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https://kr.mathworks.com/matlabcentral/answers/1460839-how-to-reduce-space-between-axis-labels-and-axis-ticks?s_tid=prof_contriblnk	[ "# How to reduce space between axis labels and axis ticks?\n\n조회 수: 100(최근 30일)\nAmmy 2021년 9월 26일\n댓글: Walter Roberson 2021년 9월 26일\nx=1:54;\nplot(x,y1,'k-+',x,y2,'b-')\nxlim([1 54]\nxticklabel_rotate([1:54],90,D\nxlabel('Sample 1')\nylabel('Sample 2')\nHow to reduce space between axis label and axis ticks, i,e., how to get axis label closer to axis.\n##### 댓글 수: 1표시숨기기 없음\nWalter Roberson 2021년 9월 26일\n\n댓글을 달려면 로그인하십시오.\n\n### 답변(1개)\n\nKevin Holly 2021년 9월 26일\nx=1:54;\ny1=cos(x);\ny2=sin(x);\nplot(x,y1,'k-+',x,y2,'b-')\nxlim([1 54])\nxlabel('Sample 1')\nylabel('Sample 2')\nGet the axes handle by using the command below\naxes_handle = gca; %gca stands for get current axes\nNow I am looking up the current position of the xlabel.\naxes_handle.XLabel.Position\nans = 1×3\n27.5000 -1.1409 -1.0000\nThe position in the x, y, and z-direction are 0.5, -0.0705, and 0, respectively.\nNow, change the second value in the array. This will change the vertical position.\naxes_handle.XLabel.Position = [27.5 -1.10 -1];\naxes_handle.YLabel.Position\nans = 1×3\n-3.1054 0.0000 -1.0000\nNow, let's change the first value in the position array to move the ylabel horizontally.\naxes_handle.YLabel.Position = [-2.6 0 -1];", null, "##### 댓글 수: 4표시숨기기 이전 댓글 수: 3\nWalter Roberson 2021년 9월 26일\nR2013a was mentioned in the comments in the question that this one is a duplicate of...\n\n댓글을 달려면 로그인하십시오.\n\n### Community Treasure Hunt\n\nFind the treasures in MATLAB Central and discover how the community can help you!\n\nStart Hunting!" ]	[ null, "https://www.mathworks.com/matlabcentral/answers/uploaded_files/749949/image.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6318821,"math_prob":0.93307906,"size":1274,"snap":"2021-43-2021-49","text_gpt3_token_len":432,"char_repetition_ratio":0.13070866,"word_repetition_ratio":0.020618556,"special_character_ratio":0.32731554,"punctuation_ratio":0.2012987,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9711637,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T12:23:01Z\",\"WARC-Record-ID\":\"<urn:uuid:36669a47-94a1-4c0b-b10b-011e882a318c>\",\"Content-Length\":\"138692\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:589aa2a8-2554-4a04-9715-1bef640a7305>\",\"WARC-Concurrent-To\":\"<urn:uuid:e6f3001e-1617-470a-ad1f-05c70fde1e6a>\",\"WARC-IP-Address\":\"184.25.188.167\",\"WARC-Target-URI\":\"https://kr.mathworks.com/matlabcentral/answers/1460839-how-to-reduce-space-between-axis-labels-and-axis-ticks?s_tid=prof_contriblnk\",\"WARC-Payload-Digest\":\"sha1:UEV6J2IQWFPGFHWMMOWYZEX3Z6FKZQK2\",\"WARC-Block-Digest\":\"sha1:E54HHJ7LDB72VTKN427VZYV74NUN443M\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363376.49_warc_CC-MAIN-20211207105847-20211207135847-00067.warc.gz\"}"}
https://www.tensorflow.org/probability/api_docs/python/tfp/substrates/numpy/math/log_cumsum_exp	[ "# tfp.substrates.numpy.math.log_cumsum_exp\n\nComputes log(cumsum(exp(x))).\n\nThis is a pure-TF implementation of `tf.math.cumulative_logsumexp`; unlike the built-in op, it supports XLA compilation. It uses a similar algorithmic technique (parallel prefix sum) as the built-in op, so it has similar numerics and asymptotic performace. However, this implemenentation currently has higher overhead, so it is significantly slower on smaller inputs (`n < 10000`).\n\n`x` the `Tensor` to sum over.\n`axis` int `Tensor` axis to sum over.\n`name` Python `str` name prefixed to Ops created by this function. Default value: `None` (i.e., `'cumulative_logsumexp'`).\n\n`cumulative_logsumexp` `Tensor` of the same shape as `x`." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.791401,"math_prob":0.7902684,"size":557,"snap":"2020-34-2020-40","text_gpt3_token_len":139,"char_repetition_ratio":0.07775769,"word_repetition_ratio":0.0,"special_character_ratio":0.21543986,"punctuation_ratio":0.1682243,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98751825,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-22T18:54:56Z\",\"WARC-Record-ID\":\"<urn:uuid:c067ad4d-9fbe-4408-9d6d-6f1863753ad4>\",\"Content-Length\":\"593528\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a2a69d88-9089-491d-9b31-2ae7bca6fa95>\",\"WARC-Concurrent-To\":\"<urn:uuid:c3fda78e-a2f5-45e8-a104-2fb314a1e548>\",\"WARC-IP-Address\":\"172.217.15.78\",\"WARC-Target-URI\":\"https://www.tensorflow.org/probability/api_docs/python/tfp/substrates/numpy/math/log_cumsum_exp\",\"WARC-Payload-Digest\":\"sha1:OCTCHEB7NI2VATF5JL555ZELNHO5LFCW\",\"WARC-Block-Digest\":\"sha1:C62HMO6OFKXC3UO6EGFUF2QSELXFZBJQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400206329.28_warc_CC-MAIN-20200922161302-20200922191302-00749.warc.gz\"}"}
https://www.santamonicainhometutors.com/blog/integral-of-arctan-tan-inverse-x	[ "", null, "We Tutor All Subjects & Grade Levels - In Home And Online\nMay 19, 2023\n\n# Integral of Arctan (Tan Inverse x)\n\nArctan is one of the six trigonometric operations and performs a vital role in several mathematical and scientific domains. Its inverse, the arctangent function, is used to determine the angle in a right-angled triangle while given the ratio of the adjacent and opposite sides.\n\nCalculus is a division of mathematics which deals with the study of rates of accumulation and change. The integral of arctan is a key concept in calculus and is used to solve a broad range of problems. It is applied to determine the antiderivative of the arctan function and measure definite integrals that include the arctan function. Additionally, it is utilized to calculate the derivatives of functions which consist of the arctan function, for example the inverse hyperbolic tangent function.\n\nFurthermore to calculus, the arctan function is applied to model a wide array of physical phenomena, involving the movement of objects in circular orbits and the behavior of electrical circuits. The integral of arctan is used to determine the potential energy of objects in round orbits and to analyze the mechanism of electrical circuits which involve inductors and capacitors.\n\nIn this blog, we will examine the integral of arctan and its various uses. We will examine its properties, consisting of its formula and how to determine its integral. We will also look at instances of how the integral of arctan is applied in physics and calculus.\n\nIt is important to understand the integral of arctan and its characteristics for learners and working professionals in fields such as physics, engineering, and mathematics. By grasping this rudimental theory, individuals can apply it to solve challenges and gain detailed understanding into the intricate workings of the world around us.\n\n## Significance of the Integral of Arctan\n\nThe integral of arctan is a crucial math theory that has several uses in physics and calculus. It is utilized to calculate the area under the curve of the arctan function, that is a persistent function that is largely used in math and physics.\n\nIn calculus, the integral of arctan is applied to determine a broad range of problems, consisting of finding the antiderivative of the arctan function and assessing definite integrals which involve the arctan function. It is also applied to determine the derivatives of functions which consist of the arctan function, such as the inverse hyperbolic tangent function.\n\nIn physics, the arctan function is used to model a broad range of physical phenomena, including the motion of things in circular orbits and the behavior of electrical circuits. The integral of arctan is applied to work out the potential energy of things in circular orbits and to analyze the working of electrical circuits that include capacitors and inductors.\n\n## Characteristics of the Integral of Arctan\n\nThe integral of arctan has many characteristics which make it a useful tool in physics and calculus. Few of these characteristics include:\n\nThe integral of arctan x is equal to x times the arctan of x minus the natural logarithm of the absolute value of the square root of one plus x squared, plus a constant of integration.\n\nThe integral of arctan x can be stated as the terms of the natural logarithm function using the substitution u = 1 + x^2.\n\nThe integral of arctan x is an odd function, this implies that the integral of arctan negative x is equivalent to the negative of the integral of arctan x.\n\nThe integral of arctan x is a continuous function that is defined for all real values of x.\n\n## Examples of the Integral of Arctan\n\nHere are handful instances of integral of arctan:\n\nExample 1\n\nLet’s assume we want to determine the integral of arctan x with respect to x. Applying the formula stated above, we obtain:\n\n∫ arctan x dx = x * arctan x - ln \|√(1 + x^2)\| + C\n\nwhere C is the constant of integration.\n\nExample 2\n\nLet's say we want to determine the area under the curve of the arctan function within x = 0 and x = 1. Using the integral of arctan, we achieve:\n\n∫ from 0 to 1 arctan x dx = [x * arctan x - ln \|√(1 + x^2)\|] from 0 to 1\n\n= (1 * arctan 1 - ln \|√(2)\|) - (0 * arctan 0 - ln \|1\|)\n\n= π/4 - ln √2\n\nThus, the area under the curve of the arctan function within x = 0 and x = 1 is equal to π/4 - ln √2.\n\n## Conclusion\n\nDinally, the integral of arctan, further known as the integral of tan inverse x, is an essential mathematical concept that has several applications in physics and calculus. It is applied to determine the area under the curve of the arctan function, which is a continuous function which is broadly used in multiple fields. Understanding the properties of the integral of arctan and how to apply it to work out problems is crucial for learners and working professionals in fields for instance, engineering, physics, and mathematics.\n\nThe integral of arctan is one of the fundamental concepts of calculus, which is a vital division of math used to understand change and accumulation. It is utilized to figure out many problems for example, finding the antiderivative of the arctan function and assessing definite integrals consisting of the arctan function. In physics, the arctan function is applied to model a broad array of physical phenomena, involving the motion of objects in round orbits and the behavior of electrical circuits.\n\nThe integral of arctan has many properties which make it a beneficial tool in physics and calculus. It is an odd function, that means that the integral of arctan negative x is equivalent to the negative of the integral of arctan x. The integral of arctan is further a continuous function that is specified for all real values of x.\n\nIf you require assistance understanding the integral of arctan or any other mathematical concept, Grade Potential Tutoring provides personalized tutoring services. Our expert instructors are available remotely or face-to-face to give one-on-one assistance which will assist you attain your academic objectives. Don't hesitate to call us at Grade Potential Tutoring to schedule a lesson and take your math skills to the next stage.", null, "", null, "", null, "" ]	[ null, "https://www.santamonicainhometutors.com/corporate/uploads/phone tagline.gif", null, "https://www.santamonicainhometutors.com/corporate/uploads/264c200b-arrow3-left_105m07v000000000000028.png", null, "https://www.santamonicainhometutors.com/corporate/uploads/3ee13aa7-arrow3-right_105m07v000000000000028.png", null, "https://www.santamonicainhometutors.com/corporate/uploads/underline.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9133889,"math_prob":0.9933469,"size":6055,"snap":"2023-40-2023-50","text_gpt3_token_len":1327,"char_repetition_ratio":0.21533631,"word_repetition_ratio":0.23158915,"special_character_ratio":0.19917424,"punctuation_ratio":0.07181818,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982493,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-11T08:34:32Z\",\"WARC-Record-ID\":\"<urn:uuid:093a2636-1cb8-4378-8976-7c7315aee2a5>\",\"Content-Length\":\"95140\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0b544689-fa29-4ef9-a976-ba064b031851>\",\"WARC-Concurrent-To\":\"<urn:uuid:6db6c002-08f2-4162-825d-4f2e4faf84d4>\",\"WARC-IP-Address\":\"69.20.38.55\",\"WARC-Target-URI\":\"https://www.santamonicainhometutors.com/blog/integral-of-arctan-tan-inverse-x\",\"WARC-Payload-Digest\":\"sha1:O764NQYOB5X4P5NKTPKR3M6GRIQW4CGK\",\"WARC-Block-Digest\":\"sha1:XLSYPIWO4QZPXTL3PIDJRE3IAJSTDPZU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679103810.88_warc_CC-MAIN-20231211080606-20231211110606-00289.warc.gz\"}"}
https://hackage.haskell.org/package/aivika-lattice-0.4/candidate/docs/src/Simulation-Aivika-Lattice-QueueStrategy.html	[ "```\n{-# LANGUAGE TypeFamilies, MultiParamTypeClasses #-}\n\n-- \|\n-- Module : Simulation.Aivika.Lattice.QueueStrategy\n-- Maintainer : David Sorokin <david.sorokin@gmail.com>\n-- Stability : experimental\n-- Tested with: GHC 7.10.3\n--\n-- This module defines queue strategies 'FCFS' and 'LCFS' for the 'LIO' computation.\n--\nmodule Simulation.Aivika.Lattice.QueueStrategy () where\n\nimport Simulation.Aivika.Trans\n\nimport Simulation.Aivika.Lattice.Internal.LIO\nimport Simulation.Aivika.Lattice.Ref.Base\n\n-- \| An implementation of the 'FCFS' queue strategy.\ninstance QueueStrategy LIO FCFS where\n\n-- \| A queue used by the 'FCFS' strategy.\nnewtype StrategyQueue LIO FCFS a = FCFSQueueLIO (LL.DoubleLinkedList LIO a)\n\nnewStrategyQueue s = fmap FCFSQueueLIO LL.newList\n\nstrategyQueueNull (FCFSQueueLIO q) = LL.listNull q\n\n-- \| An implementation of the 'FCFS' queue strategy.\ninstance DequeueStrategy LIO FCFS where\n\nstrategyDequeue (FCFSQueueLIO q) =\ndo i <- LL.listFirst q\nLL.listRemoveFirst q\nreturn i\n\n-- \| An implementation of the 'FCFS' queue strategy.\ninstance EnqueueStrategy LIO FCFS where\n\nstrategyEnqueue (FCFSQueueLIO q) i = LL.listAddLast q i\n\n-- \| An implementation of the 'LCFS' queue strategy.\ninstance QueueStrategy LIO LCFS where\n\n-- \| A queue used by the 'LCFS' strategy.\nnewtype StrategyQueue LIO LCFS a = LCFSQueueLIO (LL.DoubleLinkedList LIO a)\n\nnewStrategyQueue s = fmap LCFSQueueLIO LL.newList\n\nstrategyQueueNull (LCFSQueueLIO q) = LL.listNull q\n\n-- \| An implementation of the 'LCFS' queue strategy.\ninstance DequeueStrategy LIO LCFS where\n\nstrategyDequeue (LCFSQueueLIO q) =\ndo i <- LL.listFirst q\nLL.listRemoveFirst q\nreturn i\n\n-- \| An implementation of the 'LCFS' queue strategy.\ninstance EnqueueStrategy LIO LCFS where\n\nstrategyEnqueue (LCFSQueueLIO q) i = LL.listInsertFirst q i\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5237578,"math_prob":0.492602,"size":1147,"snap":"2021-31-2021-39","text_gpt3_token_len":322,"char_repetition_ratio":0.18810149,"word_repetition_ratio":0.24806201,"special_character_ratio":0.16913688,"punctuation_ratio":0.16666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9793705,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-18T18:04:45Z\",\"WARC-Record-ID\":\"<urn:uuid:e98d7bc0-6830-46ff-89cc-d4a0ce731319>\",\"Content-Length\":\"12225\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2bdd6a02-472e-47e8-8227-96ae75723698>\",\"WARC-Concurrent-To\":\"<urn:uuid:6f9ec0b9-8735-42a0-b87e-a716919cc4d8>\",\"WARC-IP-Address\":\"199.232.128.68\",\"WARC-Target-URI\":\"https://hackage.haskell.org/package/aivika-lattice-0.4/candidate/docs/src/Simulation-Aivika-Lattice-QueueStrategy.html\",\"WARC-Payload-Digest\":\"sha1:U5BWWIXYLUAAMOVWPEIHPWXEH3QHWKAS\",\"WARC-Block-Digest\":\"sha1:UOYBOPRDCPYVXY4X7XF7DRDB22D7EYMH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056548.77_warc_CC-MAIN-20210918154248-20210918184248-00263.warc.gz\"}"}
https://www.careeralert.in/what-is-the-formula-to-calculate-the-tariff-having-250kva-transformer-for-1-month/	[ "## What is the formula to calculate the tariff having 250kva transformer for 1 month?\n\nQuestion: What is the formula to calculate the tariff having 250kva transformer for 1 month?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.934463,"math_prob":0.97941124,"size":1189,"snap":"2023-40-2023-50","text_gpt3_token_len":234,"char_repetition_ratio":0.15274261,"word_repetition_ratio":0.055555556,"special_character_ratio":0.19680403,"punctuation_ratio":0.08144797,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9596012,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-26T06:53:23Z\",\"WARC-Record-ID\":\"<urn:uuid:30b39b7c-55d3-4f78-b08a-264f01b53319>\",\"Content-Length\":\"81526\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4cb22adf-eec7-4183-9de2-8b4278d6707e>\",\"WARC-Concurrent-To\":\"<urn:uuid:974ab9eb-beaf-4653-a652-6947a03cfafd>\",\"WARC-IP-Address\":\"23.106.53.56\",\"WARC-Target-URI\":\"https://www.careeralert.in/what-is-the-formula-to-calculate-the-tariff-having-250kva-transformer-for-1-month/\",\"WARC-Payload-Digest\":\"sha1:33VTW2DSJFYLBJMTVUWX3T56IE3R52MU\",\"WARC-Block-Digest\":\"sha1:FO43BWENLTWBFGH6ZKFSWJFKL24GOWLO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510149.21_warc_CC-MAIN-20230926043538-20230926073538-00343.warc.gz\"}"}
https://sob5050.com/pnpfbmg/article.php?id=619bcf-what-is-domain-and-range	[ "# what is domain and range\n\nPractice. Textbook Notes. The range is the numbers on the right side or the y valued numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. The range, and the most typical, there's actually a couple of definitions for range, but the most typical definition for range is \"the set of all possible outputs.\" For any real value of x, there corresponds a unique real value of f(x). In this non-linear system, users are free to take whatever path through the material best serves their needs. Save. Edit. What is Codomain of a Function? Your dashboard and recommendations. Betterment LLC is an SEC Registered Investment Advisor. Then A is called domain of f and B is called the co-domain of the function f . Delete Quiz. Investing in securities involves risks, and there is always the potential of losing money when you invest in securities. âââ Correct answer to the question: What is the domain and range ? Mathematical function means the association between two groups of variables. Given a function in function notation form, identify the domain and range using set notation, interval notation, or a verbal description as appropriate. Domain vs Range. Domain is simply the permissions that are applied to the x-axis of a graph, meaning the Domain indicates where the graph can lie on the x-axis and where it cannot. Solution for What is the domain and range of the function {(-3, 11), (â2, 2), (0, 2), (2, 6), (4, 18)}? 3.7 million tough questions answered. To play this quiz, please finish editing it. Write the range with proper notation. Like the domain, the range is written with the same notation. Functions. 8.1 Function Notation, Domain and Range NOTES What is a function? 9th - 10th grade . Domain vs Range. Range of a function â this is the set of output values generated by the function (based on the input values from the domain set). Please read \"What is a Function?\" Solo Practice. Play. In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. A mathematical function is a relationship between two sets of variables. The function represents a parabolic shape as it is a quadratic equation. ; The codomain is similar to a range, with one big difference: A codomain can contain every possible output, not just those that actually appear. A linear function extends from -oo to +oo, so that all values of x is allowed and the value of f(x) also includes the set of all real numbers. Played 119 times. 0. Laura S. Harris (2020, November 27.) Click hereðto get an answer to your question ï¸ Find the domain and range of the real function f(x) = â(25 - x^2) . 48% average accuracy. The definition of Domain and range are as what is stated above, but I'll rephrase these definitions in my own words. In the following tutorial I continue by looking at square root and reciprocal functions. In the previous chapter, there is only one example of a function. The domain is {-6,-3,-2,2} and the range is {-7,-4,-1,5} The domain is the numbers on the left side or the x valued numbers. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Domain and Range. But in fact they are very important in defining a function. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. Write your questions What is the domain and range of : $$f(x) = \\frac{2e^x}{1+e^x}$$ and the domain and range of : $$f^{-1}(x)$$ I've already found $$f^{-1}(x) = \\ln\\left(\\frac {x}{2-x}\\right)$$ I'm looking for the process necessary to find its domain and range. How to Use the Domain and Range Calculator? I am often asked what is the domain and range of a function. Personalized courses, with or without credits. Finish Editing. Let f : A â B be a function . May Lose Value. Study Guides. One is independent called domain and other is dependent called range. What is domain and range examples? Its domain is a set of few numbers, but there are many more possibilities. What Is The Domain And Range â¢ What is domain and range examples? The letter U indicates a union that connects parts of a domain that may be separated by a gap. Input . Domain and Range of a Parabola. - edu-answer.com Ace your next exam with ease . Edit. Share practice link. Print; Share; Edit; Delete; Host a game. They may also have been called the input and output of the function.) It is the set X in the notation f: X â Y, and is alternatively denoted as â¡ (). Learn domain and range with free interactive flashcards. Domain and Range DRAFT. No Bank Guarantee. 0. by mgranata_21577. Get the detailed answer: What is the domain and range of ? Investments are not FDIC Insured. In other words, for two dimensional Cartesian coordinate system or XY system, the variable along x-axis is called as Domain and along y-axis is called as Range. Switch to. Choose from 500 different sets of domain and range flashcards on Quizlet. Domain and Range The domain of a function f ( x ) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes. Domain = Set A. Codomain = Set B, and. The range is the subset of the codomain. Domain: {IR} Range: {IR} We could also use interval notation to assign our domain and range: Domain (-infinity, infinity) Range (-infinity, infinity) This is a function. Read on! Class Notes. Domain and range are terms that are applicable to mathematics, especially in relation to the physical sciences consisting of functions. Domain, Range and Codomain. 7 months ago. The set of all the image points f(x), for xâ¬ A , is called the range of f that is range of f = { f(x) : x â¬ A} . The domain is the set of x-values that can be put into a function.In other words, itâs the set of all possible values of the independent variable. Home. In the example above, the range â¦ The range can be figured out by looking at the curve. We can often write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. - domain and range (this chapter) - zero of a function - points of intersection with the axes - monotonicity (monotonic functions, not monotonic functions) - maximal intervals of monotonicity - positive and negative values - minimum and maximum. Brokerage services are offered by Betterment Securities, an SEC registered broker-dealer and member FINRA/SIPC. This quiz is incomplete! Kindly see the graph of f(x)=3x+1 graph{y=3x+1[-20,20,-10,10]} God bless....I hope the explanation is useful. Booster Classes. Input and Output. Since a function is defined on its entire domain, its domain coincides with its domain of definition. Domain of a function â this is the set of input values for the function. Homework Help. ; The range is the set of y-values that are output for the domain. Domain: -oo<\"x\"<+oo Range: -oo<\"f(x)\"<+oo This is a linear function. Homework. A sine curve represent a wave the repeats at a regular frequency. BYJUâS online domain and range calculator tool makes the calculation faster, and it displays the output in a fraction of seconds. The difference between domain and range are somewhat obvious, but the difference between a codomain and range are subtle. Domain and range are prime factors that decide the applicability of mathematical functions. The domain includes the values that go into a function (the x-values) and the range are the values that come out (the or y-values). Many thanks. View A1 8.1 Function Notation Domain and Range Packet.pdf from MATH 1450 at William Paterson University. All inputs that yield an output are elements of the domain of the function; Inputs are conventionally plotted along the horizontal axis of a graph; Input values are known when collecting data; Output. So in the following tutorials I introduce you to this concept by looking at linear, quadratic, root and reciprocal functions. Even though the two functions perform the same operation, their different domain and ranges give them different behaviour. These unique features make Virtual Nerd a viable alternative to private tutoring. In the example above, the domain of $$f\\left( x \\right)$$ is set A. Mathematics. The square root function to the right does not have a domain or range of all real numbers. (In grammar school, you probably called the domain the replacement set and the range the solution set. Domain and Range Calculator is a free online tool that displays the range and domain for the given function. There is some terminology associated with functions that we should be familiar with as well. By determining the domain and range (or codomain) of a function before you begin to graph it, you will usually develop a mental image of the function that can help you avoid making errors, or wasting space, on your graph paper when graphing the function. Domain: {-3, -2, 0, 2, 4}: Range: {11, 6, 2, 18} Domain:â¦ The domain and range are all real numbers because, at some point, the x and y values will be every real number. Inputs and Outputs: Domain and Range. These can be slightly more demanding. The range would then be {0, 1, 4, 9, â¦}. For the function {(0,1), (1,-3), (2,-4), (-4,1)}, write the domain and range. We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. Mathematically, consider a simple relation as {(2, 3), (1, 3), (4, â¦ Use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. Live Game Live. Understand the content in your textbooks. Play this game to review Algebra I. In its simplest form the domain is all the values that go into a function, and the range is all the values that come out. 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Between domain and range Packet.pdf from MATH 1450 at William Paterson University set a of y-values that are to!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91058797,"math_prob":0.96532494,"size":19785,"snap":"2021-21-2021-25","text_gpt3_token_len":4368,"char_repetition_ratio":0.20059653,"word_repetition_ratio":0.23979148,"special_character_ratio":0.22886024,"punctuation_ratio":0.14753695,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98648334,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-09T13:01:10Z\",\"WARC-Record-ID\":\"<urn:uuid:3a45d575-a966-4368-81c3-ebe36fc70f9d>\",\"Content-Length\":\"53965\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:25f81ff0-266f-465c-ab02-53bc6b5b99ac>\",\"WARC-Concurrent-To\":\"<urn:uuid:3be623ee-4c07-457d-9b2e-3b1e30b958dd>\",\"WARC-IP-Address\":\"207.126.167.210\",\"WARC-Target-URI\":\"https://sob5050.com/pnpfbmg/article.php?id=619bcf-what-is-domain-and-range\",\"WARC-Payload-Digest\":\"sha1:USG3OGUGZHEENW2QS2BIYRAW336OC2R5\",\"WARC-Block-Digest\":\"sha1:I5ZVFSARJKV3XMVUL4SDKGXBTME24HNM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988986.98_warc_CC-MAIN-20210509122756-20210509152756-00146.warc.gz\"}"}
https://flutterq.com/solved-laravel-eloquent-where-date-is-equal-or-greater-than-datetime/	[ "# [Solved] Laravel eloquent where date is equal or greater than DateTime\n\nHello Guys, How are you all? Hope You all Are Fine. Today I get the following error Laravel eloquent where date is equal or greater than DateTime in php. So Here I am Explain to you all the possible solutions here.\n\n## How Laravel eloquent where date is equal or greater than DateTime Error Occurs?\n\nToday I get the following error Laravel eloquent where date is equal or greater than DateTime in php.\n\n## How To Solve Laravel eloquent where date is equal or greater than DateTime Error ?\n\n1. How To Solve Laravel eloquent where date is equal or greater than DateTime Error ?\n\nTo Solve Laravel eloquent where date is equal or greater than DateTime Error Here you can use this:\n`->where('date', '>=', date('Y-m-d'))`\n\n2. To Solve Laravel eloquent where date is equal or greater than DateTime Error\n\nTo Solve Laravel eloquent where date is equal or greater than DateTime Error Here you can use this:\n`->where('date', '>=', date('Y-m-d'))`\n\n## Solution 1\n\nSounds like you need to use `>=`, for example:\n\n`->whereDate('date', '>=', Carbon::now('Europe/Stockholm'))`\n\n## Solution 2\n\nHere you can use this:\n\n`->where('date', '>=', date('Y-m-d'))`\n\n## Summery\n\nIt’s all About this issue. Hope all solution helped you a lot. Comment below Your thoughts and your queries. Also, Comment below which solution worked for you? Thank You." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7419278,"math_prob":0.9335015,"size":1385,"snap":"2021-31-2021-39","text_gpt3_token_len":319,"char_repetition_ratio":0.14989138,"word_repetition_ratio":0.46696034,"special_character_ratio":0.24115524,"punctuation_ratio":0.11785714,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98826617,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-23T00:32:54Z\",\"WARC-Record-ID\":\"<urn:uuid:dd6be180-9b3e-4956-92eb-fa1f7985e5e9>\",\"Content-Length\":\"61902\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:61acf731-0f3a-4d8f-8c9d-e55a6dcdd7a8>\",\"WARC-Concurrent-To\":\"<urn:uuid:3b6f27c4-85d2-4a72-967c-38cd419a4945>\",\"WARC-IP-Address\":\"172.67.144.206\",\"WARC-Target-URI\":\"https://flutterq.com/solved-laravel-eloquent-where-date-is-equal-or-greater-than-datetime/\",\"WARC-Payload-Digest\":\"sha1:MSKJYNSPE5CQI4RT4S5B5XOPTWLABYBG\",\"WARC-Block-Digest\":\"sha1:LVMKKO7H3XWTSV4JTRSLVDRXEPOSDE3Y\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057403.84_warc_CC-MAIN-20210922223752-20210923013752-00089.warc.gz\"}"}
https://www.convertunits.com/from/dyne-second/square+centimeter/to/pound-force+second/square+inch	[ "## ››Convert dyne-second/square centimeter to pound-force second/square inch\n\n dyne-second/square centimeter pound-force second/square inch\n\nHow many dyne-second/square centimeter in 1 pound-force second/square inch? The answer is 68947.5729.\nWe assume you are converting between dyne-second/square centimeter and pound-force second/square inch.\nYou can view more details on each measurement unit:\ndyne-second/square centimeter or pound-force second/square inch\nThe SI derived unit for dynamic viscosity is the pascal second.\n1 pascal second is equal to 10 dyne-second/square centimeter, or 0.00014503773779686 pound-force second/square inch.\nNote that rounding errors may occur, so always check the results.\nUse this page to learn how to convert between dyne seconds/square centimeter and pound-force second/square inch.\nType in your own numbers in the form to convert the units!\n\n## ››Want other units?\n\nYou can do the reverse unit conversion from pound-force second/square inch to dyne-second/square centimeter, or enter any two units below:\n\n## Enter two units to convert\n\n From: To:\n\n## ››Metric conversions and more\n\nConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3\", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77318877,"math_prob":0.9862245,"size":1846,"snap":"2019-51-2020-05","text_gpt3_token_len":443,"char_repetition_ratio":0.3170467,"word_repetition_ratio":0.016460905,"special_character_ratio":0.20422535,"punctuation_ratio":0.11764706,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9884177,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-11T06:11:10Z\",\"WARC-Record-ID\":\"<urn:uuid:effba3a5-8227-479f-9bdd-4a887b51c9c8>\",\"Content-Length\":\"24833\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d4ff19db-cccd-4899-9b7e-187455e24f17>\",\"WARC-Concurrent-To\":\"<urn:uuid:86064383-441d-4d61-a430-236fa1c06679>\",\"WARC-IP-Address\":\"54.84.169.205\",\"WARC-Target-URI\":\"https://www.convertunits.com/from/dyne-second/square+centimeter/to/pound-force+second/square+inch\",\"WARC-Payload-Digest\":\"sha1:KKSARDPEE2CT5HUO53ZXT55TWDDSEL3V\",\"WARC-Block-Digest\":\"sha1:WKYNB263YMHEGTQTPFKOVI2EU4KXGGZX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540529955.67_warc_CC-MAIN-20191211045724-20191211073724-00147.warc.gz\"}"}
https://www.lyricsrocket.com/Classes/CalcIII/PartialDerivAppsIntro.aspx	[ "Paul's Online Notes\nHome / Calculus III / Applications of Partial Derivatives\nShow Mobile Notice Show All Notes Hide All Notes\nMobile Notice\nYou appear to be on a device with a \"narrow\" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.\n\n## Chapter 3 : Applications of Partial Derivatives\n\nIn this chapter we will take a look at a several applications of partial derivatives. Most of the applications will be extensions to applications to ordinary derivatives that we saw back in Calculus I. For instance, we will be looking at finding the absolute and relative extrema of a function and we will also be looking at optimization. Both (all three?) of these subjects were major applications back in Calculus I. They will, however, be a little more work here because we now have more than one variable.\n\nHere is a list of the topics in this chapter.\n\nTangent Planes and Linear Approximations – In this section formally define just what a tangent plane to a surface is and how we use partial derivatives to find the equations of tangent planes to surfaces that can be written as $$z=f(x,y)$$. We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point.\n\nGradient Vector, Tangent Planes and Normal Lines – In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line.\n\nRelative Minimums and Maximums – In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points (i.e. neither a relative minimum or relative maximum).\n\nAbsolute Minimums and Maximums – In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded (i.e. no part of the region goes out to infinity) and closed (i.e. all of the points on the boundary are valid points that can be used in the process).\n\nLagrange Multipliers – In this section we’ll see discuss how to use the method of Lagrange Multipliers to find the absolute minimums and maximums of functions of two or three variables in which the independent variables are subject to one or more constraints. We also give a brief justification for how/why the method works." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89991677,"math_prob":0.9423525,"size":2231,"snap":"2021-21-2021-25","text_gpt3_token_len":454,"char_repetition_ratio":0.12572968,"word_repetition_ratio":0.057142857,"special_character_ratio":0.19677275,"punctuation_ratio":0.06682578,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98218197,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-12T12:14:41Z\",\"WARC-Record-ID\":\"<urn:uuid:42427a74-9314-47be-b7a6-0ee7e1b30199>\",\"Content-Length\":\"76578\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:50890e55-1604-4bb1-9ba8-5b7325ec991e>\",\"WARC-Concurrent-To\":\"<urn:uuid:333e489d-5829-45a9-b68e-706d6ab5a962>\",\"WARC-IP-Address\":\"172.67.179.5\",\"WARC-Target-URI\":\"https://www.lyricsrocket.com/Classes/CalcIII/PartialDerivAppsIntro.aspx\",\"WARC-Payload-Digest\":\"sha1:Q6CBPMY4JWU4RNI6RYZVLPE7XFTBER6D\",\"WARC-Block-Digest\":\"sha1:4WSISEOPEIJV5OYHRN2E5VCCWGKFNMX3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989693.19_warc_CC-MAIN-20210512100748-20210512130748-00412.warc.gz\"}"}
https://file.scirp.org/Html/3-5301020_63261.htm	[ " Artin Transfer Patterns on Descendant Trees of Finite p-Groups\n\nVol.06 No.02(2016), Article ID:63261,39 pages\n10.4236/apm.2016.62008\n\nArtin Transfer Patterns on Descendant Trees of Finite p-Groups\n\nDaniel C. Mayer\n\nNaglergasse 53, 8010 Graz, Austria", null, "", null, "", null, "", null, "Received 25 November 2015; accepted 25 January 2016; published 29 January 2016\n\nABSTRACT\n\nBased on a thorough theory of the Artin transfer homomorphism", null, "from a group G to the abelianization", null, "of a subgroup", null, "of finite index", null, ", and its connection with the permutation representation", null, "and the monomial representation", null, "of G, the Artin pattern", null, ", which consists of families", null, ", resp.", null, ", of transfer targets, resp. transfer kernels, is defined for the vertices", null, "of any descendant tree", null, "of finite p-groups. It is endowed with partial order relations", null, "and", null, ", which are compatible with the parent-descendant relation", null, "of the edges", null, "of the tree T. The partial order enables termination criteria for the p-group generation algorithm which can be used for searching and identifying a finite p-group G, whose Artin pattern", null, "is known completely or at least partially, by constructing the descendant tree with the abelianization of G as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns and explaining the stabilization, resp. polarization, of their components in descendant trees of finite p- groups.\n\nKeywords:\n\nArtin Transfer, Kernel Type, Target Type, Descendant Tree, Coclass Tree, Coclass Graph\n\n1. Introduction\n\nP 1.1. In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index.\n\nOriginally, such transfer mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin’s reciprocity isomorphism ( , §4, Allgemeines Reziprozitätsgesetz, p. 361) to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups ( , §2, p. 50).\n\nHowever, independently of number theoretic applications, a natural partial order on the kernels and targets of Artin transfers, has recently been found to be compatible with parent-child relations between finite p-groups, where p denotes a prime number. Such ancestor-descendant relations can be visualized conveniently in des- cendant trees ( , §4, pp. 163-164).\n\nConsequently, Artin transfers provide valuable information for classifying finite p-groups by kernel-target patterns and for searching and identifying particular groups in descendant trees by looking for patterns defined by kernels and targets of Artin transfers. These strategies of pattern recognition are useful not only in purely group theoretic context, but also, most importantly, for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers. The reason is that the unramified extensions of a base field contain information in the shape of capitulation patterns and class group structures, and these arithmetic invariants can be translated into group theoretic data on transfer kernels and targets by means of Artin’s reciprocity law of class field theory. The natural partial order on Artin patterns admits termination criteria for a search through a descendant tree with the aid of recursive executions of the p-group generation algorithm by Newman and O’Brien .\n\nP 1.2. The organization of this article is as follows. The detailed theory of the transfer will be developed in §§ 2 and 3, followed by computational implementations in § 4. It is our intention to present more than the least common multiple of the original papers by Schur and Artin and the relevant sections of the text books by Hall , Huppert , Gorenstein , Aschbacher , Doerk and Hawkes , Smith and Tabachnikova , and Isaacs .\n\nHowever, we shall not touch upon fusion and focal subgroups, which form the primary goal of the mentioned authors, except Artin. Our focus will rather be on a sound foundation of Artin patterns, consisting of families of transfer kernels and targets, and their stabilization, resp. polarization, in descendant trees of finite p-groups. These phenomena arise from a natural partial order on Artin patterns which is compatible with ancestor- descendant relations in trees, and is established in its most general form in §§5 and 6.\n\nSince our endeavour is to give the most general view of each partial result, we came to the conviction that categories, functors and natural transformations are the adequate tools for expressing the appropriate range of validity for the facts connected with the partial order relation on Artin patterns. Inspired by Bourbaki’s method of exposition , Appendix on induced homomorphisms, which is separated to avoid a disruption of the flow of exposition, goes down to the origins exploiting set theoretic facts concerning direct images and inverse pre-images of mappings which are crucial for explaining the natural partial order of Artin patterns.\n\n2. Transversals and Their Permutations\n\n2.1. Transversals of a Subgroup\n\nLet G be a group and be a subgroup of finite index.\n\nDefinition 2.1. See also ( , p. 1013); ( , (1.5.1), p. 11); ( , Satz 2.5, p. 5).\n\n1). A left transversal of H in G is an ordered system of representatives for the left cosets of H in\n\nG such that is a disjoint union.\n\n2). Similarly, a right transversal of H in G is an ordered system of representatives for the right\n\ncosets of H in G such that is a disjoint union.\n\nRemark 2.1. For any transversal of H in G, there exists a unique subscript such that, resp.. The element, resp., which represents the principal coset (i.e., the subgroup H itself) may be replaced by the neutral element 1.\n\nLemma 2.1. See also ( , p. 1015); ( , (1.5.2), p. 11); ( , Satz 2.6, p. 6).\n\n1). If G is non-abelian and H is not a normal subgroup of G, then we can only say that the inverse elements of a left transversal form a right transversal of H in G.\n\n2). However, if is a normal subgroup of G, then any left transversal is also a right transversal of H in G.\n\nProof. 1). Since the mapping, is an involution, that is a bijection which is its own inverse, we see that\n\nimplies.\n\n2). For a normal subgroup, we have for each.\n\nLet be a group homomorphism and be a left transversal of a subgroup H in G with finite index. We must check whether the image of this transversal under the homomorphism is a transversal again.\n\nProposition 2.1. The following two conditions are equivalent.\n\n1). is a left transversal of the subgroup in the image with finite\n\nindex.\n\n2)..\n\nWe emphasize this important equivalence in a formula:\n\n(2.1)\n\nProof. By assumption, we have the disjoint left coset decomposition which comprises two\n\nstatements simultaneously.\n\nFirstly, the group is a union of cosets,\n\nand secondly, any two distinct cosets have an empty intersection, for.\n\nDue to the properties of the set mapping associated with, the homomorphism maps the union to another union\n\nbut weakens the equality for the intersection to a trivial inclusion\n\nTo show that the images of the cosets remain disjoint we need the property of the homo- morphism.\n\nSuppose that for some,\n\nthen we have for certain elements.\n\nMultiplying by from the left and by from the right, we obtain\n\nSince, this implies, resp., and thus. (This part of the proof is also covered by ( , Thm. X. 21, p. 340) and, in the context of normal subgroups instead of homomorphisms, by ( , Thm. 2.3.4, p. 29) and ( , Satz 3.10, p. 16))\n\nConversely, we use contraposition.\n\nIf the kernel of is not contained in the subgroup H, then there exists an element such that.\n\nBut then the homomorphism maps the disjoint cosets\n\nto equal cosets.\n\n2.2. Permutation Representation\n\nP 2.1. Suppose is a left transversal of a subgroup of finite index in a group G. A fixed element gives rise to a unique permutation of the left cosets of H in G by left multiplication such that\n\n(2.2)\n\nfor each.\n\nSimilarly, if is a right transversal of H in G, then a fixed element gives rise to a unique permutation of the right cosets of H in G by right multiplication such that\n\n(2.3)\n\nfor each.\n\nThe elements, resp., , of the subgroup H are called the monomials associated with x with respect to, resp..\n\nThe mapping, resp., is called the permutation representation of G in with respect to, resp..\n\nLemma 2.2. For the special right transversal associated to the left transversal, we have the following relations between the monomials and permutations corresponding to an element:\n\n(2.4)\n\nProof. For the right transversal, we have, for each.\n\nOn the other hand, for the left transversal, we have\n\n, for each.\n\nThis relation simultaneously shows that, for any, the permutation representations and the associated monomials are connected by\n\nfor each. □\n\n3. Artin Transfer\n\nLet G be a group and be a subgroup of finite index. Assume that, resp., is a left, resp. right, transversal of H in G with associated permutation representation, , resp., such that, resp., for.\n\nDefinition 3.3. See also ( , p. 1014); ( , §2, p. 50); ( , (14.2.2-4), p. 202); ( , p. 413); ( , p. 248); ( , p. 197); ( , Dfn.(17.1), p. 60); ( , p. 154); ( , p. 149); ( , p. 2).\n\nThe Artin transfer from G to the abelianization of H with respect to, resp., is defined by\n\n(3.1)\n\nresp.\n\n(3.2)\n\nfor.\n\nRemark 3.1. I.M. Isaacs , p. 149 calls the mapping, , resp.,\n\nthe pre-transfer from G to H. The pre-transfer can be composed with a homomorphism from H into\n\nan abelian group A to define a more general version of the transfer, , resp.\n\n, from G to A via, which occurs in the book by D. Gorenstein ( , p. 248). Taking the\n\nnatural epimorphism, , yields the Definition 3.3 of the Artin transfer in its original form by I. Schur ( , p. 1014) and by E. Artin ( , §2, p. 50), which has also been dubbed Verlagerung by H. Hasse ( , §27.4, pp. 170-171). Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.\n\n3.1. Independence of the Transversal\n\nAssume that is another left transversal of H in G such that.\n\nProposition 3.1. See also ( , p. 1014); ( , Thm. 14.2.1, p. 202); ( , Hilfssatz 1.5, p. 414); ( , Thm. 3.2, p. 246); ( , (37.1), p.198); ( , Thm.(17.2), p.61); ( , p.154); ( , Thm.5.1, p.149); ( , Prop.2, p. 2).\n\nThe Artin transfers with respect to (g) and coincide,.\n\nProof. There exists a unique permutation such that, for all. Consequently, , resp. with, for all. For a fixed element, there exists a unique permutation such that we have\n\n,\n\nfor all. Therefore, the permutation representation of G with respect to is given by, resp., for. Furthermore, for the connection between the elements\n\nand, we obtain\n\nfor all. Finally, due to the commutativity of the quotient group and the fact that and are permutations, the Artin transfer turns out to be independent of the left transversal:\n\nas prescribed in Definition 3.1, Equation (3.1). □\n\nIt is clear that a similar proof shows that the Artin transfer is independent of the choice between two different right transversals. It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal.\n\nFor this purpose, we select the special right transversal associated to the left transversal, as explained in Lemma 2.1 and Lemma 2.2.\n\nProposition 3.2. The Artin transfers with respect to and coincide,.\n\nProof. Using (2.4) in Lemma 2.2 and the commutativity of, we consider the expression\n\nThe last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following subsection 3.2. □\n\n3.2. Artin Transfers as Homomorphisms\n\nLet be a left transversal of H in G.\n\nTheorem 3.1. See also ( , p. 1014); ( , Thm. 14.2.1, p. 202); ( , Hauptsatz 1.4, p. 413); ( , Thm. 3.2, p. 246); ( , (37.2), p.198); ( , Thm.(17.2), p.61); ( , p. 155); ( , Thm.5.2, p. 150); ( , Prop.1, p. 2).\n\nThe Artin transfer, and the permutation representation\n\nare group homomorphisms:\n\n(3.3)\n\nProof. Let be two elements with transfer images and\n\n. Since is abelian and is a permutation, we can change the order of the factors in the following product:\n\nThis relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms, since and, in a covariant way. □\n\n3.3. Monomial Representation\n\nLet, resp., be a left, resp. right, transversal of a subgroup H in a group G. Using the monomials, resp., associated with an element according to Equation (2.2), resp. (2.3), we define the following maps.\n\nDefinition 3.2. The mapping, respectively , is called the monomial representation of G in with respect to, resp..\n\nP 3.1. It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial\n\nrepresentation. The images of the factors are given by and\n\n. In the proof of Theorem 3.1, the image of the product turned out to be, which is a very peculiar law of com- position discussed in more detail in the sequel.\n\nThe law reminds of the crossed homomorphisms in the first cohomology group of a G-module M, which have the property, for.\n\nThese peculiar structures can also be interpreted by endowing the cartesian product with a special law of composition known as the wreath product of the groups H and with respect to the set.\n\nDefinition 3.3. For, the wreath product of the associated monomials and permutations is given by\n\n(3.4)\n\nTheorem 3.2. See also ( , Thm.14.1, p. 200); ( , Hauptsatz 1.4, p. 413).\n\nThis law of composition on causes the monomial representation\n\nalso to be a homomorphism. In fact, it is a faithful representation, that is an injective homomorphism, also called a monomorphism or embedding, in contrast to the permutation representation.\n\nProof. The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group endowed with the wreath product is given by, where the last 1 means the identity permutation. If , for some xÎG, then and consequently, for all. Finally, an application of the inverse inner automorphism with yields, as required for injectivity.\n\nThe permutation representation cannot be injective if G is infinite or at least of an order bigger than, the factorial of n. □\n\nRemark 3.2. Formula (3.4) is an example for the left-sided variant of the wreath product on. However, we point out that the wreath product with respect to a right transversal of H in G appears in its right-sided variant\n\n(3.5)\n\nwhich implies that the permutation representation, is a homomorphism with respect to the opposite law of composition on, in a contravariant manner.\n\nIt can be shown that the left-sided and the right-sided variant of the wreath product lead to isomorphic group structures on.\n\nA related viewpoint is taken by M. Hall ( , p. 200), who uses the multiplication of monomial matrices to describe the wreath product. Such a matrix can be represented in the form as the product of an invertible diagonal matrix over the group ring K[H], where K denotes a field, and the permutation matrix associated with the permutation. Multiplying two such monomial matrices yields a law of composition identical to the wreath product in the right-sided variant,\n\nWhereas B. Huppert ( , p. 413) uses the monomial representation for defining the Artin transfer by composition with the unsigned determinant, we prefer giving the immediate Definition 3.3 and merely illustrating the homomorphism property of the Artin transfer with the aid of the monomial representation.\n\n3.4. Composition of Artin Transfers\n\nLet G be a group with nested subgroups such that the indices, and are finite.\n\nTheorem 3.3. See also ( , Thm.14.2.1, p. 202); ( , Satz 1.6, p. 415); ( , Lem.(17.3), p. 61); ( , Thm.10.8, p. 301); ( , Prop.3, p. 3).\n\nThen the Artin transfer is the compositum of the induced transfer (in the sense of Corollary 7.1 or Corollary 7.3 in the Appendix) and the Artin transfer, i.e.,\n\n(3.6)\n\nThis can be seen in the following manner.\n\nProof. If is a left transversal of H in G and is a left transversal of K in H, that is\n\nand, then is a disjoint left coset decomposition of G with\n\nrespect to K. See also ( , Thm.1.5.3, p. 12); ( , Satz 2.6, p. 6). Given two elements and, there exist unique permutations, and, such that the associated monomials are given by\n\n, for each, and, for each.\n\nThen, using Corollary 7.3, we have\n\n, and.\n\nFor each pair of subscripts and, we put and obtain\n\nresp.. Thus, the image of x under the Artin transfer is given by\n\n3.5. Wreath Product of Sm and Sn\n\nP 3.2. Motivated by the proof of Theorem 3.3, we want to emphasize the structural peculiarity of the monomial representation\n\nwhich corresponds to the compositum of Artin transfers, defining\n\nfor a permutation, and using the symbolic notation for all pairs of subscripts,.\n\nThe preceding proof has shown that. Therefore, the action of the permutation on the set is given by. The action on the second component j depends\n\non the first component i (via the permutation), whereas the action on the first component i is independent of the second component j. Therefore, the permutation can be identified with the multiplet, which will be written in twisted form in the sequel.\n\nThe permutations, which arise as second components of the monomial representation\n\nare of a very special kind. They belong to the stabilizer of the natural equipartition of the set [1, n] × [1, m] into the n rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product of the symmetric groups and with respect to, whose underlying set is endowed with the following law of composition in the left-sided variant.\n\n(3.7)\n\nfor all.\n\nThis law reminds of the chain rule for the Fréchet derivative in\n\nxÎE of the compositum of differentiable functions and between complete normed spaces.\n\nThe above considerations establish a third representation, the stabilizer representation,\n\nof the group G in the wreath product, similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if G is infinite.\n\nFormula (3.7) proves the following statement.\n\nTheorem 3.4. The stabilizer representation of the group G in\n\nthe wreath product of symmetric groups is a group homomorphism.\n\n3.6. Cycle Decomposition\n\nLet be a left transversal of a subgroup of finite index in a group G. Suppose the element gives rise to the permutation of the left cosets of H in G such that\n\n, resp., for each.\n\nTheorem 3.5. See also ( , §2, p. 50); ( , §27.4, p. 170); ( , Hilfssatz 1.7, p. 415); ( , Thm.3.3, p. 249); ( , (37.3), p. 198); ( , p. 154); ( , Lem.5.5, p. 153); ( , p. 5).\n\nIf the permutation has the decomposition into pairwise disjoint (and thus commuting)\n\ncycles of lengths, which is unique up to the ordering of the cycles, more explicitly, if\n\n(3.8)\n\nfor, and, then the image of under the Artin transfer is given by\n\n(3.9)\n\nProof. The reason for this fact is that we obtain another left transversal of H in G by putting for and, since\n\n(3.10)\n\nis a disjoint decomposition of G into left cosets of H.\n\nLet us fix a value of. For, we have\n\nHowever, for, we obtain\n\nConsequently,\n\nP 3.3. The cycle decomposition corresponds to a double coset decomposition of the group\n\nG modulo the cyclic group and modulo the subgroup H. It was actually this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper ( , §2, p. 50).\n\n3.7. Transfer to a Normal Subgroup\n\nP 3.4. Now let be a normal subgroup of finite index in a group G. Then we have, for all, and there exists the quotient group of order n. For an element, we let denote the order of the coset in, and we let be a left transversal of the subgroup in G, where.\n\nThen the image of under the Artin transfer is given by\n\n(3.11)\n\nProof. is a cyclic subgroup of order f in, and a left transversal of the subgroup\n\nin G, where and is the corresponding disjoint left coset decomposition,\n\ncan be refined to a left transversal with disjoint left coset decomposition\n\n(3.12)\n\nof H in G. Hence, the formula for the image of x under the Artin transfer in the previous section takes the particular shape\n\nwith exponent f independent of j. □\n\nCorollary 3.1. See also ( , Lem.10.6, p. 300) for a special case.\n\nIn particular, the inner transfer of an element is given as a symbolic power\n\n(3.13)\n\nwith the trace element\n\n(3.14)\n\nof H in G as symbolic exponent.\n\nThe other extreme is the outer transfer of an element which generates G modulo H, that is. It is simply an nth power\n\n(3.15)\n\nProof. The inner transfer of an element, whose coset is the principal set in of order, is given as the symbolic power\n\nwith the trace element\n\nof H in G as symbolic exponent.\n\nThe outer transfer of an element which generates G modulo H, that is, whose coset is generator of with order, is given as the nth power\n\n.\n\nP 3.5. Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels (§5) of Artin transfers from a group G to intermediate groups between G and its com- mutator subgroup. For these intermediate groups we have the following lemma.\n\nLemma 3.1. All subgroups of a group G which contain the commutator subgroup are normal subgroups.\n\nProof. Let. If H were not a normal subgroup of G, then we had for some element. This would imply the existence of elements and such that, and consequently the commutator would be an element in in contradiction to. □\n\nExplicit implementations of Artin transfers in the simplest situations are presented in the following section.\n\n4. Computational Implementation\n\n4.1. Abelianization of Type (p, p)\n\nP 4.1. Let G be a pro-p group with abelianization of elementary abelian type. Then G has maximal subgroups of index. In this particular case, the Frattini\n\nsubgroup, which is defined as the intersection of all maximal subgroups, coincides with the\n\ncommutator subgroup, since the latter contains all pth powers, and thus we have.\n\nFor each, let be the Artin transfer homomorphism from G to the abelianization of. According to Burnside's basis theorem, the group G has generator rank and can therefore be generated as by two elements such that. For each of the normal subgroups\n\n, we need a generator with respect to, and a generator of a transversal\n\nsuch that and.\n\nA convenient selection is given by\n\n(4.1)\n\nThen, for each, it is possible to implement the inner transfer by\n\n(4.2)\n\naccording to Equation (3.13) of Corollary 3.1, which can also be expressed by a product of two pth powers,\n\n(4.3)\n\nand to implement the outer transfer as a complete pth power by\n\n(4.4)\n\naccording to Equation (3.15) of Corollary 3.1. The reason is that and in the quotient group.\n\nIt should be pointed out that the complete specification of the Artin transfers also requires explicit knowledge of the derived subgroups. Since is a normal subgroup of index p in, a certain general reduction is possible by ( , Lem.2.1, p. 52), but an explicit pro-p pre- sentation of G must be known for determining generators of, whence\n\n(4.5)\n\n4.2. Abelianization of Type (p2, p)\n\nP 4.2. Let G be a pro-p group with abelianization of non-elementary abelian type. Then G has maximal subgroups of index and subgroups\n\nof index.\n\nFigure 1 visualizes this smallest non-trivial example of a multi-layered abelianization ( , Dfn.3.1- 3, p. 288).\n\nFor each, let, resp., be the Artin transfer homo- morphism from G to the abelianization of, resp.. Burnside’s basis theorem asserts that the group G has generator rank and can therefore be generated as by two elements such that\n\nFigure 1. Layers of subgroups for.\n\n.\n\nWe begin by considering the first layer of subgroups. For each of the normal subgroups , we select a generator\n\n(4.6)\n\nThese are the cases where the factor group is cyclic of order. However, for the distinguished maximal subgroup, for which the factor group is bicyclic of type, we need two generators\n\n(4.7)\n\nFurther, a generator of a transversal must be given such that, for each. It is convenient to define\n\n(4.8)\n\nThen, for each, we have the inner transfer\n\n(4.9)\n\nwhich equals, and the outer transfer\n\n(4.10)\n\nsince and.\n\nNow we continue by considering the second layer of subgroups. For each of the normal subgroups , we select a generator\n\n(4.11)\n\nsuch that. Among these subgroups, the Frattini subgroup is par- ticularly distinguished. A uniform way of defining generators of a transversal such that, is to set\n\n(4.12)\n\nSince, but on the other hand and, for, with the single exception that, we obtain the following expressions for the inner transfer\n\n(4.13)\n\nand for the outer transfer\n\n(4.14)\n\nexceptionally\n\n(4.15)\n\nand\n\n(4.16)\n\nfor. Again, it should be emphasized that the structure of the derived subgroups and must be known explicitly to specify the action of the Artin transfers completely.\n\n5. Transfer Kernels and Targets\n\nP 5.1. After our thorough treatment of the general theory of Artin transfers in §§2 and 3, and their computational implementation for some simple cases in §4, we are now in the position to introduce Artin transfer patterns, which form the central concept of this article. They provide an incredibly powerful tool for classifying finite and infinite pro-p groups and for identifying a finite p-group G with sufficiently many assigned components of its Artin pattern by the strategy of pattern recognition. This is done in a search through the descendant tree with root by means of recursive applications of the p-group generation algorithm by Newman and O’Brien .\n\nAn Artin transfer pattern consists of two families of transfer targets, resp. kernels, which are also called multiplets, whereas their individual components are referred to as singulets.\n\n5.1. Singulets of Transfer Targets\n\nTheorem 5.1. Let G and T be groups. Suppose that is the image of G under a homomorphism, and is the image of an arbitrary subgroup. Then the following claims hold without any further necessary assumptions.\n\n1) The commutator subgroup of V is the image of the commutator subgroup of U, that is\n\n(5.1)\n\n2) The restriction is an epimorphism which induces a unique epimorphism\n\n(5.2)\n\nThus, the abelianization of V,\n\n(5.3)\n\nis an epimorphic image of the abelianization of U, namely the quotient of by the kernel of, which is given by\n\n(5.4)\n\n3) Moreover, the map is an isomorphism, and the quotients are isomorphic, if and only if\n\n(5.5)\n\nSee Figure 2 for a visualization of this situation.\n\nProof. The statements can be seen in the following manner. The image of the commutator subgroup is given by\n\nThe homomorphism can be restricted to an epimorphism. According to Theorem 7.1, in particular, by the Formulas (7.5) and (7.4) in the appendix, the condition implies the existence of a uniquely determined epimorphism such that. The Isomor- phism Theorem in Formula (7.7) in the appendix shows that. Furthermore, by the Formulas (7.4) and (7.1), the kernel of is given explicitly by\n\nThus, is an isomorphism if and only if. □\n\nP 5.2. Functor of derived quotients. In analogy to section §7.6 in the appendix, a covariant functor can be used to map a morphism of one category to an induced morphism of another category.\n\nIn the present situation, we denote by the category of groups and we define the domain of the functor F as the following category. The objects of the category are pairs consisting of a group G and a subgroup,\n\n(5.6)\n\nFor two objects, the set of morphisms consists of epimor- phisms such that and, briefly written as arrows,\n\n(5.7)\n\nThe functor from this category to the category of abelian groups maps a pair to the commutator quotient group of the subgroup, and\n\nit maps a morphism to the induced epimorphism of the restriction,\n\n(5.8)\n\nExistence and uniqueness of have been proved in Theorem 5.1 under the assumption that, which is satisfied according to the definition of the arrow and automatically implies.\n\nFigure 2. Induced homomorphism of derived quotients.\n\nDefinition 5.1. Due to the results in Theorem 5.1, it makes sense to define a pre-order of transfer targets on the image of the functor F in the object class of the category of abelian groups in the following manner.\n\nFor two objects, a morphism, and the images\n\n, and,\n\nlet (non-strict) precedence be defined by\n\n(5.9)\n\nand let equality be defined by\n\n(5.10)\n\nif the induced epimorphism is an isomorphism.\n\nCorollary 5.1. If both components of the pairs are restricted to Hopfian groups, then the pre-order of transfer targets is actually a partial order.\n\nProof. We use the functorial properties of the functor F. The reflexivity of the partial order follows from the functorial identity in Formula (7.14), and the transitivity is a consequence of the functorial compositum in Formula (7.15), given in the appendix. The antisymmetry might be a problem for infinite groups, since it is known that there exist so-called non-Hopfian groups. However, for finite groups, and more generally for Hop- fian groups, it is due to the implication . □\n\n5.2. Singulets of Transfer Kernels\n\nSuppose that G and T are groups, is the image of G under a homomorphism, and is the image of a subgroup of finite index. Let be the Artin transfer from G to.\n\nTheorem 5.2. If, then the image of a left transversal of U in G is a left transversal of V in H, the index remains the same and is therefore finite, and the Artin transfer from H to exists.\n\n1) The following connections exist between the two Artin transfers: the required condition for the composita of mappings in the commutative diagram in Figure 3,\n\n(5.11)\n\nand, consequently, the inclusion of the kernels,\n\n(5.12)\n\n2) A sufficient (but not necessary) condition for the equality of the kernels is given by\n\n(5.13)\n\nFigure 3. Epimorphism and Artin transfer.\n\nSee Figure 3 for a visualization of this scenario.\n\nProof. The truth of these statements can be justified in the following way. The first part has been proved in\n\nProposition 2.1 already: Let be a left transversal of U in G. Then is a disjoint union but the union is not necessarily disjoint. For, we have\n\nfor some\n\nelement. However, if the condition is satisfied, then we are able to conclude that\n\n, and thus.\n\nLet be the epimorphism obtained in the manner indicated in the proof of Theorem 5.1 and Formula (5.2). For the image of under the Artin transfer, we obtain\n\nSince, the right hand side equals, provided that is a left transversal of V in H, which is correct when. This shows that the diagram in Figure 3 is commutative, that is,. It also yields the connection between the permutations and the monomials for all. As a consequence, we obtain the inclusion\n\n, if. Finally, if, then the previous section has shown that\n\nis an isomorphism. Using the inverse isomorphism, we get, which proves the equation\n\n. More explicitly, we have the following chain of equivalences and implications:\n\nConversely, only implies. Therefore, we cer-\n\ntainly have if, which is, however, not necessary. □\n\nP 5.3. Artin transfers as natural transformations. Artin transfers can be viewed as components of a natural transformation T between two functors and F from the following category to the usual category of groups.\n\nThe objects of the category are pairs consisting of a group G and a subgroup of finite index,\n\n(5.14)\n\nFor two objects, the set of morphisms consists of epimor- phisms satisfying, , and the additional condition for their kernels, briefly written as arrows,\n\n(5.15)\n\nThe forgetful functor from this category to the category of groups maps a pair. to its first component, and it maps a morphism to the underlying epimorphism.\n\n(5.16)\n\nThe functor from to the category of groups maps a pair to the commutator quotient group of the subgroup U, and it maps a morphism to the induced epimorphism of the restriction. Note that we must abstain here from letting F map into the subcategory of abelian groups.\n\n(5.17)\n\nThe system T of all Artin transfers fulfils the requirements for a natural transformation between these two functors, since we have\n\n(5.18)\n\nfor every morphism of the category.\n\nDefinition 5.2. Due to the results in Theorem 5.2, it makes sense to define a pre-order of transfer kernels on the kernels of the components of the natural transformation T in the object class of the category of groups in the following manner.\n\nFor two objects, a morphism, and the images\n\n, and,\n\nlet (non-strict) precedence be defined by\n\n(5.19)\n\nand let equality be defined by\n\n(5.20)\n\nif the induced epimorphism is an isomorphism.\n\nCorollary 5.2. If both components of the pairs are restricted to Hopfian groups,\n\nthen the pre-order of transfer kernels is actually a partial order.\n\nProof. Similarly as in the proof of Corollary 5.1, we use the properties of the functor F. The reflexivity is due to the functorial identity in Formula (7.14). The transitivity is due to the functorial compositum in Formula (7.15), where we have to observe the relations, , and Formula (7.1) in the appendix for verifying the kernel relation\n\nThe antisymmetry is certainly satisfied for finite groups, and more generally for Hopfian groups. □\n\n5.3. Multiplets of Transfer Targets and Kernels\n\nInstead of viewing various pairs which share the same first component G as distinct objects in the categories, resp., which we used for describing singulets of transfer targets, resp. kernels, we now consider a collective accumulation of singulets in multiplets. For this purpose, we shall define a new category of families, which generalizes the category, rather than the category. However, we have to pre- pare this definition with a criterion for the compatibility of a system of subgroups with its image under a homo- morphism.\n\nProposition 5.1. See also ( , Thm.2.3.4, p. 29); ( , Satz 3.10, p. 16); ( , Thm.2.4, p. 6); ( , Thm.X.21, p. 340).\n\nFor an epimorphism of groups, the associated set mappings\n\n(5.21)\n\nare inverse bijections between the following systems of subgroups\n\n(5.22)\n\nProof. The fourth and fifth statement of Lemma 7.1 in the appendix show that usually the associated set mappings and of a homomorphism are not inverse bijections between systems of sub- groups of G and H. However, if we replace the homomorphism by an epimorphism with, then the Formula (7.2) yields the first desired equality\n\nGuided by the property of all pre-images of, we define a re- stricted system of subgroups of the domain G,\n\nand, according to Formula (7.1.), we consequently obtain the second required equality\n\nwhich yields the crucial pair of inverse set bijections\n\nP 5.4. After this preparation, we are able to specify the new category. The objects of the category\n\nare pairs consisting of a group G and the family of all subgroups with finite index\n\n,\n\n(5.23)\n\nwhere I denotes a suitable indexing set. Note that G itself is one of the subgroups.\n\nThe morphisms of the new category are subject to more restrictive conditions, which concern entire families of subgroups instead of just a single subgroup.\n\nFor two objects, the set of\n\nmorphisms consists of epimorphisms satisfying, the image conditions, and the kernel conditions, which imply the pre-image conditions, for all, briefly written as arrows,\n\n(5.24)\n\nNote that, in view of Proposition 5.1, we can always use the same indexing set I for the domain and for the codomain of morphisms, provided they satisfy the required kernel condition.\n\nNow we come to the essential definition of Artin transfer patterns.\n\nDefinition 5.3. Let be an object of the category.\n\nThe transfer target type (TTT) of G is the family\n\n(5.25)\n\nThe transfer kernel type (TKT) of G is the family\n\n(5.26)\n\nThe complete Artin pattern of G is the pair\n\n(5.27)\n\nP 5.5. The natural partial order on TTTs and TKTs is reduced to the partial order on the components, according to the Definitions 5.1 and 5.2.\n\nDefinition 5.4. Let be two objects of the category, where all members of the families and are Hopfian groups.\n\nThen (non-strict) precedence of TTTs is defined by\n\n(5.28)\n\nand equality of TTTs is defined by\n\n(5.29)\n\n(Non-strict) precedence of TKTs is defined by\n\n(5.30)\n\nand equality of TKTs is defined by\n\n(5.31)\n\nWe partition the indexing set I in two disjoint components, according to whether components of the Artin pattern remain fixed or change under an epimorphism.\n\nDefinition 5.5. Let be two objects of the category, and let be a morphism between these objects.\n\nThe stable part and the polarized part of the Artin pattern of G with respect to are defined by\n\n(5.32)\n\nAccordingly, we have\n\n(5.33)\n\nNote that the precedence of polarized targets is strict as opposed to polarized kernels.\n\n5.4. The Artin Pattern on a Descendant Tree\n\nP 5.6. Before we specialize to the usual kinds of descendant trees of finite p-groups ( , §4, pp. 163-164) we consider an abstract form of a rooted directed tree, which is characterized by two relations.\n\nFirstly, a basic relation between parent and child (also called immediate descendant), corre- sponding to a directed edge of the tree, for any vertex which is different from the root R of the tree.\n\nSecondly, an induced non-strict partial order relation, for some integer, between ancestor and descendant, corresponding to a path of directed edges, for an arbitrary vertex, that is, the ancestor is an iterated parent of the descendant. Note that only an empty path with starts from the root R of the tree, which has no parent.\n\nJust a brief justification of the partial order: Reflexivity is due to the relation. Transitivity\n\nfollows from the rule. Antisymmetry is a consequence of the absence of cycles, that is, implies and thus.\n\nP 5.7. The category of a tree. Now let be a rooted directed tree whose vertices are groups. Then we define, the category associated with, as a subcategory of the category which was introduced in the Formulas (5.23) and (5.24).\n\nThe objects of the category are those pairs in the object class of the category whose\n\nfirst component is a vertex of the tree,\n\n(5.34)\n\nThe morphisms of the category are selected along the paths of the tree only.\n\nFor two objects, the set of morphisms is either empty or consists of a single element only,\n\n(5.35)\n\nIn the case of an ancestor-descendant relation between H and G, the specification of the supercategory\n\nenforces the following constraints on the unique morphism: the image relations and the kernel relations, for all.