\n• de\n• yǎn\n• zhōng\n• qīng\n• xiè\n• ér\n• chū\n•\n• 叫“快乐”的东西从她的眼中倾泻而出，\n• sàn\n• chū\n• yào\n• yǎn\n• de\n• guāng\n• máng\n•\n•\n• 散发出耀眼的光芒……\n•\n•\n• zhè\n• jiù\n• shì\n• céng\n• jīng\n• de\n• me\n•\n• 　　这就是曾经的我么？\n•\n•\n• zuò\n• zài\n• xuán\n• zhuǎn\n• shàng\n•\n• zuǐ\n• jiǎo\n• de\n• bào\n• le\n• 　　坐在旋转木马上，嘴角的弧度暴露了\n• nèi\n• xīn\n• de\n• xiǎng\n• ?\n•\n• xiàn\n• zài\n• de\n• hěn\n• kuài\n• hěn\n• kuài\n•\n• 内心的想法，现在的我很快乐很快乐，比\n•\n• nián\n• qián\n• de\n• hái\n• yào\n• kuài\n•\n• ６年前的我还要快乐。\n•\n•\n•\n•\n•\n•\n• shì\n• shuí\n• zài\n• chàng\n• zhe\n• huān\n• de\n• 　　“哒哒哒……”是谁在唱着欢乐的歌\n•\n• xún\n• shēng\n• wàng\n•\n• yuán\n• lái\n• shì\n• zhì\n• zuò\n• mián\n• ?g\n• táng\n• de\n• ？循声望去，原来是一个制作棉花糖的机\n• zài\n• chū\n•\n• hōng\n• hōng\n•\n• de\n• shēng\n• yīn\n•\n• xiǎng\n• yǐn\n• men\n• de\n• 器在发出“轰轰”的声音，想吸引我们的\n• zhù\n•\n• 注意力。\n•\n•\n•\n•\n•\n• jiā\n• xùn\n• wéi\n• le\n• shàng\n•\n•\n• 　　“呼啦。”大家迅速地围了上去。“\n• shū\n• shū\n• yào\n• yuán\n• wèi\n• de\n•\n•\n•\n• shū\n• shū\n• yào\n• cǎo\n• méi\n• wèi\n•\n• 叔叔我要原味的！”“叔叔我要草莓味…\n•\n•\n•\n• shū\n• shū\n• yào\n• qīng\n• píng\n• guǒ\n• wèi\n• de\n•\n•\n•\n• shū\n• shū\n•\n• …”“叔叔我要青苹果味的！”“叔叔…\n•\n•\n• …”\n•\n•\n•\n•\n• 　　……\n•\n•\n• céng\n• jīng\n• shí\n•\n• men\n• xiǎng\n• xiàn\n• zài\n• zhè\n• bān\n•\n• fēng\n• kuáng\n• 　　曾经何时，我们也想现在这般“疯狂\n•\n• guò\n•\n• xiàn\n• zài\n• de\n• jiā\n• hái\n• shì\n• huì\n• xiàng\n• qián\n• yàng\n• mǎn\n• ”过。现在的大家还是会像以前那样不满\n• juē\n• zuǐ\n•\n•\n• wéi\n• shí\n• me\n• de\n• duō\n•\n•\n•\n• 地撅起嘴巴。“为什么她的比我多？”“\n• shū\n• shū\n• yào\n• gèng\n• duō\n• de\n•\n•\n• zhī\n• shì\n• qīng\n• qīng\n• xiào\n• 叔叔我要比她更多的！”我只是轻轻地笑\n•\n•\n•\n• wàng\n• zhe\n• shǒu\n• zhōng\n• de\n• mián\n• ?g\n• táng\n•\n• tián\n• tián\n• de\n• xiāng\n• zài\n• xiàng\n• 　　望着手中的棉花糖，甜甜的香气在向\n• zhōu\n• wéi\n• kuò\n• sàn\n•\n• xìng\n• de\n• wēi\n• xiào\n• chū\n• xiàn\n• zài\n• liǎn\n• shàng\n•\n• gāng\n• 周围扩散，幸福的微笑出现在我脸上，刚\n• zhǔn\n• bèi\n• chī\n• diào\n•\n• xiǎo\n• yún\n• jiù\n• chuǎn\n• pǎo\n• le\n• guò\n• lái\n• 准备吃掉它，小云就气喘吁吁地跑了过来\n•\n• de\n• mián\n• ?g\n• táng\n• shì\n• cǎo\n• méi\n• wèi\n• de\n•\n• fěn\n• fěn\n• de\n• yán\n• ràng\n• ，她的棉花糖是草莓味的，粉粉的颜色让\n• yǎn\n• zhōng\n• de\n• xiào\n• gèng\n• shēn\n• le\n•\n•\n• shuā\n•\n•\n• shēn\n• chū\n•\n• 我眼中的笑意更深了。“唰！”我伸出“\n• zhǎo\n•\n•\n• cǎo\n• méi\n• mián\n• ?g\n• táng\n• de\n• fèn\n• zhī\n• jiù\n• jìn\n• le\n• 魔爪”，草莓棉花糖的四分之一就进了我\n• de\n•\n• xiàng\n• zhēng\n• xìng\n• le\n•\n• mǎn\n• diǎn\n• 的肚子，我象征性地打了个嗝，满足地点\n• diǎn\n• tóu\n•\n•\n• sōu\n•\n•\n• dào\n• yǐng\n• shǎn\n• guò\n•\n• yòng\n• lèi\n• wāng\n• 点头。“嗖！”一道影子闪过，我用泪汪\n• wāng\n• de\n• yǎn\n• jīng\n• zhù\n• shì\n• zhe\n• zhī\n• shèng\n• xià\n• sān\n• fèn\n• zhī\n• èr\n• de\n• mián\n• ?g\n• táng\n• 汪的眼睛注视着只剩下三分之二的棉花糖\n•\n• tóu\n• shàng\n• mào\n• chū\n• le\n• zhèn\n• qīng\n• yān\n•\n•\n• xiāo\n•\n•\n• yún\n•\n• ，头上冒出了一阵青烟。“肖！育！云！\n•\n•\n• bài\n• huài\n•\n• jiào\n• zhe\n•\n• huān\n• xiào\n• shēng\n• zhōng\n•\n• ”我“气急败坏”地大叫着。欢笑声中，\n• liǎng\n• jiàn\n• jiàn\n• de\n• shēng\n• yǐng\n• zài\n• xiàng\n• zhuī\n• zhú\n•\n• dòng\n• tīng\n• de\n• 两个渐渐模糊的声影在互相追逐，动听的\n• xiào\n• shēng\n• zài\n• gōng\n• yuán\n• shàng\n• kōng\n• jiǔ\n• jiǔ\n• huí\n• dàng\n• zhe\n•\n• yuàn\n•\n• 笑声在公园上空久久回荡着，不愿离去。\n•\n•\n• zài\n• guò\n• shēng\n• shí\n•\n• shōu\n• dào\n• le\n• duō\n• tóng\n• xué\n• men\n• sòng\n• lái\n• 　　在过生日时，收到了许多同学们送来\n• de\n•\n• zhè\n• shí\n• de\n•\n• huì\n• chū\n• dān\n• chún\n• zhì\n• 的礼物，这时的我，也会露出一个单纯至\n• de\n• wēi\n• xiào\n•\n• 极的微笑。\n•\n•\n• zhè\n• yàng\n• de\n• kuài\n• duì\n•\n• míng\n• rén\n•\n• lái\n• shuō\n• huò\n• shì\n• 　　这样的快乐对于“名人”来说或许是\n• wēi\n• dào\n• de\n•\n• duì\n• lái\n• shuō\n•\n• zhè\n• xiē\n• huí\n• dōu\n• shì\n• 微不足道的，可对我来说，这些回忆都是\n• jià\n• zhī\n• bǎo\n•\n• yīn\n• wéi\n• jiàn\n• xiǎo\n• xiǎo\n• de\n• shì\n• qíng\n• ér\n• kuài\n• ér\n• 无价之宝。因为一件小小的事情而快乐而\n• wēi\n• xiào\n• jīng\n• chéng\n• le\n• de\n•\n• jiā\n• cháng\n• biàn\n• fàn\n•\n•\n• 微笑似乎已经成了我的“家常便饭”。希\n• wàng\n• hòu\n•\n• hái\n• néng\n• xiàng\n• xiàn\n• zài\n• yàng\n• dān\n• chún\n• xiào\n•\n• xìng\n• 望以后，我还能像现在一样单纯地笑，幸\n• xiào\n•\n• zài\n• fán\n• máng\n• zhōng\n• xún\n• zhǎo\n• níng\n• jìng\n• de\n• tiān\n• kōng\n•\n• zài\n• níng\n• 福地笑，在繁忙中寻找宁静的天空，在宁\n• jìng\n• de\n• tiān\n• kōng\n• zhōng\n• xún\n• zhǎo\n• shǔ\n• de\n• kuài\n• shì\n• zhe\n• 静的天空中寻找属于自己的快乐和喻示着\n• xìng\n• de\n• wēi\n• xiào\n•\n•\n•\n•\n• 幸福的微笑。\n\n无注音版：\n我喜欢这句话。\n它是我们的小学班主任在毕业前夕送给大家的话，寓意深刻。我不知道还有几个人记得这句话，可至少我记得。\n以前大家在一起的时候，欢笑声总是伴着泪水齐飞，幸福的事情与悲伤的事情交织在一起。周末，大家偶尔会出去玩一玩，走一走，把快乐的与大家分享，而悲伤的，一起分担。那天，小云说要去滨湖公园走一走，我说好，肩上的担子太重，我有些疲惫了，去滨湖公园看看，把曾经的那份纯真找回来，还有，我是不是把某些记忆遗落在那里了，我想找回来。\n大家浩浩荡荡地来到滨湖公园，看着只属于小孩子的旋转木马，看着只属于小孩子的赛车，看着只属于小孩子的蹦极……我们有些迷茫——真的不再属于这里了吗？真的真的不再是小孩子了吗？那份单纯的快乐，也再找不回来了吗……\n那又是谁，抹杀了我们的快乐。记忆中，残酷的声音在耳边久久回荡。\n“这么大了还来公园玩，害不害羞呀……”\n我只是想要一份单纯的快乐，我只是想拥有一个幸福的微笑。这样有错么？\n所以，大家抛开了一切烦恼，莫名的迷惘也好、父母的反对也罢，义无反顾地冲上了“happy之旅”。曾经被遗忘的记忆慢慢浮出水面。开心的大笑声，湿透的衣物，紧贴着额头的黑发如缎子一般光滑，一种叫“快乐”的东西从她的眼中倾泻而出，散发出耀眼的光芒……\n这就是曾经的我么？\n坐在旋转木马上，嘴角的弧度暴露了内心的想法，现在的我很快乐很快乐，比６年前的我还要快乐。\n“哒哒哒……”是谁在唱着欢乐的歌？循声望去，原来是一个制作棉花糖的机器在发出“轰轰”的声音，想吸引我们的注意力。\n“呼啦。”大家迅速地围了上去。“叔叔我要原味的！”“叔叔我要草莓味……”“叔叔我要青苹果味的！”“叔叔……”\n……\n曾经何时，我们也想现在这般“疯狂”过。现在的大家还是会像以前那样不满地撅起嘴巴。“为什么她的比我多？”“叔叔我要比她更多的！”我只是轻轻地笑。\n望着手中的棉花糖，甜甜的香气在向周围扩散，幸福的微笑出现在我脸上，刚准备吃掉它，小云就气喘吁吁地跑了过来，她的棉花糖是草莓味的，粉粉的颜色让我眼中的笑意更深了。“唰！”我伸出“魔爪”，草莓棉花糖的四分之一就进了我的肚子，我象征性地打了个嗝，满足地点点头。“嗖！”一道影子闪过，我用泪汪汪的眼睛注视着只剩下三分之二的棉花糖，头上冒出了一阵青烟。“肖！育！云！”我“气急败坏”地大叫着。欢笑声中，两个渐渐模糊的声影在互相追逐，动听的笑声在公园上空久久回荡着，不愿离去。\n在过生日时，收到了许多同学们送来的礼物，这时的我，也会露出一个单纯至极的微笑。\n这样的快乐对于“名人”来说或许是微不足道的，可对我来说，这些回忆都是无价之宝。因为一件小小的事情而快乐而微笑似乎已经成了我的“家常便饭”。希望以后，我还能像现在一样单纯地笑，幸福地笑，在繁忙中寻找宁静的天空，在宁静的天空中寻找属于自己的快乐和喻示着幸福的微笑。\n\n### 我有一个幸福的家\n\n六年级作文561字\n作者：叶晓雅\n•\n• yǒu\n• xìng\n• de\n• jiā\n•\n• de\n• jiā\n• chéng\n• yuán\n• yǒu\n• 　我有一个幸福的家，我的家里成员有四\n•\n• jiù\n• shì\n• zhōng\n• de\n•\n• shì\n• zuì\n• xiǎo\n• de\n• 个，我就是其中的一个，也是最小的一个\n• chéng\n• yuán\n•\n• yòu\n• shì\n• zuì\n• xìng\n• de\n•\n• jiě\n• jiě\n• de\n• dōng\n• ràng\n• 成员。又是最幸福的一个，姐姐的东西让\n• 阅读全文\n\n### 单纯的快乐幸福的微笑\n\n六年级作文1207字\n作者：康刘安怡\n• huān\n• zhè\n• huà\n•\n• 我喜欢这句话。\n•\n•\n• shì\n• men\n• de\n• xiǎo\n• xué\n• bān\n• zhǔ\n• rèn\n• zài\n• qián\n• sòng\n• 　　它是我们的小学班主任在毕业前夕送\n• gěi\n• jiā\n• de\n• huà\n•\n• shēn\n•\n• zhī\n• dào\n• hái\n• yǒu\n• 给大家的话，寓意深刻。我不知道还有几\n• 阅读全文\n\n### 幸福的滋味\n\n六年级作文479字\n作者：陈艺文\n•\n•\n• hóng\n• yàn\n• wài\n• liù\n• nián\n•\n• chén\n• wén\n• 　　鸿雁外语六年级　陈艺文\n•\n•\n• měi\n• rén\n• de\n• shēng\n• huó\n• huán\n• jìng\n• tóng\n•\n• me\n• měi\n• rén\n• 　　每个人的生活环境不同，那么每个人\n• duì\n• xìng\n• de\n• wèi\n• de\n• yàn\n• shì\n• xiàng\n• tóng\n• de\n•\n•\n• 对幸福的滋味的体验也是各不相同的。\n• 阅读全文\n\n### 幸福的滋味\n\n六年级作文569字\n作者：王建莹\n•\n•\n• hóng\n• yàn\n• wài\n• liù\n• nián\n•\n• wáng\n• ?\n• yíng\n• 　　鸿雁外语六年级　王建莹\n•\n•\n• xìng\n• de\n• wèi\n• shì\n• shí\n• me\n•\n• měi\n• rén\n• de\n• jiě\n• 　　幸福的滋味是什么？每个人的理解不\n• yàng\n•\n• shān\n• de\n• hái\n• wàng\n• fēi\n• chū\n• shān\n•\n• xìng\n• de\n• 一样。山区的孩子希望飞出大山，幸福的\n• 阅读全文\n\n### 最幸福的一日\n\n六年级作文：最幸福的一日\n作文字数：419\n作者：颜の之ぎ…\n•\n•\n• yǒu\n• xiē\n• rén\n• zǒng\n• shuō\n• de\n• shēn\n• biān\n• méi\n• yǒu\n• xìng\n• 　　有些人总说我的身边没有幸福可我不\n• jiào\n• nián\n• de\n• jiào\n• de\n• xìng\n• shí\n• dōu\n• zài\n• men\n• de\n• shēn\n• 觉年的我觉的幸福无时不刻都在我们的身\n• biān\n•\n•\n• kǎo\n• shì\n• kǎo\n• hǎo\n• le\n•\n• dài\n• chū\n• wán\n•\n• 边。