نشریه پژوهشهای ریاضی
سال هفتم شماره 1 (پیاپی 16، بهار 1400)

We shall give a description of the commutant of a particular multiplication operator in the weighted Bergman spaces on n-dimensional complex polydisk../files/site1/files/71/1.pdf

Many mathematical formulations of physical phenomena contain integro-differential equations. These equations arise in fluid dynamics, biological models, chemical kinetics, ecology, control theory of financial mathematics, aerospace systems, industrial mathematics etc. It is worth mentioning that integro-differential equations are usually difficult to solve analytically, and so it is required to obtain an efficient approximate solution for them.Fractional calculus deals with derivatives and integrals of arbitrary real or complex orders.  This subject has attracted attention of many scientists in mathematics, physics and engineering. So, it has become a hot issue in recent years.Fractional integro-differential equations arise in the mathematical modelling of various physical phenomena, such as heat conduction in materials with memory. Moreover, these equations are encountered in combined conduction, convection and radiation problems. There are only a few techniques for the solution of fractional integro-differential equations, since it is relatively a new subject in mathematics. Some of these methods are Legendre spectral tau method, Adomian decomposition method, piecewise polynomial collocation methods, spline collocation method, hybrid collocation method, hybrid functions approximation by block-pulse functions and Bernoulli polynomials, Taylor expansion approach, differential transform method and wavelet methods.In recent years many problems in mathematics, physics and engineering have been numerically solved by radial basis functions (RBFs) methods.  In this paper, we focus on the Gaussian and inverse multiquardic RBFs as two of the most important tools in engineering and sciences to solve a class of fractional parabolic integro-differential equations. This class of equations describes some phenomena in compression of viscoelastic media and nuclear reactor dynamics.

In the proposed method, first the fractional derivative operator is transformed into a non-singular equivalent.  Then, the Gaussian and inverse multiquardic RBFs together with the collocation method and Gauss-Legendre quadrature formula are used to transform the problem under consideration into the corresponding system of linear algebraic equations, which can be simply solved to achieve an approximate solution of the problem.

Some numerical examples are examined to demonstrate the efficiency and high accuracy of the present method. The obtained results demonstrate that there is a good agreement between the approximate solutions and the exact ones. Also we hope that the proposed method can provide numerical solutions with high accuracy for the problems under study for all fractional orders. Meanwhile, the best value for the shape parameter in the Gaussian and inverse multiquardic RBFs method can be obtained by employing an appropriate optimization method.

The following conclusions were extracted from this research.The established method transforms such problems into equivalent systems of algebraic equations by expanding the solution of the problem in terms of the RBFs and applying Gauss-Legendre integration formula.Only a few number of the RBFs is needed to obtain a high accurate numerical solution for such problems.The presented method can easily be developed for other classes of fractional partial integro-differential equations.

An R-module M is called Almost uniserial module, if any two non-isomorphic submodules of M are linearly ordered by inclusion. In this paper, we investigate some properties of Almost uniserial modules. We show that every finitely generated Almost uniserial module over a Noetherian ring, is torsion or torsionfree. Also the construction of a torsion Almost uniserial modules whose first nonzero Fitting ideal is a product of maximal ideals, is invetigated and torsion Almost uniserial modules over an integral domain and a UFD are characterized.

Throughout this paper,  is a commutative ring with non-zero identity and  is an  -module. The study of injective modules is very important in commutative algebra and homological Algebra. Any product of (even finitely many) injective modules is injective; conversly, if a direct product of modules is injective, then each module is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sum of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple. Also every factor module of every injective module is injective if and only if the ring is Hereditary. Finally every infinite direct sum of injective modules is injective if and only if the ring is Noetherian. In this paper we study some new  propertis of this modules.

The main tool used in the proofs of the main results of this paper is the properties of injective modules and injective envelopes.

We present some new properties of injective envelopes, injective modules, prime submodules and maximal submodules.

We prove the following

Over finitely generated multiplication modules, every prime submodule is irreducible.If  N is a prime submodule of finitely generated multiplication R-module M such that E(M/N) is finitely generated, then N is a maximal submodule of M. Also we give several  corollaries for this note. Also we find relations beetwen Artinian ring, Noetherian ring, indecomposable injective modules and injective cogenerators of  modules.

 Given the important role convex and quasi-convex functions play in many areas of mathematics and especially in optimization, one of the inequalities that has attracted the attention of many mathematicians in recent decades is Hermit-Hadamard’s famous inequality. Significant generalizations and refinements have been obtained for this inequality in a diverse variety of convexity, including convex operator functions of self adjoint operators on Hilbert spaces, matrix functions, quasi-convex, s-convex and log-convex functions.In this paper, we generalize this inequality to differentiable functions whose norm of their derivatives are convex functions.

