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Present Study as a "meta-analysis" research aims to determine the role and methods of communication, regarding verbal and nonverbal learning in mathematics is provided. Research conducted for this purpose (18 Researches) on mathematic teaching, in the form of research papers published in the Journal of Educational Innovations, as the main concept was considered. Out of the 8 research papers published in terms of attention to communication variables has more extension of the statistical society of this research. Then, according to the variables performance methods, questions and results showed in them was descriptive-qualitative and get “meta-analysis. The main question of this study is to research on mathematics teaching is based on the method of communication, what are the similarities and differences in the use of effective communication? During the investigation, it was determined that the communication patterns of multiple and sometimes conflicting have more attentions and proposed recommendations, which are as follows: emotional, participatory, supporting, visual, objective and balanced in a form of speech, physical, visual and audio that leads to any progress performance of learners and their learning mathematics and problem solving.

This paper presents a new stenographic scheme based on matrix embedding using BCH syndrome coding. The proposed method embeds massage into cover by changing some coefficients of cover. In this paper defining a number :::as char:::acteristic of the syndrome, which is invariant with respect to the cyclic shift, we propose a new embedding algorithm base on BCH syndrome coding, without finding roots of quadratic and cubic polynomials in Galois field. The computational complexity of the proposed method is linear and space complexity is polynomial of order 2.

In this paper, we solve the multi-period portfolio optimization problem under new assumptions. Recently, the authors examined some distributions instead of Gaussian to fit returns to improve the optimization problem and indicated, by Kolmogorov-Smirnov test, that the Kernel density estimator is the best one. In the present paper, we consider the most appropriate distribution of each asset in each period to fit return. Moreover, in order to optimize the solutions, we use piecewise linear interpolation of the empirical distribution. Furthermore, we show that how one may plan for future investments using historical data and AR(1)-GARCH(1,1) model under new assumptions.

In this paper we consider two type of mathematical models for the novel coronavirus (2019-nCov), which are in the form of a nonlinear differential equations system. In the first model the contact rate, , and transition rate of symptomatic infected indeviduals to the quarantined infected class, , are constant. And in the second model these quantities are time dependent. These models are the SEIR one, where are Susceptible, Exposed, Infected and Recovered classes of human population respectively. We establish the Newton-Taylor polynomial solutions for these system, so that the nonlinear systems are solvable by an iterative and progressive process with a good accuracy. We completely describe the algorithm of such systems in another paper and here we express briefly. This algorithm action on the interval , where is the length of partial intervals, and is the number of intervals. In every partial interval, we linearize the problem by the Newton's method and then solve the linear problem by the Taylor polynomial solutions technique. We extensively investigate the numerical analysis of the method.

In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.