如，考试考好了、妈妈带我出去玩…\n• 阅读全文\n\n### 幸福的意义\n\n六年级作文：幸福的意义\n作文字数：432\n作者：未知\n•\n•\n• xìng\n• shì\n• shí\n• me\n•\n• xìng\n• shì\n• nóng\n• nóng\n• de\n• qīn\n• qíng\n•\n• yǒu\n• 　　幸福是什么？幸福是浓浓的亲情、友\n• qíng\n•\n• xìng\n• shì\n• chū\n• zhī\n• hòu\n• de\n• huí\n• bào\n•\n• xìng\n• shì\n• shēng\n• huó\n• 情。幸福是付出之后的回报，幸福是生活\n• zhōng\n• diǎn\n• diǎn\n• de\n•\n• xìng\n• jiù\n• zài\n• men\n• shēn\n• biān\n•\n• 中点点的乐趣，幸福就在我们身边。\n• 阅读全文" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8945486,"math_prob":0.60567766,"size":5807,"snap":"2020-24-2020-29","text_gpt3_token_len":6459,"char_repetition_ratio":0.1033948,"word_repetition_ratio":0.24016282,"special_character_ratio":0.2922335,"punctuation_ratio":0.0012919897,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99226665,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-14T16:18:59Z\",\"WARC-Record-ID\":\"<urn:uuid:f42f0a28-3935-451d-aff0-342a4b0d694f>\",\"Content-Length\":\"30495\",\"Content-Type\":\"application/http; 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https://www.tutoringhour.com/worksheets/volume/hemispheres/	[ "# Volume of a Hemisphere Worksheets\n\n1. Math >\n2. Geometry >\n3. Volume >\n4. Hemispheres\n\nMaster finding the volume of hemispheres with this bundle of free printable volume of hemisphere worksheets. Help children further their practice on determining the volume of hemispheres and get acquainted with the fact that a hemisphere is half a sphere and hence its volume is also half the volume of a sphere. The pdf exercises here provide kids a great way to learn and perfect skills in finding the volume of hemispheres.\n\nThis bunch of printable worksheets is appropriate for 8th grade and high school students.\n\nCCSS: 8.G.9, G-GMD.3\n\nIntegers", null, "Decimals", null, "" ]	[ null, "https://www.tutoringhour.com/images/worksheet-th-all-before.svg", null, "https://www.tutoringhour.com/images/worksheet-th-all-before.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9083522,"math_prob":0.5117927,"size":571,"snap":"2023-40-2023-50","text_gpt3_token_len":118,"char_repetition_ratio":0.20458554,"word_repetition_ratio":0.0,"special_character_ratio":0.18739054,"punctuation_ratio":0.08411215,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95101225,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-27T09:37:11Z\",\"WARC-Record-ID\":\"<urn:uuid:4e8f2105-37e6-4b4e-90aa-1ae6e2bd6387>\",\"Content-Length\":\"43920\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ec3e1524-a045-475c-86b6-d75b93dd824f>\",\"WARC-Concurrent-To\":\"<urn:uuid:52c23f1a-d5c6-40ea-893a-8b47cc058cde>\",\"WARC-IP-Address\":\"67.227.190.101\",\"WARC-Target-URI\":\"https://www.tutoringhour.com/worksheets/volume/hemispheres/\",\"WARC-Payload-Digest\":\"sha1:ZS4IAMXVM5G7DLAMZKXCUWRBTR7DTX6N\",\"WARC-Block-Digest\":\"sha1:Q7EO7EZUN2EUOYFA3XQ5JJ6AXKC43R3X\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510284.49_warc_CC-MAIN-20230927071345-20230927101345-00250.warc.gz\"}"}
https://www.nuclear-power.net/nuclear-power/reactor-physics/atomic-nuclear-physics/atomic-theory/bohr-model-of-atom/	[ "# Bohr Model of Atom\n\n## Bohr Model of the Atom", null, "In atomic physics, the Bohr model if the atom (also known as the Rutherford-Bohr model) is modern model of the hydrogen atom introduced by Danish physicist Niels Bohr working with Ernest Rutherford at the University of Manchester in 1913.\n\nThis model provides especially the solution to the problem of the failure of classical physics in the field of atomic physics. Classical electromagnetic theory, makes three entirely wrong predictions about atoms:\n\n• atoms should emit light continuously,\n• atoms should be unstable,\n• the light they emit should have a continuous spectrum.\n\nThe Bohr model adopted Planck’s quantum hypothesis and he proposed a model in which the electrons of an atom were assumed to orbit the nucleus but could only do so in a finite set of orbits. Planck’s quantum hypothesis (Planck’s law) is named after a German theoretical physicist Max Planck, who proposed it in 1900. Planck’s law is a pioneering result of modern physics and quantum theory. Planck’s hypothesis that energy is radiated and absorbed in discrete “quanta” (or energy packets) precisely matched the observed patterns of blackbody radiation and resolved the ultraviolet catastrophe.", null, "Based on this hypothesis, Bohr postulated that an atom emits or absorbs energy only in discrete quanta corresponding to absorption or radiation of a photon. During the emission of a photon, the internal energy of the atom changes by an amount equal to the energy of the photon. Therefore, each atom must be able to exist with only certain specific values of internal energy. Each of these stationary states is characterised by a specific amount of energy called its energy level. An atom can have an amount of internal energy equal to any one of these possible energy levels, but it cannot have an energy intermediate between two levels.\n\nThe success of Bohr model is in explaining the spectral lines of atomic hydrogen and the Rydberg formula for the spectral emission lines. Bohr’s theory was the first to successfully account for the discrete energy levels of this radiation as measured in the laboratory. The emission line spectrum of hydrogen tells us that atoms of that element emit photons with only certain specific frequencies ƒ and hence certain specific energies equal to E = hƒ, where h = Planck’s constant = 6.63 x 10-34 J.s and f = frequency of the photon.\n\nAlthough Bohr’s atomic model is designed specifically to explain the hydrogen atom, his theories apply generally to the structure of all atoms. Subsequently, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. Heavier atoms, like carbon or oxygen, have more protons in the nucleus, and more electrons to cancel the charge. Bohr’s idea was that each discrete orbit could only hold a certain number of electrons. In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons corresponded to stable “closed shells”. After that orbit is full, the next energy level would have to be used. This defines an electron shell, which is the set of allowed states and it gives the atom an electron shell structure.\n\n## Bohr’s Postulates\n\nAll these features of Bohr model of the atom can be summarized in Bohr’s Postulates:\n\n1. Electrons in atoms orbit the nucleus.\n2. An atom can exist only in certain specific energy states, in which an electron can reside without the emission of radiant energy.\n3. Transmission between stationary states can occur only by jumping from one allowed orbit to another. This process produces/absorbs only a discrete quanta of electromagnetic radiation.\n4. The angular momentum of a stationary electron is also quantised.