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On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)} 在具有交集阵列 {mn - 1, (m - 1)(n + 1), n - m + 1; 1, 1, (m - 1)(n + 1)} 的小型距离不规则图上
IF 0.5 Q4 Mathematics Pub Date : 2023-10-01 DOI: 10.1515/dma-2023-0025
A. Makhnev, M. P. Golubyatnikov
Abstract Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ3, where Γ3 is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ. Computing the parameters of Γ3 by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ3 is referred to as the inverse problem. The inverse problem for Γ3 has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ3 is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c2 = 1 and c2 = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively. In this paper in the class of graphs with the intersection arrays {mn − 1, (m − 1)(n + 1)}, {n − m + 1}; 1, 1, (m − 1)(n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.
摘要 假设Γ是一个直径为 3 的距离正则图,它有一个强正则图Γ3,其中Γ3 是顶点集与图Γ的顶点集重合的图,并且只要两个顶点在图Γ中的距离为 3,它们就是相邻的。通过图 Γ 的交点阵列计算 Γ3 的参数被视为直接问题。通过 Γ3 的参数恢复图 Γ 的交点阵列称为逆问题。A. A. Makhnev 和 M. S. Nirova 早先已经解决了 Γ3 的逆问题。在 Γ3 是一个网的伪几何图形的情况下,得到了一系列可接受的交点阵列:{c2(u2 - m2) + 2c2m - c2 - 1, c2(u2 - m2), (c2 - 1)(u2 - m2) + 2c2m - c2; 1, c2, u2 - m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov)。A. A. Makhnev, M. P. Golubyatnikov 和 A. A. Makhnev, M. S. Nirova 分别研究了 c2 = 1 和 c2 = 2 的情况。在本文中,在具有交集阵列 {mn - 1, (m - 1)(n + 1)}, {n - m + 1}; 1, 1, (m - 1)(n + 1)} 的一类图形中,发现了 {3 ≤ m ≤ 13} 的所有可容许交集阵列:{20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} 和 {350,336,15; 1, 1,336}.事实证明,不存在交集数组为 {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} 和 {90,84,7; 1, 1,84} 的图形。
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引用次数: 0
Nonlinearity of functions over finite fields 有限域上函数的非线性
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0021
V. G. Ryabov
Abstract The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity.
摘要将函数的非线性和可加性非线性分别定义为到所有仿射映射集合的Hamming距离和到具有非平凡可加转换的所有映射集合的Hamming距离。在揭示非线性与函数特征的傅里叶系数关系的基础上,找到了在任意有限域上实用的重要函数类的非线性评价的方便公式。在偶特征场的情况下,自相关系数的可加性非线性得到了类似的结果。所得到的公式可以表示具有极大可能和高度非线性和可加非线性的特定函数类。
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引用次数: 0
Short complete diagnostic tests for circuits with two additional inputs in some basis 在某些基础上对具有两个附加输入的电路进行完整的短路诊断测试
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0020
K. A. Popkov
Abstract We prove that any Boolean function in n variables can be modeled by a testable Boolean circuit with two additional inputs in the basis “conjunction, oblique conjunction, disjunction, negation” so that the circuit admits a complete diagnostic test of the length at most 2n + 3 with respect to stuck-at faults of the type 1 at gate outputs.
摘要我们证明了n个变量中的任何布尔函数都可以用一个可测试的布尔电路来建模,该电路在“连接、斜连接、析取、否定”的基础上有两个额外的输入,因此该电路允许对门输出处的1型卡在故障进行长度最多为2n+3的完整诊断测试。
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引用次数: 0
On properties of multiaffine predicates on a finite set 关于有限集上多仿射谓词的性质
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0023
S. Selezneva
Abstract We consider predicates on a finite set that are invariant with respect to an affine operation fG, where G is some Abelian group. Such predicates are said to be multiaffine for the group G. Special attention is paid to predicates that are affine for a group Gq of addition modulo q=ps, where p is a prime number and s=1. We establish the predicate multiaffinity criterion for a group Gq. Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group Gq. Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group Gq, if predicates are specified by DNF.
摘要考虑有限集合上关于仿射操作fG的不变量谓词,其中G是某个阿贝尔群。这样的谓词被称为群g的多仿射谓词。特别注意的是对加模q=ps的群Gq的仿射谓词,其中p是素数且s=1。我们建立了群Gq的谓词多亲和准则。然后引入有限集合上谓词的析取范式(DNF),得到了群Gq上多仿射谓词的析取范式的性质。最后,我们展示了如何使用这些属性来设计一个多项式算法,该算法决定了一个多仿射的谓词系统对于一个群Gq的可满足性,如果谓词是由DNF指定的。
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引用次数: 0
Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case Galton最大树大小的极限定理 – 危急情况下的Watson森林
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0019
Elena V. Khvorostianskaia
Abstract We consider a critical Galton – Watson branching process starting with N particles; the number of offsprings is supposed to have the distribution pk=(k + 1)−τ−(k + 2)−τ, k=0, 1, 2, … Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with N trees and n non-root vertices as N, n → ∞, n/Nτ ⩾ C > 0.