Introduction The notion of weak amenability for commutative Banach algebras was introduced and studied for the first time by Bade, Curtis and Dales. Johnson extended this concept to the non commutative case and showed that group algebras of all locally compact groups are weakly amenable. A Banach algebra A is called weakly amenable if every continuous derivation D:A→A* is inner. It is often useful to restrict one's attention to derivations D:A→A* satisfying the property Dac+Dca=0 for all a,c∈A. Such derivations are called cyclic. Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable if every continuous cyclic derivations D:A→A*is inner. This notion was presented by Gronbaek. He investigated the hereditary properties of this concept, found some relations between cyclic amenability of a Banach algebra and the trace extension property of its ideals. Ghahramani and Loy introduced several approximate notions of amenability by requiring that all bounded derivations from a given Banach algebra A into certain Banach A-bimodules to be approximately inner. In the same paper and the subsequent one, the authors showed the distinction between each of these concepts and the corresponding classical notions and investigated properties of algebras in each of these new classes. Motivated by this notions, Esslamzadeh and Shojaee defined the concept of approximate cyclic amenability for Banach algebras and investigated the hereditary properties for this new notion. Periliminaries Let A be a Banach algebra and let X be an A-bimodule. Then the l1-direct sum A×X under the multiplication a,xb,y=ab,ay+xb (a,b∈A,x,y∈X), is a Banach algebra called the module extension of A by X and denoted by A⊕X. The class of module extension Banach algebras contains a wide variety of Banach algebras includes a triangular Banach algebra Tri(A,X,B). Every triangular Banach algebra Tri(A,X,B).can be identified with the module extension Banach algebra (A×B)⊕X. On the other hand, for two Banach algebra A and B with ∆(B)≠∅ and for θ∈∆(B), the set of all non-zero multiplicative linear functionals on B, the θ-Lau product A×θB is a Banach algebra which is defined as the l1-direct sum A×B equipped with the algebra multiplication a1,b1a2,b2=a1a2+θb2a1+θb1a2,b1b2 a1,a2∈A,b1,b2∈B. This type of product was introduced by Lau for certain class of Banach algebras known as Lau algebras and was extended by Sangani Monfared for arbitrary Banach algebras. The unitization A♯ of A can be regarded as the ι-Lau product A×ιC, where ι∈Δ(C) is the identity map. This product provides not only new examples of Banach algebras by themselves, but it can also serve as a source of (counter) examples for various purposes in functional and harmonic analysis. From the homological algebra point of view A×θB is a strongly splitting Banach algebra extension of B by A. The Lau product of Banach algebras enjoys some properties that are not shared in general by arbitrary strongly splitting extensions. For instance, commutativity is not preserved by a generally strongly splitting extension. However, A×θB is commutative if and only if both A and B are commutative. Results and discussion Many basic properties of A♯, some notions of amenability and some homological properties are extended to A×θB by many authors. In particular, Ghaderi, Nasr-Isfahani and Nemati extended some results on (approximate) cyclic amenability of A♯, obtained by Esslamzadeh and Shojaee, to A×θB. They showed that if A2 is dense in A then the cyclic amenability A×θB is equivalent to the cyclic amenability of both A and B. In this paper, by characterizing of cyclic derivations on Lau product A×θB and module extension A⊕X, we present general necessary and sufficient conditions for those to be (approximate) cyclic amenable. This not only provides new results on (approximate) cyclic amenability of these type of Banach algebras but also improves some main results in this topic. In particular we show that, under mild condition, the cyclic amenability of Tri(A,X,B) is equivalent to the cyclic amenability of the corner algebras A and B.

Introduction In population dynamics, discrete-time dynamical systems have been used to describe interaction between ecological species. Comparing to continuous-time dynamical systems, discrete-time models are more suitable to describe populations with non overlapping generations. These models in general produce rich and complex dynamical behaviors. Among various population interaction, predator-prey models play a fundamental rule in mathematical ecology. The dynamics of predator-prey system is greatly depend on the implementation of the functional response, the availability of prey for predation. In this paper we consider a planar system which describes a predator-prey model. In order to reveal comprehensive dynamics of the system, we employee theoretical tools such as center manifold theorem along with numerical tools based on numerical continuation method. Material and methods Our analysis is based on theoretical and numerical techniques. We first determine all fixed points of the system and conditions under which these points may undergo different bifurcations. To reveal more dynamics of the system, we also use numerical bifurcation methods and numerical simulations, which further examine the obtained analytical results. Results and discussion For the resented discrete-time predator-prey system, we compute several bifurcation curves, all possible codimension-1 and codimension-2 bifurcations on thses curves along with their corresponding normal form coefficients. By branch switching technique and employing software package MatcontM, we compute stability boundaries for several cycles up to period 32. We also use numerical simulation, to compute basin of attraction and strange attractor emerging around a Neimark-Sacker bifurcation. Conclusion We can highlight the following results from this paper. Detection and location of all fixed points of a discrete-time predator pray system. Computing all possible codimension-1 and -2 bifurcation and their corresponding normal form coefficients which in turn reveal criticality of the bifurcation points and determine if extra bifurcation curves can emanate from each detected bifurcation. Computing orbits up to period 32 which determine stability thresholds for different cycles. Computing basin of attraction and strange of attractor which emerge around a Neimark-Sacker bifurcation by means of numerical simulation technique.