\n\nReferences:\nNuclear and Reactor Physics:\n1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).\n2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.\n3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.\n4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317\n5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467\n6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965\n7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.\n8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.\n9. Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414." ]	[ null, "https://www.nuclear-power.net/wp-content/uploads/Faulure-of-Classical-Physics-Atomic-Nucleus-283x300.png", null, "https://www.nuclear-power.net/wp-content/uploads/Bohr-model-atom-261x300.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9255753,"math_prob":0.8689342,"size":3658,"snap":"2021-04-2021-17","text_gpt3_token_len":740,"char_repetition_ratio":0.12944718,"word_repetition_ratio":0.0,"special_character_ratio":0.19300164,"punctuation_ratio":0.075987846,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96392524,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-19T09:10:15Z\",\"WARC-Record-ID\":\"<urn:uuid:f70e3f07-71b5-49c3-8b7f-122049dfac91>\",\"Content-Length\":\"56371\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c7f34c1e-db42-49da-84bf-98fe4de65da9>\",\"WARC-Concurrent-To\":\"<urn:uuid:e0ed806a-cd29-4dd1-88d2-44f1167770fa>\",\"WARC-IP-Address\":\"104.21.66.209\",\"WARC-Target-URI\":\"https://www.nuclear-power.net/nuclear-power/reactor-physics/atomic-nuclear-physics/atomic-theory/bohr-model-of-atom/\",\"WARC-Payload-Digest\":\"sha1:EO7TLDLZALURX37NMJD7EOYLV2XA6UCU\",\"WARC-Block-Digest\":\"sha1:WDMVVY7O6PL6UTG52OMOAJUGGWNWMJSL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703518201.29_warc_CC-MAIN-20210119072933-20210119102933-00566.warc.gz\"}"}
https://mizar.uwb.edu.pl/version/current/html/ndiff_4.html	[ ":: The Differentiable Functions from $\\mathbbR$ into ${\\mathbbR}^n$\n:: by Keiko Narita , Artur Korni\\l owicz and Yasunari Shidama\n::\n:: Copyright (c) 2011-2021 Association of Mizar Users\n\nset ZR = [#] REAL;\n\nLm1: the carrier of () = REAL 1\nby REAL_NS1:def 4;\n\ntheorem Th1: :: NDIFF_4:1\nfor m being Element of NAT\nfor f1, f2 being PartFunc of REAL,(REAL m) holds f1 - f2 = f1 + (- f2)\nproof end;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of REAL,(REAL n);\nlet x be Real;\npred f is_differentiable_in x means :: NDIFF_4:def 1\nex g being PartFunc of REAL,() st\n( f = g & g is_differentiable_in x );\nend;\n\n:: deftheorem defines is_differentiable_in NDIFF_4:def 1 :\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor x being Real holds\n( f is_differentiable_in x iff ex g being PartFunc of REAL,() st\n( f = g & g is_differentiable_in x ) );\n\ntheorem :: NDIFF_4:2\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor h being PartFunc of REAL,()\nfor x being Real st h = f holds\n( f is_differentiable_in x iff h is_differentiable_in x ) ;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of REAL,(REAL n);\nlet x be Real;\nfunc diff (f,x) -> Element of REAL n means :Def2: :: NDIFF_4:def 2\nex g being PartFunc of REAL,() st\n( f = g & it = diff (g,x) );\nexistence\nex b1 being Element of REAL n ex g being PartFunc of REAL,() st\n( f = g & b1 = diff (g,x) )\nproof end;\nuniqueness\nfor b1, b2 being Element of REAL n st ex g being PartFunc of REAL,() st\n( f = g & b1 = diff (g,x) ) & ex g being PartFunc of REAL,() st\n( f = g & b2 = diff (g,x) ) holds\nb1 = b2\n;\nend;\n\n:: deftheorem Def2 defines diff NDIFF_4:def 2 :\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor x being Real\nfor b4 being Element of REAL n holds\n( b4 = diff (f,x) iff ex g being PartFunc of REAL,() st\n( f = g & b4 = diff (g,x) ) );\n\ntheorem Th3: :: NDIFF_4:3\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor h being PartFunc of REAL,()\nfor x being Real st h = f holds\ndiff (f,x) = diff (h,x)\nproof end;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of REAL,(REAL n);\nlet X be set ;\npred f is_differentiable_on X means :: NDIFF_4:def 3\n( X c= dom f & ( for x being Real st x in X holds\nf \| X is_differentiable_in x ) );\nend;\n\n:: deftheorem defines is_differentiable_on NDIFF_4:def 3 :\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor X being set holds\n( f is_differentiable_on X iff ( X c= dom f & ( for x being Real st x in X holds\nf \| X is_differentiable_in x ) ) );\n\ntheorem Th4: :: NDIFF_4:4\nfor X being set\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_on X holds\nX is Subset of REAL by XBOOLE_1:1;\n\ntheorem Th5: :: NDIFF_4:5\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) holds\n( f is_differentiable_on Z iff ( Z c= dom f & ( for x being Real st x in Z holds\nf is_differentiable_in x ) ) )\nproof end;\n\ntheorem Th6: :: NDIFF_4:6\nfor n being non zero Element of NAT\nfor Y being Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_on Y holds\nY is open\nproof end;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of REAL,(REAL n);\nlet X be set ;\nassume A1: f is_differentiable_on X ;\nfunc f \| X -> PartFunc of REAL,(REAL n) means :Def4: :: NDIFF_4:def 4\n( dom it = X & ( for x being Real st x in X holds\nit . x = diff (f,x) ) );\nexistence\nex b1 being PartFunc of REAL,(REAL n) st\n( dom b1 = X & ( for x being Real st x in X holds\nb1 . x = diff (f,x) ) )\nproof end;\nuniqueness\nfor b1, b2 being PartFunc of REAL,(REAL n) st dom b1 = X & ( for x being Real st x in X holds\nb1 . x = diff (f,x) ) & dom b2 = X & ( for x being Real st x in X holds\nb2 . x = diff (f,x) ) holds\nb1 = b2\nproof end;\nend;\n\n:: deftheorem Def4 defines \| NDIFF_4:def 4 :\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor X being set st f is_differentiable_on X holds\nfor b4 being PartFunc of REAL,(REAL n) holds\n( b4 = f \| X iff ( dom b4 = X & ( for x being Real st x in X holds\nb4 . x = diff (f,x) ) ) );\n\ntheorem :: NDIFF_4:7\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st Z c= dom f & ex r being Element of REAL n st rng f = {r} holds\n( f is_differentiable_on Z & ( for x being Real st x in Z holds\n(f \| Z) /. x = 0* n ) )\nproof end;\n\ntheorem :: NDIFF_4:8\nfor n being non zero Element of NAT\nfor x0 being Real\nfor f being PartFunc of REAL,(REAL n)\nfor g being PartFunc of REAL,()\nfor N being Neighbourhood of x0 st f = g & f is_differentiable_in x0 & N c= dom f holds\nfor h being non-zero 0 -convergent Real_Sequence\nfor c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= N holds\n( (h \") (#) ((g /* (h + c)) - (g /* c)) is convergent & diff (f,x0) = lim ((h \") (#) ((g /* (h + c)) - (g /* c))) )\nproof end;\n\ntheorem Th9: :: NDIFF_4:9\nfor x0, r being Real\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_in x0 holds\n( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) )\nproof end;\n\ntheorem Th10: :: NDIFF_4:10\nfor x0 being