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Ergodicity of the probabilistic converter, a serial connection of two automata 概率转换器的遍历性,两个自动机的串行连接
IF 0.5 Q4 Mathematics Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0034
I. A. Kruglov
Abstract The paper provides necessary and sufficient conditions for the the ergodicity of a serial connection of automata under which the output sequence of a substitution Mealy automaton is fed to the input of an output-free substitution automaton. It is shown that the condition of complete indecomposability of the state transition probability matrix of a Mealy automaton provides a sufficient condition for ergodicity of the probabilistic converter as a serial connection of automata. It is also shown that if the partial state transition functions of a Mealy automaton commute, then the condition of ergodicity of a serial connection is equivalent to that of both original probabilistic converters.
摘要本文给出了自动机串行连接遍历性的充要条件,在该条件下,替换Mealy自动机的输出序列被馈送到无输出替换自动机的输入。结果表明,Mealy自动机的状态转移概率矩阵完全不可分解的条件为作为自动机串行连接的概率变换器的遍历性提供了一个充分条件。还证明了如果Mealy自动机的部分状态转移函数是可交换的,那么串行连接的遍历性条件与两个原始概率转换器的遍历性等价。
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引用次数: 0
Multiaffine polynomials over a finite field 有限域上的多仿射多项式
IF 0.5 Q4 Mathematics Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0038
S. Selezneva
Abstract We consider polynomials f(x1, …, xn) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f(x1, …, xn) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.
摘要我们考虑有限域上的多项式f(x1,…,xn),它具有以下性质:对于域的某个元素b,方程f(x1…,xn)=b的解集与该域上某个线性方程组的解集一致。这样的多项式相对于右手边b是多仿射的。我们得到了有限域上多仿射多项式的性质。然后,我们证明了检查相对于给定右手边的多仿射可以通过具有多项式(根据变量的数量和输入多项式的和数)复杂性的算法来完成。除此之外,我们还证明了在正判决的情况下,可以恢复相应的线性方程组,其复杂性也是相同参数下的多项式。
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引用次数: 1
Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues 仿射类似物逼近q值逻辑函数对线性流形的限制
IF 0.5 Q4 Mathematics Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0037
V. G. Ryabov
Abstract For a finite q-element field Fq, we established a relation between parameters characterizing the measure of affine approximation of a q-valued logic function and similar parameters for its restrictions to linear manifolds. For q > 2, an analogue of the Parseval identity with respect to these parameters is proved, which implies the meaningful upper estimates qn−1(q − 1) − qn/2−1 and qr−1(q − 1) − qr/2−1, for the nonlinearity of an n-place q-valued logic function and of its restrictions to manifolds of dimension r. Estimates characterizing the distribution of nonlinearity on manifolds of fixed dimension are obtained.
摘要对于有限q元域Fq,建立了表征q值逻辑函数仿射逼近测度的参数与其约束线性流形的类似参数之间的关系。对于q > 2,证明了关于这些参数的Parseval恒等式的一个类比,得到了n位q值逻辑函数的非线性及其对r维流形的限制的有意义的上估计qn−1(q−1)−qn/2−1和qr−1(q−1)−qr/2−1。得到了表征固定维流形上非线性分布的估计。
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引用次数: 4
Large deviations of branching process in a random environment. II 随机环境中分支过程的大偏差。2
IF 0.5 Q4 Mathematics Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0039
A. V. Shklyaev
Abstract We consider the probabilities of large deviations for the branching process Zn in a random environment, which is formed by independent identically distributed variables. It is assumed that the associated random walk Sn = ξ1 + … + ξn has a finite mean μ and satisfies the Cramér condition E ehξi < ∞, 0 < h < h+. Under additional moment constraints on Z1, the exact asymptotic of the probabilities P (ln Zn ∈ [x, x + Δn)) is found for the values x/n varying in the range depending on the type of process, and for all sequences Δn that tend to zero sufficiently slowly as n → ∞. A similar theorem is proved for a random process in a random environment with immigration.