In this paper, we consider differentiable mappings which norm of the induced maps by them on the set of self adjoint operators is convex, quasi convex or s-convex. We show that if  is an operator monotone function on ,  is a strictly positive operator and  a unitarily invariant norm, then  for all positive integers . We also prove that is a quasi-convex function on the set of all strictly positive operators in B(H). Examples and applications for particular cases of interest are also illustrated. Finally, an error estimate for the Simpson formula is addressed.

The following conclusions were drawn from this research.As an important application of the results in this paper, we find bounds for in terms of , which is one of the central problems in perturbation theory.We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or s- convex.

FitzHugh-Nagumo-Rinzel model is an important model for the dynamics of single neuron. We observed some patterns of Bursting specially Mixed Mode Oscillation (MMO) in this model. In some ranges of parameters we can find a robust canard in this system. These phenomena have been observed by many researchers in systems with one fast variable and two slow variables. The interesting point of our work is that our system has one slow variable and two fast variables.

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metric spaces.

Let  be the quotient ring    where  is the finite field of size   and  is a positive integer. A Gray map  of length  over  is a special map from  to ( . The Gray map   is said to be a ( )-Gray map if the image of any -constacyclic code over    is a -constacyclic code over the field   . In this paper we investigate the existence of   ( )-Gray maps over . In this direction, we find an equivalent condition for such maps. Then we prove that if  is a    ( )-Gray map of length  over  and also  then we must have   ,   and  divides the length of  that is . We also prove that if   then we must have  . Moreover, we determine all of such maps over . Finally, we introduce a   ( , )-Gray map of length    over  where .

Let X be a compact Hausdorff space, E be a normed space, A(X,E)  be a regular Banach function algebra on X , and A(X,E) be a subspace of C(X,E) . In this paper, first we introduce the notion of localness of an additive map S:A(X,E) → C(X,E) with respect to  additive maps T1,...,Tn: A(X) → C(X) and then we characterize the general form of such maps for a certain class of subspaces A(X,E) of C(X,E) having  A(X)-module structure.

In this paper, we investigate and study the notion of left -biflatness of abstract Segal algebras, where  is a character on Banach algebra. Precisely, we give a necessary and sufficient condition for left -biflatness of abstract Segal algebras equipped with a left approximate identity. As an application, we show that if  is a Segal algebra on the locally group  and  is a character, then  is left -biflat if and only if  is amenable. Indeed, this is a generalization of [4, Theorem 3.4]. Moreover, we study the relationship between left -biflatness and inner -amenability and show that if the Banach algebra  is inner -amenable, then the notions of left -biflatness and left -amenability are equivalent.

Nonnegative Matrix Factorization is a new approach to reduce data dimensions. In this method, by applying the nonnegativity of the matrix data, the matrix is ​​decomposed into components that are more interrelated and divide the data into sections where the data in these sections have a specific relationship. In this paper, we use the nonnegative matrix factorization to decompose the user ratings matrix in recommender systems. The user ratings matrix is factorized in a way that the users with similar interests can be identified.
In this paper, we used a regularization method to minimize the difference between the main matrix and the factorized components. To this end we insert the coefficients which are defined as the norm of the decomposition factors in the factorization equation. The coefficients control the entries of the decomposition factors in a multiplication update process. Our numerical results on the MovieLens data set represent the greater accuracy of our proposed method in predicting user ratings for items.

For an f-ring  with bounded inversion property, we show that  , the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever  is a semiprimitive ring, , the set of all basic -ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring  with bounded inversion property, we prove that  is a complemented lattice and  is a semiprimitive ring if and only if  is a complemented lattice and  is a reduced ring if and only if the base elements for closed sets in the space  are open and  is semiprimitive if and only if the base elements for closed sets in the space  are open and  is reduced. As a result, whenever  (i.e., the ring of continuous functions), we have  is a complemented lattice if and only if  is a complemented lattice if and only if  is a -space.

The aim of this study is studying coloring graph by colorable functions and explicating the conditions and their performance on known graphs. Fixing barriers of using the method such as the conditions of creating a function, defining the type of functions, etc. are among the main purposes of the study. To this end, at first, colorability of graphs is defined and then its elements are analyzed.Definition: Let f : X → X be a graph without fixed point. f is colorable with k colors, if there is , where all Ci s do not include {f(x, x)}. Or similarly, for every  , there is the equation ∅ = Ci ∩ f(Ci).Then, to determine the chromatic number of graphs by these functions, all various theorems and lemmas are stated. It is shown that every graph is colored with a specific number by one of the colored functions

At first, coloring function is defined and its performance is stated. When the colored theorem is stated, which is the main element in coloring functions, the performance of the functions in coloring a graph is explored. Finally, the chromatic number of every graph by the functions is determined through some theorems and lemmas.

