Abstract We show that triangular families of Boolean functions comprise an exponentially small fraction of proper families of a given order. We prove that if F is a proper family of Boolean functions, then the number of solutions of an equation F(x) = A is even. Finally, we describe a new class of proper families of Boolean functions.
{"title":"Properties of proper families of Boolean functions","authors":"K. Tsaregorodtsev","doi":"10.1515/dma-2022-0030","DOIUrl":"https://doi.org/10.1515/dma-2022-0030","url":null,"abstract":"Abstract We show that triangular families of Boolean functions comprise an exponentially small fraction of proper families of a given order. We prove that if F is a proper family of Boolean functions, then the number of solutions of an equation F(x) = A is even. Finally, we describe a new class of proper families of Boolean functions.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44855576","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 With each positive integer one can naturally associate a graph in the form of a tree. This paper is concerned with the average values of the number of edges, the number of leaves and the height of trees corresponding to positive integers not greater than a given boundary.
{"title":"On the “tree” structure of natural numbers","authors":"V. Iudelevich","doi":"10.1515/dma-2022-0027","DOIUrl":"https://doi.org/10.1515/dma-2022-0027","url":null,"abstract":"Abstract With each positive integer one can naturally associate a graph in the form of a tree. This paper is concerned with the average values of the number of edges, the number of leaves and the height of trees corresponding to positive integers not greater than a given boundary.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48583510","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 Boolean majority function and the generalized Boolean majority function of an even number n of variables are considered. For these functions exact values of the Walsh coefficients and the curvature are calculated.
{"title":"Curvature of the Boolean majority function","authors":"Aleksandr S. Tissin","doi":"10.1515/dma-2022-0029","DOIUrl":"https://doi.org/10.1515/dma-2022-0029","url":null,"abstract":"Abstract The Boolean majority function and the generalized Boolean majority function of an even number n of variables are considered. For these functions exact values of the Walsh coefficients and the curvature are calculated.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45802349","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 Explicit recurrent formulas for the numbers of sequences containing a given pattern given number of times are constructed. These formulas depend on the length of the sequence, the length of the pattern and its period only. By means of these results one may find the distribution of statistics of the NIST overlapping matching test for binary sequences and arbitrary pattern parameters.
{"title":"Formulas for the numbers of sequences containing a given pattern given number of times","authors":"A. A. Serov","doi":"10.1515/dma-2022-0020","DOIUrl":"https://doi.org/10.1515/dma-2022-0020","url":null,"abstract":"Abstract Explicit recurrent formulas for the numbers of sequences containing a given pattern given number of times are constructed. These formulas depend on the length of the sequence, the length of the pattern and its period only. By means of these results one may find the distribution of statistics of the NIST overlapping matching test for binary sequences and arbitrary pattern parameters.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42907915","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 An approach to the construction of efficient algorithms for the exact computation of distributions of statistics by means of the Markov chains is described. The Pearson statistic, the number of empty cells for random allocations of particles, and the Kolmogorov – Smirnov statistic are considered as examples. Possibilities of extending the approach are discussed, in particular to the computation of the joint distributions of statistics.
{"title":"Computation of distributions of statistics by means of Markov chains","authors":"A. M. Zubkov, M. Filina","doi":"10.1515/dma-2022-0024","DOIUrl":"https://doi.org/10.1515/dma-2022-0024","url":null,"abstract":"Abstract An approach to the construction of efficient algorithms for the exact computation of distributions of statistics by means of the Markov chains is described. The Pearson statistic, the number of empty cells for random allocations of particles, and the Kolmogorov – Smirnov statistic are considered as examples. Possibilities of extending the approach are discussed, in particular to the computation of the joint distributions of statistics.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41514133","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":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":"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}
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