Introduction Analytic hierarchy process (AHP) is a method of multiple criteria decision making (MCDM) that is used to select an alternative from a set of alternatives or to rank a set of alternatives, while data envelopment analysis (DEA) is a nonparametric method that is used based on linear programming to evaluate the performance of decision making units (DMUs) that have multiple inputs and multiple outputs. The relation between methods of MCDM and DEA is a topic of interest to researchers in this part of MCDM, e.g., one of the first works done in this field is the relation between data envelopment analysis and multiple objective linear programming by Golany. Ramanathan proposed a method (DEAHP method) based on DEA for weight generation in the AHP that his method had three main drawbacks: (1) producing irrational weights for inconsistent pairwise comparison matrices; (2) non-use all the information of the inconsistent pairwise comparison matrix; and (3) insensitivity to changing elements in some matrices of pairwise comparison. To solve the problems of DEAHP method, several methods were proposed that each one produces a weight vector in the AHP, e.g., we can mentioned to data envelopment analysis method of wang and chin (DEA method) and data envelopment analysis method with assurance region of wang and et al. (DEA/AR method). In this paper, we propose a new method, which is called E-DEAHP method for short, based on DEA and Shannon entropy, a concept used in information theory, to produce a weight vector in the AHP that does not have the problems of DEAHP method and is different from the mentioned methods. Material and methods In this approach, each row of the pairwise comparison matrix is considered as a decision making unit (DMU), so that in the normalized pairwise comparison matrix the arithmetic mean of the ith row and the entropy of ith column is considered as, respectively, output and input of the ith DMU and then with employed data envelopment analysis, we find the local weight vector of the elements (decision criteria or alternatives). Also, to aggregate the obtained local weights, we use the simple additive weighting (SAW) method in multiple criteria decision making. Results and discussion It is proved that if a pairwise comparison matrix is perfectly consistent, the entropy of all its columns are the same, so in this case all decision making units will have the same input and the method will produce true weight vector. The results of the examined numerical examples show that the proposed method of this paper produces perfectly rational weights in comparison with the results of the methods known in the subject literature and can estimate a robust priority (weight) vector for a pairwise comparison matrix. Also, the results of the hierarchical problem survey show that the weights obtained from the method and their aggregation to obtain the global weight vector confirm the potential validity of the method. Conclusion In this paper, in relation to E-DEAHP method, we have achieved the following conclusions. Generating true weight vector for perfectly consistent pairwise comparison matrices. The method for ranking and selecting alternatives has a high resolution. The weight vector obtained from this method is robust, In other words, it is not affected by possible errors, unusual and false observations (UFO) that appear because of inaccurate data entry random errors, in the pairwise comparison matrix. In practice, the E-DEAHP method can be applied without the need to solve linear programming by using a simple relative relation.

Introduction ‎‎Censored sample arises in a life-testing experiment whenever the experimenter does not observe the failure times of all units placed on a life-test‎. ‎In medical or industrial studies‎, ‎researchers have to treat the censored data because they usually do not have sufficient time to observe the lifetime of all subjects in the study‎. ‎There are different types of censoring‎. ‎The most common censoring schemes are type I and type II censoring schemes‎. Progressively type II censoring is also one of the most important methods of censoring. One of the most common questions any statistician gets asked is "How large a sample size do I need?"‎. ‎Researchers are often surprised to find out that the answer depends on a number of‎ factors and they have to give the statistician some information before they can get an answer‎. ‎So far different answers have been given to respond this question by considering different criteria‎. ‎Cost criterion is one of the criteria that has always been of interest to researchers‎. ‎So far‎, ‎many researchers have used this criterion for determining the size of samples in different censoring methods‎. ‎ In some applications‎, ‎such as clinical trials and quality control‎, ‎it is almost impossible to have a fixed sample size all the time because some observations may be missing for various reasons‎. ‎In other words‎, ‎the sample size is a random variable‎. Material and methods In this paper, a cost function is introduced. Then, assuming that the sample size of progressively type II censoring is a random variable from the truncated binomial distribution, the optimal parameter of sample size distribution in progressively type II censoring, is determined. This optimal parameter is determined so that the introduced cost function does not exceed a pre-determined value, say . In this article, the exponential distribution is considered for lifetimes of observations. A simulation study is also provided to evaluate the obtained results. Finally, the conclusion of the article is presented. Results and discussion ‎We have computed the values of the expected cost function by considering three different censoring schemes‎. ‎The results show that the expected cost function is an increasing function of m but a decreasing function of θ, ‎when other components are fixed‎, ‎as we expected‎. Also, we can find that considering type II censoring leads to better results than other censoring schemes‎. On the other hand, we can conclude that type II censoring provides the minimum cost among two other censoring schemes. In the sequel, by assuming an upper bound for the cost function, say , the optimal parameter of sample size distribution is obtained. Conclusion Determining the optimal sample size is one of the issues that has been studied by many researchers. In some cases, it is not possible for the sample size to be a fixed and pre-determined value. In other words, the sample size is a random variable. In this paper, assuming that the sample size of progressively type II censoring is a random variable from the truncated binomial distribution, the optimal parameter of the sample size distribution is determined. The criterion used in this research is the cost criterion. Next, the optimal parameter of the sample size distribution is determined so that the value of the cost function is less than the specified and predetermined value, say . The results of the paper show that the type II censoring provides less values for the cost function. For all three censorsing schemes, the cost function is an increasing function of m but a decreasing function of θ, ‎when other components are fixed‎, ‎as we expected‎. As a result, the best case scenario is taking into account the type II censoring scheme, selecting smaller values for m, larger values ​​for θ, and smaller values for the parameter of sample size distribution.

