The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $sum_{n=0}^infty d_n r_n(omega)varphi_n(y)$ in orthogonal polynomials $varphi_n(y)$ with random variables $r_n(omega).$ The random coefficients $r_n(omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $gamma,delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.
{"title":"On the Convergence of Random Fourier-Jacobi Series of Continuous functions","authors":"Partiswari Maharana, Sabita Sahoo","doi":"10.46298/cm.10412","DOIUrl":"https://doi.org/10.46298/cm.10412","url":null,"abstract":"The interest in orthogonal polynomials and random Fourier series in numerous\u0000branches of science and a few studies on random Fourier series in orthogonal\u0000polynomials inspired us to focus on random Fourier series in Jacobi\u0000polynomials. In the present note, an attempt has been made to investigate the\u0000stochastic convergence of some random Jacobi series. We looked into the random\u0000series $sum_{n=0}^infty d_n r_n(omega)varphi_n(y)$ in orthogonal\u0000polynomials $varphi_n(y)$ with random variables $r_n(omega).$ The random\u0000coefficients $r_n(omega)$ are the Fourier-Jacobi coefficients of continuous\u0000stochastic processes such as symmetric stable process and Wiener process. The\u0000$varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants\u0000depending on the random variables associated with the kind of stochastic\u0000process. The convergence of random series is established for different\u0000parameters $gamma,delta$ of the Jacobi polynomials with corresponding choice\u0000of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class\u0000of continuous functions. The sum functions of the random Fourier-Jacobi series\u0000associated with continuous stochastic processes are observed to be the\u0000stochastic integrals. The continuity properties of the sum functions are also\u0000discussed.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45930378","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 the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter $alpha$ and study their trajectory behaviors. We find all fixed points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We construct some Lyapunov functions. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.
{"title":"A family of non-Volterra quadratic operators corresponding to permutations","authors":"U. Jamilov","doi":"10.46298/cm.10135","DOIUrl":"https://doi.org/10.46298/cm.10135","url":null,"abstract":"In the present paper we consider a family of non-Volterra quadratic\u0000stochastic operators depending on a parameter $alpha$ and study their\u0000trajectory behaviors. We find all fixed points for a non-Volterra quadratic\u0000stochastic operator on a finite-dimensional simplex. We construct some Lyapunov\u0000functions. A complete description of the set of limit points is given, and we\u0000show that such operators have the ergodic property.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47713817","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 define a homology theory for pre-crossed modules that specifies to rack homology in the case when the pre-crossed module is freely generated by a rack.
我们定义了预交叉模块的同调理论，该理论规定当预交叉模块由机架自由生成时，该模块为机架同调。
{"title":"Pre-crossed modules and rack homology","authors":"J. Mostovoy","doi":"10.46298/cm.10153","DOIUrl":"https://doi.org/10.46298/cm.10153","url":null,"abstract":"We define a homology theory for pre-crossed modules that specifies to rack\u0000homology in the case when the pre-crossed module is freely generated by a rack.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44967725","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}
For a fixed integer $k$, we define the multiplicative function [D_{k,omega}(n) := frac{d(n)}{k^{omega(n)}}, ]where $d(n)$ is the divisor function and $omega (n)$ is the number of distinct prime divisors of $n$. The main purpose of this paper is the study of the mean value of the function $D_{k,omega}(n)$ by using elementary methods.
{"title":"Asymptotic formula for the multiplicative function $frac{d(n)}{k^{omega(n)}}$","authors":"Meselem Karras","doi":"10.46298/cm.10104","DOIUrl":"https://doi.org/10.46298/cm.10104","url":null,"abstract":"For a fixed integer $k$, we define the multiplicative function\u0000[D_{k,omega}(n) := frac{d(n)}{k^{omega(n)}}, ]where $d(n)$ is the divisor\u0000function and $omega (n)$ is the number of distinct prime divisors of $n$. The\u0000main purpose of this paper is the study of the mean value of the function\u0000$D_{k,omega}(n)$ by using elementary methods.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45210909","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 obtain asymptotic expansion for the geometric mean of the values of positive strongly multiplicative function $f$ satisfying $f(p)=alpha(d),p^d+O(p^{d-delta})$ for any prime $p$ with $d$ real and $alpha(d),delta>0$.
{"title":"On the geometric mean of the values of positive multiplicative arithmetical functions","authors":"M. Hassani, M. Esfandiari","doi":"10.46298/cm.10133","DOIUrl":"https://doi.org/10.46298/cm.10133","url":null,"abstract":"In this paper we obtain asymptotic expansion for the geometric mean of the\u0000values of positive strongly multiplicative function $f$ satisfying\u0000$f(p)=alpha(d),p^d+O(p^{d-delta})$ for any prime $p$ with $d$ real and\u0000$alpha(d),delta>0$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49026786","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}
I. Blanco-Chacón, Ra'ul Dur'an-D'iaz, Rahinatou Yuh Njah Nchiwo, Beatriz Barbero-Lucas
We describe a decisional attack against a version of the PLWE problem in which the samples are taken from a certain proper subring of large dimension of the cyclotomic ring $mathbb{F}_q[x]/(Phi_{p^k}(x))$ with $k>1$ in the case where $qequiv 1pmod{p}$ but $Phi_{p^k}(x)$ is not totally split over $mathbb{F}_q$. Our attack uses the fact that the roots of $Phi_{p^k}(x)$ over suitable extensions of $mathbb{F}_q$ have zero-trace and has overwhelming success probability as a function of the number of input samples. An implementation in Maple and some examples of our attack are also provided.
