Let $d(n)$ and $d^{ast}(n)$ be the numbers of divisors and the numbers of unitary divisors of the integer $ngeq1$. In this paper, we prove that [ underset{ninmathcal{B}}{underset{nleq x}{sum}}frac{d(n)}{d^{ast}% (n)}=frac{16pi% %TCIMACRO{U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }{123}underset{p}{prod}(1-frac{1}{2p% %TCIMACRO{U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }+frac{1}{2p^{3}})x+mathcal{O}left( x^{frac{ln8}{ln10}+varepsilon }right) ,~left( xgeqslant1,~varepsilon>0right) , ] where $mathcal{B}$ is the set which contains any integer that is not a multiple of $5,$ but some permutations of its digits is a multiple of $5.$
{"title":"On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5","authors":"M. Bouderbala","doi":"10.46298/cm.10467","DOIUrl":"https://doi.org/10.46298/cm.10467","url":null,"abstract":"Let $d(n)$ and $d^{ast}(n)$ be the numbers of divisors and the numbers of\u0000unitary divisors of the integer $ngeq1$. In this paper, we prove that [\u0000underset{ninmathcal{B}}{underset{nleq x}{sum}}frac{d(n)}{d^{ast}%\u0000(n)}=frac{16pi% %TCIMACRO{U{b2}}% %BeginExpansion {{}^2}% %EndExpansion\u0000}{123}underset{p}{prod}(1-frac{1}{2p% %TCIMACRO{U{b2}}% %BeginExpansion\u0000{{}^2}% %EndExpansion }+frac{1}{2p^{3}})x+mathcal{O}left(\u0000x^{frac{ln8}{ln10}+varepsilon }right) ,~left(\u0000xgeqslant1,~varepsilon>0right) , ] where $mathcal{B}$ is the set which\u0000contains any integer that is not a multiple of $5,$ but some permutations of\u0000its digits is a multiple of $5.$","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45767591","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}
M. J. Santos, C. Raposo, L. Miranda, B. Feng, C. Alberto, Raposo da
In this work, we consider the existence of global solution and the exponential decay of a nonlinear porous elastic system with time delay. The nonlinear term as well as the delay acting in the equation of the volume fraction. In order to obtain the existence and uniqueness of a global solution, we will use the semigroup theory of linear operators and under a certain relation involving the coefficients of the system together with a Lyapunov functional, we will establish the exponential decay of the energy associated to the system.
{"title":"Exponential stability for a nonlinear porous-elastic system with delay","authors":"M. J. Santos, C. Raposo, L. Miranda, B. Feng, C. Alberto, Raposo da","doi":"10.46298/cm.10439","DOIUrl":"https://doi.org/10.46298/cm.10439","url":null,"abstract":"In this work, we consider the existence of global solution and the\u0000exponential decay of a nonlinear porous elastic system with time delay. The\u0000nonlinear term as well as the delay acting in the equation of the volume\u0000fraction. In order to obtain the existence and uniqueness of a global solution,\u0000we will use the semigroup theory of linear operators and under a certain\u0000relation involving the coefficients of the system together with a Lyapunov\u0000functional, we will establish the exponential decay of the energy associated to\u0000the system.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42206523","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}
Let $x$ be a positive real number, and $mathcal{P} subset [2,lambda(x)]$ be a set of primes, where $lambda(x) in o(x^{1/2})$ is a monotone increasing function. We examine $Q_{mathcal{P}}(x)$ for different sets $mathcal{P}$, where $Q_{mathcal{P}}(x)$ is the element count of the set containing those positive square-free integers, which are smaller than-, or equal to $x$, and which are only divisible by the elements of $mathcal{P}$.
{"title":"On square-free numbers generated from given sets of primes","authors":"G. Román","doi":"10.46298/cm.10527","DOIUrl":"https://doi.org/10.46298/cm.10527","url":null,"abstract":"Let $x$ be a positive real number, and $mathcal{P} subset [2,lambda(x)]$\u0000be a set of primes, where $lambda(x) in o(x^{1/2})$ is a monotone increasing\u0000function. We examine $Q_{mathcal{P}}(x)$ for different sets $mathcal{P}$,\u0000where $Q_{mathcal{P}}(x)$ is the element count of the set containing those\u0000positive square-free integers, which are smaller than-, or equal to $x$, and\u0000which are only divisible by the elements of $mathcal{P}$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49262435","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}
In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.
{"title":"APPLICATION OF HOMOTOPY ANALYSIS METHOD (HAM) TO THE NON-LINEAR KDV EQUATIONS","authors":"A. Chauhan, R. Arora","doi":"10.46298/cm.10336","DOIUrl":"https://doi.org/10.46298/cm.10336","url":null,"abstract":"In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45620312","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}
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued fraction expansions of the error function erf(A) where A is a matrix. At the end, some numerical examples illustrating the theoretical results are discussed.
{"title":"Matrix continued fractions and Expansions of the Error Function","authors":"S. Mennou, A. Chillali, A. Kacha","doi":"10.46298/cm.10395","DOIUrl":"https://doi.org/10.46298/cm.10395","url":null,"abstract":"In this paper we recall some results and some criteria on the convergence of\u0000matrix continued fractions. The aim of this paper is to give some properties\u0000and results of continued fractions with matrix arguments. Then we give\u0000continued fraction expansions of the error function erf(A) where A is a matrix.\u0000At the end, some numerical examples illustrating the theoretical results are\u0000discussed.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49137772","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}
In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $mathbb{R}^n$ where $ngeq1$, containing all constant vector fields and the Euler vector field, under some conditions on this Lie sub-algebra.
