Showing papers in "Journal of The Korean Mathematical Society in 2014"
TL;DR: In this article, the existence of extremalsolutions for a class of fractional partial differential equations with or-der 1 < α < 2 by upper and lower solution method is investigated.
Abstract: . In this work, we investigate the existence of the extremalsolutions for a class of fractional partial differential equations with or-der 1 < α < 2 by upper and lower solution method. Using the theoryof Hausdorff measure of noncompactness, a series of results about thesolutions to such differential equations is obtained. 1. IntroductionFractionalorderdifferentialequationhasbroadapplicationsin resolvingreal-world problems, and as such it attracted researchers’ attention from differentareas. In orderto evaluate the behaviorsoffractional orderdifferential equationbased models, one need to know the properties of such equation systems, inparticular, the existence of solutions to such equations. Recently, the existenceof solutions to different forms of fractional differential equation systems hasbeen investigated [1, 2, 3, 4, 5, 9, 10, 11, 15, 16, 19, 20, 21, 23, 26, 27, 29, 30,32, 33, 34].Using the upper and lower solution method to study the existence of ex-tremal solutions for fractional differential equations is an interesting topic ofresearch, which has been gaining increasing attention recently [1, 15, 16, 20,23, 27, 30, 32, 33, 34]). Presently, the upper and lower solution methodis widely used to investigate fractional ordinary differential equations (see[1, 15, 20, 23, 30, 32, 33, 34]). However, this method is seldom used to studysemilinear fractional evolution equations. [27] considered the existence of ex-tremal solutions to the following semilinear fractional evolution equation(1.1)(
34 citations
TL;DR: In this paper, a spectral Jacobi-collocation approximation for fractional order integro-dierenti al equations of the Fredholm- Volterra type is proposed.
Abstract: We propose and analyze a spectral Jacobi-collocation approx- imation for fractional order integro-dierenti al equations of Fredholm- Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L1 norm and weighted L2-norm. The numerical examples are given to illustrate the theoretical results.
33 citations
TL;DR: In this article, it was shown that a graph G is fractional (g,f,n)-critical if t(G) � b 2 −1+bn a.
Abstract: A graph G is called a fractional (g,f,n)-critical graph if any n vertices are removed from G, then the resulting graph admits a fractional (g,f)-factor. In this paper, we determine the new toughness condition for fractional (g,f,n)-critical graphs. It is proved that G is fractional (g,f,n)-critical if t(G) � b 2 −1+bn a . This bound is sharp in some sense. Furthermore, the best toughness condition for fractional (a, b,n)-critical graphs is given. All graphs considered in this paper are finite, loopless, and without multiple edges. The notation and terminology used but undefined in this paper can be found in (2). Let G be a graph with the vertex set V (G) and the edge set E(G). For a vertex x ∈ V (G), we use dG(x) and NG(x) to denote the degree and the neighborhood of x in G, respectively. Let �(G) denote the minimum degree of G. For any S ⊆ V (G), the subgraph of G induced by S is denoted by G(S). Suppose that g and f are two integer-valued functions on V (G) such that 0 ≤ g(x) ≤ f(x) for all x ∈ V (G). A spanning subgraph F of G is called a (g,f)-factor if g(x) ≤ dF(x) ≤ f(x) for each x ∈ V (G). A fractional (g,f)- factor is a function h that assigns to each edge of a graph G a number in (0,1) so that for each vertex x we have g(x) ≤ P e∈E(x) h(e) ≤ f(x). If g(x) = a, f(x) = b for all x ∈ V (G), then a fractional (g,f)-factor is a fractional (a,b)- factor. Moreover, if g(x) = f(x) = k (k ≥ 1 is an integer throughout this paper, and we will not reiterate it again) for all x ∈ V (G), then a fractional (g,f)-factor is just a fractional k-factor.
32 citations
TL;DR: In this article, the Cauchy problem for a Keller-Segel-fluid model with degenerate diffusion for cell density is considered and the existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotactic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.
Abstract: We consider the Cauchy problem for a Keller-Segel-fluid model with degenerate diffusion for cell density, which is mathematically formulated as a porus medium type of Keller-Segel equations coupled to viscous incompressible fluid equations. We establish the global-in-time existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotactic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.
27 citations
TL;DR: In this paper, the authors introduced the concept of w-injective modules and studied some basic properties of WIMs in commutative rings with identity, and showed that a ring R is w-semi-hereditary if and only if the total quotient ringT(R) of R is a von Neumann regular ring and R m is a valuation domain for any maximal w-ideal mof R.