{"title":"Trace-based cryptanalysis of cyclotomic $R_{q,0}times R_q$-PLWE for the non-split case","authors":"I. Blanco-Chacón, Ra'ul Dur'an-D'iaz, Rahinatou Yuh Njah Nchiwo, Beatriz Barbero-Lucas","doi":"10.46298/cm.11153","DOIUrl":"https://doi.org/10.46298/cm.11153","url":null,"abstract":"We describe a decisional attack against a version of the PLWE problem in\u0000which the samples are taken from a certain proper subring of large dimension of\u0000the cyclotomic ring $mathbb{F}_q[x]/(Phi_{p^k}(x))$ with $k>1$ in the case\u0000where $qequiv 1pmod{p}$ but $Phi_{p^k}(x)$ is not totally split over\u0000$mathbb{F}_q$. Our attack uses the fact that the roots of $Phi_{p^k}(x)$ over\u0000suitable extensions of $mathbb{F}_q$ have zero-trace and has overwhelming\u0000success probability as a function of the number of input samples. An\u0000implementation in Maple and some examples of our attack are also provided.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48609365","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 propose a method for solving some classes of the singular fractional nonlinear Lane-Emden type equations. The method is proposed by utilizing the second-kind Chebyshev wavelets in conjunction with the quasilinearization technique. The operational matrices for the second-kind Chebyshev wavelets are used. The method is tested on the fractional standard Lane-Emden equation, the fractional isothermal gas spheres equation, and some other examples. We compare the results produced by the present method with some well-known results to show the accuracy and efficiency of the method.
{"title":"Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equations","authors":"A. Mohammadi, Ghader Ahmadnezhad, N. Aghazadeh","doi":"10.46298/cm.9895","DOIUrl":"https://doi.org/10.46298/cm.9895","url":null,"abstract":"In this paper, we propose a method for solving some classes of the singular fractional nonlinear Lane-Emden type equations. The method is proposed by utilizing the second-kind Chebyshev wavelets in conjunction with the quasilinearization technique. The operational matrices for the second-kind Chebyshev wavelets are used. The method is tested on the fractional standard Lane-Emden equation, the fractional isothermal gas spheres equation, and some other examples. We compare the results produced by the present method with some well-known results to show the accuracy and efficiency of the method.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44095007","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}
Bekzat K. Zhakhayev, A. Dzhumadil'daev, Saule A. Abdykassymova
An algebra with identities (a, b, c) = (a, c, b) = (b, a, c) is called assosymmetric, where (x, y, z) = x(yz) − (xy)z is associator. We establish that operad of assosymmetric algebras is not Koszul. We study Sn-module, An-module and GLn-module structures on multilinear parts of assosymmetric operad.
{"title":"Assosymmetric Operad","authors":"Bekzat K. Zhakhayev, A. Dzhumadil'daev, Saule A. Abdykassymova","doi":"10.46298/cm.9740","DOIUrl":"https://doi.org/10.46298/cm.9740","url":null,"abstract":"An algebra with identities (a, b, c) = (a, c, b) = (b, a, c) is called assosymmetric, where (x, y, z) = x(yz) − (xy)z is associator. We establish that operad of assosymmetric algebras is not Koszul. We study Sn-module, An-module and GLn-module structures on multilinear parts of assosymmetric operad.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45669441","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}
A Rota-Baxter Leibniz algebra is a Leibniz algebra $(mathfrak{g},[~,~]_{mathfrak{g}})$ equipped with a Rota-Baxter operator $T : mathfrak{g} rightarrow mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.
{"title":"Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras","authors":"B. Mondal, R. Saha","doi":"10.46298/cm.10295","DOIUrl":"https://doi.org/10.46298/cm.10295","url":null,"abstract":"A Rota-Baxter Leibniz algebra is a Leibniz algebra\u0000$(mathfrak{g},[~,~]_{mathfrak{g}})$ equipped with a Rota-Baxter operator $T :\u0000mathfrak{g} rightarrow mathfrak{g}$. We define representation and dual\u0000representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology\u0000theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and\u0000formal deformation theory of Rota-Baxter Leibniz algebras and show that our\u0000cohomology is deformation cohomology. Moreover, We define an abelian extension\u0000of Rota-Baxter Leibniz algebras and show that equivalence classes of such\u0000extensions are related to the cohomology groups.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48330346","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 the first part of this article we discuss the relative cases of Quillen-Suslin's local-global principle for the general quadratic (Bak's unitary) groups, and its applications for the (relative) stable and unstable $mathrm{K}_1$-groups. The second part is dedicated to the graded version of the local-global principle for the general quadratic groups and its application to deduce a result for Bass' nil groups.
{"title":"Results on $mathrm{K}_1$ of general quadratic groups","authors":"R. Basu, Kuntal Chakraborty","doi":"10.46298/cm.9855","DOIUrl":"https://doi.org/10.46298/cm.9855","url":null,"abstract":"In the first part of this article we discuss the relative cases of\u0000Quillen-Suslin's local-global principle for the general quadratic (Bak's\u0000unitary) groups, and its applications for the (relative) stable and unstable\u0000$mathrm{K}_1$-groups. The second part is dedicated to the graded version of\u0000the local-global principle for the general quadratic groups and its application\u0000to deduce a result for Bass' nil groups.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42981712","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：