Pub Date : 2022-06-01DOI: 10.1515/dma-2022-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/dma-2022-frontmatter3","DOIUrl":"https://doi.org/10.1515/dma-2022-frontmatter3","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46624457","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}
Abstract A hypergraph H = (V, E) has the property Bk if there exists an assignment of two colors to V such that each edge contains at least k vertices of each color. A hypergraph is called simple if every two edges of it have at most one common vertex. We obtain a new lower bound for the minimal number of edges of n-uniform simple hypergraph without the property Bk.
{"title":"New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk","authors":"Y. Demidovich","doi":"10.1515/dma-2022-0015","DOIUrl":"https://doi.org/10.1515/dma-2022-0015","url":null,"abstract":"Abstract A hypergraph H = (V, E) has the property Bk if there exists an assignment of two colors to V such that each edge contains at least k vertices of each color. A hypergraph is called simple if every two edges of it have at most one common vertex. We obtain a new lower bound for the minimal number of edges of n-uniform simple hypergraph without the property Bk.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44725509","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}
Abstract In the present paper, we consider random variables with the power series distribution which is often used in the study of the generalized allocation scheme. We establish some asymptotic properties which include law of large numbers, moderate deviation principle, almost sure central limit theorem and the rate of convergence in the local limit theorem. These results supplements results obtained by A. V. Kolchin.
{"title":"On some limit properties for the power series distribution","authors":"Yu Miao, Yanyan Tang, Xiaoming Qu, Guangyu Yang","doi":"10.1515/dma-2022-0017","DOIUrl":"https://doi.org/10.1515/dma-2022-0017","url":null,"abstract":"Abstract In the present paper, we consider random variables with the power series distribution which is often used in the study of the generalized allocation scheme. We establish some asymptotic properties which include law of large numbers, moderate deviation principle, almost sure central limit theorem and the rate of convergence in the local limit theorem. These results supplements results obtained by A. V. Kolchin.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47211210","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}
Abstract The functional system of Boolean vector functions with the naturally defined superposition operation is considered. It is shown that every closed class of this system admits a finite basis.
摘要考虑了具有自然定义的叠加运算的布尔向量函数的函数系统。证明了该系统的每一个闭类都有一个有限基。
{"title":"On bases of all closed classes of Boolean vector functions","authors":"V. A. Taimanov","doi":"10.1515/dma-2023-0017","DOIUrl":"https://doi.org/10.1515/dma-2023-0017","url":null,"abstract":"Abstract The functional system of Boolean vector functions with the naturally defined superposition operation is considered. It is shown that every closed class of this system admits a finite basis.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44952732","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}
O. A. Logachev, Sergey N. Fedorov, V. V. Yashchenko
Abstract Let G be the extension of a general affine group by the group of affine functions. We study the action of G on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of G. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.
{"title":"On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions","authors":"O. A. Logachev, Sergey N. Fedorov, V. V. Yashchenko","doi":"10.1515/dma-2022-0016","DOIUrl":"https://doi.org/10.1515/dma-2022-0016","url":null,"abstract":"Abstract Let G be the extension of a general affine group by the group of affine functions. We study the action of G on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of G. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47276080","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}
Abstract In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here we give a better estimate for the cardinality of the set of exceptional lattices for which the above estimates are not valid. In the case of dimension 4 we improve the upper estimate for an arbitrary chosen parameter that controls the accuracy of these lower estimates and for the number of exceptions. In this (first) part of the paper, we also prove some auxiliary results, which will be used in the second (main) part of the paper, in which an analogue of A. Friese et al. result will be given for dimension 5.
{"title":"Estimates of lengths of shortest nonzero vectors in some lattices. I","authors":"A. S. Rybakov","doi":"10.1515/dma-2022-0018","DOIUrl":"https://doi.org/10.1515/dma-2022-0018","url":null,"abstract":"Abstract In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here we give a better estimate for the cardinality of the set of exceptional lattices for which the above estimates are not valid. In the case of dimension 4 we improve the upper estimate for an arbitrary chosen parameter that controls the accuracy of these lower estimates and for the number of exceptions. In this (first) part of the paper, we also prove some auxiliary results, which will be used in the second (main) part of the paper, in which an analogue of A. Friese et al. result will be given for dimension 5.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44995322","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}
Abstract The enumeration closure operator (the Π-operator) is considered on the set Pk of functions of the k-valued logic. It is proved that, for any k ⩾ 2, any positively precomplete class in Pk is also Π-precomplete. It is also established that there are no other Π-precomplete classes in the three-valued logic.
