Abstract We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.
{"title":"Linear recurrent relations, power series distributions, and generalized allocation scheme","authors":"A. N. Timashev","doi":"10.1515/dma-2022-0004","DOIUrl":"https://doi.org/10.1515/dma-2022-0004","url":null,"abstract":"Abstract We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"47 - 58"},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42855566","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 prove that an arbitrary Boolean function may be implemented by an irredundant Boolean circuit over an arbitrary finite complete basis so that the circuit admits a single diagnostic test of length at most 4 with respect to inversion faults at gate outputs.
{"title":"Single diagnostic tests for inversion faults of gates in circuits over arbitrary bases","authors":"Iľya G. Liubich, Dmitriy S. Romanov","doi":"10.1515/dma-2022-0001","DOIUrl":"https://doi.org/10.1515/dma-2022-0001","url":null,"abstract":"Abstract We prove that an arbitrary Boolean function may be implemented by an irredundant Boolean circuit over an arbitrary finite complete basis so that the circuit admits a single diagnostic test of length at most 4 with respect to inversion faults at gate outputs.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"1 - 9"},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46323274","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 A generalization of the method of C. Carlet for constructing differentially 4-uniform permutations of binary vector spaces in even dimension 2k is suggested. It consists in restricting APN-functions in 2k+1 variables to a linear manifold of dimension 2k. The general construction of the method is proposed and a criterion for its applicability is established. Power permutations to which this construction is applicable are completely described and a class of suitable not one-to-one functions is presented.
{"title":"A method of construction of differentially 4-uniform permutations over Vm for even m","authors":"Stepan A. Davydov, I. A. Kruglov","doi":"10.1515/dma-2021-0033","DOIUrl":"https://doi.org/10.1515/dma-2021-0033","url":null,"abstract":"Abstract A generalization of the method of C. Carlet for constructing differentially 4-uniform permutations of binary vector spaces in even dimension 2k is suggested. It consists in restricting APN-functions in 2k+1 variables to a linear manifold of dimension 2k. The general construction of the method is proposed and a criterion for its applicability is established. Power permutations to which this construction is applicable are completely described and a class of suitable not one-to-one functions is presented.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"383 - 388"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45054072","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 the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex σ̃ = (σ1, …, σn), σ1, …, σn ∈ {0, 1}, we mean the Boolean function φσ~(n) $begin{array}{} varphi^{(n)}_{tildesigma} end{array} $(x1, …, xn) which is equal to 1 on those and only those tuples of values that differ from the tuple σ̃ only in one digit. It is shown that the number 3n − 2 of contacts is necessary and sufficient for implementation of φσ~(n) $begin{array}{} varphi^{(n)}_{tildesigma} end{array} $(x̃) by a contact circuit.
{"title":"Minimal contact circuits for characteristic functions of spheres","authors":"N. P. Redkin","doi":"10.1515/dma-2021-0036","DOIUrl":"https://doi.org/10.1515/dma-2021-0036","url":null,"abstract":"Abstract We study the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex σ̃ = (σ1, …, σn), σ1, …, σn ∈ {0, 1}, we mean the Boolean function φσ~(n) $begin{array}{} varphi^{(n)}_{tildesigma} end{array} $(x1, …, xn) which is equal to 1 on those and only those tuples of values that differ from the tuple σ̃ only in one digit. It is shown that the number 3n − 2 of contacts is necessary and sufficient for implementation of φσ~(n) $begin{array}{} varphi^{(n)}_{tildesigma} end{array} $(x̃) by a contact circuit.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"403 - 408"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42505948","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 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.
{"title":"Ergodicity of the probabilistic converter, a serial connection of two automata","authors":"I. A. Kruglov","doi":"10.1515/dma-2021-0034","DOIUrl":"https://doi.org/10.1515/dma-2021-0034","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"389 - 396"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46221079","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 Let Γ be a distance-regular graph of diameter 3 with c2 = 2 (any two vertices with distance 2 between them have exactly two common neighbors). Then the neighborhood Δ of the vertex w in Γ is a partial line space. In view of the Brouwer–Neumaier result either Δ is the union of isolated (λ + 1)-cliques or the degrees of vertices k ≥ λ(λ + 3)/2, and in the case of equality k = 5, λ = 2 and Γ is the icosahedron graph. A. A. Makhnev, M. P. Golubyatnikov and Wenbin Guo have investigated distance-regular graphs Γ of diameter 3 such that Γ3 is the pseudo-geometrical network graph. They have found a new infinite set {2u2 −2m2 + 4m − 3,2u2 −2m2,u2 − m2 + 4m −2;1, 2, u2 − m2} of feasible intersection arrays for such graphs with c2 = 2. Here we prove that some distance-regular graphs from this set do not exist. It is proved also that distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist.
{"title":"On distance-regular graphs with c2 = 2","authors":"A. Makhnev, M. S. Nirova","doi":"10.1515/dma-2021-0035","DOIUrl":"https://doi.org/10.1515/dma-2021-0035","url":null,"abstract":"Abstract Let Γ be a distance-regular graph of diameter 3 with c2 = 2 (any two vertices with distance 2 between them have exactly two common neighbors). Then the neighborhood Δ of the vertex w in Γ is a partial line space. In view of the Brouwer–Neumaier result either Δ is the union of isolated (λ + 1)-cliques or the degrees of vertices k ≥ λ(λ + 3)/2, and in the case of equality k = 5, λ = 2 and Γ is the icosahedron graph. A. A. Makhnev, M. P. Golubyatnikov and Wenbin Guo have investigated distance-regular graphs Γ of diameter 3 such that Γ3 is the pseudo-geometrical network graph. They have found a new infinite set {2u2 −2m2 + 4m − 3,2u2 −2m2,u2 − m2 + 4m −2;1, 2, u2 − m2} of feasible intersection arrays for such graphs with c2 = 2. Here we prove that some distance-regular graphs from this set do not exist. It is proved also that distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"397 - 401"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44336702","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 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.
{"title":"Multiaffine polynomials over a finite field","authors":"S. Selezneva","doi":"10.1515/dma-2021-0038","DOIUrl":"https://doi.org/10.1515/dma-2021-0038","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"421 - 430"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44799650","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 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.
{"title":"Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues","authors":"V. G. Ryabov","doi":"10.1515/dma-2021-0037","DOIUrl":"https://doi.org/10.1515/dma-2021-0037","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"409 - 419"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47237726","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 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。
{"title":"Large deviations of branching process in a random environment. II","authors":"A. V. Shklyaev","doi":"10.1515/dma-2021-0039","DOIUrl":"https://doi.org/10.1515/dma-2021-0039","url":null,"abstract":"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.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"431 - 447"},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45763053","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 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. $begin{array}{} displaystyle leftlceil frac{n}{2} rightrceil leq L_{tilde{gamma}}^{rm shifts, diagn}(n) leq n. end{array}$
{"title":"Diagnostic tests under shifts with fixed filling tuple","authors":"Grigorii V. Antiufeev","doi":"10.1515/dma-2021-0027","DOIUrl":"https://doi.org/10.1515/dma-2021-0027","url":null,"abstract":"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. $begin{array}{} displaystyle leftlceil frac{n}{2} rightrceil leq L_{tilde{gamma}}^{rm shifts, diagn}(n) leq n. end{array}$","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"309 - 313"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42813652","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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