Pub Date : 2023-12-30DOI: 10.15330/cmp.15.2.608-613
Y. V. Lavrenyuk, A.S. Oliynyk
For an arbitrary odd prime $p$, we consider groups of all $p$-automata and all finite $p$-automata. We construct minimal generating sets in both the groups of all $p$-automata and its subgroup of finite $p$-automata. The key ingredient of the proof is the lifting technique, which allows the construction of a minimal generating set in a group provided a minimal generating set in its abelian quotient is given. To find the required quotient, the elements of the groups of $p$-automata and finite $p$-automata are presented in terms of tableaux introduced by L. Kaloujnine. Using this presentation, a natural homomorphism on the additive group of all infinite sequences over the field $mathbb{Z}_p$ is defined and examined.
{"title":"Minimal generating sets in groups of $p$-automata","authors":"Y. V. Lavrenyuk, A.S. Oliynyk","doi":"10.15330/cmp.15.2.608-613","DOIUrl":"https://doi.org/10.15330/cmp.15.2.608-613","url":null,"abstract":"For an arbitrary odd prime $p$, we consider groups of all $p$-automata and all finite $p$-automata. We construct minimal generating sets in both the groups of all $p$-automata and its subgroup of finite $p$-automata. The key ingredient of the proof is the lifting technique, which allows the construction of a minimal generating set in a group provided a minimal generating set in its abelian quotient is given. To find the required quotient, the elements of the groups of $p$-automata and finite $p$-automata are presented in terms of tableaux introduced by L. Kaloujnine. Using this presentation, a natural homomorphism on the additive group of all infinite sequences over the field $mathbb{Z}_p$ is defined and examined.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":" 35","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139137175","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}
Pub Date : 2023-12-30DOI: 10.15330/cmp.15.2.594-607
M.R. Kuryliak, O. Skaskiv
Let $(lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=sumlimits_{n=0}^{+infty} a_ne^{zlambda_n}$, $zinmathbb{C}$, we denote $G_{mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $mu(z,F)=maxbig{|a_n|e^{Re(zlambda_n)} : ngeq 0big}$, respectively. It is well known that $G_mu(F), G_a(F)$ are convex domains. Let us denote $mathcal{N}_1(z):={n : Re(zlambda_n)>0}$, $mathcal{N}_2(z):={n : Re(zlambda_n)<0}$ and [alpha^{(1)}(theta) :=varliminflimits_{genfrac{}{}{0pt}{2}{nto +infty}{ninmathcal{N}_1(e^{itheta})}}frac{-ln|a_n|}{Re(e^{itheta}lambda_n)},qquad alpha^{(2)}(theta) :=varlimsuplimits_{genfrac{}{}{0pt}{2}{nto +infty}{ninmathcal{N}_2(e^{itheta})}}frac{-ln|a_n|}{Re(e^{itheta}lambda_n)}.] Assume that $a_nto 0$ as $nto +infty$. In the article, we prove the following statements. $1)$ If $alpha^{(2)}(theta)
{"title":"On the domain of convergence of general Dirichlet series with complex exponents","authors":"M.R. Kuryliak, O. Skaskiv","doi":"10.15330/cmp.15.2.594-607","DOIUrl":"https://doi.org/10.15330/cmp.15.2.594-607","url":null,"abstract":"Let $(lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=sumlimits_{n=0}^{+infty} a_ne^{zlambda_n}$, $zinmathbb{C}$, we denote $G_{mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $mu(z,F)=maxbig{|a_n|e^{Re(zlambda_n)} : ngeq 0big}$, respectively. It is well known that $G_mu(F), G_a(F)$ are convex domains. Let us denote $mathcal{N}_1(z):={n : Re(zlambda_n)>0}$, $mathcal{N}_2(z):={n : Re(zlambda_n)<0}$ and [alpha^{(1)}(theta) :=varliminflimits_{genfrac{}{}{0pt}{2}{nto +infty}{ninmathcal{N}_1(e^{itheta})}}frac{-ln|a_n|}{Re(e^{itheta}lambda_n)},qquad alpha^{(2)}(theta) :=varlimsuplimits_{genfrac{}{}{0pt}{2}{nto +infty}{ninmathcal{N}_2(e^{itheta})}}frac{-ln|a_n|}{Re(e^{itheta}lambda_n)}.] Assume that $a_nto 0$ as $nto +infty$. In the article, we prove the following statements. $1)$ If $alpha^{(2)}(theta)<alpha^{(1)}(theta)$ for some $thetain [0,pi)$ then [big{te^{itheta} : tin (alpha^{(2)}(theta),alpha^{(1)}(theta))big}subset G_mu(F)] as well as [big{te^{itheta} : tin (-infty,alpha^{(2)}(theta))cup (alpha^{(1)}(theta),+infty)big}cap G_mu(F)=emptyset.] $2)$ $G_mu(F)=bigcuplimits_{thetain [0,pi)}{z=te^{itheta} : tin (alpha^{(2)}(theta),alpha^{(1)}(theta))}.