{"title":"Probability","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-5","DOIUrl":"https://doi.org/10.1201/9781351215824-5","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131369860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Basic Techniques","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-2","DOIUrl":"https://doi.org/10.1201/9781351215824-2","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"218 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116281182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Calculus","authors":"Michael Z. Spivey","doi":"10.2174/9781681087115118010011","DOIUrl":"https://doi.org/10.2174/9781681087115118010011","url":null,"abstract":"The values xi are thus all equal at an extrema. The constraint equation tells us that xi = 1/n, from which we deduce the desired result. This extrema corresponds to a maximum because the continuous function h must achieve both its maximum and minimum value on the compact set [0, 1] and we have already eliminated the minimum by introducing the restriction that the xi are nonzero. (b) Use part (a) to prove, for any n positive numbers ai, i = 1, . . . , n, that","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"359 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123551688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mechanical Summation","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-10","DOIUrl":"https://doi.org/10.1201/9781351215824-10","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123898477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Recurrence Relations and Finite Differences","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-7","DOIUrl":"https://doi.org/10.1201/9781351215824-7","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131889170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Introducing the Binomial Coefficients","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-1","DOIUrl":"https://doi.org/10.1201/9781351215824-1","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"17 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122930205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generating Functions","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-6","DOIUrl":"https://doi.org/10.1201/9781351215824-6","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126483861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Combinatorics","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-3","DOIUrl":"https://doi.org/10.1201/9781351215824-3","url":null,"abstract":"units and their combinatorial potential. Syntagmatics is treated as an aspect of language research, which involves the study of the rules of compatibility of the language units and their realization in speech. The focus is made on two linguistic phenomena: (1) valence, which is manifested at the language level and represents a potential combinability of language units, (2) compatibility, which is manifested at the level of speech and represents the realization of valency. Combinatorics is treated as making combinations of words that are subordinate to specific communicative tasks under the conditions of their implementation. The author argues that in the framework of combinatorial linguistics the syntagmatics includes forming language units in a linear sequence according to the ru- les of combinatorics. The limitations are commented: (1) in solving of communicative tasks (givenness of sense) (2) in terms of the implementation of this task, (3) in selecting a specific set of language units that express a given meaning. The author comes to the con-clusion that syntagmatics and combinatorics equally determine the combinability of lan- guage units and are relative to each other in equipollently opposition.","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131382054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Miscellaneous Techniques","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-9","DOIUrl":"https://doi.org/10.1201/9781351215824-9","url":null,"abstract":"","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129151447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Special Numbers","authors":"Michael Z. Spivey","doi":"10.1201/9781351215824-8","DOIUrl":"https://doi.org/10.1201/9781351215824-8","url":null,"abstract":"Iverson notation Iverson (the deviser of APL) invented the following handy bracket notation: [statement] = 1 if statement is true; 0 if statement is false. Thus, for example, [i = j] = 1 if i = j and [i = j] = 0 if i = j: [i = j] = δ ij , the Kronecker delta. Binomial Coefficients Here is the general definition of the binomial coefficient r k : r k := r k k! for real r and integer k ≥ 0; it is defined to be zero for real r and integer k < 0. For integer k ≥ 0, r k = [x k ](1 + x) r := coefficient of x k in the expansion of (1 + x) r. For integers k and n with 0 ≤ k ≤ n, n k , read \" n choose k, \" counts the number of ways to select a subset of k objects from a set of n objects. Stirling Numbers of the Second Kind For integers k and n ≥ 0, n k = coefficient of x k in x n written as a factorial polynomial. Thus, n k = 0 for integer k < 0 as well as for integer k > n, and for integer n ≥ 0, x n = n k=0 n k x k = k∈Z n k x k. Stirling numbers of the second kind satisfy the recurrence relation n k = k n − 1 k + n − 1 k − 1 for integer n > 0 and integer k with boundary conditions n 0 = [n = 0] and n n = 1. The number n k , read \" n subset k, \" counts the number of partitions of a set of n elements into k nonempty subsets. Other notations are used for Stirling numbers of the second kind. The classic handbook AMS 55 [1] uses a notation like S (k) n but with a fancy calligraphic \" S. \" Spiegel [3, p. 7] uses the notation S n k , while Combinatorics [4, p. 37] uses the notation S(n, k).","PeriodicalId":194932,"journal":{"name":"The Art of Proving Binomial Identities","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117185411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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