Abstract A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.
{"title":"Group service system with three queues and load balancing","authors":"M. P. Savelov","doi":"10.1515/dma-2022-0019","DOIUrl":"https://doi.org/10.1515/dma-2022-0019","url":null,"abstract":"Abstract A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"219 - 231"},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42383717","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 We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.
{"title":"Invertible matrices over some quotient rings: identification, generation, and analysis","authors":"V. Vysotskaya, L. Vysotsky","doi":"10.1515/dma-2022-0022","DOIUrl":"https://doi.org/10.1515/dma-2022-0022","url":null,"abstract":"Abstract We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form xr − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"263 - 278"},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48503393","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 class of one-dimensional discrete power series distributions containing negative binomial distributions is considered. Properties of distributions of this class are investigated. Limit theorems generalizing similar theorems for the negative binomial distributions are proved. The proofs are based both on elementary asymptotic methods and on a modification of saddle-point method.
{"title":"On a generalization of class of negative binomial distributions","authors":"Alexander N. Timashev","doi":"10.1515/dma-2022-0021","DOIUrl":"https://doi.org/10.1515/dma-2022-0021","url":null,"abstract":"Abstract A class of one-dimensional discrete power series distributions containing negative binomial distributions is considered. Properties of distributions of this class are investigated. Limit theorems generalizing similar theorems for the negative binomial distributions are proved. The proofs are based both on elementary asymptotic methods and on a modification of saddle-point method.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"247 - 261"},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48611853","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 Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such a circuit implementing a map f:Z2n→Z2n, $fcolon mathbb Z_2^n to mathbb Z_2^n,$we study the Shannon complexity function L(n, q) under the condition that the number of additional inputs is q = O(n2). For this range of q, it is shown that L(n,q)≍n2n/log2n. $L(n,q) asymp n2^n mathop / log_2 n.$We show that L(n,q)≍n2n/log2(n+q) $L(n,q) asymp n2^n mathop / log_2 (n+q)$for all q≲n2n−⌈n/ϕ(n)⌉, $q lesssim n2^{n-lceil n mathop / phi(n)rceil},$where ϕ(n)→∞andn/ϕ(n)−log2n→∞asn→∞. $phi(n) to infty {text {and}} ,n mathop / phi(n) - log_2 n to infty ,{text {as}}, n to infty.$
{"title":"On synthesis of reversible circuits consisting of NOT, CNOT, 2-CNOT gates with small number of additional inputs","authors":"D. Zakablukov","doi":"10.1515/dma-2022-0023","DOIUrl":"https://doi.org/10.1515/dma-2022-0023","url":null,"abstract":"Abstract Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such a circuit implementing a map f:Z2n→Z2n, $fcolon mathbb Z_2^n to mathbb Z_2^n,$we study the Shannon complexity function L(n, q) under the condition that the number of additional inputs is q = O(n2). For this range of q, it is shown that L(n,q)≍n2n/log2n. $L(n,q) asymp n2^n mathop / log_2 n.$We show that L(n,q)≍n2n/log2(n+q) $L(n,q) asymp n2^n mathop / log_2 (n+q)$for all q≲n2n−⌈n/ϕ(n)⌉, $q lesssim n2^{n-lceil n mathop / phi(n)rceil},$where ϕ(n)→∞andn/ϕ(n)−log2n→∞asn→∞. $phi(n) to infty {text {and}} ,n mathop / phi(n) - log_2 n to infty ,{text {as}}, n to infty.$","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"279 - 284"},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49231641","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 paper is concerned with sources of faults associated with commutative principal ideal rings. Tables of faults of such sources are known to correspond to Cayley multiplication tables in rings, whose elements are replaced by the values of a Boolean function of these elements. For such rings, the concepts of a diagnostic test and the Shannon function for the length of a diagnostic test are introduced in a natural way. It is shown that if A is a principal ideal ring with only one prime ideal p ≠ A, and if pn = 0 for some n ∈ ℕ, then, for this ring, the Shannon length function of a diagnostic test has the form Ldiagn(A, n) = Θ(n). We also define an easily testable functions, i.e., a function with respect to which the order of growth of the length of a diagnostic test with respect to this function is equal to the logarithm of the number of pairwise distinct columns of the table of faults. A link between easily testable functions and column separation of tables of faults for two concrete sources of faults is established.
{"title":"Diagnostic tests for discrete functions defined on rings","authors":"G. V. Antyufeev","doi":"10.1515/dma-2022-0014","DOIUrl":"https://doi.org/10.1515/dma-2022-0014","url":null,"abstract":"Abstract The paper is concerned with sources of faults associated with commutative principal ideal rings. Tables of faults of such sources are known to correspond to Cayley multiplication tables in rings, whose elements are replaced by the values of a Boolean function of these elements. For such rings, the concepts of a diagnostic test and the Shannon function for the length of a diagnostic test are introduced in a natural way. It is shown that if A is a principal ideal ring with only one prime ideal p ≠ A, and if pn = 0 for some n ∈ ℕ, then, for this ring, the Shannon length function of a diagnostic test has the form Ldiagn(A, n) = Θ(n). We also define an easily testable functions, i.e., a function with respect to which the order of growth of the length of a diagnostic test with respect to this function is equal to the logarithm of the number of pairwise distinct columns of the table of faults. A link between easily testable functions and column separation of tables of faults for two concrete sources of faults is established.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"147 - 153"},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47325590","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":"32 1","pages":"155 - 176"},"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":"32 1","pages":"193 - 205"},"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":"33 1","pages":"189 - 198"},"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":"32 1","pages":"177 - 192"},"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}
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