Introduction Let p(x,ζ) be the set of parametric probability distribution with parameter ζ=ζ1,…,ζn∊Rn. This set is called a statistical model or manifold. The distance between two points is measured by the Fisher metric. In general, statistical manifolds are Riemannian manifolds of distributions endowed with the Fisher information metric. On the other hand, one of the most important structures on odd dimensional Riemannian manifolds is the almost contact structure. Recently, statistical manifolds equipped with almost contact structures are studied by many authors. In this paper, we introduce statistical almost contact-like and statistical cosymplectic manifolds on a Riemannian manifold. We recall the basic definitions and define statistical cosymplectic manifolds and their invariant submanifolds. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also, we prove if the structure vector field be normal to the submanifold then the submanifold is a statistical Keahler-like manifold. Finally, we construct two examples to illustrate some results of the paper. Statistical almost contact-like manifolds Let M,g be a Riemannian manifold with the Levi-Civita connection ∇. M,g is called a statistical manifold if there exists an affine and torsion free connection ∇ such that for all U,V,W∊τM ∇UgV,W=∇VgU,W. Moreover, an affine and torsion free connection ∇* is called a dual connection with respect to g, if UgV,W=g∇UV,W+gV,∇*U W. An almost contact manifold (M,φ,ξ,η) with Riemannian metric g is an almost contact-like manifold if it has another (1,1)-tensor field φ* satisfying gφU,V=-gU,φ*V, gU, ξ= ηU. Let (M,φ,ξ,η) be an almost contact-like manifold, then for all U,V∊τ(M) the following relations hold gφU, φ*V=gU,V- ηUηV, φ*2U=-U+ηUξ. Definition. An almost contact-like manifold (M,∇,φ,ξ,η,g) with statistical structure (∇,g) is a statistical almost contact-like manifold. Moreover, M,∇,φ,ξ,η,g is called a statistical cosymplectic manifold if ∇UφV=0. M is an invariant submanifold of a statistical cosymplectic manifold M,∇,φ,ξ,η,g, if for all U∊τ(M) we have φU∊τM, φ*U∊τM. Submanifolds of statistical cosymplectic manifolds We show that the manifold M,∇,φ,ξ,η,g is a statistical cosymplectic manifold if and only if (M,∇*,φ*,ξ,η,g) is a statistical cosymplectic manifold. Moreover we prove the following theorems. Theorem. Any invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field ξ, is a statistical cosymplectic and minimal-like submanifold. Theorem. Let M be a submanifold of statistical cosymplectic manifold (M,∇,φ,ξ,η,g) such that the structure vector field ξ is normal to M. Then for any vector field U∊τ(M) we have A*ξU=0, ∇⊥Uξ=η∇Uξξ. Theorem. Let M,∇,φ,ξ,η,g be a statistical cosymplectic manifold. If M is a submanifold of M and the structure vector field ξ is normal to M then R⊥U,Vξ=0, ∀U,V∊τM. Theorem. Let M be an invariant submanifold of statistical cosymplectic manifold M,∇,φ,ξ,η,g and ξ is normal to M. Then M is a statistical Keahler-like manifold. Conclusion We introduce statistical cosymplectic manifolds and investigate some properties of their tensors. We define invariant and anti-invariant submanifolds and study invariant submanifolds with normal and tangent structure vector fields. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also we show if the structure vector field is normal to the submanifold then that is a statistical Keahler-like manifold

