M. B. D. Ngaha, Salomon Joseph Mbatakou, Romain Nimpa Pefoukeu
: In this paper, Szabó’s algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some 4 -dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.
{"title":"Szabos algorithm and applications","authors":"M. B. D. Ngaha, Salomon Joseph Mbatakou, Romain Nimpa Pefoukeu","doi":"10.55730/1300-0098.3419","DOIUrl":"https://doi.org/10.55730/1300-0098.3419","url":null,"abstract":": In this paper, Szabó’s algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some 4 -dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45571953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: The results on Γ -hypersemigroups are obtained either as corollaries of corresponding results on ∨ e or poe - semigroups or on the line of the corresponding results on le -semigroups. It has come to our attention that Theorem 3.22 in [4] cannot be obtained as corollary to Theorem 2.2 of the same paper as for a Γ -hypersemigroup, ( P ∗ ( M ) , Γ , ⊆ ) is a ∨ e -semigroup and not an le -semigroup. Also on p. 1850, l. 12 in [4], the “ le -semigroup” should be changed to “ ∨ e -semigroup”. In the present paper we prove Theorems 3.26 and 3.28 stated without proof in [4]. On this occasion, some further results are given to emphasize what we say. The results on Γ -hypersemigroups are obtained from the more abstract structure of the poe -semigroups. Further investigation on poe -semigroups and le -semigroups is interesting.
{"title":"On $Gamma$-hypersemigroups","authors":"N. Kehayopulu","doi":"10.55730/1300-0098.3414","DOIUrl":"https://doi.org/10.55730/1300-0098.3414","url":null,"abstract":": The results on Γ -hypersemigroups are obtained either as corollaries of corresponding results on ∨ e or poe - semigroups or on the line of the corresponding results on le -semigroups. It has come to our attention that Theorem 3.22 in [4] cannot be obtained as corollary to Theorem 2.2 of the same paper as for a Γ -hypersemigroup, ( P ∗ ( M ) , Γ , ⊆ ) is a ∨ e -semigroup and not an le -semigroup. Also on p. 1850, l. 12 in [4], the “ le -semigroup” should be changed to “ ∨ e -semigroup”. In the present paper we prove Theorems 3.26 and 3.28 stated without proof in [4]. On this occasion, some further results are given to emphasize what we say. The results on Γ -hypersemigroups are obtained from the more abstract structure of the poe -semigroups. Further investigation on poe -semigroups and le -semigroups is interesting.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46205260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: We develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks
{"title":"Discontinuous Galerkin method for blow-up solutions of nonlinear wave equations","authors":"Asma Azaiez, M. Benjemaa, Aida Jrajria, H. Zaag","doi":"10.55730/1300-0098.3408","DOIUrl":"https://doi.org/10.55730/1300-0098.3408","url":null,"abstract":": We develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43484263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the p -adic Volkenborn integral to the Catalan-type polynomials, some combinatorial sums and identities involving the Bernoulli numbers, the Stirling numbers and the Catalan-type numbers are derived.
{"title":"Generating functions for reciprocal Catalan-type sums: approach to linear differentiation equation and ($p$-adic) integral equations","authors":"Damla Gün, Yilmaz Simsek","doi":"10.55730/1300-0098.3396","DOIUrl":"https://doi.org/10.55730/1300-0098.3396","url":null,"abstract":": This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the p -adic Volkenborn integral to the Catalan-type polynomials, some combinatorial sums and identities involving the Bernoulli numbers, the Stirling numbers and the Catalan-type numbers are derived.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43893427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: We prove that Bergman projections on weighted mixed norm spaces on smoothly bounded domains in R n are bounded for a certain range of parameters of such spaces and assuming certain conditions on weights. The proof relies on estimates of integral means of M p ( P γ f, r ) in terms of integral means of f . This result complements earlier result on boundedness of P γ on a closely related space L p,qα (Ω)
{"title":"Boundedness of Bergman projections acting on weighted mixed norm spaces","authors":"Ivana Savković","doi":"10.55730/1300-0098.3387","DOIUrl":"https://doi.org/10.55730/1300-0098.3387","url":null,"abstract":": We prove that Bergman projections on weighted mixed norm spaces on smoothly bounded domains in R n are bounded for a certain range of parameters of such spaces and assuming certain conditions on weights. The proof relies on estimates of integral means of M p ( P γ f, r ) in terms of integral means of f . This result complements earlier result on boundedness of P γ on a closely related space L p,qα (Ω)","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42093079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abdullah Açikel, Amrouche Said, H. Belbachir, N. Irmak
: In this paper, we investigate several identities of k -generalized Lucas numbers with k -generalized Fibonacci numbers. We also establish a link between generalized s -Lucas triangle and bi s nomial coefficients given by the coefficients of the development of a power of (1 + x + x 2 + · · · + x s ) , with s ∈ N
{"title":"On $k$-generalized Lucas sequence with its triangle","authors":"Abdullah Açikel, Amrouche Said, H. Belbachir, N. Irmak","doi":"10.55730/1300-0098.3416","DOIUrl":"https://doi.org/10.55730/1300-0098.3416","url":null,"abstract":": In this paper, we investigate several identities of k -generalized Lucas numbers with k -generalized Fibonacci numbers. We also establish a link between generalized s -Lucas triangle and bi s nomial coefficients given by the coefficients of the development of a power of (1 + x + x 2 + · · · + x s ) , with s ∈ N","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49326343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: A ring R is called semiclean if every element of R can be expressed as sum of a periodic element and a unit. In this paper, we introduce a new class of ring, which is the ∗ -version of the semiclean ring, i.e. the ∗ -semiclean ring. A ∗ -ring is ∗ -semiclean if each element is a sum of a ∗ -periodic element and a unit. The term ∗ -semiclean is a stronger notion than semiclean. In this paper, many properties of ∗ -semiclean rings are discussed. It is proved that if p ∈ P ( R ) such that pRp and (1 − p ) R (1 − p ) are ∗ -semiclean rings, then R is also a ∗ -semiclean ring. As a result, the matrix ring M n ( R ) over a ∗ -semiclean ring is ∗ -semiclean. A characterization that when the group rings RC r and RG are ∗ -semiclean is done, where R is a finite commutative local ring, C r is a cyclic group of order r , and G is a locally finite abelian group. We have also found sufficient conditions when the group rings RC 3 , RC 4 , RQ 8 , and RQ 2 n are ∗ -semiclean, where R is a commutative local ring. We have also demonstrated that the group ring Z 2 D 6 is a ∗ -semiclean ring (which is not a ∗ -clean ring).