Real\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_in x0 holds\n( - f is_differentiable_in x0 & diff ((- f),x0) = - (diff (f,x0)) )\nproof end;\n\ntheorem Th11: :: NDIFF_4:11\nfor x0 being Real\nfor n being non zero Element of NAT\nfor f1, f2 being PartFunc of REAL,(REAL n) st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds\n( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) )\nproof end;\n\ntheorem :: NDIFF_4:12\nfor x0 being Real\nfor n being non zero Element of NAT\nfor f1, f2 being PartFunc of REAL,(REAL n) st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds\n( f1 - f2 is_differentiable_in x0 & diff ((f1 - f2),x0) = (diff (f1,x0)) - (diff (f2,x0)) )\nproof end;\n\ntheorem Th13: :: NDIFF_4:13\nfor r being Real\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st Z c= dom f & f is_differentiable_on Z holds\n( r (#) f is_differentiable_on Z & ( for x being Real st x in Z holds\n((r (#) f) \| Z) . x = r * (diff (f,x)) ) )\nproof end;\n\ntheorem Th14: :: NDIFF_4:14\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st Z c= dom f & f is_differentiable_on Z holds\n( - f is_differentiable_on Z & ( for x being Real st x in Z holds\n((- f) \| Z) . x = - (diff (f,x)) ) )\nproof end;\n\ntheorem Th15: :: NDIFF_4:15\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f1, f2 being PartFunc of REAL,(REAL n) st Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds\n( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds\n((f1 + f2) \| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) )\nproof end;\n\ntheorem :: NDIFF_4:16\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f1, f2 being PartFunc of REAL,(REAL n) st Z c= dom (f1 - f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds\n( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds\n((f1 - f2) \| Z) . x = (diff (f1,x)) - (diff (f2,x)) ) )\nproof end;\n\ntheorem :: NDIFF_4:17\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st Z c= dom f & f \| Z is constant holds\n( f is_differentiable_on Z & ( for x being Real st x in Z holds\n(f \| Z) . x = 0* n ) )\nproof end;\n\ntheorem Th18: :: NDIFF_4:18\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n)\nfor r, p being Element of REAL n st Z c= dom f & ( for x being Real st x in Z holds\nf /. x = (x * r) + p ) holds\n( f is_differentiable_on Z & ( for x being Real st x in Z holds\n(f \| Z) . x = r ) )\nproof end;\n\ntheorem :: NDIFF_4:19\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor x0 being Real st f is_differentiable_in x0 holds\nf is_continuous_in x0 by ;\n\ntheorem :: NDIFF_4:20\nfor X being set\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_on X holds\nf \| X is continuous\nproof end;\n\ntheorem Th21: :: NDIFF_4:21\nfor X being set\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being PartFunc of REAL,(REAL n) st f is_differentiable_on X & Z c= X holds\nf is_differentiable_on Z\nproof end;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of REAL,(REAL n);\nattr f is differentiable means :Def5: :: NDIFF_4:def 5\nf is_differentiable_on dom f;\nend;\n\n:: deftheorem Def5 defines differentiable NDIFF_4:def 5 :\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) holds\n( f is differentiable iff f is_differentiable_on dom f );\n\nregistration\nlet n be non zero Element of NAT ;\ncluster REAL --> (0* n) -> differentiable for Function of REAL,(REAL n);\ncoherence\nfor b1 being Function of REAL,(REAL n) st b1 = REAL --> (0* n) holds\nb1 is differentiable\nproof end;\nend;\n\nregistration\nlet n be non zero Element of NAT ;\nexistence\nex b1 being Function of REAL,(REAL n) st b1 is differentiable\nproof end;\nend;\n\ntheorem :: NDIFF_4:22\nfor n being non zero Element of NAT\nfor Z being open Subset of REAL\nfor f being differentiable PartFunc of REAL,(REAL n) st Z c= dom f holds\nf is_differentiable_on Z by ;\n\ntheorem Th23: :: NDIFF_4:23\nfor n being non zero Element of NAT\nfor R being PartFunc of REAL,() st R is total holds\n( R is RestFunc-like iff for r being Real st r > 0 holds\nex d being Real st\n( d > 0 & ( for z being Real st z <> 0 & \|.z.\| < d holds\n() * \|\|.(R /. z).\|\| < r ) ) )\nproof end;\n\nreconsider jj = 1 as Element of REAL by XREAL_0:def 1;\n\ntheorem Th24: :: NDIFF_4:24\nfor i being Element of NAT\nfor n being non zero Element of NAT\nfor g being PartFunc of REAL,()\nfor x0 being Real st 1 <= i & i <= n & g is_differentiable_in x0 holds\n( (Proj (i,n)) * g is_differentiable_in x0 & (Proj (i,n)) . (diff (g,x0)) = diff (((Proj (i,n)) * g),x0) )\nproof end;\n\ntheorem Th25: :: NDIFF_4:25\nfor n being non zero Element of NAT\nfor g being PartFunc of REAL,()\nfor x0 being Real holds\n( g is_differentiable_in x0 iff for i being Element of NAT st 1 <= i & i <= n holds\n(Proj (i,n)) * g is_differentiable_in x0 )\nproof end;\n\ntheorem :: NDIFF_4:26\nfor i being Element of NAT\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor x0 being Real st 1 <= i & i <= n & f is_differentiable_in x0 holds\n( (Proj (i,n)) * f is_differentiable_in x0 & (Proj (i,n)) . (diff (f,x0)) = diff (((Proj (i,n)) * f),x0) )\nproof end;\n\ntheorem :: NDIFF_4:27\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n)\nfor x0 being Real holds\n( f is_differentiable_in x0 iff for i being Element of NAT st 1 <= i & i <= n holds\n(Proj (i,n)) * f is_differentiable_in x0 )\nproof end;\n\ntheorem Th28: :: NDIFF_4:28\nfor X being set\nfor i being Element of NAT\nfor n being non zero Element of NAT\nfor g being PartFunc of REAL,() st 1 <= i & i <= n & g is_differentiable_on X holds\n( (Proj (i,n)) * g is_differentiable_on X & (Proj (i,n)) * (g \| X) = ((Proj (i,n)) * g) \| X )\nproof end;\n\ntheorem Th29: :: NDIFF_4:29\nfor X being set\nfor i being Element of NAT\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) st 1 <= i & i <= n & f is_differentiable_on X holds\n( (Proj (i,n)) * f is_differentiable_on X & (Proj (i,n)) * (f \| X) = ((Proj (i,n)) * f) \| X )\nproof end;\n\ntheorem Th30: :: NDIFF_4:30\nfor X being set\nfor n being non zero Element of NAT\nfor g being PartFunc of REAL,() holds\n( g is_differentiable_on X iff for i being Element of NAT st 1 <= i & i <= n holds\n(Proj (i,n)) * g is_differentiable_on X )\nproof end;\n\ntheorem :: NDIFF_4:31\nfor X being set\nfor n being non zero Element of NAT\nfor f being PartFunc of REAL,(REAL n) holds\n( f is_differentiable_on X iff for i being Element of NAT st 1 <= i & i <= n holds\n(Proj (i,n)) * f is_differentiable_on X )\nproof end;\n\ntheorem Th32: :: NDIFF_4:32\nfor J being Function of (),REAL\nfor x0 being Point of () st J = proj (1,1) holds\nJ is_continuous_in x0\nproof end;\n\ntheorem Th33: :: NDIFF_4:33\nfor x0 being Real\nfor I being Function of REAL,() st I = (proj (1,1)) \" holds\nI is_continuous_in x0\nproof end;\n\ntheorem Th34: :: NDIFF_4:34\nfor S, T being RealNormSpace\nfor f1 being PartFunc of S,REAL\nfor f2 being PartFunc of REAL,T\nfor x0 being Point of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds\nf2 * f1 is_continuous_in x0\nproof end;\n\nLm2: ( dom (proj (1,1)) = REAL 1 & rng (proj (1,1)) = REAL & ( for x being Element of REAL holds\n( (proj (1,1)) . <x> = x & ((proj (1,1)) \") . x = <x> ) ) )\n\nproof end;\n\ntheorem :: NDIFF_4:35\nfor n being non zero Element of NAT\nfor J being Function of (),REAL\nfor x0 being Point of ()\nfor y0 being Element of REAL\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st J = proj (1,1) & x0 in dom f & y0 in dom g & x0 = <y0> & f = g * J holds\n( f is_continuous_in x0 iff g is_continuous_in y0 )\nproof end;\n\ntheorem :: NDIFF_4:36\nfor n being non zero Element of NAT\nfor I being Function of REAL,()\nfor x0 being Point of ()\nfor y0 being Element of REAL\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st I = (proj (1,1)) \" & x0 in dom f & y0 in dom g & x0 = <y0> & f * I = g holds\n( f is_continuous_in x0 iff g is_continuous_in y0 )\nproof end;\n\ntheorem :: NDIFF_4:37\nfor x0 being Real\nfor I being Function of REAL,() st I = (proj (1,1)) \" holds\n( I is_differentiable_in x0 & diff (I,x0) = <1> )\nproof end;\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of (),REAL;\nlet x be Point of ();\npred f is_differentiable_in x means :: NDIFF_4:def 6\nex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & g is_differentiable_in y );\nend;\n\n:: deftheorem defines is_differentiable_in NDIFF_4:def 6 :\nfor n being non zero Element of NAT\nfor f being PartFunc of (),REAL\nfor x being Point of () holds\n( f is_differentiable_in x iff ex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & g is_differentiable_in y ) );\n\ndefinition\nlet n be non zero Element of NAT ;\nlet f be PartFunc of (),REAL;\nlet x be Point of ();\nfunc diff (f,x) -> Function of (),REAL means :Def7: :: NDIFF_4:def 7\nex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & it = diff (g,y) );\nexistence\nex b1 being Function of (),REAL ex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & b1 = diff (g,y) )\nproof end;\nuniqueness\nfor b1, b2 being Function of (),REAL st ex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & b1 = diff (g,y) ) & ex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & b2 = diff (g,y) ) holds\nb1 = b2\n;\nend;\n\n:: deftheorem Def7 defines diff NDIFF_4:def 7 :\nfor n being non zero Element of NAT\nfor f being PartFunc of (),REAL\nfor x being Point of ()\nfor b4 being Function of (),REAL holds\n( b4 = diff (f,x) iff ex g being PartFunc of (REAL n),REAL ex y being Element of REAL n st\n( f = g & x = y & b4 = diff (g,y) ) );\n\ntheorem Th38: :: NDIFF_4:38\nfor J being Function of (REAL 1),REAL\nfor x0 being Element of REAL 1 st J = proj (1,1) holds\n( J is_differentiable_in x0 & diff (J,x0) = J )\nproof end;\n\ntheorem :: NDIFF_4:39\nfor J being Function of (),REAL\nfor x0 being Point of () st J = proj (1,1) holds\n( J is_differentiable_in x0 & diff (J,x0) = J )\nproof end;\n\ntheorem Th40: :: NDIFF_4:40\nfor n being non zero Element of NAT\nfor I being Function of REAL,() st I = (proj (1,1)) \" holds\n( ( for R being RestFunc of (),() holds R * I is RestFunc of () ) & ( for L being LinearOperator of (),() holds L * I is LinearFunc of () ) )\nproof end;\n\ntheorem Th41: :: NDIFF_4:41\nfor n being non zero Element of NAT\nfor J being Function of (),REAL st J = proj (1,1) holds\n( ( for R being RestFunc of () holds R * J is RestFunc of (),() ) & ( for L being LinearFunc of () holds L * J is Lipschitzian LinearOperator of (),() ) )\nproof end;\n\ntheorem Th42: :: NDIFF_4:42\nfor n being non zero Element of NAT\nfor I being Function of REAL,()\nfor x0 being Point of ()\nfor y0 being Element of REAL\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st I = (proj (1,1)) \" & x0 in dom f & y0 in dom g & x0 = <y0> & f * I = g & f is_differentiable_in x0 holds\n( g is_differentiable_in y0 & diff (g,y0) = (diff (f,x0)) . <1> & ( for r being Element of REAL holds (diff (f,x0)) . <r> = r * (diff (g,y0)) ) )\nproof end;\n\ntheorem Th43: :: NDIFF_4:43\nfor n being non zero Element of NAT\nfor I being Function of REAL,()\nfor x0 being Point of ()\nfor y0 being Real\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st I = (proj (1,1)) \" & x0 in dom f & y0 in dom g & x0 = <y0> & f * I = g holds\n( f is_differentiable_in x0 iff g is_differentiable_in y0 )\nproof end;\n\ntheorem Th44: :: NDIFF_4:44\nfor n being non zero Element of NAT\nfor J being Function of (),REAL\nfor x0 being Point of ()\nfor y0 being Element of REAL\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st J = proj (1,1) & x0 in dom f & y0 in dom g & x0 = <y0> & f = g * J holds\n( f is_differentiable_in x0 iff g is_differentiable_in y0 )\nproof end;\n\ntheorem :: NDIFF_4:45\nfor n being non zero Element of NAT\nfor J being Function of (),REAL\nfor x0 being Point of ()\nfor y0 being Element of REAL\nfor g being PartFunc of REAL,()\nfor f being PartFunc of (),() st J = proj (1,1) & x0 in dom f & y0 in dom g & x0 = <y0> & f = g * J & g is_differentiable_in y0 holds\n( f is_differentiable_in x0 & diff (g,y0) = (diff (f,x0)) . <1> & ( for r being Element of REAL holds (diff (f,x0)) . <r> = r * (diff (g,y0)) ) )\nproof end;\n\ntheorem Th46: :: NDIFF_4:46\nfor n being non zero Element of NAT\nfor R being RestFunc of () st R /. 0 = 0. () holds\nfor e being Real st e > 0 holds\nex d being Real st\n( d > 0 & ( for h being Real st \|.h.\| < d holds\n\|\|.(R /. h).\|\| <= e * \|.h.\| ) )\nproof end;\n\ntheorem Th47: :: NDIFF_4:47\nfor n, m being non zero Element of NAT\nfor R being RestFunc of ()\nfor L being Lipschitzian LinearOperator of (),() holds L * R is RestFunc of ()\nproof end;\n\ntheorem Th48: :: NDIFF_4:48\nfor n, m being non zero Element of NAT\nfor R1 being RestFunc of () st R1 /. 0 = 0. () holds\nfor R2 being RestFunc of (),() st R2 /. (0. ()) = 0. () holds\nfor L being LinearFunc of () holds R2 * (L + R1) is RestFunc of ()\nproof end;\n\ntheorem Th49: :: NDIFF_4:49\nfor n, m being non zero Element of NAT\nfor R1 being RestFunc of () st R1 /. 0 = 0. () holds\nfor R2 being RestFunc of (),() st R2 /. (0. ()) = 0. () holds\nfor L1 being LinearFunc of ()\nfor L2 being Lipschitzian LinearOperator of (),() holds (L2 * R1) + (R2 * (L1 + R1)) is RestFunc of ()\nproof end;\n\ntheorem :: NDIFF_4:50\nfor n, m being non zero Element of NAT\nfor x0 being Element of REAL\nfor g being PartFunc of REAL,() st g is_differentiable_in x0 holds\nfor f being PartFunc of (),() st f is_differentiable_in g /. x0 holds\n( f * g is_differentiable_in x0 & diff ((f * g),x0) = (diff (f,(g /. x0))) . (diff (g,x0)) )\nproof end;" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7171366,"math_prob":0.9972653,"size":860,"snap":"2022-27-2022-33","text_gpt3_token_len":332,"char_repetition_ratio":0.17757009,"word_repetition_ratio":0.35643566,"special_character_ratio":0.41511628,"punctuation_ratio":0.13461539,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997197,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-04T03:18:14Z\",\"WARC-Record-ID\":\"<urn:uuid:4f9920bf-4a61-4507-822c-db1044bee380>\",\"Content-Length\":\"202907\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ebb6ef09-323b-4f87-84ec-4fb9f0fdfb83>\",\"WARC-Concurrent-To\":\"<urn:uuid:f557e444-88c7-4936-9cbb-f285a05fdc69>\",\"WARC-IP-Address\":\"212.33.73.131\",\"WARC-Target-URI\":\"https://mizar.uwb.edu.pl/version/current/html/ndiff_4.html\",\"WARC-Payload-Digest\":\"sha1:DSK5JT5ULEGBV7Y5TKEPVS2MGFDJJRFD\",\"WARC-Block-Digest\":\"sha1:NI2O2BDLHQ2V57BR2VUGLA5EUHXO4L7A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104293758.72_warc_CC-MAIN-20220704015700-20220704045700-00394.warc.gz\"}"}