$ $3)$ If $h:=varliminflimits_{nto +infty}frac{-ln |a_n|}{ln n}in (1,+infty)$, then [Big(frac{h}{h-1}cdot G_a(F)Big)supset G_mu(F)supset G_c(F).] If $h=+infty$ then $G_a(F)=G_c(F)=G_mu(F)$, therefore $G_c(F)$ is also a convex domain.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":" 16","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139141982","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}
Pub Date : 2023-12-30DOI: 10.15330/cmp.15.2.576-593
H.A. Ganie, B. Rather, M. Aouchiche
Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.
{"title":"Reciprocal distance Laplacian spectral properties double stars and their complements","authors":"H.A. Ganie, B. Rather, M. Aouchiche","doi":"10.15330/cmp.15.2.576-593","DOIUrl":"https://doi.org/10.15330/cmp.15.2.576-593","url":null,"abstract":"Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":" 11","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139140957","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}
Pub Date : 2023-12-28DOI: 10.15330/cmp.15.2.559-562
O. Bezushchak
We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $mathfrak{sl}_{infty}(V|W)$, $mathfrak{o}_{infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides description of derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures.
{"title":"Derivations of Mackey algebras","authors":"O. Bezushchak","doi":"10.15330/cmp.15.2.559-562","DOIUrl":"https://doi.org/10.15330/cmp.15.2.559-562","url":null,"abstract":"We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $mathfrak{sl}_{infty}(V|W)$, $mathfrak{o}_{infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides description of derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"37 5","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151185","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}
Pub Date : 2023-12-26DOI: 10.15330/cmp.15.2.552-558
M. Aijaz, K. Rani, S. Pirzada
Let $R$ be a commutative ring with unity $1ne 0$. In this paper, we completely describe the vertex and the edge chromatic number of the compressed zero divisor graph of the ring of integers modulo $n$. We find the clique number of the compressed zero divisor graph $Gamma_E(mathbb Z_n)$ of $mathbb Z_n$ and show that $Gamma_E(mathbb Z_n)$ is weakly perfect. We also show that the edge chromatic number of $Gamma_E(mathbb Z_n)$ is equal to the largest degree proving that $Gamma_E(mathbb Z_n)$ resides in class 1 family of graphs.
{"title":"On compressed zero divisor graphs associated to the ring of integers modulo $n$","authors":"M. Aijaz, K. Rani, S. Pirzada","doi":"10.15330/cmp.15.2.552-558","DOIUrl":"https://doi.org/10.15330/cmp.15.2.552-558","url":null,"abstract":"Let $R$ be a commutative ring with unity $1ne 0$. In this paper, we completely describe the vertex and the edge chromatic number of the compressed zero divisor graph of the ring of integers modulo $n$. We find the clique number of the compressed zero divisor graph $Gamma_E(mathbb Z_n)$ of $mathbb Z_n$ and show that $Gamma_E(mathbb Z_n)$ is weakly perfect. We also show that the edge chromatic number of $Gamma_E(mathbb Z_n)$ is equal to the largest degree proving that $Gamma_E(mathbb Z_n)$ resides in class 1 family of graphs.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"95 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139155878","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}
Pub Date : 2023-12-25DOI: 10.15330/cmp.15.2.543-551
N. Prevysokova
The paper deals with the factorization of the matrices of discrete wavelet transform based on the Galois functions of different orders. It is used the known method of factorization of the matrices of the discrete Haar transform. Factorized matrices of transforms are presented in the form of a product of sparse matrices. This representation is the basis for building fast transforms algorithms.