{"title":"$ast$-Semiclean rings","authors":"Shefali Gupta, Dinesh Udar","doi":"10.55730/1300-0098.3437","DOIUrl":"https://doi.org/10.55730/1300-0098.3437","url":null,"abstract":": A ring R is called semiclean if every element of R can be expressed as sum of a periodic element and a unit. In this paper, we introduce a new class of ring, which is the ∗ -version of the semiclean ring, i.e. the ∗ -semiclean ring. A ∗ -ring is ∗ -semiclean if each element is a sum of a ∗ -periodic element and a unit. The term ∗ -semiclean is a stronger notion than semiclean. In this paper, many properties of ∗ -semiclean rings are discussed. It is proved that if p ∈ P ( R ) such that pRp and (1 − p ) R (1 − p ) are ∗ -semiclean rings, then R is also a ∗ -semiclean ring. As a result, the matrix ring M n ( R ) over a ∗ -semiclean ring is ∗ -semiclean. A characterization that when the group rings RC r and RG are ∗ -semiclean is done, where R is a finite commutative local ring, C r is a cyclic group of order r , and G is a locally finite abelian group. We have also found sufficient conditions when the group rings RC 3 , RC 4 , RQ 8 , and RQ 2 n are ∗ -semiclean, where R is a commutative local ring. We have also demonstrated that the group ring Z 2 D 6 is a ∗ -semiclean ring (which is not a ∗ -clean ring).","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43048230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: In this paper, the main identities on locally product-like statistical submersions are obtained with the aid of statistical structures and their Riemannian curvature tensors. Some examples of locally product-like statistical submersions are presented. Some results on F -invariant, F ∗ -invariant and antiinvariant locally product-like statistical submersions are given
{"title":"Locally product-like statistical submersions","authors":"K. Takano, Esra Erkan, M. Gülbahar","doi":"10.55730/1300-0098.3397","DOIUrl":"https://doi.org/10.55730/1300-0098.3397","url":null,"abstract":": In this paper, the main identities on locally product-like statistical submersions are obtained with the aid of statistical structures and their Riemannian curvature tensors. Some examples of locally product-like statistical submersions are presented. Some results on F -invariant, F ∗ -invariant and antiinvariant locally product-like statistical submersions are given","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44946746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this present study, we pay attention to a class of nonlinear neutral type systems (NNSs) with periodic coefficients and construct some assumptions guaranteeing the exponential stability (ES) of the trivial solutions of the system considered. To get specific conditions guaranteeing the ES, we use a modified Lyapunov functional. In conclusion, we get some estimates for the exponential decay of the solutions at infinity with the constructed sufficient conditions. We give two examples to demonstrate the applicability of the results obtained with the constructed assumptions.
{"title":"Some estimates on the exponential stability of solutions of nonlinear neutral type systems with periodic coefficients","authors":"Y. Altun","doi":"10.55730/1300-0098.3444","DOIUrl":"https://doi.org/10.55730/1300-0098.3444","url":null,"abstract":"In this present study, we pay attention to a class of nonlinear neutral type systems (NNSs) with periodic coefficients and construct some assumptions guaranteeing the exponential stability (ES) of the trivial solutions of the system considered. To get specific conditions guaranteeing the ES, we use a modified Lyapunov functional. In conclusion, we get some estimates for the exponential decay of the solutions at infinity with the constructed sufficient conditions. We give two examples to demonstrate the applicability of the results obtained with the constructed assumptions.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45249099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: In this work, we investigate generalized coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms and initial-boundary value conditions, in a bounded domain. We obtain decay of solutions by use of Nakao inequality. The blow up of solutions with negative initial energy is also established
{"title":"Energy decay and blow-up of solutions for a class of system of generalized nonlinear Klein-Gordon equations with source and damping terms","authors":"Z. S. Çelik, S. Gür, E. Pişkin","doi":"10.55730/1300-0098.3428","DOIUrl":"https://doi.org/10.55730/1300-0098.3428","url":null,"abstract":": In this work, we investigate generalized coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms and initial-boundary value conditions, in a bounded domain. We obtain decay of solutions by use of Nakao inequality. The blow up of solutions with negative initial energy is also established","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46409680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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