https://www.programminglogic.com/quicksort-and-binary-search-algorithms-in-cc/	[ "# Quicksort and Binary Search Algorithms in C/C++\n\nBeing able to sort and search efficiently is one of the most important and most studied aspects of computer science. In my opinion any programmer worth his salt should be able to, in a matter of 10 minutes or so, be able to write the algorithms of binary search as well as those of the most important sorts (e.g., insertion sort, selection sort, bubble sort, merge sort, heap sort, and quicksort).\n\nOnce you master those, though, you might wanna use a template to speed up development. Below are the templates I use for binary search and quicksort.\n\nNotice that start and end in the bSearch function below refer to the first and last element, so end is the size of the array minus one. I use this approach because I think it makes easier to figure the limits you need to work with inside the function.\n\n``````int bSearch(int n,int array[], int start, int end){\nint middle = (start+end)/2;\n\nif (start>end)\nreturn 0;\n\nif (n==array[middle])\nreturn 1;\n\nif (n>array[middle])\nreturn bSearch(n,array,middle+1, end);\nelse\nreturn bSearch(n,array,start,middle-1);\n}``````\n\nAnd now the quicksort algorithm, where end is actually the size of the array.\n\n``````void swap(int v[], int i, int j){\nint temp = v[i];\nv[i]=v[j];\nv[j]=temp;\nreturn;\n}\n\nint partition(int v[], int start, int end){\nint x,l,j;\n\nx = v[end-1];\nl = start-1;\n\nfor (j=start;j<end-1;j++){\nif (v[j]<x){\nl++;\nswap(v,j,l);\n}\n}\nl++;\nswap(v,l,end-1);\nreturn l;\n}\n\nvoid quickSort(int v[],int start, int end){\nint q;\n\nif (end>start){\nq = partition(v,start,end;\nquickSort(v,start,q);\nquickSort(v,q+1,end);\n}\nreturn;\n}``````\n\n## One thought on “Quicksort and Binary Search Algorithms in C/C++”\n\n1.", null, "sookwan\n\nthanks for the code~" ]	[ null, "https://secure.gravatar.com/avatar/415b53cc6902686b1302eb4a8e58844d", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7371687,"math_prob":0.9458827,"size":1553,"snap":"2020-34-2020-40","text_gpt3_token_len":417,"char_repetition_ratio":0.1220142,"word_repetition_ratio":0.008,"special_character_ratio":0.28525436,"punctuation_ratio":0.1930295,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9952735,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-06T22:27:08Z\",\"WARC-Record-ID\":\"<urn:uuid:8dfa2afc-c05d-414e-89d4-57de702146d7>\",\"Content-Length\":\"25228\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b7924e95-563e-49d4-bdf8-f063056323e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:49991efb-787e-4c9b-ae09-0d965c659fde>\",\"WARC-IP-Address\":\"50.116.102.205\",\"WARC-Target-URI\":\"https://www.programminglogic.com/quicksort-and-binary-search-algorithms-in-cc/\",\"WARC-Payload-Digest\":\"sha1:LBPFZBACRAMHOQTM2NNDTOMHLHTT3AKK\",\"WARC-Block-Digest\":\"sha1:ZORT3WPKD4TS6MN4BDBJ7NQXSKMGDEWV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737039.58_warc_CC-MAIN-20200806210649-20200807000649-00351.warc.gz\"}"}
https://answers.everydaycalculation.com/compare-fractions/15-49-and-35-70	[ "Solutions by everydaycalculation.com\n\n## Compare 15/49 and 35/70\n\n15/49 is smaller than 35/70\n\n#### Steps for comparing fractions\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 49 and 70 is 490\n\nNext, find the equivalent fraction of both fractional numbers with denominator 490\n2. For the 1st fraction, since 49 × 10 = 490,\n15/49 = 15 × 10/49 × 10 = 150/490\n3. Likewise, for the 2nd fraction, since 70 × 7 = 490,\n35/70 = 35 × 7/70 × 7 = 245/490\n4. Since the denominators are now the same, the fraction with the bigger numerator is the greater fraction\n5. 150/490 < 245/490 or 15/49 < 35/70\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85607976,"math_prob":0.9846968,"size":788,"snap":"2023-40-2023-50","text_gpt3_token_len":260,"char_repetition_ratio":0.17857143,"word_repetition_ratio":0.0,"special_character_ratio":0.41751269,"punctuation_ratio":0.06832298,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969605,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-22T17:22:41Z\",\"WARC-Record-ID\":\"<urn:uuid:6b8601a7-d7a1-4759-b168-c83bfd6a8504>\",\"Content-Length\":\"7133\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:02a104b0-a531-4afe-af05-c094b891240c>\",\"WARC-Concurrent-To\":\"<urn:uuid:8b85d432-db01-40b2-a129-591b4d466b33>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/compare-fractions/15-49-and-35-70\",\"WARC-Payload-Digest\":\"sha1:5LUY3TM7Q4HMD2NUFTITKKE23REMEWZD\",\"WARC-Block-Digest\":\"sha1:Z6BVG65XON23D76CD3TE45ICMU4GRBE7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506421.14_warc_CC-MAIN-20230922170343-20230922200343-00340.warc.gz\"}"}
https://bsci-ch.org/how-can-you-tell-one-element-from-another/	[ "The Atom and also Electromagnetic Radiation\n\nYou are watching: How can you tell one element from another\n\n Fundamental Subatomic Particles Electromagnetic Radiation Light and Other forms of Electromagnetic Radiation\n\n Particle Symbol Charge Mass electron e- -1 0.0005486 amu proton p+ +1 1.007276 amu neutron no 0 1.008665 amu\n\nThe number of protons, neutrons, and also electrons in one atom have the right to be figured out from a collection of an easy rules.\n\nThe variety of protons in the cell core of the atom is same to the atomic number (Z). The variety of electrons in a neutral atom is equal to the variety of protons. The mass variety of the atom (M) is equal to the sum of the variety of protons and also neutrons in the nucleus. The variety of neutrons is equal to the difference in between the mass variety of the atom (M) and also the atom number (Z).\n\nExamples: Let\"s identify the number of protons, neutrons, and electrons in the following isotopes.\n\n 12C 13C 14C 14N\n\nThe various isotopes of an facet are established by composing the mass variety of the atom in the top left corner of the symbol for the element. 12C, 13C, and 14C room isotopes that carbon (Z = 6) and therefore contain 6 protons. If the atoms are neutral, they likewise must contain 6 electrons. The just difference in between these isotopes is the number of neutrons in the nucleus.