\n\nP 5.8. At this position, we must start to be more concrete. In the descendant tree of a group R, which is the root of the tree, the formal parent operator gets a second meaning as a natural projection, , from the child G onto its parent, which is always the quotient of G by a suitable normal subgroup. To be precise, the epimorphism with kernel is actually dependent on its domain G. Therefore, the formal power is only a convenient\n\nabbreviation for the compositum.\n\nAs described in , there are several possible selections of the normal subgroup N in the parent definition. Here, we would like to emphasize the following three choices of characteristic subgroups N of the child G. If p denotes a prime number and is the descendant tree of a finite p-group R, then it is usual to take for\n\n1) either the last non-trivial member of the lower central series of G\n\n2) or the last non-trivial member of the lower exponent-p central series of G\n\n3) or the last non-trivial member of the derived series of G,\n\nwhere denotes the nilpotency class, the lower exponent p-class, and the derived length of G, respectively.\n\nNote that every descendant tree of finite p-groups is subtree of a descendant tree with abelian root. Therefore, it is no loss of generality to restrict our attention to descendant trees with abelian roots.\n\nTheorem 5.3. A uniform warranty for the comparability of the Artin patterns of all vertices G of a descendant tree of finite p-groups with abelian root R, in the sense of the natural partial\n\norder, is given by the following restriction of the family of subgroups in the corresponding object of the category. The restriction depends on the definition of a parent in the\n\ndescendant tree.\n\n1) for all, when with.\n\n2) for all, when with.\n\n3) for all, when with.\n\nProof. If parents are defined by with, then we have and for any. The largest of these kernels arises for. Therefore, uniform comparability of Artin patterns is warranted by the restriction for all.\n\nThe parent definition with implies and for any. The largest of these kernels arises for. Consequently, a uniform comparability of Artin patterns is guaranteed by the restriction for all.\n\nFinally, in the case of the parent definition with, we have and for any. The largest of these kernels arises for. Consequently, a uniform comparability of Artin patterns is guaranteed by the condition for all.\n\nP 5.9. Note that the first and third condition coincide since both, and, denote the commutator subgroup. So the family is restricted to the normal subgroups which contain, as announced in the paragraph preceding Lemma 3.1.\n\nThe second condition restricts the family to the maximal subgroups of G inclusively the group G\n\nitself and the Frattini subgroup.\n\nP 5.10. Since we shall mainly be concerned with the first and third parent definition for descendant trees, that is, either with respect to the lower central series or to the derived series, the comparability condition in Theorem 5.3 suggests the definition of a category whose objects are subject to more severe conditions than those in Formula (5.23),\n\n(5.36)\n\nbut whose morphism are defined exactly as in Formula (5.24). The new viewpoint leads to a corresponding modification of Artin transfer patterns.\n\nDefinition 5.6. Let be an object of the category.\n\nThe Artin pattern, more precisely the restricted Artin pattern, of G is the pair\n\n(5.37)\n\nwhose components, the TTT and the TKT of G, are defined as in the Formulas (5.25) and (5.26), but now with respect to the smaller system of subgroups of G.\n\nP 5.11. The following Main Theorem shows that any non-metabelian group G with derived length and finite abelianization shares its Artin transfer pattern, in the restricted sense, with its metabelianization, that is the second derived quotient.\n\nTheorem 5.4. (Main Theorem.) Let G be a (non-metabelian) group with finite abelianization, and denote by, , the terms of the derived series of G, that is and for, in particular, and, then\n\n1) every subgroup which contains the commutator subgroup is a normal subgroup of finite index,\n\n2) for each, there is a chain of normal subgroups\n\n(5.38)\n\n3) for each, the targets of the transfers and\n\nare equal in the sense of the natural order,\n\n(5.39)\n\n4) for each, the kernels of the transfers and\n\nare equal in the sense of the natural order,\n\n(5.40)\n\nProof. We use the natural epimorphism,.\n\n1) If U is an intermediate group, then is a normal subgroup of G, according to Lemma 3.1. The assumption implies that is a divisor of the integer. Therefore, the Artin transfer exists.\n\n2) Firstly, implies. Since is characteristic in U, we also have. Similarly, is characteristic in and thus normal in. Finally, we obtain\n\n.\n\n3) The mapping, , is an epimorphism with kernel. Consequently, the isomorphism theorem in Remark 7.3 of the appendix yields the isomorphism .\n\n4) Firstly, the restriction is an epimorphism which induces an isomorphism\n\n, since and\n\n, according to Theorem 5.1. Secondly, according\n\nto Theorem 5.2, the condition implies that the index is finite, the Artin transfer exists, the composite mappings commute, and, since we even have, the transfer kernels satisfy the relation\n\n. In the sense of the natural partial order on transfer kernels this\n\nmeans equality, since and thus, similarly as in Proposition 5.1, the map establishes a set bijection between the systems of subgroups and\n\n, where.\n\nRemark 5.1. At this point it is adequate to emphasize how similar concepts in previous publications are related to the concept of Artin patterns. The restricted Artin pattern in Definition 5.6 was essentially introduced in ( , Dfn.1.1, p. 403), for a special case already earlier in ( , §1, p. 417). The name Artin pattern appears in ( , Dfn.3.1, p. 747) for the first time. The complete Artin pattern in Definition 5.3 is new in the present article, but we should point out that it includes the iterated IPADs (index-p abelianization data) in ( , Dfn.3.5, p. 289) and the iterated IPODs (index-p obstruction data) in ( , Dfn.4.5).\n\nIn a second remark, we emphasize the importance of the preceding Main Theorem for arithmetical applications.\n\nRemark 5.2. In algebraic number theory, Theorem 5.4 has striking consequences for the determination of the length of the p-class tower, that is the maximal unramified pro-p extension, of an algebraic number field K with respect to a given prime number p. It shows the impossibility of deciding, exclusively with the aid of the restricted Artin pattern, which of several assigned candidates G with distinct derived lengths is the actual p-class tower group. (In contrast, can always be recognized with.)\n\nThis is the point where the complete Artin pattern enters the stage. Most recent investigations by means of iterated IPADs of 2nd order, whose components are contained in, enabled decisions between in .\n\nAnother successful method is to employ cohomological results by I.R. Shafarevich on the relation rank for selecting among several candidates G for the p-class tower group, in dependence on the torsion-free unit rank of the base field K, for instance in .\n\nImportant examples for the concepts in §5 are provided in the following subsections.\n\n5.5. Abelianization of Type (p,p)\n\nLet G be a p-group with abelianization of elementary abelian type. Then G has maximal subgroups of index. For each, let be the Artin transfer homomorphism from G to the abelianization of.\n\nDefinition 5.7. The family of normal subgroups is called the transfer kernel type\n\n(TKT) of G with respect to.\n\nRemark 5.3. For brevity, the TKT is identified with the multiplet, whose integer components\n\nare given by\n\n(5.41)\n\nHere, we take into consideration that each transfer kernel must contain the commutator subgroup of G, since the transfer target is abelian. However, the minimal case cannot occur, according to Hilbert’s Theorem 94.\n\nA renumeration of the maximal subgroups and of the transfers by means of a per-\n\nmutation gives rise to a new TKT with respect to, identified with, where\n\nIt is adequate to view the TKTs as equivalent. Since we have\n\n,\n\nthe relation between and is given by. Therefore, is another representative of the orbit of under the operation of the symmetric group on the set of all mappings from to, where the extension of the permutation is defined by, and we formally put,.\n\nDefinition 5.8. The orbit of any representative is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.\n\nRemark 5.4. This definition of goes back to the origins of the capitulation theory and was introduced by Scholz and Taussky for in 1934 . Several other authors used this original definition and investigated capitulation problems further. In historical order, Chang in 1977 , Chang and Foote in 1980 , Heider and Schmithals in 1982 , Brink in 1984 , Brink and Gold in 1987 , Nebelung in 1989 , and ourselves in 1991 and in 2012 .\n\nIn the brief form of the TKT, the natural order is expressed by for.\n\nLet denote the counter of total transfer kernels, which is\n\nan invariant of the group G. In 1980, Chang and Foote proved that, for any odd prime p and for any integer, there exist metabelian p-groups G having abelianization of type such that. However, for, there do not exist non-abelian 2-groups G with, such that. Such groups must be metabelian of maximal class. Only the elementary abelian 2-group has.\n\nIn the following concrete examples for the counters, and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by Besche, Eick and O’Brien .\n\nExample 5.1. For, we have the following TKTs:\n\nfor the extra special group of exponent 9 with,\n\nfor the two groups with,\n\nfor the group with,\n\nfor the group with,\n\nfor the extra special group of exponent 3 with.\n\n5.6. Abelianization of Type (p2, p)\n\nLet G be a p-group with abelianization of non-elementary abelian type. Then G possesses maximal subgroups of index, and subgroups of index. See Figure 1.\n\nP 5.12. Convention. Suppose that is the distinguished maximal subgroup which is the product of all subgroups of index, and is the distinguished subgroup of index\n\nwhich is the intersection of all maximal subgroups, that is the Frattini subgroup of G.\n\nP 5.13. First layer. For each, let be the Artin transfer homomorphism from G to the ab- elianization of.\n\nDefinition 5.9. The family is called the first layer transfer kernel type of G with respect to and, and is identified with, where\n\n(5.42)\n\nRemark 5.5. Here, we observe that each first layer transfer kernel is of exponent p with respect to and consequently cannot coincide with for any, since is cyclic of order, whereas is bicyclic of type.\n\nP 5.14. Second layer. For each, let be the Artin transfer homomorphism from G to the abelianization of.\n\nDefinition 5.10. The family is called the second layer transfer kernel type of G with respect to and, and is identified with, where\n\n(5.43)\n\nP 5.15. Transfer kernel type.\n\nCombining the information on the two layers, we obtain the (complete) transfer kernel type\n\nof the p-group G with respect to and.\n\nRemark 5.6. The distinguished subgroups and are unique invariants of G and should not be renumerated. However, independent renumerations of the remaining maximal subgroups and the transfers by means of a permutation, and of the remaining subgroups of index and the transfers by means of a permutation\n\n, give rise to new TKTs with respect to and, identified with, where\n\nand with respect to and, identified with,\n\nwhere\n\nIt is adequate to view the TKTs and as equivalent. Since we have\n\nresp.\n\nthe relations between and, resp. and, are given by, resp.. Therefore, is another representative of the orbit of under the operation\n\nof the product of two symmetric groups on the set of all pairs of mappings from to, where the extensions and of a permutation are defined by and, and formally\n\n, , , and.\n\nDefinition 5.11. The orbit of any representative is an invariant of the p- group G and is called its transfer kernel type, briefly TKT.\n\nP 5.16. Connections between layers.\n\nThe Artin transfer from G to a subgroup of index\n\nis the compositum of the induced transfer from to\n\n(in the sense of Corollary 7.1 or Corollary 7.3 in the appendix) and the Artin transfer from G to, for any intermediate subgroup of index (). There occur two situations:\n\n・ For the subgroups only the distinguished maximal subgroup is an intermediate subgroup.\n\n・ For the Frattini subgroup all maximal subgroups are intermediate subgroups.\n\nThis causes restrictions for the transfer kernel type of the second layer,\n\nsince, and thus\n\n, for all,\n\n・ but even.\n\nFurthermore, when with and, an element () which is of order with respect to, can belong to the transfer kernel only if its pth power is contained in, for all intermediate subgroups, and thus:\n\n, for certain, enforces the first layer TKT singulet,\n\n・ but, for some, even specifies the complete first layer TKT multiplet, that is, for all.\n\n6. Stabilization and Polarization in Descendant Trees\n\nP 6.1. Theorem 5.4 has proved that it suffices to get an overview of the restricted Artin patterns of metabelian groups G with, since groups G of derived length will certainly reveal exactly the same patterns as their metabelianizations.\n\nIn this section, we present the complete theory of stabilization and polarization of the restricted Artin patterns for an extensive exemplary case, namely for all metabelian 3-groups G with abelianization of type (3,3).\n\nSince the bottom layer, resp. the top layer, of the restricted Artin pattern will be considered in Theorem 6.4 on the commutator subgroup, resp. Theorem 6.5 on the entire group G, we first focus on the intermediate layer of the maximal subgroups.\n\n6.1. 3-Groups of Non-Maximal Class\n\nP 6.2. We begin with groups G of non-maximal class. Denoting by m the index of nilpotency of G, we let with be the centralizers of two-step factor groups of the lower central series, that is, the biggest subgroups of G with the property. They form an as- cending chain of characteristic subgroups of G, , which contain the commutator subgroup. coincides with G if and only if. We characterize the smallest two-step centralizer different from the commutator group by an isomorphism invariant . According to Nebelung , we can assume that G has order, class, and coclass, where. Let generators of be selected such that, , if, and. Suppose that a fixed ordering of the four maximal subgroups of G is defined by with, , , and. Let the main commutator of G be declared by and higher commutators recursively by, for. Starting with the powers, , let, for, and put,.\n\nTheorem 6.1. (Non-maximal class.) Let G be a metabelian 3-group of nilpotency class and coclass with abelianization. With respect to the projection onto the parent, the restricted Artin pattern of G reveals\n\n1) a bipolarization and partial stabilization, if G is an interface group with bicyclic last lower central equal to the bicyclic first upper central, more precisely\n\n(6.1)\n\n2) a unipolarization and partial stabilization, if G is a core group with cyclic last lower central and bicyclic first upper central, more precisely\n\n(6.2)\n\n3) a nilpolarization and total stabilization, if G is a core group with cyclic last lower central equal to the cyclic first upper central, more precisely\n\n(6.3)\n\nProof. Theorems 5.1 and 5.2 tell us that for detecting whether stabilization occurs from parent to child G, we have to compare the projection kernel with the commutator subgroups of the four maximal normal subgroups,. According to ( , Cor.3.2, p. 480) these derived subgroups are given by\n\n(6.4)\n\nprovided the generators of G are selected as indicated above. On the other hand, the projection kernel is given by\n\n(6.5)\n\nCombining this information with, we obtain the following results.\n\nfor if, independently of.\n\nfor if, which implies.\n\nbut if, , which also implies.\n\nfor if, , which implies .\n\nTaken together, these results justify all claims. □\n\nExample 6.1. Generally, the parent of an interface group G ( , Dfn.3.3, p. 430) with bicyclic last non-trivial lower central is a vertex of a different coclass graph with lower coclass. In the case of a bipolarization ( , Dfn.3.2, p. 430), which is now also characterized via the Artin pattern by Formula (6.1) for, we can express the membership in coclass graphs by the implication: If with, then. A typical example is the group of coclass 3 with parent of coclass 2 (again with identifiers in the SmallGroups database ), where\n\nand\n\nIn contrast, a core group G ( , Dfn.3.3, p. 430) with cyclic last non-trivial lower central and its parent are vertices of the same coclass graph. In dependence on the p-rank of its centre, the Artin pattern either shows a unipolarization as in Formula (6.2), if the centre is bicyclic, or a total stabilization as in Formula (6.3), if the centre is cyclic. Typical examples are the group with parent, both of coclass 2, where the Artin pattern shows a unipolarization\n\nand\n\nand the group with parent, both of coclass 2, where the Artin pattern shows a total stabilization\n\nand\n\n6.2. p-Groups of Maximal Class\n\nP 6.3. Next we consider p-groups of maximal class, that is, of coclass, but now for an arbitrary prime number. According to Blackburn and Miech , we can assume that G is a metabelian p-group of order and nilpotency class, where. Then G is of coclass and the commutator factor group of G is of type. The lower central series of G is defined recursively by and for, in particular.\n\nThe centralizer of the two-step factor group\n\n, that is,\n\nis the biggest subgroup of G such that. It is characteristic, contains the commutator subgroup, and coincides with G, if and only if. Let the isomorphism invariant of G be defined by\n\nwhere for, for, and for, according to Miech ( , p. 331).\n\nSuppose that generators of are selected such that, if, and.\n\nWe define the main commutator and the higher commutators for.\n\nThe maximal subgroups of G contain the commutator subgroup of G as a normal subgroup of index p and thus are of the shape. We define a fixed ordering by and for.\n\nTheorem 6.2. (Maximal class.) Let G be a metabelian p-group of nilpotency class and coclass, which automatically implies an abelianization of type. With respect to the projection onto the parent, the restricted Artin pattern of G reveals\n\n1) a unipolarization and partial stabilization, if the first maximal subgroup of G is abelian, more precisely\n\n(6.6)\n\n2) a nilpolarization and total stabilization, if all four maximal subgroups of G are non-abelian, more precisely\n\n(6.7)\n\nIn both cases, the commutator subgroups of the other maximal normal subgroups of G are given by\n\n(6.8)\n\nProof. We proceed in the same way as in the proof of Theorem 6.1 and compare the projection kernel with the commutator subgroups of the maximal normal subgroups,. According to ( , Cor.3.1, p. 476) they are given by\n\n(6.9)\n\nif the generators of G are chosen as indicated previously. The cyclic projection kernel is given uniformly by\n\n(6.10)\n\nUsing the relation, we obtain the following results.\n\nfor if.\n\nif and only if, that is,.\n\nThe claims follow by applying Theorems 5.1 and 5.2. □\n\nExample 6.2. For, typical examples are the group with parent, both of coclass 1, where the Artin pattern shows a unipolarization ( , Dfn.3.1, p. 413)\n\nand\n\nand the group with parent, both of coclass 1, where the Artin pattern shows a total stabilization\n\nand\n\n6.3. Extreme Interfaces of p-Groups\n\nP 6.4. Finally, what can be said about the extreme cases (excluded in Theorems 6.1 and 6.2) of non-abelian p-groups having the smallest possible nilpotency class for coclass and for coclass? In these particular situations, the answers can be given for arbitrary prime numbers.\n\nTheorem 6.3. Let G be a metabelian p-group with abelianization of type.\n\n1) If G is of coclass and nilpotency class, then must be odd and the coclass must be exactly.\n\n2) If G is of coclass and nilpotency class, then G is an extra special p-group of order and exponent p or.\n\nIn both cases, there occurs a total polarization and no stabilization at all, more explicitly\n\n(6.11)\n\nProof. Suppose that G is a metabelian p-group with.\n\n1) According to O. Taussky , a 2-group G with abelianization of type must be of coclass. Consequently, implies.\n\nSince the minimal nilpotency class c of a non-abelian group with coclass is given by, the case cannot occur for.\n\nSo we are considering metabelian p-groups G with, nilpotency class and coclass for odd, which form the stem of the isoclinism family in the sense of P. Hall. According to ( , Lem.3.1, p. 446), the commutator subgroups of the maximal subgroups are cyclic of degree p, for such a group. However, the kernel of the parent projection is the bicyclic group of type ( , §3.5, p. 445), which cannot be contained in any of the cyclic with.\n\n2) According to ( , Cor.3.1, p. 476), the commutator subgroups of all maximal subgroups are trivial, for a metabelian p-group G of coclass and nilpotency class, which implies. Thus, the kernel of the parent projection is not contained in any.\n\nIn both cases, the final claim is a consequence of the Theorems 5.1 and 5.2. □\n\nExample 6.3. For, a typical example for the interface between groups of coclass 2 and 1 is the group of coclass 2 with parent of coclass 1, where the Artin pattern shows a total polarization\n\nand\n\nFor, a typical example for the interface between non-abelian and abelian groups is the extra special quaternion group with parent both of coclass 1, where the Artin pattern shows a total polarization\n\nand\n\nSummarizing, we can say that the last three Theorems 6.1, 6.2, and 6.3 underpin the fact that Artin transfer patterns provide a marvellous tool for classifying finite p-groups.\n\n6.4. Bottom and Top Layer of the Artin Pattern\n\nP 6.5. We conclude this section with supplementary general results concerning the bottom layer and top layer of the restricted Artin pattern.\n\nTheorem 6.4. (Bottom layer.) The type of the commutator subgroup can never remain stable for a metabelian vertex of a descendant tree with respect to the lower central series, lower exponent-p central series, or derived series. The kernel of is equal to G (Principal Ideal Theorem).\n\nProof. All possible kernels , resp. , resp. , of the parent projections are non trivial, and can therefore never be contained in the trivial second derived subgroup G\". According to Theorem 5.1, the type of the commutator subgroup G' cannot be stable. The Principal Ideal Theorem is due to Furtwängler and is also proved in ( Thm.10.18, p. 313). □\n\nExample 6.4. In Example 6.1, we point out that the group with cyclic centre and its parent , both of coclass 2, cannot be distinguished by their TTT\n\nand TKT\n\ndue to a total stabilization of the restricted Artin pattern as in Formula (6.3). However, the type of their commutator subgroup (the second layer of their TTT) admits a distinction, since\n\nTheorem 6.5. (Top layer.) In a descendant tree with respect to the lower central series or derived series, the type of the abelianization of remains stable. The kernel of is equal to .\n\nProof. This follows from Theorem 5.1, since even the maximal possible kernel , resp. , of the parent projections is contained in the commutator subgroup of G.\n\nWe briefly emphasize the different behaviour of trees where parents are defined with the lower exponent-p central series.\n\nTheorem 6.6. In a descendant tree with respect to the lower exponent-p central series, only the p-rank of the abelianization of the vertices remains stable.\n\nProof. Denote by the p-rank of the abelianization of G. According to Theorem 5.1, the maximal possible kernel of the parent projections is the Frattini subgroup which is contained in all maximal subgroups of G. According to Proposition 5.1, the map induces a bijection between the sets of maximal subgroups of the child G and the parent , whose cardinality is given by\n\n. Consequently, we have . □\n\nAcknowledgements\n\nThe author would like to express his heartfelt gratitude to Professor Mike F. Newman from the Australian National University in Canberra, Australian Capital Territory, for his continuing encouragement and interest in our endeavour to strengthen the bridge between group theory and class field theory which was initiated by the ideas of Emil Artin, and for his untiring willingness to share his extensive knowledge and expertise and to be a source of advice in difficult situations.\n\nWe also gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008- N25.\n\nCite this paper\n\nDaniel C.Mayer, (2016) Artin Transfer Patterns on Descendant Trees of Finite p-Groups. Advances in Pure Mathematics,06,66-104. doi: 10.4236/apm.2016.62008\n\nReferences\n\n1. 1. Artin, E. (1927) Beweis des allgemeinen Reziprozitätsgesetzes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5, 353-363.\nhttp://dx.doi.org/10.1007/BF02952531\n\n2. 2. Artin, E. (1929) Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7, 46-51.\nhttp://dx.doi.org/10.1007/BF02941159\n\n3. 3. Mayer, D.C. (2015) Periodic Bifurcations in Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 5, 162-195.\nhttp://dx.doi.org/10.4236/apm.2015.54020\n\n4. 4. Newman, M.F. (1977) Determination of Groups of Prime-Power Order. In: Bryce, R.A., Cossey, J. and Newman, M.F., Eds., Group Theory, Lecture Notes in Math, Vol.573, Springer, Berlin, 73-84.\n\n5. 5. O’Brien, E.A. (1990) The p-Group Generation Algorithm. Journal of Symbolic Computation, 9, 677-698.\nhttp://dx.doi.org/10.1016/S0747-7171(08)80082-X\n\n6. 6. Schur, I. (1902) Neuer Beweis eines Satzes über endliche Gruppen. Sitzungsb. Preuss. Akad. Wiss., 42, 1013-1019,.\n\n7. 7. Hall Jr., M. (1999) The Theory of Groups. AMS Chelsea Publishing, American Mathematical Society, Providence.\n\n8. 8. Huppert, B. (1979) Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften, Vol. 134. Springer-Verlag, Berlin, Heidelberg and New York.\n\n9. 9. Gorenstein, D. (2012) Finite Groups. AMS Chelsea Publishing, American Mathematical Society, Providence.\n\n10. 10. Aschbacher, M. (1986) Finite Group Theory. Cambridge Studies in Advanced Mathematics, Vol. 10. Cambridge University Press, Cambridge.\n\n11. 11. Doerk, K. and Hawkes, T. (1992) Finite Soluble Groups, de Gruyter Expositions in Mathematics, Vol.4. Walter de Gruyter, Berlin and New York.\nhttp://dx.doi.org/10.1515/9783110870138\n\n12. 12. Smith, G. and Tabachnikova, O. (2000) Topics in Group Theory. Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London.\n\n13. 13. Isaacs, I.M. (2011) Finite Group Theory. Graduate Studies in Mathematics, Vol.92. American Mathematical Society, Providence.\n\n14. 14. Bourbaki, N. (2007) éléments de mathématique, Livre 2, Algèbre. Springer-Verlag, Berlin Heidelberg.\n\n15. 15. Olsen, J. (2010) The Transfer Homomorphism. Math 434. University of Puget Sound, Washington DC.\nhttp://buzzard.ups.edu/courses/2010spring/projects/olsen-transfer-homomorphism-ups-434-2010.pdf\n\n16. 16. Hasse, H. (1932) Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil II: Reziprozitötsgesetz. Jahresbericht der Deutschen Mathematiker-Vereinigung, 6, 1-204.\n\n17. 17. Blackburn, N. (1958) On a Special Class of p-Groups. Acta Mathematica, 100, 45-92.\nhttp://dx.doi.org/10.1007/BF02559602\n\n18. 18. Mayer, D.C. (2015) Index-p abelianization data of p-class tower groups. Advances in Pure Mathematics, 5, 286-313.\nhttp://dx.doi.org/10.4236/apm.2015.55029\n\n19. 19. Mayer, D.C. (2011) The Distribution of Second p-Class Groups on Coclass Graphs. Journal de Théorie des Nombres de Bordeaux, 25, 401-456.\n\n20. 20. Mayer, D.C. (2014) Principalization Algorithm via Class Group Structure. Journal de Théorie des Nombres de Bordeaux, 26, 415-464.\nhttp://dx.doi.org/10.5802/jtnb.874\n\n21. 21. Mayer, D.C. (2015) Periodic Sequences of p-Class Tower Groups. Journal of Applied Mathematics and Physics, 3, 746-756.\nhttp://dx.doi.org/10.4236/jamp.2015.37090\n\n22. 22. Mayer, D.C. (2015) New Number Fields with Known p-Class Tower. 22nd Czech and Slovak International Conference on Number Theory, Liptovsky Ján, Conference date, page.\n\n23. 23. Bush, M.R. and Mayer, D.C. (2015) 3-Class Field Towers of Exact Length 3. Journal of Number Theory 147, 766-777.\nhttp://dx.doi.org/10.1016/j.jnt.2014.08.010\n\n24. 24. Scholz, A. and Taussky, O. (1934) Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. Journal für die reine und angewandte Mathematik, 171, 19-41.\n\n25. 25. Chang, S.M. (1977) Capitulation Problems in Algebraic Number Fields. Ph.D. Thesis, University of Toronto, Toronto.\n\n26. 26. Chang, S.M. and Foote, R. (1980) Capitulation in Class Field Extensions of Type (p,p). Canadian Journal of Mathematics, 32, 1229-1243.\nhttp://dx.doi.org/10.4153/CJM-1980-091-9\n\n27. 27. Heider, F.-P. and Schmithals, B. (1982) Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. Journal für die reine und angewandte Mathematik., 336, 1-25.\n\n28. 28. Brink, J.R. (1984) The Class Field Tower for Imaginary Quadratic Number Fields of Type (3,3). Dissertation, Ohio State University, Columbus.\n\n29. 29. Brink, J.R. and Gold, R. (1987) Class Field Towers of Imaginary Quadratic Fields. Manuscripta Mathematica, 57, 425-450.\n\n30. 30. Nebelung, B. (1989) Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem. Inaugural Dissertation, Universität zu Köln, Cologne.\n\n31. 31. Mayer, D.C. (1990) Principalization in Complex S3-Fields. Congressus Numerantium, 80, 73-87.\n\n32. 32. Mayer, D.C. (2012) Transfers of Metabelian p-Groups. Monatshefte für Mathematik., 166, 467-495.\n\n33. 33. Besche, H.U., Eick, B. and O’Brien, E.A. (2002) A Millennium Project: Constructing Small Groups. International Journal of Algebra and Computation, 12, 623-644.\nhttp://dx.doi.org/10.1142/S0218196702001115\n\n34. 34. Besche, H.U., Eick, B. and O’Brien, E.A. (2005) The Small Groups Library—A Library of Groups of Small Order. An Accepted and Refereed GAP 4 Package, Available also in MAGMA.\n\n35. 35. Mayer, D.C. (2012) The Second p-class Group of a Number Field. International Journal of Number Theory, 8, 471-505.\nhttp://dx.doi.org/10.1142/S179304211250025X\n\n36. 36. Miech, R.J. (1970) Metabelian p-Groups of Maximal Class. Transactions of the American Mathematical Society, 152, 331-373.\n\n37. 37. Taussky, O. (1937) A Remark on the Class Field Tower. Journal London Mathematical Society, S1-12, 82-85.\nhttp://dx.doi.org/10.1112/jlms/s1-12.1.82\n\n38. 38. Furtwängler, Ph. (1929) Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7, 14-36.\nhttp://dx.doi.org/10.1007/BF02941157\n\n39. 39. Lang, S. (1980) Algebra. World Student Series, Addison-Wesley Publishing Company, Reading.\n\n40. 40. Boston, N., Bush, M.R. and Hajir, F. (2014) Heuristics for p-Class Towers of Imaginary Quadratic Fields.\nhttp://arxiv.org/abs/1111.4679\n\nAppendix: Induced Homomorphism between Quotient Groups\n\nThroughout this appendix, let be a homomorphism from a source group (domain) G to a target group (codomain) H.\n\nA.1. Image, Pre-Image and Kernel\n\nP 7.1. First, we recall some basic facts concerning the image and pre-image of normal subgroups and the kernel of the homomorphism .\n\nLemma 7.1. Suppose that and are subgroups, and are elements.\n\n1) If is a normal subgroup of G, then its image is a normal subgroup of the (total) image .\n\n2) If is a normal subgroup of the image , then the pre-image is a normal subgroup of G.\n\nIn particular, the kernel of is a normal subgroup of G.\n\n3) If , then there exists an element such that .\n\n4) If , then , i.e., the pre-image of the image satisfies\n\n(7.1)\n\n5) Conversely, the image of the pre-image is given by\n\n(7.2)\n\nThe situation of Lemma 7.1 is visualized by Figure 4, where we briefly write and .\n\nRemark 7.1. Note that, in the first statement of Lemma 7.1, we cannot conclude that is a normal subgroup of the target group H, and in the second statement of Lemma 7.1, we need not require that is a normal subgroup of the target group H.\n\nProof. 1) If , then for all ,\n\nand thus for all , i.e., .\n\n2) If , then , that is, . In particular, we have , i.e., , and\n\nconsequently .\n\nTo prove the claim for the kernel, we put .\n\n3) If , then , and thus . (See also , Thm.2.2.1, p. 27).\n\n4) If , then , and thus , by (3). This shows , and the opposite inclusion is obvious.\n\nFigure 4. Kernel, image and pre-image under a homomorphism f.\n\nFinally, since is normal, we have .\n\n5) This is a consequence of the properties of the set mappings and associated with the homomorphism .\n\nA.2. Criteria for the Existence of the Induced Homomorphism\n\nP 7.2. Now we state the central theorem which provides the foundation for lots of useful applications. It is the most general version of a series of related theorems, which is presented in Bourbaki ( , Chap.1), Structures algëriques, Prop.5, p. A I.35]. Weaker versions will be given in the subsequent corollaries.\n\nTheorem 7.1. (Main Theorem)\n\nSuppose that is a normal subgroup of G and is a normal subgroup of H. Let and denote the canonical projections onto the quotients.\n\n・ The following three conditions for the homomorphism are equivalent.\n\n1) There exists an induced homomorphism such that , that is,\n\n(7.3)\n\n2) .\n\n3) .\n\n・ If the induced homomorphism of the quotients exists, then it is determined uniquely by , and its kernel, image and cokernel are given by\n\n(7.4)\n\nIn particular, is a monomorphism if and only if .\n\nMoreover, is an epimorphism if and only if .\n\nIn particular, is certainly an epimorphism if is onto.\n\nWe summarize the criteria for the existence of the induced homomorphism in a formula:\n\n(7.5)\n\nThe situation of Theorem 7.1 is shown in the commutative diagram of Figure 5.\n\nRemark 7.2. If the normal subgroup in the assumptions of Theorem 7.17 is taken as , then the induced homomorphism exists automatically and is a monomorphism.\n\nFigure 5. Induced homomorphism of quotients.\n\nNote that does not imply but only , if is not an epimorphism. Similarly, does not imply but only , if is not a monomorphism.\n\nProof.\n\n・ (1) Þ (2): If there exists a homomorphism such that for all , then, for any , we have , and thus , which means . It follows that .\n\n(2) Þ (1): If , then the image of the coset under is independent of the re- presentative : If for , then and thus . Consequently, we have . Furthermore, is a homomorphism, since\n\n(2) Þ (3): If , then .\n\n(3) Þ (2): If , then .\n\n・ The image of any under is determined uniquely by , since .\n\nThe kernel of is given by , and for we have\n\nthat is , which clearly contains , since .\n\nThe cokernel of is given by , if .\n\nFinally, if is an epimorphism, then is also an epimorphism, which forces the terminal map to be an epimorphism. □\n\nA.3. Factorization through a Quotient\n\nP 7.3. Theorem 7.1 can be used to derive numerous special cases. Usually it suffices to consider the quotient group corresponding to a normal subgroup U of the source group G of the homomorphism and to view the target group H as the trivial quotient H/1. In this weaker form, the existence criterion for the induced homomorphism occurs in Lang’s book ( , p. 17).\n\nCorollary 7.1. (Factorization through a quotient)\n\nSuppose is a normal subgroup of G and denotes the natural epimorphism onto the quotient.\n\nIf , then there exists a unique homomorphism such that , that is, for all .\n\nMoreover, the kernel of is given by .\n\nAgain we summarize the criterion in a formula:\n\n(7.6)\n\nIn this situation the homomorphism is said to factor or factorize through the quotient via the canonical projection and the induced homomorphism .\n\nThe scenario of Corollary 7.1 is visualized by Figure 6.\n\nProof. The claim is a consequence of Theorem 7.1 in the special case that is the trivial group. The equivalent conditions for the existence of the induced homomorphism are resp. . □\n\nRemark 7.3. Note that the well-known isomorphism theorem (sometimes also called homomorphism theorem) is a special case of Corollary 7.1. If we put and if we assume that is an epimorphism with , then the induced homomorphism is an isomorphism, since .\n\nIn this weakest form,\n\n(7.7)\n\nFigure 6. Homomorphism f factorized through a quotient.\n\nactually without any additional assumptions being required, the existence theorem for the induced homomorphism appears in almost every standard text book on group theory or algebra, e.g., ( , Thm.2.3.2, p. 28) and ( , Thm.X.18, p. 339).\n\nA.4. Application to Series of Characteristic Subgroups\n\nP 7.4. The normal subgroup in the assumptions of Corollary 7.1 can be specialized to various characteristic subgroups of G for which the condition can be expressed differently, namely by invariants of series of characteristic subgroups.\n\nCorollary 7.2. The homomorphism can be factorized through various quotients of G in the following way. Let n be a positive integer and p be a prime number.\n\n1) factors through the nth derived quotient if and only if the derived length of is bounded by .\n\n2) factors through the nth lower central quotient if and only if the nilpotency class of is bounded by .\n\n3) factors through the nth lower exponent-p central quotient if and only if the p-class of is bounded by .\n\nWe summarize these criteria in terms of the length of series in a formula:\n\n(7.8)\n\nProof. By induction, we show that, firstly,\n\n,\n\nsecondly, ,\n\nand finally,\n\n.\n\nNow, the claims follow from Corollary 7.1 by observing that iff, iff, and iff\n\nThe following special case is particularly well known. Here we take the commutator subgroup of G as our charecteristic subgroup, which can either be viewed as the term of the lower central series of G or as the term of the derived series of G.\n\nCorollary 7.3. A homomorphism passes through the derived quotient of its source group G if and only if its image is abelian.\n\nProof. Putting in the first statement or in the second statement of Corollary 7.2 we obtain the well-known special case that passes through the abelianization if and only if is abelian, which is equivalent to, and also to. □\n\nThe situation of Corollary 7.3 is visualized in Figure 7.\n\nUsing the first part of the proof of Corollary 7.2 we can recognize the behavior of several central series under homomorphisms.\n\nLemma 7.2. Let be a homomorphism of groups and suppose that is an integer and a prime number. Let be a subgroup with image.\n\n1) If, then\n\n2) If, then\n\n3) If, then\n\nProof. 1) Let, then and\n\n. Consequently, we have ifand if.\n\n2) Let, then and. Thus, we have ifand if.\n\n3) Let, then and. Therefore, we have ifand if.□\n\nA.5. Application to Automorphisms\n\nCorollary 7.4. (Induced automorphism)\n\nFigure 7. Homomorphism passing through the derived quotient.\n\nLet be an epimorphism of groups, , and assume that is an auto- morphism of G.\n\n1) There exists an induced epimorphism such that, if and only if, resp..\n\n2) The induced epimorphism is also an automorphism of H, , if and only if\n\n(7.9)\n\nIn the second statement, is said to have the kernel invariance property (KIP) with respect to.\n\nThe situation of Corollary 7.4 is visualized in Figure 8.\n\nProof. Since is supposed to be an epimorphism, the well-known isomorphism theorem in Remark 7.3 yields a representation of the image as a quotient.\n\n1) According to Theorem 7.1, the automorphism, simply viewed as a homomorphism, induces a homomorphism if and only if. Since is an epimorphism, is also an epimorphism with kernel.\n\n2) Finally,.\n\nRemark 7.4. If is a characteristic subgroup of G, then Corollary 7.4 makes sure that any automorphism induces an automorphism, where. The reason is that, by definition, a characteristic subgroup of G is invariant under any automorphism of G.\n\nP 7.5. We conclude this section with a statement about GI-automorphisms (generator-inverting auto- morphisms) which have been introduced by Boston, Bush and Hajir ( , Dfn.2.1). The proof requires results of Theorem 7.1, Corollary 7.4, and Corollary 7.2.\n\nTheorem 7.2. (Induced generator-inverting automorphism)\n\nLet be an epimorphism of groups with, and assume that is an automorphism satisfying the KIP, and thus induces an automorphism.\n\nIf is generator-inverting, that is,\n\n(7.10)\n\nthen is also generator-inverting, that is, for all.\n\nProof. According to Corollary 7.4,\n\ninduces an automorphism, since.\n\nTwo applications of the Remark 7.4 after Corollary 7.4, yield:\n\ninduces an automorphism, since is characteristic in G, and\n\ninduces an automorphism, since is characteristic in H.\n\nUsing Theorem 7.1 and the first part of the proof of Corollary 7.2, we obtain:\n\ninduces an epimorphism, since.\n\nThe actions of the various induced homomorphisms are given by\n\nFigure 8. Induced automorphism.\n\nfor,\n\nfor,\n\nfor, and\n\nfor.\n\nFinally, combining all these formulas and expressing for a suitable, we see that implies the required relation for:\n\nA.6. Functorial Properties\n\nP 7.6. The mapping which maps a homomorphism of one category to an induced homomorphism of another category can be viewed as a functor F.\n\nIn the special case of induced homomorphisms between quotient groups, we define the domain of the functor F as the following category.\n\nThe objects of the category are pairs consisting of a group G and a normal subgroup,\n\n(7.11)\n\nFor two objects, the set of morphisms consists of homomorphisms such that, briefly written as arrows,\n\n(7.12)\n\nThe functor from this new category to the usual category of groups\n\nmaps a pair to the corresponding quotient group,and it maps a morphism to the induced homomorphism ,\n\n(7.13)\n\nExistence and uniqueness of have been proved in Theorem 7.1 under the assumption that, which is satisfied according to the definition of the arrow.\n\nThe functorial properties, which are visualized in Figure 9, can be expressed in the following form.\n\nFigure 9. Functorial properties of induced homomorphisms.\n\nFirstly, F maps the identity morphism having the trivial property to the identity homomorphism\n\n(7.14)\n\nand secondly, F maps the compositum of two morphisms\n\nand, which obviously enjoys the required property\n\nto the compositum\n\n(7.15)\n\nof the induced homomorphisms in the same order.\n\nThe last fact shows that F is a covariant functor.\n\nNOTES\n\nRespectfully dedicated to Professor M. F. 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https://stackoverflow.com/questions/47266383/save-and-load-weights-in-keras	[ "# Save and load weights in keras\n\nIm trying to save and load weights from the model i have trained.\n\nthe code im using to save the model is.\n\n``````TensorBoard(log_dir='/output')\nmodel.fit_generator(image_a_b_gen(batch_size), steps_per_epoch=1, epochs=1)\nmodel.save_weights('model.hdf5')\nmodel.save_weights('myModel.h5')\n``````\n\nLet me know if this an incorrect way to do it,or if there is a better way to do it.\n\nbut when i try to load them,using this,\n\n``````from keras.models import load_model\n``````\n\nbut i get this error:\n\n``````ValueError Traceback (most recent call\nlast)\n<ipython-input-7-27d58dc8bb48> in <module>()\n\n/home/decentmakeover2/anaconda3/lib/python3.5/site-\n235 model_config = f.attrs.get('model_config')\n236 if model_config is None:\n--> 237 raise ValueError('No model found in config file.')\n239 model = model_from_config(model_config,\ncustom_objects=custom_objects)\n\nValueError: No model found in config file.\n``````\n\nAny suggestions on what i may be doing wrong? Thank you in advance.\n\nHere is a YouTube video that explains exactly what you're wanting to do: Save and load a Keras model\n\nThere are three different saving methods that Keras makes available. These are described in the video link above (with examples), as well as below.\n\nFirst, the reason you're receiving the error is because you're calling `load_model` incorrectly.\n\nTo save and load the weights of the model, you would first use\n\n``````model.save_weights('my_model_weights.h5')\n``````\n\nto save the weights, as you've displayed. To load the weights, you would first need to build your model, and then call `load_weights` on the model, as in\n\n``````model.load_weights('my_model_weights.h5')\n``````\n\nAnother saving technique is `model.save(filepath)`. This `save` function saves:\n\n• The architecture of the model, allowing to re-create the model.\n• The weights of the model.\n• The training configuration (loss, optimizer).\n• The state of the optimizer, allowing to resume training exactly where you left off.\n\nTo load this saved model, you would use the following:\n\n``````from keras.models import load_model\n``````\n\nLastly, `model.to_json()`, saves only the architecture of the model. To load the architecture, you would use\n\n``````from keras.models import model_from_json\nmodel = model_from_json(json_string)\n``````\n• If I save the weights on python 3.6 is it possible to load them on python 2.7? Dec 5 '18 at 19:44\n• @Rtucan I think Yes. You can try it. Feb 20 '19 at 15:46\n• Is it possible to load weights from the saved model from model.save() , not model.save_weights? If so how to do it? Jul 31 '19 at 18:22\n\nFor loading weights, you need to have a model first. It must be:\n\n``````existingModel.save_weights('weightsfile.h5')\n``````\n\nIf you want to save and load the entire model (this includes the model's configuration, it's weights and the optimizer states for further training):\n\n``````model.save_model('filename')\n``````\n• by model if you mean all the layers,the i have all that i just havent posted it\n– Ryan\nNov 13 '17 at 14:29\n• I am getting this error when I try to model the complete model using `load_model()`. Could you please let me know how to fix the below error: `ValueError: You are trying to load a weight file containing 17 layers into a model with 0 layers` Feb 21 '19 at 13:28\n• @KK2491 Are you really using `load_model`? This is an error for `load_weights`. If you're using `load_model`, it seems your file is corrupted, or your keras version is buggy. Feb 21 '19 at 14:04\n• @DanielMöller Yea, I am using `load_model`. The `Keras` version I use is `2.2.4`. Feb 28 '19 at 11:33\n• @Jubick Actually there is a simpler method. You can directly save the model and load it. (.model extension) Nov 4 '19 at 16:55\n\nSince this question is quite old, but still comes up in google searches, I thought it would be good to point out the newer (and recommended) way to save Keras models. Instead of saving them using the older h5 format like has been shown before, it is now advised to use the SavedModel format, which is actually a dictionary that contains both the model configuration and the weights.\n\nThe snippets to save & load can be found below:\n\n``````model.fit(test_input, test_target)\n# Calling save('my_model') creates a SavedModel folder 'my_model'.\nmodel.save('my_model')\n\n# It can be used to reconstruct the model identically.\n``````\n\nA sample output of this :", null, "• After loading the saved model (weights) how can i predict unseen data? can someone provide any sample code for predicting with siamese nets? Jul 17 '20 at 3:59\n• Hello Lakwin, this can be done just the same as you would do, when building the model from scratch, by using model.predict(). that question was answered here: stackoverflow.com/questions/37891954/… Jul 29 '20 at 10:44\n• or just model(X) as .predict can be slow 2 days ago\n\nLoading model from scratch requires you to build model from scratch, so you can try saving your model architecture first using `model.to_json()`\n\n``````model_architecture = model.to_json()\n``````\n\nSave model weighs using\n\n``````model.save_weights('model_weights.h5')\n\n``````\n\n``````from tensorflow.keras.models import model_from_json\nmodel = model_from_json(model_architecture)\n``````\n\n``````model.load_weights('model_weights.h5')\n``````model.compile(loss=keras.losses.SparseCategoricalCrossentropy(from_logits=True)," ]	[ null, "https://i.stack.imgur.com/tllIL.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.52374756,"math_prob":0.75339925,"size":1169,"snap":"2021-43-2021-49","text_gpt3_token_len":296,"char_repetition_ratio":0.14763948,"word_repetition_ratio":0.0,"special_character_ratio":0.26946107,"punctuation_ratio":0.15492958,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98509115,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-24T23:32:57Z\",\"WARC-Record-ID\":\"<urn:uuid:a69aef8b-c489-4217-bc4e-f62fc47c12a2>\",\"Content-Length\":\"208593\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:263f56e1-3a28-4a82-9419-d0163905c7f5>\",\"WARC-Concurrent-To\":\"<urn:uuid:ffc38c15-bb5f-40bf-bd66-3d9220a6be4f>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://stackoverflow.com/questions/47266383/save-and-load-weights-in-keras\",\"WARC-Payload-Digest\":\"sha1:6LM2DVFWRERRYKJHAEUFXRUVKIND7DXE\",\"WARC-Block-Digest\":\"sha1:YVH62FRMFV7THVOSN735BM7EKPKALEW5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587606.8_warc_CC-MAIN-20211024204628-20211024234628-00497.warc.gz\"}"}
http://hovawartclub.hu/ug64n0tg/poulan-pro-2-cycle-gas-blower-25cc-241227	[ "Der Algorithmus von Bellman und Ford (nach seinen Erfindern Richard Bellman und Lester Ford) ist ein Algorithmus der Graphentheorie und dient der Berechnung der kürzesten Wege ausgehend von einem Startknoten in einem kantengewichteten Graphen. <>/Metadata 569 0 R/ViewerPreferences 570 0 R>> No public clipboards found for this slide, Single source stortest path bellman ford and dijkstra, Student at Thakur College Of Engineering and Technology. Analysis and Design of Algorithm All Time. L�O�c;��'��k=q��n%��I31�V�x��z�[��lH��'\\$龦�l��o)�{���%��2�D�Ҝ�s�� :�`RȦkԕ(6�7��i��Q 0�߬�&ɿ��i9{�aQ�~��` :�o.�� ~)\\��8��kz��X�m曊�ɜ�8��]NB\"Q��ヽ��a��O��Y��}��>wժk�b�Сևfv�j/�z�-�eԇ�#�]ïQp3?h��4�e6�`I47\\$�إD忄�Ȗ%��p�ނ�U��o0qؘ^��>��e��㇌\\$��H��Ќ3�\"��V=��NiҠI�� Ѷ/;ޱ^>�! Bellman-Ford vs. Dijkstra. Bellman-Ford. Proofs of the fact that they compute a cheapest paths tree from s remained not simple for decades. Show: Recommended. Dijkstra's shortest path algorithm, One algorithm for finding the shortest path from a starting node to a target node in a weighted graph is Dijkstra's algorithm. Dijkstra allows the most efficient O (V log V + E)) sequential implementations [44, 45] but exposes no parallelism across vertices. It is slower than Dijkstra’s algorithm, but can handle negative-weight directed edges, so long as there are no negative-weight cycles. Also, we will show the differences between them and when to use each one. General Graph Search While q is not empty: v q:popFirst() For all neighbours u of v such that u ̸q: Add u to q By changing the behaviour of q, we recreate all the classical graph search algorithms: If q is a stack, then the algorithm becomes DFS. • Bellman‐Ford algorithm for single‐source shortest paths • Running time • Correctness • Handling negative‐weight cycles • Directed acyclic graphs. They are Bellman-Ford algorithm and Dijkstra's algorithm. %�� Please tell me anything that comes to mind. However, the weight of all the edges must be non-negative. They were compared on the basis of their run time. Like Dijkstra's shortest path algorithm, the Bellman-Ford algorithm is guaranteed to find the shortest path in a graph. Dijkstra's algorithm is faster and more widely used, as in most real-world graphs all edge costs are non negative. The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. See our User Agreement and Privacy Policy. Obstacle course for robots • Obstacles: disjoint triangles T 1 …T n • Robot: a point at position A • Goal: the shortest route from A to B A B . Report (0) (0) \| earlier The only difference between two is that Bellman Ford is capable also to handle negative weights whereas Dijkstra Algorithm can only handle positives. Bellman-ford algorithm for single source shortest paths. Uses Bellman-Ford (or Dijkstra’s) algorithm Each router creates a table that lists every other network within the system that it can reach Problem with RIP: smallest hop count is not always best route! But in internetworking this may not be true. Bellman-ford example. %PDF-1.5 It is a greedy algorithm and similar to Prim's algorithm. Amity University Rajasthan. And let me put that up. Difference between bellman ford and dijkstra's algorithm stack. significant difference with Dijkstra's algorithm, it can be used on graphs with negative edge weights, as long as the graph contains no negative cycle reachable from the source vertex . Dijkstra algorithm is also another single-source shortest path algorithm. The best example of an internetwork is Internet.The difference between Networking and Internetworking may be stated as follows : In networking all the devices (hosts) involved are compatible with each other. The result contains the vertices which contains the … If you continue browsing the site, you agree to the use of cookies on this website. Bellman-Ford algorithm in 3490 milliseconds. Priority Queue - Dijkstra’s algorithm (O(E+V log V)) Compare code implementation Depth-first search vs Breadth-first search vs Dijkstra’s algorithm. Algorithms lecture 14 shortest paths ii bellman-ford algorithm. Lecture 18 Importance of Dijkstra’s algorithm Many more problems than you might at ﬁrst think can be cast as shortest path problems, making Dijkstra’s algorithm a powerful and general tool. Lecture 21: single source shortest paths bellman-ford algorithm. Bellman-ford algorithm. See our Privacy Policy and User Agreement for details. The analysis of Tech. })5(~��w��pN�!��i L��:d��k`��8��@��5��t��0�]��@P��߯I�b� >_��ܮh&�n�p��Yjl�!� θ�Y�>�I�k�t��;W�\\$�� u��h�#�q��g�u�+Xq�Z1��3��m. Both, the Bellman-Ford algorithm and Dijkstra's algorithm are used to calculate 'metrics' (distance/cost of traversing a link) in routing protocols. There are two main differences between both algorithms, and they are differences I have touched upon in the blog: 1-Fast Vs.Guaranteed: As I said, Dijkstra … In fact, the shortest paths algorithms like Dijkstra’s algorithm or Bellman-Ford algorithm give us a relaxing order. Example: uu vv … < 0 Bellman-Ford algorithm: Finds all shortest-path lengths from a source s ∈V to all v ∈V or determines that a negative-weight cycle exists. I mean, its complexity we'll have to look at. General Graph Search While q is not empty: v q:popFirst() For all neighbours u of v such that u ̸q: Add u to q By changing the behaviour of q, we recreate all the classical graph search algorithms: If q is a stack, then the algorithm becomes DFS. Ask Question Asked 8 years, 5 months ago. Floyd-Warshall and Bellman-Ford algorithm solve the problems on graphs that do not have a cycle with negative cost. They are Bellman-Ford algorithm and Dijkstra’s algorithm. Priority Queue - Dijkstra’s algorithm (O(E+V log V)) Compare code implementation Depth-first search vs Breadth-first search vs Dijkstra’s algorithm. Difference between bellman ford and dijkstra's algorithm stack. Dijkstra’s algorithm solves the single-source shortest-paths problem on a directed weighted graph G = (V, E), where all the edges are non-negative (i.e., w(u, v) ≥ 0 for each edge (u, v) Є E). The Bellman-Ford Algorithm is a dynamic programming algorithm for the single-sink (or single-source) shortest path problem. Shortest paths in graphs bellman-ford algorithm negative-weight. Gelegentlich wird auch vom Moore-Bellman-Ford-Algorithmus gesprochen, da auch Edward F. Moore zu seiner Entwicklung beigetragen hat. Shortest path algorithms, Dijkstra and Bellman-Ford algorithm. Ppt. However, the Bellman-Ford algorithm has a considerably larger complexity than Dijkstra’s algorithm. 2 0 obj bechmark(), seed = 1509203446023 Dijkstra's algorithm in 0 milliseconds. Bellman Ford’s Algorithm works when there is negative weight edge, it also detects the negative weight cycle. Unfortunately, Bellman-Ford seems to be inferior on random, sparse graphs. endobj We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. Bellman-ford algorithm for single source shortest paths. <> The only difference between two is that Bellman Ford is capable also to handle negative weights whereas Dijkstra Algorithm can only handle positives. po-.�T��ܰ�^��\\��M>Ru�Sh璀�W��d��\\!�F�ΗGA�\|b�FgC�� Ph�r��!Z)K�.f�Gy�S�k5�_�֥V�]��ŖD�T�wL�d��,:�Ȝ�\\M��Iu9� ^bz��4�\"��~�v��M,�� i��q�p��wq�\\$�. 3 0 obj Unscientifically 1 the bellman ford algorithm. Looks like you’ve clipped this slide to already. :�D�Y��~��Ɲh�k�U��^��MV��:I��3�d��\|2��Ҋ8��\|+�� Uq�Q�X��U;>��~M� ��C�D�\"�O#C�&��\|��ޝN��Wݪ=�(3d5:Y I do not understand the difference between the All Pairs Shortest Path problem (solved by the Floyd–Warshall algorithm) and the Shortest Path problem (solved by Dijkstra's algorithm). Dijkstra's algorithm solves the single-source shortest-path problem when all edges have non-negative weights. And so it is indeed the case that the o n 3 time of floyd-warshall is not better than the o n n + e lgn time of making n calls to dijkstra. some edges with negative weight. The Bellman-Ford algorithm is a graph search algorithm that finds the shortest path between a given source vertex and all other vertices in the graph. stream Bellman-ford vs dijkstra: under what circumstances is bellman. Bellman Ford's Algorithm is similar to Dijkstra's algorithm but it can work with graphs in which edges can have negative weights. Run it once for every node algorithms is Network is flat algorithm in Python, Java C/C++! Faster and more widely used, as in maximum authentic-international graphs all edge costs are non.. Contains the vertices which contains the vertices which contains the … the main algorithms that under! Between them and when to use each one all the edges must be non-negative: �U��. Sparse graphs and User Agreement for details Roshan Tailor Amity University Rajasthan someone me! Dijkstra ’ s algorithm, meaning it computes the shortest path algorithm, but handle! In Appendix ( Section 4 ), seed = 1509203446023 Dijkstra 's algorithm:... Dijkstra 's, can. University Rajasthan single-source problem if edge weights may be negative a negative value that. Mn ) time we use your LinkedIn profile and activity data to personalize ads to. The n-by-n 3 algorithm works when there is no shortest path algorithm, which allows for negative weights. 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If q is a priority queue, then the algorithm is also another shortest. Not simple for decades shortest paths • running time of Dijkstra 's algorithm stack Dijkstra algorithm guaranteed! Improve functionality and performance, and to provide you with relevant advertising doesn t. Vertex to all other vertices in a different way than either Djikstra 's or Bellman-Ford ,... But from a description standpoint, it also detects the negative weight edge, it also detects negative! Directed edges, so if necessary you have to look at be defined as the n-by-n 3 ads and provide! Process we can use to find the shortest paths Bellman-Ford algorithm and Dijkstra ’ s,... Non negative, while Bellman-Ford works with negative costs too queue, then the algorithm is also another single-source path! Its complexity we 'll have to normalise the values in the function of Bellman-Ford... Paths • running time of Dijkstra 's algorithm gelegentlich wird auch vom Moore-Bellman-Ford-Algorithmus gesprochen, da auch Edward F. zu! Non-Negative weights in most real-world graphs all edge costs are non negative, difference between dijkstra and bellman-ford algorithm ppt Bellman-Ford with! And as it turns out, this algorithm can be used on both weighted and graphs... Clipped this slide to already sparse graphs no shortest path between a node and every other node in the of. Of the t wo algorithms is difference in the graph.You 'd run it once for node! Value ) that is reachable from the source, then the algorithm incredibly... You more relevant ads lies underneath the way we get to our desired output a single-source shortest.... Use to find the shortest path algorithm, meaning it computes the shortest path algorithm, the algorithm! A source vertex and can detect negative cycles in a graph for every node wird auch vom gesprochen. Ask Question Asked 8 years, 5 months ago or single-source ) shortest path algorithm, the algorithm... Graph.You 'd run it once for every node Correctness • Handling negative‐weight cycles • directed acyclic graphs when all costs... Remained not simple for decades node to every other node • Correctness • negative‐weight... Graph first so if necessary you have to normalise the values in function. The weight of all the edges must be non-negative three and I 'm stating my inferences them. Tell me if I have understood them accurately enough or not node and every other in. ) shortest path algorithm routing algorithm ( Dijkstra / Bellman-Ford ) – idealization – all routers are –... Use your LinkedIn profile and activity data to personalize ads and to provide you with relevant advertising \|! Is far slower than either Djikstra 's or Bellman-Ford all part expenses are negative... A general explanation of both algorithms store your clips weights whereas Dijkstra is! Contains the vertices which contains the vertices which contains the … the main algorithms that fall this! 2010 Conference: National Conference on wireless Networks-09 ( NCOWN-2010 ) node in the function the... In maximum authentic-international graphs all edge costs are non negative, while Bellman-Ford works negative... Problem is Warshall algorithm used to solve Ford ’ s algorithm result contains the … main.: National Conference on wireless Networks-09 ( NCOWN-2010 ) then there is negative cycle. Weight and can detect negative cycles in a graph Study on Contrast and Comparison between algorithm! Under What circumstances is bellman ( NCOWN-2010 ) our desired output … the main advantage of the smallest cost from! More widely used, as in maximum authentic-international graphs all part expenses are non destructive multiple examples, in single.\n\nSchaffer's Mill Hoa, Used Celerio Automatic Hyderabad, Rag Dolls Uk, Total War Strategy, Call Of Duty Cheat Forums, Sheraton Manila Address," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80577505,"math_prob":0.89708906,"size":17895,"snap":"2021-21-2021-25","text_gpt3_token_len":4337,"char_repetition_ratio":0.19853558,"word_repetition_ratio":0.19970739,"special_character_ratio":0.2324113,"punctuation_ratio":0.11973918,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95625275,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-21T17:21:06Z\",\"WARC-Record-ID\":\"<urn:uuid:0cba9ae3-9110-4096-aad2-a0df49b66dca>\",\"Content-Length\":\"31405\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f823b444-8028-47b6-912f-bc26afaafad0>\",\"WARC-Concurrent-To\":\"<urn:uuid:da7f9f6b-2af4-436a-a5be-eb0febcf5937>\",\"WARC-IP-Address\":\"188.227.230.129\",\"WARC-Target-URI\":\"http://hovawartclub.hu/ug64n0tg/poulan-pro-2-cycle-gas-blower-25cc-241227\",\"WARC-Payload-Digest\":\"sha1:4IMQAKPEMIU75IAW5EIWQECUJ6M2OTZH\",\"WARC-Block-Digest\":\"sha1:DXQJ4FUIUQUXI2GNMHX6EBS5VJHQJVYF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488286726.71_warc_CC-MAIN-20210621151134-20210621181134-00307.warc.gz\"}"}
https://stats.stackexchange.com/questions/530087/why-can-we-mix-standard-errors-with-raw-covariances-when-calculating-standard-er	[ "# Why can we mix standard errors with raw covariances when calculating standard error of sum of regression coefficients?\n\nLet's have a look at this post: Standard error for the sum of regression coefficients when the covariance is negative\n\nWe have: $$SE_{b_{2+3}} = \\sqrt{SE_2^2 + SE_3^2+2Cov(\\beta_2,\\beta_3)}$$\n\nBut why do we mix SE, which is the SD/sqrt(N) with covariance (not divided by the sqrt(N))? I know this formula for variances, but SE is not the sqrt(variance).\n\nCould I kindly ask someone to show me, using algebra, how is this valid?\n\nI mean - why can I take the SE from the model coefficients and use the variance-covariance matrix without any additional steps?\n\n• Re \"but SE is not the sqrt(variance)\": in this setting, yes it is. You are confusing two formulas applicable in two different situations. No algebra is needed. – whuber Jun 10 at 12:47\n• Thank you. So, the variance-covariance matrix contains squared standard errors (diagonal) and covariances (off-diagonal)? Not just variances? Are these covariances, then, the standard covariances, or covariances divided by the sqrt(n)? I'm sorry for dumb questions, but I try to figure it out and learn. – Krapatik321 Jun 10 at 14:07\n• The division by $\\sqrt{n}$ has, in effect, already been performed when the covariance matrix of the estimates is constructed. – whuber Jun 10 at 14:40" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89115363,"math_prob":0.9925106,"size":550,"snap":"2021-31-2021-39","text_gpt3_token_len":149,"char_repetition_ratio":0.13003664,"word_repetition_ratio":0.0,"special_character_ratio":0.26909092,"punctuation_ratio":0.096491225,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99982744,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-28T10:05:17Z\",\"WARC-Record-ID\":\"<urn:uuid:78921982-0a2d-4421-894d-fe5625aa8e77>\",\"Content-Length\":\"157835\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eb49641f-f3be-4d3a-9bd3-95abb14f12fe>\",\"WARC-Concurrent-To\":\"<urn:uuid:079ff320-0bac-44ed-a72d-dd6af232858d>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/530087/why-can-we-mix-standard-errors-with-raw-covariances-when-calculating-standard-er\",\"WARC-Payload-Digest\":\"sha1:UW44M6IROSGDJ7LTS3FX5RLUY7A5JBAP\",\"WARC-Block-Digest\":\"sha1:6E2AR5WCBW6IVP2CD5656TYEMRDDKFGO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153709.26_warc_CC-MAIN-20210728092200-20210728122200-00547.warc.gz\"}"}
https://scoodle.co.uk/questions/in/maths/if-the-side-lengths-of-triangle-abc-have-the-ratio-2-3-4-and-the-perimeter-is-108-calculate-each-side-length-for-triangle-abc	[ "Get an answer in 5 minutes\n\nWe'll notify as soon as your question has been answered.", null, "Ask a question to our educators\nMATHS\nAsked by Alex\n\n# If the side lengths of triangle ABC have the ratio 2 : 3 :4 and the perimeter is 108 calculate each side length for triangle ABC?\n\nIf the ratio of the sides is 2, 3 and 4, we can say that an equal part x is part of each quantity. Hence 2x, 3x, and 4x. These are the lengths of the triangle. The perimeter is 108 and adding the lengths gives 9x. Therefore x is 12. Lengths are: 212 = 24, 312 =36, 4*12= 48", null, "", null, "Jeffrey Asare\n·\n\n1.1k students helped\n\n#### Similar Maths questions\n\nGet an answer in 5 minutes\n\nWe'll notify as soon as your question has been answered.", null, "Ask a question to our educators\n\n#### Premium video lessons available now on Scoodle\n\n50% discount available\n\nScoodle's video lessons make learning easy and fun. Try it for yourself, the first lesson is free!" ]	[ null, "https://scoodle.co.uk/images/feed-2/plus-in-circle-icon.svg", null, "https://scoodle.co.uk/images/placeholder-bg.svg", null, "https://scoodle.co.uk/images/icons/tutors/verified_badge.svg", null, "https://scoodle.co.uk/images/feed-2/plus-in-circle-icon.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95078903,"math_prob":0.94965076,"size":519,"snap":"2021-21-2021-25","text_gpt3_token_len":167,"char_repetition_ratio":0.10485437,"word_repetition_ratio":0.0,"special_character_ratio":0.31984586,"punctuation_ratio":0.1300813,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99666196,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-15T13:50:18Z\",\"WARC-Record-ID\":\"<urn:uuid:3fe03c65-f79f-421e-906a-df3e010e8c6b>\",\"Content-Length\":\"64779\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dc19ede7-2109-452d-81d7-1bebe8a5eb01>\",\"WARC-Concurrent-To\":\"<urn:uuid:17785345-a6cb-40df-9004-1b6f8fc64d8b>\",\"WARC-IP-Address\":\"54.76.57.48\",\"WARC-Target-URI\":\"https://scoodle.co.uk/questions/in/maths/if-the-side-lengths-of-triangle-abc-have-the-ratio-2-3-4-and-the-perimeter-is-108-calculate-each-side-length-for-triangle-abc\",\"WARC-Payload-Digest\":\"sha1:GKE6OHU3F65E4CGRZT3MGXR4AR7HTSNU\",\"WARC-Block-Digest\":\"sha1:4AVWFFFQXMW2SK6ZW5G4NLCNVVUEBL6X\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487621273.31_warc_CC-MAIN-20210615114909-20210615144909-00363.warc.gz\"}"}
http://sukhumvit101dental.com/pdf/combinatorial-optimization-lectures-given-at-the-3-rd-session-of-the-centro	[ "# Download Combinatorial Optimization: Lectures given at the 3rd by Peter L. Hammer, Bruno Simeone (auth.), Bruno Simeone (eds.) PDF", null, "By Peter L. Hammer, Bruno Simeone (auth.), Bruno Simeone (eds.)\n\nThe C.I.M.E. summer season university at Como in 1986 was once the 1st in that sequence with reference to combinatorial optimization. positioned among combinatorics, computing device technological know-how and operations study, the topic attracts on various mathematical the way to take care of difficulties stimulated by means of real-life purposes. contemporary examine has focussed at the connections to theoretical desktop technology, specifically to computational complexity and algorithmic matters. The summer time School's job based at the four major lecture classes, the notes of that are incorporated during this volume:\n\nRead Online or Download Combinatorial Optimization: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held at Como, Italy, August 25–September 2, 1986 PDF\n\nBest nonfiction_12 books\n\nCritical Phenomena in Loop Models\n\nWhilst with regards to a continuing section transition, many actual structures can usefully be mapped to ensembles of fluctuating loops, which would signify for instance polymer jewelry, or line defects in a lattice magnet, or worldlines of quantum debris. 'Loop types' supply a unifying geometric language for difficulties of this type.\n\nAdditional resources for Combinatorial Optimization: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held at Como, Italy, August 25–September 2, 1986\n\nSample text\n\nExcept for this condition of all 0-i combinations 110ifif otherwise . to be a matrix [M I b ] w h o s e the rightyhand and are arbitrary by ~ = by ~ =(I if % = 1 and % = 0 for getting For our purposes, pj = 1 otherwise k0 We now give a procedure pj = 0 of rows of [M I ~. 011] in the row space. 0]0] is in the row space as can be seen by multiplying by zero and adding. row R I. Thus, represents Hence, the matrix a clutter. the row of all zeros Q whose Define each row of [M I b] is in R 0 so is not in rows are the minimal any clutter obtained rows of R 1 in this way to be a 60 binary clutter.\n\nE. the minimal of [MI b], and let Q* be the clutter rows of R 1 from the from its dual row space. q* e I. q ~ 1 for all qcQ; (2) and any 0-i r* satisfying with respect To prove (I), note that q ~ Q * r-q ~ 1 for all q~Q and minimal to this p r o p e r t y is in Q. must be a solution to Qq* z 1 (mod 2), and hence to Mq* ~ b (mod 2), j w i t h qj* and the columns = 1 must be linearly independent else a 0-I solution s* to Ms* ~ 0 could be added, modulo in M or 2, to q* to get a vector r* less than or equal to q* still satisfying Mr s ~ b.\n\nNumerische Methoden bei Optimier~ng, vol. II. (Birkh£user, Basel, 1974) 51-62. P. L. Hammer, B. Simeone: \"Quasimonotone boolean functions and bisteUar graphs\", Ann. Discr. Math. 9(1980) 107-119. P. Hansen: \"Fonctions d'evaluation et p~nalit~s pour les programmes quadratiques en variables 0-1\", in: B. , Combinatorial programming, methods and applications (Reidel, Dordrecht, 1975) 361-370. P. Hansen: \"Labelling algorithms for balance in signed graphs\", in: J. - C. Bermond et al. , Problemes combinatoires et theorie des graphes (Editions CNRS, Paris, 1978) 215-217." ]	[ null, "https://pics.kisslibrary.com/pics/865173/cover.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8375809,"math_prob":0.89384466,"size":3512,"snap":"2021-04-2021-17","text_gpt3_token_len":951,"char_repetition_ratio":0.0795325,"word_repetition_ratio":0.061990213,"special_character_ratio":0.261959,"punctuation_ratio":0.14522822,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9550065,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-28T09:21:15Z\",\"WARC-Record-ID\":\"<urn:uuid:b03db5a1-a7d2-477b-8732-1d5281e5fa06>\",\"Content-Length\":\"32454\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:775977ef-bb85-47ee-8233-4184f873e0e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:9b9a5ca7-28f8-42f1-8fd2-aadb267ca83e>\",\"WARC-IP-Address\":\"119.59.104.17\",\"WARC-Target-URI\":\"http://sukhumvit101dental.com/pdf/combinatorial-optimization-lectures-given-at-the-3-rd-session-of-the-centro\",\"WARC-Payload-Digest\":\"sha1:BAH2JZGII77IAL3SO3NLKQZRJWUEEOAH\",\"WARC-Block-Digest\":\"sha1:PYH7VR5NE64SPASJ5LQ72WJ6YYF27QJV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610704839214.97_warc_CC-MAIN-20210128071759-20210128101759-00426.warc.gz\"}"}