Abstract: . Let Rbe a commutative ring with identity. An R-module Mis said to be w-projective if Ext 1R (M,N) is GV-torsion for any torsion-freew-module N. In this paper, we define a ring R to be w-semi-hereditaryif every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules andstudy some basic properties of w-injective modules. Using these concepts,we show that Ris w-semi-hereditary if and only if the total quotient ringT(R) of R is a von Neumann regular ring and R m is a valuation domainfor any maximal w-ideal mof R. It is also shown that a connected ring Ris w-semi-hereditary if and only if Ris a Pru¨fer v-multiplication domain. 1. IntroductionThroughout, R denotes a commutative ring with identity 1 and E(M) de-notes the injective hull (or envelope) of an R-module M. And let us regardthat the v-, t- and w-operation are well-known star-operations on domains. Forunexplained terminologies and notations, we refer to [3, 14, 15].Pru¨fer v-multiplication domains (PVMD for short) have received a good dealof attention in much literature. A domain R is called a PVMD if every nonzerofinitely generated ideal I is t-invertible, that is, there is a fractional ideal B ofR such that (IB)
23 citations
TL;DR: In this paper, a conditional version of the classical central limit theorem is derived rigorously by using conditional characteristic functions, and a more general version of conditional central limit for the case of conditionally independent but not necessarily conditionally identically distributed random variables is established.
Abstract: A conditional version of the classical central limit theorem is derived rigorously by using conditional characteristic functions, and a more general version of conditional central limit theorem for the case of conditionally independent but not necessarily conditionally identically distributed random variables is established. These are done anticipating that the field of conditional limit theory will prove to be of significant applicability.
23 citations
TL;DR: In this article, a two-scale product approximation for semilinear heat equations in the mixed finite element method is proposed and analyzed, where the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system.
Abstract: We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorithm. With this scheme, the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system. We derive optimal order L ∞ ((0,T);L 2 ())-error estimates for the rele- vant variables. Numerical results are presented to support the theory developed in this paper.
20 citations
TL;DR: In this article, the notions of harmonic Alexander operator and harmonic Libera operator are intro-duced and their properties are investigated. And a new method- ology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalented analytic functions.
Abstract: Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new method- ology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent analytic functions. The notions of harmonic Alexander operator and harmonic Libera operator are intro- duced and their properties are investigated.
19 citations
TL;DR: Hong and Lee as discussed by the authors developed a combinatorial description of the crystal B(1) for finite-dimensional simple Lie algebras in terms of certain Young tableaux, and established an explicit bijection between these tableaux and canonical bases indexed by Lusztig's parametrization.
Abstract: A combinatorial description of the crystal B(1) for finite- dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindi- kin-Karpelevich formula as a sum over the set of Young tableaux.
17 citations
TL;DR: In this article, Li et al. showed that c-dense!-scrambled sets are present in every transitive system with two-sided limit shadowing property (TSLmSP) and that every map on topological graph has a dense Mycielski!-Scrambled set.
Abstract: We consider !-chaos as defined by S. H. Li in 1993. We show that c-dense !-scrambled sets are present in every transitive system with two-sided limit shadowing property (TSLmSP) and that every transitive map on topological graph has a dense Mycielski !-scrambled set. As a preliminary step, we provide a characterization of dynamical properties of maps with TSLmSP.
15 citations
TL;DR: In this article, a fitted mesh finite difference method (FMFDM) is proposed to solve singularly perturb-ed turning point problems, which is robust with respect to the singular perturbation parameter.
Abstract: Investigation of the numerical solution of singularly perturb- ed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.
TL;DR: In this article, the authors established several kinds of ex- tended fixed point theorems in ordered partial metric spaces with higher dimension under generalized notions of mixed monotone mappings.
Abstract: A partial metric, also called a nonzero self-distance, is moti- vated by experience from computer science. Besides a lot of properties of partial metric analogous to those of metric, fixed point theorems in partial metric spaces have been studied recently. We establish several kinds of ex- tended fixed point theorems in ordered partial metric spaces with higher dimension under generalized notions of mixed monotone mappings.
TL;DR: In this article, the authors characterized the supercyclicity of unilateral weighted shifts on the spaces of positive invertible diagonal operators on a separable complex Hilbert space and gave necessary and sufficient conditions for the super cyclicity of those weighted shifts, which extends some previous results of H. Salas.
Abstract: 【In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})$ (or $L^2(\mathbb{Z},\mathcal{K})$ ) with weight sequence { $A_n$ } of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$ . Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.】
TL;DR: In this paper, Anderson and Mulay generalized the results of Anderson and Livingston on the classic zero-divisor graph to the case of R-modules and showed that a R-module is connected with diam(Γ(M)) ≤ 3.