\n\n12C: 6 electrons, 6 protons, and 6 neutrons\n\n13C: 6 electrons, 6 protons, and 7 neutrons\n\n14C: 6 electrons, 6 protons, and also 8 neutrons\n\n Practice problem 1:Calculate the number of electrons in the Cl- and Fe3+ ions. Click below to inspect your answer to Practice trouble 1\n\nMuch of what is known about the framework of the electrons in one atom has been obtained by studying the interaction between matter and different creates of electromagnetic radiation. Electromagnetic radiation has some the the properties of both a particle and also a wave.\n\nParticles have actually a identify mass and also they accounting space. Waves have no mass and also yet they bring energy as they travel through space. In addition to their capacity to carry energy, tide have four other properties properties: speed, frequency, wavelength, and also amplitude. The frequency (v) is the variety of waves (or cycles) per unit the time. The frequency the a tide is reported in units of cycles per second (s-1) or hertz (Hz).\n\nThe idealized drawing of a tide in the figure listed below illustrates the meanings of amplitude and wavelength. The wavelength (l) is the smallest distance between repeating points on the wave. The amplitude of the tide is the distance between the highest (or lowest) allude on the wave and also the center of heaviness of the wave.", null, "If us measure the frequency (v) the a wave in cycles per 2nd and the wavelength (l) in meters, the product that these 2 numbers has actually the units of meters every second. The product that the frequency (v) time the wavelength (l) the a tide is as such the speed (s) at which the wave travels through space.\n\nvl = s\n\n Practice problem 2:What is the speed of a wave that has actually a wavelength of 1 meter and a frequency the 60 cycles every second? Click here to inspect your answer to Practice problem 2\n\n Practice problem 3:Orchestras in the United claims tune their tools to an \"A\" that has a frequency of 440 cycles every second, or 440 Hz. If the speed of sound is 1116 feet per second, what is the wavelength the this note? Click right here to examine your answer come Practice trouble 3 Click right here to watch a systems to Practice difficulty 3\n\nSee more: Find Out How Many Ribs Are In A Full Rack, Barbecue Time: How Many Ribs Are In A Rack\n\nLight and also Other forms of Electromagnetic Radiation\n\nLight is a wave with both electric and also magnetic components. It is therefore a kind of electromagnetic radiation.\n\nVisible light includes the small band that frequencies and also wavelengths in the part of the electro-magnetic spectrum the our eyes deserve to detect. It has radiation v wavelengths between around 400 nm (violet) and also 700 nm (red). Due to the fact that it is a wave, irradiate is bent once it enters a glass prism. When white light is focused on a prism, the irradiate rays of various wavelengths room bent by differing quantities and the irradiate is transformed into a spectrum the colors. Beginning from the side of the spectrum where the irradiate is bent by the the smallest angle, the colors space red, orange, yellow, green, blue, and violet.\n\nAs we deserve to see indigenous the following diagram, the energy carried by light increases as us go native red come blue throughout the clearly shows spectrum.", null, "Because the wavelength that electromagnetic radiation can be as lengthy as 40 m or as brief as 10-5 nm, the clearly shows spectrum is only a small portion of the total variety of electromagnetic radiation.", null, "The electromagnetic spectrum has radio and TV waves, microwaves, infrared, clearly shows light, ultraviolet, x-rays, g-rays, and cosmic rays, as presented in the number above. These different forms of radiation all travel at the speed of light (c). Castle differ, however, in your frequencies and wavelengths. The product that the frequency times the wavelength of electromagnetic radiation is constantly equal come the speed of light.\n\nvl = c\n\nAs a result, electromagnetic radiation that has actually a long wavelength has actually a low frequency, and radiation through a high frequency has actually a brief wavelength.\n\n Practice difficulty 4:Calculate the frequency of red irradiate that has a wavelength of 700.0 nm if the rate of irradiate is 2.998 x 108 m/s. .tags a { color: #fff; background: #909295; padding: 3px 10px; border-radius: 10px; font-size: 13px; line-height: 30px; white-space: nowrap; } .tags a:hover { background: #818182; } Home Contact - Advertising Copyright © 2021 bsci-ch.org #footer {font-size: 14px;background: #ffffff;padding: 10px;text-align: center;} #footer a {color: #2c2b2b;margin-right: 10px;}" ]	[ null, "https://bsci-ch.org/how-can-you-tell-one-element-from-another/imager_1_9323_700.jpg", null, "https://bsci-ch.org/how-can-you-tell-one-element-from-another/imager_2_9323_700.jpg", null, "https://bsci-ch.org/how-can-you-tell-one-element-from-another/imager_3_9323_700.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93153757,"math_prob":0.9705207,"size":5658,"snap":"2021-43-2021-49","text_gpt3_token_len":1288,"char_repetition_ratio":0.1558189,"word_repetition_ratio":0.022657055,"special_character_ratio":0.22304702,"punctuation_ratio":0.09300184,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95204115,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T06:48:31Z\",\"WARC-Record-ID\":\"<urn:uuid:b24ca430-ca50-48bc-9d92-b64e58edca06>\",\"Content-Length\":\"16219\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d8dcbdab-d407-41c2-ad80-e711d3be954f>\",\"WARC-Concurrent-To\":\"<urn:uuid:1242c6be-185e-4d4f-af95-85f1f44f0d7a>\",\"WARC-IP-Address\":\"172.67.195.230\",\"WARC-Target-URI\":\"https://bsci-ch.org/how-can-you-tell-one-element-from-another/\",\"WARC-Payload-Digest\":\"sha1:YUR6ALJYRYR4BACXMKRFGIQTT42NQJ3R\",\"WARC-Block-Digest\":\"sha1:YVCDATNGQ2A5BSWTQ6VDWDVIJHYTIW3P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363336.93_warc_CC-MAIN-20211207045002-20211207075002-00209.warc.gz\"}"}