Introduction ‎‎ The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal I in a polynomial ring, the regularity of powers of I is asymptotically linear. In other words, there exist integers a(I) and b(I) such that regIs=aIs+b(I) for every integer s≫0. It is known that a(I) is bounded above by the maximum degree of generators of I. Moreover, if I is generated in a single degree d, then aI=d. But in general, it is not so much known about b(I) even if I is monomial ideal. However, when I is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that bI≤regI-2. In fact, they conjectured that the inequality regIs≤2s+regI-2 holds for any integer s≥1, when I is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for s=2. In this paper, we provide an alternative proof for their result. While the proof in [10] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic. Material and methods To every simple graph G one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of G. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of G to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the sth power of an edge ideal I(G) has an upper bound which is defined in terms of the regularity of its (s-1)th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the G. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph G, with edge ideal I(G), we have regIG2≤reg(IG)+2. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most regIG. To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than G. Conclusion The following conclusions were drawn from this research. The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of s=2. It is known that for every graph G with edge ideal I(G) and induced matching number ν(G), we have 2s+νG-1≤reg(IGs), for every integer s≥1. Thus, our result implies that if regIG=νG+1, then regIG2=νG+3. The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of s=3 .

Introduction ‎Ricci solitons as a generalization of Einstein manifolds introduced by Hamilton in mid 1980s‎. ‎In the last two decades‎, ‎a lot of researchers have been done on Ricci solitons‎. ‎Currently‎, ‎Ricci solitons have became a crucial tool in studding Riemannian manifolds‎, ‎especially for manifolds with positive urvature‎. ‎Ricci ‎solitons ‎also ‎serve ‎as ‎similar‎ ‎solutions ‎for‎ ‎the ‎Ricci ‎flow ‎which ‎is ‎an ‎evolutionary ‎equation ‎for ‎the‎ ‎metric‎s ‎of a‎ ‎Riemannian ‎manifold. ‎It ‎is ‎clear ‎that ‎the ‎Ricci ‎flow ‎describes ‎the ‎heat ‎character ‎of ‎the ‎metrics ‎and ‎curvatures ‎of ‎manifolds. On ‎the ‎other ‎hand, ‎hyperbolic ‎Ricci ‎flow ‎was ‎first ‎study ‎by ‎Kong ‎and ‎Liu. This ‎flow ‎is a‎ ‎system ‎of ‎non-linear ‎evolution ‎partial ‎differential ‎equation‎s of second order. ‎The ‎short ‎time ‎existence ‎and ‎uniqueness‎ ‎theorem ‎of ‎hyperbolic ‎geometric ‎flow ‎has ‎been ‎proved ‎in. ‎It ‎is ‎s‎hown ‎that ‎the ‎hyperbolic ‎Ricci ‎flow ‎carries ‎many ‎interesting‎ ‎properties ‎of ‎both ‎Ricci ‎flow ‎as ‎well ‎as ‎the ‎Einstein ‎equation. ‎‎ According to these notions and their applications in both geometry and physics, in this paper we introduce a new hyperbolic flow and study its geometric quantities along to this flow. Self-similar solution of this flow may create interesting geometries on the underlying manifold. Results In this paper, we consider the hyperbolic Gradient-Bourguignon flow on a compact manifold M and show that this flow has a unique solution on short-time with imposing on initial conditions. After then, we find evolution equations for Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of M under this flow. In the final section, we give some examples of this flow on some compact manifolds.