{"title":"Completeness criterion with respect to the enumeration closure operator in the three-valued logic","authors":"S. Marchenkov, V. A. Prostov","doi":"10.1515/dma-2022-0010","DOIUrl":"https://doi.org/10.1515/dma-2022-0010","url":null,"abstract":"Abstract The enumeration closure operator (the Π-operator) is considered on the set Pk of functions of the k-valued logic. It is proved that, for any k ⩾ 2, any positively precomplete class in Pk is also Π-precomplete. It is also established that there are no other Π-precomplete classes in the three-valued logic.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45194701","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}
Abstract Estimates for the cardinality of the set of correlation-immune n-ary Boolean functions with fixed weight are obtained.
摘要给出了一组具有固定权值的相关免疫n元布尔函数集的基数估计。
{"title":"Some cardinality estimates for the set of correlation-immune Boolean functions","authors":"E. Karelina","doi":"10.1515/dma-2022-0008","DOIUrl":"https://doi.org/10.1515/dma-2022-0008","url":null,"abstract":"Abstract Estimates for the cardinality of the set of correlation-immune n-ary Boolean functions with fixed weight are obtained.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45534550","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}
Abstract Some results of classical statistical decision theory are generalized by means of the theory of fuzzy sets. The concepts of an admissible decision in the restricted sense, an admissible decision in the broad sense, a Bayes decision in the restricted sense, and a Bayes decision in the broad sense are introduced. It is proved that any Bayes decision in the broad sense with positive prior discrete density is admissible in the restricted sense. The class of Bayes decisions is shown to be complete under certain conditions on the loss function. Problems with a finite set of possible states are considered.
{"title":"Admissible and Bayes decisions with fuzzy-valued losses","authors":"A. S. Shvedov","doi":"10.1515/dma-2022-0013","DOIUrl":"https://doi.org/10.1515/dma-2022-0013","url":null,"abstract":"Abstract Some results of classical statistical decision theory are generalized by means of the theory of fuzzy sets. The concepts of an admissible decision in the restricted sense, an admissible decision in the broad sense, a Bayes decision in the restricted sense, and a Bayes decision in the broad sense are introduced. It is proved that any Bayes decision in the broad sense with positive prior discrete density is admissible in the restricted sense. The class of Bayes decisions is shown to be complete under certain conditions on the loss function. Problems with a finite set of possible states are considered.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42891262","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}
Abstract A period of a Boolean function f(x1, …, xn) is a binary n-tuple a = (a1, …, an) that satisfies the identity f(x1 + a1, …, xn + an) = f(x1, …, xn). A Boolean function is periodic if it admits a nonzero period. We propose an algorithm that takes the Zhegalkin polynomial of a Boolean function f(x1, …, xn) as the input and finds a basis of the space of all periods of f(x1, …, xn). The complexity of this algorithm is nO(d), where d is the degree of the function f. As a corollary we show that a basis of the space of all periods of a Boolean function specified by the Zhegalkin polynomial of a bounded degree may be found with complexity which is polynomial in the number of variables.
{"title":"Finding periods of Zhegalkin polynomials","authors":"S. Selezneva","doi":"10.1515/dma-2022-0012","DOIUrl":"https://doi.org/10.1515/dma-2022-0012","url":null,"abstract":"Abstract A period of a Boolean function f(x1, …, xn) is a binary n-tuple a = (a1, …, an) that satisfies the identity f(x1 + a1, …, xn + an) = f(x1, …, xn). A Boolean function is periodic if it admits a nonzero period. We propose an algorithm that takes the Zhegalkin polynomial of a Boolean function f(x1, …, xn) as the input and finds a basis of the space of all periods of f(x1, …, xn). The complexity of this algorithm is nO(d), where d is the degree of the function f. As a corollary we show that a basis of the space of all periods of a Boolean function specified by the Zhegalkin polynomial of a bounded degree may be found with complexity which is polynomial in the number of variables.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44438184","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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