Pub Date : 2022-03-09DOI: 10.1108/ajms-03-2021-0057
M. Aslam, M. Siddiqi, A. Siddiqui
PurposeIn 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsic Gauss curvature, and squared mean curvature of any surface in a Euclidean 4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen’s inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.Design/methodology/approachThe authors used standard technique for obtaining generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.FindingsThe authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated.Originality/valueThe research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of submanifolds in that ambient space with several connections.
1979年，P. Wintgen得到了欧几里得4空间中任意曲面的外在法曲率、内在高斯曲率与平均平方曲率之间的基本关系，当且仅当曲率椭圆为圆时成立。1999年，P. J. De Smet, F. Dillen, L. Verstraelen和L. Vrancken给出了实空间形式下一般黎曼子流形的一个关于Wintgen不等式的猜想，称为ddvv猜想。后来，这一猜想分别被吕志和葛、唐志分别证明为真。从那时起，对Wintgen不等式和Wintgen理想子流形的研究吸引了许多研究者，并在过去的15年中发现了许多有趣的结果。本文的主要目的是推广具有四分之一对称度量连接的保形Sasakian空间形式双斜子流形的Wintgen不等式的这个猜想。设计/方法/方法作者利用标准技术得到了具有四分之一对称度量连接的保形Sasakian空间形式双斜子流形的广义Wintgen不等式。建立了具有四分之一对称度量连接的保形Sasakian空间形式双斜子流形的广义Wintgen不等式，并给出了该不等式成立的条件。文中还列举了一些特殊情况。独创性/价值研究可能是一个挑战，因为新的发展集中在不同的不变量方面的新关系，对于不同类型的子流形在环境空间中有几个连接。
{"title":"Generalized Wintgen inequality for BI-SLANT submanifolds in conformal Sasakian space form with quarter-symmetric connection","authors":"M. Aslam, M. Siddiqi, A. Siddiqui","doi":"10.1108/ajms-03-2021-0057","DOIUrl":"https://doi.org/10.1108/ajms-03-2021-0057","url":null,"abstract":"PurposeIn 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsic Gauss curvature, and squared mean curvature of any surface in a Euclidean 4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen’s inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.Design/methodology/approachThe authors used standard technique for obtaining generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.FindingsThe authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated.Originality/valueThe research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of submanifolds in that ambient space with several connections.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42876709","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 : 2022-03-08DOI: 10.1108/ajms-09-2021-0235
Riyajur Rahman, N. Saikia
PurposeLet p[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.Design/methodology/approachUsing q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.FindingsIn the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.Originality/valueThe partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.
{"title":"Congruences modulo prime for fractional colour partition function","authors":"Riyajur Rahman, N. Saikia","doi":"10.1108/ajms-09-2021-0235","DOIUrl":"https://doi.org/10.1108/ajms-09-2021-0235","url":null,"abstract":"PurposeLet p[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.Design/methodology/approachUsing q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.FindingsIn the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.Originality/valueThe partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48750630","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 : 2022-03-08DOI: 10.1108/ajms-07-2020-0024
Chems Eddine Berrehail, Zineb Bouslah
PurposeThis study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.Design/methodology/approachThe authors shall use the averaging theory, more precisely Theorem $6$.FindingsThe main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.Originality/valueIn this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.
{"title":"Periodic solutions for a class of fifth-order differential equations","authors":"Chems Eddine Berrehail, Zineb Bouslah","doi":"10.1108/ajms-07-2020-0024","DOIUrl":"https://doi.org/10.1108/ajms-07-2020-0024","url":null,"abstract":"PurposeThis study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.Design/methodology/approachThe authors shall use the averaging theory, more precisely Theorem $6$.FindingsThe main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.Originality/valueIn this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42174523","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 : 2022-02-04DOI: 10.1108/ajms-10-2020-0103
D. Dey, P. Majhi
PurposeCotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds.Design/methodology/approachThe authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations.FindingsFirst the authors consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next the authors assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to.Originality/valueThe results of this paper are new and interesting. Also, the Proposition 3.2 will be helpful in further study of this space.
{"title":"Almost Kenmotsu 3-h-metric as a cotton soliton","authors":"D. Dey, P. Majhi","doi":"10.1108/ajms-10-2020-0103","DOIUrl":"https://doi.org/10.1108/ajms-10-2020-0103","url":null,"abstract":"PurposeCotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds.Design/methodology/approachThe authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations.FindingsFirst the authors consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next the authors assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to.Originality/valueThe results of this paper are new and interesting. Also, the Proposition 3.2 will be helpful in further study of this space.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49302182","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 : 2022-01-13DOI: 10.1108/ajms-07-2021-0164
A. Molano
Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.
{"title":"Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials","authors":"A. Molano","doi":"10.1108/ajms-07-2021-0164","DOIUrl":"https://doi.org/10.1108/ajms-07-2021-0164","url":null,"abstract":"Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688446","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 : 2021-12-31DOI: 10.1108/ajms-08-2021-0189
Rishab Ranjan, P. N. Pandey, Ajit Paul
PurposeIn this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.Design/methodology/approachFor, the authors have used the notion of conformal transformation and Douglas space.FindingsThe authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.Originality/valueThe authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.
{"title":"Conformal transformation of Douglas space of second kind with special (α, β)-metric","authors":"Rishab Ranjan, P. N. Pandey, Ajit Paul","doi":"10.1108/ajms-08-2021-0189","DOIUrl":"https://doi.org/10.1108/ajms-08-2021-0189","url":null,"abstract":"PurposeIn this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.Design/methodology/approachFor, the authors have used the notion of conformal transformation and Douglas space.FindingsThe authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.Originality/valueThe authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42989663","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 : 2021-12-29DOI: 10.1108/ajms-07-2021-0173
M’hamed El-Louh, M. El Allali, F. Ezzaki
PurposeIn this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true.Design/methodology/approachIn this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space.FindingsThe existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space.Originality/valueThe purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.
