Abstract We present a class of irregular languages defined by means of the change of number systems.
摘要我们提出了一类由数制变化定义的不规则语言。
{"title":"On a class of irregular languages","authors":"Kirill I. Groshev","doi":"10.1515/dma-2022-0031","DOIUrl":"https://doi.org/10.1515/dma-2022-0031","url":null,"abstract":"Abstract We present a class of irregular languages defined by means of the change of number systems.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"379 - 382"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42106751","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.
得到了一组具有固定权值的相关免疫n元布尔函数集的基数的抽象估计。
{"title":"Some cardinality estimates for the set of correlation-immune Boolean functions","authors":"E. Karelina","doi":"10.1515/dma-2022-0032","DOIUrl":"https://doi.org/10.1515/dma-2022-0032","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":"32 1","pages":"383 - 388"},"PeriodicalIF":0.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43018702","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 consider local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{xi }_{1}}+cdots +{{xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−
{"title":"Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants","authors":"Konstantin Yu. Denisov","doi":"10.1515/dma-2022-0026","DOIUrl":"https://doi.org/10.1515/dma-2022-0026","url":null,"abstract":"Abstract We consider local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{xi }_{1}}+cdots +{{xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−<h<0 $mathbf{E}exp left( h{{xi }_{i}} right)<infty text{ if }{{h}^{-}}<h<0$for some h−<−1. ${{h}^{-}}<-1.$Under these assumptions, we find the asymptotic representation of local probabilities P(Zn=⌊ exp(θn) ⌋) for θ∈[ θ1,θ2 ]⊂(μ−;μ) $mathbf{P}left( {{Z}_{n}}=leftlfloor exp (theta n) rightrfloor right)text{ for }theta in left[ {{theta }_{1}},{{theta }_{2}} right]subset left( {{mu }^{-}};mu right)$for some non-negative μ−.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"313 - 323"},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46693708","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 by means of results obtained in part of the paper we show that these estimates also hold in the dimension 5.
{"title":"Estimates of lengths of shortest nonzero vectors in some lattices, II","authors":"A. S. Rybakov","doi":"10.1515/dma-2022-0028","DOIUrl":"https://doi.org/10.1515/dma-2022-0028","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 by means of results obtained in part of the paper we show that these estimates also hold in the dimension 5.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"341 - 358"},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46127729","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 For Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.
{"title":"Classification of Hadamard products of one-codimensional subcodes of Reed–Muller codes","authors":"I. Chizhov, M. Borodin","doi":"10.1515/dma-2022-0025","DOIUrl":"https://doi.org/10.1515/dma-2022-0025","url":null,"abstract":"Abstract For Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"297 - 311"},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591335","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 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":"32 1","pages":"369 - 378"},"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":"32 1","pages":"325 - 340"},"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":"32 1","pages":"359 - 367"},"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":"32 1","pages":"233 - 245"},"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":"32 1","pages":"285 - 295"},"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}
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