http://www.celsiusfahrenheit.co/6800	[ "# 🌡6800 C to F\n\n🔆🌡6800 C to F. How many degrees Fahrenheit in a degree Celsius. °C to F Calculator.\n\n### Celsius to Fahrenheit\n\n Celsius Fahrenheit You can edit any of the fields below: = Detailed result here", null, "Change to Fahrenheit to Celsius\n\n## How to convert from Celsius to Fahrenheit\n\nIt is ease to convert a temperature value from Celsius to Fahrenheit by using the formula below:\n\n [°F] = [°C] × 9⁄5 + 32\nor\n Value in Fahrenheit = Value in Celsius × 9⁄5 + 32\n\nTo change 6800° Celsius to Fahrenheit, just need to replace the value [°C] in the formula below and then do the math.\n\n### Step-by-step Solution:\n\n1. Write down the formula: [°F] = [°C] × 9⁄5 + 32\n2. Plug the value in the formula: 6800 × 9⁄5 + 32\n3. Multiply by 9: 61200⁄5 + 32\n4. Divide by 5: 12240 + 32\n\n### Values around 6800 Celsius(s)\n\nCelsiusFahrenheitCelsiusFahrenheit\n6799.13759.56799.23759.6\n6799.33759.66799.43759.7\n6799.53759.76799.63759.8\n6799.73759.86799.83759.9\n6799.93759.968003760.0\n6800.13760.16800.23760.1\n6800.33760.26800.43760.2\n6800.53760.36800.63760.3\n6800.73760.46800.83760.4\n6800.93760.568013760.6" ]	[ null, "http://www.celsiusfahrenheit.co/images/calculator16x20.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51482666,"math_prob":0.93571496,"size":1061,"snap":"2020-34-2020-40","text_gpt3_token_len":429,"char_repetition_ratio":0.21381268,"word_repetition_ratio":0.011363637,"special_character_ratio":0.533459,"punctuation_ratio":0.19305019,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9943871,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-20T06:44:56Z\",\"WARC-Record-ID\":\"<urn:uuid:b5ace8dc-1d7c-4f1a-b7ca-d6ea5f1236cf>\",\"Content-Length\":\"25129\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f6ce88f4-0054-45ab-93c4-073a94d9f4a2>\",\"WARC-Concurrent-To\":\"<urn:uuid:29a3c2ac-d8f1-41a8-9c06-f22fca2d5e87>\",\"WARC-IP-Address\":\"67.205.30.187\",\"WARC-Target-URI\":\"http://www.celsiusfahrenheit.co/6800\",\"WARC-Payload-Digest\":\"sha1:MTSEJLFAZ4KZOLZCUC3ON4RIIDJH5CN7\",\"WARC-Block-Digest\":\"sha1:FG5UZNKJ766ZW43LEGMC3AVJS342OQEE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400196999.30_warc_CC-MAIN-20200920062737-20200920092737-00003.warc.gz\"}"}
https://socratic.org/questions/how-do-you-solve-8-7-3-5m-2-5-5-4-6m	[ "# How do you solve 8.7=3.5m-2.5(5.4-6m)?\n\nFeb 25, 2017\n\nSee the entire solution process below:\n\n#### Explanation:\n\nStep 1) Expand the terms in parenthesis on the right side of the equation by multiplying each term within the equation by the term outside the equation:\n\n$8.7 = 3.5 m - \\left(2.5 \\times 5.4\\right) + \\left(2.5 \\times 6 m\\right)$\n\n$8.7 = 3.5 m - 13.5 + 15 m$\n\nStep 2) Group and combine like terms on the right side of the equation:\n\n$8.7 = 3.5 m + 15 m - 13.5$\n\n$8.7 = \\left(3.5 + 15\\right) m - 13.5$\n\n$8.7 = 18.5 m - 13.5$\n\nStep 3) Add $\\textcolor{red}{13.5}$ to each side of the equation to isolate the $m$ term while keeping the equation balanced:\n\n$8.7 + \\textcolor{red}{13.5} = 18.5 m - 13.5 + \\textcolor{red}{13.5}$\n\n$22.2 = 18.5 m - 0$\n\n$22.2 = 18.5 m$\n\nStep 4) Divide each side of the equation by $\\textcolor{red}{18.5}$ to solve for $m$ while keeping the equation balanced:\n\n$\\frac{22.2}{\\textcolor{red}{18.5}} = \\frac{18.5 m}{\\textcolor{red}{18.5}}$\n\n$1.2 = \\frac{\\textcolor{red}{\\cancel{\\textcolor{b l a c k}{18.5}}} m}{\\cancel{\\textcolor{red}{18.5}}}$\n\n$1.2 = m$\n\n$m = 1.2$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80063003,"math_prob":1.0000064,"size":431,"snap":"2023-40-2023-50","text_gpt3_token_len":111,"char_repetition_ratio":0.19437939,"word_repetition_ratio":0.08571429,"special_character_ratio":0.24825986,"punctuation_ratio":0.0989011,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999034,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-25T13:05:42Z\",\"WARC-Record-ID\":\"<urn:uuid:2bbd3229-3e44-41d1-a11a-6f46dfde9ada>\",\"Content-Length\":\"34545\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:24cef8a5-28e8-4472-9309-399e7463fe9e>\",\"WARC-Concurrent-To\":\"<urn:uuid:8410c57e-f9ad-4316-9763-b2f9dc2b7608>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-solve-8-7-3-5m-2-5-5-4-6m\",\"WARC-Payload-Digest\":\"sha1:42RAGKRBDC6LVVUALCXABGXXNALMGQBO\",\"WARC-Block-Digest\":\"sha1:F34N6VERJVDVCZ6CR3NQC3EHW46D6PSR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233508977.50_warc_CC-MAIN-20230925115505-20230925145505-00213.warc.gz\"}"}
https://www.reference.com/web?q=surface+area+examples&qo=relatedSearchBing&o=600605&l=dir	[ "https://www.shmoop.com/basic-geometry/surface-area-examples.html\n\nPre-Algebra giving you a hard time? Shmoop's free Basic Geometry Guide has all the explanations, examples, and exercises you've been craving.\n\nhttps://www.onlinemathlearning.com/surface-area-formula.html\n\nSurface area formula for solid cylinder, hollow cylinder, prism, cone, pyramid, sphere, hemisphere, cube, cuboid, rectangular prism and triangular prism, ...\n\nhttps://study.com/academy/lesson/what-is-surface-area-definition-formulas-quiz.html\n\n... of the 3-D shape. Here is a list of basic shape formulas to help with finding the surface area of the 3-D shapes: ... Examples of Surface Areas: Triangular Prism ...\n\nhttp://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx\n\nMay 30, 2018 ... In this section we'll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a ... Now let's work a couple of examples.\n\nhttp://www.math.com/tables/geometry/surfareas.htm\n\nIn general, the surface area is the sum of all the areas of all the shapes that cover ... Be careful!! Units count. Use the same units for all measurements. Examples ...\n\nhttps://www.aaamath.com/geo79_x9.htm\n\nAn interactive math lesson to teach the surface area of a rectangular prism. ... Add the three areas together to find the surface area; Example: The surface area of ...\n\nhttps://www.khanacademy.org/science/ap-biology/cell-structure-and-function/cell-size/v/surface-area-of-a-box\n\nSurface area is the sum of the areas of all faces (or surfaces) on a 3D shape. A cuboid has 6 rectangular faces. To find the surface area of a cuboid, add the ...\n\nhttps://www.youtube.com/watch?v=sskf3tF2heU\n\nNov 6, 2007 ... For a complete lesson on surface area, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside ...\n\nhttp://www.mathguide.com/lessons/SurfaceArea.html\n\nSurface Area: Learn how to calculate the surface area of common solids. ... Example 1: Given l = 4 yds, w = 2 yds, and h = 5 yds, the surface area would be.\n\nhttps://www.khanacademy.org/math/basic-geo/basic-geo-volume-sa/basic-geometry-surface-area/v/surface-area-word-problem-example\n\nSal solves a word problem that involves surface area of a square pyramid." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83185136,"math_prob":0.7792379,"size":2219,"snap":"2019-13-2019-22","text_gpt3_token_len":554,"char_repetition_ratio":0.17471783,"word_repetition_ratio":0.017793594,"special_character_ratio":0.2410996,"punctuation_ratio":0.25301206,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9948711,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-22T07:37:26Z\",\"WARC-Record-ID\":\"<urn:uuid:7981f5ac-37b6-401d-921a-9690efb8866d>\",\"Content-Length\":\"73520\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0ee10f33-3161-4578-91a0-ad7fd2108e9a>\",\"WARC-Concurrent-To\":\"<urn:uuid:3121a5c2-1b28-4787-896c-649051b81e7d>\",\"WARC-IP-Address\":\"151.101.250.114\",\"WARC-Target-URI\":\"https://www.reference.com/web?q=surface+area+examples&qo=relatedSearchBing&o=600605&l=dir\",\"WARC-Payload-Digest\":\"sha1:PPWNURCUFBGIJSYNIUEICQAFMZMVNASS\",\"WARC-Block-Digest\":\"sha1:5PNSWVA5CGCXALLGVNJ266BMMACPK536\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232256764.75_warc_CC-MAIN-20190522063112-20190522085112-00447.warc.gz\"}"}