{"title":"Hopf-Galois structures on Galois extensions of fields, regular subgroups, and skew braces","authors":"","doi":"10.1090/surv/260/02","DOIUrl":"https://doi.org/10.1090/surv/260/02","url":null,"abstract":"","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123311376","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":"Ridge Functions and Applications in Neural Networks","authors":"V. Ismailov","doi":"10.1090/surv/263","DOIUrl":"https://doi.org/10.1090/surv/263","url":null,"abstract":"","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125943658","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":"Epsilon-nets: An algorithm","authors":"","doi":"10.1090/surv/265/07","DOIUrl":"https://doi.org/10.1090/surv/265/07","url":null,"abstract":"","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125244784","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":"Hopf orders in group rings","authors":"","doi":"10.1090/surv/260/12","DOIUrl":"https://doi.org/10.1090/surv/260/12","url":null,"abstract":"","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132594597","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":"Recovery Methodologies: Regularization and Sampling","authors":"W. Freeden, M. Nashed","doi":"10.1090/surv/274","DOIUrl":"https://doi.org/10.1090/surv/274","url":null,"abstract":"","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131519263","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 Sequences","authors":"G. Everest, A. Poorten, I. Shparlinski, T. Ward","doi":"10.1090/surv/104","DOIUrl":"https://doi.org/10.1090/surv/104","url":null,"abstract":"Introduction The importance of recurrence sequences hardly needs to be explained. Their study is plainly of intrinsic interest and has been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. For example, the theory of power series representing rational functions [55], pseudo-random number generators ([48], [49], [50], [74]), k-regular [8] and automatic sequences [34], and cellular automata [36]. Sequences of solutions of classes of interesting Diophantine equations form linear recurrence sequences — see [67], [68], [75], [76]. A great variety of power series, for example zeta-functions of algebraic varieties over finite fields [33], dynamical zeta functions of many dynamical systems [12], [30], [35], generating functions coming from group theory [60], [61], Hilbert series in commutative algebra [39], Poincaré series [11], [21], [60] and the like — are all known to be rational in many interesting cases. The coefficients of the series representing such functions are linear recurrence sequences, so many powerful results from the present study may be applied. Linear recurrence sequences even participated in the proof of Hilbert's Tenth Problem over Z ([38], [79], [80]). In the proceedings [22], the problem is resolved for many other rings. The article [51] by Pheidas suggests using the arithmetic of bilinear recurrence sequences to settle the still open rational case. Recurrence sequences also appear in many parts of the mathematical sciences in the wide sense (which includes applied mathematics and applied computer science). For example, many systems of orthogonal polynomials, including the Tchebychev polynomials and their finite field analogues, the Dickson polynomials, satisfy recurrence relations. Linear recurrence sequences are also of importance in approximation theory and cryptography and they have arisen in computer graphics [40] and time series analysis [13]. We survey a selection of number-theoretic properties of linear recurrence sequences together with their direct generalizations. These include non-linear recurrence sequences and exponential polynomials. Applications are described to motivate the material and to show how some of the problems arise. In many sections we concentrate on particular properties of linear recurrence sequences which are important for a variety of applications. Where we are able, we try to consider properties that are particularly instructive in suggesting directions for future study. Several surveys of properties of linear recurrence sequences have been given However, they do not cover as wide a range of important features and applications as we attempt here. We have relied …","PeriodicalId":311524,"journal":{"name":"Mathematical Surveys and Monographs","volume":"13 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123800938","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}