Abstract We analyse closed classes in k-valued logics containing all linear functions modulo k. The classes are determined by divisors d of a number k and canonical formulas for functions. We construct the lattice of all such classes for k = p2, where p is a prime, and construct fragments of the lattice for other composite k.
{"title":"Some families of closed classes in Pk defined by additive formulas","authors":"D. G. Meshchaninov","doi":"10.1515/dma-2022-0011","DOIUrl":"https://doi.org/10.1515/dma-2022-0011","url":null,"abstract":"Abstract We analyse closed classes in k-valued logics containing all linear functions modulo k. The classes are determined by divisors d of a number k and canonical formulas for functions. We construct the lattice of all such classes for k = p2, where p is a prime, and construct fragments of the lattice for other composite k.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45586835","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 show that in multivalued logic there exist a continual family of pairwise incomparable closed sets with minimal logarithmic growth rate and a continual chain of nested closed sets with minimal logarithmic growth rate. As a corollary we prove that any subset-preserving class in multivalued logic contains a continual chain of nested closed sets and a continual family of pairwise incomparable closed sets such that none of the sets is a subset of any other precomplete class.
{"title":"Continuality of classes of functions in multivalued logic with minimal logarithmic growth rate","authors":"Stepan Alekseevich Komkov","doi":"10.1515/dma-2022-0009","DOIUrl":"https://doi.org/10.1515/dma-2022-0009","url":null,"abstract":"Abstract We show that in multivalued logic there exist a continual family of pairwise incomparable closed sets with minimal logarithmic growth rate and a continual chain of nested closed sets with minimal logarithmic growth rate. As a corollary we prove that any subset-preserving class in multivalued logic contains a continual chain of nested closed sets and a continual family of pairwise incomparable closed sets such that none of the sets is a subset of any other precomplete class.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49590785","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 Compositions of n are finite sequences of positive integers (σi)i=1k $begin{array}{} (sigma_i)_{i = 1}^k end{array} $ such that σ1+σ2+⋯+σk=n. $$begin{array}{} sigma_1+sigma_2+cdots +sigma_k = n. end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.
{"title":"The site-perimeter of compositions","authors":"A. Blecher, C. Brennan, A. Knopfmacher","doi":"10.1515/dma-2022-0007","DOIUrl":"https://doi.org/10.1515/dma-2022-0007","url":null,"abstract":"Abstract Compositions of n are finite sequences of positive integers (σi)i=1k $begin{array}{} (sigma_i)_{i = 1}^k end{array} $ such that σ1+σ2+⋯+σk=n. $$begin{array}{} sigma_1+sigma_2+cdots +sigma_k = n. end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48468093","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 random permutations having uniform distribution on the set of all permutations of the n-element set with lengths of cycles belonging to a fixed set A (so-called A-permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.
{"title":"Variance of the number of cycles of random A-permutation","authors":"A. L. Yakymiv","doi":"10.1515/dma-2022-0005","DOIUrl":"https://doi.org/10.1515/dma-2022-0005","url":null,"abstract":"Abstract We consider random permutations having uniform distribution on the set of all permutations of the n-element set with lengths of cycles belonging to a fixed set A (so-called A-permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43821489","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 polynomial scheme with N outcomes. The Pearson statistic is the classical one for testing the hypothesis that the probabilities of outcomes are given by the numbers p1, …, pN. We suggest a couple of N − 2 statistics which along with the Pearson statistics constitute a set of N − 1 asymptotically jointly independent random variables, and find their limit distributions. The Pearson statistics is the square of the length of asymptotically normal random vector. The suggested statistics are coordinates of this vector in some auxiliary spherical coordinate system.
{"title":"A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic","authors":"M. P. Savelov","doi":"10.1515/dma-2022-0003","DOIUrl":"https://doi.org/10.1515/dma-2022-0003","url":null,"abstract":"Abstract We consider a polynomial scheme with N outcomes. The Pearson statistic is the classical one for testing the hypothesis that the probabilities of outcomes are given by the numbers p1, …, pN. We suggest a couple of N − 2 statistics which along with the Pearson statistics constitute a set of N − 1 asymptotically jointly independent random variables, and find their limit distributions. The Pearson statistics is the square of the length of asymptotically normal random vector. The suggested statistics are coordinates of this vector in some auxiliary spherical coordinate system.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49632954","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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 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":null,"pages":null},"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}
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