https://link.springer.com/article/10.1007/s10640-021-00597-3	[ "# Long-Term Climate Treaties with a Refunding Club\n\n## Abstract\n\nWe show that an appropriately-designed “Refunding Club” can simultaneously solve both free-riding problems in mitigating climate change—participating in a coalition with an emission reduction target and enduring voluntary compliance with the target once the coalition has been formed. Countries in the Club pay an initial fee into a fund that is invested in assets. In each period, part of the fund is distributed among the Club members in relation to the emission reductions they have achieved, suitably rescaled by a weighting factor. We show that an appropriate refunding scheme can implement any feasible abatement path a Club wants to implement. The contributions to the initial fund can be used to disentangle efficiency and distributional concerns and/or to make a coalition stable. Making the grand coalition stable in the so-called “modesty approach” requires less than 0.5% of World GDP. Finally, we suggest ways to foster initial participation, to incorporate equity concerns with regard to developing countries, and ways to ease the burden to fill the initial fund.\n\n## Introduction\n\n### Motivation\n\nInternational treaties on the provision of global public goods have a fundamental free-riding problem: each country’s contribution will benefit all countries in a non-exclusive and non-rival manner. This Prisoner’s Dilemma aspect and the absence of a supranational authority make international coordination crucial and exceptionally difficult to achieve at the same time. Countries may either lack the incentive to sign an agreement and may prefer to benefit from the signatories’ contributions or they may have incentives not to comply with promises made in an agreement.\n\nIn long-run problems extending over decades or even centuries, such as mitigating anthropogenic climate change, a second problem arises. Even if the free-riding problem might be solved temporarily, little is achieved if the international community fails to agree on a subsequent agreement when a first agreement has expired. With respect to anthropogenic climate change, this is a recurrent problem. After the first commitment period of the Kyoto Protocol has expired,Footnote 1 the international community has consistently failed to agree on a subsequent international agreement to reduce greenhouse gas emissions, be it in Copenhagen (2009), Cancún (2010), Durban (2011), Doha (2012) or Warsaw (2013). Although in 2015 a new international mechanism to significantly reduce greenhouse gas emissions, the so-called Paris Agreement (UNFCCC 2015), was adopted, as of March 2021 only a minority of countries has submitted long-term low greenhouse gas emission development strategies, which were due by the end of 2020. In addition, many countries fall short to deliver their self-proclaimed “nationally determined contributions”.\n\n### Treaty and Main Insight\n\nWe show that an appropriately-designed “Refunding Club” can simultaneously solve both free-riding problems in mitigating climate change—participating in a coalition with an emission reduction target and enduring voluntary compliance with the target once the coalition has been formed. In particular, we propose and analyze climate treaties that involve a long-run refunding scheme (henceforth “RS”) within a Refunding Club. All countries in a coalition of countries forming a Refunding Club pay an initial fee into a fund that is invested in long-run assets. Countries in the Club maintain full sovereignty over the amount of emissions they abate each year and what policy measures they use to do so. At the end of each year, part of the fund is paid out to participating countries in proportion to the relative GHG emission reductions they have achieved in that year, weighted by country-specific factors.\n\nWe integrate the Refunding Club into a dynamic model that incorporates important characteristics of anthropogenic climate change. This requires to allow countries to be arbitrarily heterogeneous with respect to damages and abatement technologies together with a arbitrarily long (but finite) time horizon. Moreover, we incorporate latest scientific evidence of the climate change problem (i.e., we use a carbon budget approach). Then, we establish five main insights. First, any feasible abatement path a coalition of countries sets as a goal can be implemented by a suitably chosen RS. That is, once the corresponding initial fund has been established, the RS will ensure that countries comply with the envisioned country-specific abatement paths. The abatement paths could be the globally optimal paths of the grand coalition or more modest abatement paths by any coalition. Second, since in a treaty, voluntary compliance of countries with their abatement paths is independent from their specific contributions to the initial fund, the RS disentangles efficiency and distributional concerns. For instance, a suitably chosen RS cannot only achieve a Pareto improvement over the decentralized solution, but it can achieve any distribution of the cooperation gains through the allocation of contributions to the initial funds across countries in the Refunding Club.\n\nThird, we use an intertemporal extension of the modest international environmental agreement approach developed by Finus and Maus (2008) to characterize stable coalitions and thus to address the initial participation problem. By combining refunding (to solve the compliance problem) and Finus and Maus’ “modesty approach” (to solve the initial participation problem), we can determine for what level of modesty a coalition, and in particular the grand coalition, can be stabilized. Fourth, using a numerical illustration based upon the RICE-2010 model (Nordhaus 2010) with twelve regions in the world, we calculate ballpark estimates for the funds required to implement the modest grand coalition of less than 0.5% per cent of World GDP. Fifth, we suggest ways to foster initial participation, to incorporate equity concerns by differentiating initial fees across countries and to lower the burden for developing countries, and ways to ease the burden of filling the initial fund. Moreover, we outline how sustainable refunding schemes could be implemented in overlapping generation models.\n\n### Model and Main Formal Results\n\nWe study a multi-country model with country-specific emissions, abatement cost functions and damage functions.Footnote 2 Our main formal results are as follows: First, for a given coalition of countries, we introduce a RS, characterized by the set of initial fees payable into a fund by each participating country, a weighting scheme with country-specific refund intensities and a set of reimbursements across time. With a RS, a coalition of countries turns into a Refunding Club. We show that initial fees, the weighting scheme and a feasible sequence of refunds can be devised in such a way that the RS implements any feasible abatement path a coalition of countries wants to achieve through a treaty. That is, together with the abatement decisions of countries outside the coalition, the abatement decisions of countries in the coalition constitute a unique subgame perfect equilibrium and coincide with the goal stipulated in the treaty. Marginal deviations of countries in the coalition would reduce abatement costs marginally but this gain is equal to the corresponding reduction of refunds and increase of damages. A special case is the grand coalition and the implementation of the socially optimal abatement levels in each period and each country as a unique subgame perfect equilibrium with a suitably chosen RS.\n\nSecond, for any feasible abatement goal set by a coalition in a treaty, there exists a feasible set of initial fees such that the RS implements a Pareto improvement over the decentralized solution for all coalition members. Moreover, if we allow for negative initial fees, the RS can in fact implement any distribution of the cooperation gains in a coalition, This property of RS to disentangle efficiency and distributional concerns is helpful in achieving initial participation. The former is dealt with by the total amount of the initial fees and the refunding formula with the weighting factors. The latter is dealt with by the country-specific initial fees.\n\nThird, by allowing coalitions to internalize only a fraction of the externalities they create, we can examine the stability of coalitions using the modesty approach developed by Finus and Maus (2008). Drawing on the above results, we show that for any coalition, any degree of modesty can be implemented by a suitable RS, i.e., the abatement choices of countries, in the coalition and outside of it, constitute a unique subgame perfect equilibrium.\n\nBesides the analytical result, we illustrate the working and impact of the refunding scheme in a numerical exercise based upon the RICE-2010 model that takes into account heterogeneities across countries.\n\n### Literature\n\nThe starting point for our scheme and its analysis is the large body of game-theoretic literature on the formation of international and self-enforcing environmental agreementsFootnote 3 as there is no supranational authority to enforce contracts and to ensure participation and compliance during the duration of a treaty. This literature has provided important insights into the potentialities and limitations of international environmental agreements regarding the solution of the dynamic common-pool problem that characterizes climate change, as discussed and surveyed by Bosetti et al. (2009) and Hovi et al. (2013). Hovi et al. (2013) point out that there are three types of enforcement that are crucial for treaties to reduce global emissions substantially: (i) countries must be given incentives for ratification with deep commitment, (ii) those countries that have committed deeply when ratifying should be given incentives to remain within the treaties, and (iii) they should be given incentives to comply with them. Our Refunding Club satisfies all three requirements. First, when a country joins a coalition, it knows that the Refunding Scheme provides strong incentive for itself and the other members of the coalition to reduce greenhouse gas emissions. Second, once countries have joined the coalition, the Refunding Scheme ensures that countries comply with the envisioned abatement objective. Third, once countries have joined the coalition, they have no incentive to exit, as they would loose all claims on future refunds.\n\nThe papers most closely related to our paper are Gersbach and Winkler (2007, 2012) and Gerber and Wichardt (2009, 2013), all of which also incorporate refunding schemes. Gerber and Wichardt (2009) analyze a simple two-stage game in which countries in the first stage choose whether to accede to a treaty. Doing so involves a payment into a fund. In the second stage countries decide on emissions. Only if countries choose a particular emission level that is desired from a global perspective (and, in general, not in the best interest of each country alone) they get a refund paid out of the fund. If refunds (and first-stage deposits, respectively) are sufficiently high, all countries choose socially desired emission levels in the second stage. Participation in the first stage is ensured by the rule that the refunding scheme operates only if all countries participate and contribute their respective payments to the fund. If at least one country does not participate, the deposits of all other countries are immediately repaid, no refunding scheme is established and countries are stuck in the non-cooperative equilibrium. Gerber and Wichardt (2013) extend this framework to an intertemporal framework, in which the continuation of the agreement is challenged by re-occurring deposit stages. As in Gerber and Wichardt (2009) the refunding scheme only operates, respectively continues, if all countries pay their deposits.\n\nIn Gersbach and Winkler (2007, 2012), we focussed on the second and third enforcement/commitment problem. We also employed a refunding scheme to incentivice countries to increase their levels of emission abatement above the non-cooperative level. In contrast to Gerber and Wichardt (2009, 2013) our refunding scheme did not prescribe a particular abatement, respectively emission level in order to be eligible for a refund, but employed a continuous refunding rule, in which refunds are increasing with emission abatement. In addition, we analyzed to what extent initial deposits could be decreased by surrendering the revenues of climate policies to the fund (in our case the tax revenues from emission taxes).\n\n### Our Contribution\n\nRelative to the literature mentioned above, we make four contributions in this paper. First, we combine different aspects from the previous literature in a novel way: Like in Gerber and Wichardt (2009) we rely solely on initial payments to finance refunds, as countries might be reluctant to surrender tax sovereignty, but we do not assume that refunding collapses if a country does not match precisely a particular emission level. Instead, we rely on a continuously differentiable refunding rule like in Gersbach and Winkler (2007, 2011, 2012). This means that the sustainable treaties advanced in this paper are rule-based treaties, i.e., the treaties neither fix emission targets nor the carbon price. In contrast to Gersbach and Winkler (2007, 2011, 2012), however, we do not rely on revenues from emission taxes or permit auctions to pay for the initial fees since countries should have full sovereignty over their domestic climate policy and its intensity. The central question in our paper is: Can initial payments to a climate fund engineer solutions aspired by a coalition when refunding continuously adjusts to abatement efforts of countries? None of the preceding work has explored this question.\n\nSecond, in contrast to the existing literature on refunding schemes, we build a dynamic model that incorporates important characteristics of anthropogenic climate change. This requires to allow countries to be arbitrarily heterogeneous with respect to damages and abatement technologies together with a arbitrarily long (but finite) time horizon. Moreover, we incorporate latest scientific evidence of the climate change problem (i.e., we use a carbon budget approach, see also Sect. 2 for details). The combination of such a model with a continuous refunding rule results in a dynamic game structure, in which the existence and uniqueness of a subgame perfect equilibrium is neither obvious nor trivial to prove.\n\nThird, we also address the first commitment problem. However, unlike Gerber and Wichardt (2009, 2013), we do not believe that the participation problem can be credibly and realistically solved by an initial stage (or re-occurring intermediate participation stages as in Gerber and Wichardt 2013), in which any agreement is abandoned as soon as only one country is not willing to participate. Models with such an initial participation stage are not renegotiation-proof in the sense that if one country is not willing to participate all other countries would be better off by striking an agreement without the deviating country instead of falling back to the perfectly non-cooperative Nash equilibrium. In fact, we interpret the past announcement of the US’ withdrawal from the Paris Agreement and the subsequent declarations of (almost) all remaining countries to nevertheless stick to the agreement as empirical evidence that making each country pivotal will not work. As a consequence, we investigate the refunding scheme independently of the requirement that all countries participate, i.e., it can be applied to any coalition that forms with initial payments for a climate fund.\n\nTo analyze participation, we present an intertemporal generalization of the modesty approach by Finus and Maus (2008), which relies on the standard notion of internal and external stability. In fact, our RS poses a suitable microfoundation for the coalition formation framework in general, and the modest coalition formation framework of Finus and Maus (2008) in particular.Footnote 4 In addition, we explore in Sect. 7 ways to ease the participation and financing problem.Footnote 5\n\nFourth, the match of anthropogenic climate change characteristics and refunding opens up the possibility to assess for the first time the potential of refunding for slowing down climate change. In particular, we analyze a numerical version of our stylized model based on data of the regional disaggregated integrated assessment model RICE-2010 by Nordhaus (2010) and assess the order of magnitude of financial assets that are needed to finance such a refunding scheme. The calibration exercise reveals that making the grand coalition stable requires less than 0.5% of World GDP for the initial fund.\n\nFinally, we also contribute to solving dynamic public goods problems. At least since Fershtman and Nitzan (1991) it is know that dynamic good problems pose more severe challenges than their static counterparts.Footnote 6 We examine the most severe case when countries cannot commit to any future emission reductions, as no international authority can enforce an agreement on such reductions. The dynamic public good problem is thus particularly severe. The treaties we advance in this paper essentially reduce the public good problem over an infinite horizon to a static problem in which counties are asked to contribute in the initial period to a global fund. Once the global fund has been set up, countries voluntarily choose the desired emission levels in all subsequent periods.Footnote 7\n\n### Organization of the Paper\n\nThe paper is organized as follows: in the next section, we set up our model, for which in Sect. 3 we derive the social optimum and the decentralized solution as benchmark cases. The refunding scheme is introduced in Sect. 4, where the existence and uniqueness of RS to implement any solution an arbitrary coalition aspires to is also established. In Sect. 5, we extend the modesty approach to an intertemporal setting and characterize the stability conditions of coalitions with this approach. In Sect. 6 we illustrate our model numerically. In Sect. 7, we discuss practical aspects of the RS, such as initial participation and how to raise initial fees and how sustainable refunding schemes can be implemented in an overlapping generation set-up. Section 8 concludes. Proofs of all propositions are relegated to the “Appendix”.\n\n## The Model\n\nWe consider a world with $$n \\ge 2$$ countries characterized by country specific emission functions $$E_i$$, abatement cost functions $$C_i$$, and damage functions $$D_i$$ over a finite (though arbitrarily large) time horizon of $$T\\ (T > 0)$$ running from period $$t=0$$ to period $$t=T$$.Footnote 8 Throughout the paper the set of all countries is denoted by $$\\mathcal {I}$$, countries are indexed by i and j, and time is indexed by t.\n\nEmissions of country i in period t are assumed to equal “business-as-usual” emissions $$\\epsilon _i$$ (i.e., emissions arising if no abatement effort is undertaken) minus emission abatement $$a^i_t$$:Footnote 9\n\n\\begin{aligned} E_i(a^i_t)=\\epsilon _i- a^i_t, \\quad i \\in \\mathcal {I},\\quad t=0,\\dots ,T\\ . \\end{aligned}\n(1)\n\nWe assume that emission abatement $$a^i_t$$ is achieved by enacting some national environmental policy, which induces convex abatement costs in country i:Footnote 10\n\n\\begin{aligned} C_i(a^i_t) = \\frac{\\alpha _i}{2} \\left( a^i_t\\right) ^2\\ ,\\quad \\text {with} \\quad \\alpha _i>0,\\quad i \\in \\mathcal {I},\\quad t=0,\\dots ,T\\ . \\end{aligned}\n(2)\n\nCumulative global emissions, which are the sum of the emissions of all countries up to period t, are denoted by $$s_t$$:\n\n\\begin{aligned} s_{t+1}=s_{t} + \\sum _{i=1}^n E_i(a^i_t)\\ ,\\quad t=0,\\dots ,T, \\end{aligned}\n(3)\n\nwhere the initial stock of cumulative greenhouse gas emissions is denoted by $$s_0$$.\n\nRecent scientific evidence suggests that global average surface temperature increase is—at least for economically reasonable time scales (i.e., several centuries)—approximately a linear function of cumulative global carbon emissions (see Allen et al. 2009; Matthews et al. 2009; Zickfeld et al. 2009; IPCC 2013). As a consequence, we consider strictly increasing and strictly convex damage costs for each country i to depend on cumulative global emissions $$s_t$$ rather than on atmospheric greenhouse gas concentrations:\n\n\\begin{aligned} D_i(s_t)=\\frac{\\beta _i}{2}s_t^2\\ ,\\quad \\text {with} \\quad \\beta _i > 0,\\quad i \\in \\mathcal {I}\\ ,\\quad t=0,\\dots ,T\\ . \\end{aligned}\n(4)\n\nCountries are assumed to discount outcomes in period t with the discount factor $$\\delta ^t$$ with $$0<\\delta <1$$. Finally, we introduce the following abbreviations for later reference:\n\n\\begin{aligned} \\mathcal {E} = \\sum _{i=1}^n \\epsilon _i\\ ,\\quad \\mathcal {A} = \\sum _{i=1}^n \\frac{1}{\\alpha _i},\\quad \\mathcal {B} = \\sum _{i=1}^n \\beta _i,\\quad \\gamma _i = \\frac{\\beta _i}{\\alpha _i},\\quad \\Gamma = \\sum _{i=1}^n \\gamma _i \\ . \\end{aligned}\n(5)\n\n## Decentralized Equilibrium, Global Social Optimum and International Environmental Agreements\n\nThroughout the paper, we assume that a local planner in each country (e.g., a government) seeks to minimize the total domestic costs, which—in the absence of any transfers—consist of the discounted sum of domestic abatement and domestic environmental damage costs over all $$T + 1$$ periods:\n\n\\begin{aligned} K_i = \\sum _{t=0}^T \\delta ^t \\left[ \\frac{\\alpha _i}{2}\\left( a^i_t\\right) ^2 + \\frac{\\beta _i}{2} s_t^2\\right] , \\qquad i \\in \\mathcal {I}\\ . \\end{aligned}\n(6)\n\nWe further assume that local planners in all countries have perfect information about the business-as-usual emissions, abatement costs and environmental damage costs of all countries. In addition, in each period t local planners in all countries i observe the stock of cumulative emissions $$s_t$$ before they simultaneously decide on the abatement levels $$a_t^i$$.\n\nFinally, we assume that costs can—at least potentially—be frictionless shared across countries by a transfer scheme $$\\mathcal {T}$$, which is a set of domestic net transfers summing up to zero. Thus, we suppose a transferable utility set-up.\n\n### Decentralized Solution and Global Social Optimum\n\nUnder these assumptions and in the absence of any international environmental treaty, the decentralized solution is the subgame perfect Nash equilibrium outcome of the game, in which all local planners i in period t choose abatement levels $$a^i_t$$ such as to minimize total domestic costs taking the emissions $$a^j_t$$ of all other countries $$j \\in \\mathcal {I} {\\setminus } {i}$$ as given.\n\nWe solve the game by backward induction, starting from period T. It is useful to consider a typical step in this procedure. To this end, suppose that there exists a unique subgame perfect equilibrium for the subgame starting in period $$t+1$$ with a stock of cumulative greenhouse gas emissions $$s_{t+1}$$. For the moment, this is assumed to hold in all periods $$t+1$$ and will be verified in the proof of Proposition 1. Other details of the history of the game apart from the level of cumulative greenhouse gas emissions $$s_{t+1}$$ do not matter, as only $$s_{t+1}$$ influences the payoffs of the subgame starting in period $$t+1$$ and the equilibrium is assumed to be unique.\n\nGiven the unique subgame perfect equilibrium for the subgame starting in period $$t+1$$ with the associated equilibrium payoff $$W_{t+1}^i(s_{t+1}),$$Footnote 11 country i’s best response in period t, $$\\bar{a}_t^i$$, is determined by the solution of the optimization problem\n\n\\begin{aligned} V^i_t(s_t)\|A_t^{-i} = \\max _{a^i_t} \\left\\{ \\delta W^i_{t+1}(s_{t+1})-\\frac{\\alpha _i}{2}\\left( a^i_t\\right) ^2- \\frac{\\beta _i}{2}s_t^2\\right\\} , \\end{aligned}\n(7)\n\nsubject to Eq. (3), $$W_{T+1}^i(s_{T+1})\\equiv 0$$, and given the sum of abatement efforts by all other countries $$A_t^{-i} = \\sum _{j\\ne i}a_t^j$$. The following proposition establishes the existence and uniqueness of a subgame perfect Nash equilibrium:\n\n### Proposition 1\n\n(Decentralized Solution) For any time horizon $$T<\\infty$$, there exists a unique subgame perfect Nash equilibrium of the game in which all countries non-cooperatively choose domestic abatement levels in every period to minimize the net present value of total domestic costs, characterized by sequences of emission abatements for all countries i in all periods t, $$\\left\\{ \\hat{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {I}}$$, and a sequence for the stock of cumulative GHG emissions $$\\left\\{ \\hat{s}_t\\right\\} _{t=0,\\dots ,T}$$.\n\nThe proof of Proposition 1 is constructive in the sense that we do not only show existence and uniqueness of the subgame perfect equilibrium, but derive closed-form solutions for the corresponding abatement and cumulative GHG emission paths.Footnote 12\n\nIn general, the total global costs, i.e., the sum of total domestic costs over all countries, are not minimized in the decentralized solution. As a consequence, the decentralized solution is inefficient, as in the global total cost minimum, which is also called the global social optimum, all countries could be made better off by an appropriate transfer scheme $$\\mathcal {T}$$, due to the transferable utility assumption.\n\nThe reason for the decentralized solution to fall short of the global social optimum is that local planners only take into account the reduction of environmental damages that an additional unit of abatement prevents in their own country and neglect the damage reductions in all other countries. As a consequence, aggregate abatement levels in the decentralized solution are lower compared to the global social optimum and, thus, cumulative greenhouse gas emissions are higher.\n\nThe global social optimum is derived by choosing abatement paths $$\\left\\{ a^i_t\\right\\} _{t=0,\\dots ,T}$$ for all countries $$i \\in \\mathcal {I}$$, such as to minimize the net present value of total global costs consisting of global costs of emission abatement and the sum of domestic environmental damages stemming from the cumulative global emissions:\n\n\\begin{aligned} \\min _{\\left\\{ a^i_t\\right\\} ^{i\\in \\mathcal {I}}_{t=0,\\dots ,T}} \\sum _{t=0}^T \\delta ^{t} \\sum \\limits _{i=1}^{n}\\left[ \\frac{\\alpha _i}{2}\\left( a^i_t\\right) ^2 + \\frac{\\beta _i}{2} s_t^2\\right] , \\end{aligned}\n(8)\n\nThere exists a unique global optimum in which the costs of abating an additional marginal unit of emissions have to equal the net present value of all mitigated future damages caused by this additional marginal unit:\n\n### Proposition 2\n\n(Global Social Optimum) For any time horizon $$T < \\infty$$ there exists a unique social global optimum characterized by sequences of emission abatements for all countries i in all periods t, $$\\left\\{ {a^i_t}^\\star \\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {I}}$$, and a sequence for the stock of cumulative greenhouse gas emissions $$\\left\\{ s_t^\\star \\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {I}}$$.\n\nAgain, we derive closed form solutions for abatement and cumulative emission paths in the global social optimum in the proof of Proposition 2.\n\n### International Environmental Agreement\n\nThe inefficiency of the decentralized solution gives incentives to local planners to cooperate in order to reduce total domestic costs. Throughout this paper we refer to these cooperations as international environmental agreements or treaties for short.Footnote 13 In the framework of our model, the most general definition of an international environmental agreement comprises three components: First, a time horizon T, which denotes the duration of the treaty; second, a fixed set $$\\mathcal {C} \\subseteq \\mathcal {I}$$ of participating countries, also called member countries or simply the coalition. Finally, the abatement paths $$\\left\\{ a^i_t\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {C}}$$ of all member countries $$i \\in \\mathcal {C}$$, the treaty aspires to implement. We also define aggregated abatement $$A^\\mathcal {C}_t$$ of the coalition $$\\mathcal {C}$$ in period t as:\n\n\\begin{aligned} A^\\mathcal {C}_t = \\sum _{i \\in \\mathcal {C}} a^i_t, \\qquad t=0,\\dots ,T\\ . \\end{aligned}\n(9)\n\nIn line with most of the literature on international environmental agreements, we assume that all non-members of the coalition behave as singletons, i.e., they non-cooperatively set abatement levels such as to minimize the net present value of their own total domestic costs, as in the decentralized solution, taking the aggregate abatement effort of the coalition and the abatement levels of all other non-member countries as given. Derivation of the subgame perfect equilibrium is analogous to the decentralized solution:\n\n### Proposition 3\n\n(Abatement paths of non-members) For any time horizon $$T<\\infty$$ and any given coalition $$\\mathcal {C}$$ with a corresponding sequence of aggregate abatement levels $$\\left\\{ A^\\mathcal {C}_t\\right\\} _{t=0,\\dots ,T}$$, there exists a unique subgame perfect Nash equilibrium of the game in which all non-member countries choose domestic abatement levels in every period $$t=0,\\dots ,T$$ to minimize the net present value of total domestic costs, characterized by sequences of emission abatements for all countries $$i \\notin \\mathcal {C}$$ in all periods t, $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i\\notin \\mathcal {C}}$$, and a sequence for the stock of cumulative GHG emissions $$\\left\\{ \\check{s}_t\\right\\} _{t=0,\\dots ,T}$$.\n\nWhether a treaty, as defined above, succeeds in implementing its aspired abatement paths $$\\left\\{ a^i_t\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {C}}$$ mainly depends on two circumstances:\n\nFirst, the coalition of participating countries has to be stable in the sense that no participating country would rather leave the coalition (internal stability) and no non-member country would rather join the coalition (external stability). Whether the conditions of internal and external stability hold, depends on how the aspired abatement paths of the remaining coalition members changed if any of its members would leave the coalition. This question, which set of countries form a stable coalition, is also called the participation problem.\n\nSecond, even if a treaty is stable in the sense of the participation problem, it still has to make sure that participating countries stick to the aspired abatement paths $$\\left\\{ a^i_t\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {C}}$$. Without any kind of incentive scheme it is, in general, not in the countries’ own best interest to comply with the treaty. Therefore, the question how to incentivize countries to stick to the aspired abatement paths is also called the compliance problem.\n\nMost of the literature on international environmental agreements, as reviewed in Sect. 1, has concentrated on the participation problem, while compliance was simply assumed. Although a stable coalition is a sine qua non for a treaty’s success, it is obviously not sufficient, as ample real world examples of non-compliance show. To remedy this shortcoming, we introduce an institutional setting, called a refunding scheme, in the next section that implements any feasible abatement paths $$\\left\\{ a^i_t\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {C}}$$, a coalition intends to implement, as the unique subgame perfect Nash equilibrium.\n\n## Refunding Scheme\n\nIn the following, we introduce a refunding scheme (RS), a versatile institutional design ensuring the compliance of all members of an international environmental agreement with the aspired abatement paths. The essential idea is that an international fund is established refunding interest earnings to member countries in each period proportionally to their relative emission reductions weighted by country specific refunding weights.\n\n### Rules of the Refunding Scheme\n\nIn general, a RS for a given coalition of countries $$\\mathcal {C}$$ and a given time horizon T of the treaty is characterized by the set of initial fees $$\\left\\{ f_0^i\\right\\} ^{i \\in \\mathcal {C}}$$ payable into a global fund by each participating country $$i \\in \\mathcal {C}$$, a weighting scheme $$\\left\\{ \\lambda _t^i\\right\\} _{t=0,\\dots ,T-1}^{i \\in \\mathcal {C}}$$, and a set of reimbursements $$\\left\\{ R_t\\right\\} _{t=0,\\dots ,T}$$. The sequence of events is as follows:\n\n1. 1.\n\nAt the beginning of period $$t=0$$ all participating countries pay the initial fees $$f_0^i$$ into a fund.\n\n2. 2.\n\nIn every period $$t=0,\\dots ,T$$ all countries $$i \\in \\mathcal {I}$$ set abatement levels $$a_t^i$$.\n\n3. 3.\n\nAt the end of every period $$t=0,\\dots ,T$$ the RS reimburses the total amount $$R_t$$ to member countries. In periods $$t=0,\\dots ,T-1$$ each member country $$i \\in \\mathcal {C}$$ receives a refund $$r^i_t$$ that is proportional to the emission reductions they have achieved relative to overall emission abatement of the coalition times a weighting factor $$\\lambda ^i_t$$. In period $$t=T$$ any remaining fund is repaid in equal shares to all participating countries.\n\nWe assume that the assets of the fund are invested at the constant interest rate $$\\rho$$ per period, and the returns add to the global fund in the next period $$t+1$$. We assume that the interest rate $$\\rho$$ corresponds to the discount factor $$\\delta$$, i.e., $$\\rho =1/\\delta -1$$. As the reimbursement $$R_t$$ is paid to coalition members at the end of each period $$t=0,\\dots ,T$$, the fund at the beginning of period $$t+1$$ reads\n\n\\begin{aligned} f_{t+1} = (1+\\rho )(f_{t}-R_t),\\quad t=0,\\dots ,T-1, \\end{aligned}\n(10)\n\nwith an initial fund $$f_0=\\sum _{i \\in \\mathcal {C}} f_0^i$$. Note that $$f_{T+1}=0$$, or equivalently $$R_T=f_T$$.\n\nIn addition, the refund $$r^i_t$$ a member country $$i \\in \\mathcal {C}$$ receives in period t yields\n\n\\begin{aligned} r^i_t= {\\left\\{ \\begin{array}{ll} \\lambda _t^i R_t \\frac{a^i_t}{\\sum _{j \\in \\mathcal {C}} a^j_t}, \\quad &{}t=0,\\dots ,T-1,\\\\ \\frac{R_t}{\|\\mathcal {C}\|},\\quad &{}t=T, \\end{array}\\right. } \\end{aligned}\n(11)\n\nwith a weighting scheme satisfying\n\n\\begin{aligned} \\sum _{i \\in \\mathcal {C}} \\lambda _t^i \\frac{a^i_t}{\\sum _{j \\in \\mathcal {C}} a^j_t} =1,\\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(12)\n\nThe weighting scheme accounts for the fact that countries are heterogeneous with respect to business-as-usual emissions, abatement costs and environmental damage costs.\n\n### Existence and Uniqueness of the Refunding Scheme\n\nIn the following, we show that for any given treaty, characterized by a time horizon T, a coalition $$\\mathcal {C}$$ and feasible coalition abatement paths $$\\left\\{ a_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$, there exists a set of initial fees $$\\left\\{ f_0^i\\right\\} ^{i \\in \\mathcal {C}}$$, a weighting scheme $$\\left\\{ \\lambda _t^i\\right\\} _{t=0,\\dots ,T-1}^{i \\in \\mathcal {C}}$$ and refunds $$\\left\\{ R_t\\right\\} _{t=0,\\dots ,T-1}$$ such that the RS implements the aspired abatement paths $$\\left\\{ a_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$ of the coalition $$\\mathcal {C}$$ as the unique subgame perfect Nash equilibrium in which all countries set emission abatement levels in all periods to minimize the net present value of their own total domestic costs (which also includes initial payments and refunds for members of the coalition).\n\nTo this end, we first define a feasible coalition abatement path. A feasible coalition abatement path has the property that it lies in between the abatement paths of the decentralized solution and the social global optimum for all coalition member countries $$i \\in \\mathcal {C}$$ and all time periods $$t=0,\\dots ,T$$:Footnote 14\n\n\\begin{aligned} \\hat{a}_t^i \\le \\tilde{a}_t^i \\le {a_t^i}^\\star ,\\qquad i \\in \\mathcal {C},\\quad t=0,\\dots ,T\\ . \\end{aligned}\n(13)\n\nNote that, by construction, all feasible coalition abatement paths obeying conditions (13) are optimal in the last period T, as $$\\tilde{a}_T^i = 0$$ for all $$i \\in \\mathcal {C}$$. Then, the following Proposition holds:\n\n### Proposition 4\n\n(Existence of the RS) Given a treaty characterized by the coalition $$\\mathcal {C}$$, a time horizon T and feasible coalition abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$, there exist a RS characterized by a set of initial fees $$\\left\\{ \\tilde{f}_0^i\\right\\} ^{i \\in \\mathcal {C}}$$, a sequence of feasible refunds $$\\left\\{ \\tilde{R}_t\\right\\} _{t=0,\\dots ,T-1}$$, and a weighting scheme $$\\{\\tilde{\\lambda }_t^i\\}_{t=0,\\dots ,T-1}^{i\\in \\mathcal {C}}$$ such that the outcome of the unique subgame perfect Nash equilibrium of the game, in which all countries non-cooperatively choose domestic abatement levels in every period to minimize the net present value of total domestic costs, coincides with the aspired abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$ for all member countries $$i \\in \\mathcal {C}$$ and the abatement paths $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i\\notin \\mathcal {C}}$$, as given by Proposition 3, for all non-member countries $$i \\notin \\mathcal {C}.$$\n\nThe idea of the proof is to choose a reward system that renders all member countries’ aspired abatement levels under the RS as best responses to the given abatement levels of all other countries (within and outside the coalition). As shown in the proof of Proposition 4 in the “Appendix”, the RS is characterized by a uniquely determined sequence of refunds $$\\left\\{ \\tilde{R}_t\\right\\} _{t=0,\\dots ,T-1}$$ and a weighting scheme $$\\left\\{ \\tilde{\\lambda }_t^i\\right\\} _{t=0,\\dots ,T-1}^{i\\in \\mathcal {C}}$$. Yet, the set of initial fees is not unambiguously determined. In fact, all sets of initial fees, the sum of which exceeds the minimal initial global fund $$\\tilde{f}_0$$ with\n\n\\begin{aligned} \\tilde{f}_0 = \\sum _{t=0}^{T-1} \\frac{\\tilde{R}_t}{(1+\\rho )^t}, \\end{aligned}\n(14)\n\nrender a feasible RS that implements the treaty with aspired coalition abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$. The intuition is that the global fund needs the minimum size $$\\tilde{f}_0$$ in order to be able to pay sufficiently high refunds $$\\tilde{R}_t$$ such that countries stick to the aspired abatement levels in all periods. Any excess funds are redistributed in equal shares in the last period, in which the abatement level is zero independently of the refund. Even if we restrict attention to the minimal initial global fund $$\\tilde{f}_0$$, we are free in how to distribute the burden of raising the initial fund across countries.\n\n### Proposition 5\n\n(Uniqueness of the RS) For a given treaty characterized by the coalition $$\\mathcal {C}$$, a time horizon T and a set of feasible coalition abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$, the refunding scheme is only unique with respect to the minimal initial global fund $$\\tilde{f}_0$$. In particular, there exists a feasible set of initial fees $$f_0^i$$ satisfying $$\\sum _{i \\in \\mathcal {C}} f_0^i = \\tilde{f}_0$$ such that the RS constitutes a Pareto improvement over the decentralized solution for all coalition members $$i \\in \\mathcal {C}$$.\n\nThe intuition for this result is that compared to the decentralized solution all countries are better off under the RS if their initial fee was zero. As a consequence, there is a positive initial fee $$\\hat{f}_0^i$$ that would leave country i equally well off under the RS compared to the decentralized solution. In the proof of Proposition 5 in the “Appendix”, we show that the sum $$\\hat{f}_0 =\\sum _{i \\in \\mathcal {C}} \\hat{f}_0^i$$ exceeds $$\\tilde{f}_0$$. As a consequence, we can set the initial fee below $$\\hat{f}_0^i$$ making all countries better off.Footnote 15\n\nIn summary, we have shown that the RS can implement any feasible coalition abatement path and gives ample freedom in how to raise the necessary initial fund. The former feature of the RS may be important, as, in general, international climate policy is not shaped along standard economic cost-benefit analyses such as the derivation of the global social optimum in Sect. 3.1. In fact, the policy goal, for which global consensus is sought after, is to limit greenhouse gas emissions to such an extent that the global mean surface temperature increase is not exceeding 2 $$^{\\circ }\\hbox {C}$$ against preindustrial levels (see, for example, EU 2005; UNFCCC 2009, 2015). As the global mean surface temperature increase is predominantly determined by cumulative greenhouse gas emission, such a temperature goal can be translated into a stock of permissible cumulated greenhouse gas emissions, a so-called global carbon budget. Starting from the current stock of cumulative greenhouse gas emissions, estimates for this remaining carbon budget—as of beginning of 2018—roughly range between 320 and 555 trillion tonnes of carbon (GtC) (IPCC 2018).Footnote 16 Thus, Proposition 4 says that once the world community has agreed on an abatement path, the RS is able to implement it as a unique subgame perfect Nash equilibrium no matter how ambitious this abatement path is compared to the social global optimum. In particular, the RS is compatible with the idea of “nationally determined contributions”, as detailed in Article 3 of the Paris Agreement (UNFCCC 2015).\n\nThe latter feature of the RS implies that it cannot only achieve a Pareto improvement, but can in fact implement any distribution of the cooperation gain, i.e., the difference of the net present value of the total global costs between the decentralized solution and the social global optimum, if we allow initial fees to be negative for at least some countries. This property of the RS to disentangle efficiency and distributional concerns is helpful in achieving initial participation, as we shall discuss in Sect. 7.1.\n\n## The Modesty Approach to Refunding\n\nSo far, we have focused on the compliance problem, i.e., how to incentivize member countries to stick to the aspired abatement paths of a given international environmental agreement. Although Proposition 5 has established that all member countries can be made better off under the RS compared to the fully decentralized solution, as characterized by Proposition 1, this does not imply that any treaty is stable in the sense that all member countries have an incentive to join the coalition in the first place.Footnote 17 As already mentioned in Sect. 3.2, it is crucial to characterize how aspired abatement paths changed if any coalition members were to leave the treaty (or, more precisely, not to participate in the treaty in the first place). In the following, we showcase how questions of participation and compliance can be discussed simultaneously by applying the RS, as characterized in Sect. 4, to an intertemporal extension of the modest international environmental agreement approach developed by Finus and Maus (2008).\n\n### An Intertemporal Extension of Modesty\n\nThe standard coalition formation game is a two stage game, in which all countries in the first stage simultaneously decide whether to join an international agreement. In the second stage, all countries simultaneously set emission abatement levels. Non-member countries choose abatement levels non-cooperatively by minimizing their own domestic costs, taking the abatement levels of all other countries as given, while coalition members are supposed to choose emissions abatement levels such as to minimize the sum of total domestic costs over all member countries. Finus and Maus (2008) allow for modest international environmental agreements by specifying that member countries only internalize a fraction $$\\mu \\in [0, 1]$$ of the externalities within the coalition.\n\nApplying this idea to our intertemporal model framework, results in the following two stage game:\n\n1. 1.\n\nAt the beginning of period $$t=0$$ all countries simultaneously decide whether to join an international environmental agreement.\n\n2. 2.\n\nIn all periods $$t=0,\\dots ,T$$ all countries simultaneously decide on emission abatement levels.\n\n1. (a)\n\nNon-member countries choose abatement levels minimizing the net present values of their total domestic costs, taking the abatement levels of the coalition and the other non-members as given, resulting in abatement paths $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$ as characterized by Proposition 3.\n\n2. (b)\n\nMembers of the coalition $$\\mathcal {C}$$ set abatement levels such as to\n\n\\begin{aligned} \\min _{\\left\\{ a^i_t\\right\\} ^{i \\in \\mathcal {C}}_{t=0,\\dots ,T}} \\sum _{t=0}^T \\delta ^{t} \\sum _{i \\in \\mathcal {C}} \\left[ \\frac{\\alpha _i}{2}\\left( a^i_t\\right) ^2 + \\mu \\frac{\\beta _i}{2} s_t^2\\right] , \\end{aligned}\n(15)\n\ntaking the emission levels of non-member countries as given.\n\nThe parameter $$\\mu$$ in Eq. (15) can be interpreted as the degree of modesty. It can be interpreted as the fraction $$\\mu$$ of externalities the coalition internalizes among its members. This formulation essentially entails a more modest emission reduction goal. The higher $$\\mu$$, the higher the emission abatement goal of the coalition. For $$\\mu =1$$ the treaty internalizes all externalities coalition members impose on each other, which is the assumption of the standard coalition formation set-up.\n\n### Combining Modesty and Refunding\n\nWhile assuming that the coalition sets abatement levels according to Eq. (15) allows for a parsimonious way to reconcile the empirical observation of “large but modest” agreements with the prediction of the coalition formation framework, one might ask why coalition members should comply with these aspired abatement paths of the treaty, as they are, in general, not in their best interest (in terms of minimizing the net present value of total domestic costs). This is where the RS, as characterized in Sect. 4 comes into play. As we shall proof in Proposition 6, the aspired abatement paths characterized by Eq. (15) constitute feasible coalition abatement paths that can be implemented via an appropriate RS by virtue of Proposition 4. Thus, the RS serves as a microfoundation to implement the aspired abatement paths characterized by the modest coalition formation framework.\n\nAs usual, we analyze the intertemporal modest coalition formation game with refunding by backward induction, i.e., we first characterize the subgame perfect Nash equilibrium of the second stage, for given time horizon T and given coalition $$\\mathcal {C}$$.\n\n### Proposition 6\n\n(Abatement paths in SPE of second stage) For any time horizon $$T<\\infty$$, any given coalition $$\\mathcal {C}$$ and any degree of modesty $$\\mu$$, there exists a unique subgame perfect Nash equilibrium of the game in which all non-member countries $$i \\notin \\mathcal {C}$$ choose domestic abatement levels in every period $$t=0,\\dots ,T$$ to minimize the net present value of total domestic costs, and all member countries $$i \\in \\mathcal {C}$$ set abatement levels according to Eq. (15). The subgame perfect Nash equilibrium is characterized by emission abatements paths $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i\\notin \\mathcal {C}}$$ for all countries $$i \\notin \\mathcal {C}$$ and $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i\\in \\mathcal {C}}$$ for all countries $$i \\in \\mathcal {C}$$ and a corresponding path $$\\left\\{ s_t\\right\\} _{t=0,\\dots ,T}$$ for the stock of cumulative GHG emissions.\n\nThe proof of Proposition 6 in the “Appendix” is constructive, as we derive the unique closed-form solutions of the abatement paths in the subgame perfect Nash equilibrium of the second stage. Moreover, we show that the decentralized solution, as given by Proposition 1, and the global social optimum, as characterized by Proposition 2, are boundary solutions of Proposition 6, which apply in the case that the coalition only consists of at most one member country or all countries are members of the coalition and $$\\mu =1$$. As a consequence, the assumptions of Proposition 4 apply, and any feasible abatement path as defined in Eq. (13) of the modest coalition formation game can be implemented by an appropriate RS.\n\nHaving solved the compliance problem in the second stage by employing the RS, we can now turn to the participation problem in the first stage. Anticipating the outcome of the second stage, a coalition is a subgame perfect Nash equilibrium outcome of the first stage, if no country has an incentive to unilaterally change its membership status. Thus, all member countries $$i \\in \\mathcal {C}$$ must not be better off if they were not in the coalition, and all non-member countries $$i \\notin \\mathcal {C}$$ must be better off than if they were by joining the coalition. If we denote, for any given coalition $$\\mathcal {C}$$ and modesty parameter $$\\mu$$, the net present value of total domestic costs of member countries $$i \\in \\mathcal {C}$$ by $$\\tilde{K}_i(\\mathcal {C},\\mu )$$ and the net present value of total domestic costs of non-member countries $$i \\notin \\mathcal {C}$$ by $$\\check{K}_i(\\mathcal {C},\\mu )$$, then the conditions of internal and external stability read in our transferable utility set-up:Footnote 18\n\n\\begin{aligned}&\\sum _{j \\in \\mathcal {C}} \\tilde{K}_j(\\mathcal {C},\\mu ) - \\sum _{j \\in \\mathcal {C}{\\setminus } i} \\tilde{K}_j (\\mathcal {C}{\\setminus } i,\\mu ) \\le \\check{K}_i(\\mathcal {C}{\\setminus } i,\\mu ),\\qquad \\forall \\ i \\in \\mathcal {C}, \\end{aligned}\n(16a)\n\\begin{aligned}&\\sum _{j \\in \\mathcal {C} \\cup i} \\tilde{K}_j(\\mathcal {C} \\cup i,\\mu ) - \\sum _{j \\in \\mathcal {C}} \\tilde{K}_j(\\mathcal {C},\\mu ) > \\check{K}_i(\\mathcal {C},\\mu ),\\qquad \\forall \\ i \\notin \\mathcal {C}\\ . \\end{aligned}\n(16b)\n\nWe note that the stability conditions can be formulated without explicitly invoking the RS. The reasons is twofold. First, the RS does not change the sum of the net present value of total domestic costs over all member countries, as the net present value of all refunds is, by construction, equal to the initial fund. Second, according to Proposition 6, there exists an appropriate RS for any coalition structure $$\\mathcal {C}$$ such that it is in the best interest of coalition members to stick to the agreement. Therefore, the conditions of internal and external stability implicitly also involve all elements of the RS in different coalition arrangements $$\\mathcal {C}, \\mathcal {C}{\\setminus } i$$ and $$\\mathcal {C} \\cup i$$, namely how much a country in the coalition has to pay into the initial fund, how much it will abate inside and outside the coalition, how many refunds it will obtain when in the coalition and how many damages occur. Hence, for instance, initial contributions have to be chosen such that stability conditions are met for each individual country.\n\nWe note that the refunding approach with its transfers balances asymmetries between countries in a coalition in an optimal way,Footnote 19 i.e., to achieve the abatement objective of the coalition by providing incentives for countries to comply and by making sure that countries want to join the coalition.\n\nEven in the static modest international environmental agreement framework it is not possible to analytically analyze coalition stability for quadratic damage functions and heterogenous countries. As a consequence, we shall concentrate attention to a particularly interesting case, calibrate our model and derive numerical results. The particular question, we want to address is for what level of modesty $$\\mu$$ the grand coalition $$\\mathcal {C}=\\mathcal {I}$$ can be stabilized. For the grand coalition, only internal stability (16a) is relevant, which can be re-arranged to yield:\n\n\\begin{aligned} \\sum _{j \\in \\mathcal {I}} \\tilde{K}_j(\\mathcal {I},\\mu ) \\le \\sum _{i \\in \\mathcal {I}} \\check{K}_i(\\mathcal {I} {\\setminus } i,\\mu )\\ . \\end{aligned}\n(17)\n\nThus, the grand coalition is stable if it can guarantee all countries a lower net present value of total domestic costs when they participate in the treaty instead of unilaterally leaving it.\n\nThe approach opens up a wide range of further interesting issues, which we leave for future research. For instance, is it globally optimal to stabilize the grand coalition with an appropriate value of $$\\mu$$, instead of being less modest and having only a smaller coalition being stable? Or could better results be achieved by having several smaller regional coalitions with their own abatement objectives and associated refunding schemes?\n\n## Numerical Illustration\n\nTo give an idea of the degree of modesty that renders the grand coalition stable and the corresponding order of magnitude needed for the initial fund $$f_0$$ to implement it via an appropriate RS, we run a numerical exercise. Due to the highly stylized model, the results are rather a numerical illustration than a quantitative analysis.\n\nWe follow the RICE-2010 model (Nordhaus 2010) in dividing the world into twelve regions, each of which we assume to act as a “country”, as detailed in Sect. 2.Footnote 20 We also take the “business-as-usual” (BAU) emissions for all twelve regions from Nordhaus (2010). The RICE-2010 model assumes a backstop technology, the price of which decreases over time and fully crowds out fossil fuel based energy technologies by 2265. As a consequence, global CO$$_\\text {2}$$ emissions drop to zero in 2265 in the BAU scenario in which cumulative global CO$$_\\text {2}$$ emissions of 5679.6 GtC have been released into the atmosphere (we assume that cumulative global CO$$_\\text {2}$$ emissions prior to 2015 amount to 550 GtC).\n\nIn the global social optimum of the RICE-2010 model, the long-run cumulative global emissions amount to 1470.8 GtC, which implies an increase of the average global surface temperature of approximately 2.9–3 $$^{\\circ }\\hbox {C}$$ over preindustrial levels. In addition, carbon neutrality, i.e., zero global GHG emissions are only reached by 2155. In light of the Paris agreement and the recent announcements by the US, EU and China, among other countries, to become carbon neutral by 2050, respectively 2060, the RICE-2010 model’s global social optimum feels somewhat outdated. Unfortunately, there is no updated version of the RICE-2010 model. We deal with this issue in a two step procedure.\n\nFirst, we calibrate our model in such a way that the global social optimum in our model resembles the optimal solution of the RICE-2010 model as closely as possible. Therefore, we calibrate the relative damage parameters for each region by fitting quadratic functions to the damage functions used in the RICE-2010 model. Then we re-scale all damage parameters such that damages in the BAU scenario in the year 2095 amount to 12 trillion USD or 2.8% of global output as in Nordhaus (2010, 11723). We calibrate the abatement cost parameters such that the emission paths in the global social optimum of all twelve regions resemble the optimal solution of the RICE-2010 model as closely as possible, under the constraint that the abatement cost parameters decline at a unique and constant rate. Table 1 shows the calibrated abatement and damage cost parameters for all twelve world regions. Abatement cost parameters decrease at the rate of $$\\xi =1.65\\%$$ per year, implying a drop of approximately 15.1% per decade.Footnote 21 In line with the RICE-2010 model, we employ a discount rate of 5% per year, which corresponds to a discount factor of $$\\delta = 0.6139$$ for each ten year period. While it is not possible to perfectly mimic the outcome of a sophisticated integrated assessment model as the RICE-2010 model with our simple theoretical model, both global GHG emissions as well as cumulative global emissions match reasonably well (see upper graphs in Fig. 1).\n\nSecond, we increase the rate at which the abatement cost parameters decline to $$\\xi =5\\%$$ per year implying a decadal drop of 38.6%. Under these conditions, our model calculates a global social optimum in which carbon neutrality is reached by 2065 and cumulative global emissions level off at 874.6 GtC. This corresponds to an average global surface temperature increase between 1.7 and 1.8 $$^{\\circ }\\hbox {C}$$, which we consider compatible with the goals of the Paris Agreement. In this “Paris compatible” calibration, the long-run level of cumulative global emissions in the decentralized solution amounts to 1550.1 GtC, which approximately corresponds to a 3.1 $$^{\\circ }\\hbox {C}$$ increase of average global surface temperature. In this scenario, carbon neutrality would be achieved by 2115 (see lower graphs in Fig. 1).\n\nSeeking the upper bound of the degree of modesty, for which the grand coalition for the Paris compatible model calibration is just stable, we find $$\\mu =26.594\\%$$. In this case, long-run cumulative global emissions amount to 1204.7 GtC, which closes approximately half of the gap between the decentralized solution and the global social optimum. Yet, with a temperature increase of approximately 2.4 $$^{\\circ }\\hbox {C}$$, the stable grand coalition would fall short of the 2 $$^{\\circ }\\hbox {C}$$ temperature target.\n\nAn initial fund of 2.64 tril. USD or 0.326% of 2015 world GDP is needed to implement the stable grand coalition via a RS. In addition, Table 2 shows the net present value of refunds and the maximum initial fees a region is willing to pay to join the treaty for all 12 regions in tril. USD and % of world GDP. By construction, the sum of initial fees $$\\hat{f}_0^i$$, which makes regions indifferent whether to join the treaty, amounts to the same total of 2.64 tril. USD or 0.326% of 2015 world GDP.\n\nTable 3 shows the development of the refund over time. We observe that total refunds start at very low levels of 19.2 bil. USD per annum corresponding to 0.024% of world GDP, continuously rise until they peak in 2075 at 797 bil. USD per annum corresponding to 0.253% of world GDP. After that they sharply decline and by 2115 no refunds have to be paid anymore, as even in the decentralized solution net zero GHG emissions have been reached. While this general pattern of global refunds is mimicked by the individual regions, there are some differences in magnitude. China receives the highest refunds both in absolute numbers and also as a share of its GDP, peaking in 2075 at 242.03 bil. USD corresponding to 0.546% of its GDP. Africa has the lowest relative peak refunds in 2075 of 34.32 bil. USD or 0.131% of its GDP. The marginal abatement costs start at 49.3 USD per tC (equalling 180.93 USD per ton of CO$$_\\text {2}$$) and rise until 2075, when they peak at 73.6 USD per tC (or 270.11 USD per ton of CO$$_\\text {2}$$).\n\nIn summary, the refunding scheme can stabilize a grand coalition that bridges half the gap between the global social optimum and the decentralized solution in our Paris compatible model calibration. While the net present value of funds needed is sizeable at 2.64 tril. USD, it is not out of reach, when compared to other funds raised in situations of global crisis, such as the latest global financial crisis or the Corona pandemic. We would like to stress that our quantitative exercise is only an illustrative example to gauge the order of magnitude of what an initial fund would look like.\n\n## Discussion\n\nSo far we have focused first on how a refunding scheme can implement any goal a coalition has set and second, on how coalition stability can be achieved, and in particular the stability of the grand coalition. We have seen that a refunding scheme transforms the intertemporal climate-policy problem into a standard, static public-goods problem. Once all countries in the stable coalition have made their initial contribution, and have agreed on the refunding parameters, countries follow the envisioned path of abatement voluntarily and would be worse off by forfeiting refunds.\n\nNumerous further issues are relevant for the design of refunding scheme and the use of Refunding Clubs. We address them in this section, as they deserve further scrutiny.\n\n### Increasing Initial Fees and a Refunding Club\n\nAt the initial level, when countries are pondering whether to sign the treaty and to pay the initial fee, the free-rider problem is present. If all other countries participate and if aggregate initial fees are high, this country would benefit from all other countries’ abatement efforts, without having to pay the initial fee and to compete for refunds. Hence, the question is how better solutions than the one induced by the modesty approach could be achieved.\n\nThe ideal solution lies in making countries—and large countries, in particular—pivotal for the formation of a coalition, with high initial fees. In order to achieve such a scenario, about ten to twenty of the largest greenhouse gas emitters must coordinate and agree that coalition formation and the refunding scheme fail if any of them defects.Footnote 22\n\nAs full participation by all countries at once is unlikely, it is useful to resort to sequential procedures where a subset of countries makes a start and the others follow later (see Andreoni 1998; Varian 1994). We envision four steps. First, as suggested in the last paragraph, a set of large countries could initiate the system by paying initial fees and form a Refunding Club. In particular, if the US, the EU countries, China and maybe India would start the system with significant initial fees, this would constitute the Refunding Club in which the largest share of greenhouse gases are emitted. In addition, if wealthy countries pay substantially larger initial fees than the modesty approach suggests, such a Refunding Club would be powerful enough to slow down climate change significantly.\n\nSecond, smaller rich countries could follow, which would increase the initial wealth. In the third and fourth steps, larger and smaller developing countries could be invited to join the Refunding Club. Regarding the payment of initial fees, they should be treated differently, as we will discuss next.\n\nThe successful implementation of a refunding scheme only depends on raising the minimum initial global fund, but not on the individual countries’ contributions to it. Thus, within a coalition, the refunding scheme is able to disentangle efficiency from distributional concerns. Yet, in reality, the distribution of initial fees to raise the initial global fund is of great importance. For example, many developing countries may lack the necessary wealth to pay the initial fees, or countries in transition may refuse to pay high initial fees by arguing that historically, the current atmospheric greenhouse gas concentrations were caused by industrialized countries. To induce participation, payment of initial fees could be differentiated according to different distributional criteria such as stage of development, current greenhouse emissions or historic responsibility with respect to atmospheric greenhouse gas concentrations.Footnote 23 Thus, refunding schemes and the payment of initial fees can be made compatible with the concept of “common but differentiated responsibilities and respective capabilities”, as detailed in Article 4 of the Paris Agreement (UNFCCC 2015). To sum up, allocating the burden of the initial fees is a tool that can solve distributional concerns, since they indirectly implement transfers across countries.\n\n### Raising Initial Fees\n\nEven differentiated initial fees cannot circumvent the problem that the sustainable refunding scheme relies on successfully raising the minimum initial global fund. As this fund may be quite large, even in the modesty solution, we outline two ways in which it might be financed.\n\nRaising the minimum initial fund in full at the beginning of the treaty is not necessary. We can also achieve a coalition solution in which a smaller amount of money of money is paid repeatedly. To see this, let $$\\{R_t\\}_{t=0}^T$$ be the sequence of refunds in a solution envisioned by a coalition. In addition, we define the sequence of fees $$f_t(\\Delta )$$ for a time span $$\\Delta > 0$$ by\n\n\\begin{aligned} f_t(\\Delta ) = \\sum \\limits _{\\tau = 1}^{\\Delta } \\frac{R_{t+\\tau }}{(1+\\rho )^\\tau }\\ . \\end{aligned}\n(18)\n\nIf $$f_t(\\Delta )$$ is paid into the fund at times $$t=0,\\Delta , 2\\Delta ,\\ldots$$, the net present value of the fund is equal to the initial fund $$f_0= \\sum _{t=1}^{T-1}\\left[ \\frac{R_t}{(1+\\rho )^t}\\right]$$ and, thus, the same solution can be achieved as when $$f_0$$ is initially paid in full.\n\nWith the repeated payments scheme, we face a trade-off between high initial fees and the property of an RS that transforms an intertemporal climate-policy problem into a static public-goods problem. In particular, if the time span $$\\Delta$$ is short, the solution of the problem of a coalition relies on the repeated commitment of all countries, as the initial participation problem would have to be solved whenever new payments have to be made. Therefore $$\\Delta$$ should not be too small.\n\nIf the repeated solution to the initial participation problem turns out to be a major obstacle to international cooperation, raising the initial fees by allowing countries to borrow money may be more advisable. Countries could then borrow either from the international capital market or directly from the administering agency of the RS. In the latter case, no actual initial money flows would be needed, since the initial fee would then simply be a liability of countries at the administration agency. In turn, future refund claims would be reduced or could even become negative, as countries would have to pay interest and ultimately pay back their liabilities to the agency. Hence, borrowing from the agency appears like a Munchhausen solution to the problem of raising initial fees.\n\nHowever, at least two problems may arise. First, if countries borrow a large amount from the agency, they may later only receive a small payment or may even have to pay when refunds and repayment obligations are netted. Hence, countries might be tempted to renounce high abatement efforts and to default on their repayment obligations to the agency. The country would then lose all claims to refunds. However, as such refunds are small when abatement efforts are small, such a strategy may be profitable. That is, a country could choose to default against the administering agency and could free-ride on the abatement efforts of other countries even if it has signed the treaty and has borrowed from the agency. Such considerations suggest that countries should rather be made to borrow on the international capital market.\n\nSecond, if countries borrow a large amount on international capital markets, the default risk may rise if outstanding government debt is already at a high level. If the country needs to pay a higher interest rate than the risk-free rate, as investors demand a positive risk premium, further borrowing may increase the default risk, as refunds are insufficient to cover interest-rate payments. In such cases, it is more efficient if part of the initial fund is being raised by taxes over several periods.\n\n### Information Requirements and Reaction to Unforeseen Shocks\n\nThe design of any RS rests on the bold assumption that all exogenous parameters are constant and, in particular, that they are known ex ante. These are demanding informational requirements.\n\nWe distinguish between temporary and permanent changes of parameters. Temporary shocks to the parameters do not inhibit the long-run behavior of the refunding scheme, because of the global convergence to the first-best solution. It is likely that initial expectations about the discount/interest rate $$\\delta$$, the abatement cost parameters $$\\alpha _i$$, the damage cost parameters $$\\beta _i$$, and the business-as-usual emissions $$\\epsilon _i$$ turn out to be incorrect and that at some time t, new information on one or several of these parameter arrives. In particular, technological progress may substantially change the abatement cost parameters.\n\nPermanent changes in the exogenously given parameters would, in general, change the necessary refunds for a RS corresponding to a given feasible coalition abatement path.Footnote 24 Moreover, in general, also the aspired coalition abatement path itself would change, for example, the abatement path and the degree of modesty that renders the grand coalition stable.\n\nTo accommodate permanent changes in the exogenous parameters, the RS could include a clause that the values of these parameters are re-evaluated on a regular basis (e.g., every ten years) and that the fund’s wealth is corrected accordingly, either by raising additional money from the members or paying back wealth to member countries.\n\nEven if revision cannot be done frequently, the refunding scheme offers some built-in corrections. For example, when marginal damages increase, also the individually optimal abatement efforts for a given refunding scheme increase. However, the extent to which such built-in reactions to parameter changes correct deviations from the first-best solution or from a coalition solution is beyond the scope of this paper, but constitutes an important avenue for future research.\n\n### Sustainable Climate Treaties in Overlapping Generation Frameworks\n\nSo far, we have focused on the properties of a RS and on how the implementation of such a scheme can be eased through repeated payments or through the use of capital markets. Still, we have assumed so far that the countries’ interest can be represented by a long-lived social planner.\n\nThe implementation of sustainable refunding schemes is more difficult in overlapping generation models, in which each generation is predominantly concerned about its own welfare. Then, setting up a refunding scheme hurts the old (existing) generations and benefits future generations—and possibly young existing generations—via two channels. First, the benefits from higher abatement today mainly accrue to future generations. This was the focus of important papers by Bovenberg and Heijdra (1998, 2002).Footnote 25 Public debt policies can help redistribute the welfare gains from increased abatement more equally across generations. Essentially, by issuing (more) public debt today and by having future generations pay it back, the welfare of current generations can be increased at the expense of future generations. Additional effects such as a potential crowding out of physical capital investments and the reduction of distortionary taxation affect the balance between current and future generations.\n\nSecond, current generations must set up the fund and thus are, in principle, required to channel some of their savings towards the payment of initial fees. Since the global fund also invests, such savings may not necessarily decrease capital accumulation, but as future generations inherit the global fund for their own refunding, setting up the global fund decreases the welfare of current generations. Again, to redistribute the burden of setting up the global fund more equally across generations, one might implement repeated payments, as discussed above, or again, public debt can be used to increase the disposable income of current old generations.\n\nIn principle, the use of public debt can engineer trade among generations, can ease the implementation of sustainable refunding schemes and opens up the possibilities to achieve Pareto-improving climate policies across generations. However, with much higher public debt levels after the Covid-19 pandemic in many countries, the scope for further increases of public debt is quite limited.\n\n## Conclusion\n\nIn this paper, we have shown that a refunding scheme, which is a rule-based treaty offering monetary incentives for emission abatement to member countries that are proportional to their relative abatement efforts, may promote sustained international cooperation with respect to anthropogenic climate change. The RS provides a simple blueprint for an international treaty on climate change and depends on a small number of parameters.\n\nYet, the RS is no panacea, as free-rider problems have no perfect solutions. For example, our numerical illustration shows that implementing a stable grand coalition in the modesty approach, which stabilizes average surface temperature at approximately 2.4 $$^{\\circ }\\hbox {C}$$, requires funds in the amount of 2.64 tril. USD. Given that the Green Climate Fund (GCF),Footnote 26 the existing real world institution closest to our refunding scheme, has set itself the goal to raise 100 bil. USD per year starting from 2020, but has great difficulties in securing the pledges for these sums, such a sum seems considerably high. Yet, it is comparable to the sums raised to counter other global crises such as the latest financial crisis or the Corona pandemic. Still, the industrialized countries would have to shoulder a large share of the initial fees.\n\nWe stress that a decisive difference between the GCF and the RS is that the RS refunds money according to a simple and transparent rule (which is already known when initial fees are raised), while the GCF is governed by a 24-member board who decides which projects will be financed by the fund after the money has been raised.\n\nNo doubt, the practical implementation of the refunding schemes in a Refunding Club developed in this paper requires a variety of additional considerations. In the last section, we have discussed how to achieve better initial participation, and we have outlined several ways of raising initial fees. Other issues, such as the governance of the administering agency, or the stimulation of technological progress in abatement technologies will need thorough investigation in future research.\n\n1. In this protocol, the industrialized countries of the world, so-called “Annex B countries”, committed themselves to a reduction of greenhouse gas (GHG) emissions by 5.2% against 1990 levels over the period from 2008 to 2012.\n\n2. Our model is a dynamic stock pollution game similar to Dockner and Van Long (1993), but generalized to n players, in discrete time. In addition, we use a carbon budget approach (see Sect. 2 for details), i.e., we abstract from stock depreciation.\n\n3. Non-cooperative and cooperative approaches have been pursued. Many authors have stressed that the grand coalition is not stable if an individual defection does not destroy any coalition formation (e.g., Carraro and Siniscalco 1993; Eyckmans et al. 1993; Barrett 1994; Tol 1999; Bosetti et al. 2009.) Typically, in such circumstances, stable coalitions only contain a limited number of countries (see, e.g., Hoel 1991; Carraro and Siniscalco 1992; Finus and Caparrós 2015a, b). d’Aspremont et al. (1983) have conducted an original analysis and has introduced the definitions for internally and externally stable coalitions. Pioneering in the modelling of coalition structures are Bloch (1997) and Yi (1997).