The propagation criterion for monotone Boolean functions with least vector support sets consisting of one or two vectors is studied. We obtain necessary and sufficient conditions for the validity of the propagation criterion for a vector in terms of the Hamming weights of vectors in least vector support set depending on whether these vectors share some nonzero components with the given vector. We find the cardinality of the set of vectors satisfying the propagation criterion for such functions.
{"title":"Propagation criterion for monotone Boolean functions with least vector support set of 1 or 2 elements","authors":"Gleb A. Isaev","doi":"10.1515/dma-2024-0006","DOIUrl":"https://doi.org/10.1515/dma-2024-0006","url":null,"abstract":"\u0000 The propagation criterion for monotone Boolean functions with least vector support sets consisting of one or two vectors is studied. We obtain necessary and sufficient conditions for the validity of the propagation criterion for a vector in terms of the Hamming weights of vectors in least vector support set depending on whether these vectors share some nonzero components with the given vector. We find the cardinality of the set of vectors satisfying the propagation criterion for such functions.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140771040","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}
Replacing the equality in the condition of the homomorphism of a multi-based universal algebra with inequality leads to the definition of an inverse homomorphism of a multi-based universal algebra. In this paper methods for constructing nontrivial inverse homomorphisms of finite groups are studied.
{"title":"Inverse homomorphisms of finite groups","authors":"Igor G. Shaposhnikov","doi":"10.1515/dma-2024-0010","DOIUrl":"https://doi.org/10.1515/dma-2024-0010","url":null,"abstract":"\u0000 Replacing the equality in the condition of the homomorphism of a multi-based universal algebra with inequality leads to the definition of an inverse homomorphism of a multi-based universal algebra. In this paper methods for constructing nontrivial inverse homomorphisms of finite groups are studied.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140782340","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}
A general cellular circuit of functional and switching elements (CCFSE) is a mathematical model of integral circuits (ICs), which takes into account peculiarities of their physical synthesis. A principal feature of this model distinguishing it from the well-known classes of circuits of gates (CGs) is the presence of additional requirements on the geometry of the circuit which ensure the accounting of the necessary routing resources for IC creation. The complexity of implementation of a multiplexer function of Boolean algebra (FBA) in different classes of circuits has been extensively studied. In the present paper, we give asymptotically sharp upper and lower estimates for the area of a CCFSE implementing a multiplexer FBA of order n. We construct a family of circuit multiplexers of order n of area equal to the halved upper estimate, and provide a method of delivering the corresponding lower estimate.
功能和开关元件通用单元电路（CCFSE）是一种积分电路（IC）数学模型，它考虑到了积分电路物理合成的特殊性。该模型有别于众所周知的门电路（CG）的一个主要特点是，它对电路的几何形状有额外的要求，以确保为集成电路的创建提供必要的路由资源。关于在不同类别电路中实现布尔代数多路复用器函数（FBA）的复杂性，已经进行了广泛的研究。在本文中，我们给出了实现 n 阶多路复用器 FBA 的 CCFSE 面积的近似尖锐的上限和下限估计值。我们构建了一个面积等于上限估计值减半的 n 阶电路多路复用器系列，并提供了提供相应下限估计值的方法。
{"title":"Asymptotically sharp estimates for the area of multiplexers in the cellular circuit model","authors":"S. A. Lozhkin, V. S. Zizov","doi":"10.1515/dma-2024-0009","DOIUrl":"https://doi.org/10.1515/dma-2024-0009","url":null,"abstract":"\u0000 A general cellular circuit of functional and switching elements (CCFSE) is a mathematical model of integral circuits (ICs), which takes into account peculiarities of their physical synthesis. A principal feature of this model distinguishing it from the well-known classes of circuits of gates (CGs) is the presence of additional requirements on the geometry of the circuit which ensure the accounting of the necessary routing resources for IC creation. The complexity of implementation of a multiplexer function of Boolean algebra (FBA) in different classes of circuits has been extensively studied. In the present paper, we give asymptotically sharp upper and lower estimates for the area of a CCFSE implementing a multiplexer FBA of order n. We construct a family of circuit multiplexers of order n of area equal to the halved upper estimate, and provide a method of delivering the corresponding lower estimate.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140769112","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}
We consider two parsimonious models of binary high-order Markov chains and discover their ability to approximate arbitrary high-order Markov chains. Two types of global measures for approximation accuracy are introduced, theoretical and experimental results are obtained for these measures and for the considered parsimonious models. New consistent statistical parameter estimator is constructed for parsimonious model based on two-layer artificial neural network.
{"title":"On the approximation of high-order binary Markov chains by parsimonious models","authors":"Yuriy S. Kharin, V. Voloshko","doi":"10.1515/dma-2024-0007","DOIUrl":"https://doi.org/10.1515/dma-2024-0007","url":null,"abstract":"\u0000 We consider two parsimonious models of binary high-order Markov chains and discover their ability to approximate arbitrary high-order Markov chains. Two types of global measures for approximation accuracy are introduced, theoretical and experimental results are obtained for these measures and for the considered parsimonious models. New consistent statistical parameter estimator is constructed for parsimonious model based on two-layer artificial neural network.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140796468","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}
We study the complexity of implementation of systems of monomials by composition circuits. Here, by the complexity we mean the smallest possible number of operations required for computation of the system of monomials from the variables; under this approach, results of intermediate computations can be used several times. The main result of this paper establishes, for an arbitrary system of three monomials of two variables without zeroth powers, a formula for the complexity of their joint implementation by composition circuits with an additive error which is either 1 or 0.