{"title":"Factorization of the matrices of discrete wavelet transform on the Galois functions base","authors":"N. Prevysokova","doi":"10.15330/cmp.15.2.543-551","DOIUrl":"https://doi.org/10.15330/cmp.15.2.543-551","url":null,"abstract":"The paper deals with the factorization of the matrices of discrete wavelet transform based on the Galois functions of different orders. It is used the known method of factorization of the matrices of the discrete Haar transform. Factorized matrices of transforms are presented in the form of a product of sparse matrices. This representation is the basis for building fast transforms algorithms.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"6 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139158713","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}
Pub Date : 2023-12-24DOI: 10.15330/cmp.15.2.529-542
V.M. Simulik
The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $textit{C}ell^{texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated.
{"title":"On the Dirac-like equation in 7-component space-time and generalized Clifford-Dirac algebra","authors":"V.M. Simulik","doi":"10.15330/cmp.15.2.529-542","DOIUrl":"https://doi.org/10.15330/cmp.15.2.529-542","url":null,"abstract":"The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $textit{C}ell^{texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"2004 21","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139160168","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}
Pub Date : 2023-12-17DOI: 10.15330/cmp.15.2.514-523
A. A. Basumatary, D.J. Sarma, B.C. Tripathy
The intention of this article is to define the concept of weakly $M$-preopen function in biminimal structure spaces. Several properties of this function have been established and its relationship with some other notions related to $M$-preopen sets in biminimal spaces have been investigated.
{"title":"Weakly $M$-preopen functions in biminimal structure spaces","authors":"A. A. Basumatary, D.J. Sarma, B.C. Tripathy","doi":"10.15330/cmp.15.2.514-523","DOIUrl":"https://doi.org/10.15330/cmp.15.2.514-523","url":null,"abstract":"The intention of this article is to define the concept of weakly $M$-preopen function in biminimal structure spaces. Several properties of this function have been established and its relationship with some other notions related to $M$-preopen sets in biminimal spaces have been investigated.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"549 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139176757","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}
Pub Date : 2023-12-11DOI: 10.15330/cmp.15.2.507-513
L.H. Gallardo
We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $omega(d) geq 9$, and $deg(d) geq 258$, where $d := gcd(Q^2,sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ sigma(A)=A$ in the polynomial ring ${mathbb{F}}_2[x]$. The function $sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].
{"title":"On the index of special perfect polynomials","authors":"L.H. Gallardo","doi":"10.15330/cmp.15.2.507-513","DOIUrl":"https://doi.org/10.15330/cmp.15.2.507-513","url":null,"abstract":"We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $omega(d) geq 9$, and $deg(d) geq 258$, where $d := gcd(Q^2,sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ sigma(A)=A$ in the polynomial ring ${mathbb{F}}_2[x]$. The function $sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"36 11","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139010243","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}
Pub Date : 2023-12-10DOI: 10.15330/cmp.15.2.495-506
A. Şahin, O. Alagöz
In this paper, we consider the class of mappings satisfying $(CSC)$-condition. Further, we prove the strong and $triangle$-convergence theorems of the $JF$-iteration process for this class of mappings in Hadamard spaces. At the end, we give a numerical example to show that the $JF$-iteration process is faster than some well known iterative processes. Our results improve and extend the corresponding recent results of the current literature.
{"title":"On the approximation of fixed points for the class of mappings satisfying $(CSC)$-condition in Hadamard spaces","authors":"A. Şahin, O. Alagöz","doi":"10.15330/cmp.15.2.495-506","DOIUrl":"https://doi.org/10.15330/cmp.15.2.495-506","url":null,"abstract":"In this paper, we consider the class of mappings satisfying $(CSC)$-condition. Further, we prove the strong and $triangle$-convergence theorems of the $JF$-iteration process for this class of mappings in Hadamard spaces. At the end, we give a numerical example to show that the $JF$-iteration process is faster than some well known iterative processes. Our results improve and extend the corresponding recent results of the current literature.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"883 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138982743","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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