Abstract: . Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F.Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in thepresent article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3.We also show that for a reduced module M with Z(M) ∗ 6= M \ {0},gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, weshow that for a finitely generated semisimple R-module M such that itshomogeneous components are simple, x,y∈ M\ {0} are adjacent if andonly if xRTyR = (0). Among other things, it is also observed thatΓ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, andZ(M) ∗ 6= M\{0}, if and only if ann(M) is prime and Z(M) ∗ 6= M\{0}. 1. IntroductionAll rings in this paper are commutative with identity and all modules areunitary right modules. Let G be an undirected graph. We say that G isconnected if there is a path between any two distinct vertices. For distinctvertices x and y in G, the distance between x and y, denoted by d(x,y), is thelength of a shortest path connecting x and y (d(x,x) = 0 and d(x,y) = ∞ ifno such path exists). The diameter of G isdiam(G) = sup{d(x,y) | x and y are vertices of G}.A cycle of length n in G is a path of the form x
TL;DR: In this article, it was shown that any robustly shadowable chain component of a C 1 vector field does not contain a hyperbolic singularity, and that the chain component is not shadowable if and only if it has no non-hyperbolic nonsmooth singularity.
Abstract: Let be a hyperbolic closed orbit of a C 1 vector field X on a compact boundaryless Riemannian manifold M, and let CX() be the chain component of X which contains . We say that CX() is C 1 ro- bustly shadowable if there is a C 1 neighborhood U of X such that for any Y 2 U, CY (Y ) is shadowable for Yt, where Y denotes the continuation of with respect to Y. In this paper, we prove that any C 1 robustly shad- owable chain component CX() does not contain a hyperbolic singularity, and it is hyperbolic if CX() has no non-hyperbolic singularity.
TL;DR: In this article, a class of delayed Nicholson's blow-flies model with a nonlinear density-dependent mortality term was considered and the authors established some criteria to ensure that the so-lutions of this model converge globally exponentially to a positive almost periodic solution.
Abstract: This paper concerns with a class of delayed Nicholson's blow- flies model with a nonlinear density-dependent mortality term. Under appropriate conditions, we establish some criteria to ensure that the so- lutions of this model converge globally exponentially to a positive almost periodic solution. Moreover, we give some examples and numerical sim- ulations to illustrate our main results.
TL;DR: In this paper, the Hopf-Ore extension of Hopf group coalgebra is introduced, and the necessary and sufficient condition for the Ore extensions to become a Hopf Group coalgebra are discussed.
Abstract: The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.
TL;DR: In this article, it was shown that a strong k-hypertournament H on n vertices, where 3 � kn 2 is vertex-pancyclic, and C is a Hamiltonian cycle in H, has at least three pancyclic arcs.
Abstract: A k-hypertournament H on n vertices, where 2 � kn, is a pair H = (V,A), where V is the vertex set of H and A is a set of k-tuples of vertices, called arcs, such that for all subsets SV with |S| = k, A contains exactly one permutation of S as an arc. Recently, Li et al. showed that any strong k-hypertournament H on n vertices, where 3 � kn 2, is vertex-pancyclic, an extension of Moon's theorem for tournaments. In this paper, we prove the following generalization of another of Moon's theorems: If H is a strong k-hypertournament on n vertices, where 3 � kn 2, and C is a Hamiltonian cycle in H, then C contains at least three pancyclic arcs.
TL;DR: In a recent systematic study, C. Sandon and F. Zanello as discussed by the authors proved 27 of the 30 conjectured identities arising from Ramanujan's formulas for mul- tipliers in the theory of modular equations.
Abstract: In a recent systematic study, C. Sandon and F. Zanello of- fered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for mul- tipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ra- manujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.
TL;DR: In this article, the authors studied the local and semilocal convergence of the relaxed Newton's method with a relaxation parameter 0 < � < 2 and gave a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces.
Abstract: In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < � < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter �. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for � = 1. In many areas related to the applied sciences one confronts the problem of solving a nonlinear equation of the form f(x) = 0. The solutions of these equa- tions can rarely be found in closed form. That is why most solution methods are iterative. There exist lots of iterative methods with different properties that allow us to solve this kind of equations, but the most well-known and used is the Newton's method, which has the following form:
TL;DR: In this paper, the authors introduced the total graph of a commutative ring with respect to identity-summand elements, denoted by T(Γ(R)), and investigated basic properties of S(R) which help to gain interesting results about T(R)) and its subgraphs.