摘要考虑由独立同分布变量构成的随机环境中分支过程Zn的大偏差概率。假设关联随机游走Sn = ξ1 +…+ ξn有一个有限均值μ,且满足cramsamir条件E ehξi <∞,0 < h < h+。在Z1的附加矩约束下,对于值x/n在取决于过程类型的范围内变化,以及对于所有在n→∞时足够缓慢地趋于零的序列Δn,找到了概率P (ln Zn∈[x, x + Δn))的确切渐近。对于随机环境下的随机过程,也证明了一个类似的定理。
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引用次数: 4
Diagnostic tests under shifts with fixed filling tuple 具有固定填充元组的轮班下的诊断测试
IF 0.5 Q4 Mathematics Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0027
Grigorii V. Antiufeev
Abstract We consider a fault source under which the fault functions are obtained from the original function f(x̃n) ∈ P2n $begin{array}{} displaystyle P_2^n end{array}$ by a left shift of values of the Boolean variables by at most n. For the vacant positions of the variables, the values are selected from a given filling tuple γ̃ = (γ1, γ2, …, γn) ∈ E2n $begin{array}{} displaystyle E^n_2 end{array}$, which also moves to the left by the number of positions corresponding to a specific fault function. The problem of diagnostic of faults of this kind is considered. We show that the Shannon function Lγ~shifts,diagn(n), $begin{array}{} displaystyle L_{tilde{gamma}}^{rm shifts, diagn}(n), end{array}$ which is equal to the smallest necessary test length for diagnostic of any n-place Boolean function with respect to a described fault source, satisfies the inequality n2≤Lγ~shifts,diagn(n)≤n. $begin{array}{} displaystyle leftlceil frac{n}{2} rightrceil leq L_{tilde{gamma}}^{rm shifts, diagn}(n) leq n. end{array}$
摘要我们考虑一个故障源,在该故障源下,通过布尔变量的值最多左移n,从原始函数f(xõn)∈P2n$bearning{array}{}displaystyle P_2^nend{array}$获得故障函数。对于变量的空位,这些值是从给定的填充元组γ=(γ1,γ2,…,γn)∈E2n$begin{array}{}displaystyle E^n2end{array}$中选择的,它也向左移动与特定故障函数对应的位置数。考虑了这类故障的诊断问题。我们证明了Shannon函数Lγ~shift,diagn(n),$bbegin{array}{}displaystyle L_$begin{array}{}displaystyleleftlceilfrac{n}{$
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引用次数: 0
On the complexity of monotone circuits for threshold symmetric Boolean functions 关于阈值对称布尔函数单调电路的复杂性
IF 0.5 Q4 Mathematics Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0031
I. Sergeev
Abstract The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( n $begin{array}{} sqrt n end{array} $), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.
摘要通过基{∧,∧}上的电路实现阈值k=O(1)的阈值对称n位布尔函数的复杂性不超过2 log2k·n+O(n)。此外,还证明了阈值-2函数的复杂度为2n+θ(n$begin{array}{}sqrt nend{array}$),并证明了阈值-3函数的复杂性为3n+O(logn),给出了相应的下界。
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引用次数: 0
Boolean analogues of the Pascal triangle with maximal possible number of ones 具有最大可能数目的帕斯卡三角形的布尔类比
IF 0.5 Q4 Mathematics Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0029
F. M. Malyshev
Abstract The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by s(s+1)2 $begin{array}{} displaystyle frac{s(s+1)}{2} end{array}$ elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 $begin{array}{} displaystyle frac{s(s+1)}{3} end{array}$⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.
摘要本文的目的是在由s元素的顶行定义的域GF(2)的s(s+1)2$begin{array}{}displaystylefrac{s(s+1)}{2}end{array}$元素形成的布尔三角数组Ts中找到单元的最大可能数ξ。除了最上面的一行之外,每行的每个元素都是上面一行的两个元素的和(如Pascal三角形)。证明了ξ⩽s(s+1)3$begin{array}{}displaystylefrac{s(s+1)}{3}end{array}$,并且该值仅在具有作为斐波那契级数mod 2的上排的三角形上获得。
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引用次数: 0
The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups 有限亏缺法与构造阿贝尔群的正态和几乎正态问题
IF 0.5 Q4 Mathematics Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0030
A. V. Menyachikhin
Abstract The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽2m (m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.
摘要描述了利用已知群的正态构造群的新正态(几乎正态)的有限亏缺方法。描述了一类变换,在该类变换下,所有正态(几乎正态)的集合保持不变。我们推测所有正态(几乎正态)集合是由有限亏缺法实现的变换生成的。这个猜想在所有不超过12阶的阿贝尔群上都得到了验证。利用谱-线性法和谱-微分法设计域𝔽2m (m = 4,…,8)加性群上的置换,构造了具有足够高值的最重要密码参数的正态。
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引用次数: 0
On the average-case complexity of Boolean functions under binomial distribution on their domains 二项分布布尔函数在其定义域上的平均情况复杂度
IF 0.5 Q4 Mathematics Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0028
A. V. Chashkin
Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.
摘要给定n维布尔立方体上的二项式概率分布,考虑了带条件停止的直线程序实现布尔函数的复杂性。顺序,如n→ ∞, 建立了几乎所有n位布尔函数的平均情况复杂度。
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引用次数: 0
Large deviations of branching process in a random environment 随机环境中分支过程的大偏差
IF 0.5 Q4 Mathematics Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0025
A. V. Shklyaev
Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Yn+1=AnYn + Bn, where A1, A2, … are independent identically distributed random variables and Bn may depend on {(Ak,Bk),0⩽k
摘要本文第一部分给出了由随机差分方程Yn+1=AnYn + Bn定义的序列的大偏差概率的渐近公式,其中A1, A2,…是独立的同分布随机变量,对于任意n≥1,Bn可能依赖于{(Ak,Bk),0≤k
{"title":"Large deviations of branching process in a random environment","authors":"A. V. Shklyaev","doi":"10.1515/dma-2021-0025","DOIUrl":"https://doi.org/10.1515/dma-2021-0025","url":null,"abstract":"Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Yn+1=AnYn + Bn, where A1, A2, … are independent identically distributed random variables and Bn may depend on {(Ak,Bk),0⩽k<n} $ {(A_k,B_k),0leqslant k lt n} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44535506","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}
引用次数: 4
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Discrete Mathematics and Applications
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