{"title":"Convergence theorem of Pettis integrable multivalued pramart","authors":"M’hamed El-Louh, M. El Allali, F. Ezzaki","doi":"10.1108/ajms-07-2021-0173","DOIUrl":"https://doi.org/10.1108/ajms-07-2021-0173","url":null,"abstract":"PurposeIn this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true.Design/methodology/approachIn this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space.FindingsThe existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space.Originality/valueThe purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44387470","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 : 2021-12-29DOI: 10.1108/ajms-08-2021-0192
Farouk Metiri, Halim zeghdoudi, Ahmed Saadoun
PurposeThis paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.Design/methodology/approachIn the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004).FindingsThe authors declare that there is no conflict of interest and the funding information is not applicable.Research limitations/implicationsThis work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners.Practical implicationsThere are several examples of actuarial science (credibility).Originality/valueIn this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.
{"title":"Some results on quadratic credibility premium using the balanced loss function","authors":"Farouk Metiri, Halim zeghdoudi, Ahmed Saadoun","doi":"10.1108/ajms-08-2021-0192","DOIUrl":"https://doi.org/10.1108/ajms-08-2021-0192","url":null,"abstract":"PurposeThis paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.Design/methodology/approachIn the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004).FindingsThe authors declare that there is no conflict of interest and the funding information is not applicable.Research limitations/implicationsThis work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners.Practical implicationsThere are several examples of actuarial science (credibility).Originality/valueIn this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45480212","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 : 2021-12-14DOI: 10.1108/ajms-07-2021-0162
S. Nayaka
PurposeLet b¯2,3(n), which enumerates the number of (2, 3)-regular overcubic bipartition of n. The purpose of the paper is to describe some congruences modulo 8 for b¯2,3(n). For example, for each α ≥ 0 and n ≥ 1, b¯2,3(8n+5)≡0(mod8),b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8).
{"title":"Arithmetic properties of (2, 3)-regular overcubic bipartitions","authors":"S. Nayaka","doi":"10.1108/ajms-07-2021-0162","DOIUrl":"https://doi.org/10.1108/ajms-07-2021-0162","url":null,"abstract":"<jats:sec><jats:title content-type=\"abstract-subheading\">Purpose</jats:title><jats:p>Let <jats:inline-formula><m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"><m:msub><m:mrow><m:mover accent=\"true\"><m:mrow><m:mi>b</m:mi></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow><m:mrow><m:mn>2,3</m:mn></m:mrow></m:msub><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow></m:math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"AJMS-07-2021-0162001.tif\" /></jats:inline-formula>, which enumerates the number of (2, 3)-regular overcubic bipartition of <jats:italic>n</jats:italic>. The purpose of the paper is to describe some congruences modulo 8 for <jats:inline-formula><m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"><m:msub><m:mrow><m:mover accent=\"true\"><m:mrow><m:mi>b</m:mi></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow><m:mrow><m:mn>2,3</m:mn></m:mrow></m:msub><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow></m:math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"AJMS-07-2021-0162002.tif\" /></jats:inline-formula>. For example, for each <jats:italic>α</jats:italic> ≥ 0 and <jats:italic>n</jats:italic> ≥ 1, <jats:inline-formula><m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"><m:msub><m:mrow><m:mover accent=\"true\"><m:mrow><m:mi>b</m:mi></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow><m:mrow><m:mn>2,3</m:mn></m:mrow></m:msub><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mn>8</m:mn><m:mi>n</m:mi><m:mo>+</m:mo><m:mn>5</m:mn></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow><m:mo>≡</m:mo><m:mn>0</m:mn><m:mspace width=\"0.3em\" /><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mi>mod</m:mi><m:mspace width=\"0.3em\" /><m:mn>8</m:mn></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow><m:mo>,</m:mo></m:math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"AJMS-07-2021-0162003.tif\" /></jats:inline-formula> <jats:inline-formula><m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"><m:msub><m:mrow><m:mover accent=\"true\"><m:mrow><m:mi>b</m:mi></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow><m:mrow><m:mn>2,3</m:mn></m:mrow></m:msub><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mn>2</m:mn><m:mo>⋅</m:mo><m:msup><m:mrow><m:mn>3</m:mn></m:mrow><m:mrow><m:mi>α</m:mi><m:mo>+</m:mo><m:mn>3</m:mn></m:mrow></m:msup><m:mi>n</m:mi><m:mo>+</m:mo><m:mn>4</m:mn><m:mo>⋅</m:mo><m:msup><m:mrow><m:mn>3</m:mn></m:mrow><m:mrow><m:mi>α</m:mi><m:mo>+</m:mo><m:mn>2</m:mn></m:mrow></m:msup></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow><m:mo>≡</m:mo><m:mn>0</m:mn><m:mspace width=\"0.3em\" /><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:mrow><m:mi>mod</m:mi><m:mspace width=\"0.3em\" /><m:mn>8</m:mn></m:mrow><m:mo stretchy=\"false\">)</m:mo></m:mrow><m:mo>.</m:mo></m:math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"AJMS-07-2021-0162004.tif\" /></jats:inline-formula></jats:p></jats:sec>","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"62021563","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 : 2021-11-02DOI: 10.1108/ajms-04-2021-0085
R. Ibrahim
PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.
{"title":"A new analytic solution of complex Langevin differential equations","authors":"R. Ibrahim","doi":"10.1108/ajms-04-2021-0085","DOIUrl":"https://doi.org/10.1108/ajms-04-2021-0085","url":null,"abstract":"PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.","PeriodicalId":36840,"journal":{"name":"Arab Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44618652","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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