Abstract: . Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graphof R with respect to identity-summand elements, denoted by T(Γ(R)),and investigate basic properties of S(R) which help us to gain interestingresults about T(Γ(R)) and its subgraphs. 1. IntroductionAssociating a graph to an algebraic structure is a research subject and hasattracted considerable attention. In fact, the research in this subject aims atexposing the relationship between algebra and graph theory and at advancingthe application of one to the other.In 1988, Beck [11] introduced the idea of a zero-divisor graph of a commu-tative ring R with identity. This notion was later redefined by Anderson andLivingston in [6]. Since then, there has been a lot of interest in this subject andvarious papers were published establishing different properties of these graphsas well as relations between graphs of various extensions. The total graph ofa commutative ring was introduced by Anderson and Badawi in [3], as thegraph with all elements of R as vertices, and two distinct vertices x,y∈ Rareadjacent if and only if x+ y∈ Z(R) where Z(R) is the set of all zero divisorof R. In [4], Anderson and Badawi studied the subgraph T
TL;DR: In this paper, it was shown that coneat submodules of any right R-module are coclosed if and only ifR is right K-ring; every right Rmodule is coneat-flat if only R is right V-ring, and coneat modules of right injective modules are exactly the modules which have no maximal submodules.
Abstract: . A submodule N of a right R-module M is called coneat iffor every simple right R-module S, any homomorphism N → S can beextended to a homomorphism M → S. M is called coneat-flat if thekernel of any epimorphism Y →M →0 is coneat in Y. It is proven that(1) coneat submodules of any right R-module are coclosed if and only ifR is right K-ring; (2) every right R-module is coneat-flat if and only ifR is right V-ring; (3) coneat submodules of right injective modules areexactly the modules which have no maximal submodules if and only ifR is right small ring. If R is commutative, then a module M is coneat-flat if and only if M + is m-injective. Every maximal left ideal of R isfinitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring Ris perfect if and only if every coneat-flatmodule is projective. We also study the rings over which coneat-flat andflat modules coincide. 1. IntroductionA subgroup A of an abelian group B is said to be neat in B if pA = A∩pBfor every prime integer p. The notion of neat subgroup was generalized tomodules by Renault (see, [12]). Namely, a submodule N of a right R-moduleM is called neat in M, if for every simple right R-module S, Hom(S,M) →Hom(S,M/N) → 0 is epic. Dually, in [8], a submodule N of a right R-moduleM is called coneat in M if Hom(M,S) → Hom(N,S) → 0 is epic for everysimple right R-module S. The notions of neat and coneat are coincide overthe ring of integers. By [8, Theorem], the commutative domains over whichneat and coneat submodules coincide are exactly the domains with finitely gen-erated maximal ideals (i.e., N-domains). This result was extended to certaincommutative rings in [5]. Recently, modules related to neat and coneat sub-modules are considered by several authors. In [5], a right R-module M is calledabsolutely neat (resp. coneat) if M is a neat (resp. coneat) submodule of anymodule containing it. According to [16], a right R-module M is m-injective
TL;DR: In this article, the traveling wave solutions of nonlocal dispersal models with nonlocal delays are investigated by the upper and lower solutions, and the asymptotic behavior of traveling wave solution is studied by the idea of contracting rectangles.
Abstract: This paper is concerned with the traveling wave solutions of nonlocal dispersal models with nonlocal delays. The existence of traveling wave solutions is investigated by the upper and lower solutions, and the asymptotic behavior of traveling wave solutions is studied by the idea of contracting rectangles. To illustrate these results, a delayed competition model is considered by presenting the existence and nonexistence of trav- eling wave solutions, which completes and improves some known results. In particular, our conclusions can deal with the traveling wave solutions of evolutionary systems which admit large time delays reflecting intra- specific competition in population dynamics and leading to the failure of comparison principle in literature.
TL;DR: In this article, it was shown that for a ring R having a complete set of centrally primitive idempotents, every non-zero left (resp. right) semicentral idemomorphism is a finite sum of orthogonal left and right (resp., e.g., 0 6 e ∈ Sr(R)).
Abstract: Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and Sl(R) (resp. Sr(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) e ∈ Sl(R) (resp. e ∈ Sr(R)) if and only if re = ere (resp. er = ere) for all nilpotent elements r ∈ R if and only if fe ∈ I(R) (resp. ef ∈ I(R)) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) which are isomorphic to e if and only if (fe) n = (efe) n (resp. (ef) n = (efe) n ) for all f ∈ I(R) which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any 0 6 e ∈ Sl(R) (resp. 0 6 e ∈ Sr(R)).