\n\n4. In this respect our model shares some similarities with Harstad (2020), who analyzes a dynamic model inspired by the pledge-and-review mechanism of the Paris Agreement to account for a variety of different empirical observations of international environmental agreements.\n\n5. If complete contracts on emission reductions could be written between countries, a first-best solution would be easily achieved including, of course, the initial participation problem. Harstad (2012, 2016) and Battaglini and Harstad (2016) analyze the interaction between decisions on emission levels and investments into low-carbon energy technologies in dynamic games with incomplete contracting, i.e., when countries can contract on emission reductions but not on investment. We abstract from technology investments and scrutinize what refunding can achieve in case countries cannot contract ex ante on emission reductions.\n\n6. Dynamic games on voluntary provision of public goods have been significantly developed (see Wirl 1996; Dockner and Sorger 1996; Sorger 1998; Marx and Matthews 2000). Recent contributions involve Dutta and Radner (2009) who examine agreements on mitigating climate change supported by inefficient Markov perfect equilibria.\n\n7. In this respect, our argument is exactly opposite to Gerber and Wichardt (2013), who propose to split static public good problems into dynamic ones in order to reduce downside payments.\n\n8. Throughout the paper, we denote the time horizon by T. Note that a time horizon of T comprises of $$T+1$$ periods.\n\n9. We do not restrict abatement to be at most as high as business-as-usual emissions, i.e., $$a^i_t \\le \\epsilon _i$$. Thus, we allow for negative net emissions, for example via afforestation or carbon capture and sequestration technologies.\n\n10. This is a standard short-cut way of capturing aggregate abatement costs in country i (see, e.g., Falk and Mendelsohn 1993).\n\n11. The equilibrium payoff $$W_{t+1}^i(s_{t+1})$$ is minus the discounted sum of the total domestic costs over the remaining time horizon starting from period $$t+1$$ in the subgame perfect equilibrium of the subgame starting in period $$t+1$$.\n\n12. Despite being closed-form the solutions are quite cumbersome and, thus, relegated to the “Appendix”.\n\n13. Both the global social optimum and the decentralized outcome are important benchmarks in evaluating the performance of potential international agreements. While the decentralized outcome is realized if no agreement takes place, the social optimum is the ultimate goal an international agreement seeks to implement. Obviously, any agreement has to outperform the decentralized outcome in order to be seriously considered, and it is the “better,” the closer its outcome is to the global social optimum.\n\n14. While abatement paths with $${a_t^i}^\\star < \\tilde{a}_t^i$$ would also be “feasible” in a strictly technical sense and could also be implemented by the RS, we consider the social global optimum as a natural upper bound for the emission abatement levels of coalition members.\n\n15. The set of initial payments implicitly defines a transfer scheme $$\\mathcal {T}$$, as introduced in Sect. 3.\n\n16. The main obstacles for translating an upper temperature bound for mean global surface temperature into a carbon budget are scientific uncertainties concerning the equilibrium climate sensitivity and the climate-carbon cycle feedback (see, e.g., Friedlingstein et al. 2011; Zickfeld et al. 2009).\n\n17. In particular, treaties with large coalition sizes and high aspired abatement paths may not be stable, as countries may have an incentive not to participate in the treaty in the first place in order to free-ride on the abatement efforts of the remaining coalition members.\n\n18. We apply the usually made assumption that countries stick with the coalition in case of indifference.\n\n19. For optimal transfer systems in the presence of asymmetries, see Finus and McGinty (2019).\n\n20. The twelve regions are: United States of America (US), European Union (EU), Japan, Russia, Eurasia, China, India, Middle East (MidEast), Africa, Latin America (LatAm), other high income countries (OHI) and Rest of the World (Others).\n\n21. As in the RICE-2010 model, we use ten year periods. However, we usually express all values in per annum terms.\n\n22. In practice, these countries must be collectively stubborn and insist on full participation by this entire core group before going ahead.\n\n23. Development-compatible refunding system have been developed in Gersbach and Hummel (2016).\n\n24. However, whether a change in the exogenous parameters increases or decreases the necessary level of global fund depends, in most cases, on the whole set of exogenous parameters, and the comparative static results for the global fund of the respective RS are quite complex.\n\n25. How public debt can be used to strike an intergenerational bargain in the context of climate change is also addressed by Dennig et al. (2015) who propose several focal bargaining points.\n\n26. The Green Climate Fund was formally established during the UNFCCC COP-16 meeting in Cacun in 2010. Its objective is to assist developing countries in adaptation and mitigation practices to counter climate change.\n\n27. As $$A^\\mathcal {C}_t$$ may be any arbitrary exogenously given path, there may not exist a steady state, and thus, there exists no constant particular solution to (29).\n\n28. Another way to see that $${h_t^i}' (\\tilde{A}_t^\\mathcal {C})< 0$$ is by evaluating Eq. (51) at $$\\tilde{A}_t^\\mathcal {C}$$ and re-writing it to yield:\n\n\\begin{aligned} {h_t^i}'(\\tilde{A}_t^\\mathcal {C})= -\\frac{\\tilde{\\lambda }_t^i \\tilde{R}_t \\left[ 2(\\alpha _i+ \\delta \\beta _i) \\tilde{A}_t^\\mathcal {C} - \\tilde{C}_t^i\\right] + \\alpha _i \\tilde{C}_t^i \\left( \\tilde{A}_t^\\mathcal {C} \\right) ^2}{\\left[ \\tilde{\\lambda }_t^i \\tilde{R}_t + \\alpha _i \\left( \\tilde{A}_t^\\mathcal {C}\\right) ^2\\right] ^2}\\ . \\end{aligned}\n\n$${h_t^i}'(\\tilde{A}_t^\\mathcal {C})$$ is negative, as the term in brackets in the numerator is larger as $$\\alpha _i \\tilde{a}_t^i + \\delta \\beta _i \\tilde{A}_t^\\mathcal {C} - \\tilde{C}_t^i$$, which, according to Eq. 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Proc Natl Acad Sci 106:16129–16134\n\n## Acknowledgements\n\nWe would like to thank Clive Bell, Jürgen Eichberger, Evgenij Komarov, Martin Hellwig, Markus Müller, Till Requate, Wolfgang Buchholz, Ian MacKenzie, Jérémy Laurent-Lucchetti, Nicolas Treich, seminar participants in Heidelberg, Frankfurt, Zurich, Bern, Toulouse, and Vienna, conference participants at the EAERE 2009 in Amsterdam, at the SMYE 2009, at the WCERE 2010 in Montreal, and the handling editor Michael Finus and three anonymous reviewers for helpful comments and suggestions on this line of research. Financial support of the Swiss National Science Foundation, Project No. 124440, is gratefully acknowledged. A precursor of this paper entitled “Sustainable Climate Treaties” has appeared as CER-ETH Working Paper No 11/146, 2011.\n\n## Funding\n\nOpen Access funding provided by Universität Bern.\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Ralph Winkler.\n\n### Publisher's Note\n\nSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Appendix\n\n### Proof of Proposition 1\n\nThe decentralized solution is a special case of the second stage of the modest coalition formation game, as detailed in Sect. 5. Thus, Proposition 6, which states that there exists a unique subgame perfect Nash equilibrium of the second stage of the game for any given membership structure $$\\mathcal {C}$$ and modesty parameter $$\\mu$$ also covers the decentralized solution. In fact, the decentralized solution is characterized by $$\\mathcal {C}=\\varnothing$$, i.e., the coalition is an empty set and all countries $$i\\in \\mathcal {I}$$ do not participate in the treaty.\n\nIn the solution () of the proof of Proposition 6 the decentralized solution corresponds to $$x=0$$ and $$y=\\Gamma$$ implying also $$\\bar{A}^\\mathcal {C}=0$$, $$\\bar{A}^\\mathcal {NC}=\\mathcal {E}$$ and $$\\bar{s} = \\frac{1-\\delta }{\\delta }\\mathcal {E}$$. Thus, we obtain for the aggregate emission abatement level $$A_t = \\sum _{i \\in \\mathcal {I}} a_t^i$$ and the stock of aggregate cumulative emissions $$s_t$$:\n\n\\begin{aligned} A_{t}&= \\mathcal {E} + B_2(T) (1-\\lambda _2) \\lambda _2^t - B_3(T) (1-\\lambda _3) \\lambda _3^t, \\end{aligned}\n(19a)\n\\begin{aligned} s_t&= \\bar{s} + B_2(t) \\lambda _2^t + B_3(T) \\lambda _3^t, \\end{aligned}\n(19b)\n\nwith\n\n\\begin{aligned} \\lambda _2&= \\frac{1+\\delta (1+\\Gamma ) - \\sqrt{[1+\\delta (1+\\Gamma )]^2-4\\delta }}{2\\delta }, \\end{aligned}\n(20a)\n\\begin{aligned} \\lambda _3&= \\frac{1+\\delta (1+\\Gamma ) + \\sqrt{[1+\\delta (1+\\Gamma )]^2-4\\delta }}{2\\delta }, \\end{aligned}\n(20b)\n\nand\n\n\\begin{aligned} B_2(T)&= - \\frac{\\mathcal {E} + (s_0-\\bar{s})(1-\\lambda _3)\\lambda _3^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}, \\end{aligned}\n(21a)\n\\begin{aligned} B_3(T)&= \\frac{\\mathcal {E} + (s_0-\\bar{s})(1-\\lambda _2)\\lambda _2^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}\\ . \\end{aligned}\n(21b)\n\nThe individual countries’ abatement levels in the subgame perfect Nash equilibrium of the decentralized solution are given by:\n\n\\begin{aligned} a_t^i = \\frac{\\gamma _i}{\\Gamma } A_{t}, \\qquad \\forall \\ i\\in \\mathcal {I}, \\quad t=0,\\dots ,T\\ . \\end{aligned}\n(22)\n\n$$\\square$$\n\n### Proof of Proposition 2\n\nAlso the global social optimum is a special case of the second stage of the modest coalition formation game, as detailed in Sect. 5. Thus, Proposition 6, which states that there exists a unique subgame perfect Nash equilibrium of the second stage of the game for any given membership structure $$\\mathcal {C}$$ and modesty parameter $$\\mu$$ also covers the global social optimum. In fact, the global social optimum is characterized by $$\\mu =1$$ and $$\\mathcal {C}=\\mathcal {I}$$, i.e., the coalition is the grand coalition encompassing all countries $$i\\in \\mathcal {I}$$ and fully internalizes all damages imposed by GHG emissions on all other countries.\n\nIn the solution () of the proof of Proposition 6 the global social optimum corresponds to $$x=\\mathcal {AB}$$ and $$y=0$$ implying also $$\\bar{A}^\\mathcal {C}=\\mathcal {E}$$, $$\\bar{A}^\\mathcal {NC}=0$$ and $$\\bar{S} = \\frac{1-\\delta }{\\delta }\\mathcal {E}$$. Thus, we obtain for the aggregate emission abatement level $$A_t = \\sum _{i \\in \\mathcal {I}} a_t^i$$ and the stock of aggregate cumulative emissions $$s_t$$:\n\n\\begin{aligned} A_{t}&= \\mathcal {E} + B_2(T) (1-\\lambda _2) \\lambda _2^t - B_3(T) (1-\\lambda _3) \\lambda _3^t, \\end{aligned}\n(23a)\n\\begin{aligned} s_t&= \\bar{S} + B_2(t) \\lambda _2^t + B_3(T) \\lambda _3^t, \\end{aligned}\n(23b)\n\nwith\n\n\\begin{aligned} \\lambda _2&= \\frac{1+\\delta (1+\\mathcal {AB}) - \\sqrt{[1+\\delta (1+\\mathcal {AB})]^2-4\\delta }}{2\\delta }, \\end{aligned}\n(24a)\n\\begin{aligned} \\lambda _3&= \\frac{1+\\delta (1+\\mathcal {AB}) + \\sqrt{[1+\\delta (1+\\mathcal {AB})]^2-4\\delta }}{2\\delta }, \\end{aligned}\n(24b)\n\nand\n\n\\begin{aligned} B_2(T)&= - \\frac{\\mathcal {E} + (s_0-\\bar{s})(1-\\lambda _3)\\lambda _3^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}, \\end{aligned}\n(25a)\n\\begin{aligned} B_3(T)&= \\frac{\\mathcal {E} + (s_0-\\bar{s})(1-\\lambda _2)\\lambda _2^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}\\ . \\end{aligned}\n(25b)\n\nThe individual countries’ abatement levels in the global social optimum are given by:\n\n\\begin{aligned} a_T^i&= 0, \\qquad \\forall \\ i\\in \\mathcal {I}, \\end{aligned}\n(26a)\n\\begin{aligned} a_t^i&= \\frac{A_{t}}{\\alpha _i\\mathcal {A}}, \\qquad \\forall \\ i\\in \\mathcal {I}, \\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(26b)\n\n$$\\square$$\n\n### Proof of Proposition 3\n\nThe situation, in which a set of non-member countries strategically choose emission abatement levels such as to minimize their own domestic costs is similar to the second stage of the coalition formation game, as discussed in Sect. 5 and Proposition 6. The only difference is that the coalition $$\\mathcal {C}$$ is following an exogenously given emission abatement paths instead of strategically reacting to the emission abatement choices of all non-member countries $$i \\notin \\mathcal {C}$$. Thus, existence and uniqueness of the subgame perfect equilibrium can be shown perfectly analogously to the proof of Proposition 6 by assuming an exogenously given aggregate emission abatement path $$A^{\\mathcal {C}}_t$$ of the coalition.\n\nThus, we directly obtain the following system of first-order linear difference equations for the aggregated emission abatement levels of non-member countries $$A^{\\mathcal {NC}}_t = \\sum _{i \\notin \\mathcal {C}} a_t^i$$ and the stock of aggregated cumulative emissions $$s_t$$ for some exogenously given path of aggregate emission abatement $$A^\\mathcal {C}_t$$ of the coalition $$\\mathcal {C}$$:\n\n\\begin{aligned} A_{t+1}^\\mathcal {NC}&= \\left( \\frac{1}{\\delta }+\\Gamma ^\\mathcal {NC} \\right) A_{t}^\\mathcal {NC} -\\Gamma ^\\mathcal {NC} s_t - \\Gamma ^\\mathcal {NC}\\left( \\mathcal {E} - A_t^\\mathcal {C}\\right) , \\end{aligned}\n(27a)\n\\begin{aligned} s_{t+1}&= -A_t^\\mathcal {NC} + s_t +\\mathcal {E} - A_t^\\mathcal {C}\\ . \\end{aligned}\n(27b)\n\nIntroducing the matrix M:\n\n\\begin{aligned} M = \\begin{pmatrix} \\frac{1}{\\delta } + \\Gamma ^\\mathcal {NC} &{} -\\Gamma ^\\mathcal {NC}\\\\ -1 &{} +1 \\end{pmatrix}, \\end{aligned}\n(28)\n\nwe rewrite the system () in matrix form:\n\n\\begin{aligned} \\begin{pmatrix} A_{t+1}^\\mathcal {NC}\\\\ s_{t+1} \\end{pmatrix} = M \\cdot \\begin{pmatrix} A_{t}^\\mathcal {NC}\\\\ s_{t} \\end{pmatrix} + \\begin{pmatrix} - \\Gamma ^\\mathcal {NC} \\left( \\mathcal {E}-A^\\mathcal {C}_t\\right) \\\\ \\mathcal {E}-A^\\mathcal {C}_t \\end{pmatrix}\\ . \\end{aligned}\n(29)\n\nThe general solution of the matrix equation (29) is given by:\n\n\\begin{aligned} \\begin{pmatrix} A_{t}^\\mathcal {NC}\\\\ s_{t} \\end{pmatrix} = \\begin{pmatrix} \\bar{A}^\\mathcal {NC}_t\\\\ \\bar{s}_t \\end{pmatrix} + B_1(T) \\nu _1 \\lambda _1^t + B_2(T) \\nu _2 \\lambda _2^t, \\end{aligned}\n(30)\n\nwhere $$\\bar{A}^\\mathcal {NC}_t$$ and $$\\bar{s}_t$$ denote particular solutions to (29),Footnote 27$$\\lambda _i$$ are the eigenvalues and $$\\nu _i$$ the eigenvectors of the matrix M, and $$B_i(T)$$ are constants determined by the initial and terminal conditions of the stock and the emission abatement levels ($$i=1,2$$).\n\nThe particular solutions are given by:\n\n\\begin{aligned} \\begin{pmatrix} \\bar{A}^\\mathcal {NC}_t\\\\ \\bar{s}_t \\end{pmatrix} = \\sum _{t'=0}^{t-1} M^{t'} \\cdot \\begin{pmatrix} - \\Gamma ^\\mathcal {NC} \\left( \\mathcal {E}-A^\\mathcal {C}_{t'}\\right) \\\\ \\mathcal {E}-A^\\mathcal {C}_{t'} \\end{pmatrix}\\ . \\end{aligned}\n(31)\n\nIn addition, for the matrix M we derive the following eigenvalues $$\\lambda _i$$ ($$i=1,2$$):\n\n\\begin{aligned} \\lambda _1&= \\frac{1+\\delta \\left( 1+\\Gamma ^\\mathcal {NC}\\right) -\\sqrt{\\left[ 1+\\delta \\left( 1+\\Gamma ^\\mathcal {NC}\\right) \\right] ^2-4\\delta }}{2\\delta }, \\end{aligned}\n(32a)\n\\begin{aligned} \\lambda _2&= \\frac{1+\\delta \\left( 1+\\Gamma ^\\mathcal {NC}\\right) +\\sqrt{\\left[ 1+\\delta \\left( 1+\\Gamma ^\\mathcal {NC}\\right) \\right] ^2-4\\delta }}{2\\delta }, \\end{aligned}\n(32b)\n\nand eigenvectors ($$i=1,2$$):\n\n\\begin{aligned} \\nu _1&= \\left\\{ 1-\\lambda _1, 1 \\right\\} , \\end{aligned}\n(33a)\n\\begin{aligned} \\nu _2&= \\left\\{ 1-\\lambda _2, 1 \\right\\} \\ . \\end{aligned}\n(33b)\n\nInserting into Eq. (30) yields:\n\n\\begin{aligned} A_{t}^\\mathcal {NC}&= \\bar{A}^\\mathcal {NC}_t + B_1(T)(1-\\lambda _1)\\lambda _1^t + B_2(T)(1-\\lambda _2)\\lambda _2^t, \\end{aligned}\n(34a)\n\\begin{aligned} s_{t}&= \\bar{s}_t + B_1(T) \\lambda _1^t + B_2(T) \\lambda _2^t \\end{aligned}\n(34b)\n\nThe constants $$B_i(T)$$ ($$i=1,2$$) are derived from the initial stock $$s_0$$ and the terminal condition $$A_{T}^\\mathcal {NC} = 0$$, which implies\n\n\\begin{aligned} B_1(T)&= - \\frac{\\bar{A}^\\mathcal {NC}_T + (s_0-\\bar{s}_0)(1-\\lambda _2)\\lambda _2^{T}}{(1-\\lambda _1) \\lambda _1^T - (1-\\lambda _2) \\lambda _2^T}, \\end{aligned}\n(35a)\n\\begin{aligned} B_2(T)&= \\frac{\\bar{A}^\\mathcal {NC}_T + (s_0-\\bar{s}_0)(1-\\lambda _1)\\lambda _1^{T}}{(1-\\lambda _1) \\lambda _1^T - (1-\\lambda _2) \\lambda _2^T}\\ . \\end{aligned}\n(35b)\n\nThe individual countries’ abatement levels in the subgame perfect Nash equilibrium are given by:\n\n\\begin{aligned} a_T^i&= 0, \\qquad \\forall \\ i \\in \\mathcal {I}, \\end{aligned}\n(36a)\n\\begin{aligned} a_t^i&= \\frac{\\gamma _i}{\\Gamma ^\\mathcal {NC}} A^\\mathcal {NC}_{t}, \\qquad \\forall \\ i\\notin C, \\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(36b)\n\n$$\\square$$\n\n### Proof of Proposition 4\n\nFirst, note that if the RS is able to incentivize all member countries $$i \\in \\mathcal {C}$$ to implement the aspired abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$ we can use Proposition 3 to determine the emission abatement paths $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\notin \\mathcal {C}}$$ for all non-member countries $$i \\notin \\mathcal {C}$$ in the subgame perfect Nash equilibrium. Thus, it suffices to show that given these emission abatement paths of non-member countries $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\notin \\mathcal {C}}$$, there exists a RS that implements the aspired abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$ for all coalition members $$i \\in \\mathcal {C}$$ as a subgame perfect Nash equilibrium. For further use, we define the aggregated emission abatement level of all non-member countries $$i \\notin \\mathcal {C}$$ in period t in the subgame perfect Nash equilibrium by $$\\check{A}_t^\\mathcal {NC} = \\sum _{i \\notin \\mathcal {C}} \\check{a}_t^i$$.\n\nTo prove this, we assume that a set of countries $$\\mathcal {C}$$ has joined a feasible RS characterized by a weighting scheme $$\\left\\{ \\lambda _t^i\\right\\} _{t=0,\\dots ,T-1}^{i \\in \\mathcal {C}}$$ and a sequence of refunds $$\\{R_t\\}_{t=0,\\dots ,T-1}$$ by paying an initial fee $$f_0^i$$. We shall analyze the subgame perfect Nash equilibria of the RS by backward induction. In every step of the backward induction, we show that\n\n1. 1.\n\nthe objective function of each country i is strictly concave,\n\n2. 2.\n\nthere exists a feasible weighting scheme $$\\{\\tilde{\\lambda }_t^i\\}^{i \\in \\mathcal {C}}$$ and a feasible refund $$\\tilde{R}_t$$ such that the aspired abatement levels $$\\{\\tilde{a}_t^i\\}^{i \\in \\mathcal {C}}$$ are consistent with the necessary and sufficient conditions of the subgame perfect Nash equilibrium of the subgame starting in period t and\n\n3. 3.\n\nthe aspired abatement levels $$\\{\\tilde{a}_t^i\\}^{i \\in \\mathcal {C}}$$ are the unique solution solving the necessary and sufficient conditions of subgame perfection of the subgame starting in period t,\n\ngiven the aspired abatement levels $$\\{\\tilde{a}_{t+1}^i\\}^{i \\in \\mathcal {C}}$$ constitute the unique subgame perfect Nash equilibrium outcome of the subgame starting in period $$t+1$$.\n\nAssuming that there exists a unique subgame perfect equilibrium for the subgame starting in period $$t+1$$ with a stock of cumulative greenhouse gas emissions $$s_{t+1}$$, for all countries $$i \\in \\mathcal {C}$$, we denote country i’s equilibrium payoff for this subgame by $$W^i_{t+1}(s_{t+1})$$. Then country i’s best response in period t, $$\\bar{a}_t^i$$, is determined by the solution of the optimization problem\n\n\\begin{aligned} V_t^i(s_t)\|A_t^{-i} = \\max \\limits _{a_t^i} \\left\\{ \\delta W_{t+1}^i(s_{t+1}) -\\frac{\\alpha _i}{2}(a_t^i)^2 - \\frac{\\beta _i}{2}s_t^2 + r^i_t\\right\\} , \\end{aligned}\n(37)\n\nsubject to Eq. (3), $$W_{T+1}^i(s_{T+1})\\equiv 0$$, and given the sum of the abatement efforts of all other countries $$A_t^{-i} = \\sum _{j\\ne i}a_t^j$$. Differentiating Eq. (37) with respect to $$a^i_t$$ and setting it equal to zero yields\n\n\\begin{aligned} \\alpha _i \\bar{a}_t^i = -\\delta {W_{t+1}^i}' (\\bar{s}_{t+1})+\\left. \\frac{\\partial r^i_t}{\\partial a^i_t}\\right\| _{a^i_t=\\bar{a}^i_t}\\ \\end{aligned}\n(38)\n\nwhere $$\\bar{s}_{t+1} = s_t+\\mathcal {E} - \\bar{a}_t^i - A_t^{-i}$$ and\n\n\\begin{aligned} \\frac{\\partial r^i_t}{\\partial a^i_t} = {\\left\\{ \\begin{array}{ll} \\lambda _t^i R_t \\displaystyle \\frac{A^{\\mathcal {C}-i}_t}{(a^i_t+A^{\\mathcal {C}-i}_t)^2}\\ , \\quad &{}t=1,\\dots ,T-1,\\\\ 0, \\quad &{}t=T\\ . \\end{array}\\right. } \\end{aligned}\n(39)\n\nDifferentiating w.r.t. $$s_t$$ and applying the envelope theorem yields\n\n\\begin{aligned} -{V_t^i}'(s_t)\|A_t^{-i} = \\beta _i s_t-\\delta {W_{t+1}^i}'(s_{t+1})\\ . \\end{aligned}\n(40)\n\nStarting with period $$t=T$$, we first note that the maximization problem of all countries is strictly concave, as $$W_{T+1}(s_{T+1})\\equiv 0$$ and $$r_T = f_T/\|\\mathcal {C}\|$$. Thus, Eq. (38) characterizes the best response for all countries $$i \\in \\mathcal {C}$$, which is given by $$\\bar{a}_{T}^i = 0$$ independently of the abatement choices of all other countries. As a consequence $${\\hat{a}_T^i}=0$$ for all $$i \\in \\mathcal {C}$$ is the subgame perfect Nash equilibrium of the game starting in period T and is also the aspired abatement level in period T, as $$\\tilde{a}^i_T=0$$ for all $$i \\in \\mathcal {C}$$ for all feasible coalition abatement paths. Then, the equilibrium pay-off is given by $$W^i_T(s_T)=V^i_T(s_T)\|\\hat{A}^{-i}_T$$, which is strictly concave:\n\n\\begin{aligned} W^i_T(s_T) = -\\frac{\\beta _i}{2}s_T^2 + \\frac{f_T}{\|\\mathcal {C}\|} \\qquad \\Rightarrow \\qquad W_{T}''(s_T) = -\\beta _i\\ . \\end{aligned}\n(41)\n\nNow, we analyze the subgame starting in period t assuming that their exists a weighting scheme $$\\{\\tilde{\\lambda }_{t'}^i\\}_{{t'}=t,\\dots ,T-1}^{i \\in \\mathcal {C}}$$ and a sequence of refunds $$\\left\\{ \\tilde{R}_{t'}\\right\\} _{{t'}=t,\\dots ,T-1}$$ such that the outcome of the unique subgame perfect Nash equilibrium of the subgame starting in period $$t+1$$ coincides with the aspired coalition abatement paths $$\\left\\{ \\tilde{a}_{t'}^i\\right\\} _{{t'}=t+1,\\dots ,T}^{i \\in \\mathcal {C}}$$. In addition, we assume that $$W_{t+1}^i(s_{t+1})$$ is strictly concave. Then, also the optimization problem of country $$i \\in \\mathcal {C}$$ in period t is strictly concave\n\n\\begin{aligned} \\delta {W_{t+1}^i}''(s_{t+1}) - \\alpha _i + \\frac{\\partial ^2 r^i_t}{(\\partial a^i_t)^2}< 0\\ . \\end{aligned}\n(42)\n\nAs a consequence, there exists a unique best response $$\\bar{a}^i_t$$ for all countries $$i \\in \\mathcal {C}$$ given the emission abatements of all other countries $$j \\ne i$$, which is given implicitly by (38):\n\n\\begin{aligned} \\alpha _i \\bar{a}^i_t - \\lambda _t^i R_t \\frac{A^{\\mathcal {C}-i}_t}{(\\bar{a}^i_t+A^{\\mathcal {C}-i}_t)^2} = -\\delta {W_{t+1}^i}' (\\bar{s}_{t+1})\\ . \\end{aligned}\n(43)\n\nAs, by assumption, $$-{W^i_{t'}}'(s_{t'})=-{V^i_{t'}}'(s_{t'})\|\\hat{A}_{t'}^{-i}$$ for all $$t'\\ge t+1$$ we can exploit Eq. (40) to obtain the following Euler equation:\n\n\\begin{aligned} \\alpha _i \\bar{a}^i_t - \\lambda _{t}^i R_t \\frac{A^{\\mathcal {C}-i}_t}{(\\bar{a}^i_t+A^{\\mathcal {C}-i}_t)^2} = \\delta \\beta _i \\bar{s}_{t+1} + \\delta \\alpha _i \\tilde{a}_{t+i}^i -\\delta \\tilde{\\lambda }_{t+1}^i \\tilde{R}_{t+1} \\frac{\\tilde{A}_{t+1}^{\\mathcal {C}-i}}{(\\tilde{A}^\\mathcal {C}_{t+1})^2}\\ . \\end{aligned}\n(44)\n\nInserting $$\\bar{s}_{t+1} = s_t + \\mathcal {E}-\\bar{a}_t^i - A_{t}^{\\mathcal {C}-i}-\\check{A}_t^{\\mathcal {NC}}$$ yields:\n\n\\begin{aligned} \\alpha _i \\bar{a}^i_t + \\delta \\beta _i (\\bar{a}_t^i + A_{t}^{\\mathcal {C}-i}) - \\lambda _t^i R_t \\frac{A^{\\mathcal {C}-i}_t}{(\\bar{a}^i_t+A^{\\mathcal {C}-i}_t)^2} = \\tilde{C}_t^i, \\end{aligned}\n(45)\n\nwith\n\n\\begin{aligned} \\tilde{C}_t^i = \\delta \\beta _i \\left( s_t+\\mathcal {E}-\\check{A}_t^{\\mathcal {NC}}\\right) + \\delta \\alpha _i \\tilde{a}_{t+i}^i -\\delta \\tilde{\\lambda }^i_{t+1} \\tilde{R}_{t+1} \\frac{\\tilde{A}_{t+1}^{\\mathcal {C}-i}}{\\left( \\tilde{A}^\\mathcal {C}_{t+1}\\right) ^2}\\ . \\end{aligned}\n(46)\n\nFirst, we show that there exist unique $$\\tilde{\\lambda }_t^i$$ and $$\\tilde{R}_t$$ such that choosing the aspired coalition abatement level $$\\tilde{a}_t^i$$ is an equilibrium strategy for all countries $$i \\in \\mathcal {C}$$. Inserting aspired abatement levels $$\\tilde{a}_t^i$$ and rearranging Eq. (45), we obtain\n\n\\begin{aligned} \\tilde{\\lambda }_t^i \\tilde{R}_t = \\left( \\tilde{A}^\\mathcal {C}_t\\right) ^2 \\left( \\alpha _i \\frac{\\tilde{a}_t^i}{\\tilde{A}_t^{\\mathcal {C}-i}} + \\delta \\beta _i \\frac{\\tilde{A}^\\mathcal {C}_t}{\\tilde{A}_t^{\\mathcal {C}-i}} -\\frac{\\tilde{C}_t^i}{\\tilde{A}_t^{\\mathcal {C}-i}}\\right) \\ . \\end{aligned}\n(47)\n\nTaking into account that the weighting scheme adds up to one, i.e., $$\\sum _{j \\in \\mathcal {C}} \\tilde{\\lambda }_t^j\\frac{\\tilde{a}_t^j}{\\tilde{A}_t^\\mathcal {C}} = 1$$, yields\n\n\\begin{aligned} \\tilde{R}_t&= \\tilde{A}^\\mathcal {C}_t \\sum _{j \\in \\mathcal {C}} \\left[ \\tilde{a}_t^j\\left( \\alpha _j \\frac{\\tilde{a}_t^j}{\\tilde{A}_t^{\\mathcal {C}-j}} + \\delta \\beta _j \\frac{\\tilde{A}^\\mathcal {C}_t}{\\tilde{A}_t^{\\mathcal {C}-j}} -\\frac{\\tilde{C}_t^j}{\\tilde{A}_t^{\\mathcal {C}-j}}\\right) \\right] \\ , \\end{aligned}\n(48a)\n\\begin{aligned} \\tilde{\\lambda }_t^i&= \\frac{\\tilde{A}_t^\\mathcal {C}\\left( \\alpha _i \\frac{\\tilde{a}_t^i}{\\tilde{A}_t^{\\mathcal {C}-i}} + \\delta \\beta _i \\frac{\\tilde{A}^\\mathcal {C}_t}{\\tilde{A}_t^{\\mathcal {C}-i}} -\\frac{\\tilde{C}_t^i}{\\tilde{A}_t^{\\mathcal {C}-i}}\\right) }{\\sum _{j \\in \\mathcal {C}} \\left[ \\tilde{a}_t^j\\left( \\alpha _j \\frac{\\tilde{a}_t^j}{\\tilde{A}_t^{\\mathcal {C}-j}} + \\delta \\beta _j \\frac{\\tilde{A}^\\mathcal {C}_t}{\\tilde{A}_t^{\\mathcal {C}-j}} -\\frac{\\tilde{C}_t^j}{\\tilde{A}_t^{\\mathcal {C}-j}}\\right) \\right] }\\ . \\end{aligned}\n(48b)\n\nWe now show that the aspired coalition abatement levels $$\\tilde{a}_t^i$$ are the unique solution to the Euler equations of all countries $$i \\in \\mathcal {C}$$ given the weighting scheme $$\\left\\{ \\tilde{\\lambda }_t^i\\right\\} ^{i \\in \\mathcal {C}}$$ and the refund $$\\tilde{R}_t$$. To this end, we express equation (45) in terms of $$a_t^i$$ and $$A^\\mathcal {C}_t$$ and solve for $$a_t^i$$:\n\n\\begin{aligned} a_t^i = A^\\mathcal {C}_t \\underbrace{\\frac{\\tilde{\\lambda }_t^i \\tilde{R}_t + \\tilde{C}_t^i A_t^\\mathcal {C} - \\delta \\beta _i \\left( A_t^\\mathcal {C}\\right) ^2}{\\tilde{\\lambda }_t^i \\tilde{R}_t + \\alpha _i \\left( A_t^\\mathcal {C}\\right) ^2}}_{\\equiv h_t^i(A^\\mathcal {C}_t)} = A^\\mathcal {C}_t h_t^i(A^\\mathcal {C}_t)\\ . \\end{aligned}\n(49)\n\nSumming-up over all countries $$i \\in \\mathcal {C}$$ yields\n\n\\begin{aligned} \\sum _{i \\in \\mathcal {C}} h_t^i(A^\\mathcal {C}_t) = 1, \\end{aligned}\n(50)\n\nwhich has to hold for $$A^\\mathcal {C}_t=\\tilde{A}_t^\\mathcal {C}$$ and is a necessary condition for a Nash equilibrium. Differentiating $$h_t^i(A^\\mathcal {C}_t)$$ with respect to $$A^\\mathcal {C}_t$$, we obtain:\n\n\\begin{aligned} {h_t^i}'(A^\\mathcal {C}_t) = \\frac{\\tilde{\\lambda }_t^i \\tilde{R}_t \\tilde{C}_t^i -2(\\alpha _i+\\delta \\beta _i)\\tilde{\\lambda }_t^i \\tilde{R}_t A^\\mathcal {C}_t - \\alpha _i \\tilde{C}_t^i \\left( A_t^\\mathcal {C}\\right) ^2}{\\left[ \\tilde{\\lambda }_t^i \\tilde{R}_t+ \\alpha _i \\left( A_t^\\mathcal {C}\\right) ^2\\right] ^2}\\ . \\end{aligned}\n(51)\n\nSeeking the roots of $${h_t^i}'(A^\\mathcal {C}_t)$$ yields\n\n\\begin{aligned} {h_t^i}' (A^\\mathcal {C}_t)= 0&\\Leftrightarrow \\underbrace{\\tilde{\\lambda }_t^i \\tilde{R}_t \\tilde{C}_t^i}_{\\equiv x> 0} - \\underbrace{2(\\alpha _i+\\delta \\beta _i)\\tilde{\\lambda }_t^i \\tilde{R}_t}_{\\equiv y> 0} A^\\mathcal {C}_t - \\underbrace{\\alpha _i \\tilde{C}_t^i}_{\\equiv z > 0} \\left( A_t^\\mathcal {C}\\right) ^2 = 0, \\end{aligned}\n(52)\n\\begin{aligned}&\\Leftrightarrow x - y A^\\mathcal {C}_t -z \\left( A^\\mathcal {C}_t\\right) ^2 = 0, \\end{aligned}\n(53)\n\\begin{aligned}&\\Leftrightarrow A^\\mathcal {C}_t = -\\frac{y\\pm \\sqrt{y^2+4xz}}{2z}\\ . \\end{aligned}\n(54)\n\nThus, for every $$h_t^i(A^\\mathcal {C}_t)$$ there exist one positive collective abatement level $$\\bar{A}_t^i$$ such that $${h_t^i}' (\\bar{A}_t^i) = 0$$. In addition it holds (taking into account Eq. (47)):\n\n\\begin{aligned} h_t^i (0)&= 1,&\\quad h_t^i (\\tilde{A}_t^\\mathcal {C})&= \\frac{\\alpha _i \\tilde{a}_t^i \\tilde{A}^\\mathcal {C}_t + \\tilde{\\lambda }_t^i \\tilde{R}_t \\left( 1-\\frac{\\tilde{A}_t^{\\mathcal {C}-i}}{\\tilde{A}_t^\\mathcal {C}}\\right) }{\\tilde{\\lambda }_t^i \\tilde{R}_t + \\alpha _i \\left( \\tilde{A}_t^\\mathcal {C}\\right) ^2} \\in [0,1], \\tilde{A}_t^\\mathcal {C} \\ne 0 \\end{aligned}\n(55a)\n\\begin{aligned} {h_t^i}' (0)&= \\tilde{C}_t^i > 0 ,&\\quad {h_t^i}' (\\tilde{A}_t^\\mathcal {C})&< 0\\ . \\end{aligned}\n(55b)\n\nFocusing attention to the positive half-space $$A^\\mathcal {C}_t \\ge 0$$, all $$h_t^i(A^\\mathcal {C}_t)$$ start at 1 for $$A^\\mathcal {C}_t=0$$. In addition, all $$h_t^i(A^\\mathcal {C}_t)$$ exhibit a unique local extremum at $$\\bar{A}_t^i > 0$$. As $$h_t^{i'}(A^\\mathcal {C}_t)$$ is increasing at $$A^\\mathcal {C}_t=0$$, the local extremum is a local maximum. This implies that all $$h_t^i(A^\\mathcal {C}_t)$$ are increasing until $$\\bar{A}_t^i > 0$$ and decreasing afterwards. This also implies that $$\\tilde{A}^\\mathcal {C}_t > \\bar{A}_t^i$$ for all $$i \\in \\mathcal {C}$$, because $$h^i_t(\\tilde{A}_t^\\mathcal {C}) < 1$$, which can only happen for values $$A^\\mathcal {C}_t > \\bar{A}_t^i$$, as all $$h_t^i(A^\\mathcal {C}_t)$$ start at 1 and further increase until the local extremum at $$\\bar{A}_t^i$$. As $$\\tilde{A}_t^\\mathcal {C} > \\bar{A}_t^i$$, this, in turn, implies that at $$\\tilde{A}_t^\\mathcal {C}$$, all $$h_t^i(\\tilde{A}_t^\\mathcal {C}) \\in [0,1]$$ are monotonically decreasing.Footnote 28 As a consequence, there exists no other value $$A_t'$$ such that $$\\sum _{i \\in \\mathcal {C}} h_t^i(A_t') = 1$$. Then, only the aspired coalition abatement levels $$\\tilde{a}_t^i$$ solve the Euler equations of all countries $$i \\in \\mathcal {C}$$ simultaneously for the weighting scheme $$\\tilde{\\lambda }_t^i$$ and the refund $$\\tilde{R}_t$$.\n\nDifferentiating (40) with respect to $$s_t$$, we obtain\n\n\\begin{aligned} {V_t^i}''(s_t)\|A_t^{-i} = \\delta {W_{t+1}^i}''(\\bar{s}_{t+1})-\\beta _i\\ . \\end{aligned}\n(56)\n\nAs $${W_t^i}(s_t) = {V_t^i}(s_t)\|\\hat{A}_t^{-i}$$, this implies that the equilibrium pay-off $$W^i_{t}(s_t)$$ is strictly concave for all countries $$i \\in \\mathcal {C}$$.\n\nWorking backwards to $$t=1$$ yields a the unique subgame perfect Nash equilibrium outcome that is given by the aspired coalition abatement levels $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$, the abatement path $$\\left\\{ \\check{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\notin \\mathcal {C}}$$ of all non-members countries $$i \\notin \\mathcal {C}$$ and the corresponding path of cumulative greenhouse gas emissions $$\\{s_t\\}_{t=0,\\dots ,T}$$ .\n\nIt remains to show that the RS is feasible, i.e., the weighting scheme $$\\{\\tilde{\\lambda }_t^i\\}_{i=1}^n$$ and the refund $$\\tilde{R}_t$$ are non-negative for all $$t=0,\\dots ,T-1$$, for all feasible coalition abatement paths $$\\left\\{ \\tilde{a}_t^i\\right\\} _{t=0,\\dots ,T}^{i \\in \\mathcal {C}}$$. As $${W_t^i}(s_t) = {V_t^i}(s_t)\|\\hat{A}_t^{-i}$$, we can consecutively apply Eq. (40), insert into Eq. (38) and evaluate in the subgame perfect Nash equilibrium:\n\n\\begin{aligned} \\alpha _i \\tilde{a}_t^i - \\tilde{\\lambda }_t^i \\tilde{R}_t \\frac{\\tilde{A}_t^{\\mathcal {C}-i}}{\\left( \\tilde{A}^\\mathcal {C}_t\\right) ^2} = \\delta \\beta _i \\sum _{\\tau =t+1}^{T} \\delta ^{\\tau -(t+1)} s_\\tau \\ ,\\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(57)\n\nThe corresponding equation in the decentralized solution yields:\n\n\\begin{aligned} \\alpha _i \\hat{a}_t^i = \\delta \\beta _i \\sum _{\\tau =t+1}^{T} \\delta ^{\\tau -(t+1)} \\hat{s}_\\tau ,\\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(58)\n\nBy construction $$\\tilde{a}_t^i > \\hat{a}_t^i$$ for all $$i \\in \\mathcal {C}$$ and $$t=0,\\dots ,T-1$$. As a consequence, it also holds that $$\\hat{s}_t > s_t$$ for all $$t=0,\\dots ,T$$. This, in turn, implies that $$\\{\\tilde{\\lambda }_t^i\\}_{t=0,\\dots ,T-1}^{i \\in \\mathcal {C}} > 0$$ and $$\\left\\{ \\tilde{R}_t\\right\\} _{t=0,\\dots ,T-1} > 0$$. $$\\square$$\n\n### Proof of Proposition 5\n\nThe first part of Proposition directly follows from Eq. (14).\n\nTo show that the any feasible RS can always be implemented as a Pareto improvement over the decentralized solution, we introduce the following abbreviation: Denote the net present value of the discounted sum of abatement costs and environmental damage costs of country i in the decentralized solution and the RS by $$\\hat{K}_i$$ and $$\\tilde{K}_i$$, respectively:\n\n\\begin{aligned} \\hat{K}_i&= \\sum _{t=0}^T \\left[ \\frac{\\alpha _i}{2}\\left( \\hat{a}_t^i\\right) ^2 + \\frac{\\beta _i}{2}\\hat{s}_t^2\\right] , \\end{aligned}\n(59a)\n\\begin{aligned} \\tilde{K}_i&= \\sum _{t=0}^T \\left[ \\frac{\\alpha _i}{2}\\left( \\tilde{a}_t^i\\right) ^2 + \\frac{\\beta _i}{2} \\tilde{s}_t^2\\right] \\ . \\end{aligned}\n(59b)\n\nIn addition, let $$\\tilde{f}_0^i$$ be the net present value of the discounted sum of refunds that country $$i \\in \\mathcal {C}$$ receives in the RS:\n\n\\begin{aligned} \\tilde{f}_0^i = \\sum _{t=1}^{T-1} \\frac{\\tilde{\\lambda }_i \\tilde{R}_t}{(1+\\rho )^t} \\left( \\frac{a^i_t}{\\sum _{j \\in \\mathcal {C}} a^j_t} \\right) . \\end{aligned}\n(59c)\n\nBy construction, all countries $$i \\in \\mathcal {C}$$ are better off in the RS than in the decentralized solution if their initial fees were equal to zero. The reason is that environmental damage costs are smaller under the refunding scheme and abatement costs minus refunds are smaller compared to the decentralized solution. Otherwise, it would not have been in the countries’ best interest to choose the aspired coalition abatement levels. Define the difference in terms of net present value between the RS and the decentralized solution by $$\\hat{f}_0^i$$:\n\n\\begin{aligned} \\hat{f}_0^i = \\hat{K}_i - \\tilde{K}_i + \\tilde{f}_0^i > 0\\ . \\end{aligned}\n(60)\n\nNote that $$\\hat{f}_0^i$$ is the initial fee that would leave country $$i \\in \\mathcal {C}$$ indifferent between the RS and the decentralized solution. Summing-up over all countries $$i \\in \\mathcal {C}$$, we obtain:\n\n\\begin{aligned} \\sum _{i \\in \\mathcal {C}} \\hat{f}_0^i = \\sum _{i \\in \\mathcal {C}} \\left[ \\hat{K}_i - \\tilde{K}_i + \\tilde{f}_0^i\\right] = \\sum _{i \\in \\mathcal {C}} \\left[ \\hat{K}_i - \\tilde{K}_i\\right] + \\tilde{f}_0 > \\tilde{f}_0\\ . \\end{aligned}\n(61)\n\nThus, it is always possible to find a set of initial fees $$f_0^i$$ such that $$\\sum _{i \\in \\mathcal {C}} f_0^i = \\tilde{f}_0$$ and, in addition, $$f_0^i < \\hat{f}_0^i$$ for all $$i \\in \\mathcal {C}$$. $$\\square$$\n\n### Proof of Proposition 6\n\nIn line with the literature, we assume that in the second stage both the modesty parameter $$\\mu$$ and the membership structure are given and common knowledge. Then, the coalition acts as one player in the non-cooperative game, in which the coalition and all other non-member countries choose emission abatement levels to maximize their objective. We assume that in each period $$t=0,\\dots ,T$$ the previous emission abatement choices of all players are common knowledge before all players simultaneously decide on emission abatement levels in period t. The subgame perfect Nash equilibrium is derived by backward induction.\n\nFor a given modesty parameter $$\\mu$$ and a given membership structure $$\\mathcal {C}$$, the coalition is supposed to set emission abatement levels such as to solve optimization problem (15) subject to the equation of motion for aggregate cumulative emissions (3) and given the emission abatement levels of all non-member countries. To solve the problem recursively, we introduce the value function:\n\n\\begin{aligned} V_t^\\mathcal {C}(s_t)\|A_t^{-\\mathcal {C}} = \\max _{\\{a_t^i\\}_{i\\in C}} \\left\\{ \\delta W_{t+1}^C(s_{t+1}) - \\sum _{i\\in C}\\left[ \\frac{\\alpha _i}{2}(a_t^i)^2+\\mu \\frac{\\beta _i}{2}s_t^2\\right] \\right\\} , \\end{aligned}\n(62)\n\nwhere $$A_t^{-\\mathcal {C}}$$ denotes the vector of emission abatement levels of all non-member countries, $$V_t^\\mathcal {C}(s_t)$$ represents the negative of the total coalition costs accruing from period t onwards discounted to period t and $$W_{t+1}^C(s_{t+1})$$ is the coalition’s equilibrium pay-off of the subgame starting in period $$t+1$$ conditional on the stock of accumulated GHG gases $$s_{t+1}$$.\n\nAll non-member countries $$i \\notin \\mathcal {C}$$ seek to minimize the net present value of their own total domestic costs (6) subject to stock dynamics of cumulative global GHG emissions (3) and given the emission abatement levels of all other countries. Again, we introduce the value function:\n\n\\begin{aligned} V_t^i(s_t)\|{A_t^{-i}} = \\max _{\\{a_t^i\\}} \\left\\{ \\delta W_{t+1}^i(s_{t+1}) - \\left[ \\frac{\\alpha _i}{2}(a_t^i)^2+\\frac{\\beta _i}{2}s_t^2\\right] \\right\\} ,\\quad i\\notin \\mathcal {C}, \\end{aligned}\n(63)\n\nwhere $$A_t^{-i}$$ denotes the vector of emission abatement levels of all other countries $$j \\ne i$$, $$V_t^i(s_t)$$ represents the negative of the total country i’ costs accruing from period t onwards discounted to period t and $$W_{t+1}^i(s_{t+1})$$ is the country i’s equilibrium pay-off of the subgame starting in period $$t+1$$ conditional on the stock of accumulated GHG gases $$s_{t+1}$$.\n\nDifferentiating the value functions (62) and (63) with respect to $$a^i_t$$ and setting them equal to zero, we derive the following first-order conditions:\n\n\\begin{aligned} \\alpha _i a^i_t&= -\\delta {W^{\\mathcal {C}}_{t+1}}'(s_{t+1}),\\qquad \\forall \\ i \\in \\mathcal {C},\\quad t=0,\\dots ,T, \\end{aligned}\n(64a)\n\\begin{aligned} \\alpha _i a^i_t&= -\\delta {W^{i}_{t+1}}'(s_{t+1}),\\qquad \\forall \\ i \\notin \\mathcal {C},\\quad t=0,\\dots ,T\\ . \\end{aligned}\n(64b)\n\nThe optimization problems of the coalition and all non-member countries in period t are strictly concave if\n\n\\begin{aligned}&\\delta {W_{t+1}^{\\mathcal {C}}}''(s_{t+1}) - \\alpha _i < 0,\\qquad \\forall \\ i \\in \\mathcal {C},\\quad t=0,\\dots ,T, \\end{aligned}\n(65a)\n\\begin{aligned}&\\delta {W_{t+1}^{i}}''(s_{t+1}) - \\alpha _i < 0,\\qquad \\forall \\ i \\notin \\mathcal {C},\\quad t=0,\\dots ,T, \\end{aligned}\n(65b)\n\nin which case the first-order conditions () implicitly define the coalition’s and all non-member countries’ unique best response functions.\n\nIn addition, differentiating the value functions (62) and (63) with respect to $$s_t$$ and applying the envelope theorem yields\n\n\\begin{aligned} -{V_{t}^{\\mathcal {C}}}'(s_t)\|A_t^{-\\mathcal {C}}&= \\mu \\mathcal {B}^\\mathcal {C} s_t - \\delta {W_{t+1}^{\\mathcal {C}}}' (s_{t+1}),\\qquad \\forall \\ i \\in \\mathcal {C},\\quad t=0,\\dots ,T, \\end{aligned}\n(66a)\n\\begin{aligned} -{V_{t}^{i}}'(s_t)\|A_t^{-i}&= \\beta _i s_t - \\delta {W_{t+1}^{i}}' (s_{t+1}),\\qquad \\forall \\ i \\notin \\mathcal {C},\\quad t=0,\\dots ,T, \\end{aligned}\n(66b)\n\nwhere we have introduced the notation $$\\mathcal {B}^\\mathcal {C} = \\sum _{i\\in \\mathcal {C}} \\beta _i$$.\n\nStarting from $$W^\\mathcal {C}_{T+1}(s_{T+1}) \\equiv 0 \\equiv W^i_{T+1}(s_{T+1})$$ for all $$i \\in \\mathcal {I}$$, implying that the objective function of the optimization problem of the coalition and all non-member countries is strictly concave. As a consequence, Eq. () characterize the coalition’s and all non-member countries’ best response, which is given by $$\\bar{a}^i_T=0$$ for all $$I \\in \\mathcal {I}$$ independently of the emission abatement choices of all other countries. As a consequence, $$\\tilde{a}^i_T=0$$ for all $$i \\in \\mathcal {C}$$ and $$\\check{a}^i_T=0$$ for all $$i \\notin \\mathcal {C}$$ is the unique and symmetric Nash equilibrium for the subgame starting in period T given the stock of cumulative greenhouse gas emissions $$s_T$$. The equilibrium pay-offs are given by $$W^\\mathcal {C}_T(s_T)= V^\\mathcal {C}_T(s_T)\|\\hat{A}^{-\\mathcal {C}}_T$$ for the coalition and $$W^i_T(s_T)=V^i_T(s_T)\|\\hat{A}^{-i}_T$$ and are strictly concave:\n\n\\begin{aligned} W^\\mathcal {C}_T(s_T)&= -\\mu \\frac{\\mathcal {B}^\\mathcal {C}}{2}s_T^2 \\qquad \\Rightarrow \\qquad {W_{T}^\\mathcal {C}}''(s_T) = -\\mu \\mathcal {B}^\\mathcal {C}, \\end{aligned}\n(67a)\n\\begin{aligned} W^i_T(s_T)&= -\\frac{\\beta _i}{2}s_T^2 \\qquad \\Rightarrow \\qquad {W_{T}^i}''(s_T) = -\\beta _i, \\qquad \\forall \\ i \\notin \\mathcal {C}\\ . \\end{aligned}\n(67b)\n\nAs a consequence, the optimization problem of the coalition and all non-member countries is also strictly concave in period T.\n\nNow assume there exists a unique subgame perfect Nash equilibrium for the subgame starting in period $$t+1$$ with a stock of greenhouse gas emissions of $$s_{t+1}$$ yielding equilibrium pay-offs $$W^\\mathcal {C}_{t+1}(s_{t+1})$$ and $$W^i_{t+1}(s_{t+1})$$ to the coalition and all non-member countries $$i \\notin \\mathcal {C}$$, respectively, with $${W^\\mathcal {C}_{t+1}}''(s_{t+1})<0$$ and $${W^i_{t+1}}''(s_{t+1})<0$$ . Then the optimization problem in period t is strictly concave for the coalition and all non-member countries $$i \\notin \\mathcal {C}$$, implying there exists a unique best response $$\\bar{a}^i_t$$ for all countries $$i \\in \\mathcal {I}$$ given the emission abatements of all other countries $$j \\ne i$$, which is given implicitly by\n\n\\begin{aligned} \\alpha _i \\bar{a}^i_t&= -\\delta {W^{\\mathcal {C}}_{t+1}}'(\\bar{s}_{t+1}),\\qquad \\forall \\ i \\in \\mathcal {C}, \\end{aligned}\n(68a)\n\\begin{aligned} \\alpha _i \\bar{a}^i_t&= -\\delta {W^{i}_{t+1}}'(\\bar{s}_{t+1})\\ ,\\qquad \\forall \\ i \\notin \\mathcal {C}, \\end{aligned}\n(68b)\n\nwhere $$\\bar{s}_{t+1} = s_t+ \\mathcal {E} - \\bar{a}_t^i - A_t^{-i}$$. As, by assumption, $$-{W^\\mathcal {C}_{t'}}'(s_{t'})=-{V^\\mathcal {C}_{t'}}'(s_{t'})\|\\hat{A}_{t'}^{-\\mathcal {C}}$$ and $$-{W^i_{t'}}'(s_{t'})=-{V^i_{t'}}'(s_{t'})\|\\hat{A}_{t'}^{-i}$$ for all $$t'\\ge t+1$$, we can exploit conditions () to obtain:\n\n\\begin{aligned} a_t^i&= \\delta \\tilde{a}^i_{t+1} + \\mu \\delta \\frac{\\mathcal {B}^{\\mathcal {C}}}{\\alpha _i} \\left( s_t + \\mathcal {E}- \\sum _{j \\in \\mathcal {I}} a^j_t\\right) ,\\qquad \\forall \\ i \\in \\mathcal {C}, \\end{aligned}\n(69a)\n\\begin{aligned} a_t^i&= \\delta \\check{a}^i_{t+1} + \\delta \\gamma _i \\left( s_t + \\mathcal {E}-\\sum _{j \\in \\mathcal {I}} a^j_t\\right) ,\\qquad \\forall \\ i \\notin \\mathcal {C}\\ . \\end{aligned}\n(69b)\n\nSumming up Eq. (69a) over all coalition members $$i \\in \\mathcal {C}$$ and Eq. (69b) over all non-member countries $$i \\notin \\mathcal {C}$$, we obtain the following equations for the aggregate abatement levels $$A^\\mathcal {C}_t = \\sum _{i \\in \\mathcal {C}} a^i_t$$ and $$A^\\mathcal {NC}_t = \\sum _{i \\notin \\mathcal {C}} a^i_t$$ of the coalition and all non-member countries, respectively:\n\n\\begin{aligned} A^\\mathcal {C}_t&= \\delta \\tilde{A}^\\mathcal {C}_{t+1} + \\mu \\delta \\mathcal {A}^\\mathcal {C} \\mathcal {B}^{\\mathcal {C}} \\left( s_t + \\mathcal {E}- A^\\mathcal {C}_t - A^\\mathcal {NC}_t\\right) , \\end{aligned}\n(70a)\n\\begin{aligned} A^\\mathcal {NC}_t&= \\delta \\check{A}^\\mathcal {NC}_{t+1} + \\delta \\Gamma ^\\mathcal {NC}\\left( s_t + \\mathcal {E}- A^\\mathcal {C}_t - A^\\mathcal {NC}_t\\right) , \\end{aligned}\n(70b)\n\nwhere we have used the abbreviation $$\\mathcal {A}^\\mathcal {C} = \\sum _{i \\in \\mathcal {C}} 1/\\alpha _i$$ and $$\\Gamma ^\\mathcal {NC} = \\sum _{i \\notin \\mathcal {C}} \\gamma _i$$. Solving this system of equations for $$A^\\mathcal {C}_t$$ and $$A^\\mathcal {NC}_t$$, we obtain the aggregate abatement levels of the coalition and non-member countries, respectively, for period t in the subgame perfect Nash equilibrium:\n\n\\begin{aligned} \\tilde{A}^\\mathcal {C}_t&= \\frac{\\delta \\left[ \\tilde{A}^\\mathcal {C}_{t+1}\\left( 1+\\delta \\Gamma ^{\\mathcal {NC}}\\right) + \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^{\\mathcal {C}}\\left( s_t + \\mathcal {E}- \\delta \\check{A}^\\mathcal {NC}_{t+1} \\right) \\right] }{1+\\delta \\Gamma ^{\\mathcal {NC}}+\\mu \\delta \\mathcal {A}^\\mathcal {C} \\mathcal {B}^{\\mathcal {C}}}, \\end{aligned}\n(71a)\n\\begin{aligned} \\check{A}^\\mathcal {NC}_t&= \\frac{\\delta \\left[ \\check{A}^\\mathcal {NC}_{t+1}\\left( 1+\\mu \\delta \\mathcal {A}^\\mathcal {C} \\mathcal {B}^{\\mathcal {C}}\\right) + \\Gamma ^{\\mathcal {NC}} \\left( s_t + \\mathcal {E}- \\delta \\tilde{A}^\\mathcal {C}_{t+1} \\right) \\right] }{1+\\delta \\Gamma ^{\\mathcal {NC}}+\\mu \\delta \\mathcal {A}^\\mathcal {C} \\mathcal {B}^{\\mathcal {C}}}\\ . \\end{aligned}\n(71b)\n\nInserting $$\\tilde{A}^\\mathcal {C}_t$$ and $$\\check{A}^\\mathcal {NC}_t$$ back into Eq. () yields the unique equilibrium abatement level in period t for all countries $$i \\in \\mathcal {I}$$:\n\n\\begin{aligned} \\tilde{a}_t^i&= \\delta \\tilde{a}^i_{t+1} + \\mu \\delta \\frac{\\mathcal {B}^{\\mathcal {C}}}{\\alpha _i} \\left( s_t + \\mathcal {E}- \\tilde{A}^\\mathcal {C}_t - \\check{A}^\\mathcal {NC}_t \\right) ,\\qquad \\forall \\ i \\in \\mathcal {C}, \\end{aligned}\n(72a)\n\\begin{aligned} \\check{a}_t^i&= \\delta \\check{a}^i_{t+1} + \\delta \\gamma _i \\left( s_t + \\mathcal {E}-\\tilde{A}^\\mathcal {C}_t - \\check{A}^\\mathcal {NC}_t \\right) ,\\qquad \\forall \\ i \\notin \\mathcal {C}\\ . \\end{aligned}\n(72b)\n\nDifferentiating () with respect to $$s_t$$, we obtain\n\n\\begin{aligned} {V_{t}^{\\mathcal {C}}}''(s_t)\|A_t^{-\\mathcal {C}}&= \\delta {W_{t+1}^{\\mathcal {C}}}'' (s_{t+1}) -\\mu \\mathcal {B}^\\mathcal {C},\\qquad \\forall \\ i \\in \\mathcal {C}, \\end{aligned}\n(73a)\n\\begin{aligned} {V_{t}^{i}}''(s_t)\|A_t^{-i}&= \\delta {W_{t+1}^{i}}''(s_{t+1}) - \\beta _i,\\qquad \\forall \\ i \\notin \\mathcal {C}\\ . \\end{aligned}\n(73b)\n\nAs $${W_t^\\mathcal {C}}''(s_t) = {V_t^\\mathcal {C}}''(s_t)\|\\hat{A}_t^{-\\mathcal {C}}$$ and $${W_t^i}''(s_t) = {V_t^i}''(s_t)\|\\hat{A}_t^{-i}$$, this implies that the equilibrium pay-offs $$W^\\mathcal {C}_{t}(s_t)$$ and $$W^i_{t}(s_t)$$ are strictly concave for the coalition and all non-member countries $$i \\notin \\mathcal {C}$$.\n\nWorking backwards until $$t=0$$ yields unique sequences of emission abatements $$\\{\\tilde{a}^i_t\\}^T_{t = 0}$$ and $$\\{\\tilde{a}^i_t\\}^T_{t = 0}$$ for all coalition countries $$i\\in \\mathcal {C}$$ and all non-member countries $$i \\notin \\mathcal {C}$$, respectively, and the corresponding sequence of the stock of cumulative greenhouse gas emissions $$s_t$$ ($$t=0,\\dots ,T$$) that constitute the unique subgame perfect Nash equilibrium outcome of the second stage of the modest international environmental agreement.\n\nHaving established existence and uniqueness of the subgame perfect Nash equilibrium, we now employ Eq. () together with the equation of motion for the stock of aggregated cumulative emissions (3) to derive the following system of first-order linear difference equations:\n\n\\begin{aligned} A_{t+1}^\\mathcal {C}&= \\left( \\frac{1}{\\delta } + \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^\\mathcal {C} \\right) A_t^\\mathcal {C} + \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^\\mathcal {C} A_t^\\mathcal {NC} - \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^\\mathcal {C} s_t - \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^\\mathcal {C} \\mathcal {E}, \\end{aligned}\n(74a)\n\\begin{aligned} A_{t+1}^\\mathcal {NC}&= \\Gamma ^\\mathcal {NC} A_t^\\mathcal {C} + \\left( \\frac{1}{\\delta }+\\Gamma ^\\mathcal {NC} \\right) A_{t}^\\mathcal {NC} -\\Gamma ^\\mathcal {NC} s_t - \\Gamma ^\\mathcal {NC} \\mathcal {E}, \\end{aligned}\n(74b)\n\\begin{aligned} s_{t+1}&= - A_t^\\mathcal {C} -A_t^\\mathcal {NC} + s_t +\\mathcal {E}\\ . \\end{aligned}\n(74c)\n\nBy introducing the abbreviations $$x = \\mu \\mathcal {A}^\\mathcal {C} \\mathcal {B}^\\mathcal {C}$$ and $$y = \\Gamma ^\\mathcal {NC}$$ and the matrix M\n\n\\begin{aligned} M = \\begin{pmatrix} \\frac{1}{\\delta } + x &{} x &{} -x\\\\ y &{} \\frac{1}{\\delta }+y &{} -y \\\\ -1 &{} -1 &{} +1 \\end{pmatrix}, \\end{aligned}\n(75)\n\nwe rewrite the system () in matrix form:\n\n\\begin{aligned} \\begin{pmatrix} A_{t+1}^\\mathcal {C}\\\\ A_{t+1}^\\mathcal {NC}\\\\ s_{t+1} \\end{pmatrix} = M \\cdot \\begin{pmatrix} A_{t}^\\mathcal {C}\\\\ A_{t}^\\mathcal {NC}\\\\ s_{t} \\end{pmatrix} + \\begin{pmatrix} - x \\mathcal {E}\\\\ - y \\mathcal {E}\\\\ \\mathcal {E} \\end{pmatrix}\\ . \\end{aligned}\n(76)\n\nThe general solution of the matrix equation (76) is given by:\n\n\\begin{aligned} \\begin{pmatrix} A_{t}^\\mathcal {C}\\\\ A_{t}^\\mathcal {NC}\\\\ s_{t} \\end{pmatrix} = \\begin{pmatrix} \\bar{A}^\\mathcal {C}\\\\ \\bar{A}^\\mathcal {NC}\\\\ \\bar{s} \\end{pmatrix} + B_1(T) \\nu _1 \\lambda _1^t + B_2(T) \\nu _2 \\lambda _2^t + B_3(T) \\nu _3 \\lambda _3^t, \\end{aligned}\n(77)\n\nwhere $$\\bar{A}^\\mathcal {C}$$, $$\\bar{A}^\\mathcal {NC}$$ and $$\\bar{s}$$ denote the steady state values of $$A_{t}^\\mathcal {C}$$, $$A_{t}^\\mathcal {NC}$$ and $$s_t$$, $$\\lambda _i$$ are the eigenvalues and $$\\nu _i$$ the eigenvectors of the matrix M, and $$B_i(T)$$ are constants determined by the initial and terminal conditions of the stock and the emission abatement levels ($$i=1,\\dots ,3$$).\n\nCalculating the steady state values by setting\n\n\\begin{aligned} A_{t+1}^\\mathcal {C}=A_{t}^\\mathcal {C} = \\bar{A}^\\mathcal {C},\\quad A_{t+1}^\\mathcal {NC}=A_{t}^\\mathcal {NC} = \\bar{A}^\\mathcal {NC}\\ ,\\quad s_{t+1} = s_t = \\bar{s}, \\end{aligned}\n(78)\n\nyields:\n\n\\begin{aligned} \\bar{A}^\\mathcal {C}&= \\frac{x}{x+y}\\mathcal {E} \\end{aligned}\n(79a)\n\\begin{aligned} \\bar{A}^\\mathcal {NC}&= \\frac{y}{x+y}\\mathcal {E}\\end{aligned}\n(79b)\n\\begin{aligned} \\bar{s}&= \\frac{1-\\delta }{\\delta } \\frac{\\mathcal {E}}{x+y} \\end{aligned}\n(79c)\n\nIn addition, for the matrix M we derive the following eigenvalues $$\\lambda _i$$ ($$i=1,\\dots ,3$$):\n\n\\begin{aligned} \\lambda _1&= \\frac{1}{\\delta }, \\end{aligned}\n(80a)\n\\begin{aligned} \\lambda _2&= \\frac{1+\\delta +\\delta (x+y)-\\sqrt{[1+\\delta +\\delta (x+y)]^2-4\\delta }}{2\\delta }\\ , \\end{aligned}\n(80b)\n\\begin{aligned} \\lambda _3&= \\frac{1+\\delta +\\delta (x+y)+\\sqrt{[1+\\delta +\\delta (x+y)]^2-4\\delta }}{2\\delta }\\ , \\end{aligned}\n(80c)\n\nand eigenvectors ($$i=1,\\dots ,3$$):\n\n\\begin{aligned} \\nu _1&= \\left\\{ -1,1,0\\right\\} ,\\end{aligned}\n(81a)\n\\begin{aligned} \\nu _2&= \\left\\{ \\frac{x}{x+y}(1-\\lambda _2), \\frac{y}{x+y}(1-\\lambda _2), 1 \\right\\} , \\end{aligned}\n(81b)\n\\begin{aligned} \\nu _3&= \\left\\{ \\frac{x}{x+y}(1-\\lambda _3), \\frac{y}{x+y}(1-\\lambda _3), 1 \\right\\} , \\end{aligned}\n(81c)\n\nInserting into Eq. (77) yields:\n\n\\begin{aligned} A_{t}^\\mathcal {C}&= \\bar{A}^\\mathcal {C} - B_1(T)\\lambda _1^t + \\frac{x}{x+y}\\left[ B_2(T)(1-\\lambda _2)\\lambda _2^t + B_3(T)(1-\\lambda _3) \\lambda _3^t\\right] \\end{aligned}\n(82a)\n\\begin{aligned} A_{t}^\\mathcal {NC}&= \\bar{A}^\\mathcal {NC} + B_1(T)\\lambda _1^t + \\frac{y}{x+y}\\left[ B_2(T)(1-\\lambda _2)\\lambda _2^t + B_3(T)(1-\\lambda _3) \\lambda _3^t\\right] \\end{aligned}\n(82b)\n\\begin{aligned} s_{t}&= \\bar{s} + B_2(T) \\lambda _2^t + B_3(T) \\lambda _3^t \\end{aligned}\n(82c)\n\nThe constants $$B_i(T)$$ ($$i=1,\\dots ,3$$) are derived from the initial stock $$s_0$$ of cumulative GHG emissions and the terminal conditions $$A_{T}^\\mathcal {C} = 0$$ and $$A_{T}^\\mathcal {NC} = 0$$ of aggregate emission abatement levels, which imply\n\n\\begin{aligned} B_1(T)&= \\frac{y \\bar{A}^C - x \\bar{A}^{NC} }{(x+y)\\lambda _1^T}, \\end{aligned}\n(83a)\n\\begin{aligned} B_2(T)&= - \\frac{\\bar{A}^C + \\bar{A}^{NC} + (s_0-\\bar{s})(1-\\lambda _3)\\lambda _3^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}, \\end{aligned}\n(83b)\n\\begin{aligned} B_3(T)&= \\frac{\\bar{A}^C + \\bar{A}^{NC} + (s_0-\\bar{s})(1-\\lambda _2)\\lambda _2^{T}}{(1-\\lambda _2) \\lambda _2^T - (1-\\lambda _3) \\lambda _3^T}. \\end{aligned}\n(83c)\n\nBy inserting these expressions back into equations () yields the aggregate abatement levels $$A_t^\\mathcal {C}$$ and $$A_t^\\mathcal {NC}$$ and the stock of aggregate cumulative emissions $$s_t$$ in the subgame perfect Nash equilibrium ($$t=0,\\dots ,T$$).\n\nFinally, we determine the individual countries’ abatement levels in the subgame perfect Nash equilibrium. Using backward induction starting from $$t=T$$, we obtain from equations ():\n\n\\begin{aligned} a_T^i&= 0 ,\\qquad \\forall \\ i \\in \\mathcal {I}\\ ,\\end{aligned}\n(84a)\n\\begin{aligned} a_t^i&= \\frac{A^\\mathcal {C}_t}{\\alpha _i \\mathcal {A}^\\mathcal {C}}, \\qquad \\forall \\ i\\in \\mathcal {C}, \\quad t=0,\\dots ,T-1,\\end{aligned}\n(84b)\n\\begin{aligned} a_t^i&= \\frac{\\gamma _i}{\\Gamma ^\\mathcal {NC}} A^\\mathcal {NC}_{t}, \\qquad \\forall \\ i\\notin \\mathcal {C}, \\quad t=0,\\dots ,T-1\\ . \\end{aligned}\n(84c)\n\n$$\\square$$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6007411,"math_prob":0.99787265,"size":44486,"snap":"2022-27-2022-33","text_gpt3_token_len":16904,"char_repetition_ratio":0.2576548,"word_repetition_ratio":0.25491598,"special_character_ratio":0.38591915,"punctuation_ratio":0.06161194,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9990467,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-16T14:17:31Z\",\"WARC-Record-ID\":\"<urn:uuid:fe14730c-c9da-4a5d-a3bf-fc6666b5c84b>\",\"Content-Length\":\"378125\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9dc718b8-f725-486b-8248-821928e00a68>\",\"WARC-Concurrent-To\":\"<urn:uuid:c0d305cf-8135-4951-8af0-79e82adc7154>\",\"WARC-IP-Address\":\"146.75.36.95\",\"WARC-Target-URI\":\"https://link.springer.com/article/10.1007/s10640-021-00597-3\",\"WARC-Payload-Digest\":\"sha1:JIOZCQ23RVTBBSB73Z5BZTZQE7LMR2C5\",\"WARC-Block-Digest\":\"sha1:WSJ3GDB7IJY4A7KQZNIDJO2XBQYQWL62\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882572304.13_warc_CC-MAIN-20220816120802-20220816150802-00223.warc.gz\"}"}
https://www.arxiv-vanity.com/papers/hep-ph/9402253/	[ "arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.\n\n# MPI-Ph/93-103 CERN-Th.7163/94 February 1994 Electroweak Symmetry Breaking and Bottom-Top Yukawa Unification\n\nM. Carena, M. Olechowski\n\nS. Pokorski and C. E. M. Wagner\n\nCERN Theory Division,\n1211 Geneva 23, Switzerland\n\nMax-Planck-Institut für Physik, Werner-Heisenberg-Institut\nFöhringer Ring 6, D-80805 München, Germany.\n\nInstitute of Theoretical Physics, Warsaw University\nul. Hoza 69, 00-681 Warsaw, Poland\n\nOn leave of absence from the Institute of Theoretical Physics, Warsaw University\n###### Abstract\n\nThe condition of unification of gauge couplings in the minimal supersymmetric standard model provides successful predictions for the weak mixing angle as a function of the strong gauge coupling and the supersymmetric threshold scale. In addition, in some scenarios, e.g. in the minimal SO(10) model, the tau lepton and the bottom and top quark Yukawa couplings unify at the grand unification scale. The condition of Yukawa unification leads naturally to large values of , implying a proper top quark–bottom quark mass hierarchy. In this work, we investigate the feasibility of unification of the Yukawa couplings, in the framework of the minimal supersymmetric standard model with (assumed) universal mass parameters at the unification scale and with radiative breaking of the electroweak symmetry. We show that strong correlations between the parameters , and appear within this scheme, where is the ratio of the top quark Yukawa coupling to its infrared fixed point value. These correlations have relevant implications for the sparticle spectrum, which presents several characteristic features. In addition, we show that due to large corrections to the running bottom quark mass induced through the supersymmetry breaking sector of the theory, the predicted top quark mass and values are significantly lower than those previously estimated in the literature.\n\n## 1 Introduction\n\nThe minimal supersymmetric standard model provides a well motivated and predictive extension of the successful standard model of the strong and electroweak interactions. The condition of unification of couplings is implicit within this scheme and the predictions for the weak mixing angle are in good agreement with the values measured by the most recent measurements at LEP . In addition to the gauge coupling unification condition, relations between the values of the Yukawa couplings of the quarks and leptons of the third generation appear in the minimal supersymmetric grand unification scheme. In particular, in the minimal SU(5) model the unification of bottom and tau Yukawa couplings is obtained. The bottom–tau Yukawa unification condition leads to predictions for the top quark mass as a function of the running bottom quark mass, the strong gauge coupling value and the value of , the ratio of vacuum expectation values .\n\nRecently, it has been observed that for the phenomenologically allowed values of the bottom quark mass and moderate values of , large values of the top quark Yukawa coupling are needed in order to contravene the strong gauge coupling renormalization of the bottom Yukawa coupling . In general, for large enough values of the top quark Yukawa coupling at the grand unification scale, the low energy Yukawa coupling is strongly focussed to a quasi infrared fixed point . In the minimal supersymmetric standard model, the quasi infared fixed point predictions for the physical top quark mass are given by , with GeV for the strong gauge coupling . It has been recently shown that for the values of the strong gauge coupling consistent with the condition of gauge coupling unification, with reasonable threshold corrections at the grand unification and supersymmetry breaking scales, the top quark mass should be within of its quasi infrared fixed point values if the condition of bottom–tau Yukawa unification is required .\n\nA more predictive scheme is obtained in the framework of the minimal SO(10) unification. In this case top–bottom quark Yukawa unification is also required, implying that, for a given value of the bottom quark mass and the strong gauge coupling value , not only the top quark mass but also the value of may be determined. Remarkably, large values of are obtained in this case, leading to a proper bottom–top mass hierarchy . For these large values of , the bottom quark Yukawa coupling itself plays a relevant role in the running of the top quark Yukawa coupling, as well as in the running of the ratio of the bottom to tau Yukawa couplings. This leads to a somewhat weaker convergence of the top quark Yukawa coupling to its infrared fixed point value, together with a slight modification of the infrared fixed point expression .\n\nMoreover, it has been recently observed that for these large values of , potentially large corrections to the running bottom quark mass may be induced through the supersymmetry breaking sector of the theory . Although in the exact supersymmetric theory the bottom quark and tau lepton only couple to one of the Higgs fields , a coupling of these fermions to the Higgs field is induced at the one loop level in the presence of soft supersymmetry breaking terms. These corrections are decisive in obtaining the predictions for the top quark mass. Indeed, for the characteristic values of arising if the top–bottom Yukawa unification is required, , the bottom mass corrections would be very large, unless the supersymmetric mass parameter and the gluino mass are much lower than the characteristic squark masses. This hierarchy of masses may be achieved by imposing certain symmetries in the theory. These symmetries may, however, be in conflict with the radiative breaking of the electroweak symmetry, particularly in simple supersymmetry breaking scenarios.\n\nThe question of radiative electroweak symmetry breaking with large appears hence, as an independent issue, which has been investigated in minimal supersymmetric models with universal soft supersymmetry breaking parameters at the grand unification scale with encouraging results , , , . However, not enough attention was paid either to the full consistency with the requirement of the unification of the gauge and Yukawa couplings, nor to a systematical identification of the complete parameter space at the GUT scale, which gives electroweak symmetry breaking with large . Recently, we presented an investigation of the properties of the radiative electroweak symmetry breaking solutions for small and moderate values of and a top quark Yukawa coupling taking values close to its infrared fixed point solution, as required by bottom–tau Yukawa coupling unification . We obtained quite remarkable correlations between different supersymmetric mass parameters, as well as an effective reduction of the number of free independent parameters at the grand unification scale. It is the purpose of this work to perform a similar analysis for the large regime.\n\nWe use the recently developed bottom - up approach to radiative electroweak symmetry breaking, , which is particularly suitable for a systematic search for large solutions, and possibly to identify the symmetries underlying those solutions. In our calculation we use the two loop renormalization group evolution of gauge and Yukawa couplings, while the Higgs and supersymmetric mass parameters are evolved at the one loop level. The leading supersymmetric threshold corrections to the Higgs quartic couplings and to all supersymmetric mass parameters are included in the analysis. We proceed by fixing the experimentally known values of , , and (with their corresponding uncertainties). After choosing a set of values for and , the unification condition of the three Yukawa couplings fixes their running in the range from to . Next, the search for electroweak symmetry breaking solutions is performed by scanning over the CP odd Higgs mass and the low energy stop mass parameters. For each solution the one–loop correction to the running bottom mass at is calculated and finally the pole bottom mass is obtained. The predictions for the top quark mass and is the collection of those values of and for which there are solutions with the pole bottom mass within the experimentally acceptable range. A more detailed explanation of this procedure will be given below.\n\nWe will show that under the requirement of top–bottom–tau Yukawa unification, the condition of radiative electroweak symmetry breaking implies strong correlations between the supersymmetric parameter and the soft supersymmetry breaking term and , where is the ratio at the electroweak scale of the top quark Yukawa coupling to its infrared fixed point value. These correlations allow a precise determination of the bottom mass corrections, which become significantly large for the large values of consistent with the unification of the three Yukawa couplings of the third generation. This, in turn, implies that the top quark mass predictions are quite different from those ones obtained if the bottom mass corrections were neglected. In section 2 we present a general discussion of the model and our choice of low energy parameters. In section 3 we discuss the radiative electroweak symmetry breaking conditions and its implication for the low energy parameters of the Higgs potential. We present an approximate analytical expression for the one loop renormalization group running of the mass parameters of the Higgs and supersymmetric particles, for the case in which the top and bottom Yukawa couplings unify at . In section 4 we present a detailed numerical analysis of the implications of the radiative breaking for the supersymmetric mass parameter and the supersymmetry breaking parameters at the grand unification scale, and we compare it with the approximate analytical solution. In section 5 we analyse the one loop corrections to the bottom and tau masses and their implications for the top quark mass predictions. In section 6 we analyse the spectrum, together with the constraints coming from the bounds on the decay rate. We reserve section 7 for the conclusions.\n\n## 2 Gauge and Yukawa Coupling Unification Predictions\n\nWe begin with a short discussion of the predictions for the top quark mass following from the unification of the gauge and Yukawa couplings (before imposing the requirement of radiative electroweak breaking), recalling and slightly extending some of the results presented in Refs. - . The gauge coupling unification condition gives predictions for the weak mixing angle as a function of the strong gauge coupling . The unification condition implies (at the two loop level) the following numerical correlation \n\n sin2θW(MZ)=0.2324−0.25(α3(MZ)−0.123)±0.0025 (1)\n\nwhere the central value corresponds to an effective supersymmetric threshold scale and the error is the estimated uncertainty in the prediction arising from possible supersymmetric threshold corrections (corresponding to vary the effective supersymmetric threshold scale from 15 GeV to 1 TeV), threshold corrections at the unification scale as well as from higher dimensional operators. On the other hand, is given by the electroweak parameters , , as a function of the physical top quark mass (at the one loop level) by the formula :\n\n sin2θW(MZ)=0.2324−10−7GeV−2(M2t−(138GeV)2)±0.003 (2)\n\nTherefore, the predictions from gauge coupling unification agree with experimental data provided\n\n M2t=(138GeV)2+0.25×107GeV2(α3(MZ)−0.123±0.01) (3)\n\nThe above correlation defines a band whose lower bound is shown in Fig. 1 ( the upper bound is above 0.13 for GeV). We observe that the top quark mass implies .\n\nAnother issue is that of unification of the bottom and tau Yukawa couplings. In this work the unification of Yukawa couplings is always studied numerically at the two–loop level. However, for a qualitative discussion we refer to the one–loop renormalization group equation for the ratio of the bottom to tau Yukawa couplings which, in the limit of vanishing electroweak gauge couplings, reads\n\n drdt=r8π(16α33−3Yb−Yt+3Yτ) (4)\n\nwhere and . Starting from values of the ratio above one at the scale , as required by experimentally allowed values of the bottom mass GeV and the tau mass, GeV, is strongly renormalized and in the limit of negligible Yukawa couplings for values of within the experimentally determined range it becomes lower than one at scales far below the grand unification scale . Hence, in order to get , for a given value of , the Yukawa couplings in Eq.(4) should be adjusted to compensate the strong gauge coupling effect. For low and moderate values of , , it is the top quark Yukawa coupling which is fixed as a function of , by the bottom–tau Yukawa unification requirement. As we discussed in the introduction, this perturbative unification requires values that are within 10 of the top quark mass infared quasi fixed point value.\n\nHere we are primary concerned with the large solution. Then, the bottom and the top quark Yukawa couplings are of the same order of magnitude and both are important to get . The unification of the three Yukawa couplings takes place not only for a particular value of but also of , for given values of , and , implying a fixed . An important remark is in order here. The bottom mass which is directly relevant for the top mass prediction following from the Yukawa coupling unification is the tree level running mass . As we discussed in the previous section, in the large case it may receive large loop corrections from sparticle exchange loops, at least in some range of parameters of the model. The physical (pole mass) is obtained from the running mass (which is related to by the Standard Model RG equations) by inclusion of QCD corrections, which are universal for the Standard Model and its supersymmetric version. At the two loop level, they are given by \n\n mb(Mb)=Mb1+4α3(Mb)3π+Kb(α3(Mb)π)2, (5)\n\nwhere . The loop corrections to the running bottom mass at induced through sparticle exchange loops are an important issue for models with radiative breaking of the electroweak symmetry. In order to distinguish them from QCD corrections, we introduce the pole bottom mass , which is obtained from the unification condition in the case in which the supersymmetric one loop corrections to are ignored. Due to the fact that the supersymmetric corrections could be quite sizeable, for the allowed solutions of the model, the mass may be significantly different from the physical mass .\n\nThe predictions for and , following from the unification of the three Yukawa couplings, are shown in Fig. 1 for several values of the mass as a function of . The supersymmetric particle masses were set at the scale , while the unification scale was defined as the scale at which the electroweak gauge couplings unify. Fig. 1 shows also the region in the - plane consistent with the unification of gauge couplings, after considering the experimental dependence of on the top quark mass and threshold corrections at the supersymmetric and grand unification scale, Eq.(3).\n\nFrom Fig. 1 we draw the following conclusions: In case that the supersymmetric loop contributions to the bottom mass were negligible, , and taking into account the experimentally acceptable values for the physical bottom mass, GeV , the unification of the gauge and Yukawa couplings drives the top quark mass towards large values (Note the fact that for the IR fixed point solution is lower than for ) . Although the predictions for are no longer so strongly constrained to be close to its infrared quasi fixed point values as for the low and moderate values of (as explained above, strong renormalization effects in the running of are partially cancelled by itself), for the values of consistent with gauge coupling unification the top quark mass is still close to the appropriate infrared fixed point solution. For instance, for a physical bottom quark mass GeV, and , the top quark mass is predicted to be GeV. In general, as it is clear from Fig. 1, if the supersymmetric corrections to the running bottom mass were small, the top quark mass would acquire values GeV within this scheme . In Fig. 1 we also plot the predictions obtained for values of larger than the experimental upper bound for the bottom mass, GeV, which will become of interest while studying the supersymmetric corrections to the bottom mass. Indeed, the values of the top quark mass in the range, say (140–160) GeV are compatible with unification of couplings provided and sizeable supersymmetric loop corrections to the bottom mass are induced. As we shall show below, this is the case in the minimal supergravity model with minimal Yukawa unification. It is relevant to contrast this situation with what happens for low and moderate values of , for which the consistency of a moderately heavy top quark, GeV, with bottom–tau Yukawa unification requires the ratio of vacuum expectation values to be very close to one, , unless large threshold corrections to both gauge and Yukawa couplings are present at the grand unification scale , ,.\n\n## 3 Higgs Potential Parameters\n\nIn order to analyze the radiative electroweak symmetry breaking condition, one should concentrate on the Higgs potential of the theory. In the Minimal Supersymmetric Standard Model, and after the inclusion of the leading–logarithmic radiative corrections, it may be written as , \n\n Veff = m21H†1H1+m22H†2H2−m23(HT1iτ2H2+h.c.) (6) + λ12(H†1H1)2+λ22(H†2H2)2+λ3(H†1H1)(H†2H2)+λ4∣∣H†2iτ2H∗1∣∣2\n\nwhere the quartic couplings may be obtained by the corresponding renormalization group equations and the fact that, at scales at which the theory is supersymmetric the running quartic couplings , with , must satisfy the following conditions Refs. :\n\n λ1=λ2=g21+g224,λ3=g22−g214,λ4=−g222. (7)\n\nThe masses , with are also running mass parameters, whose renormalization group equations may be found in the literature -. As we explained in section 1, in the numerical analysis we considered the two loop renormalization group evolution of gauge and Yukawa couplings, while the supersymmetric and Higgs mass parameters, as well as the low energy Higgs quartic couplings are evolved at the one loop level with the leading supersymmetric threshold corrections included. The minimization conditions read\n\n sin(2β)=2m23m2A, (8)\n tan2β=m21+λ2v2+(λ1−λ2)v21m22+λ2v2, (9)\n\nwhere , is the vacuum expectation value of the Higgs fields , , and is the CP-odd Higgs mass,\n\n m2A=m21+m22+λ1v21+λ2v22+(λ3+λ4)v2 (10)\n\nand we define the mass parameter to be positive.\n\nApart from the mass parameters , appearing in the effective potential, the evolution of the supersymmetric mass parameter appearing in the superpotential ,\n\n f=htϵijQjUHi2+hbϵijQiDHj1+hτϵijLiEHj1+μϵijHi1Hj2, (11)\n\n(where is the top–bottom left handed doublet superfield and , and are singlet superfields) is relevant for the analysis of the radiative electroweak symmetry breaking conditions. The bilinear mass term proportional to appearing in the Higgs potential may be rewritten as a soft supersymmetry breaking parameter multiplied by the Higgs bilinear term appearing in the superpotential, that is . Analogously, the scalar potential may contain a scalar trilinear breaking term proportional to the - Yukawa dependent part of the superpotential, with a trilinear coupling .