{"title":"On generalized derivations of polynomial vector fields Lie algebras","authors":"P. Randriambololondrantomalala, Sania Asif","doi":"10.46298/cm.10386","DOIUrl":"https://doi.org/10.46298/cm.10386","url":null,"abstract":"In this paper, we study the generalized derivation of a Lie sub-algebra of\u0000the Lie algebra of polynomial vector fields on $mathbb{R}^n$ where $ngeq1$,\u0000containing all constant vector fields and the Euler vector field, under some\u0000conditions on this Lie sub-algebra.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44287719","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}
Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a non negative integer, then we have the congruence $C_{r+np}equiv (X_1^p-X_p)^nC_r mod{pZ_p[X_1,cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-adic integers. We prove that for $pneq 2$, the preceding congruence holds modulo $npZ_p[X_1,cdots,X_{r+np}]$. This allows us to prove a Junod's conjecture for Meixner polynomials.
{"title":"Congruences for the cycle indicator of the symmetric group","authors":"Abdelaziz Bellagh, Assia Oulebsir","doi":"10.46298/cm.10391","DOIUrl":"https://doi.org/10.46298/cm.10391","url":null,"abstract":"Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the\u0000symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a\u0000non negative integer, then we have the congruence $C_{r+np}equiv\u0000(X_1^p-X_p)^nC_r\u0000 mod{pZ_p[X_1,cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-adic\u0000integers. We prove that for $pneq 2$, the preceding congruence holds modulo\u0000$npZ_p[X_1,cdots,X_{r+np}]$. This allows us to prove a Junod's conjecture for\u0000Meixner polynomials.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45567724","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}
Without imposing any conditions on dependence structure, we give a seemingly overlooked simple sufficient condition for $L^{p}$ random variables $X_{1}, X_{2}, dots$ with given $1 leq p leq +infty$ to satisfy [frac{1}{a_{n}}sum_{i=1}^{b_{n}}(X_{i} - mathbb{E} X_{i}) overset{L^{p}}to 0 ,,, mathrm{as}, n to infty,]where $(a_{n})_{n in mathbb{N}}, (b_{n})_{n in mathbb{N}}$ are prespecified unbounded sequences of positive integers.Some unexpected convergences of sample means follow.
在不对依赖结构施加任何条件的情况下，我们给出了一个看似被忽略的$L^{p}$随机变量$X_{1}, X_{2}, dots$的简单充分条件，给定$1 leq p leq +infty$满足[frac{1}{a_{n}}sum_{i=1}^{b_{n}}(X_{i} - mathbb{E} X_{i}) overset{L^{p}}to 0 ,,, mathrm{as}, n to infty,]，其中$(a_{n})_{n in mathbb{N}}, (b_{n})_{n in mathbb{N}}$是预先指定的无界正整数序列。随后出现了一些意想不到的样本均值收敛。
{"title":"A General Weak Law of Large Numbers for Sequences of $L^{p}$ Random Variables","authors":"Yu-Lin Chou","doi":"10.46298/cm.10292","DOIUrl":"https://doi.org/10.46298/cm.10292","url":null,"abstract":"Without imposing any conditions on dependence structure, we give a seemingly overlooked simple sufficient condition for $L^{p}$ random variables $X_{1}, X_{2}, dots$ with given $1 leq p leq +infty$ to satisfy [frac{1}{a_{n}}sum_{i=1}^{b_{n}}(X_{i} - mathbb{E} X_{i}) overset{L^{p}}to 0 ,,, mathrm{as}, n to infty,]where $(a_{n})_{n in mathbb{N}}, (b_{n})_{n in mathbb{N}}$ are prespecified unbounded sequences of positive integers.Some unexpected convergences of sample means follow.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48107413","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}
In this work, we determine the general solution of the quinquevigintic functional equation and also investigate its stability of this equation in the setting of matrix normed spaces and the framework of matrix non-Archimedean fuzzy normed spaces by using the fixed point method.
{"title":"Stability on Quinquevigintic Functional Equation in Different Spaces","authors":"R. Murali, S. Pinelas, V. Vithya","doi":"10.46298/cm.10341","DOIUrl":"https://doi.org/10.46298/cm.10341","url":null,"abstract":"In this work, we determine the general solution of the quinquevigintic\u0000functional equation and also investigate its stability of this equation in the\u0000setting of matrix normed spaces and the framework of matrix non-Archimedean\u0000fuzzy normed spaces by using the fixed point method.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42771735","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}
We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on standard Bernoulli polynomials, as the addition formula and the binomial formula. We also prove a result that allows us to obtain poly-Bernoulli polynomial identities from polynomial identities, and we use this result to obtain several identities involving products of poly-Bernoulli and/or standard Bernoulli polynomials. We prove two generalized recurrences for bi-variate poly-Bernoulli polynomials, and obtain some corollaries from them.
{"title":"On bi-variate poly-Bernoulli polynomials","authors":"C. Pita-Ruiz","doi":"10.46298/cm.10327","DOIUrl":"https://doi.org/10.46298/cm.10327","url":null,"abstract":"We introduce poly-Bernoulli polynomials in two variables by using a\u0000generalization of Stirling numbers of the second kind that we studied in a\u0000previous work. We prove the bi-variate poly-Bernoulli polynomial version of\u0000some known results on standard Bernoulli polynomials, as the addition formula\u0000and the binomial formula. We also prove a result that allows us to obtain\u0000poly-Bernoulli polynomial identities from polynomial identities, and we use\u0000this result to obtain several identities involving products of poly-Bernoulli\u0000and/or standard Bernoulli polynomials. We prove two generalized recurrences for\u0000bi-variate poly-Bernoulli polynomials, and obtain some corollaries from them.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45359221","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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