摘要我们认为一个关键的Galton – 从N粒子开始的Watson分支过程;子代的数量应该具有分布pk=(k + 1) -τ−(k + 2) -τ,k=0,1,2,…得到了相应Galton的最大树大小的极限分布 – 具有N棵树和N个非根顶点为N,N的Watson森林 → ∞, n/nτ⩾C > 0
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引用次数: 0
The limit joint distributions of statistics of three tests of the NIST package NIST软件包三次测试统计的极限联合分布
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0022
Maksim P. Savelov
Abstract For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.
摘要对于具有伯努利分布的独立随机变量序列,得到了NIST统计包的三个检验(“单位检验”、“块内频率检验”和“累积和检验”)统计量的联合分布和极限联合分布。在«一个块内的频率测试»中使用两个块的情况下,给出了这些统计数据的成对协方差。
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引用次数: 0
On distance-regular graphs Γ of diameter 3 for which Γ3 is a triangle-free graph 关于直径为3的距离正则图Γ,其中Γ3是无三角形图
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0018
A. Makhnev, Wenbin Guo
Abstract There exist well-known distance-regular graphs Γ of diameter 3 for which Γ3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ3 is a triangle-free graph.
存在已知的直径为3的距离正则图Γ,其中Γ3是一个无三角形图。Johnson图J(8,3)给出了一个例子,其中交叉数组{15,8,3;1,4,9}。本文讨论了具有交数组{78,50,9;1,15,60}和{174110,18,130132}的距离正则图Γ的存在性问题,其中Γ3是无三角形图。
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引用次数: 0
On the universality of product for classes of linear functions of two variables 关于两变量线性函数类乘积的普适性
IF 0.5 Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-0024
A. A. Voronenko
Abstract Earlier, the author introduced the concept of a universal function and proved the existence of universal functions for classes of linear k-valued functions of two variables for k ≥ 5. In this paper, we show that the product modulo k is a universal function for the class of linear k-valued functions of two variables if and only if k = 6l ± 1.
摘要早先,作者引入了泛函数的概念,并证明了k≥5的两变量线性k值函数类的泛函数的存在性。本文证明了乘积模k是一类二元线性k值函数的泛函数,当且仅当k=6l±1。
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引用次数: 0
Frontmatter 头版头条
Q4 Mathematics Pub Date : 2023-08-01 DOI: 10.1515/dma-2023-frontmatter4
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引用次数: 0
On a number of particles in a marked set of cells in a general allocation scheme 关于一般分配方案中标记单元集中的多个粒子
IF 0.5 Q4 Mathematics Pub Date : 2023-06-01 DOI: 10.1515/dma-2023-0014
A. Chuprunov
Abstract In a generalized allocation scheme of n particles over N cells we consider the random variable ηn,N(K) which is the number of particles in a given set consisting of K cells. We prove that if n, K, N → ∞, then under some conditions random variables ηn,N(K) are asymptotically normal, and under another conditions ηn,N(K) converge in distribution to a Poisson random variable. For the case when N → ∞ and n is a fixed number, we find conditions under which ηn,N(K) converge in distribution to a binomial random variable with parameters n and s = KN $begin{array}{} displaystyle frac{K}{N} end{array}$, 0 < K < N, multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of n particles over N cells with random variables having a power series distribution defined by the function B(β) = ln(1 − β) the conditions n, N, K → ∞, KN $begin{array}{} displaystyle frac{K}{N} end{array}$ → s, N = γ ln(n) + o(ln(n)), where 0 < s < 1, 0 < γ < ∞, are satisfied, then distributions of random variables ηn,N(K)n $begin{array}{} displaystyle frac{eta_{n,N}(K)}{n} end{array}$ converge to a beta-distribution with parameters sγ and (1 − s)γ.
摘要在n个单元上的n个粒子的广义分配方案中,我们考虑随机变量ηn,n(K),它是由K个单元组成的给定集合中的粒子数。我们证明了如果n,K,n→ ∞, 则在某些条件下,随机变量ηn,n(K)是渐近正态的,在另一个条件下,ηn,n(K)在分布上收敛为泊松随机变量。对于N→ ∞ 当n是一个固定数时,我们发现ηn,n(K)在分布中收敛于一个参数为n的二项式随机变量的条件,s=KN$beagin{array}{}displaystylefrac{K}{n}end{array}$,0<K<n,乘以一个整数系数。结果表明,对于具有幂级数分布的随机变量的n个单元上的n个粒子的广义分配方案,如果函数B(β)=ln(1−β)定义了条件n,n,K→ ∞, KN$begin{array}{}displaystylefrac{K}{→ s、 N=γln(N)+o(ln(N。
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Discrete Mathematics and Applications
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