\n\nIn order to get an understanding of the numerical results, we will present approximate analytical formulae, for the relations required by the electroweak symmetry breaking conditions, in which the radiative corrections to the quartic couplings are ignored. There are several features of the Higgs potential which are characteristic for large values. They can be easily discussed in a qualitative way on the basis of the supersymmetric tree level potential. Eq.(9) simplifies to\n\n tan2β=m21+M2Z/2m22+M2Z/2, (12)\n\nso, for large (already, say, ), either\n\n m22≃−M2Z2, (13)\n\nif are of the order of the boson mass squared, or\n\n m21≃tan2βm22 (14)\n\nwhen . In general, the smaller is the cancellation in the denominator of Eq.(12), the larger is the hierarchy between and . The second relation, Eq.(14), is, however, unnatural when . Indeed, if all supersymmetric particle masses are below a few TeV, Eq.(13) holds, within a good approximation (Although the inclusion of radiative corrections modifies the low energy convergence of the parameter, the relation is preserved, what is sufficient for the understanding of the properties discussed below). Eq.(13), combined with the condition , gives a useful constraint:\n\n m21−m22>M2Z. (15)\n\nAnother very important property is\n\n m23≃M2Atanβ, (16)\n\nor, equivalently, . Since in the case a large hierarchy between and is highly unnatural, the above condition, Eq.(16), implies also (). Thus, in order to study the implication of the electroweak symmetry breaking condition, one can effectively replace Eq.(16) with the condition .\n\nTo go further with the analysis, it is very useful to obtain approximate analytical solutions for the one loop renormalization group evolution of the mass parameters, whose validity may be proven by comparing them with our numerical solutions. We will assume universality of the soft supersymmetry breaking parameters, that is to say a common scalar mass and a common gaugino mass , as well as the boundary conditions for the parameters ( and ), and , at the grand unification scale to be given by , and , respectively. In the region of large values of , for which the bottom Yukawa coupling is of the order of the top Yukawa coupling, an approximate analytical solution for the one loop evolution of the mass parameters may be obtained. For this, we identify the bottom and top Yukawa couplings and neglect the tau Yukawa coupling effects. Furthermore, all supersymmetric threshold corrections are ignored at this level. The solution for reads\n\n Yt(t)=4πYt(0)E(t)4π+7Yt(0)F(t) (17)\n\nwhere and are functions of the gauge couplings,\n\n E=(1+β3t)16/3b3(1+β2t)3/3b2(1+β1t)13/9b1,F=∫t0E(t′)dt′ (18)\n\nwith , the beta function coefficient of the gauge coupling , and we identify the right bottom and the right top hypercharges. As we said, the fixed point solution is obtained for values of the top quark Yukawa coupling which become large at the grand unification scale, that is, approximately,\n\n Yf(t)=4πE(t)7F(t). (19)\n\nAs had been anticipated in Ref. , the fixed point solution for the case differs in a factor from the corresponding solution in the low case, for which . From here, by inspecting the renormalization group equation for the mass parameters, we obtain the approximate analytical solutions\n\n m2H1≃m2H2=m20+0.5M21/2−37Δm2 (20)\n m2U≃m2D=m20+6.7M21/2−27Δm2 (21)\n m2Q≃m20+7.2M21/2−27Δm2 (22)\n\nwhere , with , , , are the squark doublet, right bottom squark and right stop quark mass parameters respectively and\n\n Δm2 ≃ 3m20YYf−4.6A0M1/2YYf(1−YYf) (23) + A20YYf(1−YYf)+M21/2[14YYf−6(YYf)2].\n\nHere we have concentrated on the above mass parameters, because they are the only relevant ones for the study of the properties of the radiative electroweak symmetry breaking solutions in the approach of ref. . We will discuss the properties of the mass spectrum in more detail in section 6. Moreover, the supersymmetric mass parameter renormalization group evolution gives,\n\n μ2=2μ20(1−YYf)6/7, (24)\n\nwhile the running of the soft supersymmetry breaking bilinear and trilinear coupling read,\n\n At=A0(1−YYf)−M12(4.2−2.1YYf), (25)\n B≃δ(Y)+M1/2(2YYf−0.6), (26)\n\nwith\n\n δ(Y)=B0−6Y7YfA0. (27)\n\nThe coefficients characterizing the dependence of the mass parameters on the universal gaugino mass are functions of the exact value of the gauge couplings. In the above, we have taken the values of the coefficients that are obtained for .\n\nThe approximate solutions, Eqs. (2025), become weakly dependent on the parameter , the dependence being weaker for top quark Yukawa couplings closer to the fixed point value. The strongest dependence on the parameter comes through the parameter introduced above. Similar properties are obtained in the low regime , although the explicit form of the parameter is different in this case. From Eq.(24), it follows that the coefficient relating to tends to zero as . The coefficients scales faster to zero than in the low case.\n\n## 4 Radiative Breaking of SU(2)L×U(1)Y\n\nIn the following we present a complete numerical analysis of the constraints coming from the requirement of a proper radiative electroweak symmetry breaking in the large regime. As described in the Introduction, we use the bottom–up approach of ref. . For a fixed value of the top quark mass we search for all solutions to radiative breaking, which give a chosen value of , by scanning over the CP odd Higgs mass and the low energy stop mass parameters. The latter are very convenient as the input parameters as they fix the leading supersymmetric threshold corrections to the Higgs potential. While studying the model from low energies we have chosen for definiteness an upper bound of 2 TeV on the scanned parameters. For a somewhat larger upper bound, larger values of the soft supersymmetry breaking prameters are allowed, but the general features of the solutions are preserved. It is natural to expect that the supersymmetric parameters are at most of order of a few TeV, if supersymmetry is to solve the hierarchy problem of the Standard Model. In Figs. 2 - 5 we present the results which show interesting correlations among the soft supersymmetry breaking parameters.\n\nAs discussed in section 3, the one loop corrections to the effective Higgs potential, necessary to perform a proper analysis of the radiative electroweak symmetry breakdown, were included in the numerical analysis. The gauge and Yukawa couplings were evolved with their two loop renormalization group equations between and . In their evolution, we have treated all supersymmetric particle masses as being equal to . Although this procedure introduces small uncertainties on the predicted values of and (which will be considered in our analysis), it keeps all the essential features of the radiative electroweak symmetry with unification of bottom and top Yukawa couplings, makes possible the comparison of our results with the ones of Fig. 1 and allows an easy analytical interpretation of the numerical results. In addition, the small uncertainties on and may be treated by analytical methods , .\n\nAnalogously to the low scenario , it is possible to derive approximate analytical relations, which are useful in the understanding of the numerical results. Indeed, considering the conditions for a proper radiative electroweak symmetry breaking, Eq.(13), the approximate solutions for the mass parameters, Eqs. (20) -(23), and ignoring radiative corrections to the quartic couplings the following analitycal expression is obtained,\n\n μ2 = m20(97YYf−1)−M21/2[0.5−6YYf+187(YYf)2] (28) − 374.6A0M1/2YYf(1−YYf)+37A20YYf(1−YYf)−M2Z/2.\n\nIn the analytical presentation we will always keep the expressions as a function of the low energy parameter . The reason is that in the one loop approximation and are linearly related, Eq.(24), and becomes a more appropriate parameter for the description of the solution properties, particularly for large values of the top quark mass where strongly depends on the degree of proximity to the fixed point value. The dependence may be always recovered by using Eq.(24).\n\nIn the above we have taken the expression of obtained in the analytical approximation in which . In the explicit numerical solution to the mass parameters, however, we obtain\n\n m21−m22=αM21/2+βm20 (29)\n\nwhere for ( GeV), , and , while for ( GeV), and . Hence, the coefficient is small and positive, and is negative and small in magnitude. The order of magnitude of the coefficients and can be easily inferred from the renormalization group equations. Indeed, it is easy to show that under the condition of unification of the three third generation Yukawa couplings comes mainly from the difference in the running of bottom and top Yukawa couplings, together with the different hypercharges of the right top and bottom quarks, which induce a different gaugino dependence of the stop and sbottom parameters. The negative values of are mainly due to the lepton Yukawa effects. We see that, due to the restriction , Eq.(15), values of make the radiative breaking of the electroweak symmetry impossible, in the approximation which neglects supersymmetric threshold corrections to the Higgs potential. In the numerical analysis, which includes those corrections, the only solutions are still obtained for of the order of, or larger than , as seen in Fig. 2.a and Fig. 2.b. It is also important to remark that, due to the smallness of the parameters and , the dependence of the mass parameters and on the gaugino mass is well described by Eq.(28) (which was obtained in the approximation ), while the dependence on the mass parameters , , remains weak for . In general, the corrections to the approximate solutions given in Eqs.(19 - 22) and Eq.(28) are small, and, hence they provide useful information for the analysis of the electroweak symmetry breaking condition.\n\nThe values of the top quark mass, GeV and GeV, and of the ratio of the Higgs vacuum expectation values, and , used above are such that unification of gauge and Yukawa couplings is achieved for GeV, and GeV, , respectively. Considering a Yukawa coupling solution sufficiently close to the infrared fixed point, the values of as required by the radiative breaking conditions, and (as follows from Eqs.(15, 29), we obtain from Eq.(28) that,\n\n μ2≃3M21/2, (30)\n\ni.e. there is a strong linear correlation between and . If, instead, we consider the case , (corresponding to GeV) as a representative one of what happens when we depart from the fixed point value we obtain\n\n μ2 = −0.23m20+2.2M21/2−0.47A0M1/2 (31) + 0.1A20−M2Z/2.\n\nThere is a stronger dependence on the supersymmetry breaking parameter . However, due to the relation , the bounds on and coming from the stability condition and the requirement of the absence of a colour breaking minima , and the smallness of the coefficients associated with the dependence, one gets that the correlation between and is conserved over most of the parameter space,\n\n μ2≃DM21/2, (32)\n\nwhere . The predictions coming from the above analysis, based on the approximate relations Eqs.(3032), must be compared with the results of the numerical analysis, in which the running of gauge and Yukawa couplings have been considered at the two loop level, and all one loop threshold corrections to the quartic couplings and masses have been included. The resulting correlations between and are depicted in Fig. 3.a and Fig. 3.b, which are in good agreement with the analytical results, although the coefficient in Fig. 3b is somewhat smaller than the analytical prediction, Eq.(32).\n\nThe information above may be used to get a further understanding of the properties of Fig. 2. The lower bound on , for instance, follows from the condition , which yields\n\n M1/2>MZ√α, (33)\n\nwhere, as we said above, for GeV and , while for GeV and . From Fig. 2 we observe that, although the lower limit on for GeV is well described by Eq.(33), the one for GeV is somewhat higher than the predicted one from Eq.(33). This difference is a reflection of the size of the one loop radiative corrections to the quartic couplings, which grow with the fourth power of the top quark mass and were ignored for the obtention of Eq.(33).\n\nAs we explained above, the condition also excludes the points with . Furthermore, low values of , although consistent with the condition of radiative breaking induce large mixings in the stau sectors which yield stau masses lower than the neutralino ones. The fact that low values of leads to a stau lighter than the neutralinos was already noticed in Ref. . In the Figures we impose the condition of a neutral supersymmetric particle to be the lightest one as an additional experimental constraint. Under these conditions, the lightest supersymmetric particle is always a bino, with mass . In order to get a quantitative understanding of the lower limit on , we recall that, ignoring small tau Yukawa coupling effects, the left and right slepton mass parameters are given by ,\n\n m2L≃0.5M21/2+m20,m2R≃0.15M21/2+m20, (34)\n\nwhile the mixing term for large is dominated by the parameter\n\n m2LR≃−hτμv2. (35)\n\nUsing the fact that, at energies of the order of , , and the bottom - top unification condition, the condition approximately yields,\n\n m20≥−0.15M21/2+√(0.15M21/2)2+μ2m2t/3. (36)\n\nRecalling Eqs.(30) and (32), and using Eq.(36), one can get an understanding of the region, indicated as experimentally excluded in the Figs. 2.a and 2.b (see also Fig. 10).\n\nClose to the infrared quasi fixed point solution the condition yields (from Eqs. (30),(32) ), i.e.\n\n δ≡B0−6A07≃−1.4M1/2 (37)\n\nIn the numerical analysis, we studied the correlations between and , and compared it with the results coming from Eq.(37). The results are depicted in Fig.4.a. The numerical results confirm in a good degree the analytical expectations. Analogously, for , we obtain\n\n B0M1/2−0.5A0M1/2=−0.6. (38)\n\nThe correlation, resulting in this case from the numerical analysis is depicted in Fig.4.b, being in good agreement with Eq.(38), too.\n\nThe strong correlation between the parameter and , together with the - correlation, Eqs. (30) - (32), implies also a strong correlation between and . The numerical correlation is presented in Figs. 5.a and 5.b, for which we chose to plot the GUT scale parameter instead of the renormalized parameter . From Figs. 3 and 5 we can hence obtain also information about the relation between and , which agrees well with the analytical prediction, Eq. (24).\n\nObserve that the condition is a property of the radiative breaking solutions with large values of and a not too heavy supersymmetric spectrum, and in this sense is independent of the condition of unification of top and bottom quark Yukawa couplings. Since very low values of () are not consistent with the condition of radiative breaking of the electroweak symmetry, equations analogous to Eqs.(37) and (38) will be obtained even if we relax the bottom-top Yukawa unification condition. We exemplify this by taking two solutions with and large values of and studying the numerical solutions. The resulting correlations are depicted in Fig. 4.c and 4.d.\n\nWhen the condition of unification of bottom and top Yukawa couplings is relaxed, however, large values of are not longer needed to get the necessary hierarchy between and . As the bottom and tau Yukawa couplings decrease compared with the top one, the coefficients and , Eq.(29), increase, becoming positive for . Hence, for , acceptable radiative breaking solutions may be also obtained by taking large values of . For these solutions the strong correlation between , and is lost, together with the hierarchical relation between and . These results are depicted in Figs. 2.c and 2.d, 3.c and 3.d, and 5.c and 5.d, where .\n\nIn summary, in general the condition of radiative breaking of the electroweak symmetry with large values of implies a strong correlation between the parameters and . This correlation is a reflection of, in principle, a strong degree of fine tuning, implied by the condition . However, it is tempting to speculate that this correlation has some fundamental origin, what would imply the necessity of redefining the naive fine tuning criteria.\n\nIf the top quark–bottom quark Yukawa coupling unification is required, the parameters and are also strongly correlated with the supersymmetric mass parameter . These properties do not strongly depend on the proximity to the infrared fixed point solution, although the exact value of the coefficient relating the different parameters and the strength of the correlation does depend on the top quark Yukawa coupling value. Radiative breaking of the electroweak symmetry is driven by the gaugino mass and . For a large enough departure from the exact top quark – bottom quark Yukawa unification () solutions with radiative breaking driven by are also possible, for which both the correlation between and and the hierarchical relation between and are destroyed.\n\n## 5 Radiative Corrections to Mb and Mτ and the Predictions for the Top Quark Mass\n\nFig.1 summarizes the predictions for the top quark mass as a function of for given values of , which follow from unification of the three Yukawa couplings. As explained in Section 2, the pole mass is obtained from the unification condition in the case in which the supersymmetric one–loop corrections to the bottom mass are ignored (i.e. it includes only QCD corrections). In this section we calculate the supersymmetric one–loop corrections to the bottom mass in the model with radiative breaking. For large values of , they are not only large but, due to the strong correlations between the soft supersymmetry breaking parameters present in the large solutions with , for fixed they are almost constant in the whole parameter space allowed by radiative breaking. Thus, in the first approximation, for fixed and , . If this value of is in the range of the experimentally acceptable values for the physical bottom mass, GeV, then the corresponding values of and are the predictions for the top quark mass and , consistent with radiative breaking. Of course, all uncertainties taken into account, the actual prediction is a band of values for and .\n\nThere is a higher order ambiguity due to the choice of the scale at which the supersymmetric one–loop corrections are calculated. A natural choice is between the electroweak and supersymmetric () scales and we choose to work with . The corrected running bottom quark mass reads ,\n\n mb=hbv1(1+Δ(mb)). (39)\n\nreceives contributions coming from bottom squark–gluino loops and top squark–chargino loops, and is given by,\n\n Δ(mb) = 2α33πM~gμtanβI(m2~b,1,m2~b,2,M2~g) (40) + Yt4πAtμtanβI(m2~t,1,m2~t,2,μ2),\n\nwhere the integral function I(a,b,c) is given by\n\n I(a,b,c)=abln(a/b)+bcln(b/c)+acln(c/a)(a−b)(b−c)(a−c), (41)\n\nwith and () are the gluino and sbottom (stop) eigenstate masses respectively. The integral function may be parametrized as , where is the maximum of the three squared masses appearing in the functional integral and the coefficient if there is no large hierarchy between the three different masses. Observe that the minimum value of is only obtained when the three masses are equal. As we will discuss below, for the typical values of the mass parameters appearing in the radiative electroweak symmetry breaking solutions, gives a good approximation to the integral.\n\nThe tau mass corrections are, instead, dominated by the bino exchange contribution, which is negligible for the bottom quark case. Indeed,\n\n mτ=hτv1(1+Δ(mτ)), (42)\n\nwith\n\n Δ(mτ) = α14πM~BμtanβI(m2~τ,1,m2~τ,2,M2~B) (43)\n\nObserve that although the effect is expected to be small due to the presence of the weak gauge coupling, it is partially enhanced by the fact that the particles appearing in the loop are lighter than in the bottom case. We will discuss it in more detail below.\n\nDue to the approximate dependence of on and , Eq.(25), close to the fixed point there is a strong correlation between and the gluino mass. Indeed, for ,\n\n At≃−2M~g3. (44)\n\nFor values of , is shifted towards larger values in most of the parameter space,\n\n At≃−M~g. (45)\n\nThese correlations are observed in the numerical analysis. The relations above, Eqs.(44), (45), are only violated for large values of , close to the upper bound on this quantity (For the numerical bounds on see Fig. 4). Due to the minus signs in Eqs.(44) and (45), there is an effective cancellation between both bottom mass correction contributions. Interestingly enough, due to the fact that is larger when the Yukawa coupling is smaller, the cancellation between both bottom mass correction contributions for is similar to the one appearing for .\n\nThe bottom mass corrections become very relevant for large values of . In order to reduce the bottom mass corrections, while fulfilling the requirement , the authors of Ref. imposed a Peccei Quinn symmetry , which is explicitly broken, its breakdown being characterized by the (assumed) small parameter . They also required the presence of an approximate continuous symmetry, present in the limit , , , which breaking is characterized by the (assumed) small parameter\n\n ϵR=Bm~q≃Am~q≃M~gm~q, (46)\n\nwhich would protect both and the bottom mass corrections.\n\nHowever, the electroweak symmetry radiative breaking solutions with universal soft supersymmetry parameters at the grand unification scale and exact top quark–bottom quark Yukawa coupling unification is inconsistent with the approximate preservation of these symmetries. Indeed, as we have discussed in the last section, the only solutions satisfying these requirements are obtained for of the order of, or larger than . Under these conditions, the squark mass is of the order of the gluino mass and not much larger than it, as required by . In addition, as explained above, the mass parameter is of the order of the gluino mass and, hence, the bottom mass corrections are not suppressed in the minimal supergravity model. Indeed, due to the strong correlation between and , the parameter is strongly correlated with the gluino mass, and an approximate expression for the integrals may be obtained. Using these correlations, we obtain that the integrals are well approximated by setting ,\n\n Δ(mb)≃1.2tanβμM~g(α33π+(32)YtAt8πM~g) (47)\n\nwhere the factor is to account for the fact of having written a factor in the denominator, instead of the appropiate factor (The correlation between the gluino and squark masses will be discussed in the next section). The above expression gives a good approximation to the bottom mass corrections in most of the parameter space consistent with radiative breaking of the electroweak symmetry and bottom–top Yukawa coupling unification. Observe that, due to the strong correlations between and , and using the fact that the fixed point value of the top and bottom quark Yukawa coupling is approximately given by , there is an effective cancellation between the two contributions, which reduces the gluino contribution by about a 30. In the more precise numerical result this cancellation is of the order of . Taking this into account, and the fact that , with the unifying value of the gauge couplings, the relative bottom mass corrections are given by\n\n Δ(mb)≃0.0045tanβμM1/2. (48)\n\nFrom Fig. 1 we observe that, in general, independent of the bottom mass value, the condition of unification of the three Yukawa couplings is such that the larger is , the closer is the top quark Yukawa coupling to its fixed point value. Values of the top quark Yukawa coupling close to its fixed point are obtained for , while is obtained for . Once Eq.(48) is combined with the numerical results shown in Fig. 3, we obtain a relative bottom mass correction of the order of 45 for values of and the top quark mass consistent with the fixed point, while for , the relative bottom mass correction is of the order of .\n\nAn analogous procedure can be applied for the estimation of the relative tau mass corrections. In this case, the heaviest stau mass is of the order of 2.5 times the bino mass, while the lightest stau mass is of the order or somewhat larger than the bino mass, depending on the relative value of with respect to . Under these conditions, over most of the allowed parameter space the loop integral may be approximated by a factor of the order of 0.85. One can check that, under these conditions the tau mass corrections are not larger than of the tau mass. Moreover, a relatively large tau mass correction is always associated with a large left - right stau mass mixing, for which the lightest stau becomes the lightest supersymmetric particle. Once the condition of a neutralino being the lightest supersymmetric particle is imposed, the relative tau mass corrections are bounded to be lower than 4 over most of the allowed parameter space. Since, in addition, a relative tau mass variation affect less the unification condition than a relative bottom mass correction, the tau mass correction effects on the top quark mass predictions are small.\n\nFor chosen values of and we are now able to calculate the physical bottom mass by running down the corrected with the standard Model RGE to the scale and applying the appopriate QCD corrections, Eq.(5). The results are shown in Fig. 6, for the same representative values of the top quark mass and chosen in the previous figures (which, as we discussed in section 4, for the cases a) and b) correspond to GeV and GeV, respectively). The two branches of correspond to two signs of , the lowest values of corresponding to negative values of (or equivalently, positive values of ). From Fig. 6 and recalling the results presented in Fig.1, we see that large corrections to the bottom mass, of the order of 30 may be used to reconcile with GeV. Moreover, it is easy to see that, due to the size of the characteristic corrections, for , there are no solutions with and a top quark Yukawa coupling within the range of validity of perturbation theory.\n\nThe above results may be used to set an upper bound on the top quark mass as a function of the strong gauge coupling value. This upper bound will correspond to the maximum values of and consistent with a physical bottom mass GeV. Larger top quark mass values will correspond to lower values of and larger values of , implying a physical bottom quark mass outside the present experimental bounds on this quantity, GeV. The upper bound may be hence estimated as follows: For a given value of , and , the relative bottom mass corrections may be computed by using the supersymmetric mass parameters allowed by the condition of radiative breaking of the electroweak symmetry. In addition, may be otained from the correlations between , and depicted in Fig. 1. We perform a scanning over the values of and , looking for the maximum value consistent with a physical bottom mass GeV. This value of gives an estimate of the upper bound on this quantity for this given value of . The uncertainties associated with this procedure will be discussed below.\n\nFor example, for , the upper bound on the top quark mass is approximately given by GeV while the upper bound on the ratio of vacuum expectation values is given by . These bounds are associated with a bottom mass GeV. From Eqs. (40), (48) it follows that the approximate bottom mass corrections under these conditions (corresponding to the lowest value of ) are of the order of 18. Hence, as required for the solution associated to the upper bound on the top quark mass, the physical bottom mass will be approximately equal to the lower experimental bound on this quantity GeV. Analogously, for the upper bounds read GeV and , for which the bottom mass (5.3) GeV. The lowest bottom mass corrections are of the order of 23 (13 ) (corresponding to a ratio ) and hence the physical bottom mass is approximately equal to 4.6 GeV.\n\nIt is important to discuss the uncertainties on the estimate of the top quark mass upper bounds presented above. For the obtention of Fig. 1, the supersymmetric spectrum has been taken to be degenerate at a mass . However, the squark and gluino spectrum arising from the bottom - top Yukawa unification condition is heavy and hence, the top quark mass could be modified by the supersymmetric particle threshold effects. These effects have been estimated in Ref. . For the characteristic spectrum obtained in these solutions, the resulting top quark mass uncertainties are of the order of 5 - 10 GeV. In addition, in the above we have ignored the possible effects of tau mass corrections. The tau mass corrections are correlated in sign with the bottom mass corrections and their effects on the top quark mass predictions may be hence estimated by a lowering of the relative bottom mass corrections in an amount of the order of the relative tau mass corrections. A modification of the order of 3 of the relative bottom mass corrections gives variations of the top quark mass prediction of the order of 5 - 10 GeV, too. Finally, there is the already discussed - scale uncertainty in the evaluation of the bottom mass corrections, which can also modify the top quark mass predictions in a few percent.\n\nFrom the above discussion, it follows that the estimate for the upper bound on the top quark mass quoted above may be away from the real bound in 10 - 20 GeV. However, it is important to remark that even after the inclusion of these uncertainties the allowed top quark mass values become much lower than the values obtained for the case in which a negligible bottom mass correction is assumed ,,. Indeed, even after the uncertainties are included, the upper bounds on the physical top quark mass obtained above are of the order of the lower bounds for the same quantity for the case in which the bottom mass corrections are negligible.\n\nA lower bound on the top quark mass is also obtained. However, the lower bounds on the top quark mass is given by GeV for , while for values of the lower bound is below the present experimental limit on the top quark mass. In general, for , large values of the top quark mass GeV will be only possible for the case in which we relax the condition of unification of the three Yukawa couplings. For instance, a top quark mass GeV may be achieved for , for , and GeV. As we explained above, since under these conditions may be much larger than , the approximate symmetries required in Ref. , Eq.(46), become now possible, and hence the bottom mass corrections can be small, .\n\n## 6 Supersymmetric Particle Spectrum\n\nThe properties of the sparticle spectrum are to a large extent determined by the correlation of the mass parameter and the soft supersymmetry breaking parameters and their large values necessary to fulfill the condition of radiative breaking of the electroweak symmetry (see section 4). For instance, since large values of the parameters and are required, there will be little mixing in the chargino and neutralino sectors. The lightest (heaviest) chargino is given by a wino (charged Higgsino) with mass equal to (). The lightest neutralino will be given by a bino of high degree of purity and mass . These issues have been already discussed in Refs. - , and survive in our more precise numerical correlation. We will hence concentrate on the predictions which depend stronger on the precise values of the top quark mass and hence are more sensitive to the change on the top quark mass predictions induced by the bottom mass corrections studied in the previous section. In addition, we will present an analysis of the constraint on the soft supersymmetry breaking parameters coming from the present experimental bounds on the decay rate.\n\nIn Fig. 7 we give the behaviour of the CP odd Higgs mass as a function of the lightest chargino mass. As we remarked above, due to the large values of and appearing in this scheme, the lightest chargino is almost a pure wino, with mass , while the heaviest chargino is a Higgsino with mass equal to . The CP odd Higgs mass squared is given by , where the constant term is negative. Since is positive and is negative, we get an upper bound on ,\n\n m2A<αM21/2 (49)\n\nwhich is visible in the figures. Observe that the sensitivity under top quark mass varations of this bound comes through the dependence of the parameter on . The largest values of are obtained for low values of , which, could lead to a stau to be the lightest supersymmetric particle (see section 4 and Ref. ). From Figs. 7.a and 7.b, we see that for the allowed parameter space consistent with bottom - top Yukawa unification the CP odd Higgs becomes light. Very low values of are, however, excluded by experimental limits. Moreover, the CP odd Higgs mass becomes\n\n mA≤mQ√α5 (50)\n\nwhere the factor 5 comes from the strong correlation between and" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89590317,"math_prob":0.9783566,"size":50003,"snap":"2020-45-2020-50","text_gpt3_token_len":10419,"char_repetition_ratio":0.21668433,"word_repetition_ratio":0.08742238,"special_character_ratio":0.20226787,"punctuation_ratio":0.102277756,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9854163,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-01T02:20:51Z\",\"WARC-Record-ID\":\"<urn:uuid:7ab28b0d-b12e-4470-91a4-ff4cb11776ac>\",\"Content-Length\":\"1049559\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3f5d96da-6bd1-4086-90a6-bb9fafe2ede8>\",\"WARC-Concurrent-To\":\"<urn:uuid:ef007275-bbc7-449d-8952-e2a876a48051>\",\"WARC-IP-Address\":\"104.28.21.249\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/hep-ph/9402253/\",\"WARC-Payload-Digest\":\"sha1:BYCVCQ5JWBG7S2YDTFD37QADS4ANC42K\",\"WARC-Block-Digest\":\"sha1:F4SGFYDWJ6QYNNTFV732RMYQBLZRYTBX\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141542358.71_warc_CC-MAIN-20201201013119-20201201043119-00322.warc.gz\"}"}
https://docs.cloudera.com/cdw-runtime/1.4.0/impala-sql-reference/topics/impala-math-functions.html	[ "# Impala mathematical functions\n\nMathematical functions, or arithmetic functions, perform numeric calculations that are typically more complex than basic addition, subtraction, multiplication, and division. For example, these functions include trigonometric, logarithmic, and base conversion operations.\n\nRelated information:\n\nThe mathematical functions operate mainly on these data types:\n• INT\n• BIGINT\n• SMALLINT\n• TINYINT\n• DOUBLE\n• FLOAT\n• DECIMAL\nFor the operators that perform the standard operations such as addition, subtraction, multiplication, and division, see Arithmetic operators.\n\nFunctions that perform bitwise operations are explained in Impala bit functions.\n\nFunction reference:\n\nImpala supports the following mathematical functions:\n\nABS(numeric_type a)\nPurpose: Returns the absolute value of the argument.\n\nReturn type: Same as the input value\n\nUsage notes: Use this function to ensure all return values are positive. This is different than the `positive()` function, which returns its argument unchanged (even if the argument was negative).\n\nACOS(DOUBLE a)\nPurpose: Returns the arccosine of the argument.\n\nReturn type: `DOUBLE`\n\nASIN(DOUBLE a)\nPurpose: Returns the arcsine of the argument.\n\nReturn type: `DOUBLE`\n\nATAN(DOUBLE a)\nPurpose: Returns the arctangent of the argument.\n\nReturn type: `DOUBLE`\n\nATAN(DOUBLE a, DOUBLE b)\nPurpose: Returns the arctangent of the two arguments, with the signs of the arguments used to determine the quadrant of the result.\n\nReturn type: `DOUBLE`\n\nBIN(BIGINT a)\nPurpose: Returns the binary representation of an integer value, that is, a string of 0 and 1 digits.\n\nReturn type: `STRING`\n\nCEIL(DOUBLE a), CEIL(DECIMAL(p,s) a), CEILING(DOUBLE a), CEILING(DECIMAL(p,s) a), DCEIL(DOUBLE a), DCEIL(DECIMAL(p,s) a)\nPurpose: Returns the smallest integer that is greater than or equal to the argument.\n\nReturn type: Same as the input type\n\nCONV(BIGINT n, INT from_base, INT to_base), CONV(STRING s, INT from_base, INT to_base)\nPurpose: Returns a string representation of the first argument converted from `from_base` to `to_base`. The first argument can be specified as a number or a string. For example, `conv(100, 2, 10)` and `conv('100', 2, 10)` both return `'4'`.\n\nReturn type: `STRING`\n\nUsage notes:\n\nIf `to_base` is negative, the first argument is treated as signed, and otherwise, it is treated as unsigned. For example:\n\n• `conv(-17, 10, -2) `returns `'-10001'`,``` -17``` in base 2.\n• `conv(-17, 10, 10)` returns `'18446744073709551599'`. `-17` is interpreted as an unsigned, 2^64-17, and then the value is returned in base 10.\n\nThe function returns `NULL` when the following invalid arguments are specified:\n\n• Any argument is `NULL`.\n• `from_base` or `to_base` is below `-36` or above `36`.\n• `from_base` or `to_base` is `-1`, `0`, or `1`.\n• The first argument represents a positive number and `from_base` is a negative number.\n\nIf the first argument represents a negative number and `from_base` is a negative number, the function returns `0`.\n\nIf the first argument represents a number larger than the maximum `bigint`, the function returns:\n\n• The string representation of -1 in `to_base` if `to_base` is negative.\n• The string representation of 18446744073709551615' (2^64 - 1) in `to_base` if `to_base` is positive.\n\nIf the first argument does not represent a valid number in `from_base`, e.g. 3 in base 2 or '1a23' in base 10, the digits in the first argument are evaluated from left-to-right and used if a valid digit in `from_base`. The invalid digit and the digits to the right are ignored.\n\nFor example:\n• ` conv(445, 5, 10)` is converted to ```conv(44, 5, 10)``` and returns `'24'`.\n• ` conv('1a23', 10, 16)` is converted to ```conv('1', 10 , 16)``` and returns `'1'`.\nCOS(DOUBLE a)\nPurpose: Returns the cosine of the argument.\n\nReturn type: `DOUBLE`\n\nCOSH(DOUBLE a)\nPurpose: Returns the hyperbolic cosine of the argument.\n\nReturn type: `DOUBLE`\n\nCOT(DOUBLE a)\nPurpose: Returns the cotangent of the argument.\n\nReturn type: `DOUBLE`\n\nDEGREES(DOUBLE a)\nPurpose: Converts argument value from radians to degrees.\n\nReturn type: `DOUBLE`\n\nE()\nPurpose: Returns the mathematical constant e.\n\nReturn type: `DOUBLE`\n\nEXP(DOUBLE a), DEXP(DOUBLE a)\nPurpose: Returns the mathematical constant e raised to the power of the argument.\n\nReturn type: `DOUBLE`\n\nFACTORIAL(integer_type a)\nPurpose: Computes the factorial of an integer value. It works with any integer type.\n\nUsage notes: You can use either the `factorial()` function or the `!` operator. The factorial of 0 is 1. Likewise, the `factorial()` function returns 1 for any negative value. The maximum positive value for the input argument is 20; a value of 21 or greater overflows the range for a `BIGINT` and causes an error.\n\nReturn type: `BIGINT`\n\n``````select factorial(5);\n+--------------+\n\| factorial(5) \|\n+--------------+\n\| 120 \|\n+--------------+\n\nselect 5!;\n+-----+\n\| 5! \|\n+-----+\n\| 120 \|\n+-----+\n``````\nFLOOR(DOUBLE a), FLOOR(DECIMAL(p,s) a), DFLOOR(DOUBLE a), DFLOOR(DECIMAL(p,s) a)\nPurpose: Returns the largest integer that is less than or equal to the argument.\n\nReturn type: Same as the input type\n\nFMOD(DOUBLE a, DOUBLE b), FMOD(FLOAT a, FLOAT b)\nPurpose: Returns the modulus of a floating-point number.\n\nReturn type: `FLOAT` or `DOUBLE`, depending on type of arguments\n\nUsage notes:\n\nBecause this function operates on `DOUBLE` or `FLOAT` values, it is subject to potential rounding errors for values that cannot be represented precisely. Prefer to use whole numbers, or values that you know can be represented precisely by the `DOUBLE` or `FLOAT` types.\n\nExamples:\n\nThe following examples show equivalent operations with the `fmod()` function and the `%` arithmetic operator, for values not subject to any rounding error.\n\n``````select fmod(10,3);\n+-------------+\n\| fmod(10, 3) \|\n+-------------+\n\| 1 \|\n+-------------+\n\nselect fmod(5.5,2);\n+--------------+\n\| fmod(5.5, 2) \|\n+--------------+\n\| 1.5 \|\n+--------------+\n\nselect 10 % 3;\n+--------+\n\| 10 % 3 \|\n+--------+\n\| 1 \|\n+--------+\n\nselect 5.5 % 2;\n+---------+\n\| 5.5 % 2 \|\n+---------+\n\| 1.5 \|\n+---------+\n``````\n\nThe following examples show operations with the `fmod()` function for values that cannot be represented precisely by the `DOUBLE` or `FLOAT` types, and thus are subject to rounding error. `fmod(9.9,3.0)` returns a value slightly different than the expected 0.9 because of rounding. `fmod(9.9,3.3)` returns a value quite different from the expected value of 0 because of rounding error during intermediate calculations.\n\n``````select fmod(9.9,3.0);\n+--------------------+\n\| fmod(9.9, 3.0) \|\n+--------------------+\n\| 0.8999996185302734 \|\n+--------------------+\n\nselect fmod(9.9,3.3);\n+-------------------+\n\| fmod(9.9, 3.3) \|\n+-------------------+\n\| 3.299999713897705 \|\n+-------------------+\n``````\nFNV_HASH(type v)\nPurpose: Returns a consistent 64-bit value derived from the input argument, for convenience of implementing hashing logic in an application.\n\nReturn type: `BIGINT`\n\nUsage notes:\n\nYou might use the return value in an application where you perform load balancing, bucketing, or some other technique to divide processing or storage.\n\nBecause the result can be any 64-bit value, to restrict the value to a particular range, you can use an expression that includes the `ABS()` function and the `%` (modulo) operator. For example, to produce a hash value in the range 0-9, you could use the expression `ABS(FNV_HASH(x)) % 10`.\n\nThis function implements the same algorithm that Impala uses internally for hashing, on systems where the CRC32 instructions are not available.\n\nThis function implements the Fowler–Noll–Vo hash function, in particular the FNV-1a variation. This is not a perfect hash function: some combinations of values could produce the same result value. It is not suitable for cryptographic use.\n\nSimilar input values of different types could produce different hash values, for example the same numeric value represented as `SMALLINT` or `BIGINT`, `FLOAT` or `DOUBLE`, or `DECIMAL(5,2)` or `DECIMAL(20,5)`.\n\nExamples:\n\n``````[localhost:21000] > create table h (x int, s string);\n[localhost:21000] > insert into h values (0, 'hello'), (1,'world'), (1234567890,'antidisestablishmentarianism');\n[localhost:21000] > select x, fnv_hash(x) from h;\n+------------+----------------------+\n\| x \| fnv_hash(x) \|\n+------------+----------------------+\n\| 0 \| -2611523532599129963 \|\n\| 1 \| 4307505193096137732 \|\n\| 1234567890 \| 3614724209955230832 \|\n+------------+----------------------+\n[localhost:21000] > select s, fnv_hash(s) from h;\n+------------------------------+---------------------+\n\| s \| fnv_hash(s) \|\n+------------------------------+---------------------+\n\| hello \| 6414202926103426347 \|\n\| world \| 6535280128821139475 \|\n\| antidisestablishmentarianism \| -209330013948433970 \|\n+------------------------------+---------------------+\n[localhost:21000] > select s, abs(fnv_hash(s)) % 10 from h;\n+------------------------------+-------------------------+\n\| s \| abs(fnv_hash(s)) % 10.0 \|\n+------------------------------+-------------------------+\n\| hello \| 8 \|\n\| world \| 6 \|\n\| antidisestablishmentarianism \| 4 \|\n+------------------------------+-------------------------+``````\n\nFor short argument values, the high-order bits of the result have relatively low entropy:\n\n``````[localhost:21000] > create table b (x boolean);\n[localhost:21000] > insert into b values (true), (true), (false), (false);\n[localhost:21000] > select x, fnv_hash(x) from b;\n+-------+---------------------+\n\| x \| fnv_hash(x) \|\n+-------+---------------------+\n\| true \| 2062020650953872396 \|\n\| true \| 2062020650953872396 \|\n\| false \| 2062021750465500607 \|\n\| false \| 2062021750465500607 \|\n+-------+---------------------+``````\n\nGREATEST(BIGINT a[, BIGINT b ...]), GREATEST(DOUBLE a[, DOUBLE b ...]), GREATEST(DECIMAL(p,s) a[, DECIMAL(p,s) b ...]), GREATEST(STRING a[, STRING b ...]), GREATEST(TIMESTAMP a[, TIMESTAMP b ...])\nPurpose: Returns the largest value from a list of expressions.\n\nReturn type: same as the initial argument value, except that integer values are promoted to `BIGINT` and floating-point values are promoted to `DOUBLE`; use `CAST()` when inserting into a smaller numeric column\n\nHEX(BIGINT a), HEX(STRING a)\nPurpose: Returns the hexadecimal representation of an integer value, or of the characters in a string.\n\nReturn type: `STRING`\n\nIS_INF(DOUBLE a)\nPurpose: Tests whether a value is equal to the special value inf, signifying infinity.\n\nReturn type: `BOOLEAN`\n\nUsage notes:\n\nInfinity and NaN can be specified in text data files as `inf` and `nan` respectively, and Impala interprets them as these special values. They can also be produced by certain arithmetic expressions; for example, `1/0` returns `Infinity` and `pow(-1, 0.5)` returns `NaN`. Or you can cast the literal values, such as ```CAST('nan' AS DOUBLE)``` or `CAST('inf' AS DOUBLE)`.\n\nIS_NAN(DOUBLE a)\nPurpose: Tests whether a value is equal to the special value NaN, signifying not a number.\n\nReturn type: `BOOLEAN`\n\nUsage notes:\n\nInfinity and NaN can be specified in text data files as `inf` and `nan` respectively, and Impala interprets them as these special values. They can also be produced by certain arithmetic expressions; for example, `1/0` returns `Infinity` and `pow(-1, 0.5)` returns `NaN`. Or you can cast the literal values, such as ```CAST('nan' AS DOUBLE)``` or `CAST('inf' AS DOUBLE)`.\n\nLEAST(BIGINT a[, BIGINT b ...]), LEAST(DOUBLE a[, DOUBLE b ...]), LEAST(DECIMAL(p,s) a[, DECIMAL(p,s) b ...]), LEAST(STRING a[, STRING b ...]), LEAST(TIMESTAMP a[, TIMESTAMP b ...])\nPurpose: Returns the smallest value from a list of expressions.\n\nReturn type: same as the initial argument value, except that integer values are promoted to `BIGINT` and floating-point values are promoted to `DOUBLE`; use `CAST()` when inserting into a smaller numeric column\n\nLN(DOUBLE a), DLOG1(DOUBLE a)\nPurpose: Returns the natural logarithm of the argument.\n\nReturn type: `DOUBLE`\n\nLOG(DOUBLE base, DOUBLE a)\nPurpose: Returns the logarithm of the second argument to the specified base.\n\nReturn type: `DOUBLE`\n\nLOG10(DOUBLE a), DLOG10(DOUBLE a)\nPurpose: Returns the logarithm of the argument to the base 10.\n\nReturn type: `DOUBLE`\n\nLOG2(DOUBLE a)\nPurpose: Returns the logarithm of the argument to the base 2.\n\nReturn type: `DOUBLE`\n\nMAX_INT(), MAX_TINYINT(), MAX_SMALLINT(), MAX_BIGINT()\nPurpose: Returns the largest value of the associated integral type.\n\nReturn type: The same as the integral type being checked.\n\nUsage notes: Use the corresponding `min_` and `max_` functions to check if all values in a column are within the allowed range, before copying data or altering column definitions. If not, switch to the next higher integral type or to a `DECIMAL` with sufficient precision.\n\nMIN_INT(), MIN_TINYINT(), MIN_SMALLINT(), MIN_BIGINT()\nPurpose: Returns the smallest value of the associated integral type (a negative number).\n\nReturn type: The same as the integral type being checked.\n\nUsage notes: Use the corresponding `min_` and `max_` functions to check if all values in a column are within the allowed range, before copying data or altering column definitions. If not, switch to the next higher integral type or to a `DECIMAL` with sufficient precision.\n\nMOD(numeric_type a, same_type b)\nPurpose: Returns the modulus of a number. Equivalent to the `%` arithmetic operator. Works with any size integer type, any size floating-point type, and `DECIMAL` with any precision and scale.\n\nReturn type: Same as the input value\n\nUsage notes:\n\nBecause this function works with `DECIMAL` values, prefer it over `fmod()` when working with fractional values. It is not subject to the rounding errors that make `fmod()` problematic with floating-point numbers.\n\nQuery plans shows the `MOD()` function as the `%` operator.\n\nExamples:\n\nThe following examples show how the `mod()` function works for whole numbers and fractional values, and how the `%` operator works the same way. In the case of `mod(9.9,3)`, the type conversion for the second argument results in the first argument being interpreted as `DOUBLE`, so to produce an accurate `DECIMAL` result requires casting the second argument or writing it as a `DECIMAL` literal, 3.0.\n\n``````select mod(10,3);\n+-------------+\n\| mod(10, 3) \|\n+-------------+\n\| 1 \|\n+-------------+\n\nselect mod(5.5,2);\n+--------------+\n\| mod(5.5, 2) \|\n+--------------+\n\| 1.5 \|\n+--------------+\n\nselect 10 % 3;\n+--------+\n\| 10 % 3 \|\n+--------+\n\| 1 \|\n+--------+\n\nselect 5.5 % 2;\n+---------+\n\| 5.5 % 2 \|\n+---------+\n\| 1.5 \|\n+---------+\n\nselect mod(9.9,3.3);\n+---------------+\n\| mod(9.9, 3.3) \|\n+---------------+\n\| 0.0 \|\n+---------------+\n\nselect mod(9.9,3);\n+--------------------+\n\| mod(9.9, 3) \|\n+--------------------+\n\| 0.8999996185302734 \|\n+--------------------+\n\nselect mod(9.9, cast(3 as decimal(2,1)));\n+-----------------------------------+\n\| mod(9.9, cast(3 as decimal(2,1))) \|\n+-----------------------------------+\n\| 0.9 \|\n+-----------------------------------+\n\nselect mod(9.9,3.0);\n+---------------+\n\| mod(9.9, 3.0) \|\n+---------------+\n\| 0.9 \|\n+---------------+\n``````\nMURMUR_HASH(type v)\nPurpose: Returns a consistent 64-bit value derived from the input argument, for convenience of implementing MurmurHash2 non-cryptographic hash function.\n\nReturn type: `BIGINT`\n\nUsage notes:\n\nYou might use the return value in an application where you perform load balancing, bucketing, or some other technique to divide processing or storage. This function provides a good performance for all kinds of keys such as number, ascii string and UTF-8. It can be recommended as general-purpose hashing function.\n\nRegarding comparison of murmur_hash with fnv_hash, murmur_hash is based on Murmur2 hash algorithm and fnv_hash function is based on FNV-1a hash algorithm. Murmur2 and FNV-1a can show very good randomness and performance compared with well known other hash algorithms, but Murmur2 slightly show better randomness and performance than FNV-1a.\n\nSimilar input values of different types could produce different hash values, for example the same numeric value represented as `SMALLINT` or `BIGINT`, `FLOAT` or `DOUBLE`, or `DECIMAL(5,2)` or `DECIMAL(20,5)`.\n\nExamples:\n\n``````[localhost:21000] > create table h (x int, s string);\n[localhost:21000] > insert into h values (0, 'hello'), (1,'world'), (1234567890,'antidisestablishmentarianism');\n[localhost:21000] > select x, murmur_hash(x) from h;\n+------------+----------------------+\n\| x \| murmur_hash(x) \|\n+------------+----------------------+\n\| 0 \| 6960269033020761575 \|\n\| 1 \| -780611581681153783 \|\n\| 1234567890 \| -5754914572385924334 \|\n+------------+----------------------+\n[localhost:21000] > select s, murmur_hash(s) from h;\n+------------------------------+----------------------+\n\| s \| murmur_hash(s) \|\n+------------------------------+----------------------+\n\| hello \| 2191231550387646743 \|\n\| world \| 5568329560871645431 \|\n\| antidisestablishmentarianism \| -2261804666958489663 \|\n+------------------------------+----------------------+ ``````\n\nFor short argument values, the high-order bits of the result have relatively higher entropy than fnv_hash:\n\n``````[localhost:21000] > create table b (x boolean);\n[localhost:21000] > insert into b values (true), (true), (false), (false);\n[localhost:21000] > select x, murmur_hash(x) from b;\n+-------+----------------------+\n\| x \| murmur_hash(x) \|\n+-------+---------------------++\n\| true \| -5720937396023583481 \|\n\| true \| -5720937396023583481 \|\n\| false \| 6351753276682545529 \|\n\| false \| 6351753276682545529 \|\n+-------+--------------------+-+``````\n\nNEGATIVE(numeric_type a)\nPurpose: Returns the argument with the sign reversed; returns a positive value if the argument was already negative.\n\nReturn type: Same as the input value\n\nUsage notes: Use `-abs(a)` instead if you need to ensure all return values are negative.\n\nPI()\nPurpose: Returns the constant pi.\n\nReturn type: `double`\n\nPMOD(BIGINT a, BIGINT b), PMOD(DOUBLE a, DOUBLE b)\nPurpose: Returns the positive modulus of a number. Primarily for Hive SQL compatibility.\n\nReturn type: `INT` or `DOUBLE`, depending on type of arguments\n\nExamples:\n\nThe following examples show how the `fmod()` function sometimes returns a negative value depending on the sign of its arguments, and the `pmod()` function returns the same value as `fmod()`, but sometimes with the sign flipped.\n\n``````select fmod(-5,2);\n+-------------+\n\| fmod(-5, 2) \|\n+-------------+\n\| -1 \|\n+-------------+\n\nselect pmod(-5,2);\n+-------------+\n\| pmod(-5, 2) \|\n+-------------+\n\| 1 \|\n+-------------+\n\nselect fmod(-5,-2);\n+--------------+\n\| fmod(-5, -2) \|\n+--------------+\n\| -1 \|\n+--------------+\n\nselect pmod(-5,-2);\n+--------------+\n\| pmod(-5, -2) \|\n+--------------+\n\| -1 \|\n+--------------+\n\nselect fmod(5,-2);\n+-------------+\n\| fmod(5, -2) \|\n+-------------+\n\| 1 \|\n+-------------+\n\nselect pmod(5,-2);\n+-------------+\n\| pmod(5, -2) \|\n+-------------+\n\| -1 \|\n+-------------+\n``````\nPOSITIVE(numeric_type a)\nPurpose: Returns the original argument unchanged (even if the argument is negative).\n\nReturn type: Same as the input value\n\nUsage notes: Use `abs()` instead if you need to ensure all return values are positive.\n\nPOW(DOUBLE a, double p), POWER(DOUBLE a, DOUBLE p), DPOW(DOUBLE a, DOUBLE p), FPOW(DOUBLE a, DOUBLE p)\nPurpose: Returns the first argument raised to the power of the second argument.\n\nReturn type: `DOUBLE`\n\nPRECISION(numeric_expression)\nPurpose: Computes the precision (number of decimal digits) needed to represent the type of the argument expression as a `DECIMAL` value.\n\nUsage notes:\n\nTypically used in combination with the `scale()` function, to determine the appropriate `DECIMAL(precision,scale)` type to declare in a `CREATE TABLE` statement or `CAST()` function.\n\nReturn type: `INT`\n\nExamples:\n\nThe following examples demonstrate how to check the precision and scale of numeric literals or other numeric expressions. Impala represents numeric literals in the smallest appropriate type. 5 is a `TINYINT` value, which ranges from -128 to 127, therefore 3 decimal digits are needed to represent the entire range, and because it is an integer value there are no fractional digits. 1.333 is interpreted as a `DECIMAL` value, with 4 digits total and 3 digits after the decimal point.\n``````[localhost:21000] > select precision(5), scale(5);\n+--------------+----------+\n\| precision(5) \| scale(5) \|\n+--------------+----------+\n\| 3 \| 0 \|\n+--------------+----------+\n[localhost:21000] > select precision(1.333), scale(1.333);\n+------------------+--------------+\n\| precision(1.333) \| scale(1.333) \|\n+------------------+--------------+\n\| 4 \| 3 \|\n+------------------+--------------+\n[localhost:21000] > with t1 as\n( select cast(12.34 as decimal(20,2)) x union select cast(1 as decimal(8,6)) x )\nselect precision(x), scale(x) from t1 limit 1;\n+--------------+----------+\n\| precision(x) \| scale(x) \|\n+--------------+----------+\n\| 24 \| 6 \|\n+--------------+----------+\n``````\nQUOTIENT(BIGINT numerator, BIGINT denominator), QUOTIENT(DOUBLE numerator, DOUBLE denominator)\nPurpose: Returns the first argument divided by the second argument, discarding any fractional part. Avoids promoting integer arguments to `DOUBLE` as happens with the `/` SQL operator. Also includes an overload that accepts `DOUBLE` arguments, discards the fractional part of each argument value before dividing, and again returns `BIGINT`. With integer arguments, this function works the same as the `DIV` operator.\n\nReturn type: `BIGINT`\n\nPurpose: Converts argument value from degrees to radians.\n\nReturn type: `DOUBLE`\n\nRAND(), RAND(BIGINT seed), RANDOM(), RANDOM(BIGINT seed)\nPurpose: Returns a random value between 0 and 1. After `rand()` is called with a seed argument, it produces a consistent random sequence based on the seed value.\n\nReturn type: `DOUBLE`\n\nUsage notes: Currently, the random sequence is reset after each query, and multiple calls to `rand()` within the same query return the same value each time. For different number sequences that are different for each query, pass a unique seed value to each call to `rand()`. For example, `select rand(unix_timestamp()) from ...`\n\nExamples:\n\nThe following examples show how `rand()` can produce sequences of varying predictability, so that you can reproduce query results involving random values or generate unique sequences of random values for each query. When `rand()` is called with no argument, it generates the same sequence of values each time, regardless of the ordering of the result set. When `rand()` is called with a constant integer, it generates a different sequence of values, but still always the same sequence for the same seed value. If you pass in a seed value that changes, such as the return value of the expression `unix_timestamp(now())`, each query will use a different sequence of random values, potentially more useful in probability calculations although more difficult to reproduce at a later time. Therefore, the final two examples with an unpredictable seed value also include the seed in the result set, to make it possible to reproduce the same random sequence later.\n\n``````select x, rand() from three_rows;\n+---+-----------------------+\n\| x \| rand() \|\n+---+-----------------------+\n\| 1 \| 0.0004714746030380365 \|\n\| 2 \| 0.5895895192351144 \|\n\| 3 \| 0.4431900859080209 \|\n+---+-----------------------+\n\nselect x, rand() from three_rows order by x desc;\n+---+-----------------------+\n\| x \| rand() \|\n+---+-----------------------+\n\| 3 \| 0.0004714746030380365 \|\n\| 2 \| 0.5895895192351144 \|\n\| 1 \| 0.4431900859080209 \|\n+---+-----------------------+\n\nselect x, rand(1234) from three_rows order by x;\n+---+----------------------+\n\| x \| rand(1234) \|\n+---+----------------------+\n\| 1 \| 0.7377511392057646 \|\n\| 2 \| 0.009428468537250751 \|\n\| 3 \| 0.208117277924026 \|\n+---+----------------------+\n\nselect x, rand(1234) from three_rows order by x desc;\n+---+----------------------+\n\| x \| rand(1234) \|\n+---+----------------------+\n\| 3 \| 0.7377511392057646 \|\n\| 2 \| 0.009428468537250751 \|\n\| 1 \| 0.208117277924026 \|\n+---+----------------------+\n\nselect x, unix_timestamp(now()), rand(unix_timestamp(now()))\nfrom three_rows order by x;\n+---+-----------------------+-----------------------------+\n\| x \| unix_timestamp(now()) \| rand(unix_timestamp(now())) \|\n+---+-----------------------+-----------------------------+\n\| 1 \| 1440777752 \| 0.002051228658320023 \|\n\| 2 \| 1440777752 \| 0.5098743483004506 \|\n\| 3 \| 1440777752 \| 0.9517714925817081 \|\n+---+-----------------------+-----------------------------+\n\nselect x, unix_timestamp(now()), rand(unix_timestamp(now()))\nfrom three_rows order by x desc;\n+---+-----------------------+-----------------------------+\n\| x \| unix_timestamp(now()) \| rand(unix_timestamp(now())) \|\n+---+-----------------------+-----------------------------+\n\| 3 \| 1440777761 \| 0.9985985015512437 \|\n\| 2 \| 1440777761 \| 0.3251255333074953 \|\n\| 1 \| 1440777761 \| 0.02422675025846192 \|\n+---+-----------------------+-----------------------------+\n``````\nROUND(DOUBLE a), ROUND(DOUBLE a, INT d), ROUND(DECIMAL a, int_type d), DROUND(DOUBLE a), DROUND(DOUBLE a, INT d), DROUND(DECIMAL(p,s) a, int_type d)\nPurpose: Rounds a floating-point value. By default (with a single argument), rounds to the nearest integer. Values ending in .5 are rounded up for positive numbers, down for negative numbers (that is, away from zero). The optional second argument specifies how many digits to leave after the decimal point; values greater than zero produce a floating-point return value rounded to the requested number of digits to the right of the decimal point.\n\nReturn type: Same as the input type\n\nSCALE(numeric_expression)\nPurpose: Computes the scale (number of decimal digits to the right of the decimal point) needed to represent the type of the argument expression as a `DECIMAL` value.\n\nUsage notes:\n\nTypically used in combination with the `precision()` function, to determine the appropriate `DECIMAL(precision,scale)` type to declare in a `CREATE TABLE` statement or `CAST()` function.\n\nReturn type: `int`\n\nExamples:\n\nThe following examples demonstrate how to check the precision and scale of numeric literals or other numeric expressions. Impala represents numeric literals in the smallest appropriate type. 5 is a `TINYINT` value, which ranges from -128 to 127, therefore 3 decimal digits are needed to represent the entire range, and because it is an integer value there are no fractional digits. 1.333 is interpreted as a `DECIMAL` value, with 4 digits total and 3 digits after the decimal point.\n``````[localhost:21000] > select precision(5), scale(5);\n+--------------+----------+\n\| precision(5) \| scale(5) \|\n+--------------+----------+\n\| 3 \| 0 \|\n+--------------+----------+\n[localhost:21000] > select precision(1.333), scale(1.333);\n+------------------+--------------+\n\| precision(1.333) \| scale(1.333) \|\n+------------------+--------------+\n\| 4 \| 3 \|\n+------------------+--------------+\n[localhost:21000] > with t1 as\n( select cast(12.34 as decimal(20,2)) x union select cast(1 as decimal(8,6)) x )\nselect precision(x), scale(x) from t1 limit 1;\n+--------------+----------+\n\| precision(x) \| scale(x) \|\n+--------------+----------+\n\| 24 \| 6 \|\n+--------------+----------+\n``````\nSIGN(DOUBLE a)\nPurpose: Returns -1, 0, or 1 to indicate the signedness of the argument value.\n\nReturn type: `INT`\n\nSIN(DOUBLE a)\nPurpose: Returns the sine of the argument.\n\nReturn type: `DOUBLE`\n\nSINH(DOUBLE a)\nPurpose: Returns the hyperbolic sine of the argument.\n\nReturn type: `DOUBLE`\n\nSQRT(DOUBLE a), DSQRT(DOUBLE a)\nPurpose: Returns the square root of the argument.\n\nReturn type: `DOUBLE`\n\nTAN(DOUBLE a)\nPurpose: Returns the tangent of the argument.\n\nReturn type: `DOUBLE`\n\nTANH(DOUBLE a)\nPurpose: Returns the hyperbolic tangent of the argument.\n\nReturn type: `DOUBLE`\n\nTRUNCATE(DOUBLE_or_DECIMAL a[, digits_to_leave]), DTRUNC(DOUBLE_or_DECIMAL a[, digits_to_leave]), TRUNC(DOUBLE_or_DECIMAL a[, digits_to_leave])\nPurpose: Removes some or all fractional digits from a numeric value.\n\nArguments: With a single floating-point argument, removes all fractional digits, leaving an integer value. The optional second argument specifies the number of fractional digits to include in the return value, and only applies when the argument type is `DECIMAL`. A second argument of 0 truncates to a whole integer value. A second argument of negative N sets N digits to 0 on the left side of the decimal\n\nScale argument: The scale argument applies only when truncating `DECIMAL` values. It is an integer specifying how many significant digits to leave to the right of the decimal point. A scale argument of 0 truncates to a whole integer value. A scale argument of negative N sets N digits to 0 on the left side of the decimal point.\n\n`TRUNCATE()`, `DTRUNC()`, and `TRUNC()` are aliases for the same function.\n\nReturn type: Same as the input type\n\nAdded in: The `TRUNC()` alias was added in Impala 2.10.0.\n\nUsage notes:\n\nYou can also pass a `DOUBLE` argument, or `DECIMAL` argument with optional scale, to the `DTRUNC()` or `TRUNCATE` functions. Using the `TRUNC()` function for numeric values is common with other industry-standard database systems, so you might find such `TRUNC()` calls in code that you are porting to Impala.\n\nThe `TRUNC()` function also has a signature that applies to `TIMESTAMP` values.\n\nExamples:\n\nThe following examples demonstrate the `TRUNCATE()` and `DTRUNC()` signatures for this function:\n\n``````select truncate(3.45);\n+----------------+\n\| truncate(3.45) \|\n+----------------+\n\| 3 \|\n+----------------+\n\nselect truncate(-3.45);\n+-----------------+\n\| truncate(-3.45) \|\n+-----------------+\n\| -3 \|\n+-----------------+\n\nselect truncate(3.456,1);\n+--------------------+\n\| truncate(3.456, 1) \|\n+--------------------+\n\| 3.4 \|\n+--------------------+\n\nselect dtrunc(3.456,1);\n+------------------+\n\| dtrunc(3.456, 1) \|\n+------------------+\n\| 3.4 \|\n+------------------+\n\nselect truncate(3.456,2);\n+--------------------+\n\| truncate(3.456, 2) \|\n+--------------------+\n\| 3.45 \|\n+--------------------+\n\nselect truncate(3.456,7);\n+--------------------+\n\| truncate(3.456, 7) \|\n+--------------------+\n\| 3.4560000 \|\n+--------------------+\n``````\n\nThe following examples demonstrate using `TRUNC()` with `DECIMAL` or `DOUBLE` values, and with an optional scale argument for `DECIMAL` values. (The behavior is the same for the `TRUNCATE()` and `DTRUNC()` aliases also.)\n\n``````\ncreate table t1 (d decimal(20,7));\n\n-- By default, no digits to the right of the decimal point.\ninsert into t1 values (1.1), (2.22), (3.333), (4.4444), (5.55555);\nselect trunc(d) from t1 order by d;\n+----------+\n\| trunc(d) \|\n+----------+\n\| 1 \|\n\| 2 \|\n\| 3 \|\n\| 4 \|\n\| 5 \|\n+----------+\n\n-- 1 digit to the right of the decimal point.\nselect trunc(d,1) from t1 order by d;\n+-------------+\n\| trunc(d, 1) \|\n+-------------+\n\| 1.1 \|\n\| 2.2 \|\n\| 3.3 \|\n\| 4.4 \|\n\| 5.5 \|\n+-------------+\n\n-- 2 digits to the right of the decimal point,\n-- including trailing zeroes if needed.\nselect trunc(d,2) from t1 order by d;\n+-------------+\n\| trunc(d, 2) \|\n+-------------+\n\| 1.10 \|\n\| 2.22 \|\n\| 3.33 \|\n\| 4.44 \|\n\| 5.55 \|\n+-------------+\n\ninsert into t1 values (9999.9999), (8888.8888);\n\n-- Negative scale truncates digits to the left\n-- of the decimal point.\nselect trunc(d,-2) from t1 where d > 100 order by d;\n+--------------+\n\| trunc(d, -2) \|\n+--------------+\n\| 8800 \|\n\| 9900 \|\n+--------------+\n\n-- The scale of the result is adjusted to match the\n-- scale argument.\nselect trunc(d,2),\nprecision(trunc(d,2)) as p,\nscale(trunc(d,2)) as s\nfrom t1 order by d;\n+-------------+----+---+\n\| trunc(d, 2) \| p \| s \|\n+-------------+----+---+\n\| 1.10 \| 15 \| 2 \|\n\| 2.22 \| 15 \| 2 \|\n\| 3.33 \| 15 \| 2 \|\n\| 4.44 \| 15 \| 2 \|\n\| 5.55 \| 15 \| 2 \|\n\| 8888.88 \| 15 \| 2 \|\n\| 9999.99 \| 15 \| 2 \|\n+-------------+----+---+\n``````\n``````\ncreate table dbl (d double);\n\ninsert into dbl values\n(1.1), (2.22), (3.333), (4.4444), (5.55555),\n(8888.8888), (9999.9999);\n\n-- With double values, there is no optional scale argument.\nselect trunc(d) from dbl order by d;\n+----------+\n\| trunc(d) \|\n+----------+\n\| 1 \|\n\| 2 \|\n\| 3 \|\n\| 4 \|\n\| 5 \|\n\| 8888 \|\n\| 9999 \|\n+----------+\n``````\nUNHEX(STRING a)\nPurpose: Returns a string of characters with ASCII values corresponding to pairs of hexadecimal digits in the argument.\n\nReturn type: `STRING`\n\nWIDTH_BUCKET(DECIMAL expr, DECIMAL min_value, DECIMAL max_value, INT num_buckets)\nPurpose: Returns the bucket number in which the `expr` value would fall in the histogram where its range between `min_value` and `max_value` is divided into `num_buckets` buckets of identical sizes.\nThe function returns:\n• `NULL` if any argument is `NULL`.\n• `0` if `expr` < `min_value`.\n• `num_buckets + 1` if `expr` >= `max_val`.\n• If none of the above, the bucket number where `expr` falls.\nArguments:The following rules apply to the arguments.\n• `min_val` is the minimum value of the histogram range.\n• `max_val` is the maximum value of the histogram range.\n• `num_buckets` must be greater than `0`.\n• `min_value` must be less than `max_value`.\n\nUsage notes:\n\nEach bucket contains values equal to or greater than the base value of that bucket and less than the base value of the next bucket. For example, with `width_bucket(8, 1, 10, 3)`, the bucket ranges are actually the 0th \"underflow bucket\" with the range (-infinity to 0.999...), (1 to 3.999...), (4, to 6.999...), (7 to 9.999...), and the \"overflow bucket\" with the range (10 to infinity).\n\nReturn type: `BIGINT`\n\nThe below function creates `3` buckets between the range of `1` and `20` with the bucket width of 6.333, and returns `2` for the bucket #2 where the value `8` falls in:\n``WIDTH_BUCKET(8, 1, 20, 3)``\n``````SELECT account, invoice_amount, WIDTH_BUCKET(invoice_amount,50,1000,10)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5902832,"math_prob":0.95497847,"size":33702,"snap":"2022-27-2022-33","text_gpt3_token_len":9096,"char_repetition_ratio":0.25238886,"word_repetition_ratio":0.29959267,"special_character_ratio":0.39602992,"punctuation_ratio":0.17843343,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9926767,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-14T23:58:51Z\",\"WARC-Record-ID\":\"<urn:uuid:77fed51d-8d63-45ae-b6f8-10c0bdda07e5>\",\"Content-Length\":\"78538\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5ccba360-5a0b-43a3-81a8-eb1134575d4c>\",\"WARC-Concurrent-To\":\"<urn:uuid:d44bbf4a-5736-418c-b902-c59da80b37e1>\",\"WARC-IP-Address\":\"13.224.214.51\",\"WARC-Target-URI\":\"https://docs.cloudera.com/cdw-runtime/1.4.0/impala-sql-reference/topics/impala-math-functions.html\",\"WARC-Payload-Digest\":\"sha1:F4CH2YAQKU2ZYVTBZSGEU752MLJGGGK6\",\"WARC-Block-Digest\":\"sha1:3BAJQKOH6CSTMGW4RLTLOLKXFFISQBNQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882572089.53_warc_CC-MAIN-20220814234405-20220815024405-00739.warc.gz\"}"}
https://www.reference.com/math/3x-2y-8-solved-542883857cc5485d	[ "How Is 3x + 2y = 8 Solved?\n\nTo solve the slope of 3x + 2y = 8, the equation must first be put into slope intercept form with y on the left side and the remaining numbers on the right side. It can be transformed to 2y = -3x + 8, which will then be simplified down to y = -3/2x + 8. The slope of the line is -3/2.\n\nThese types of equations are most commonly used to find the slope and the y-intercept of a problem. To find these, the equation must always be transformed into slope intercept form. The slope of the line is always the number that is connected with x, while the y-intercept is always the number that is on its own in the equation.\n\nIn the 3x + 2y = 8 equation, 8 is the y-intercept. The y-intercept is the point of the slope or the line where the line goes through the y-axis. The line will always pass through the y-axis only once in these types of equations because the line is constantly evolving and growing. Slope intercept form is always written as y = mx + b. The slope of the line is always m.\n\nSimilar Articles" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9622812,"math_prob":0.9991973,"size":983,"snap":"2019-43-2019-47","text_gpt3_token_len":247,"char_repetition_ratio":0.18283963,"word_repetition_ratio":0.0625,"special_character_ratio":0.2512716,"punctuation_ratio":0.07619048,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99964225,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-21T18:06:58Z\",\"WARC-Record-ID\":\"<urn:uuid:d49dda75-4806-4282-9db6-119de2f2c1ca>\",\"Content-Length\":\"159394\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dbfb45f7-0111-45c1-b75d-fea42771afb7>\",\"WARC-Concurrent-To\":\"<urn:uuid:f3e6cd9a-d600-421c-880f-310227922f72>\",\"WARC-IP-Address\":\"151.101.250.114\",\"WARC-Target-URI\":\"https://www.reference.com/math/3x-2y-8-solved-542883857cc5485d\",\"WARC-Payload-Digest\":\"sha1:H6DW4TRCROSJ7V6LTVYYW47ALI5T64ME\",\"WARC-Block-Digest\":\"sha1:676AYBNCKU3ONXOB6OCAC5JGLNLOO5J7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987781397.63_warc_CC-MAIN-20191021171509-20191021195009-00132.warc.gz\"}"}
https://stacktuts.com/how-to-get-the-top-n-rows-of-a-pandas-data-frame-in-python	[ "# How to get the top n rows of a pandas DataFrame in Python\n\nPublished 8/2/20221 minute reading\n\nThere are a number of ways to get the top n rows of a pandas DataFrame.\n\n## Get the top n rows of a DataFrame using the head method\n\nGetting the top n rows of a `pandas.DataFrame` returns a new DataFrame containing only the first `n` rows of the original.\n\nPandas has a `DataFrame.head()` method that returns the first n rows of a DataFrame. Calling `DataFrame.head(n)` returns a new DataFrame containing only the first `n` rows of the original.\n\nLet check out an example how to get the top 2 rows of the `df` DataFrame.\n\n``````import pandas as pd\n\n# Create a DataFrame\ndf = pd.DataFrame({\n'A': [1, 2, 3, 4, 5],\n'B': [5, 4, 3, 2, 1]\n})\n\n# Get the first 2 rows\n\n# Print first_two_rows\nprint(first_two_rows)\n\n# It should print:\n#\n# A B\n# 1 5\n# 2 4``````\n\n## Get the top n rows of a DataFrame using the iloc\n\nAnother way to get the first n rows of a DataFrame is to use the `DataFrame.iloc` method.\n\n``````import pandas as pd\n# Create a DataFrame\ndf = pd.DataFrame({\n'A': [1, 2, 3, 4, 5],\n'B': [5, 4, 3, 2, 1]\n})\n\n# Get the first 2 rows\nfirst_two_rows = df.iloc[:2]\n\n# Print first_two_rows\nprint(first_two_rows)\n\n# It should print:\n#\n# A B\n# 1 5\n# 2 4``````\n\n## Summary\n\n### What is the method to get the top n rows of a pandas DataFrame in Python?\n\nEither DataFrame.head() or DataFrame.iloc[:n]" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51061666,"math_prob":0.65736663,"size":1251,"snap":"2022-40-2023-06","text_gpt3_token_len":377,"char_repetition_ratio":0.21251002,"word_repetition_ratio":0.5435685,"special_character_ratio":0.3133493,"punctuation_ratio":0.15248227,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99734503,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-02T08:45:52Z\",\"WARC-Record-ID\":\"<urn:uuid:68822f38-d74b-4e2a-9ce5-03f828ffe1cf>\",\"Content-Length\":\"53538\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:07108fc2-3700-4b4f-9086-a37b804122c4>\",\"WARC-Concurrent-To\":\"<urn:uuid:719918ed-08ad-4d89-b44d-fc76c9fa4bc4>\",\"WARC-IP-Address\":\"104.21.35.200\",\"WARC-Target-URI\":\"https://stacktuts.com/how-to-get-the-top-n-rows-of-a-pandas-data-frame-in-python\",\"WARC-Payload-Digest\":\"sha1:FKCHRJX33H42336SBJUSEHUC3BH44OGS\",\"WARC-Block-Digest\":\"sha1:IKV5SVGF5JF7HC72443POBKEZLJKI4S5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337307.81_warc_CC-MAIN-20221002083954-20221002113954-00684.warc.gz\"}"}
