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.
{"title":"Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case","authors":"Elena V. Khvorostianskaia","doi":"10.1515/dma-2023-0019","DOIUrl":"https://doi.org/10.1515/dma-2023-0019","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"205 - 217"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47910452","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 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.
{"title":"On distance-regular graphs Γ of diameter 3 for which Γ3 is a triangle-free graph","authors":"A. Makhnev, Wenbin Guo","doi":"10.1515/dma-2023-0018","DOIUrl":"https://doi.org/10.1515/dma-2023-0018","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"199 - 204"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47785169","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 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.
{"title":"On the universality of product for classes of linear functions of two variables","authors":"A. A. Voronenko","doi":"10.1515/dma-2023-0024","DOIUrl":"https://doi.org/10.1515/dma-2023-0024","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"269 - 271"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42137330","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 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.
{"title":"On a number of particles in a marked set of cells in a general allocation scheme","authors":"A. Chuprunov","doi":"10.1515/dma-2023-0014","DOIUrl":"https://doi.org/10.1515/dma-2023-0014","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"157 - 165"},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885290","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 scatter properties of the modular addition operation for imprimitivity systems of the translation group of the binary vector space Vn = {0, 1}n. We describe all the subspaces of the space Vn that induce imprimitivity systems with worst possible scatter by the modular addition operation.
{"title":"On scatter properties of modular addition operation over imprimitivity systems of the translation group of the binary vector space","authors":"D. A. Burov","doi":"10.1515/dma-2023-0013","DOIUrl":"https://doi.org/10.1515/dma-2023-0013","url":null,"abstract":"Abstract We study scatter properties of the modular addition operation for imprimitivity systems of the translation group of the binary vector space Vn = {0, 1}n. We describe all the subspaces of the space Vn that induce imprimitivity systems with worst possible scatter by the modular addition operation.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"127 - 156"},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45068167","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 class EPℕ of exponentially-polynomial functions which can be obtained by arbitrary superpositions of the constants 0, 1 and arithmetic operations of addition, multiplication, and powering. For this class, we solve the algorithmic equality problem of two functions that assume a finite number of values. Next, this class is restricted to the class PEPℕ, in which the function xy is replaced by a sequence of functions { pix $begin{array}{} displaystyle p_i^x end{array}$}, where p0, p1, … are all prime numbers. For the class PEPℕ, the problem of membership of a function to a finitely generated class is effectively reduced to the equality problem of two functions. In turn, the last problem is effectively solved for the set of all one-place PEPℕ-functions.
{"title":"On the equality problem of finitely generated classes of exponentially-polynomial functions","authors":"S. Marchenkov","doi":"10.1515/dma-2023-0015","DOIUrl":"https://doi.org/10.1515/dma-2023-0015","url":null,"abstract":"Abstract We consider the class EPℕ of exponentially-polynomial functions which can be obtained by arbitrary superpositions of the constants 0, 1 and arithmetic operations of addition, multiplication, and powering. For this class, we solve the algorithmic equality problem of two functions that assume a finite number of values. Next, this class is restricted to the class PEPℕ, in which the function xy is replaced by a sequence of functions { pix $begin{array}{} displaystyle p_i^x end{array}$}, where p0, p1, … are all prime numbers. For the class PEPℕ, the problem of membership of a function to a finitely generated class is effectively reduced to the equality problem of two functions. In turn, the last problem is effectively solved for the set of all one-place PEPℕ-functions.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"167 - 175"},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45958357","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 fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.
{"title":"Fault-tolerant resolvability of some graphs of convex polytopes","authors":"S. Sharma, H. Raza, V. K. Bhat","doi":"10.1515/dma-2023-0016","DOIUrl":"https://doi.org/10.1515/dma-2023-0016","url":null,"abstract":"Abstract The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"177 - 187"},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46614530","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 this paper, we give a method to determine a complete set of mutually orthogonal Latin squares of order m, where m is an odd prime or power of a prime, as a group transversal of a Frobenius group.
{"title":"Mutually Orthogonal Latin Squares as Group Transversals","authors":"R. Pradhan, V. K. Jain","doi":"10.1515/dma-2023-0010","DOIUrl":"https://doi.org/10.1515/dma-2023-0010","url":null,"abstract":"Abstract In this paper, we give a method to determine a complete set of mutually orthogonal Latin squares of order m, where m is an odd prime or power of a prime, as a group transversal of a Frobenius group.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"99 - 103"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42166851","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 n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.
{"title":"Criteria for maximal nonlinearity of a function over a finite field","authors":"V. G. Ryabov","doi":"10.1515/dma-2023-0012","DOIUrl":"https://doi.org/10.1515/dma-2023-0012","url":null,"abstract":"Abstract An n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"33 1","pages":"117 - 126"},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42932067","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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