TL;DR: The identity-summand graph Γ(R) as mentioned in this paper is a generalization of the zero-divisor graph to include the non-identity identity graph of R with two distinct vertices joint by an edge.
Abstract: . An element r of a commutative semiring R with identity issaid to be identity-summand if there exists 1 6= a∈Rsuch that r+a= 1.In this paper, we introduce and investigate the identity-summand graphof R, denoted by Γ(R). It is the (undirected) graph whose vertices arethe non-identity identity-summands of Rwith two distinct vertices jointby an edge when the sum of the vertices is 1. The basic properties andpossible structures of the graph Γ(R) are studied. 1. IntroductionOne of the associated graphs to a ring R is the zero-divisor graph; it is asimple graph with vertex set Z(R)\{0}, and two vertices x and y are adjacentif and only if xy = 0 which is due to Anderson and Livingston [8]. This graphwas first introduced by Beck, in [11], where all the elements of R are consideredas the vertices. Since then, there has been a lot of interest in this subject andvarious papers were published establishing different properties of these graphsas well as relations between graphs of various extensions [3, 8-9, 18, 20-21].Recently, the study of graphs of rings are extended to include semirings as in[16].As a generalization of rings, structure of semirings have proven to be usefultools in various disciplines. They have important applications in mathematicsand theoretical computer science [19, 22]. From now on let R be a commutativesemiring with identity. We define another graph on R, Γ(R), with vertices aselements of S
TL;DR: In this article, a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting is presented.
Abstract: n´ Abstract. We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as (1-3,7-14,16,20,21) at least for the cases of Newton's method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study.
TL;DR: By using the ring isomorphism, the structures and Hamming distances of all (�+u�)-constacyclic codes and ( �+u.�+u 2 )-constacyClic codes of length p s over R are obtained.
Abstract: Constacyclic codes of length p s over R = Fpm + uFpm + u2Fpm are precisely the ideals of the ring R(x) hxp s −1i . In this paper, we investigate constacyclic codes of length p s over R. The units of the ring R are of the forms , �+u�, �+u� +u 2 and �+u 2 , where �, � and are nonzero elements of Fpm. We obtain the structures and Hamming distances of all (�+u�)-constacyclic codes and (�+u�+u 2 )-constacyclic codes of length p s over R. Furthermore, we classify all cyclic codes of length p s over R, and by using the ring isomorphism we characterize -constacyclic codes of length p s over R.
TL;DR: In this article, the structure of polynomial rings and power series rings without identity was studied, and it was proved that these two ring proper-ties pass to power-series rings and polynomials without identity.
Abstract: We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian
TL;DR: In this paper, the existence of a nontrivial hyperinvariant subspace of k-quasi class A ∗ operators was shown, and the spectrum continuity of this class of operators was proved.
Abstract: An operator T 2 L(H), is said to belong to k-quasi class A ∗ operator if T ∗k � |T n+1 | 2 n+1 | T ∗ | 2 � T kO for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class A ∗. Second, we consider the tensor product for k-quasi class A ∗, giving a necessary and sufficient condition for T S to be a k-quasi class A ∗, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class A∗ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and (B ∗ ) −1 are k-quasi class A∗ operators such that AX = XB, then A∗X = XB∗. Finally, we will prove the spectrum continuity of this class of operators.
TL;DR: In this paper, it was shown that R 0 can be taken R 0 =5.68371 using Kadiri's method together with Platt's numerical verification of Riemann Hypothesis.
Abstract: . In 2005 Kadiri proved that the Riemann zeta function ζ(s)does not vanish in the regionRe(s) ≥ 1 −1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 5.69693. In this paper we will show that R 0 can be taken R 0 =5.68371 using Kadiri’smethod together with Platt’s numerical verificationof Riemann Hypothesis. 1. IntroductionIt is well known that the distribution of prime numbers is deeply relatedwith the zeros of the Riemann zeta function ζ(s). Let π(x) be the number ofprimes up to x. Hadamard and de la Vall´ee Poussin proved the prime numbertheorem which states that π(x) is asymptotic to Li(x) =R x2dtlogt . In 1899, dela Vall´ee Poussin proved that ζ(s) does not vanish in the regionRe(s) ≥ 1−1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 34.82. From this zero-free region for ζ(s) the error term π(x)−Li(x)can be estimated:π(x) −Li(x) = Ox exp−rlogxR 0 when x → ∞.See [18] and Section 1 of [7]. The constant R 0 has been improved by manyauthors, particularly by Rosser [13, 14, 15], Schoenfeld [14, 15], Stechkin [16],Ford [1, 2], and Kadiri [7]. Recently Kadiri improved significantly on R