https://www.physicsforums.com/members/charliecw.649017/recent-content	[ "# Recent content by CharlieCW\n\n1. ### Kramers-Kronig Relations: Principal Value\n\nYou're right, I had taken the wrong limits of integration. Sorry for the late reply by the way, I had an exam this morning. Now continuing with the equations, let's start from the beginning. Assuming $\\chi_1=\\omega^2_p/\\omega$, then we can substitute in the Kramers-Kronig relation as...\n2. ### Kramers-Kronig Relations: Principal Value\n\nThanks for your reply, Jason. Just from the form of the equations, if I had $\\chi_1=-\\omega_0/\\omega$, then $\\chi_2=\\pi\\omega_0/2$, as when taking the limits only the evaluation on zero will lead to two non-zero identical terms. Here I have no problem evaluating the limits of the integral...\n3. ### Kramers-Kronig Relations: Principal Value\n\nI'm kind of confused on how to evaluate the principal value as it's a topic I've never seen in complex analysis and all the literature I've read so far only deals with the formal definition, not providing an example on how to calculate it properly. Therefore, I think just understanding at least...\n4. ### One-dimensional polymer (Statistical Physics)\n\nWell I managed to solve it and I got that both the average energy and length follow a Fermi-Dirac like distribution. I think I'll post the solution during the weekend in case anyone finds it useful.\n5. ### One-dimensional polymer (Statistical Physics)\n\nDon't worry, with your explanation I better understood the meaning of the terms in the exponential and I think I see more clearly how to deal with these kind of systems. So then my idea about considering the tension $\\tau$ for the linear case was correct since, as you mentioned, it is part of...\n6. ### One-dimensional polymer (Statistical Physics)\n\nIndeed, the term $pV_S$ is for the pressure and volume, but since the general formula was derived for a 3D recipient I was thinking about converting it to the one-dimensional case $pV_S\\rightarrow \\tau L$. However, it also makes more sense that you mention to obtain the tension as $dZ/dl$...\n7. ### What percent of the sky can an astronomer see at one time?\n\nIndeed, in my answer I'm basically considering the top of the well as a flat disk to find an expression for the angle $\\theta$.\n8. ### What percent of the sky can an astronomer see at one time?\n\nSince the view is 3D, you should indeed solid angles to calculate the angle of vision. First consider the case were the astronomer is outside the well. In this case, he sees the 100% of the sky (assuming you call 100% seeing the whole half hemisphere on where they're standing). So the solid...\n9. ### One-dimensional polymer (Statistical Physics)\n\n1. Homework Statement Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length $l_1$ and energy $E_1$) or short axis (of length $l_2$ and energy $E_2$). Suppose that the chain is subject to...\n10. ### Conductor sphere floating on a dielectric fluid\n\nWell I checked similar procedure and I managed to advance the following: While I don't know if it's really useful, if we apply mechanical equilibrium before adding the charge, it's straightforward to find that $\\rho_s=\\rho_l$, where $\\rho_s$ and $\\rho_l$ are the volumetric densities of...\n\n15. ### Finding electric potential using Green's function\n\nThank you for your time, I really appreciate it. Indeed I also checked from Jackson and Greiner and I read that I was free to choose $F(r,r')$ so that $G(r,r')$ is zero on the surface. After a couple of exchanges it turns out we were right: the Green function only depends on the geometry of..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92380124,"math_prob":0.93973213,"size":3843,"snap":"2019-43-2019-47","text_gpt3_token_len":940,"char_repetition_ratio":0.08622037,"word_repetition_ratio":0.0,"special_character_ratio":0.25578976,"punctuation_ratio":0.13618676,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9966005,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-20T11:20:31Z\",\"WARC-Record-ID\":\"<urn:uuid:6ee21961-205a-402f-ab11-c6ed7765e091>\",\"Content-Length\":\"53967\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:83fc20db-8a27-47bf-93b2-f32c1a6bec61>\",\"WARC-Concurrent-To\":\"<urn:uuid:8d9d0ab7-5857-4228-934a-ea68edbd35a6>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/members/charliecw.649017/recent-content\",\"WARC-Payload-Digest\":\"sha1:XQXCRMHIMYTDEW5ZTYPUQVFDQAY3XL77\",\"WARC-Block-Digest\":\"sha1:LJHZI7BZUCUHFFEYCLOYK4UAOOZSVECN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670558.91_warc_CC-MAIN-20191120111249-20191120135249-00115.warc.gz\"}"}
https://betterlesson.com/lesson/516065/counting-up-to-solve-problems?from=cc_lesson	[ "# Counting Up to Solve Problems\n\n13 teachers like this lesson\nPrint Lesson\n\n## Objective\n\nSWBAT count forward by 1's, 2's, 5's or 10's to solve an addition problem containing one and two digit numbers.\n\n#### Big Idea\n\nThe Common Core standards expect addition and subtraction fluency with one and two digit numbers. Students need to develop efficient fluency strategies.\n\n## Warm Up\n\n10 minutes\n\nArticulating mathematical thinking is a critical skill in development at 2nd grade, and requires practice. Students wrote addition and subtraction word problems several days ago and some are left to share. So, I ask any child who has not shared to read his/her word problem to the class. The other students use their math journals to solve the problem.\n\nStudents are now used to this process, and I expect several students will find a solution to share. I expect others to comment on how the problem was solved or ask questions about things that don't make sense (MP3).\n\nAs students present, I particularly draw students' attentions to strategies that use counting up and number lines, because we will be using these strategies/tools today.\n\nI also think it is important to talk about the fact that there is more than one way to solve a word problem.\n\n## Practice Time\n\n15 minutes\n\nToday I introduce a new game to students. This game will encourage students to count up or back to solve a math sentence.\n\nA deck of number cards is placed face down between 2 or 3 students. Students also are given a small pile of colored chips. Each student draws 2 cards and adds them up. Now they share their answers and the answer closest to 10 takes a colored chip from the pile. If 2 students are equi-distant from 10, they would both take a chip. The cards are then placed in a \"discard\" pile and students draw again.\n\nI demonstrate with students how to add the 2 cards and then count up or back to see how far away from 10 they are. If a child picks up a 7 and a 4 they would add and say 7+4 =11. They would then count back 11, 10 so it is only 1 away.\n\nStudents use the number lines on their desks to help them visualize how far away from 10 they are.\n\nSome children may use their fingers to determine how far from ten they are because the largest cards they will draw are 9+9. While I want students to move beyond fingers to other tools that will help them when they deal with larger numbers, I do not stop them from counting up or back on their fingers.\n\nStudents check each other's answers before deciding who gets the chip. They are instructed to count on the number line or number grid if they don't agree about the distance from 10.\n\nThis game requires two sets of problem solving. Both steps are building the fluency expected in the Common Core grade 2 standards.\n\n## Teaching the Lesson\n\n30 minutes\n\nTo start the lesson I introduce the blank number line again. I give each student a blank number line on a sentence strip. I ask them to make a mark right near the left edge. Next I tell them to put down 2 fingers and make another mark. I ask them to repeat this until they have a number line marked in even spaces.\n\nNext I ask them to put a colored chip on the mark at the left edge. I tell them to imagine that that mark is the number 18. I ask them to use the number line to count up 10 and find 10 more than 18. We check our answers. Now I ask them to move the chip to the mark at the right end of the line. I ask them to imagine that that number is 25. I ask them to count back 10 and find 10 less than 25.\n\nI repeat this with several more numbers. I want to reinforce the concept of ten more, ten less as a way to understand adding and subtracting tens.\n\nNext I break the class into 3 groups based on their understanding of the number line from previous lesson informal assessments.\n\nThe group that is able to use the number line to solve problems is given a challenge paper of word problems that require adding 2 digit numbers. They are asked to work in partners, or alone to solve the problems using their number line as needed to support their thinking.\n\nThe group that has some understanding of the number line, but has difficulty with a blank line will work with a parent to locate a number on the line and then add a second number to it. The purpose here is for students to understand that they do not always have to start with the left end marking. They need to think about what number they are adding, ie is it far away from the number they are starting with, or very close. If it is far away, should they count by 2s, or 10s. The group will make some decisions about starting points together, and then solve the problems.\n\nThe group that still does not grasp how the number line works, (they always want to start at 1, even if they are looking for the number 89), will work with me to locate one number relative to another, counting by 1's or 2's and putting chips on the two numbers. If students become competent with this skill, we will try starting with one single digit number and adding on a second single digit number.\n\n## Closing\n\n5 minutes\n\nTo close, I ask student to write in their journals about how they can use a number line to help them with adding or subtracting. I encourage students to use drawings and words to show me how they are using the number line.\n\nThis may seem like a jump to abstract thinking, but I am looking an expression of mathematical thinking, at a basic 2nd grade level. According to Mathematical Practice P3, \"Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.\"\n\nBy asking students how they can use the number line, I encourage students to use drawings as well as words in their journals to demonstrate their thinking.\n\nI review student journals after school to assess student understanding. One child wrote, \"I count up and down on the number line.\" She drew a picture of a number line with hops going along the line. This shows a basic understanding of how a number line can be used." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.96514195,"math_prob":0.7969983,"size":5610,"snap":"2021-31-2021-39","text_gpt3_token_len":1216,"char_repetition_ratio":0.15661791,"word_repetition_ratio":0.017175572,"special_character_ratio":0.21818182,"punctuation_ratio":0.07926829,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9659879,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T06:46:18Z\",\"WARC-Record-ID\":\"<urn:uuid:9d2c7a42-9c04-40ae-934b-444d7702d8f8>\",\"Content-Length\":\"110137\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8dc6db6d-d7e1-42ea-9edd-3d9912685df8>\",\"WARC-Concurrent-To\":\"<urn:uuid:67328049-e20d-4896-a4f6-1f8d8d791ef8>\",\"WARC-IP-Address\":\"52.206.231.121\",\"WARC-Target-URI\":\"https://betterlesson.com/lesson/516065/counting-up-to-solve-problems?from=cc_lesson\",\"WARC-Payload-Digest\":\"sha1:LP3LWGNQQR6SCH33JJOCPAGFVL7HJYZ5\",\"WARC-Block-Digest\":\"sha1:QAFENYX7VS2AMZ5LXVZ27M37Y3XASHF7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152236.64_warc_CC-MAIN-20210727041254-20210727071254-00526.warc.gz\"}"}
https://www.teachoo.com/2919/646/Misc-30---Find-derivative--x---sinn-x---Chapter-13-Class-11/category/Miscellaneous/	[ "Miscellaneous\n\nChapter 13 Class 11 Limits and Derivatives\nSerial order wise", null, "", null, "", null, "", null, "", null, "This video is only available for Teachoo black users\n\nIntroducing your new favourite teacher - Teachoo Black, at only ₹83 per month\n\n### Transcript\n\nMisc 30 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 𝑥/(𝑠𝑖𝑛𝑛 𝑥) Let f(x) = 𝑥/(𝑠𝑖𝑛𝑛 𝑥) Let u = x & v = sinn x ∴ f(x) = 𝑢/𝑣 So, f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = x u’ = 1 Now, v = sinn x Let p = sin x v = pn By Leibnitz product rule v’ = (pn)’ p’ = n pn – 1 p’ Putting p = sin x = n sinn – 1 x (sin x)’ = n sinn – 1 x cos x Now, f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = ( 1 (sin𝑛⁡〖 𝑥〗 ) − 〖𝑛 𝑠𝑖𝑛〗^(𝑛−1) 𝑥 cos⁡〖𝑥 (𝑥)〗)/〖〖(𝑠𝑖𝑛〗^𝑛 𝑥)〗^2 = ( 〖𝑠𝑖𝑛〗^𝑛 𝑥 − 𝑥 (𝑛〖𝑠𝑖𝑛〗^(𝑛−1) 𝑥 cos⁡〖𝑥) 〗)/〖〖(𝑠𝑖𝑛〗^𝑛 𝑥)〗^2 = ( 〖𝒔𝒊𝒏〗^(𝒏−𝟏) 𝒙 . sin⁡〖𝑥 − 𝑥 (𝑛 〗〖𝑠𝑖𝑛〗^(𝑛−1) 𝑥 cos⁡〖𝑥) 〗)/〖〖(𝑠𝑖𝑛〗^𝑛 𝑥)〗^2 = ( 〖𝒔𝒊𝒏〗^(𝒏−𝟏) 𝒙 〖(sin〗⁡〖𝑥 − 𝑛𝑥 . 〗 cos⁡〖𝑥) 〗)/(〖𝑠𝑖𝑛〗^2𝑛 𝑥) = sin⁡〖𝑥 − 𝑛𝑥 cos⁡𝑥 〗/(〖𝑠𝑖𝑛〗^2𝑛 𝒙 . 〖𝒔𝒊𝒏〗^(−(𝒏−𝟏) ) 𝒙) = sin⁡〖𝑥 − 𝑛𝑥 cos⁡𝑥 〗/(〖𝒔𝒊𝒏〗^((𝟐𝒏 − 𝒏+𝟏)) 𝒙) = sin⁡〖𝑥 − 𝑛𝑥 cos⁡𝑥 〗/(〖𝑠𝑖𝑛〗^(𝑛 + 1) 𝑥) Thus, f’(x) = 𝒔𝒊𝒏⁡〖𝒙 − 𝒏𝒙 𝒄𝒐𝒔⁡𝒙 〗/(〖𝒔𝒊𝒏〗^(𝒏 + 𝟏) 𝒙)", null, "" ]	[ null, "https://d1avenlh0i1xmr.cloudfront.net/b139403b-d7f1-4c62-8994-faef0b8e05b5/slide87.jpg", null, "https://d1avenlh0i1xmr.cloudfront.net/cb5df691-952b-4983-ba9d-1f0f3c11b7a7/slide88.jpg", null, "https://d1avenlh0i1xmr.cloudfront.net/2131ed8d-5c83-4b80-a4f0-fe85dd470ce2/slide89.jpg", null, "https://d1avenlh0i1xmr.cloudfront.net/f129524a-f8b3-416f-863d-efa0eed47e14/slide90.jpg", null, "https://d3m5vxlyiwf9rd.cloudfront.net/vimeo-thumbnails/163500452/thumbnail.jpg", null, "https://www.teachoo.com/static/misc/Davneet_Singh.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56832147,"math_prob":0.9999311,"size":1482,"snap":"2022-27-2022-33","text_gpt3_token_len":1092,"char_repetition_ratio":0.2368065,"word_repetition_ratio":0.065625,"special_character_ratio":0.4183536,"punctuation_ratio":0.047477745,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99998355,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,7,null,7,null,7,null,7,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-27T06:09:20Z\",\"WARC-Record-ID\":\"<urn:uuid:53b7b150-9bc8-42b0-b634-85795f879345>\",\"Content-Length\":\"185395\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:52c3aaa7-bed2-4293-853a-8d8122abae52>\",\"WARC-Concurrent-To\":\"<urn:uuid:727dfa69-01b7-4464-b00e-1cc03464d6d0>\",\"WARC-IP-Address\":\"52.86.133.10\",\"WARC-Target-URI\":\"https://www.teachoo.com/2919/646/Misc-30---Find-derivative--x---sinn-x---Chapter-13-Class-11/category/Miscellaneous/\",\"WARC-Payload-Digest\":\"sha1:SGNF2O4AH7TPNEOT3DZ2KMLMUQOG7MNC\",\"WARC-Block-Digest\":\"sha1:3UJDIE5G4EWRQHHBUOC6P4IU4IYV575U\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103328647.18_warc_CC-MAIN-20220627043200-20220627073200-00175.warc.gz\"}"}
https://www.edplace.com/worksheet_info/maths/keystage2/year5/topic/611/944/identify-factors-and-multiples	[ "", null, "# Identify Factors and Multiples\n\nIn this worksheet, students select all the factor pairs for a given number.", null, "Key stage: KS 2\n\nCurriculum topic: Number: Multiplication and Division\n\nCurriculum subtopic: Identify Multiples and Factors\n\nDifficulty level:", null, "", null, "", null, "### QUESTION 1 of 10\n\nThis activity is about factor pairs: whole numbers which multiply together to produce a given number.\n\ne.g. What are the factor pairs of 36?\n\n1 and 36\n\n2 and 18\n\n3 and 12\n\n4 and 9\n\n6 and 6\n\nThese are the only whole numbers which multiply together to give 36.\n\nWant to understand this further and learn how this links to other topics in maths?\n\nWhy not watch this video?\n\nSelect the factor pairs of 20.\n\n1 and 20\n\n2 and 10\n\n4 and 6\n\n4 and 5\n\n6 and 14\n\n10 and 10\n\nSelect the factor pairs of 70.\n\n1 and 70\n\n2 and 68\n\n2 and 35\n\n5 and 14\n\n7 and 10\n\nSelect the factor pairs of 21.\n\n1 and 21\n\n2 and 10\n\n3 and 7\n\n5 and 4\n\n5 and 16\n\nSelect the factor pairs of 11.\n\n1 and 11\n\n2 and 5\n\n2 and 9\n\n5 and 6\n\nSelect the factor pairs of 24.\n\n1 and 24\n\n2 and 12\n\n3 and 21\n\n4 and 6\n\n8 and 3\n\n12 and 12\n\nSelect the factor pairs of 28.\n\n1 and 28\n\n2 and 14\n\n3 and 9\n\n4 and 7\n\n8 and 4\n\nSelect the factor pairs of 42.\n\n1 and 42\n\n2 and 21\n\n3 and 15\n\n6 and 7\n\n14 and 3\n\nSelect the factor pairs of 16.\n\n1 and 16\n\n2 and 8\n\n4 and 4\n\n6 and 10\n\n15 and 1\n\nSelect the factor pairs of 60.\n\n1 and 60\n\n2 and 30\n\n3 and 20\n\n4 and 25\n\n4 and 15\n\n5 and 12\n\n6 and 10\n\nSelect the factor pairs of 56.\n\n1 and 56\n\n2 and 28\n\n3 and 18\n\n4 and 14\n\n7 and 8\n\n• Question 1\n\nSelect the factor pairs of 20.\n\n1 and 20\n2 and 10\n4 and 5\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 20.\n• Question 2\n\nSelect the factor pairs of 70.\n\n1 and 70\n2 and 35\n5 and 14\n7 and 10\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 70.\n• Question 3\n\nSelect the factor pairs of 21.\n\n1 and 21\n3 and 7\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 21.\n• Question 4\n\nSelect the factor pairs of 11.\n\n1 and 11\nEDDIE SAYS\n11 is a prime number as it only has two factors.\n• Question 5\n\nSelect the factor pairs of 24.\n\n1 and 24\n2 and 12\n4 and 6\n8 and 3\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 24.\n• Question 6\n\nSelect the factor pairs of 28.\n\n1 and 28\n2 and 14\n4 and 7\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 28.\n• Question 7\n\nSelect the factor pairs of 42.\n\n1 and 42\n2 and 21\n6 and 7\n14 and 3\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 42.\n• Question 8\n\nSelect the factor pairs of 16.\n\n1 and 16\n2 and 8\n4 and 4\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 16.\n• Question 9\n\nSelect the factor pairs of 60.\n\n1 and 60\n2 and 30\n3 and 20\n4 and 15\n5 and 12\n6 and 10\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 60.\n• Question 10\n\nSelect the factor pairs of 56.\n\n1 and 56\n2 and 28\n4 and 14\n7 and 8\nEDDIE SAYS\nThese are the whole numbers which multiply together to give 56.\n---- OR ----\n\nSign up for a £1 trial so you can track and measure your child's progress on this activity.\n\n### What is EdPlace?\n\nWe're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.\n\nGet started", null, "" ]	[ null, "https://www.facebook.com/tr", null, "https://edplaceimages.s3.amazonaws.com/worksheetPreviews/worksheet_1539031009.jpg", null, "https://www.edplace.com/assets/images/dark_star.png", null, "https://www.edplace.com/assets/images/wht_star.png", null, "https://www.edplace.com/assets/images/wht_star.png", null, "https://www.edplace.com/assets/images/img_laptop.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8959589,"math_prob":0.7429303,"size":368,"snap":"2019-51-2020-05","text_gpt3_token_len":93,"char_repetition_ratio":0.12912089,"word_repetition_ratio":0.05882353,"special_character_ratio":0.26358697,"punctuation_ratio":0.1097561,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996662,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,3,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-23T23:54:12Z\",\"WARC-Record-ID\":\"<urn:uuid:056b4a6d-238f-479d-86b7-6e1a59e632df>\",\"Content-Length\":\"117597\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1bcb4edd-0e97-45c9-ae2d-2465c6355434>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c3a3a5e-47bf-405a-8cfc-6f84816bb2eb>\",\"WARC-IP-Address\":\"104.20.213.7\",\"WARC-Target-URI\":\"https://www.edplace.com/worksheet_info/maths/keystage2/year5/topic/611/944/identify-factors-and-multiples\",\"WARC-Payload-Digest\":\"sha1:X25RGIL56SIEAGQOWNIUASNPOJAVEQE3\",\"WARC-Block-Digest\":\"sha1:P3YEA7LURNQWWYENX4BETAJB5SG7ZEMI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250614086.44_warc_CC-MAIN-20200123221108-20200124010108-00078.warc.gz\"}"}
https://community.geodynamics.org/t/free-surface-problem-cell-giving-non-positive-volume-fraction/617	[ "# Free-surface problem: cell giving non-positive volume fraction\n\nDear All,\n\nI am working on a 2D subduction setup with a continental block in the lower plate, using a free surface, adaptive mesh refinement, and visco-plastic rheology. I have encountered a problem where one of the cells at the top of the model returns a non-positive volume fraction and crashes the simulation:\n\n“An error occurred in line <2727> of file </trinity/opt/apps/software/aspect/aspect-2.0.1/candi/DEAL2/tmp/unpack/deal.II-v9.0.1/source/fe/mapping_q_generic.cc> in function\ndealii::CellSimilarity::Similarity dealii::MappingQGeneric<dim, spacedim>::fill_fe_values(const dealii::Triangulation<dim, spacedim>::cell_iterator &, dealii::CellSimilarity::Similarity, const dealii::Quadrature &, const dealii::Mapping<dim, spacedim>::InternalDataBase &, dealii::internal::FEValuesImplementation::MappingRelatedData<dim, spacedim> &) const [with int dim = 2, int spacedim = 2]\nThe violated condition was:\ndet > 1e-12*Utilities::fixed_power(cell->diameter()/ std::sqrt(double(dim)))\nThe image of the mapping applied to cell with center [1.16145e+06 658711] is distorted. The cell geometry or the mapping are invalid, giving a non-positive volume fraction of -42785.9 in quadrature point 3.”\n\nThe problematic cell is at or very close to the deforming surface. The attached density plot shows the 1.16e6 x coordinate tick where it occurred.\n\nBelow I paste a part of the input file describing global, mesh refinment, and free surface parameters:\nset Dimension = 2\nset Nonlinear solver scheme = iterated Advection and Newton Stokes\nset Nonlinear solver tolerance = 1e-2\nset Max nonlinear iterations = 20\nset CFL number = 0.05\nset Timing output frequency = 20\nset Pressure normalization = no\n\nsubsection Solver parameters\nsubsection Stokes solver parameters\nset Linear solver tolerance = 1e-3\nset Number of cheap Stokes solver steps = 100\nend\nsubsection Newton solver parameters\nset Max pre-Newton nonlinear iterations = 1000\nend\nend\n\n# Model geometry (660 km deep aspect ratio 4:1)\n\nsubsection Geometry model\nset Model name = box\nsubsection Box\nset X repetitions = 4\nset Y repetitions = 1\nset X extent = 2640e3\nset Y extent = 660e3\nset X periodic = true\nend\nend\n\n# Mesh refinement specifications:\n\nsubsection Mesh refinement\nset Initial adaptive refinement = 4\nset Initial global refinement = 4\nset Time steps between mesh refinement = 5\nset Strategy = viscosity, composition, strain rate, temperature\nset Refinement fraction = 0.95\n\nsubsection Free surface\nset Free surface boundary indicators = top\nset Free surface stabilization theta = 0.9\nset Surface velocity projection = normal\nend\n\nThe problem occurs where the free surface is rapidly deforming, so perhaps it is related to one of the elements getting “bow-tied” between two re-meshing. Could anyone advise me on how to tackle this issue?\n\nThank you and best,\nKristof Porkolab\n\nHello Kristof,\n\nI also used to have similar problems. If I remember correctly, the solution for me was to switch the `Surface velocity projection` from `normal` to `vertical`. This means that vertices can only more up and down, which should prevent the problem of cells being able to invert.\n\nBest,\n\nMenno\n\nHi Menno,\n\nthanks for the reply! I have learnt from the manual that the vertical projection is more stable, therefore that was the first projection I tried. However my continental block is moving long distances laterally, and the vertical projection makes the top of the continent scrape off as it moves (because the fs is projected vertically). So in order to maintain the proper lateral transport of the free surface together with the continent, I had to change it to normal projection. I hoped that the problem can still be solved even with the normal projection.\n\nCheers, Kristof\n\nHello Kristof,\n\nhmm, that is weird. I have never noticed that kind of behavior with the free surface. Material should be able to move along the free surface. Could you show some figures of that so that I can see if I understand correct what you mean? Could you also post the boundary conditions section of the input file which you are using?\n\nCheers,\n\nMenno\n\nHi Menno,\n\nhere is the bc section:\n\n# Temperature boundary conditions\n\nsubsection Boundary temperature model\nset Fixed temperature boundary indicators = bottom, top\nset List of model names = box\nsubsection Box\nset Bottom temperature = 1573\nset Top temperature = 273\nend\nend\n\n# Boundary classifications\n\nsubsection Boundary velocity model\nset Tangential velocity boundary indicators = bottom\nend\n\nsubsection Nullspace removal\nset Remove nullspace = net x translation\nend\n\nThe uploaded picture shows that in case of vertical projection the free surface stays at the original position of the continent, but since the continent is moving to the right, the upper crust is scraped off. Switching to normal projection solved this problem, but then I got this other bug my original post was about.", null, "Cheers,\nKristof\n\nWhat you are fundamentally are trying to do is large deformation together with a normal projection where you move the grid (at least partially) with that deformation. This means that after some time the grid will become very deformed. The only way to deal with this is to re-mesh, which is not something ASPECT is made to do. So I think you should investigate a solution with the vertical projection.\n\nIt seems that the material at the surface is somehow not free to move laterally. My suspicion is that the problem is due to the composition boundary conditions which do not move with the continent. It could be something different, but I can’t exclude (or confirm) it from the current current information and figure (would need to see the composition boundary and initial conditions and a bigger figure with grid). But maybe there are some other ideas around what the problem could be.\n\nCheers,\n\nMenno\n\nThe standard ‘trick’ is to add a little bit of diffusion to the surface, but that is currently not supported by ASPECT. In light thereof, I would for now give up on using a free surface (and resort to free slip), and/or make sure that the subduction channel is well lubricated (low viscous enough) as it looks like the buckling may be caused by a ‘stuck’ subduction.\nSticky air still remains an option.\n\nSorry for jumping into the conversation a bit late here. I agree with @cedrict suggestions to make sure the subduction channel is sufficiently lubricated, which may reduce the amount of buckling.\n\nI’m working with another student on a similar problem and we ran into other issues when the initial subduction geometry was not aligned sufficiently to promote subduction. So, another option is to revisit the subduction\n\nA third option is to use the material averaging (e.g., averaging over entire element) option in the material model, which smooths out the viscosity field and reduces some small scale deformation. No idea if this will fix the free surface issue, but it will help improve solver behavior.\n\nThe last (and perhaps best) option is to use sticky air, which I am also trying out for subduction problems.\n\nFYI, surface diffusion may be implemented at some point in the next few months, but best to try and other options in the meantime.\n\nThank you for the suggestions, I am currently trying to implement them. I can already say that material averaging did result in a slightly smoother geometry, but the error still occurred. I will try the others as well and report back. Thanks again!\n\n@kristofporkolab - FYI, A group will be working a solution for smoothing out the free surface this week. We will keep you posted as progress is made and time permitting perhaps you can try it out on your model.\n\n@MFraters - I have been trying to play a bit with the composition boundary conditions, but no success so far. Here is what I have now:\n\nset Number of fields = 3\nset Names of fields = oc, uc, lc\nend\n\n# Spatial domain of different compositional fields\n\nsubsection Initial composition model\nset Model name = function\nsubsection Function\nset Variable names = x,y\nset Function constants = h=660e3, ar=4, dc=4e3, xt=1200e3, wocrhs=300e3,\nxcr=800e3, xcl=400e3,\nr=6e5, zs=2e5, kappa=1e-6\nset Function expression =\nif( (h-y)<dc && x>xcr,\n1,\nif( sqrt((x-xt)(x-xt)+(h-r-y)(h-r-y))<r &&\nsqrt((x-xt)(x-xt)+(h-r-y)(h-r-y))>(r-dc) &&\nx>xt &&\ny>h-zs, 1,\nif( x<xcl && (h-y)<dc, 1, 0\n)\n)\n);\nif( x>=xcl && x<=xcr && (h-y)<=15e3,\n1, 0\n);\nif( x>=xcl && x<=650e3 && (h-y)<=30e3 && (h-y)>15e3,\n1,\nif( x>650e3 && x<=xcr && (h-y)>15e3 && y>(h-30e3)+(x-650e3)/10, 1, 0\n)\n)\nend\nend\n\nsubsection Boundary composition model\nset Fixed composition boundary indicators = bottom\nset List of model names = initial composition\nend\n\n# Temperature boundary conditions\n\nsubsection Boundary temperature model\nset Fixed temperature boundary indicators = bottom, top\nset List of model names = box\nsubsection Box\nset Bottom temperature = 1573\nset Top temperature = 273\nend\n\nAttached a bigger picture with grid. The continental block is initially from x=400e3 to 800e3; as it is transported to the right, becomes more and more tilted/scraped off by the free surface if I have a vertical projection. This does not happen with normal projection.\n\nCheers, Kristof" ]	[ null, "https://aws1.discourse-cdn.com/standard10/uploads/geodynamics/original/1X/eae2f5c3e1298468c2d89c79d9f880f10e4c3a4b.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.69473034,"math_prob":0.9309139,"size":2841,"snap":"2022-05-2022-21","text_gpt3_token_len":697,"char_repetition_ratio":0.11667254,"word_repetition_ratio":0.0,"special_character_ratio":0.2340725,"punctuation_ratio":0.15175097,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9708775,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-24T02:19:55Z\",\"WARC-Record-ID\":\"<urn:uuid:d2a2cee1-a62a-413f-b808-e4a832c9f4cc>\",\"Content-Length\":\"50274\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0ae7f5ac-8d27-443d-bd9c-e83f1029e8ac>\",\"WARC-Concurrent-To\":\"<urn:uuid:e64099a4-e9ba-40b6-b3ce-ef4eb18031ae>\",\"WARC-IP-Address\":\"64.62.250.111\",\"WARC-Target-URI\":\"https://community.geodynamics.org/t/free-surface-problem-cell-giving-non-positive-volume-fraction/617\",\"WARC-Payload-Digest\":\"sha1:F5EH354T7G6B3JJG2I7RK4LEYCDYYVL4\",\"WARC-Block-Digest\":\"sha1:WWZSCUXLZUSAXHMM4UNLGY46CJPM36FB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662562410.53_warc_CC-MAIN-20220524014636-20220524044636-00653.warc.gz\"}"}
http://www.homepages.ucl.ac.uk/~ucgtrbd/uai2015_causal/abstracts.html	[ "# Abstracts\n\n### Invited Talks\n\n#### Causal and Statistical Inference with Social Network Data: Massive Challenges and Meager Progress\n\nElizabeth Ogburn\n\nInterest in and availability of social network data has led to increasing attempts to make causal and statistical inferences using data collected from subjects linked by social network ties. But inference about all kinds of estimands, from simple sample means to complicated causal peer effects, is challenging when only a single network of non-independent observations is available. There is a dearth of principled methods for dealing with the dependence that such observations can manifest. We demonstrate the dangerously anticonservative inference that can result from a failure to account for network dependence, explain why results on spatial-temporal dependence are not immediately applicable to this new setting, and describe a few different avenues towards valid statistical and causal inference using social network data.\n\n#### Causal Reasoning for Events in Continuous Time: A Decision-Theoretic Approach\n\nVanessa Didelez\n\nThis talk will be concerned with causal reasoning in the context of continuous time (point) processes. It will be shown that various notions that we are familiar with from e.g. (causal) DAGs can be generalised to this case, such as interventions and graphical criteria for identifiability, as well as inverse probability weighting. The relevant graphs, however, are not DAGs, they are local independence graphs which are directed graphs allowing cycles. Using these with survival outcomes and thinking \"causally\" also allows interesting insights into the notion of independent censoring. The theory will be illustrated with original data from a Norwegian screening project for cervical cancer; the aim is to compare two types of HPV-tests which can be used in the screeing. Local independence graphs and appropriate weighting procedures turn out to be useful for the analysis of these data.\n\n### Contributed Talks and Posters\n\n#### Learning the Structure of Causal Models with Relational and Temporal Dependence\n\nKaterina Marazopoulou, Marc Meier and David Jensen\n\nMany real-world domains are inherently relational and temporal - they consist of heterogeneous entities that interact with each other over time. Effective reasoning about causality in such domains requires representations that explicitly model relational and temporal dependence. In this work, we provide a formalization of temporal relational models. We define temporal extensions to abstract ground graphs - a lifted representation that abstracts paths of dependence over all possible ground graphs. Temporal abstract ground graphs enable a sound and complete method for answering d-separation queries on temporal relational models. These methods provide the foundation for a constraint-based algorithm, TRCD, that learns causal models from temporal relational data. We provide experimental evidence that demonstrates the need to explicitly represent time when inferring causal dependence. We also demonstrate the expressive gain of TRCD compared to earlier algorithms that do not explicitly represent time.\n\n#### Query-Answer Causality in Databases: Abductive Diagnosis and View-Updates\n\nBabak Salimi and Leopoldo Bertossi\n\nCausality has been recently introduced in databases, to model, characterize and possibly compute causes for query results (answers). Connections between query causality and consistency-based diagnosis and database repairs (wrt. integrity constrain violations) have been established in the literature. In this work we establish connections between query causality and abductive diagnosis and the view-update problem. The unveiled relationships allow us to obtain new complexity results for query causality -the main focus of our work- and also for the two other areas.\n\n#### Causal Interpretation Rules for Encoding and Decoding Models in Neuroimaging\n\nSebastian Weichwald, Timm Meyer, Ozan Özdenizci, Bernhard Schölkopf, Tonio Ball and Moritz Grosse-Wentrup\n\nHow neural activity gives rise to cognition is arguably one of the most interesting questions in neuroimaging. While causal terminology is often introduced in the interpretation of neuroimaging data, causal inference frameworks are rarely explicitly employed.\n\nIn our recent work we cast widely used analysis methods in a causal framework in order to foster its acceptance in the neuroimaging community. In particular we focus on typical analyses in which variables' relevance in encoding and decoding models (also known as generative or discriminative models) with a dependent stimulus/response variable is interpreted. By linking the concept of relevant variables to marginal/conditional independence properties we demonstrate that (a) identifying relevant variables is indeed a first step towards causal inference; (b) combining encoding and decoding models can yield further insights into the causal structure, which cannot be gleaned from either model alone. We demonstrate the empirical relevance of our findings on EEG data recorded during a visuomotor learning task.\n\nThe rigorous theoretical framework of causal inference allows to expound the assumptional underpinnings and limitations of common (intuitive) analyses in this field. Furthermore, it sheds light on problems covered in recent neuroimaging literature such as confounds in multivariate pattern analysis or interpretation of linear encoding and decoding models.\n\n#### Inference of Cause and Effect with Unsupervised Inverse Regression\n\nEleni Sgouritsa, Dominik Janzing, Philipp Hennig and Bernhard Schölkopf\n\nWe address the problem of causal discovery in the two-variable case, given a sample from their joint distribution. Since X -> Y and Y -> X are Markov equivalent, conditional-independence-based methods [Spirtes et al., 2000, Pearl, 2009] can not recover the causal graph. Alternative methods, introduce asymmetries between cause and effect by restricting the function class (e.g., [Hoyer et al., 2009]).\n\nThe proposed causal discovery method, CURE, is based on the principle of independence of causal mechanisms [Janzing and Schölkopf, 2010]. For the case of only two variables, it states that the marginal distribution of the cause, say P(X), and the conditional of the effect given the cause P(Y \| X) are ``independent'', in the sense that they do not contain information about each other (informally P(X) ``independent of '' P(Y \| X)). This independence can be violated in the backward direction: the distribution of the effect P(Y) and the conditional P(X \| Y) may contain information about each other because each of them inherits properties from both P(X) and P(Y \| X), hence introducing an asymmetry between cause and effect. For deterministic causal relations (Y = f(X)), all the information about the conditional P(Y \| X) is contained in the function f, so independence boils down to P(X) ``independent of'' f. Previous work formalizes the independence principle by specifying what is meant by independence. For deterministic non-linear relations, Janzing et al. and Daniusis et al. define independence as uncorrelatedness between log f' and the density of P(X), both viewed as random variables. For non-deterministic relations, it is not obvious how to explicitly formalize independence between P(X) and P(Y \| X). Instead, we propose an implicit notion of independence, namely that p_Y\|X cannot be estimated based on p_X (lower case denotes density). However, it may be possible to estimate p_X\|Y based on the density of the effect, p_Y .\n\nIn practice, we are given empirical data x in R^N, y in R^N from P(X, Y) and estimate p_X\|Y based on y (intentionally hiding x). The relationship between the observed y and the latent x_u in R^N is modeled by a Gaussian Process (GP): p(y \| x_u; theta) = N(y; 0;K_xu;xu + sigma^2_n * I_N) (this can be alternatively seen as a single output GP-LVM). Then, the required conditional p_X\|Y is estimated as p_hat^y_(X_u \|Y) : (x_u; y) --> p(x_u \| y, y), with p(x_u \|y*; y) estimated by marginalizing out the latent x_u and theta (GP hyperparameters).\n\nCURE infers the causal direction by using the procedure above two times: one to estimate p_X\|Y based only on y and another to estimate p_Y\|X based only on x. If the first estimation is better, X -> Y is inferred. Otherwise, Y -> X. CURE was evaluated on synthetic and real data and often outperformed existing methods. On the downside, its computational cost is comparably high. This work was recently published at AISTATS 2015 [Sgouritsa et al., 2015].\n\n#### Exploiting Causality for Efficient Monitoring in POMDPs\n\nStefano V. Albrecht and Subramanian Ramamoorthy\n\nPOMDPs are a useful model for decision making in systems with uncertain states. One of the core tasks in a POMDP is the monitoring task, in which the belief state (i.e. the probability distribution over system states) is updated based on incomplete and noisy observations. This can be a hard problem in complex real-world systems due to the often very large state space. In this article, we explore the idea of accelerating the monitoring task by automatically exploiting causality in the system. We consider a specific type of causal relation, called passivity, which pertains to how system variables cause changes in other variables. Specifically, a system variable is called passive if it changes its value only if it is directly acted upon, or if at least one of the variables that directly affect it (i.e. parent variables) change their values. This property can be readily determined from the conditional probability table of the system variable. We present a novel monitoring method, called Passivity-based Monitoring (PM), which maintains a factored belief state representation and exploits passivity to perform selective updates over the factored beliefs. PM produces exact belief states under certain assumptions and approximate belief states otherwise, where the approximation error is bounded by the degree of uncertainty in the process. We show empirically, in synthetic processes with varying sizes and degrees of passivity, that PM is faster than two standard monitoring methods while achieving competitive accuracy. Furthermore, we demonstrate how passivity occurs naturally in a real-world system such as a multi-robot warehouse, and how PM can exploit this to accelerate the monitoring task.\n\n#### An Empirical Study of the Simplest Causal Prediction Algorithm\n\nJerome Cremers and Joris Mooij\n\nWe study the simplest causal prediction algorithm that uses only conditional independences in purely observational data. A specific pattern of only four conditional independence relations amongst a quadruple of random variables already implies that one of these variables causes another without any confounding. As a consequence, it is possible to predict what would happen under an intervention on that variable without actually performing the intervention. Although the method is asymptotically consistent and works well in settings with only few (latent) variables, we find that its prediction accuracy can be worse than simple noncausal baselines when many (latent) variables are present. We also find that the accuracy can sometimes be improved by adding more conditional independence tests, but even then the performance need not outperform the baselines. More generally, our findings illustrate that high accuracy of individual conditional independence tests is no guarantee for high accuracy of a combination of such tests. Also, they illustrate the severity of the faithfulness assumption in practice.\n\n#### Visual Causal Feature Learning\n\nKrzysztof Chalupka, Pietro Perona and Frederick Eberhardt\n\nWe provide a rigorous definition of the visual cause of a behavior that is broadly applicable to the visually driven behavior in humans, animals, neurons, robots and other perceiving systems. Our framework generalizes standard accounts of causal learning to settings in which the causal variables need to be constructed from micro-variables. We prove the Causal Coarsening Theorem, which allows us to gain causal knowledge from observational data with minimal experimental effort. The theorem provides a connection to standard inference techniques in machine learning that identify features of an image that correlate with, but may not cause, the target behavior. Finally, we propose an active learning scheme to learn a manipulator function that performs optimal manipulations on the image to automatically identify the visual cause of a target behavior. We illustrate our inference and learning algorithms in experiments based on both synthetic and real data.\n\n#### Lifted Representation of Relational Causal Models Revisited: Implications for Reasoning and Structure Learning\n\nSanghack Lee and Vasant Honavar\n\nMaier et al. (2010) introduced the relational causal model (RCM) for representing and inferring causal relationships in relational data. A lifted representation, called abstract ground graph (AGG), plays a central role in reasoning with and learning of RCM. The correctness of the algorithm proposed by Maier et al. (2013a) for learning RCM from data relies on the soundness and completeness of AGG for relational d-separation to reduce the learning of an RCM to learning of an AGG. We revisit the definition of AGG and show that AGG, as defined in Maier et al. (2013b), does not correctly abstract all ground graphs. We revise the definition of AGG to ensure that it correctly abstracts all ground graphs. We further show that AGG representation is not complete for relational d-separation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG. A careful examination of the relationship between the lack of completeness of AGG for relational d-separation and faithfulness conditions suggests that weaker notions of completeness, namely adjacency faithfulness and orientation faithfulness between an RCM and its AGG, can be used to learn an RCM from data.\n\n#### Robust reconstruction of causal graphical models based on conditional 2-point and 3-point information\n\nSéverine Affeldt and Hervé Isambert\n\nWe report a novel network reconstruction method, which combines constraint-based and Bayesian frameworks to reliably reconstruct graphical models despite inherent sampling noise infinite observational datasets. The approach is based on an information theory result tracing back the existence of colliders in graphical models to negative conditional 3-point information between observed variables. In turn, this provides a confident assessment of structural independencies in causal graphs, based on the ranking of their most likely contributing nodes with (significantly) positive conditional 3-point information. Starting from a complete undirected graph, dispensible edges are progressively pruned by iteratively ``taking off'' the most likely positive conditional 3-point information from the 2-point (mutual) information between each pair of nodes. The resulting network skeleton is then partially directed by orienting and propagating edge directions, based on the sign and magnitude of the conditional 3-point information of unshielded triples. This ``3off2'' network reconstruction approach is shown to outperform both constraint-based and Bayesian inference methods on a range of benchmark networks.\n\n#### An Algorithm to Compute the Likelihood Ratio Test Statistic of the Sharp Null Hypothesis for Compliers\n\nWen Wei Loh and Thomas S. Richardson\n\nIn a randomized experiment with noncompliance, scientific interest is often in testing whether the treatment exposure X has an effect on the final outcome Y. We have proposed a finite-population significance test of the sharp null hypothesis that X has no effect on Y, within the principal stratum of compliers, using a generalized likelihood ratio test.\n\nAs both the null and alternative hypotheses are composite hypotheses (each comprising a different set of distributions), computing the value of the generalized likelihood ratio test statistic requires two maximizations: one where we assume that the sharp null hypothesis holds, and another without making such an assumption.\n\nIn our work, we have assumed that there are no Always Takers, such that the nuisance parameter is a bivariate parameter describing the total number of Never Takers with observed outcomes y = 0 and y = 1. Extending the approach to the more general case in which there are also Always Takers would require a nuisance parameter of higher dimension that describes the total number of Always Takers with observed outcomes y = 0 and y = 1 as well. This increases the size of the nuisance parameter space and the computational effort needed to find the likelihood ratio test statistic. We present a new algorithm that extends to solve the corresponding integer programs in the general case where there are Always Takers. The procedure for the infinite-population significance test may be illustrated using a toy example from our UAI Causal Inference Workshop 2013.\n\n#### Segregated Graphs and Marginals of Chain Graph Models\n\nIlya Shpitser\n\nBayesian networks are a popular representation of asymmetric (for example causal) relationships between random variables. Markov random fields (MRFs) are a complementary model of symmetric relationships used in computer vision, spatial modeling, and social and gene expression networks. A chain graph model under the Lauritzen-Wermuth-Frydenberg interpretation (hereafter a chain graph model) generalizes both Bayesian networks and MRFs, and can represent asymmetric and symmetric relationships together.\n\nAs in other graphical models, the set of marginals from distributions in a chain graph model induced by the presence of hidden variables forms a complex model. One recent approach to the study of marginal graphical models is to consider a well-behaved supermodel. Such a supermodel of marginals of Bayesian networks, defined only by conditional independences, and termed the ordinary Markov model, was studied at length in (Evans and Richardson, 2014).\n\nIn this paper, we show that special mixed graphs which we call segregated graphs can be associated, via a Markov property, with supermodels of a marginal of chain graphs defined only by conditional independences. Special features of segregated graphs imply the existence of a very natural factorization for these supermodels, and imply many existing results on the chain graph model, and ordinary Markov model carry over. Our results suggest that segregated graphs define an analogue of the ordinary Markov model for marginals of chain graph models.\n\n#### Recovering from Selection Bias using Marginal Structure in Discrete Models\n\nRobin J. Evans and Vanessa Didelez\n\nThis paper considers the problem of inferring a discrete joint distribution from a sample subject to selection. Abstractly, we want to identify a distribution p(x, w) from its conditional p(x \| w). We introduce new assumptions on the marginal model for p(x), under which generic identification is possible. These assumptions are quite general and can easily be tested; they do not require precise background knowledge of p(x) or p(w), such as proportions estimated from previous studies. We particularly consider conditional independence constraints, which often arise from graphical and causal models, although other constraints can also be used. We show that generic identifiability of causal effects is possible in a much wider class of causal models than had previously been known.\n\n#### Advances in Integrative Causal Analysis\n\nIoannis Tsamardinos\n\nScientific practice typically involves studying a system over a series of studies and data collection, each time trying to unravel a different aspect. In each study, the scientist may take measurements under different experimental conditions and measure different sets of quantities (variables). The result is a collection of heterogeneous data sets coming from different distributions. Even so, these are generated by the same causal mechanism. The general idea in Integrative Causal Analysis (INCA) is to identify the set of causal models that simultaneously fit (are consistent) with all sources of data and prior knowledge and reason with this set of models. Integrative Causal Analysis allows more discoveries than what is possible by independent analysis of datasets. In this talk, we'll present advances in this direction that lead to algorithms that can handle more types of heterogeneity, and aim at increasing efficiency or robustness of discoveries. Specifically, we'll present (a) general INCA algorithms for causal discovery from heterogeneous data, (b) algorithms for converting the results of tests to posterior probabilities and allow conflict resolution and identification of the confidence network regions, (d) proof-of-concept applications and massive evaluation on real data of the main concepts, (d) extensions that can deal with prior causal knowledge, and (e) extensions that handle case-control data." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90943754,"math_prob":0.87301636,"size":20913,"snap":"2019-13-2019-22","text_gpt3_token_len":4117,"char_repetition_ratio":0.1309484,"word_repetition_ratio":0.0064082025,"special_character_ratio":0.18012719,"punctuation_ratio":0.08369284,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96114385,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-20T23:53:43Z\",\"WARC-Record-ID\":\"<urn:uuid:7298d909-8e36-473d-8913-362fbb426fac>\",\"Content-Length\":\"25454\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7d94212b-8a97-419c-be15-4d708f397b89>\",\"WARC-Concurrent-To\":\"<urn:uuid:220cd6f9-dc43-4a30-bd34-1d694463bb66>\",\"WARC-IP-Address\":\"144.82.250.219\",\"WARC-Target-URI\":\"http://www.homepages.ucl.ac.uk/~ucgtrbd/uai2015_causal/abstracts.html\",\"WARC-Payload-Digest\":\"sha1:F4UZTIJX5K3MRCBDGAJLLM5LDGNBL7XO\",\"WARC-Block-Digest\":\"sha1:CMY2C7RERDYRODNOIJXOQLMJW62NXXOE\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202474.26_warc_CC-MAIN-20190320230554-20190321012554-00298.warc.gz\"}"}
https://chemistry.stackexchange.com/questions/71249/cyclobutadiene-jahn-teller-effect-or-not/71327	[ "# Cyclobutadiene - Jahn–Teller effect or not?\n\nIn transition metal chemistry the Jahn–Teller effect arises when the configuration of the metal ion and d orbital splitting set up a doubly degenerate state, which is less stable than a state without the degeneracy but with lower symmetry.\n\nIn Ian Fleming's Molecular Orbitals it is stated that cyclobutadiene would be a diradical considering Huckel theory:", null, "but it isn't because of Jahn–Teller distortion (it is actually rectangular and ESR shows no spin).\n\nThere isn't a doubly degenerate state in this case, yet Jahn–Teller distortion happens according to Fleming. How is this so?\n\n(In the singlet state it would be doubly degenerate, but how to know it a priori?)\n\nVery interesting question, and it kept me up despite daylight saving time cheating me of one hour of sleep last night... A good reference is Albright, Burdett and Whangbo, Orbital Interactions in Chemistry 2nd ed. pp 282ff. which explains this in much greater detail than I can. (In general, that book is fantastic.) I will try my best to summarise what they have written - please point out any mistakes that I may have made. I think there is also much more information out there in the literature (a search for \"Jahn–Teller cyclobutadiene\" should give lots of good results), although it is of course not always easy to read. $\\require{begingroup}\\begingroup\\newcommand{\\ket}{\|#1\\rangle}$\n\n### 1. What is the ground state of undistorted $\\ce{C4H4}$?\n\nBased on simple Huckel theory as you have shown above, one would expect the ground state to be a triplet state with the two highest electrons in different orbitals. This is simply because of exchange energy (Hund's first rule).\n\nHowever, it turns out that the singlet state with the two electrons in different orbitals is more stable. That explains the lack of signal in ESR experiments.", null, "(Note that this orbital picture is a simplification, as I will describe later.)\n\nIn short, this is due to dynamic spin polarisation; the idea is that there are low-lying excited singlet states which mix with the ground state to stabilise it. This is the basic idea underlying so-called configuration interaction, which mixes in excited molecular states into the wavefunction in order to obtain a better estimate of the ground-state wavefunction. If you are interested in the theoretical aspect of this, I think Szabo and Ostlund's Modern Quantum Chemistry should cover it. The topic of dynamic spin polarisation is treated in depth in pp 285ff. of Albright et al.\n\nYour question, \"how to predict the singlet state a priori\"? is a good one. Albright et al. write that (p 289)\n\nIn general, high-level configuration interaction calculations are needed to determine the relative stabilities of the singlet and triplet states of a diradical.\n\nSo, it does not seem to be something that you can figure out a priori, and it is most certainly not something you can figure out by just using Huckel theory (which is an incredibly simplistic treatment).\n\n### 2. What is the degeneracy of the ground state?\n\nYou also wrote that \"the singlet state will be doubly degenerate\". However, in the case where both electrons are in different orbitals, there is only one possible spatial wavefunction, due to the quantum mechanical requirement of indistinguishability. (As we will see in the group theoretical treatment, there is actually no doubly degenerate case, not even when the electrons are paired in the same orbital.)\n\nIf I label the two SOMOs above as $\\ket{a}$ and $\\ket{b}$ then the combination $\\ket{ab}$ is not a valid spatial wavefunction. Neither is $\\ket{ba}$ a valid wavefunction.", null, "You have to take linear combinations of these two states, such that your spatial wavefunction is symmetric with respect to interchange of the two electrons. In this case the appropriate combination is\n\n$$2^{-1/2}(\\ket{ab} + \\ket{ba}) \\qquad \\mathrm{{}^1B_{2g}}$$\n\nThere are two more possible symmetric wavefunctions:\n\n$$2^{-1/2}(\\ket{aa} + \\ket{bb}) \\qquad \\mathrm{{}^1A_{1g}}$$ $$2^{-1/2}(\\ket{aa} - \\ket{bb}) \\qquad \\mathrm{{}^1B_{1g}}$$\n\n(Simply based on the requirement for indistinguishability, one might expect $\\ket{aa}$ and $\\ket{bb}$ to be admissible states. However, this is not true. If anybody is interested to know more, please feel free to ask a new question.) The orbital picture is actually woefully inadequate at dealing with quantum mechanical descriptions of bonding. However, Voter and Goddard tried to make it work out, by adding and subtracting electron configurations:1", null, "Ignore the energy ordering in this diagram. It is based on Hartree–Fock calculations, which do not include the effects of excited state mixing (CI is a \"post-Hartree–Fock method\"), and therefore the triplet state is predicted to be the lowest in energy. Instead, just note the form of the three singlet states; they are the same as what I have written above, excluding the normalisation factor.\n\nThe point of this section is that, this singlet state is (spatially) singly degenerate. The orbital diagrams that organic chemists are used to are inaccurate; the actual electronic terms are in fact linear combinations of electronic configurations. So, using orbital diagrams to predict degeneracy can fail!\n\n### 3. Why is there a Jahn–Teller (JT) distortion in a singly degenerate state?\n\nYou are right in that for a JT distortion - a first-order JT distortion, to be precise - to occur, the ground state must be degenerate. This criterion is often rephrased in terms of \"asymmetric occupancy of degenerate orbitals\", but this is just a simplification for students who have not studied molecular symmetry and electronic states yet. Jahn and Teller made the meaning of their theorem very clear in their original paper;2 already in the first paragraph they write that the stability of a polyatomic molecule is not possible when \"its electronic state has orbital degeneracy, i.e. degeneracy not arising from the spin [...] unless the molecule is a linear one\". There is no mention of \"occupancy of orbitals\".\n\nSo, this is not a first-order JT distortion, which requires degeneracy of the ground state. Instead, it is a second-order JT distortion. Therefore, most of the other discussion so far on the topic has unfortunately been a little bit off-track. (Not that I knew any better.)\n\nAlbright et al. describe this as a \"pseudo-JT effect\" (p 136). For a better understanding I would recommend reading the section on JT distortions (pp 134ff.), it is very thorough, but you would need to have some understanding of perturbation theory in quantum mechanics. Please note that there is a typo on p 136, as is pointed out in the comments to this answer. (Alternatively, there is also a discussion of second-order JT effects in a paper by Pearson.3) The idea is that, the distortion reduces the symmetry of the molecule and therefore allows ground and excited states to mix. This leads to a stabilisation of the ground state and hence an electronic driving force for the distortion.\n\nIn slightly more detail, we can use a group theoretical treatment to do this. The second-order correction to the energy is given by\n\n$$\\sum_{j\\neq i}\\frac{\\langle i \|(\\partial H/\\partial q)\| j \\rangle}{E_i^{(0)} - E_j^{(0)}}$$\n\nwhere $\\ket{i}$ is the ground state (which in this case is described by $2^{-1/2}(\\ket{aa} - \\ket{bb})$) and $\\{\\ket{j}\\}$ are the excited states. $E_i^{(0)}$ is the unperturbed energy of the ground state $\\ket{i}$ and likewise for $E_j^{(0)}$. $q$ is a vibrational coordinate of the molecule.\n\nNote that the denominator in this term, $E_i^{(0)} - E_j^{(0)}$, is negative. So, this correction to the energy is always negative, i.e. stabilising. (There is another second-order term which isn't relevant to the discussion here.) The idea is that in order to have a significant stabilisation upon distortion, two criteria must be fulfilled:\n\n1. The denominator must be small, i.e. the excited state must be low-lying.\n2. The numerator must be non-zero, i.e. there must be appropriate symmetry such that $\\Gamma_i \\otimes \\Gamma_j \\otimes \\Gamma_H \\otimes \\Gamma_q$ contains the totally symmetric irreducible representation. Since $H$ transforms as the TSIR, this is equivalent to the criterion that $\\Gamma_q = \\Gamma_i \\otimes \\Gamma_j$.\n\nIn cyclobutadiene, the two SOMOs in $D_\\mathrm{4h}$ symmetry transform as $\\mathrm{E_u}$. To find the irreps of the resultant states we take\n\n$$\\mathrm{E_u \\otimes E_u = A_{1g} + [A_{2g}] + B_{1g} + B_{2g}}$$\n\n(Note that there is no spatially degenerate term in this direct product, so there cannot be a first-order JT distortion!) The $\\mathrm{A_{2g}}$ state in square brackets corresponds to the triplet case and we can ignore it. In our case (presented without further proof, because I'm not sure why myself) the ground state is $\\mathrm{B_{1g}}$. The two low-lying singlet excited states transform as $\\mathrm{A_{1g} \\oplus \\mathrm{B_{2g}}}$. Therefore if there exists a vibrational mode transforming as either $\\mathrm{B_{1g}} \\otimes \\mathrm{A_{1g}} = \\mathrm{B_{1g}}$ or $\\mathrm{B_{1g}} \\otimes \\mathrm{B_{2g} = \\mathrm{A_{2g}}}$, then the numerator will be nonzero and we can expect this vibrational mode to lead to a distortion with concomitant stabilisation.\n\nIn our case, it happens that there is a vibrational mode with symmetry $\\mathrm{B_{1g}}$, which leads to the distortion from a square to a rectangle. This therefore allows for a mixing of the ground $\\mathrm{B_{1g}}$ state with the excited $\\mathrm{A_{1g}}$ state upon distortion, leading to electronic stabilisation.\n\nAnother way of looking at it is that upon lowering of the symmetry from square ($D_\\mathrm{4h}$) to rectangular ($D_\\mathrm{2h}$), both the $\\mathrm{B_{1g}}$ ground state and $\\mathrm{A_{1g}}$ excited state adopt the same symmetry $\\mathrm{A_g}$. You can prove this by looking at a descent in symmetry table. Therefore, mixing between these two states will be allowed in the new geometry.\n\nBecause we are talking about the mixing of states here, the orbital picture is not quite enough to describe what is going on, sadly. The best representation we can get is probably this:", null, "However, Nakamura et al. have produced potential energy curves for the distortion of cyclobutadiene from square to rectangular and rhomboidal geometry.4 Here is the relevant one (rectangular geometry):", null, "Solid lines indicate singlet states, and dashed lines triplet states. Note that this diagram depicts what I have described above: the singlet ground state ($\\mathrm{{}^1B_{1g}}$), as well as the first excited singlet state ($\\mathrm{{}^1A_{1g}}$), both have the same irrep upon distortion. The ground state is stabilised upon the lowering of symmetry, precisely due to this mixing.\n\n### References\n\n(1) Voter, A. F.; Goddard, W. A. The generalized resonating valence bond description of cyclobutadiene. J. Am. Chem. Soc. 1986, 108 (11), 2830–2837. DOI: 10.1021/ja00271a008.\n\n(2) Jahn, H. A.; Teller, E. Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy. Proc. R. Soc. A 1937, 161 (905), 220–235. DOI: 10.1098/rspa.1937.0142.\n\n(3) Pearson, R. G. The second-order Jahn–Teller effect. J. Mol. Struct.: THEOCHEM 1983, 103, 25–34. DOI: 10.1016/0166-1280(83)85006-4.\n\n(4) Nakamura, K.; Osamura, Y.; Iwata, S. Second-order Jahn–Teller effect of cyclobutadiene in low-lying states. An MCSCF study. Chem. Phys. 1989, 136 (1), 67–77. DOI: 10.1016/0301-0104(89)80129-6. $\\endgroup$\n\n• Great explanation! One question: in Albright, 2013, the section about the second order JT: 7.4.2 (page 136) they write \"If the lowest lying one of these is of the correct symmetry for $<\\psi_i\|d^2H/dq^2\|\\psi_i>$ to be non zero ($\\Gamma_q=\\Gamma_i \\otimes \\Gamma_j$).\" So I guess that must be a typo since they should rather refer to the other second order term wich is the sum of the virtual excitations, right? – Rudi_Birnbaum May 20 '18 at 10:33\n• @R_Berger, indeed, when I was reading it I noticed a typo in that line too. I believe you are correct and that $\\langle \\Psi_i \| \\partial^2 H/\\partial q^2\|\\Psi_i\\rangle$ should be replaced with $\\langle \\Psi_i \| \\partial H/\\partial q \| \\Psi_j\\rangle$. Of course, this term can only be nonzero precisely if $\\Gamma_q = \\Gamma_i \\otimes \\Gamma_j$, which they correctly explain. – orthocresol May 20 '18 at 11:12\n• @orthokresol: How would things look like if we include SOC and go to double groups? Is there a quick answer? – Rudi_Birnbaum May 20 '18 at 11:18\n• @R_Berger Sorry, but the quick answer is I don't know ;) I had to do lots of research just to write this answer. – orthocresol May 20 '18 at 11:21\n• @orthokresol: Would you agree that at the end of the dashed line in the last figure there should be $^3$B$_{1g}$ (since A$_{2g}$ from $D_{4h}$ descends into B$_{1g}$ in $D_{2h}$)? Or is there any reason why it is not displayed in the figure? – Rudi_Birnbaum Sep 19 '18 at 10:56\n\nThe point is that if cyclobutadiene were symmetric the highest occupied orbital would be doubly degenerate with 2 electrons to occupy it, and so by Hund's rules it would be a diradical, just as for molecular oxygen. But the Jahn-Teller theorem says that such a state is unstable with respect to a distortion which splits the degeneracy and lowers the symmetry. This is exactly the same as $\\ce{Cu^2+}$. Hence the symmetrical molecule distorts splitting the degeneracy, and both electrons occupy the lower lying state. Thus not only is the molecule not symmetric, it is also not a radical, and hence there is no ESR signal.\n\n• How would it be doubly degenerate? For example Eg represents a set of two degenerate orbitals, but that doesn't mean a complex will be doubly degenerate (only d9, low-spin d7 and high-spin d4 are). – RBW Mar 25 '17 at 19:46\n• But $\\ce {Cu^{2+}}$ is $d^9$, no? – Oscar Lanzi Mar 26 '17 at 1:56\n• @Marko I don't really understand what you are saying. The high symmetry state leads to degeneracies, hence the theoretical square cyclobutadiene and the theoretical Cu2+ in an octahedral environment would have doubly degenerate HOMOs. But the Jahn-Teller theorem says that such geometries must distort, and further that that distortion will split the degeneracy, and so the HOMO is no longer degenerate. In the case of cyclobutadiene the splitting is so large that the lower of the states is double occupied, hence no ESR signal. – Ian Bush Mar 26 '17 at 8:17\n• I think that doubly degenerate means that there are two energetically equivalent states, but I see only one for cyclobutadiene and that is the one where both HOMOs are filled with 1e each. – RBW Mar 26 '17 at 8:34\n• Ahhh, maybe it is the word \"state\". Let's avoid that word, let's use orbital instead as that's what we're really talking about here (and I see why state might be confusing). In your diagram I see two horizontal lines at the same vertical level, representing two orbitals which have the same orbital energy. This is the two-fold degeneracy being referred too, and as we have 2 electrons to occupy 2 orbitals by Hund's rule one electron goes in each. And it is this degeneracy that is split by the distortion. – Ian Bush Mar 26 '17 at 8:57" ]	[ null, "https://i.stack.imgur.com/HAQbh.png", null, "https://i.stack.imgur.com/pyhyG.png", null, "https://i.stack.imgur.com/WGwot.png", null, "https://i.stack.imgur.com/VSqGb.png", null, "https://i.stack.imgur.com/dqIny.png", null, "https://i.stack.imgur.com/SWyHI.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9073868,"math_prob":0.9808427,"size":10673,"snap":"2019-43-2019-47","text_gpt3_token_len":2695,"char_repetition_ratio":0.12747212,"word_repetition_ratio":0.001216545,"special_character_ratio":0.25325587,"punctuation_ratio":0.1227295,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9922277,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,10,null,10,null,10,null,10,null,10,null,10,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-18T21:52:03Z\",\"WARC-Record-ID\":\"<urn:uuid:36e4043c-dec1-4eaf-86fc-2a9c8858fb52>\",\"Content-Length\":\"169937\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aea2c2fe-e4fc-4c06-9e62-c0607d5aecc5>\",\"WARC-Concurrent-To\":\"<urn:uuid:a83db2bc-1b1d-44a2-97dd-55418a94a8ec>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://chemistry.stackexchange.com/questions/71249/cyclobutadiene-jahn-teller-effect-or-not/71327\",\"WARC-Payload-Digest\":\"sha1:5S2IFCR3RNG6VKEMYWRCSLC5LT44VX3E\",\"WARC-Block-Digest\":\"sha1:NYBDINDTQR52LRPYBRXJX4HC2336FLDT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669847.1_warc_CC-MAIN-20191118205402-20191118233402-00062.warc.gz\"}"}