In this study the chromatic number of some graphs such as simple graph, triangular graph, complete graph, etc. was determined by these functions. It was shown that the performance of the functions are complicated at first, but they have a good performance and simplicity in every point.

The study concluded that:- Except when you cannot define a function for a graph at all, the chromatic number of every graph is determinable in this way,The efficiency of this method in finding the chromatic number of graphs of many vertices is much easier. Users’ probability of mistake in estimating the chromatic number of graphs of many vertices is very low..

In this article, we introduce the weak Armendariz ideals as a generalization of the Armendariz ideals and we examine its properties, its relation to other structures. Also by giving numerous examples and diverse, we evaluate the behavior of weak Armendariz ideals under some ring extensions..

Throughout this paper,  is a commutative Noetherian ring with non-zero identity,  is an ideal of ,  is a finitely generated -module, ‎and  is an arbitrary -module which is not necessarily finitely generated. Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that  is finitely generated for all . In [12], ‎Hartshorne gave a counter-example and raised the question whether  is finitely generated for all  and . The th generalized local cohomology module of  and  with respect to ,was introduced by Herzog in [14]. It is clear that  is just the ordinary local cohomology module  of  with respect to . As a generalization of Hartshornechr('39')s question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]).Question. When is  finitely generated for all  and ? In this paper, we study  in general for a finitely generated -module  and an arbitrary -module .

The main tool used in the proofs of the main results of this paper is the spectral sequences.

We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers , , , and  with ,  and the -modules  and  are in a Serre subcategory of the category of -modules (i.e. the class of   -modules which is closed under taking submodules, quotients, and extensions).

We apply the main results of this paper to some Serre subcategories (e.g. the class of zero  -modules and the class of finitely generated -modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an -module  is said to be -cofinite (resp. minimax) if  and  is finitely generated for all  [12] (resp. there is a finitely generated submodule  of  such that  is Artinian [26]) where We show that if  is finitely generated for all  and  is minimax for all , then  is -cofinite for all  and  is finitely generated (Corollary 3.5). We prove that if  is finitely generated for all , where  is the arithmetic rank of , and  is -cofinite for all , then  is also an -cofinite -module (Corollary 3.6). We show that if  is local, , and  is finitely generated for all , then  is -cofinite for all  if and only if  is finitely generated for all  (Corollary 3.7). We also prove that if  is local, ,  is finitely generated for all , and  (or ) is -cofinite for all , then  is -cofinite for all  (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules../files/site1/files/71/15.pdf

Representation as well as central extension are two of the most important concepts in the theory of Lie (super)algebras. Apart from the interest of mathematicians, the attention of physicist are also drawn to these two subjects because of the significant amount of their applications in Physics. In fact, for physicists, the study of projective representations of Lie (super)algebras  are very important.Projective representations of a Lie superalgebra  are representations of the central extensions of. So the study of projective representations has two steps; at first, one needs to know the central extensions and then to study their representations. The first question in the study of central extensions is finding the universal one (if it exists). In 1984, universal central extensions of the algebras of the form  for a unital commutative associative algebra  and a simple finite dimensional Lie algebra , were identified. Then in 2011, the case when  is a basic classical simple Lie superalgebra was studied by K. Iohara and Y. Koga.  They first study the case for Lie superalgebras  of rank 1; then they study -forms of  and prove the existence of a Chevalley base type for  using its structure as a basic classical simple Lie superalgebra. This in particular helps them to define an even nondegenerate symmetric invariant bilinear form on 

In this work, we study universal central extensions of Lie superalgebras of the form ,  where  is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all  derivations and   is a unital commutative associative algebra. Our techniques are totally different from the ones done before; in fact to get our results we use the materials of the previous work of the author (joint with Karl-Hermann Neeb) regarding central extensions of .

We find the universal central extensions of Lie superalgebras of the form ,  where  is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all derivations and   is a unital commutative associative algebra. 

Universal central extensions of Lie superalgebras of the form A ⊗  as above are identified. Our main result covers the results of the previous works in this regard and moreover, since odd nondegenerate invariant bilinear forms on   are allowed, we get something more, e.g., the uinversal central extension of A ⊗  for the queer Lie superalgebra   = (n) is also covered by our main theorem.