{"title":"On the complexity of implementation of a system of three monomials of two variables by composition circuits","authors":"Sergey A. Korneev","doi":"10.1515/dma-2024-0008","DOIUrl":"https://doi.org/10.1515/dma-2024-0008","url":null,"abstract":"\u0000 We study the complexity of implementation of systems of monomials by composition circuits. Here, by the complexity we mean the smallest possible number of operations required for computation of the system of monomials from the variables;\u0000under this approach, results of intermediate computations can be used several times. The main result of this paper establishes, for an arbitrary system of three monomials of two variables without zeroth powers, a formula for the complexity of their joint implementation by composition circuits with an additive error which is either 1 or 0.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140775246","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 the sequence {ξn,t}t≥1 of controlled critical branching processes with immigration, where n = 1, 2, … is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n $begin{array}{} sqrt{n} end{array} $ converge to the distribution of a random variable whose square has a gamma distribution.
摘要 我们考虑了有移民的受控临界分支过程序列{ξn,t}t≥1，其中n = 1, 2, ...是限制种群规模的整数参数。研究表明，对于 n → ∞，所考虑的分支过程的静态分布以 n $begin{array}{} 归一化。sqrt{n}end{array} $ 收敛到其平方具有伽马分布的随机变量的分布。
{"title":"Limit theorem for stationary distribution of a critical controlled branching process with immigration","authors":"Vladimir I. Vinokurov","doi":"10.1515/dma-2023-0030","DOIUrl":"https://doi.org/10.1515/dma-2023-0030","url":null,"abstract":"Abstract We consider the sequence {ξn,t}t≥1 of controlled critical branching processes with immigration, where n = 1, 2, … is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n $begin{array}{} sqrt{n} end{array} $ converge to the distribution of a random variable whose square has a gamma distribution.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139328386","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 the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values 1 and –1 with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic TFourier, the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of TFourier. A hypothesis about the limit distribution of TFourier is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.
摘要 我们考虑的问题是测试这样一个假设，即被测序列是一个独立随机变量序列，其取值 1 和 -1 的概率相等。为了解决这个问题，NIST 软件包中的离散傅立叶变换（频谱）检验使用了统计量 TFourier，但其确切的极限分布是未知的。本文提出了一种新的统计量，并确定了其极限分布。这一新统计量是对 TFourier 统计量的轻微修改。Pareschi F., Rovatti R. 和 Setti G. 提出的数值实验证实了这一假设。
{"title":"Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence","authors":"M. P. Savelov","doi":"10.1515/dma-2023-0029","DOIUrl":"https://doi.org/10.1515/dma-2023-0029","url":null,"abstract":"Abstract We consider the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values 1 and –1 with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic TFourier, the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of TFourier. A hypothesis about the limit distribution of TFourier is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330293","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 class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H, and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2m, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.
摘要 索引为 2 的循环子群的非阿贝尔 2 群 H 类包括二面群、广义四元组、半二面群和模态最大循环群，它们在离散数学和密码学中有着广泛的应用。我们介绍了群 H 上的片断四元数变换，并提出了它们的双射性标准。对于阶数为 2m 的广义四元数群，我们在所考虑的片断-夸西亚芬变换类中获得了正交变换、完全变换及其左类似物的完整分类。我们还评估了它们的心数。
{"title":"Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions","authors":"B. Pogorelov, M. Pudovkina","doi":"10.1515/dma-2023-0028","DOIUrl":"https://doi.org/10.1515/dma-2023-0028","url":null,"abstract":"Abstract The class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H, and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2m, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330860","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 recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $begin{array}{} displaystyle [sqrt{x}]. end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.
{"title":"On polynomial-modular recursive sequences","authors":"S. Marchenkov","doi":"10.1515/dma-2023-0027","DOIUrl":"https://doi.org/10.1515/dma-2023-0027","url":null,"abstract":"Abstract We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $begin{array}{} displaystyle [sqrt{x}]. end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139328931","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 a nonempty set ω of primes, V. A. Vedernikov had constructed ω-fibred formations of groups via function methods. We study lattice properties of ω-fibred formations of finite groups with direction δ satisfying the condition δ0 ≤ δ. The lattice ωδFθ of all ω-fibred formations with direction δ and θ-valued ω-satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ωδF, ωδFτ, τωδF, ωδnF of ω-fibred formations of groups are shown to be algebraic.
摘要 对于素数的非空集ω，V. A. Vedernikov 通过函数方法构造了群ω-纤维形式。我们研究了满足δ0 ≤ δ条件的有限群方向为δ的ω纤维网格的网格性质。在网格的网格θ是代数的条件下，所有方向为δ的ω纤维网格和θ值ω卫星的网格ωδFθ被证明是代数的。作为一个推论，ωδF，ωδFτ，τωδF，ωδnF 的ω-纤维形式群的网格被证明是代数的。
{"title":"On algebraicity of lattices of ω-fibred formations of finite groups","authors":"Serafim P. Maksakov, M. Sorokina","doi":"10.1515/dma-2023-0026","DOIUrl":"https://doi.org/10.1515/dma-2023-0026","url":null,"abstract":"Abstract For a nonempty set ω of primes, V. A. Vedernikov had constructed ω-fibred formations of groups via function methods. We study lattice properties of ω-fibred formations of finite groups with direction δ satisfying the condition δ0 ≤ δ. The lattice ωδFθ of all ω-fibred formations with direction δ and θ-valued ω-satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ωδF, ωδFτ, τωδF, ωδnF of ω-fibred formations of groups are shown to be algebraic.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139330317","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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