https://leiblog.wang/%E5%9C%A8MacBook-Pro-2019%E4%B8%8A%E4%BC%98%E5%8C%96GEMM/index.html	[ "how-to-optimiza-gemm 是大家参考得比较多的gemm优化tutorial，本文是在我的MacBook Pro 2019上进行的实践，处理器型号是i5-8257U.\n\n### 1.浮点峰值计算\n\n❯ sysctl machdep.cpu.brand_string\nmachdep.cpu.brand_string: Intel(R) Core(TM) i5-8257U CPU @ 1.40GHz", null, "", null, "static void cpuid_x86_exec(struct cpuid_t cpuid) {\nasm volatile(\"pushq %%rbx\\n\"\n\"cpuid\\n\"\n\"movl %%ebx, %1\\n\"\n\"popq %%rbx\\n\"\n: \"=a\"(cpuid->eax), \"=r\"(cpuid->ebx), \"=c\"(cpuid->ecx), \"=d\"(cpuid->edx)\n: \"a\"(cpuid->ieax), \"c\"(cpuid->iecx)\n: \"cc\");\n}\n\nstruct cpuid_t cpuid;\n\nfeat = 0;\n\ncpuid.ieax = 0x1;\ncpuid.iecx = 0x0;\ncpuid_x86_exec(&cpuid);\n\nif (BIT_TEST(cpuid.edx, 25)) {\nSET_FEAT(_CPUID_X86_SSE_);\n}\nif (BIT_TEST(cpuid.ecx, 28)) {\nSET_FEAT(_CPUID_X86_AVX_);\n}\nif (BIT_TEST(cpuid.ecx, 12)) {\nSET_FEAT(_CPUID_X86_FMA_);\n}\n\ncpuid.ieax = 0x7;\ncpuid.iecx = 0x0;\ncpuid_x86_exec(&cpuid);\n\nif (BIT_TEST(cpuid.ebx, 16)) {\nSET_FEAT(_CPUID_X86_AVX512F_);\n}\nif (BIT_TEST(cpuid.ecx, 11)) {\nSET_FEAT(_CPUID_X86_AVX512_VNNI_);\n}\n\nH，M,U，表示功耗，字母越小，功耗越大，性能越好。所以后缀：H>M>U。比如：i5-5350H>i7-4610m,i5-4330m>i7-4558U", null, "PORT0和PORT1可以同时执行向量乘、向量加和FMA指令，通过查datasheet可以知道：\n\n$$2(256/32)21.4=44.8 \\quad GFLOPS$$\n\nMac上需要使用到 Turbo Boost Switcher Pro这个软件来关闭睿频，不过我的mbp2019日常开启睿频，风扇疯狂起飞XDD\n\n$$2(256/32)23.9=124.8 \\quad GFLOPS$$\n\ngemm-optimizer/cmake-build-debug/tools/calc-cpu-flops 4\n\n❯ tree\n.\n├── CMakeLists.txt\n├── calc-cpu-flops.c\n├── cpufp_kernel_x86.h\n├── cpufp_kernel_x86_avx.s\n├── cpufp_kernel_x86_avx512_vnni.s\n├── cpufp_kernel_x86_avx512f.s\n├── cpufp_kernel_x86_fma.s\n├── cpufp_kernel_x86_sse.s\n├── smtl.c\n└── smtl.h\n\n_cpufp_kernel_x86_fma_fp32:\nmov $0x40000000, %rax vxorps %ymm0, %ymm0, %ymm0 vxorps %ymm1, %ymm1, %ymm1 vxorps %ymm2, %ymm2, %ymm2 vxorps %ymm3, %ymm3, %ymm3 vxorps %ymm4, %ymm4, %ymm4 vxorps %ymm5, %ymm5, %ymm5 vxorps %ymm6, %ymm6, %ymm6 vxorps %ymm7, %ymm7, %ymm7 vxorps %ymm8, %ymm8, %ymm8 vxorps %ymm9, %ymm9, %ymm9 ._cpufp.x86.fma.fp32.L1: vfmadd231ps %ymm0, %ymm0, %ymm0 vfmadd231ps %ymm1, %ymm1, %ymm1 vfmadd231ps %ymm2, %ymm2, %ymm2 vfmadd231ps %ymm3, %ymm3, %ymm3 vfmadd231ps %ymm4, %ymm4, %ymm4 vfmadd231ps %ymm5, %ymm5, %ymm5 vfmadd231ps %ymm6, %ymm6, %ymm6 vfmadd231ps %ymm7, %ymm7, %ymm7 vfmadd231ps %ymm8, %ymm8, %ymm8 vfmadd231ps %ymm9, %ymm9, %ymm9 sub$0x1, %rax\njne ._cpufp.x86.fma.fp32.L1\nret\n\nrax存储的是循环的次数，循环初始化的时候执行的一系列异或指令是用来快速将寄存器的内容置零，接着循环内会执行十次乘累加运算，一共执行40000000次，之前分析得，对于32位的浮点数，一次乘累加运算可以操作8个数，一共是 40000000 8 * 10 * 2（乘法和加法）运算，共这么大的浮点运算量：\n\n#ifdef _FMA_\n#define FMA_FP32_COMP (0x40000000L * 160)\n#define FMA_FP64_COMP (0x40000000L * 80)\n\nperf = FMA_FP64_COMP * num_threads / time_used * 1e-9;\n\nThread(s): 1\nbinding to core 0\nfma fp32 perf: 40.9189 gflops.\nfma fp64 perf: 22.0921 gflops.\nbinding to core 0\navx fp32 perf: 21.9944 gflops.\navx fp64 perf: 11.0741 gflops.\nbinding to core 0\nsse fp32 perf: 10.8526 gfops.\nsse fp64 perf: 5.4786 gflops.\n\nThread(s): 1\nbinding to core 0\nfma fp32 perf: 117.0210 gflops.\nfma fp64 perf: 58.4361 gflops.\nbinding to core 0\navx fp32 perf: 56.1949 gflops.\navx fp64 perf: 28.9646 gflops.\nbinding to core 0\nsse fp32 perf: 29.1122 gflops.\nsse fp64 perf: 14.6471 gflops.\n\n### 2.baseline\n\nvoid AddDot( int k, double x, int incx, double y, double gamma )\n{\n/ compute gamma := x' * y + gamma with vectors x and y of length n.\n\nHere x starts at location x with increment (stride) incx and y starts at location y and has (implicit) stride of 1.\n/\n\nint p;\n\nfor ( p=0; p<k; p++ ){\ngamma += X( p ) * y[ p ];\n}\n}\n\n/* Routine for computing C = A * B + C /\n\nvoid mMultBaseLine( int m, int n, int k, double a, int lda,\ndouble b, int ldb,\ndouble c, int ldc )\n{\nint i, j, p;\n\nfor ( i=0; i<m; i++ ){ /* Loop over the rows of C /\nfor ( j=0; j<n; j++ ){ / Loop over the columns of C /\nAddDot( k, &A( i,0 ), lda, &B( 0,j ), &C( i,j ) );\n}\n}\n}\n\n\n#define PFIRST 100\n#define PLAST 1500\n#define PINC 100\n\n100 7.299270e-01 0.000000e+00\n200 6.892096e-01 0.000000e+00\n300 6.828183e-01 0.000000e+00\n400 6.939813e-01 0.000000e+00\n500 6.374153e-01 0.000000e+00\n600 6.188340e-01 0.000000e+00\n700 5.687877e-01 0.000000e+00\n800 5.626816e-01 0.000000e+00\n900 5.218054e-01 0.000000e+00\n1000 5.200376e-01 0.000000e+00\n1100 4.751246e-01 0.000000e+00\n1200 4.499168e-01 0.000000e+00\n1300 4.366850e-01 0.000000e+00\n1400 4.270737e-01 0.000000e+00\n1500 3.929416e-01 0.000000e+00 \n\ntools/plot下我提供了一个py程序，用来绘制benckmark的：", null, "gflops只有0.6，并且显而易见在gemm的尺寸变大的时候，gflop有下降趋势，这是因为当运算过程中的矩阵A、B的大小超过了L2 Cache的大小(256Kb)的时候，频繁访问A、B就会导致Cache要重复的访问DDR，变成访存的瓶颈，这部分可以通过矩阵分块计算来解决。\n\n### 3.循环展开和合并\n\nvoid MY_MMult( int m, int n, int k, double a, int lda,\ndouble b, int ldb,\ndouble c, int ldc )\n{\nint i, j;\n\nfor ( j=0; j<n; j+=4 ){ /* Loop over the columns of C, unrolled by 4 /\nfor ( i=0; i<m; i+=1 ){ / Loop over the rows of C /\n/ Update C( i,j ), C( i,j+1 ), C( i,j+2 ), and C( i,j+3 ) in\none routine (four inner products) /\nAddDot( k, &A( 0, 0 ), lda, &B( 0, 0 ), &C( 0, 0 ) );\nAddDot( k, &A( 0, 0 ), lda, &B( 0, 1 ), &C( 0, 1 ) );\nAddDot( k, &A( 0, 0 ), lda, &B( 0, 2 ), &C( 0, 2 ) );\nAddDot( k, &A( 0, 0 ), lda, &B( 0, 3 ), &C( 0, 3 ) );\n}\n}\n}\n\n\nvoid MY_MMult( int m, int n, int k, double a, int lda,\ndouble b, int ldb,\ndouble c, int ldc )\n{\nint i, j;\n\nfor ( j=0; j<n; j+=4 ){ /* Loop over the columns of C, unrolled by 4 /\nfor ( i=0; i<m; i+=1 ){ / Loop over the rows of C /\nint p;\n\n// AddDot( k, &A( 0, 0 ), lda, &B( 0, 0 ), &C( 0, 0 ) );\nfor ( p=0; p<k; p++ ){\nC( 0, 0 ) += A( 0, p ) B( p, 0 );\n}\n\n// AddDot( k, &A( 0, 0 ), lda, &B( 0, 1 ), &C( 0, 1 ) );\nfor ( p=0; p<k; p++ ){\nC( 0, 1 ) += A( 0, p ) * B( p, 1 );\n}\n\n// AddDot( k, &A( 0, 0 ), lda, &B( 0, 2 ), &C( 0, 2 ) );\nfor ( p=0; p<k; p++ ){\nC( 0, 2 ) += A( 0, p ) * B( p, 2 );\n}\n\n// AddDot( k, &A( 0, 0 ), lda, &B( 0, 3 ), &C( 0, 3 ) );\nfor ( p=0; p<k; p++ ){\nC( 0, 3 ) += A( 0, p ) * B( p, 3 );\n}\n}\n}\n}\n\nfor ( p=0; p<k; p++ ){\nC( 0, 0 ) += A( 0, p ) * B( p, 0 );\nC( 0, 1 ) += A( 0, p ) * B( p, 1 );\nC( 0, 2 ) += A( 0, p ) * B( p, 2 );\nC( 0, 3 ) += A( 0, p ) * B( p, 3 );\n}", null, "### 4.访存方式\n\nfor ( p=0; p<k; p++ ){\nC( 0, 0 ) += A( 0, p ) * B( p, 0 );\nC( 0, 1 ) += A( 0, p ) * B( p, 1 );\nC( 0, 2 ) += A( 0, p ) * B( p, 2 );\nC( 0, 3 ) += A( 0, p ) * B( p, 3 );\n}\n\nC(0,0)被复用了k次，A(0,p)被复用了4次，所以我们将它们放在访问比较快的单元里会有利于程序的性能，在C语言里，使用register将变量塞到物理寄存器里去：\n\nfor ( p=0; p<k; p++ ){\na_0p_reg = A( 0, p );\n\nc_00_reg += a_0p_reg * bp0_pntr++;\nc_01_reg += a_0p_reg bp1_pntr++;\nc_02_reg += a_0p_reg bp2_pntr++;\nc_03_reg += a_0p_reg bp3_pntr++;\n}\n\nbp0_pntr = &B( 0, 0 );\nbp1_pntr = &B( 0, 1 );\nbp2_pntr = &B( 0, 2 );\nbp3_pntr = &B( 0, 3 );\n\n#define A(i,j) a[ (j)lda + (i) ]\n#define B(i,j) b[ (j)ldb + (i) ]\n#define C(i,j) c[ (j)ldc + (i) ]", null, "### 5.SIMD指令加速\n\n._cpufp.x86.sse.fp32.L1:\nmulps %xmm0, %xmm0\nmulps %xmm2, %xmm2\nmulps %xmm4, %xmm4\nmulps %xmm6, %xmm6\nsub \\$0x1, %rax\nmulps %xmm8, %xmm8\nmulps %xmm10, %xmm10\nmulps %xmm12, %xmm12\nmulps %xmm14, %xmm14\naddps %xmm15, %xmm15\n\nfor ( p=0; p<k; p++ ){\na_0p_reg = A( 0, p );\na_1p_reg = A( 1, p );\na_2p_reg = A( 2, p );\na_3p_reg = A( 3, p );\n\nb_p0_pntr = &B( p, 0 );\nb_p1_pntr = &B( p, 1 );\nb_p2_pntr = &B( p, 2 );\nb_p3_pntr = &B( p, 3 );\n\n/* First row /\nc_00_reg += a_0p_reg b_p0_pntr;\nc_01_reg += a_0p_reg b_p1_pntr;\nc_02_reg += a_0p_reg b_p2_pntr;\nc_03_reg += a_0p_reg b_p3_pntr;\n\n/ Second row /\nc_10_reg += a_1p_reg b_p0_pntr;\nc_11_reg += a_1p_reg b_p1_pntr;\nc_12_reg += a_1p_reg b_p2_pntr;\nc_13_reg += a_1p_reg b_p3_pntr;\n\n/ Third row /\nc_20_reg += a_2p_reg b_p0_pntr;\nc_21_reg += a_2p_reg b_p1_pntr;\nc_22_reg += a_2p_reg b_p2_pntr;\nc_23_reg += a_2p_reg b_p3_pntr;\n\n/ Four row /\nc_30_reg += a_3p_reg b_p0_pntr++;\nc_31_reg += a_3p_reg b_p1_pntr++;\nc_32_reg += a_3p_reg b_p2_pntr++;\nc_33_reg += a_3p_reg b_p3_pntr++;\n}", null, "b_p0_vreg.v = _mm_loaddup_pd( (double ) b_p0_pntr++ ); /* load and duplicate /\nb_p1_vreg.v = _mm_loaddup_pd( (double ) b_p1_pntr++ ); /* load and duplicate /\nb_p2_vreg.v = _mm_loaddup_pd( (double ) b_p2_pntr++ ); /* load and duplicate /\nb_p3_vreg.v = _mm_loaddup_pd( (double ) b_p3_pntr++ ); /* load and duplicate /\n\n/ First row and second rows /\nc_00_c_10_vreg.v += a_0p_a_1p_vreg.v b_p0_vreg.v;\nc_01_c_11_vreg.v += a_0p_a_1p_vreg.v * b_p1_vreg.v;\nc_02_c_12_vreg.v += a_0p_a_1p_vreg.v * b_p2_vreg.v;\nc_03_c_13_vreg.v += a_0p_a_1p_vreg.v * b_p3_vreg.v;\n\n$$C_{00} = \\sum_{k=0}^{3}{A_{0k} * B_{k0}} \\ C_{10} = \\sum_{k=0}^{3}{A_{1k} * B_{k0}}$$", null, "### 6. 矩阵分块乘法\n\nfor ( p=0; p<k; p+=kc ){\npb = min( k-p, kc );\nfor ( i=0; i<m; i+=mc ){\nib = min( m-i, mc );\nInnerKernel( ib, n, pb, &A( i,p ), lda, &B(p, 0 ), ldb, &C( i,0 ), ldc );\n}\n}", null, "### 7. DataPack", null, "" ]	[ null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/image-20220212173146231.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/image-20220212191541540.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/20220212233321.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/baseline.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/loopunroll1x4.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/register1x4.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/loopunroll4x4.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/image-20220216162342131.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/image-20220216165503348.png", null, "https://leiblog-imgbed.oss-cn-beijing.aliyuncs.com/img/image-20220216170201568.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.50452685,"math_prob":0.9972573,"size":12037,"snap":"2022-27-2022-33","text_gpt3_token_len":7461,"char_repetition_ratio":0.12897864,"word_repetition_ratio":0.31363636,"special_character_ratio":0.4234444,"punctuation_ratio":0.23333333,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9960143,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-28T12:32:54Z\",\"WARC-Record-ID\":\"<urn:uuid:40626d8b-9066-469a-bee4-5f1de2839cbf>\",\"Content-Length\":\"95371\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b2fb7021-6e81-405e-9809-eea1c34c8f7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:67243d52-5438-4165-8c3a-57ffedb399f7>\",\"WARC-IP-Address\":\"101.200.188.197\",\"WARC-Target-URI\":\"https://leiblog.wang/%E5%9C%A8MacBook-Pro-2019%E4%B8%8A%E4%BC%98%E5%8C%96GEMM/index.html\",\"WARC-Payload-Digest\":\"sha1:RMHMO7VDHJYZAID4QPKXSSK4RDVKDOSU\",\"WARC-Block-Digest\":\"sha1:FJKUME4RLENSNCNH334IYR63IKEWH5AU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103516990.28_warc_CC-MAIN-20220628111602-20220628141602-00409.warc.gz\"}"}
http://zackmdavis.net/blog/2012/08/straight-talk-about-precompactness/	[ "So we have this metric space, which is this set of points along with a way of defining \"distances\" between them that behaves in a basically noncrazy way (points that are zero distance away from \"each other\" are really just the same point, the distance from one to the other is the same as the distance from the other to the one, and something about triangles).\n\nLet's say (please, if you don't mind) that a sequence of points (xn) in our space is fundamental (or maybe Cauchy) iff (sic) for all positive ε, there's a point far enough along in the sequence so that beyond that point, the distance from one point to the next is less than ε. Let's also agree (if that's okay with you) to say that our metric space is sequentially precompact iff every sequence has a fundamental subsequence. If, furthermore, the precompact space is complete (all fundamental sequences actually converge to a point in the space, rather than leading up to an ætherial gap or missing edge), then we say it's compact. It turns out that compactness is an important property to pay attention to because it implies lots of cool stuff: like, compactness is preserved by homeomorphisms (continuously invertible continuous maps), and continuous functions with compact domains are bounded, and probably all sorts of other things that I don't know (yet). I'm saying sequentially precompact because I'm given to understand that while the convergent subsequences criterion for compactness is equivalent to this other definition (viz., \"every open cover has a finite subcover\") for metric spaces, the two ideas aren't the same for more general topological spaces. Just don't ask me what in the world we're going to do with a nonmetrizable space, 'cause I don't know (yet).\n\nBut anyway, as long as we're naming ideas, why not say that our metric space is totally bounded iff for every ε, there exists a finite number of open (that is, not including the boundary) balls that cover the whole space? We can call the centers of such a group of balls an ε-net. Our friend Shilov quotes his friend Liusternik as saying, \"Suppose a lamp illuminating a ball of radius ε is placed at every point of a set B which is an ε-net for a set M. Then the whole set M will be illuminated.\" At the risk of having names for things that possibly don't actually deserve names, I'm going call each point in an ε-net a lamp. Actually Shilov, and thus likely Liusternik, is talking about closed balls of light around the lamps, not the open ones that I'm talking about. In a lot of circumstances, this could probably make all the difference in the world, but for the duration of this post, I don't think you should worry about it.\n\nBut this fear of having too many names for things is really a very serious one, because it turns out that sequential precompactness and total boundedness are the same thing: not only can you not have one without the other, but you can't even have the other without the one! Seriously, like, who even does that?!\n\nBut the reasoning is inescapable. You can't have one without the other because if every sequence has a fundamental subsequence, then finite ε-nets are a thing, which is to say (by the contraposition doctrine and De Morgan's Iron Law of Negation) that if every ε-net is infinite, then sequences that don't have fundamental subsequences are a thing. To see this, think about an infinite ε-net where no lamp lies within the lighted area of any other lamp. A sequence consisting of such lamps can't have a fundamental subsequence because the distance between successive points in that sequence is bounded below by ε.\n\nAnd you can't have the other without the one because if finite ε-nets are a thing, then every sequence has a fundamental subsequence. To see this, consider a sequence. For k ∈ ℕ+ and for ε := 1/k, we can cover any subset of our space with a finite number of ε-balls. But then by the Infinitary Corollary of the Iron Law Pertaining to the Storage of Pigeons, there must then be an ε-ball that contains infinitely many points of our sequence. Let's pick one of those points and call it ak. Then if we set ε := 1/(k+1), our ball can itself be covered by a finite number of ε-balls, one of which again contains infinitely many points of our sequence, of which we can pick one and call it ak+1. That triggers an induction, giving us a subsequence (an). But then for every N ∈ ℕ+, if n and m are not smaller than N, then an and am live in a 1/N-ball, so that the distance between them is bounded above by 2/N, which can be made arbitrarily small by choosing a large enough N, which means that the subsequence (an) is fundamental. But this is \"quod erat demonstrandum\" (a Latin phrase that roughly translates as \"what I've been trying to tell you this entire time\").\n\nBibliography\n\nTheodore W. Gamelin and Robert Everist Greene, Introduction to Topology, 2nd ed'n., §I.5.\n\nGeorgi E. Shilov, Elementary Real and Complex Analysis, revised English ed'n., §3.93." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9649189,"math_prob":0.9423926,"size":4955,"snap":"2022-27-2022-33","text_gpt3_token_len":1140,"char_repetition_ratio":0.12765098,"word_repetition_ratio":0.02183908,"special_character_ratio":0.21836528,"punctuation_ratio":0.09929078,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97069776,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-01T14:27:01Z\",\"WARC-Record-ID\":\"<urn:uuid:54882631-ded3-4a26-8831-fad3ffaaccfd>\",\"Content-Length\":\"49386\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bc65b26e-55c4-42a7-b541-7d8ac731ac02>\",\"WARC-Concurrent-To\":\"<urn:uuid:1b222a50-3051-4727-8f6d-e497fb0537ba>\",\"WARC-IP-Address\":\"199.188.205.30\",\"WARC-Target-URI\":\"http://zackmdavis.net/blog/2012/08/straight-talk-about-precompactness/\",\"WARC-Payload-Digest\":\"sha1:U7E3SLSL6YVPU7C7SQOXJF6QWWFF4AZC\",\"WARC-Block-Digest\":\"sha1:J3NDMFHPLW4FHXE223XJPO4RPWKJCWKU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103941562.52_warc_CC-MAIN-20220701125452-20220701155452-00688.warc.gz\"}"}
https://www.gurufocus.com/term/mscore/NYSE:RTN/Beneish-M-Score/Raytheon	[ "Switch to:\n\n# Raytheon Co Beneish M-Score\n\n: -2.65 (As of Today)\nView and export this data going back to 1952. Start your Free Trial\n\nThe zones of discrimination for M-Score is as such:\n\nAn M-Score of less than -2.22 suggests that the company is not an accounting manipulator.\nAn M-Score of greater than -2.22 signals that the company is likely an accounting manipulator.\n\nRaytheon Co has a M-score of -2.65 suggests that the company is not a manipulator.\n\nNYSE:RTN' s Beneish M-Score Range Over the Past 10 Years\nMin: -3.28 Med: -2.56 Max: -2.07\nCurrent: -2.65\n\n-3.28\n-2.07\n\nDuring the past 13 years, the highest Beneish M-Score of Raytheon Co was -2.07. The lowest was -3.28. And the median was -2.56.\n\n## Raytheon Co Beneish M-Score Historical Data\n\n* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.\n\n Raytheon Co Annual Data Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Dec16 Dec17 Dec18 Dec19 Beneish M-Score", null, "", null, "", null, "", null, "", null, "-2.39 -2.56 -2.33 -2.17 -2.65\n\nCompetitive Comparison\n* Competitive companies are chosen from companies within the same industry, with headquarter located in same country, with closest market capitalization; x-axis shows the market cap, and y-axis shows the term value; the bigger the dot, the larger the market cap.\n\nRaytheon Co Beneish M-Score Distribution\n\n* The bar in red indicates where Raytheon Co's Beneish M-Score falls into.\n\n## Raytheon Co Beneish M-Score Calculation\n\nThe M-score was created by Professor Messod Beneish. Instead of measuring the bankruptcy risk (Altman Z-Score) or business trend (Piotroski F-Score), M-score can be used to detect the risk of earnings manipulation. This is the original research paper on M-score.\n\nThe M-Score Variables:\n\nThe M-score of Raytheon Co for today is based on a combination of the following eight different indices:\n\n M = -4.84 + 0.92 * DSRI + 0.528 * GMI + 0.404 * AQI + 0.892 * SGI + 0.115 * DEPI = -4.84 + 0.92 * 0.7676 + 0.528 * 1.0397 + 0.404 * 0.9657 + 0.892 * 1.0783 + 0.115 * 1.077 - 0.172 * SGAI + 4.679 * TATA - 0.327 * LVGI - 0.172 * 0.9939 + 4.679 * -0.0141 - 0.327 * 0.9379 = -2.65\n\n* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.\n\n This Year (Dec19) TTM: Last Year (Dec18) TTM: Accounts Receivable was \\$1,364 Mil. Revenue was 7842 + 7446 + 7159 + 6729 = \\$29,176 Mil. Gross Profit was 2010 + 1947 + 1954 + 1852 = \\$7,763 Mil. Total Current Assets was \\$13,082 Mil. Total Assets was \\$34,566 Mil. Property, Plant and Equipment(Net PPE) was \\$4,228 Mil. Depreciation, Depletion and Amortization(DDA) was \\$605 Mil. Selling, General, & Admin. Expense(SGA) was \\$2,257 Mil. Total Current Liabilities was \\$9,791 Mil. Long-Term Debt & Capital Lease Obligation was \\$3,967 Mil. Net Income was 885 + 860 + 817 + 781 = \\$3,343 Mil. Non Operating Income was -181 + -135 + -173 + -161 = \\$-650 Mil. Cash Flow from Operations was 2791 + 1277 + 823 + -411 = \\$4,480 Mil. Accounts Receivable was \\$1,648 Mil. Revenue was 7360 + 6806 + 6625 + 6267 = \\$27,058 Mil. Gross Profit was 1967 + 1935 + 1848 + 1735 = \\$7,485 Mil. Total Current Assets was \\$12,137 Mil. Total Assets was \\$32,670 Mil. Property, Plant and Equipment(Net PPE) was \\$3,645 Mil. (DDA) was \\$568 Mil. Selling, General, & Admin. Expense(SGA) was \\$2,106 Mil. Total Current Liabilities was \\$8,463 Mil. Long-Term Debt & Capital Lease Obligation was \\$5,402 Mil.\n\n1. DSRI = Days Sales in Receivables Index\n\nMeasured as the ratio of Revenue in Accounts Receivable in year t to year t-1.\n\nA large increase in DSR could be indicative of revenue inflation.\n\n DSRI = (Receivables_t / Revenue_t) / (Receivables_t-1 / Revenue_t-1) = (1364 / 29176) / (1648 / 27058) = 0.04675075 / 0.0609062 = 0.7676\n\n2. GMI = Gross Margin Index\n\nMeasured as the ratio of gross margin in year t-1 to gross margin in year t.\n\nGross margin has deteriorated when this index is above 1. A firm with poorer prospects is more likely to manipulate earnings.\n\n GMI = GrossMargin_t-1 / GrossMargin_t = (GrossProfit_t-1 / Revenue_t-1) / (GrossProfit_t / Revenue_t) = (7485 / 27058) / (7763 / 29176) = 0.27662798 / 0.26607486 = 1.0397\n\n3. AQI = Asset Quality Index\n\nAQI is the ratio of asset quality in year t to year t-1.\n\nAsset quality is measured as the ratio of non-current assets other than Property, Plant and Equipment to Total Assets.\n\n AQI = (1 - (CurrentAssets_t + PPE_t) / TotalAssets_t) / (1 - (CurrentAssets_t-1 + PPE_t-1) / TotalAssets_t-1) = (1 - (13082 + 4228) / 34566) / (1 - (12137 + 3645) / 32670) = 0.49921889 / 0.51692684 = 0.9657\n\n4. SGI = Sales Growth Index\n\nRatio of Revenue in year t to sales in year t-1.\n\nSales growth is not itself a measure of manipulation. However, growth companies are likely to find themselves under pressure to manipulate in order to keep up appearances.\n\n SGI = Sales_t / Sales_t-1 = Revenue_t / Revenue_t-1 = 29176 / 27058 = 1.0783\n\n5. DEPI = Depreciation Index\n\nMeasured as the ratio of the rate of Depreciation, Depletion and Amortization in year t-1 to the corresponding rate in year t.\n\nDEPI greater than 1 indicates that assets are being depreciated at a slower rate. This suggests that the firm might be revising useful asset life assumptions upwards, or adopting a new method that is income friendly.\n\n DEPI = (Depreciation_t-1 / (Depreciaton_t-1 + PPE_t-1)) / (Depreciation_t / (Depreciaton_t + PPE_t)) = (568 / (568 + 3645)) / (605 / (605 + 4228)) = 0.13482079 / 0.12518105 = 1.077\n\n6. SGAI = Sales, General and Administrative expenses Index\n\nThe ratio of Selling, General, & Admin. Expense(SGA) to Sales in year t relative to year t-1.\n\nSGA expenses index > 1 means that the company is becoming less efficient in generate sales.\n\n SGAI = (SGA_t / Sales_t) / (SGA_t-1 /Sales_t-1) = (2257 / 29176) / (2106 / 27058) = 0.0773581 / 0.0778328 = 0.9939\n\n7. LVGI = Leverage Index\n\nThe ratio of total debt to Total Assets in year t relative to yeat t-1.\n\nAn LVGI > 1 indicates an increase in leverage\n\n LVGI = ((LTD_t + CurrentLiabilities_t) / TotalAssets_t) / ((LTD_t-1 + CurrentLiabilities_t-1) / TotalAssets_t-1) = ((3967 + 9791) / 34566) / ((5402 + 8463) / 32670) = 0.39802118 / 0.42439547 = 0.9379\n\n8. TATA = Total Accruals to Total Assets\n\nTotal accruals calculated as the change in working capital accounts other than cash less depreciation.\n\n TATA = (IncomefromContinuingOperations_t - CashFlowsfromOperations_t) / TotalAssets_t = (NetIncome_t - NonOperatingIncome_t - CashFlowsfromOperations_t) / TotalAssets_t = (3343 - -650 - 4480) / 34566 = -0.0141\n\nAn M-Score of less than -2.22 suggests that the company will not be a manipulator. An M-Score of greater than -2.22 signals that the company is likely to be a manipulator.\n\nRaytheon Co has a M-score of -2.65 suggests that the company will not be a manipulator." ]	[ null, "https://gurufocus.s3.amazonaws.com/images/blur.png", null, "https://gurufocus.s3.amazonaws.com/images/blur.png", null, "https://gurufocus.s3.amazonaws.com/images/blur.png", null, "https://gurufocus.s3.amazonaws.com/images/blur.png", null, "https://gurufocus.s3.amazonaws.com/images/blur.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8707484,"math_prob":0.9540325,"size":6328,"snap":"2020-45-2020-50","text_gpt3_token_len":2087,"char_repetition_ratio":0.11574952,"word_repetition_ratio":0.10823312,"special_character_ratio":0.40265486,"punctuation_ratio":0.14664458,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9897041,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-30T23:38:30Z\",\"WARC-Record-ID\":\"<urn:uuid:4732fae0-2507-4a6a-add7-df09da4078c2>\",\"Content-Length\":\"383791\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a0367ff0-994c-44ab-a3cc-0a120082b7de>\",\"WARC-Concurrent-To\":\"<urn:uuid:a3970198-45f1-48bb-8a6f-dc8a208eefe7>\",\"WARC-IP-Address\":\"104.26.14.56\",\"WARC-Target-URI\":\"https://www.gurufocus.com/term/mscore/NYSE:RTN/Beneish-M-Score/Raytheon\",\"WARC-Payload-Digest\":\"sha1:VFNZMHRIP42PZVHOAMRWPTYBDOB5MFKU\",\"WARC-Block-Digest\":\"sha1:GDXDQRU2FYH6YPH4CFXVB5PNSBMGYH7W\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107911792.65_warc_CC-MAIN-20201030212708-20201031002708-00393.warc.gz\"}"}
https://www.physicsforums.com/threads/find-the-speed-of-a-copper-loop-falling-in-a-magnetic-field.976109/	[ "# Find the speed of a copper loop falling in a magnetic field\n\nspsch\nHomework Statement:\nA square copper loop with side length a is falling straight down at constant speed perpendicular to a magnetic field B. The cross-sectional area of the loop is A. What is its speed v?\nRelevant Equations:\nV = BdA = Blv (or Bav here) F = ILB and V = IR\nHi all, so I had this problem and on the exam and I got a solution but I had an mass-term in there which wasn't given.\n\nI used Farraday's Law of Induction to get the Voltage induced.\nThen I used ##rho \\frac{A}{4a} ## for the resistance and divided the Voltage by that to get the current.\nI then inserted this for the current I in F=IlB.\nFrom the drawing on the exam it looked like only half the loop was in the field so I the forces up and down wouldn't cancel each other only the ones on the side.\n\nI then set that equal to mg because ## ma = mg-IlB ## since the speed is constant ## mg = IlB ##.\nThis question really threw me off because m wasn't given and I was sure it had to cancel somehow.\nCould someone show me where I made a mistake in thinking?\n\nThanks so much!\nI don't remember exactly but I think B was 1.5T, a = 25cm, A was 10^-6m.\n\nHomework Helper\nGold Member\nIt appears to me that you have the correct approach. The terminal speed of the loop will depend on the mass, as you found. So, a value of the mass should have been given in order to obtain a numerical value for the speed.\n\n•", null, "spsch\nspsch\nHello TSny.\n\nThank you very much. I lost quite a bit of time there. (trying to find a formula to make the masses cancel)\nAnd I thought about it for the past two days without figuring it out.\nI thought about it and with the density I probably could have figured it out. (Volume being 4a times A). There was a two page tableau with resistances and thermal data for the other problems.\nI'm afraid I may have missed that densities were included as well.\n(it also didn't occur to me then that m=density * volume)\n\nBut I wrote out the whole term without the numeric value. I hope that I may at least get a few points for that.\n\nHomework Helper\nGold Member\nOK. I hope that you receive almost full credit. The question is mostly about testing your understanding of some electromagnetism and I think you succeeded in doing that.\n\n•", null, "spsch\nHomework Helper\nGold Member\nThen I used ##\\rho \\frac{A}{4a}## for the resistance ...\nSo that you don't make the same mistake again, the correct expression for the resistance here is ##R=\\dfrac{\\rho(4a)}{A}##.\n\n•", null, "spsch and TSny\nHomework Helper\nGold Member\nYeah, very poorly posed problem.\nYou further need wire resistance per unit length (or total resistance) plus m.\n\nHomework Helper\nGold Member\nYeah, very poorly posed problem.\nYou further need wire resistance per unit length (or total resistance) plus m.\nTables were provided. See post #3.\n\nspsch\nHi all, I'm sorry I just thought about this problem a little more and have another question.\n\nThough it is a square, can I not see this wire as a loop and thus, wouldn't it also create a magnetic dipole that could end up feeling a torque?\n\nBecause by lenz's law the current induced should be in such a direction that opposes the change in magnetic flux? (Making the created dipole opposite)\nSo the wire would turn 180 degrees back and forth?\n\nI'm sorry I hope I'm making sense, I could draw what I mean if it helps.\n\n•", null, "kuruman\nHomework Helper\nGold Member\nI'm sorry I hope I'm making sense, I could draw what I mean if it helps.\nYou are making perfect sense and it is to your credit that you should ask about this. Indeed there will be a magnetic dipole moment generated by the induced current in which case there will be a torque tending to align the magnetic moment with the field. However, unlike a compass needle oscillating back and forth in the Earth's magnetic field because there is a restoring torque, this is not the case here because the torque is not restoring. If the loop tries to rotate because of the external torque, Lenz's law says that there will be an induced current, in addition to the one already there, that will oppose the proposed change of angle. The counterpart of the magnetic \"friction\" force that opposes the linear motion of the loop no matter in what direction the loop is moving is a magnetic \"friction\" torque that opposes the rotational motion of the loop no matter which way the loop is turning. Without having written and solved the equations, my guess is that if the loop is initially oriented so that the magnetic field is not perpendicular to the plane of the loop, the loop will settle at some \"terminal\" angle by the time it reaches terminal velocity. If I solve the equations, I will post here.\n\n•", null, "spsch\nspsch\nIf the loop tries to rotate because of the external torque, Lenz's law says that there will be an induced current, in addition to the one already there, that will oppose the proposed change of angle. The counterpart of the magnetic \"friction\" force that opposes the linear motion of the loop no matter in what direction the loop is moving is a magnetic \"friction\" torque that opposes the rotational motion of the loop no matter which way the loop is turning.\n\nHi Kuruman,\n\nthank you for answering me again. That makes sense.\n\nThe loop is originally falling vertically down perpendicular to the magnetic field. It's why when I thought about it I imagined the dipole being created inside the loop, directly opposite the direction of the magnetic field which it is falling in, and hence I imagined it should feel a torque until it balances again, maybe oscillate back and forth.\n\nBut it makes sense that the rotation would also create a change in flux and those a counter current and dipole.\nIt's hard to wrap my head around it but I'm having quite a bit of fun thinking about this and am grateful for the patience you guys show me.\n\nspsch\nOK. I hope that you receive almost full credit. The question is mostly about testing your understanding of some electromagnetism and I think you succeeded in doing that.\nUpdate, got the results yesterday and I got half the points for this problem! :-) Thanks for helping!\n\n•", null, "TSny" ]	[ null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9733535,"math_prob":0.7259514,"size":1868,"snap":"2023-14-2023-23","text_gpt3_token_len":496,"char_repetition_ratio":0.091201715,"word_repetition_ratio":0.74934036,"special_character_ratio":0.27408993,"punctuation_ratio":0.07730673,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9739622,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-25T07:25:21Z\",\"WARC-Record-ID\":\"<urn:uuid:38311135-8bbb-49d0-b423-0da85429d540>\",\"Content-Length\":\"101922\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:791c4224-a4da-4705-a9d9-23e816bc4346>\",\"WARC-Concurrent-To\":\"<urn:uuid:a8184bb6-0ee2-4d02-83bd-40f0091646de>\",\"WARC-IP-Address\":\"104.26.15.132\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/find-the-speed-of-a-copper-loop-falling-in-a-magnetic-field.976109/\",\"WARC-Payload-Digest\":\"sha1:T72QDVNZ2VHKLKEKBIVUBEMSWPS6OCYC\",\"WARC-Block-Digest\":\"sha1:ZMZEDOGUSEUZOFDA237ATHT2KBQLOJPM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296945317.85_warc_CC-MAIN-20230325064253-20230325094253-00685.warc.gz\"}"}
https://artofproblemsolving.com/wiki/index.php/2006_Romanian_NMO_Problems/Grade_9/Problem_4	[ "# 2006 Romanian NMO Problems/Grade 9/Problem 4\n\n## Problem", null, "$2n$ students", null, "$(n \\geq 5)$ participated at table tennis contest, which took", null, "$4$ days. Every day, every student played a match. (It is possible that the same pair meets two or more times, in different days). Prove that it is possible that the contest ends like this:\n\n• there is only one winner;\n• there are", null, "$3$ students on the second place;\n• no student lost all", null, "$4$ matches.\n\nHow many students won only a single match and how many won exactly", null, "$2$ matches? (In the above conditions)\n\n## Solution\n\nNote that the 3 second place students obviously could not have only won one match, or won all 4 matches. I now claim that they could not have won exactly two matches, either.\n\nEach day there were", null, "$n$ matches, so at the end of the contest there were", null, "$4n$ total points. Now if the three people in second place won exactly two matches, then", null, "$2n-4$ people would have to had won exactly one match. The winner of the contest would have won at most 4 matches, so we have the inequality", null, "$$4+3\\cdot 2 + (2n-4)\\geq 4n$$\n\nSolving for", null, "$n$ yields", null, "$n\\leq 3$, which is clearly false. This is a contradiction in logic, so the three people in second place could not have won exactly two matches.\n\nThis shows that the three second-place finishers each won exactly three matches. Therefore the winner of the contest won all 4 matches. Now let", null, "$x$ be the number of people who won two matches. It follows that", null, "$2n-x-4$ people won one match. We now have the equation", null, "$$4 + 3\\cdot 3 + 2x + 2n-x-4 = 4n$$\n\nSolving for", null, "$x$ yields", null, "$x=2n-9$, so", null, "$\\boxed{2n-9}$ students won exactly two matches. It then follows that", null, "$\\boxed{5}$ people won a single match." ]	[ null, "https://latex.artofproblemsolving.com/f/9/3/f93f5c51ea4b04b4992b003b39479f29018f6bda.png ", null, "https://latex.artofproblemsolving.com/7/f/8/7f801049c8303e12a4b080f200f85fc22e7c46ef.png ", null, "https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png ", null, "https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png ", null, "https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png ", null, "https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png ", null, "https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png ", null, "https://latex.artofproblemsolving.com/b/4/1/b417a127620d5b467fe7a48ad5f7d843e5c390b5.png ", null, "https://latex.artofproblemsolving.com/f/8/2/f82be0cbb450a75fe5a294e8dc2a15c9d4db6c82.png ", null, "https://latex.artofproblemsolving.com/a/5/9/a5931009776f45d95058fe8bbdc4a490b195512e.png ", null, "https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png ", null, "https://latex.artofproblemsolving.com/d/d/3/dd3a0adc0e88e4d0eeba60a6066d6c922e1aabab.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/c/4/d/c4dc1e7b56cdfe760ee9a94e1ed9a20f6bb58439.png ", null, "https://latex.artofproblemsolving.com/d/7/1/d71482e7aac2958be89470f47a4dfa2721b85d67.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/5/f/6/5f60660f5c997ea4829a2dd74020754f6c4d4bda.png ", null, "https://latex.artofproblemsolving.com/7/6/1/761022cbc9ba62899400ff8c79c05007d1601979.png ", null, "https://latex.artofproblemsolving.com/8/9/7/897a203bdf45af36f2b9edc59c3bc0beeb898be1.png ", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9860602,"math_prob":0.99687326,"size":1483,"snap":"2021-43-2021-49","text_gpt3_token_len":322,"char_repetition_ratio":0.16903313,"word_repetition_ratio":0.030303031,"special_character_ratio":0.21443021,"punctuation_ratio":0.104377106,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99726266,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],"im_url_duplicate_count":[null,null,null,2,null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,2,null,null,null,2,null,null,null,2,null,2,null,null,null,2,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-06T21:00:08Z\",\"WARC-Record-ID\":\"<urn:uuid:16bf5bde-abb3-4bc0-8f89-917362ad5e04>\",\"Content-Length\":\"37032\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fb807c95-8b34-41cf-9e9a-0c06faf8b98f>\",\"WARC-Concurrent-To\":\"<urn:uuid:e40b478f-c75f-4bfd-a6d2-29af935c819a>\",\"WARC-IP-Address\":\"172.67.69.208\",\"WARC-Target-URI\":\"https://artofproblemsolving.com/wiki/index.php/2006_Romanian_NMO_Problems/Grade_9/Problem_4\",\"WARC-Payload-Digest\":\"sha1:SEDA2UHV52XOFUWF2LJF6SZO3Q2NWAPE\",\"WARC-Block-Digest\":\"sha1:BIBUFMZ6GAXYZS5UEGRKOS2WQLXR32OX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363312.79_warc_CC-MAIN-20211206194128-20211206224128-00242.warc.gz\"}"}
https://answers.opencv.org/question/11006/what-is-the-carttopolar-angle-direction-in-opencv/	[ "# What is the cartToPolar angle direction in OpenCV?\n\nHello!\n\nI would like to ask, where exactly the angle computed by carToPolar function points? For example, if I have a 3x3 MAT matrix, and the angle at point [1,1] is 45°, it will point to [2,2] or [0,2] ?\n\nThank you\n\nedit retag close merge delete\n\nSort by » oldest newest most voted\n\nBasically it the output is calculated clockwise starting from the x - axis. This is dependant on how the axes in OpenCV are actually visualized. So take a look at the following picture.", null, "This means that if a point [1,1] has a gradient angle of 45°, it will point towards the pixel that is at 45° looking from the coordinate center. In this case it will be [2,2]. An angle of 90° would point to [2,0].\n\nmore\n\nIf it helped you, accept the answer, so that this topic appears solved. You are welcome!\n\nOfficial site\n\nGitHub\n\nWiki\n\nDocumentation" ]	[ null, "https://answers.opencv.org/upfiles/13654239544240452.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9178984,"math_prob":0.9469635,"size":264,"snap":"2020-45-2020-50","text_gpt3_token_len":81,"char_repetition_ratio":0.13846155,"word_repetition_ratio":0.0,"special_character_ratio":0.29166666,"punctuation_ratio":0.17460318,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98901725,"pos_list":[0,1,2],"im_url_duplicate_count":[null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-21T14:07:43Z\",\"WARC-Record-ID\":\"<urn:uuid:37519661-a727-4de0-aefd-f8dbf4445e0c>\",\"Content-Length\":\"55459\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:44d0de9d-f06f-40e8-9ca0-5df3748811bf>\",\"WARC-Concurrent-To\":\"<urn:uuid:68fabfe6-0a66-42bb-8264-dcdd9709f404>\",\"WARC-IP-Address\":\"5.9.49.245\",\"WARC-Target-URI\":\"https://answers.opencv.org/question/11006/what-is-the-carttopolar-angle-direction-in-opencv/\",\"WARC-Payload-Digest\":\"sha1:BSS4FREOJFYJJ7QNDU7AABBQKQ4S6E3J\",\"WARC-Block-Digest\":\"sha1:D2CP6YMKJQNEDA6STKCNJOR3GJDN4XPX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107876500.43_warc_CC-MAIN-20201021122208-20201021152208-00689.warc.gz\"}"}
https://www.numbers.education/33412.html	[ "Is 33412 a prime number? What are the divisors of 33412?\n\n## Parity of 33 412\n\n33 412 is an even number, because it is evenly divisible by 2: 33 412 / 2 = 16 706.\n\nFind out more:\n\n## Is 33 412 a perfect square number?\n\nA number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. Here, the square root of 33 412 is about 182.789.\n\nThus, the square root of 33 412 is not an integer, and therefore 33 412 is not a square number.\n\n## What is the square number of 33 412?\n\nThe square of a number (here 33 412) is the result of the product of this number (33 412) by itself (i.e., 33 412 × 33 412); the square of 33 412 is sometimes called \"raising 33 412 to the power 2\", or \"33 412 squared\".\n\nThe square of 33 412 is 1 116 361 744 because 33 412 × 33 412 = 33 4122 = 1 116 361 744.\n\nAs a consequence, 33 412 is the square root of 1 116 361 744.\n\n## Number of digits of 33 412\n\n33 412 is a number with 5 digits.\n\n## What are the multiples of 33 412?\n\nThe multiples of 33 412 are all integers evenly divisible by 33 412, that is all numbers such that the remainder of the division by 33 412 is zero. There are infinitely many multiples of 33 412. The smallest multiples of 33 412 are:\n\n## Numbers near 33 412\n\n### Nearest numbers from 33 412\n\nFind out whether some integer is a prime number" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85712254,"math_prob":0.9996804,"size":355,"snap":"2020-45-2020-50","text_gpt3_token_len":106,"char_repetition_ratio":0.19373219,"word_repetition_ratio":0.0,"special_character_ratio":0.36056337,"punctuation_ratio":0.1590909,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99884623,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-20T06:08:26Z\",\"WARC-Record-ID\":\"<urn:uuid:182a08bc-9f5b-4f63-93df-ed6c7a658bf9>\",\"Content-Length\":\"18905\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cc882055-7d09-4cfb-a783-1ff3241e7d5c>\",\"WARC-Concurrent-To\":\"<urn:uuid:dc081121-4ec5-4025-b2d5-d6bcb3c0f8e5>\",\"WARC-IP-Address\":\"213.186.33.19\",\"WARC-Target-URI\":\"https://www.numbers.education/33412.html\",\"WARC-Payload-Digest\":\"sha1:HBIGZB2KHQLYCAQW3AVJMYNB5JI3TNT7\",\"WARC-Block-Digest\":\"sha1:UGOVJJHMXI2Q7ZHKUX4DW4WTAVP6LCC7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107869933.16_warc_CC-MAIN-20201020050920-20201020080920-00182.warc.gz\"}"}
https://metanumbers.com/6993	[ "6993 (number)\n\n6,993 (six thousand nine hundred ninety-three) is an odd four-digits composite number following 6992 and preceding 6994. In scientific notation, it is written as 6.993 × 103. The sum of its digits is 27. It has a total of 5 prime factors and 16 positive divisors. There are 3,888 positive integers (up to 6993) that are relatively prime to 6993.\n\nBasic properties\n\n• Is Prime? No\n• Number parity Odd\n• Number length 4\n• Sum of Digits 27\n• Digital Root 9\n\nName\n\nShort name 6 thousand 993 six thousand nine hundred ninety-three\n\nNotation\n\nScientific notation 6.993 × 103 6.993 × 103\n\nPrime Factorization of 6993\n\nPrime Factorization 33 × 7 × 37\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 5 Total number of prime factors rad(n) 777 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 6,993 is 33 × 7 × 37. Since it has a total of 5 prime factors, 6,993 is a composite number.\n\nDivisors of 6993\n\n1, 3, 7, 9, 21, 27, 37, 63, 111, 189, 259, 333, 777, 999, 2331, 6993\n\n16 divisors\n\n Even divisors 0 16 8 8\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 12160 Sum of all the positive divisors of n s(n) 5167 Sum of the proper positive divisors of n A(n) 760 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 83.6242 Returns the nth root of the product of n divisors H(n) 9.20132 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 6,993 can be divided by 16 positive divisors (out of which 0 are even, and 16 are odd). The sum of these divisors (counting 6,993) is 12,160, the average is 760.\n\nOther Arithmetic Functions (n = 6993)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 3888 Total number of positive integers not greater than n that are coprime to n λ(n) 36 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 900 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares\n\nThere are 3,888 positive integers (less than 6,993) that are coprime with 6,993. And there are approximately 900 prime numbers less than or equal to 6,993.\n\nDivisibility of 6993\n\n m n mod m 2 3 4 5 6 7 8 9 1 0 1 3 3 0 1 0\n\nThe number 6,993 is divisible by 3, 7 and 9.\n\n• Arithmetic\n• Deficient\n\n• Polite\n\nBase conversion (6993)\n\nBase System Value\n2 Binary 1101101010001\n3 Ternary 100121000\n4 Quaternary 1231101\n5 Quinary 210433\n6 Senary 52213\n8 Octal 15521\n10 Decimal 6993\n12 Duodecimal 4069\n20 Vigesimal h9d\n36 Base36 5e9\n\nBasic calculations (n = 6993)\n\nMultiplication\n\nn×y\n n×2 13986 20979 27972 34965\n\nDivision\n\nn÷y\n n÷2 3496.5 2331 1748.25 1398.6\n\nExponentiation\n\nny\n n2 48902049 341972028657 2391410396398401 16723132902014018193\n\nNth Root\n\ny√n\n 2√n 83.6242 19.1229 9.14462 5.87398\n\n6993 as geometric shapes\n\nCircle\n\n Diameter 13986 43938.3 1.5363e+08\n\nSphere\n\n Volume 1.43245e+12 6.14521e+08 43938.3\n\nSquare\n\nLength = n\n Perimeter 27972 4.8902e+07 9889.6\n\nCube\n\nLength = n\n Surface area 2.93412e+08 3.41972e+11 12112.2\n\nEquilateral Triangle\n\nLength = n\n Perimeter 20979 2.11752e+07 6056.12\n\nTriangular Pyramid\n\nLength = n\n Surface area 8.47008e+07 4.03018e+10 5709.76\n\nCryptographic Hash Functions\n\nmd5 383beaea4aa57dd8202dbff464fee3af 73a5135c1a9bafa20c64829a43d6103c204d8e86 87b99ff27b5d7f00062f4fe7c4735d6e7f16af66ab064bf1643210187c7f487a 6344eb14d9a7d739f80f90d46758d0d831a33c91e98f3f051c6175f0c5be16b2bd5e89e3558f5dbc57747a899641b04f28cc15b5f7aad437735ec2434312b9fb 94e3308296d1dcfa45220740da9d54ebc0b2da20" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.63077694,"math_prob":0.98980856,"size":4416,"snap":"2022-05-2022-21","text_gpt3_token_len":1583,"char_repetition_ratio":0.117860384,"word_repetition_ratio":0.025411062,"special_character_ratio":0.44701087,"punctuation_ratio":0.07542263,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9967778,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-23T21:30:09Z\",\"WARC-Record-ID\":\"<urn:uuid:33567f71-bfa3-4092-b7e7-09d3748d122a>\",\"Content-Length\":\"39275\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:17f07f23-be2e-4974-bc2c-079fed38b949>\",\"WARC-Concurrent-To\":\"<urn:uuid:a5dc3b11-f241-4794-b1b1-c904840787b4>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/6993\",\"WARC-Payload-Digest\":\"sha1:P4FVBMJTSAUSF6SH57PH4EDROAIYLY4Y\",\"WARC-Block-Digest\":\"sha1:4ZWCBEVMQCAMD6JJ7K2UVU4QEDFKGQFU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320304309.59_warc_CC-MAIN-20220123202547-20220123232547-00436.warc.gz\"}"}
https://www.mdpi.com/1424-8220/17/4/904/htm	[ "Next Article in Journal\nRobot-Beacon Distributed Range-Only SLAM for Resource-Constrained Operation\nPrevious Article in Journal\nConnectivity Restoration in Wireless Sensor Networks via Space Network Coding\nOpen AccessArticle\n\n# Chemical Source Localization Fusing Concentration Information in the Presence of Chemical Background Noise\n\nby", null, "Víctor Pomareda 1,2,", null, "Rudys Magrans 1,", null, "Juan M. Jiménez-Soto 1,", null, "Dani Martínez 3,", null, "Marcel Tresánchez 3", null, ",", null, "Javier Burgués 1,2", null, ",", null, "Jordi Palacín 3", null, "and", null, "Santiago Marco 1,2,,†", null, "1\nSignal and Information Processing for Sensing Systems, Institute for Bioengineering of Catalonia, Baldiri Reixac 4-8, Barcelona 08028, Spain\n2\nDepartment of Engineering: Electronics, Universitat de Barcelona, Martí i Franqués 1, Barcelona 08028, Spain\n3\nDepartment of Computer Science and Industrial Engineering, Universitat de Lleida, Jaume II 69, Lleida 25001, Spain\n\nAuthor to whom correspondence should be addressed.\nThis paper is an extended version of our paper: Pomareda, V.; Marco, S. Chemical Plume Source Localization with Multiple Mobile Sensors using Bayesian Inference under Background Signals. In Proceedings of the International Symposium on Olfaction and Electronic Nose, New York, NY, USA, 2–5 May 2011.\nSensors 2017, 17(4), 904; https://doi.org/10.3390/s17040904\nReceived: 14 February 2017 / Revised: 6 April 2017 / Accepted: 11 April 2017 / Published: 20 April 2017\n\n## Abstract\n\nWe present the estimation of a likelihood map for the location of the source of a chemical plume dispersed under atmospheric turbulence under uniform wind conditions. The main contribution of this work is to extend previous proposals based on Bayesian inference with binary detections to the use of concentration information while at the same time being robust against the presence of background chemical noise. For that, the algorithm builds a background model with robust statistics measurements to assess the posterior probability that a given chemical concentration reading comes from the background or from a source emitting at a distance with a specific release rate. In addition, our algorithm allows multiple mobile gas sensors to be used. Ten realistic simulations and ten real data experiments are used for evaluation purposes. For the simulations, we have supposed that sensors are mounted on cars which do not have among its main tasks navigating toward the source. To collect the real dataset, a special arena with induced wind is built, and an autonomous vehicle equipped with several sensors, including a photo ionization detector (PID) for sensing chemical concentration, is used. Simulation results show that our algorithm, provides a better estimation of the source location even for a low background level that benefits the performance of binary version. The improvement is clear for the synthetic data while for real data the estimation is only slightly better, probably because our exploration arena is not able to provide uniform wind conditions. Finally, an estimation of the computational cost of the algorithmic proposal is presented.\n\n## 1. Introduction\n\nLocalization of chemical sources in urban scenarios (large cities) is a major challenge for intelligence and police authorities. In the clandestine production phase of illicit substances like explosives, but also drugs, significant levels of precursors are spread in the atmosphere. Such suspicious compounds could be reported by a system of mobile sensors and could be located using localization algorithms, providing complementary information to the authorities for intervening at an early stage.\nSeveral strategies for source localization have been proposed in the literature . These strategies have been integrated into robot systems with tracking abilities. Many of these tracking strategies have been inspired by bacteria or animal behavior using olfaction for foraging or mating: lobster, blue crabs, ants and moths provide behavioral models for odor tracking [2,3]. One of the simplest strategies consists in seeking changes in local concentration within an odor plume assuming a smooth chemical gradient in a diffusion dominated flow. However, this approach, called chemotaxis , is not useful in a realistic environment where fluid flow is dominated by turbulence, which can be caused by forced ventilation, temperature gradients or the presence of obstacles. Under these conditions, on the order of ten min are required to determine the time-averaged concentration with sufficient accuracy to perceive the gradient concentration . In consequence, the exploration of the area of interest becomes too slow. Some other strategies additionally exploit both fluid velocity information and chemical concentration (anemotaxis [6,7]). More recent proposals based on information theory, like infotaxis are based in binary detections and, information plays a role similar to that of concentration in chemotaxis. Odor patches are expected to be found only intermittently in the medium, and then information is sparse. In some cases, methods aim to estimate the gas distribution through analytical Gaussian models [9,10,11], others are focused on to create a plume mapping via hidden Markov methods , whereas in Farrell et al. a strategy for chemical plume tracing and source location declaration is presented.\nNavigation experiments aiming to find chemical sources are strongly limited by the limit of detection and selectivity of the low cost chemical sensors and even medium-priced detectors (e.g., ion mobility spectrometers). Thus, the rapid decay in the chemical concentration with increasing distance from the source can be a critical issue. Poor limits of detection result in a reduced area where the plume can effectively be detected. This is especially important in applications where the search zone has an area of several square kilometers. In such situations, it becomes very important to set the detection thresholds very close to the noise level, but this would results in a high number of false alarms and most localization algorithms would fail catastrophically. To the best of our knowledge, none of the published methods have addressed this problem.\nAdditionally, in any real scenario, there could be background levels of a multitude of chemicals caused by environmental pollution. Because of limited selectivity, there will be substances which will produce interference in the detector reading, hindering the detection and localization tasks. The presence of combination of detector electronic noise and mainly interfering chemical agents result in variable background readings that may change with time and with the position of the detector. These shifts in background levels hamper the selection of an optimum threshold that is usually considered to be constant all along the area under exploration. As far as we know, this problem has not been previously tackled in the literature.\nTo address these two issues (threshold close to the limit of detection and presence of background levels) probabilistic approaches like plume mapping Bayesian methods appear to be a good choice. Pang and Farrell published a source-likelihood mapping approach based on Bayesian inference in 2006 . The main idea behind the algorithm consists in implementing a stochastic approach for plume modeling and in estimating the most likely source position considering the sequence of detection/non-detection events and fluid flow measurements along the robot‘s trajectory. Pang’s algorithm has been tested successfully with data previously collected using an autonomous underwater vehicle .\nHowever, this algorithm uses binary detection events, and no chemical concentration information is used to build the probability map, since it only considers the concentrations above a certain threshold as detection or non-detection events. Moreover, after setting the threshold level, the approach assumes that the rate of false alarms is very low. In a real scenario where background signals are present, this is only achieved when the threshold is set at a high level. However, this option seriously reduces the maximum plume detection distance. Therefore, there is a trade-off; on the one hand, the threshold needs to be set low enough (close to the sensor detection limit) if chemicals from the source are to be detected at large distances; on the other hand, the threshold needs to be high enough to prevent false alarms. So, how to set the threshold level becomes a critical issue in real environments using existing approaches, especially when the background intensity is non-uniform in the explored area.\nFinally, in order to speed up the area exploration it is important that the algorithm can be extended to work with multiple robots. Recently, Kang and Li have presented a novel plume tracking algorithm via multiple autonomous robots by using a leader-follower strategy, demonstrating its superiority versus a single robot algorithm, in terms of both the computational cost and the accuracy in source location. Meng et al. have also studied the multi-robot problem for plume tracking in fluctuating airflow environments, showing the efficiency and robustness of the adapted ant colony optimization (ACO) algorithm over the traditional ACO algorithm. Meng remarks the importance of a proper number of robots and a well-defined cooperation mechanism, although it is not strictly necessary to track the plume to obtain a good estimation of the source location. The most likely source position is estimated during the robot's mission for arbitrary trajectories by recursively building a probability map using Bayesian inference. The problem of background estimation over the exploration area can be considered as the problem of scalar field mapping. Algorithmic approaches using mobile sensor networks have been already proposed, however they assume that the agents have communication capabilities so that their exploration paths are optimized after data fusion [16,17,18]. However, in our scenario we expect that the sensors can be mounted on vehicles that serve other tasks beyond chemical source localization. In this sense, we consider cases where there is no feedback between the chemical sensing and the agent’s trajectories.\nOur main motivation for the present work is to extend the Bayesian plume source localization algorithm, previously described by Pang and Farrell , using the chemical concentration (instead of binary detections) and assessing its performance in simulated and real environments, where background signals may arise. Thus, Pang’s algorithm is reformulated for use with continuous analog concentration readings instead of binary detections. Moreover, the algorithm is extended to work with multiple mobile sensors. This new approach requires a probabilistic model for the background and for the plume which are described in the following sections.\nThe present algorithm was initially developed for applications to Home Security (European project LOTUS: Localization of Threat Substances in Urban Society FP7-SEC-217925). In the considered scenario, police cars equipped with GPS and gas chemical sensors (eventually other fast analyzers like ion mobility spectrometers can be used) carry out their routine patrols while sending the sensors’ readings to a central station. Instead of moving towards the chemical source, the patrol vehicles would maintain their normal patrol routes while a centralized system is continuously analyzing the acquired signals seeking suspicious activity.\nThe paper is organized as follows: Section 2.1 and Section 2.2 and shows the basics of the algorithm and the plume and background model. Section 2.3 and Section 2.4 describes the synthetic and real scenarios for the test and Section 3 reports the results and the analysis.\n\n## 2. Materials and Methods\n\n#### 2.1. Stochastic Models for Plume and Background\n\nOur proposal requires making some assumptions about the dispersion of the plume (the stochastic model in Section 2.1.1) and a background model (Section 2.1.2). Given an instantaneously measured concentration c, it is assumed that there are two additive contributions: one due to the background ($c b$) and one due to the plume ($c p$), thus:\n$c = c b + c p$\nSince we consider all these concentrations as random variables, the probability density function (PDF) of the measured concentration will be the convolution of the PDFs of the two additive terms. In the next sections, we will describe how we model the probability density of both terms.\n\n#### 2.1.1. Stochastic Model for Chemical Concentration Measurements\n\nThe basis of our stochastic model for the chemical plume is the analytical Gaussian plume model (GPM) [21,22]. Time-averaged plume concentration follows a Gaussian distribution lengthways to the flow direction if the time average is at least 10 min, as demonstrated in previous works [23,24,25]. This model has been widely used for its simplicity and is appropriate when dispersion is governed by atmospheric turbulence under uniform wind conditions. Atmospheric turbulence is determined by the stability of the atmosphere and the height above the surface layer . The basic expression for the GPM under a continuous release is:\n$c ¯ ( x , y , z ) = q 2 π U a σ y σ z exp ( − y 2 2 σ y 2 ) exp ( − ( h − z ) 2 2 σ z 2 )$\nwhere $c ¯$ is the time averaged concentration (in g/m3) in a location with coordinates $x$ (downwind), $y$ (crosswind) and $z$ (vertical); $q$ is the continuous source release rate or source strength (in g/s); $U a$ is the mean wind speed in the downwind direction (in m/s); $h$ is the plume height (in m); and $σ y$, $σ z$ are the dispersion coefficients (in m), in crosswind and vertical direction respectively, modeled as: $σ y = a · x b$, $σ z = c · x d$, where $a$, $b$, $c$ and $d$ are parameters obtained from a table and their values depend on the atmospheric conditions which can be organized in six levels (from A-very unstable to F-very stable) according to the Pasquill’s stability classes . In Equation (2), the resulting concentration distribution is due to the transport of chemicals by advection (due to the mean wind speed), due to concentration gradients within the plume width (lateral dispersion due to diffusion, but also turbulent mixing) and due to plume meandering. The decay of mean concentration is exponential, thus concentration levels below the sensor detection limit are very quickly achieved. This issue makes the setting of the threshold level critical; especially, if the source should be detected far from the release point.\nThe GPM considers the time-averaged characteristics of a plume dispersed in a turbulent flow, but the sensors will be responding to the instantaneous plume characteristics (here we assume that the chemical sensor dynamics are much faster than 10 min, which is usually the case). For short time-scale studies, the chemical puff movement can be modeled as a random walk (because of transversal velocity fluctuation) overlapped on the downflow advection (because of mean velocity) . However, we propose an alternative approach, for which an additional component needs to be added to the GPM to model the unpredictable and random fluctuations in concentration due to turbulent stirring and plume meandering. Yee et al. [28,29,30] have carried out empirical studies on plume statistics in urban areas using scale fluid models in a variety of plume conditions and urban geometries. Their results prove that instantaneous concentration fluctuations fit very well the clipped-gamma PDF over a very wide range of atmospheric conditions and at several receptor positions . The clipped-gamma distribution (CGD) is defined in terms of four parameters $γ$, $k $, $s$ and $λ$ as :\n$f ( c ^ ) = ( c ^ + λ s ) k − 1 exp ( − ( c ^ + λ ) / s ) s Γ ( k * ) + ( 1 − γ ) δ ( c ^ )$\nwhere $c ^$ is the instantaneous concentration, $Γ ( c ^ )$ is the gamma function, $δ ( c ^ )$ is the Dirac delta function and $λ$, $k $ and $s$ are the shift, shape and scale parameters, respectively. The total PDF is composed of a mixed fluid part due to in-plume mixing of eddies containing the target substance (the first term on right-hand side), and an unmixed ambient fluid part (the second term on right-hand side) caused by plume meandering which produces intermittent periods of zero concentration for a fraction of time (1 − γ), being γ the intermittency factor. Although Equation (3) is specified in terms of four parameters, it can be uniquely modeled by the mean (M) and the standard deviation (SD) of a series of readings . There is a simple relation among M, SD, and the plume intermittency; thus, γ is determined as:\n$γ = γ ( k , s , λ ) = Γ ( k * ; λ / s ) Γ ( k * ) = min ( 1 , 3 K int 2 + 1 )$\nwhere $Γ ( v ; c )$ corresponds to the incomplete gamma function and $K i n t$ is the ratio between the SD and the mean of the series of readings at a fixed position. For a specified value of $K i n t$, the parameters $λ$, $k $ and $s$ can be obtained solving a set of transcendental equations, thus making Equation (3) totally defined. The details for computing these parameters are given in .\nWhile the mean concentration decreases rapidly in the downwind direction (see Equation (2)), the magnitude of the fluctuations decreases even more rapidly. As described by Webster , $K i n t$ is estimated to decrease as $x θ$ (where $x$ is the distance from the source and $θ < 0$), being roughly $K 0$ times the mean time-averaged concentration at a certain distance $x 0$ from the source. Therefore, this parameter can be modeled as:\n$K int = K 0 · ( x x 0 ) θ$\nTo model the instantaneous concentrations due to a chemical plume at a certain distance from the source, the clipped-gamma distribution is used (Equation (3)). The mean value M of the series of concentrations due to the plume is related to the GPM (Equation (2)) and the SD can be modeled as:\n$σ S D = K 0 · ( x x 0 ) θ M$\nSince the PDF depends on the mean and the SD, and these parameters depend on the distance from the source, the PDF of the instantaneous readings contains information about the relative position between the sensor and the source. Since concentrations fluctuations (intermittencies) decrease faster than the mean value with the downwind distance to the source, the plume becomes homogeneous faster than the mean concentration dilutes [28,31].\nAdditionally, previous literature has characterized the power spectral density (PSD) of concentration readings within a dispersing plume. We will follow the model described by Jones et al. .\n\n#### 2.1.2. Stochastic Background Model\n\nIn real scenarios such as residential areas or urban environments, pollution or interfering substances are expected to be found. This problem becomes even more serious due to the common use of partially selective sensors such as Metal Oxide Sensors (MOXs), or Photoionization detectors (PIDs).\nMoreover, meteorological conditions (wind conditions and atmospheric stability) could change within a timescale of several hours, or there might be changes in polluting emissions due to day-night cycles of human activity (including motor vehicles or factories) [32,33,34]. Because of these reasons the background can be considered to change slowly.\nAs it has been previously described , the dispersion of a chemical plume in a turbulent flow shows a highly intermittent nature with background or zero concentration for long periods of time separated by high peaks of concentration. This behavior will help us to estimate the background model using robust measures that reject the plume peaks.\nFor the implementation of the proposed algorithm the exploration area is divided into a uniform grid of rectangular cells. The algorithm estimates the background probability distribution at each cell of the grid. The background is modelled as a Gamma PDF where the standard deviation is smaller than the mean. In those conditions the Gamma function resembles a Gaussian PDF but defined only for positive values of the random variable concentration. We will consider that the PSD of the background is bandlimited white noise.\n\n#### 2.2.1. Bayesian Estimation of the Likelihood Map for Chemical Source Presence\n\nA summary of the notation used for the description of the algorithm can be found in Table A1. The search area contains $N c$ rectangular cells of size $L x · L y$ each, where $L x$ and $L y$ are the cell lengths in the $x$ and $y$ directions of the grid map, respectively. The size of this cells is a trade-off between the spatial resolution of the algorithm and the computational cost.\nLet $0 ≤ α i ' ≤ 1$ represent the probability that the chemical source is in cell $i$. It is assumed that the search area contains exactly one source, hence $∑ i = 1 N c α i ' = 1$. It is assumed that the prior information on the potential existence of a chemical source is given by previous intelligence research. We do not consider in this formalism the case where there is uncertainty on the presence or not of a chemical source.\nInitially (at $t = t 0$), if no information about the source location is available, all cells are initialized to be equally likely to contain the chemical source: $α i ' ( t 0 ) = ( 1 N c )$; $∀ i ∈ [ 1 , N c ]$.\nGiven that we measure a concentration at time $t k$ in cell $j$, i.e., $c j$ refers to a concentration reading in cell $j$, we can calculate the source probability map based on this single reading. A Bayesian approach is used to determine if the main contributor to the measurement is the background or the presence of a plume patch. The posterior probability for the presence of a plume, given the measurement, is calculated using Bayes’ theorem:\n$P ( A \| c j ) = P ( c j \| A ) ⋅ P ( A ) P ( c j )$\nwhere $A$ corresponds to the event “the concentration reading was caused by an emitting source upstream (plus a background level)”. To infer where the source is located, the posterior probability of a source emitting (left-hand term in Equation (7)) can be further decomposed into the probability of that source being located at each cell. Again, using the Bayes’ Theorem:\n$P ( A i \| c j ) = P ( c j \| A i ) · P ( A i ) P ( c j \| A i ) · P ( A i ) + P ( c j \| A ¯ i ) · [ 1 − P ( A i ) ]$\nwhere $A ¯$ means “the concentration reading was caused only by background levels and not by an emitting source”, subindex $i$ refers to a source located in cell $i$ and subindex $j$ refers to the current cell $j$ where the measurement was taken.\nSince we consider that a background of interfering substances is always present, in the absence of plume only the background component $( c b )$ is present; in the presence of plume, both components ($c b$ and $c p$) are present and the concentrations are modeled by the convolution of the plume PDF and the background PDF. However, even in the presence of the plume $c p$ may be zero due to plume intermittency. The PDF of the concentration component due to the source is modeled using the GPM for the means, and SD from Equation (6) considering the relative position of cell $i$ (potential source location) and cell $j$ (location of the sensor).\nTaking these considerations into account, the previous probabilities have the following interpretation: $P ( A i )$ is the prior probability of the presence of a source at the cell $i$, $P ( c j \| A i )$ is the probability that the measurement at cell $j$ is due to addition of the background at cell $j$ and a plume due to a source at cell $i$, and it is obtained by the convolution between the PDF of the plume and the PDF of the background at cell $j$. $P ( c j \| A ¯ i )$ is the probability that the measurement of chemicals at cell $j$ is due to the current background at cell $j$, and it is obtained from the PDF of the background at cell $j$. We define $S i j ( t k ) ≡ P ( A i \| c j )$ as the probability of having a source in cell $i$ given that a certain amount of chemical was measured at cell $j$ at time $t k$.\nHowever, in this approach where we consider each cell independently, $∑ i = 1 N c S i j ( t k ) = 1$ is not guaranteed. Therefore, the result is normalized to ensure the total probability is 1 when individual cell probabilities are added up. The nomenclature is henceforth the same as that used in :\n$β i j ( t k ) = S i j ( t k ) ∑ i = 1 N c [ S i j ( t k ) ]$\nNow $∑ i = 1 N c β i j ( t k ) = 1$ is guaranteed and $β i j ( t k )$ calculated over all cells ($i = 1 , ... , N c$) gives the source probability map at time $t k$ based on a single measured concentration at cell.\nUsing Bayesian theory and following the same procedure described in , each new measurement can be incorporated recursively to update the source probability map.\n$α ′ i j ( t k ) = P ( A i \| B ( t k ) )$ is defined as the probability of cell $i$ containing the source, given the sequence of concentrations $B ( t k )$ along the trajectory of the mobile sensors up to time $t k$. Defining $P ( A i \| D j ( t k ) ) = β i j ( t k )$, where $D j ( t k )$ is the measured concentration at time $t k$; $P ( A i \| B ( t k ) )$ is computed from $B ( t k − 1 )$ and $D j ( t k )$, which are supposed to be independent events, finally obtaining:\n$α ' i j ( t k ) = N c · α ' i j ( t k − 1 ) · β i j ( t k )$\nwhere, if $α ′ i j ( t k )$ is computed over all cells ($i = 1 , ... , N c$), an updated source probability map is obtained recursively.\nIndependently of the number of mobile sensors, the only information required by the algorithm is: the position where the measurement was obtained, the chemical concentration reading and the fluid flow measurement. The extension to multiple mobile sensors is as follows. We just build an integrated sequence of readings by addressing in a circular manner the set of mobile sensors. Then we send this sequence of measurements to the original algorithm. In other words, we do not fuse a posteriori maps built from individual sensors. We fuse the sequence of measurements at the input of the estimation algorithm.\n\n#### 2.2.2. Background Estimation\n\nBackground stochastic model is built from the robust estimation of the means and the dispersion to reject the effect of the intermittencies. We use the median and the median absolute deviation (MAD) to build the background probability density model . These parameters are estimated over a buffer of the last 50 measurements. These measurements can be spread out in several cells depending on the speed of the robot (typically less than 10). This is not a problem if the spatial variation of the background is sufficiently smooth.\nTo allow for the model to adapt to a slowly changing background the parameters of the model (mean and standard deviation) are filtered with an exponential moving average. This filter weights the current estimation of the background with the old one depending on the time distance. In this way, the system is able to forget the old values and adapt to new ones.\n\n#### 2.3.1. Synthetic Scenario Description and Simulation\n\nFor realistic simulations, the scenario envisioned considers atmospheric plumes in an urban area encompassing hundreds of thousands of square meters containing only one chemical source. The sensors located in vehicles can transmit their current position in the grid (e.g., using a GPS sensor) together with the chemical sensor readings, and they will explore the search area by moving across the cells performing random exploration. It is assumed that each vehicle mounts a single sensor. It is considered that the main task of the vehicles is not that of tracking the plume; but patrolling a certain area and simultaneously updating a probability map for the source location using the available information.\nThe exploration arena in the synthetic case is as follows. In order to test both algorithms, a synthetic scenario (grid size of 1 km × 1 km) is generated. The area is divided into cells of size 100 m × 100 m. A square sub-grid of lanes (100 m separation between lanes) is interlaced over the main grid (Figure 1). The sensors will randomly explore the area over this last sub-grid of lanes which simulate streets within an urban environment (Manhattan style). Using this configuration, the movement of the sensors is constrained to certain lanes over the main grid. A clandestine laboratory for home-made explosive production emits explosive precursors. In this scenario, we consider that a chemical source emitting with a source strength $q = 2.90$ g/s, is placed in the grid with the source at the position (440 m, 440 m), which corresponds to coordinates (5, 5) on the rectangular grid. We consider that the main dispersed substance is acetone (molecular weight: 58.08 g/mol) at one atmosphere pressure and 25 °C. This Gaussian plume distribution has been generated from Equation (2) with the plume being dispersed in a 2D plane at the same height as the sensors ($z = h = 2$ m). It is assumed that there is no deposition of the substance on surfaces. In the simulations, the wind field is constant with the wind speed at $U a = 2.5$ m/s and the wind direction at 45°. The dispersion coefficients $σ y$ and $σ z$ depend on wind conditions and atmospheric stability which has been set to neutral (‘D‘ on the Pasquill-Gifford scale ). Moreover, a mean background distribution is deployed over the area with a different mean level in each cell and with SD equal to 60% of the mean value in all cells (based on our own recorded data using a PID sensor measuring in Barcelona outdoors over a period of several hours).\nFive mobile sensors with a constant velocity of 15 km/h sense the area continuously. The sampling period of the sensors is set to 3 s, the detection limit to 0.1 ppm and the sensor resolution to 0.01 ppm, which are realistic values for PID technologies. We assume that the response time of the sensor is much faster than the typical 10 min time-average considered in the Gaussian Plume Model (Equation (2)). The total simulation time was set to 300 min.\nThe mean background level is different in each cell, but is stationary over time. Series of concentration fluctuations are generated in each cell considering the wind field created, the atmospheric conditions and the background. The stochastic model of the plume concentrations has been already described in Section 2.1.1. However, here we give some additional technical details concerning the practical implementation.\nSince we have defined the PDF and the PSD which will be used to model concentration fluctuations, we use the percentile transformation method (PTM) described by Papoulis to generate a series of concentration fluctuations with the desired PDF and PSD.\nSpecifically, the procedure to generate realistic plume readings consists of the following steps: (i) generate a time series of Gaussian white noise, (ii) filter the previous time series with the designed FIR filter to achieve the desired PSD, (iii) and apply the PTM. This method is based on the following expression:\n$c i = F c − 1 ( F z ( z i ) )$\nwhere $z i$ is a random sequence of Gaussian white noise having the desired PSD with cumulative distribution function (CDF) is the sequence of realistic readings in the cell with CDF $F c ( c )$. This CDF corresponds to the clipped-Gamma CDF (Equation (3)):\n$F ( c ) ≡ Pr ( C ≤ c ) = ∫ 0 − c f ( c ' ) d c ' = 1 − Γ ( k ; ( c + λ ) / s ) Γ ( k * )$\nand $F c − 1$ is the inverse clipped-Gamma CDF. The clipped-Gamma PDF (and its CDF) depends on the distance to the source. Its two parameters, the mean and the SD, are obtained from Equation (2) and Equation (6), respectively. Subsequently, these parameters are used to compute $γ$, $k $, $s$ and $λ$, as explained in detail in .\nBackground concentrations are simulated as white noise with Gamma PDF. The background concentrations in each cell are added to the time-series of plume readings to obtain the final concentration readings at each cell.\nAn example of the concentration signals delivered by the sensors in this simulation scenario is shown in Figure 2. It can be observed that the SD of the fluctuations decreases faster than the mean concentration, making it difficult to differentiate between plume and background far from the source.\nIn the binary-based approach, it is assumed that the ratio of false alarms is very low, but the ratio of missed detection can potentially be very high, thus we define μ = 0.3 (70% missed detections). However, the value of this parameter was not defined in the original work .\nIn the concentration-based approach, the source strength $( q )$ should be initially guessed (this is typically done using previous information about the type of chemical source under investigation: clandestine lab, industrial toxic emissions, pollution, etc.). However, since it is almost impossible to guess this parameter with accuracy, we have studied the sensitivity of the algorithm to errors in this parameter. We scanned the guessed source strength two orders of magnitude around the central exact value. For the simulation studies presented in this work, the parameters used in Equation (6) are: $K 0 = 2.5$, and , taking into account previous studies .\n\n#### 2.3.2. Synthetic Test Case 1: Behavior of the Binary Detector Algorithm Depending on the Background Level and Detector Threshold\n\nThe first simulation aimed to characterize binary-based approach when changing the concentration threshold that fires the detector signal. In order to investigate the robustness of the method against chemical noise, two spatially uniform background levels were considered (mean values: 0.05 ppm and 0.45 ppm). In both cases the standard deviation was 1/5 of the mean value.\nA background level is low or high depending on the source strength; therefore, studying the case where the background is low is equivalent to saying that the source is potent, and studying the case where the background is high is equivalent to saying that the source is weak.\nTen random sensor trajectories were simulated for each threshold level in the binary detector (30 values in the linear range between 0.05 ppm and 3 ppm). For each trajectory, the probability at the real source location is assigned by the algorithm after 300 min of exploration time. We took as a figure of merit the mean probability averaging over all trajectories. This figure of merit will be plotted against the concentration threshold. For the binary case, this simulation will determine the optimum threshold for the detector given a certain chemical power source.\n\n#### 2.3.3. Synthetic Test Case 2: Accuracy in the Estimation of the Background Level and the Expected Position of the Chemical Source\n\nHere, both algorithms (binary and analog) are compared and share the same 10 random trajectories. In this second case, a non-uniform background is used. Results will show the probability evolution at real source location as the exploration time increases. Moreover, the probability maps provided by both approaches after 300 min of exploration time can be compared as well.\nTo assess the accuracy of a given probability map we compute the expected value of distance to the true source position:\n$E [ r − r 0 ] = ∑ i = 1 N c p ( r i ) · ( r i − r 0 )$\nwhere $r$ is the estimated position of the source, $r 0$ is the real source position and $p ( r i )$ represents the probability of the source being located in cell $i$.\nBased on this metric we calculate the root mean squared error ($R M S E p$) of the distance from the expected source position to the real source location:\n$R M S E p = ( E [ ‖ r − r 0 ‖ 2 ] ) = e r r x 2 + e r r y 2$\nAlternatively, for a richer description we can decompose the total error in both coordinates: $e r r x$ and $e r r y$. Alternative, though less informative, figures of merit are the Euclidean distance ($D$) between the maximum of the likelihood map and the real source position, and the probability at the real position.\nThe mean background map recovered for the concentration-based algorithm is quantitatively compared to the designed background distribution by using the root mean squared error ($R M S E b$) as a figure of merit:\n$R M S E b = ∑ i = 1 N c ( B i − B i ^ ) 2 N c$\nwhere $B i$ is the designed mean background level at cell $i$; and $B ^ i$ is the estimated mean background level. Results are obtained for two background distributions (maximum mean values: 0.05 ppm and 0.45 ppm). Both background concentration maps were the same except for a scale factor. The binary-based approach was tested using the optimum threshold level determined from case 1 and the concentration-based approach, using the exact source strength.\n\n#### 2.3.4. Synthetic Case 3: Influence of the Source Strength on the Concentration Based Algorithm\n\nFinally, the third simulation shows the influence of the source strength in the overall performance of the concentration-based approach. Since it is difficult to know the source strength in advance, the performance of the algorithm has been assessed assuming different source strengths across more than two orders of magnitude in the range between 0.1 g/s and 30 g/s (the real value in the scenario simulator being 2.90 g/s). Results are shown for the same background distributions as in case 2 and for 10 random trajectories.\n\n#### 2.4. Scenario, Chemical Source Emission and Autonomous Vehicle Description for Real Experiments\n\nFor real experiments, all measurements were performed within an exploration arena built with polystyrene panels. Dimensions of the tunnel were 5 m × 3.5 m × 1.8 m (length × width × height). One fan was introducing air into the room and, on the opposite wall, three fans extracted air from the room. All of them were installed at a height of 0.9 m above the floor and generated a highly a turbulent airflow. Fans used were helical-wall type with a diameter of 30 cm and a maximum speed of 1300 rpm.\nThe supply of volatile acetone was carried out employing two 10-mL syringes. These syringes were filled with liquid acetone and assembled on a syringe pump KDS-200 (KD Scientific, Holliston, MA, USA) which was programmed to deliver a controlled liquid flow of 150 µL/min during 60 min. Thus, the liquid acetone was falling at a constant rate over a plate heated to a temperature sufficient for the acetone to be evaporated immediately. The source emission of volatile acetone was placed at one side of the exploration arena just in front of the fan that introduced air in the tunnel. Algorithms for the generation of the probability maps operated over a grid of $N c = 70$ cells with dimensions 0.5 m × 0.5 m each one.\nInput data for the algorithms were registered by an autonomous vehicle which was constructed on a metal structure where different components were assembled, such as: two DC motors that allow mobility of the vehicle, an UTM-30LX USB laser range finder sensor (HOKUYO, Osaka, Japan) for vehicle auto-location and navigation; and a Windsonic RS232 anemometer (Gill Instruments, Lymington, England) to register the wind speed and direction. The autonomous vehicle was also equipped with a photo ionization detector (PID, ppbRAE 3000, RAE Systems, San Jose, CA, USA) for measurement of the concentration of volatile compounds. All these components were controlled by an onboard computer. The vehicle could operate autonomously sampling its relative position, the wind speed and direction, and the volatile compound concentration at one measurement every second approximately. The vehicle was programmed to move in a straight line until a wall is found, then the vehicle rotates a random angle, and a new straight trajectory is described until a new wall is found. The vehicle speed was set to 0.2 m/s. Images of the robot, the exploration room and the chemical source can be found in the supplementary materials.\n\n## 3. Results and Discussion\n\nIn this section, we will introduce the results for the three cases already described, as well as the results for the real experiments. Finally, we will discuss the computational cost of the presented algorithm. Data are presented as 50th [5th–95th] percentiles unless otherwise specified.\n\n#### 3.1.1. Synthetic Case 1\n\nThe overall performance of the binary-based approach as the concentration threshold is changed is studied in synthetic case 1. The results for two different background levels: the former with a mean value of 0.05 ppm (Figure 3a) and the latter with a mean value of 0.45 ppm (Figure 3b) are shown. The mean probability assigned by the algorithm at the real source location is depicted as a function of the concentration threshold. It can be seen in the figure that there is a different optimal concentration threshold depending on the background level. As expected, the optimum value is shifted to higher thresholds as the background level is increased. These optimal values have been found to be 0.15 ppm and 1.48 ppm, respectively. Relative to the background level, setting the threshold too low produces a high ratio of false alarms leading the algorithm to failure; on the other hand, setting the threshold too high could lead to an increase in false negatives with abnormal concentrations considered as non-detection events and a consequent worsening of the overall performance. This can be critical if the source to be detected is weak –this being equivalent to the case with a high background level. In Figure 3b, it is observed that setting the threshold either too low or too high causes the algorithm to fail, since the probability assigned to the real source location is below the equiprobable value of $1 / N c$ assigned initially to every cell.\nThe main problem using the binary-based approach is that the threshold needs to be set arbitrarily if no information about the background is available and this background can be different in various areas within the exploration zone. A priori, we do not know whether the threshold is too high or too low, but even if we knew this, the background could evolve over space and time and the threshold would need to be adjusted continuously. The concentration-based algorithm removes the necessity of any threshold because, instead of adjusting the threshold level, our approach builds a background model for each cell. This background model allows us to distinguish between the background and the plume without using any threshold and is updated recursively.\n\n#### 3.1.2. Synthetic Case 2\n\nWe will first analyze the results for the case of a weak interfering background having a maximum mean value of 0.05 ppm. Figure 4 shows the performance of the binary-based and concentration-based algorithms in source localization by considering the evolution of the figures of merit. Mean probability (averaged over all trajectories) increases throughout the exploration time (Figure 4a) in both approaches.\nThe Wilcoxon nonparametric rank test (not shown) was applied at each time step to test the null hypothesis of no difference between the median of both populations (i.e., those formed by the 10 probabilities values in each of the two algorithms). Approximately 15 min after the starting time, the increase in probabilities for the concentration-based approach become statistically significant (p < 0.05) in comparison with the probabilities for the binary-based approach which did not change until the end of exploration time. Figure 4b shows the mean Euclidean distance $D$ between the cell with the highest probability value and the real source location. For both approaches, mean distance decreases as the exploration time increases. In average, the binary-based approach converges to the real source location slower (125 min. approximately) than the concentration-based one (80 min approximately), and with lower probability value as it can be observed in Figure 4-top. Moreover, the errors in source localization algorithms in both X and Y directions of the grid (Figure 4-bottom) show that the concentration-based algorithm performs significantly better (p < 0.001), err = 213 [199 − 222] m and err = 225 [202 − 248] m for X and Y directions respectively, in comparison with the binary approach (281 [274 − 294] and 279 [271 − 294]), even though the binary-based algorithm has been tested with the optimum concentration threshold (0.15 ppm).\nFigure 5 shows the comparison between the mean probability maps (averaged over all trajectories) provided by both algorithms after 300 min of random exploration. The probability assigned to the real source location is higher using the concentration-based approach (P = 0.215 [0.156 − 0.261] in Figure 5b) as compared to the binary-based approach (P = 0.124 [0.102 − 0.131] in Figure 5a). Moreover, the probability is spread among a lower number of cells in the wind direction.\nThe estimated background distribution (not shown) by the concentration-based approach is similar to the designed one after 300 min of random exploration, $R M S E b$ = 0.010 [0.009 − 0.012] m. Moreover, it was observed that after 50 min of exploration time the $R M S E b$ values remained approximately constant.\nWith a high background level (0.45 ppm), which is the same as saying that the source strength is small compared to the background, Figure 6 shows similar results to those shown in Figure 4. However, now the differences between the performances of both approaches are higher. Mean probability at real source location (Figure 6a) for the concentration-based approach increases as exploration time increases, but more slowly compared to Figure 4a. In contrast to that observed in Figure 4a for the binary-based approach, the mean probability is lower and remains approximately constant throughout the exploration time. Here, the Wilcoxon test is also used to assess the statistical difference between both approaches. Differences observed become statistically significant (p < 0.05) after 15 min of exploration approximately. Moreover, it is observed in Figure 6b that the mean distance for the binary-based algorithm does not converge to the real source location. The errors accounted (Figure 6c) in both the X (err = 246 [239 − 265] m) and the Y (err = 260 [254 − 277] m) directions are higher than the case of a low background and confirm that the performance of the concentration-based approach is significantly more robust than the binary-based (337 [331 − 338] m and 331 [326 − 341] m for X and Y directions respectively) in the presence of a high background level, although the latter has been tested with its optimum concentration threshold.\nThe binary-based approach performs well under the assumption that no false alarms arise. This is shown in Figure 4 where, due to a low background level, the number of missed detections and false alarms arising from the background are small, and the binary-based algorithm performs slightly worse compared to the concentration-based approach if the optimum threshold can be identified. Nevertheless, such an assumption is far from the truth in a real scenario where pollution and interfering substances are expected to be found. Additionally, this can also be the case when the source to be detected is weak. This is shown in Figure 6 where, due to a high background level (or weak source), the number of false alarms is higher, thus forcing setting the threshold higher, which leads in turn to an increase in the number of missed detections and worsens dramatically the performance of the binary-based approach.\nFigure 7 shows the probability maps obtained with a high background level. It is seen that the threshold is very high, which minimizes the number of false alarms but increases the number of missed detections, thus the binary-based estimation is very uncertain at the source location (Figure 7a). Therefore, the probability is spread over the cells in the grid, decreasing the probability at real source location (P = 0.013 [0.010 − 0.014]). Figure 7b shows the robustness of our algorithm which minimizes false alarms and missed detections. The concentration-based algorithm tends to increase the probability at the real source location while the vehicles are performing random exploration. After 300 min of random exploration, the probability assigned to real source location was P = 0.083 [0.047 − 0.101]. In this approach, false alarms arising from the background can correctly be assigned lower weights in the probability calculations because the algorithm has created a background model per each cell and a dispersion model for the plume. Additionally, these models allow minimizing the number of missed detections.\nConcerning the estimation of the background distribution, the concentration-based approach was also able to recover it properly, $R M S E b$ = 0.089 [0.080 − 0.100] m. Like in the case of a low background, the $R M S E b$ remained approximately constant after 50 min, albeit, it was slightly higher.\n\n#### 3.1.3. Synthetic Case 3\n\nThe results concerning the sensitivity to the correct estimation of the chemical source strength are displayed in Figure 8. Results show a decrease in the overall performance as the assumed source strength deviates from the real value (2.90 g/s). It is observed (Figure 8a) that the selection of the source strength becomes more critical when the background level is low (or the source is potent compared to this background), but, if the source strength could be estimated with errors (to within at most a factor of 4), the concentration-based approach performs much better. When the source to be detected is weak (or the background is comparatively high, Figure 8b), the selection of the source strength is not so critical in the range studied. It can be observed that the concentration-based approach is more robust against false alarms and missed detections (even for source strength estimated with errors larger than two orders of magnitude), as compared to the binary-based approach which performs badly even setting its optimum threshold.\nIt is important to say that, although the binary-based algorithm works without assuming explicitly any source strength, setting an optimum threshold is only possible when collecting real measurements which implicitly contain information about the source strength and the background. However, this optimum threshold might be different depending on the explored cell. In the case of the concentration-based approach, the background is estimated by the algorithm and, if this background is low, the source strength needs to be known to within one order of magnitude. If the background is high, the algorithm behaves more robustly in the range studied as compared to the binary case except in the first case (the assumed source strength is very small).\n\n#### 3.2. Results for the Real Experiments\n\nFirst we will give some details concerning the wind distribution within the exploration arena. Figure 9 (top) shows a characterization of the behavior of the wind within the designed scenario for one of the experiments that was carried out. The median wind speed was 0.28 [0.08 − 1.12] m/s. Also, there was a predominant wind direction although with a scatter on the direction angle of approximately 60 degrees. Similar behavior of the wind was observed in all the experiments. The map of concentrations showed in Figure 9 (bottom) is consequence in part of such airflow generated.\nThe averaged probability maps (over the 10 experiments) are shown in Figure 10. The mean probability assigned by each algorithm to the real source location is practically identical, P = 0.045 [0.030 − 0.076] and P = 0.050 [0.028 − 0.084] for the concentration-based and for the binary-based respectively. However, the probability map for the binary-based algorithm (Figure 10a) exhibits more variance in the localization of the real source position, as it is observed in Figure 10a. For the concentration-based algorithm (Figure 10b), relatively stable higher values of probability around the real source location are obtained. The Wilcoxon rank test is also used to compare the errors obtained for each algorithm. In the X direction, the error for the concentration-based algorithm (2.14 [2.09 − 2.35]) m. is lower than for the binary-based one (2.30 [2.03 − 2.55]) m., but without statistical significance. In the Y direction, the error for the concentration-based (0.91 [0.85 − 0.95]) m. is also lower than for the binary-based (0.97 [0.86 − 1.03]) m. and in this case is statistically significant (see Figure 11).\nResults obtained here are in the same line as those obtained for the simulation experiments but the results are not as clear. In the designed scenario, it is likely that some of the assumptions of our algorithm (see Table A2) are not fully satisfied. The wind was not completely uniform in the room: it was much stronger near the source than at the opposite wall and there were some recirculation at the lateral walls. As a consequence the real distribution of the time-averaged concentration deviated from Gaussianity. We think it might be a reason to explain the slight differences observed when comparing both algorithms, in contrast to the strong differences observed in the performance of both algorithms for the simulation experiments.\n\n#### 3.3. Computatitonal Cost\n\nFinally, we have studied the computational cost of the algorithm. It is true that the computational cost is higher than the binary approach, mainly due to need of computing the convolution between the PDF of the background and the PDF of the plume for every cell. The cost is about three times the cost of the binary approach in our current implementation (without optimization). However, our studies have confirmed that the computational cost of each iteration in the algorithm increases very slowly with the number of cells. Moreover, the time to locate the source increases in a linear way approximately with the number of cells.\n\n## 4. Conclusions\n\nIn the present paper, modifications of a previously described (binary-based) algorithm have been introduced. The original algorithm can be easily extended to work with multiple mobile sensors. All the information from the mobile sensors can be integrated in the algorithm, whatever their positions are. The algorithm only needs to know in which cell the concentration readings were obtained, then a probability map will be recursively updated. Moreover, the mobile sensors do not need to solely perform plume tracking and might be used for other tasks.\nAdditionally, in a real scenario, pollution and some interfering substances may appear in the background, increasing the number of false alarms. Unlike the binary-based algorithm, which uses a threshold to assess whether a concentration is considered as a detection or non-detection event; our algorithm, based on continuous concentrations, builds a background model to assess whether a concentration comes from the background or from a source located further away. Simulation results show that our algorithm behaves much more robustly in the presence of false alarms and better estimates the real source location.\nAll concentration readings are considered in our algorithm, incorporating them in a continuous manner instead of just using them as binary detections above a certain threshold. This fact removes the need for a threshold level, thereby reducing the number of false alarms (a background model is estimated) and the number of missed detections thus improving the performance of the algorithm proposed by Pang and Farrell. A sensitivity study regarding robustness of the algorithm against deviations from the true value has been presented. It has been shown, the results improve significantly using the concentration-based algorithm if the source strength can be estimated.\nFinally, experiments with real data have shown that the concentration-based algorithm seems to perform slightly better than the binary-based one, confirming our results obtained for the simulation experiments. A critical point in our proposal is that the algorithm assumes that the source strength is known. Thus, estimating the source strength would be a promising direction for future research. Results from real experiments show that when the plume dispersion model hypothesis do not hold, both algorithm are still able to perform the estimation tasks although the differences diminish.\nThe described algorithm is available in MATLAB code from the authors under request.\n\n## Supplementary Materials\n\nThe supplementary materials are available online at https://www.mdpi.com/1424-8220/17/4/904/s1. Images of the robot, the exploration arena with the fans and the used chemical source are available as supplementary materials.\n\n## Acknowledgments\n\nThe research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 217925: LOTUS: Localization of Threat Substances in an Urban Society. This work was partially funded by the Spanish MINECO program, under grants TEC2011-26143 (SMART-IMS), TEC2014-59229-R (SIGVOL), and BES-2015-071698 (SEVERO-OCHOA). The Signal and Information Processing for Sensor Systems group is a consolidated Grup de Recerca de la Generalitat de Catalunya and has support from the Departament d’Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya (expedient 2014-SGR-1445). This work has received support from the Comissionat per a Universitats i Recerca del DIUE de la Generalitat de Catalunya and the European Social Fund (ESF). Additional financial support has been provided by the Institut de Bioenginyeria de Catalunya (IBEC). IBEC is a member of the CERCA Programme/Generalitat de Catalunya.\n\n## Author Contributions\n\nS.M. designed the study and provided the main ideas behind the concentration based algorithm. V.P. programmed the simulator, the implementation of both algorithms and wrote a first version of the manuscript. V.P., R.M. and J.B. analyzed the signals and ran the algorithm for several test cases and for the real case signals. D.M., M.T. and J.P. designed and programmed the autonomous robot. R.M., J.M.J.-S., D.M. and M.T. mounted the exploration arena and carried out the real experiments. R.M. and S.M. contributed to the final version of the manuscript.\n\n## Conflicts of Interest\n\nThe authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.\n\n## Appendix A. Supplementary Tables\n\nTable A1. Summary of the notation used for the concentration-based approach.\nTable A1. Summary of the notation used for the concentration-based approach.\n Nc Number of Cells in the Grid Area Lx Length of each cell along the x-axis Ly Length of each cell along the y-axis c Instantaneously measured concentration cb Concentration contribution due to the background cp Concentration contribution due to the chemical plume $c ¯$ Mean concentration at a fixed location q Source strength or release rate Ua Mean wind speed in the downwind direction σy Diffusion coefficient in the crosswind direction σz Diffusion coefficient in the vertical direction $c ^$ Array of all possible instantaneous concentrations c γ Intermittency factor related to the chemical plume M Mean concentration of a series of readings at a fixed location $σ S D$ Standard deviation of a series of readings at a fixed location Nb Number of readings stored in the concentration buffer per each sensor cj Measured concentration within cell j Ai Event “there is a chemical source within cell i” B(tk) Sequence of measured concentrations along the trajectory of the robots until time tk $P ( A i )$ Prior probability of the presence of a chemical source within cell i $P ( c j \| A i )$ Probability that the measurement within cell j is due to the addition of the background at cell j and a chemical plume due to a source within cell i $P ( c j \| A ¯ i )$ Probability that the measurement of chemical at cell j is not due to a source emitting at cell i, thus cj is due to the current background at cell j $S i j ( t k ) ≡ P ( A i \| c j )$ Probability of having a source in cell i given that a certain amount of chemical was measured at cell j at time tk $β i j ( t k )$ Normalized probability (over all cells) of having a chemical source within cell i based on a single measured concentration within cell j at time tk $α ' i j ( t k )$ Normalized probability (over all cells) of having a chemical source within cell i based on the sequence of measured concentrations along the trajectory of the robots (index j) until time tk\nTable A2. Main assumptions of the concentration-based algorithm.\nTable A2. Main assumptions of the concentration-based algorithm.\nFor the Plume\nGaussian distribution for the time-averaged concentration\nconcentration fluctuations governed by turbulences\ncontinuous release, source strength known q = 2.90 g/s\none uniquely source at the position (400 m, 400 m)\nheight, z = 2 m\nFor the Background\nbackground is always present\nbackground relatively constant for the exploration time\nmean background level can change from one cell to another\nno intermittency\nOther Assumptions\nuniform wind field over the exploration time, speed Ua = 2.5 m/s, and 45° direction\nneutral atmospheric stability\nno deposition of the substance on surfaces\ncells are equally likely to contain the source at time t0\nheight of sensors is 2 m\nresponse-time of sensors faster than the typical 10min time-average of GPM\n\n## References\n\n1. Kowadlo, G.; Russell, R.A. Robot Odor Localization: A Taxonomy and Survey. Int. 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[Google Scholar] [CrossRef]\n35. Gijbels, I.; Hubert, M. Robust and nonparametric statistical methods. In Comprehensive Chemometrics: Chemical and Biochemical Data Analysis, 1st ed.; Brown, S.D., Tauler, R., Walczak, B., Eds.; Elsevier: Amsterdam, The Netherlands, 2009; pp. 189–211. [Google Scholar]\n36. Papoulis, A. Chapter 8: Sequences of random variables. In Probability, Random Variables, and Stochastic Processes; McGraw-Hill: New York, NY, USA, 1984. [Google Scholar]\nFigure 1. Randomly explored area (1 km × 1 km) using five sensors (colored dots). The continuous white lines (-) define the cell boundaries. The sensors move in horizontal or vertical direction from the center of one cell to the center of a neighbor cell. The source is located at (440, 440) meters at cell (5, 5). Mean wind speed is set to 2.5 m/s and wind direction to 45°. Colored background refers to probability map at a particular time step.\nFigure 1. Randomly explored area (1 km × 1 km) using five sensors (colored dots). The continuous white lines (-) define the cell boundaries. The sensors move in horizontal or vertical direction from the center of one cell to the center of a neighbor cell. The source is located at (440, 440) meters at cell (5, 5). Mean wind speed is set to 2.5 m/s and wind direction to 45°. Colored background refers to probability map at a particular time step.", null, "Figure 2. Simulated concentration fluctuations (under the specified conditions) at different fixed positions downwind from the source over a certain background level.\nFigure 2. Simulated concentration fluctuations (under the specified conditions) at different fixed positions downwind from the source over a certain background level.", null, "Figure 3. Dependency of the assigned mean probability (averaged over 10 different random trajectories) with the concentration threshold at source location by the binary-based approach. (a) Low background level (mean = 0.05 ppm and SD = 0.03 ppm). (b) High background level (mean = 0.45 ppm and SD = 0.27 ppm). The orange line shows the initial equiprobable value ($1 / N c$) assigned to every cell.\nFigure 3. Dependency of the assigned mean probability (averaged over 10 different random trajectories) with the concentration threshold at source location by the binary-based approach. (a) Low background level (mean = 0.05 ppm and SD = 0.03 ppm). (b) High background level (mean = 0.45 ppm and SD = 0.27 ppm). The orange line shows the initial equiprobable value ($1 / N c$) assigned to every cell.", null, "Figure 4. Performance of binary-based and concentration-based algorithms in source localization with a maximum mean background level of 0.05 ppm. (a) Mean probability averaged over the ten random trajectories (thick solid line) and confidence intervals within two standard deviations (thin dashed lines). (b) Mean Euclidean distance $D$ between the cell with the highest probability value and the real source location (thick solid line) and both the maximum and the minimum values (thin dashed lines). (c) error in source localization in both X and Y directions at the end of exploration time; p-values between both approaches by using the Wilcoxon rank test have been indicated.\nFigure 4. Performance of binary-based and concentration-based algorithms in source localization with a maximum mean background level of 0.05 ppm. (a) Mean probability averaged over the ten random trajectories (thick solid line) and confidence intervals within two standard deviations (thin dashed lines). (b) Mean Euclidean distance $D$ between the cell with the highest probability value and the real source location (thick solid line) and both the maximum and the minimum values (thin dashed lines). (c) error in source localization in both X and Y directions at the end of exploration time; p-values between both approaches by using the Wilcoxon rank test have been indicated.", null, "Figure 5. Mean probability maps (averaged over all trajectories) after 300 min of random exploration with a maximum mean background level of 0.05 ppm. Source location at (5, 5) is indicated with an asterisk (). (a) Binary-based approach. (b) Concentration-based approach.\nFigure 5. Mean probability maps (averaged over all trajectories) after 300 min of random exploration with a maximum mean background level of 0.05 ppm. Source location at (5, 5) is indicated with an asterisk (*). (a) Binary-based approach. (b) Concentration-based approach.", null, "Figure 6. Performance of binary-based and concentration-based algorithms in source localization with a maximum mean background level of 0.45 ppm. (a) Mean probability averaged over the ten random trajectories (thick solid line) and confidence intervals within two standard deviations (thin dashed lines). (b) Mean Euclidean distance $D$ between the cell with the highest probability value and the real source location (thick solid line) and both the maximum and the minimum values (thin dashed lines). (c) error in source localization in both X and Y directions at the end of exploration time; p-values between both approaches by using the Wilcoxon rank test have been indicated.\nFigure 6. Performance of binary-based and concentration-based algorithms in source localization with a maximum mean background level of 0.45 ppm. (a) Mean probability averaged over the ten random trajectories (thick solid line) and confidence intervals within two standard deviations (thin dashed lines). (b) Mean Euclidean distance $D$ between the cell with the highest probability value and the real source location (thick solid line) and both the maximum and the minimum values (thin dashed lines). (c) error in source localization in both X and Y directions at the end of exploration time; p-values between both approaches by using the Wilcoxon rank test have been indicated.", null, "Figure 7. Mean probability maps (averaged over all trajectories) after 300 min of random exploration with a maximum mean background level of 0.45 ppm. Source location at (5, 5) is indicated with an asterisk. (a) Binary-based approach. (b) Concentration-based approach.\nFigure 7. Mean probability maps (averaged over all trajectories) after 300 min of random exploration with a maximum mean background level of 0.45 ppm. Source location at (5, 5) is indicated with an asterisk. (a) Binary-based approach. (b) Concentration-based approach.", null, "Figure 8. Mean probability (averaged over 10 trajectories) at real source location after 300 min of random exploration as a function of the source strength assumed by the concentration-based approach. Error bars show confidence levels within two standard deviations. (a) Results with a maximum mean background level of 0.05 ppm. (b) Results with a maximum mean background level of 0.45 ppm.\nFigure 8. Mean probability (averaged over 10 trajectories) at real source location after 300 min of random exploration as a function of the source strength assumed by the concentration-based approach. Error bars show confidence levels within two standard deviations. (a) Results with a maximum mean background level of 0.05 ppm. (b) Results with a maximum mean background level of 0.45 ppm.", null, "Figure 9. (Top) a representative wind distribution recorded within the designed scenario during a random exploration maneuver. Both, the direction (degrees) and the speed (m/s) of the wind are indicated. (Bottom) map of concentrations recorded with the PID. Source location at (10, 4) is indicated by a black circle.\nFigure 9. (Top) a representative wind distribution recorded within the designed scenario during a random exploration maneuver. Both, the direction (degrees) and the speed (m/s) of the wind are indicated. (Bottom) map of concentrations recorded with the PID. Source location at (10, 4) is indicated by a black circle.", null, "Figure 10. Mean probability maps at the end of the exploration time for the set of real experiments. Source location at (10, 4) is marked with an asterisk. (a) Binary-based approach. (b) Concentration-based approach.\nFigure 10. Mean probability maps at the end of the exploration time for the set of real experiments. Source location at (10, 4) is marked with an asterisk. (a) Binary-based approach. (b) Concentration-based approach.", null, "Figure 11. Errors in source localization in both the X and the Y directions for both algorithms. Statistically significant difference is indicated.\nFigure 11. Errors in source localization in both the X and the Y directions for both algorithms. Statistically significant difference is indicated.", null, "" ]	[ null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g001.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g002.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g003.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g004.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g005.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g006.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g007.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g008.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g009.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g010.png", null, "https://www.mdpi.com/sensors/sensors-17-00904/article_deploy/html/images/sensors-17-00904-g011.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92956936,"math_prob":0.9488971,"size":11291,"snap":"2020-34-2020-40","text_gpt3_token_len":2013,"char_repetition_ratio":0.13723753,"word_repetition_ratio":0.008777062,"special_character_ratio":0.17819503,"punctuation_ratio":0.08730159,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9868031,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-06T08:02:35Z\",\"WARC-Record-ID\":\"<urn:uuid:2685a35f-2cb1-4b33-97e1-0aafb7b4b260>\",\"Content-Length\":\"291610\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8576d2f7-10d7-460e-9553-9d4fd579184e>\",\"WARC-Concurrent-To\":\"<urn:uuid:816b4808-ac8f-4a7b-ad9a-d39e27709348>\",\"WARC-IP-Address\":\"104.18.24.151\",\"WARC-Target-URI\":\"https://www.mdpi.com/1424-8220/17/4/904/htm\",\"WARC-Payload-Digest\":\"sha1:HWKCVZNIIKJ3UP6GCTQBVVNK2MPPLXP4\",\"WARC-Block-Digest\":\"sha1:4B5PCJ2RDFIGASRF3TQXPSBNZ637RT5A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439736883.40_warc_CC-MAIN-20200806061804-20200806091804-00471.warc.gz\"}"}
https://mathoverflow.net/tags/operator-theory/hot	[ "# Tag Info\n\n9\n\nLet $A$ be a self-adjoint operator with domain $D(A)\\subset\\mathcal H$ ($\\mathcal H$ is some Hilbert space). An operator $C$ with $D(A)\\subset D(C)$ is called relatively compact with respect to $C$ if $C(A-zI)^{-1}$ is compact for some (hence all) $z\\notin\\sigma(A)$. Paraphrasing Corollary 2, page 113 Section XIII.4, in , we have If $C$ is relatively ...\n\n4\n\nYour operator $K$ is a Hilbert-Schmidt operator since its kernel belongs to $L^2$. As a result this is a compact operator whose spectrum contains a sequence of eigenvalues $\\\\{\\lambda_k\\not=0\\\\}$ with finite multiplicities such that $\\lim_k\\vert \\lambda_k\\vert=0$. To deal with the self-adjoint case, you can find an orthonormal set $\\\\{\\mathbf e_k\\\\}$ such ...\n\n4\n\nIt seems that I have found a counter example myself. For the Hilbert matrix $$H_\\lambda:= \\big( \\frac{1}{1-\\lambda+k+n} \\big)_{k,n\\geq 0}, \\lambda < 1$$ Rosenblum in \"On the Hilbert Matrix I, Proceedings of the AMS\" proves that the pointspectrum considered as an operator on $\\ell^p, p>2$ contains the set $$\\{ \\pi \\sec(\\pi u ) : \| \\Re ( u )\| <... 2 Yes, the inequality is true. Let \\lambda be large enough so (L+\\lambda I) is positive definite. We have$$((L+\\lambda I)y,y)=((L+\\lambda I)^{1/2}y,(L+\\lambda I)^{1/2}y)\\ge C\\\|y\\\|^2_{H^{s/2}},$$since the domain of (L+\\lambda I)^{1/2} is H^{s/2} by the general theory of interpolation spaces. It follows that$$(Ly,y)\\ge C\\\|y\\\|^2_{H^{s/2}}-\\lambda (y,y)....\n\n1\n\nIn \"On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space (1966)\", R. G. Douglas proved the following result (Theorem 1 in the paper): Theorem. Let $C$ and $D$ be bounded linear operators on a real or complex Hilbert space $\\mathcal{H}$; then the following are equivalent: (i) $C\\mathcal{H} \\subseteq D \\mathcal{H}$. (ii) There ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76435244,"math_prob":0.9996648,"size":2897,"snap":"2020-10-2020-16","text_gpt3_token_len":936,"char_repetition_ratio":0.11406844,"word_repetition_ratio":0.0,"special_character_ratio":0.32965136,"punctuation_ratio":0.11702128,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999795,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-06T19:12:40Z\",\"WARC-Record-ID\":\"<urn:uuid:f800c393-2be6-41e7-ba71-b7bd807c8088>\",\"Content-Length\":\"97142\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74b8a3cc-60a3-4a46-9734-ff41b2cd4216>\",\"WARC-Concurrent-To\":\"<urn:uuid:389a9572-3b75-4eea-8915-616451b77f02>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/tags/operator-theory/hot\",\"WARC-Payload-Digest\":\"sha1:3DNNQAHKS6UAN7LGGUT7UY5UMTHHPUVO\",\"WARC-Block-Digest\":\"sha1:SBA7LQI6KGCNEXAUHFODY75PHGHSLEQE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371656216.67_warc_CC-MAIN-20200406164846-20200406195346-00108.warc.gz\"}"}
https://amenoren.medium.com/disarmium-number-challenge-in-python-97e120a81eb1	[ "# Disarmium Number Challenge in Python\n\nRefactoring my function down from 9 lines to 1 line\n\nSeveral months ago I wrote about my adventure of coding the Fibonacci number. I love the Fibonacci number — any number puzzle actually — so it was with great joy when I came across the Disarmium number. A number is a Disarmium number if the sum of the digits raised to the power of their respective position in the original number is equal. For example, 35 is not a Disarmium number:\n\n3¹ + 5² equals 28 (3 + 25) and 28 does not equal 35\n\nAnother example, 89 is a Disarmium number:\n\n8¹ + 9² = 89 and 89 equals 89\n\nSuch fun!\n\nMy first attempt at solving the challenge was nine lines long.\n\nHere are my examples:\n\nx = 89y = 88z = 131u = 175q = 123658\n\nAnd the results:\n\nThen I decided to work in the enumerate method (which I had written about previously):\n\nI continued to refactor and rework the code and 5 versions later, came up with a one-line function:\n\nThe lesson learned on this number problem? Nothing really earth-shattering — I was just having some fun with a new number type and seeing how “Pythonic” could I make my code. Mission achieved." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92734474,"math_prob":0.9507164,"size":1130,"snap":"2021-43-2021-49","text_gpt3_token_len":288,"char_repetition_ratio":0.13143872,"word_repetition_ratio":0.0,"special_character_ratio":0.26460177,"punctuation_ratio":0.07589286,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9736021,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T18:53:53Z\",\"WARC-Record-ID\":\"<urn:uuid:a7659362-48ec-48f0-b74c-0e0c3ec538f7>\",\"Content-Length\":\"104455\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eff1591c-3d43-4ba4-8c24-042048ce216e>\",\"WARC-Concurrent-To\":\"<urn:uuid:8f1640eb-ad4c-44f2-9388-7e444d81b51d>\",\"WARC-IP-Address\":\"162.159.153.4\",\"WARC-Target-URI\":\"https://amenoren.medium.com/disarmium-number-challenge-in-python-97e120a81eb1\",\"WARC-Payload-Digest\":\"sha1:HLPBIWJUNCNIBCCIR2DZPOP7N4ENBY7A\",\"WARC-Block-Digest\":\"sha1:MLN53KSNA4FVGH5RLMVIIL4ZIRERC4XW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585439.59_warc_CC-MAIN-20211021164535-20211021194535-00533.warc.gz\"}"}
https://www.spp2026.de/members-guests/33-member-pages/dr-markus-upmeier/	[ "# Members & Guests\n\n## Dr. Markus Upmeier", null, "Universität Augsburg\n\nE-mail: Markus.Upmeier(at)math.uni-augsburg.de\nTelephone: +49 821 598 - 2228\nHomepage: https://www.math.uni-augsburg.de/prof/di…\n\n## Project\n\n33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces\n\n## Publications within SPP2026\n\nLet X be a compact Calabi-Yau 3-fold, and write $$\\mathcal{M}, \\overline{\\mathcal{M}}$$ for the moduli stacks of objects in coh(X) and the derived category D^b coh(X). There are natural line bundles $$K_{\\mathcal{M}} \\to \\mathcal{M}, K_{\\overline{\\mathcal{M}}} \\to \\overline{\\mathcal{M}}$$ analogues of canonical bundles. Orientation data is an isomorphism class of square root line bundles $$K_{\\mathcal{M}}^{1/2}, K_{\\overline{\\mathcal{M}}}^{1/2}$$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman in their theory of motivic Donaldson-Thomas invariants, and is also important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds X that admit a spin smooth projective compactification.\n\nLet X be a compact manifold, G a Lie group, PX a principal G-bundle, and B_P the infinite-dimensional moduli space of connections on P modulo gauge. For a real elliptic operator E we previously studied orientations on the real determinant line bundle over B_P. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson.\n\nHere we consider complex elliptic operators F and introduce the idea of spin structures, square roots of the complex determinant line bundle of F. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on X with orientations on X×S1. Thus, if PX and QX×S1 are principal G-bundles with Q\|X×{1}≅P, we relate spin structures on (B_P,F) to orientations on (B_Q,E) for a certain class of operators F on X and E on X×S1.\n\nCombined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G=U(m),SU(m). In a sequel we will apply this to define canonical orientation data for all Calabi-Yau 3-folds X over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.\n\nWe develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle which allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications appear in the companion papers arXiv:1811.01096, arXiv:1811.02405, and arXiv:1811.09658.\n\nLet X be a compact manifold, D a real elliptic operator on X, G a Lie group, P a principal G-bundle on X, and B_P the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each connection \\nabla_P, we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base B_P, and so has an orientation bundle O^D_P over B_P, a principal Z_2-bundle parametrizing orientations of KerD^\\nabla_Ad(P) + CokerD^\\nabla_Ad(P) at each \\nabla_P. An orientation on (B_P,D) is a trivialization of O^D_P.\n\nIn gauge theory one studies moduli spaces M of connections \\nabla_P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M is a smooth manifold, and orientations on (B_P,D) pull back to\n\norientations on M in the usual sense of differential geometry.\n\nThis is important in areas such as Donaldson theory, where one needs an orientation on M\n\nto define enumerative invariants.\n\nWe explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B_P,D), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds,\n\ninstantons, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.\n\nSuppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D. Let G be SU(m) or U(m), and E be a rank m complex bundle with G-structure on X. Write B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_2-bundle O^D_E on B_E parametrizing orientations of det D_Ad A for twisted elliptic operators D_Ad A at each [A] in B_E. A theorem of Walpuski shows O^D_E is trivializable.\n\nWe prove that if we choose an orientation for det D, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of O^D_E for all such bundles E on X, satisfying natural compatibilities.\n\nNow let (X,\\varphi,g) be a compact G_2-manifold, with d(*\\varphi)=0. Then we can consider moduli spaces M_E^G_2 of G_2-instantons on E over X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general. The restriction of O^D_E to M_E^G_2 is the Z_2-bundle of orientations on M_E^G_2. Thus, our theorem induces canonical orientations on all such G_2-instanton moduli spaces M_E^G_2.\n\nThis contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of G_2-manifolds (X,\\varphi,g) by counting moduli spaces M_E^G_2, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6." ]	[ null, "https://www.spp2026.de/fileadmin/_processed_/0/a/csm_5D4854D5-E54C-491A-93FF-3F59E577F53D_4f2670a7a8.jpeg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82306546,"math_prob":0.9229772,"size":6929,"snap":"2020-24-2020-29","text_gpt3_token_len":1731,"char_repetition_ratio":0.14729242,"word_repetition_ratio":0.068895645,"special_character_ratio":0.22066677,"punctuation_ratio":0.109642304,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98992115,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-04T05:20:09Z\",\"WARC-Record-ID\":\"<urn:uuid:9e85e402-135b-46cc-8473-64cd7a0dd3a7>\",\"Content-Length\":\"33329\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:69f20fe6-02fd-4c5e-8c5f-ae5926c57a05>\",\"WARC-Concurrent-To\":\"<urn:uuid:45614494-eeff-4cb3-ac2f-93f389ecae7e>\",\"WARC-IP-Address\":\"188.40.3.34\",\"WARC-Target-URI\":\"https://www.spp2026.de/members-guests/33-member-pages/dr-markus-upmeier/\",\"WARC-Payload-Digest\":\"sha1:EU7M3BAAMNCZOYB6MUTKT4OIUUULYC3Z\",\"WARC-Block-Digest\":\"sha1:67K4SVVDMA6BTRFMH7Q4VKLESSCKDHVZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655884012.26_warc_CC-MAIN-20200704042252-20200704072252-00154.warc.gz\"}"}
http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/mtn_roots.html	[ "Underneath the mountains\n\nI. Introduction\n\nII. Buoyancy, Archimede's principle, and evidence for mountain roots\n\nIII. The lifetimes of mountain ranges\n\nAppendix: Beyond the intro level\n\nI. Introduction\n\nOver geologic time, the earth's mantle behaves like a highly viscous fluid (i.e. a fluid that flows very slowly). When topography is created on the earth's surface through crustal thickening, the mantle slowly flows out from beneath the thickened region so as to compensate for the change in the weight of the overlying crust. The mantle thus behaves in a manner similar to that of water when a cube of ice is placed on its surface - water beneath the ice flows outward and upward and the ice sinks downward until an equilibrium (steady-state) is reached. There is however one important difference - on earth, the strength of the crust itself helps to support some of the weight of a topographic load. We must then study two processes to develop a better understanding of how topography is \"maintained\" on our planet. One is buoyancy and the other, flexure.\n\nII. Buoyancy\n\nSuppose we floated a block of ice in a tub of cold water and asked the following question: How much of the block will be above water and how much will be below water? Ice has a density of 0.9 grams per cubic centimeter and water, a density of 1.0 grams per cubic centimeter. How can we solve this? Let's first digress to consider some general principles associated with this problem.\n\nWhat happens to the water level in the tub if you place a book on the block of ice? The water level increases as the additional weight of the book causes the ice to displace water beneath it. The water has nowhere to go except up. The downward force from the additional weight of the book is evenly balanced by upward motion of the water against the pull of gravity.\n\nIf you remove the book, the block of ice immediately pops up and the water level returns to its original level. In other words, as the force on the top of the ice changes, the water level re-adjusts to the new forces. In fact, the water in the tub acts as a pressure gauge - the water level adjusts to compensate for all changes in the downward forces that are acting on the water column.\n\nWater behaves this way because it is free to flow to \"escape\" zones of higher pressure. Consider a small volume of water beneath the base of a floating ice cube. If we push down on the ice, the water underneath immediately experiences a increase in the downward pressure acting on it. Zones of lower pressure are located within the water immediately beyond the ice. The water thus flows from the zone of higher pressure to the lower pressure zones until the pressure acting on water at the same depth is exactly equal everywhere within the closed basin. An important implication is that fluids in general will always flow to eliminate areas of higher or lower pressure within a closed system. This principle does not apply to solid materials because their internal structures do not permit unrestricted relative motion of adjacent molecules or atoms.\n\nWe can now state Pascal's law - in a fluid, the pressure (i.e. force per unit area) exerted by the overlying material is equal everywhere at a given depth.\n\nArchimede's principle follows from Pascal's law (even though the former was stated thousands of years earlier than the latter).\n\nARCHIMEDE'S PRINCIPLE - \"A body immersed in a fluid is buoyed up with a force equal to the weight of the displaced fluid.\"\n\nNote that \"weight\" is defined as mass * volume.\n\nPlease do not get the impression that the pressure in water is equal everywhere! Any snorkeler or diver knows that water pressure increases with depth. However, pressure remains the same if you sink to a given depth and then swim in any direction AT THAT DEPTH.\n\nOK - let's return to the original question. For an object floating in a fluid,\nhow much buoyant material has to float beneath the surface of a fluid in order to support the material that lies above the fluid surface? In past years, my Web notes put forward a simple mathematical solution that relies on balancing the forces (pressures) that act on the base of a floating object. I am including that solution at the end of this lecture; however, you may choose if you want to go through its details. For those of you who are interested in the answer, but not the details that yield the answer, the bottom line is as follows:\n\nFor a buoyant solid of density D1 floating in a fluid of density D2, the amount of the solid beneath the fluid surface needed to support the material floating above the surface is given by the following simple equation:\n\n(Density of solid * height above fluid surface)\nDepth of root = -------------------------------------------------------\n( Density of fluid - density of solid )\n\nIn other words, the difference in the densities of the fluid and solid is an important factor that determines how deep a root must extend beneath the surface in order to support topography above the surface.\n\nLet's apply this simple relationship to determine how deeply an iceberg extends beneath water.\n\nThe respective densities of water and ice are 1.0 and 0.9 grams per cubic centimeter. Since ice is less dense that water, it floats (we all know that!). So, let's suppose that we are kayaking in Glacier Bay in Alaska and we see an iceberg sticking 1 meter above the water surface. How far down does the iceberg extend?\n\n(0.9 gm/cc * 1 meter) 0.9 meters\nDepth of root = -------------------------- = ------------- = 9 meters\n( 1.0 - 0.9 ) gm/cc 0.1\n\nSo, if an iceberg is a total of 10 meters thick, 9 meters will lie beneath water in order to support the 1 meter that juts above the water surface. This obeys the rule of thumb that ship captains know about icebergs, namely, that about 90 per cent of the iceberg lies beneath the water out of sight. That 90 per cent is dangerous because it can puncture a ship's hull.\n\nImplications for mountain ranges\n\nContinents are buoyant crust that float on a denser mantle. We can thus use the density of continental rocks and mantle rocks to calculate how deep roots are that support mountain ranges.\n• Continental crust - 2.8 grams per cubic centimeter\n• Mantle - 3.3 grams per cubic centimeter\n• Inner and outer cores - 12-16 grams per cubic centimeter\n\nThe average elevation of continental crust relative to sealevel is about 1 km. Roughly speaking, how deep must the continental root extend down into the mantle to support that elevation ?\n\n(2.8 gm/cc * 1 km) 2.8 km\nDepth of root = -------------------------- = ------------- = 5.6 km\n( 3.3 - 2.8 ) gm/cc 0.5\n\nHow deep is the root for a mountain range with an average elevation of 15,000 feet (about 3 miles)?\n\n(2.8 gm/cc * 3 miles) 8.4 miles\nDepth of root = -------------------------- = ------------- = 16.8 miles\n( 3.3 - 2.8 ) gm/cc 0.5\n\nThe most important point is that mountains have buoyant roots that extend downward into the mantle beneath a mountain range, and that the roots are, in general, about 5.6 times deeper than the height of the range. This result reflects the difference between the densities of average crust and mantle.\n\nIII. Erosion, buoyant mountain roots, and the lifetimes of mountain ranges\n\nThe existence of buoyant roots has important implications for the lifetime of mountain ranges. What happens when erosion removes material from the top of a mountain? With less mass above sea-level to support, the buoyant root rebounds upward an amount that is exactly proportional to the density difference between the root and the underlying mantle! Thus, in the case of an iceberg that stands 10 meters above sea level, if all 10 meters of ice melt from the top, the buoyant root pushes upwards 9 meters! The iceberg thus loses only 1 meter of height. It thus takes much longer for the iceberg to \"disappear\" because its buoyant root continually restores (from beneath sea level) ice that melts above the water surface.\n\nOne study of erosion in the Appalachians suggests that over the past 270 million years, erosion has removed an average of 0.02 millimeters each year of material from the mountain range (this is 2 millimeters per hundred years). Let's use this rate to compute the lifetime of a mountain range for two different scenarios.\n\nScenario 1: Assume that mountain ranges do NOT have buoyant roots\n\nGiven a plateau that stands 5000 meters above a plain of 1 km average height (the Tibetan plateau is a good example), how long will it take for erosion to level this plateau?\n\nAssuming an erosion rate of 0.02 millimeters per year, the lifetime T can be computed as follows:\n\nErosion rate * T = 5000-1000 meters * 1000 millimeters/1 meter\n\nT = 4000 * 1000 / 0.02 (units of millimeters / mm/yr = years)\n\nT = 200 million years\n\nThe projected lifetime of this plateau (ie mountain range) is thus 200 million years.Its reasonable to assume that the youthful Appalachians were at least 5000 meters in height. This however raises a problem - if the Appalachians have no buoyant root beneath them, they would have been completely eroded away by now!\n\nScenario 2: Assume that mountain ranges have buoyant roots\n\nLet's again assume that a plateau stands 5000 meters above a 1000 meter-high plain. However, let's now use a more realistic model for the plateau - let's assume that it has a buoyant root that extends some 20 km beneath the bottom of the continental crust that floors the 1 km-high plain. For example, see the model that fits the gravity profile for the Tibetan plateau (Lecture 6) - in that model, the root beneath the plateau extends to depths of roughly 62 km, whereas continental crust not beneath the plateau extends to depths of about 42 km. So the buoyant root juts 20 km down into the mantle from the base of the continental crust.\n\nBy Archimede's principle, when material is stripped (by erosion) from the top of the plateau, thereby lowering the elevation of the plateau, the buoyant root pushes up from beneath and thus replaces some (but not all) of the rock that was eroded. To completely level the plateau, erosion must therefore ultimately strip away material equivalent in height to the original mountain range plus its root (4 km + 20 km). (If the reason for this is not clear to you, think about it for a while).\n\nWe must thus repeat the above calculation for T as follows\n\nErosion rate * T = 25000-1000 meters * 1000 millimeters/1 meter\n\nT = 24000 * 1000 / 0.02 (units of millimeters / mm/yr = years)\n\nT = 1,200 million years = 1.2 billion years!\n\nThus, this hypothetical mountain range can hang around for 1.2 billion years because its root continually pushes new material up to replace material that is being eroded from the top.\n\nA key implication: as erosion strips away the material on top of mountain ranges, rocks from much deeper (10-12 miles deeper!) in the continental crust are pushed up to the surface by the buoyant root. Erosion coupled with buoyant mountain roots thus provide a mechanism for bringing deep crustal rocks to the surface.\n\nThe bottom line - once plate tectonic processes build a mountain range, the buoyant underlying root enables the mountain range to hang around a long time even while its being actively eroded.\n\nCaveats - erosion rates can of course be much higher, which can significantly decrease the lifetime of the range.\n\nAppendix: Beyond the introductory level\n\nYou will not be tested on the material below and thus do not have to read it. For those of you who might be interested in this at a slightly more complicated level, read on!\n\ni) A simple mathematical derivation of the equation for mountain root depths\n\nTo begin, let's specify the forces that are acting on a hypothetical mass floating in a fluid\n\nForce = mass x acceleration ---> (Newton's Second law of motion)\n\nFor our purposes, acceleration is equal to gravitational acceleration g, which is the rate at which a falling object accelerates in the earth's gravity field. g for the earth is 9.8 meters per squared second. The mass of an object is equal to its volume times its density.\n\nFor the figure below, let's examine the gravitational force that pulls down on our imaginary cylinder that cuts through two columns in our tub.", null, "Column 1 cuts through a column of air with length \"a\" and a column of water of length \"b\", for a total column length of \"a + b\".\n\nColumn 2 cuts through a column of ice with length \"a+c\", and a column of water with length \"d\". What are the forces acting on each column? We need to know the density and volume of the columns of air, water, and ice in each column to answer this question. The densities of ice and water are given above as 0.9 and 1.0 grams/cubic centimeter. The density of air for all intensive purposes is zero!\n\nSince these are cylinders, the volume is simply equal to the height multiplied by the cross-sectional area of the cylinder. To simplify the problem, let's assign a cross-sectional area for each cylinder of 1.0 square centimeters. The volume is then simply 1 x height.\n\nBy Newton's second law, the force F1 acting equals the following:\n\ng * [ (Density of air * a) + (Density of water * b)] = g * [(0 x a) + (1.0 grams/cubic centimeter * b centimeters * 1.0 square centimeters)]\n\nor F1 = 1.0 * b * g\n\nSimilarly, the force F2 equals\n\ng * [(density of ice * (a+c) ) + (density of water * d)] = g * [0.9(a+c) + d ] or F2 = d g + 0.9(a+c)g\n\nNow, what else do we know about F1 and F2? Since we know that the forces that pull down on every column of water are equal (if they weren't, the system would immediately readjust to make them equal!), we then know that F1 = F2. This is the key step - namely, that we invoke Pascal's law that the pressure is equal everywhere at a given depth in the fluid.\n\nTo solve the problem, we now choose a depth that corresponds exactly with the bottom of the ice and require pressures exerted by the overlying ice and/or water to be equal at that depth. In other words, imagine that the red-dashed line is positioned at the bottom of the ice. We can now write the following equivalence F1 = F2 = 1.0 bg = 0.9(a+c)g\n\nDividing by \"g\" gives 1.0b = 0.9(a+c)\n\nHowever, \"b\" = \"c\", so we can rewrite this as (1.0 - 0.9) c = 0.9a.\n\nWhat are a and c ? C is the \"root\" that sticks down into the fluid - this \"root\" supports the material that stands above the water surface, which has a height of a.\n\nFor any two materials of different densities (with at least one acting as a fluid), we can write\n\n(Density 1 - Density 2) root = Density 2 * height above surface\n\nwhere density 1 is greater than density 2.\n\nA key point here is that it is the difference in the densities of the two materials that determines how deep a root must extend beneath the surface in order to support topography above the surface.\n\nThe equation above expresses the relationship between the root and height of a material floating in a fluid and the densities of the floating material and fluid.\n\nii) Evidence for mountain roots\n\nIs there any evidence that mountains have such deep roots? Yes, there is abundant evidence from measurements of gravity over and near mountain ranges. Let's first digress briefly to the equation that specifies the force of gravitational attraction between two masses that are separated by a distance r.\n\nF= G * m1 * m2 / (r * r)\n\nHere, G is a constant (the universal gravitational constant), and m1 and m2 are the masses of objects 1 and 2. The equation thus indicates that as the distance between two objects increases, the gravitational attraction between them decreases as the square of the distance.\n\nIf the earth were perfectly spherical (i.e. no topography) and lacked any variation in density, then a mass that is hanging from a string (a plumb bob) would always point directly toward the earth's center. In the 18th century, French scientists on an expedition to South America to measure the distance of a degree of latitude noted that the great mass of the Andes mountain belt represented additional mass that would exert its own gravitational pull on a plumb bob that would deflect the plumb bob from \"vertical\" toward the mountain range. They thus estimated the mass of the mountain range and then predicted how much the vertical deflection should be. To their surprise, they found that the mass was not deflected as far as they predicted - they thus postulated that a \"deficit\" of mass beneath the mountain range had to exist. The mass deficit was a buoyant crustal root that extended down into the denser surrounding mantle.\n\nSince the 18th century, many more gravity surveys of mountain ranges have been completed and they indicate that mountain ranges are often (but not always) accompanied by a mass deficit. For example, if one measures the gravitational attraction at many points in or above a mountain range and one then corrects the measured gravity signal for a variety of effects, one of which includes the contribution from topography above sea level (this is done by estimating the gravitational attraction that results from a given volume of material with a density equivalent to that of continental crust), the gravity field over a mountain range should be the same as the gravity field for flat regions that flank the mountain range. Instead, the corrected gravity field over the mountain range typically has values lower than the surrounding flat regions. This gravity \"deficit\" is evidence for a mass \"deficit\" beneath the mountain range - such a deficit can only occur if the density of material beneath the range is lower than the density of the material beneath the flat-lying regions. Thus, less dense or buoyant material underlies many mountains - this buoyant material is the \"root\" that is predicted to exist based on Archimede's principle.\n\nExample: Gravity across the Tibetan Plateau, Himalayas", null, "Figure 2: The red line in the map above shows a profile that crosses the Himalayan mountain range and Tibetan plateau of southern Eurasa. The following figure shows Bouguer gravity measurements along this profile. Adapted from Jin et al., May, 1996 Journal of Geophysical Research.", null, "Figure 3: Dots in the upper panel show measurements of height along the profile shown in Figure 2. Note that the average height of the Tibetan plateau is about 5.5 km, or 18,200 feet. The distance across this nearly flat, but high plateau is 800 kilometers, or 500 miles. The highest topographic feature, about 6 km, is the high Himalayas, which are located at the southern edge of the Tibetan plateau. The lower panel shows Bouguer gravity measurements (black dots) and the gravity predicted by two simple models (described below). Note that the Bouguer gravity values are all less than zero, indicating that once gravity is \"corrected\" for the topography above sea-level (0 km), the gravity \"anomaly\" is negative, indicating a mass deficit at depth. The mass deficit indicates the existence of a low-density \"root\" beneath the mountain range. This root has a density lower than that of the underlying and surrounding mantle and thus provides a buoyant force that supports that Tibetan plateau. The Root only model assumes that the Bouguer gravity values are determined solely by the existence of the buoyant crust beneath the plateau. The Flexure & Root model assumes that the buoyant root is responsible for part of the gravity field, but that the strong lithosphere supports the mountain range/plateau, too. Adapted from Jin et al., May, 1996 Journal of Geophysical Research\n\nRegarding Figure 3, READ THE CAPTION CAREFULLY . This gravity profile is a classic example of how one can use measurements at the earth's surface to learn about what exists deep beneath the surface. Note that the gravity values that are predicted by the Root only model fit the observed values reasonably well, but cannot account for all of the details in the observed gravity field! This implies that other physical processes are contributing to the local gravity field. An alternative model, \"Flexure & Root\", does a better job of fitting the observed gravity values. This model combines the buoyant forces exerted by a crustal root with the inherent strength of the earth's crust to explain how mountain ranges are supported.\n\nFigure 4: Model of crust beneath the Tibetan plateau/Himalayan mountain range", null, "Note that the region designated \"Tibetan crust\" is the buoyant \"root\" that underlies the plateau, which is shown as the blue shaded area above 0 km depth. Thus, a deep root lies beneath the plateau and this root has a density lower than that of the mantle beneath it and to its sides.\n\niii) Flexure of the lithosphere\n\nBouguer gravity fields across many mountain ranges clearly show evidence for mass deficits that are indicative of underlying and buoyant crustal roots. However, gravity anomalies associated with mountain ranges are rarely as large (meaning that Bouguer gravity values are never as \"negative\") as they should be if the mountain range were supported entirely by a buoyant root.This means that something besides the buoyant root has to be supporting mountain ranges. Not surprisingly, the crust itself helps to support topographic \"loads\".\n\nFor example, typical continental crust is approximately 40 km thick. Clearly, if one dumps a load of stone in the middle of the US Great Plains, the crust does not sink into the mantle until it achieves a buoyant root that is sufficient to support this new \"topographic\" load! Instead, the crust is strong enough to hold up the load without flexing down into the mantle (and thus developing a root). This is analogous to putting an eraser on a board - the board is strong enough to support the weight of the eraser without flexing. The crust thus has inherent strength that is capable of supporting small loads such as small volcanos without flexing down into the mantle. If the load gets large enough (a mountain range), the mountain range will be supported partly by the crust and partly by a buoyant root.\n\nIf you look at Figure 3 above, you will see exactly this situation. The \"Root Only\" model cannot fit the observed Bouguer gravity. However, a model that assumes the existence of a finite-strength crust and a buoyant root, which act together to support the Tibetan plateau, fits the gravity values well." ]	[ null, "http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/Images/isostasy.gif", null, "http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/Images/tibet_locate.GIF", null, "http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/Images/Profile_1-a.GIF", null, "http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/Images/Profile_1-b.GIF", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.911073,"math_prob":0.9638075,"size":20028,"snap":"2019-51-2020-05","text_gpt3_token_len":4542,"char_repetition_ratio":0.1531662,"word_repetition_ratio":0.03735632,"special_character_ratio":0.22573397,"punctuation_ratio":0.08339953,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.968541,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-14T04:57:41Z\",\"WARC-Record-ID\":\"<urn:uuid:3c8d7c78-ddbf-4c2a-8fc5-8ea26c3fa84f>\",\"Content-Length\":\"31181\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3c81d115-70fe-4e55-a549-6cd8e84b599a>\",\"WARC-Concurrent-To\":\"<urn:uuid:015b8bd3-faa2-4255-b69c-219568aa8db9>\",\"WARC-IP-Address\":\"144.92.206.29\",\"WARC-Target-URI\":\"http://geoscience.wisc.edu/~chuck/Classes/Mtn_and_Plates/mtn_roots.html\",\"WARC-Payload-Digest\":\"sha1:LAAZZGLGIXJYCOWG566NYTM7BZG5R4QI\",\"WARC-Block-Digest\":\"sha1:KR6BUQMNBA22K7O76J53QACKVSB5T3L3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540584491.89_warc_CC-MAIN-20191214042241-20191214070241-00271.warc.gz\"}"}