Introduction Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach fixed point theorem. There exists a vast literature on the topic field and this is a very active field of research at present. Fixed point theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (integral and partial equations, variational inequalities, etc). Its core subject is concerned with the conditions for the existence of one or more fixed points of a mapping or multi-valued mapping T from a topological space X into itself that is, we can find x ∈ X such that Tx = x (for mapping) or x ∈ Tx (for multi-valued mapping). In a wide range of mathematical, computational, economic, modeling, and engineering problems, the existence of a solution to a theoretical or real-world problem is equivalent to the existence of a fixed point for a suitable map or operator. Fixed points are therefore of paramount importance in many areas of mathematics, sciences, and engineering. In 1922 Stefan Banach proved a famous theorem which under suitable conditions stated the existence and uniqueness of a mapping. The result of the fixed point theorem or Banach contraction principle was obtained by Stefan Banach. In 1985, V. Popa proved common fixed point theorems for multi-valued mappings that verify rational inequalities, which contain the Hausdorff metric in their expressions. In 2010, A. Petcu proved other common fixed point theorems for two or more multi-valued mappings without using the Hausdorff metric. In this paper, by using some completely different conditions, we study the existence of common fixed points for multi-valued mappings with applying inequalities on binomials and trinomials. Material and methods The content of this paper is organized as follows. First, we present some definitions, lemmas, and basic results that will be used in the proofs of our theorems. Then, we study the existence of common fixed points for multi-valued mappings by applying inequalities on binomials and trinomials. Results and discussion Let F be all multi-valued mappings of X into Pb,cl(X). First, we define an equivalence relation for the elements of F as follows; F ∼ G if and only if fixF = fixG, (F,G ∈ F). Where fixF= x∈X:x∈Fx. Denote the equivalence class of F by F, and define it as follows: F=F~={F:F∈F}. Also define d on F with dF,G=H(fixF,fixG). F,d is a metric space. In this article, by considering some conditions on the maps F and G in complete metric space we conclude that F=G. Conclusion The well known Banach contraction principle ensures the existence and uniqueness of the fixed point of a contraction on a complete metric space. After this interesting principle, several authors generalized this principle by introducing the various contractions on metric spaces. Thereafter, Popa and Petcu obtained some results in about common fixed points of multi-valued mappings. This paper studies the existence of common fixed points for multi-valued mappings by applying inequalities on binomials and trinomials and using different conditions.

Introduction A class of random variables called negatively orthant dependent random variables defined in the classical setting of probability spaces by Lehman in 1966. Joag-Dev and Proschan extended this class of random variables and showed every sequence of negatively associated random variables is negatively orthant dependent. In 2008, acceptable random variables are defined by Antonini in the classical setting of probability spaces. Let {Xn}n∈N be a sequence of random variables in a probability space (Ω,F,P), where Ω is a sample space, F is a σ-algebra of the subsets of Ω, and P is a probability measure in F. Furthermore let E be the expectation of random variables in (Ω,F,P). In the proof of limit theorems it is so important to obtain an exponential bound for a partial sum i=1n(Xi-EXi) . Sung et. al., obtained an exponential bound for i=1n(Xi-EXi) and proved it for the cacceptable random variables class. Kim, Nooghabi and Azarnoosh, Roussas and Xing, obtained exponential bound for negatively associated random variables. In this paper, we define a class of acceptable random variables in noncommutative (quantum) probability spaces. In fact this paper transfers some of probability inequalities from the commutative probability spaces into the noncommutative probability spaces. Material and methods In this scheme, first for the convenience of the reader we repeat the main definitions of noncommutative probability spaces. In the second step we define acceptable random variables in the noncommutative probability spaces and prove some probability inequalities for this class of random variables. Results and discussion In general case, probability inequalities determine upper and lower bounds for the expectation of a random variable or the probability measure of an event. Sometimes we can not obtain the expectation or the probability measure exactly. In these situations, these bounds are important for control the expectation of a random variable or the probability measure of an event. In this paper we want to answer this question: Can we obtain these bounds for probability inequalities in von Neumann algebras? Methods are used in this paper to compare classical setting are different. One of difficulties of the noncommutative setting is this fact that the product of two positive elements is not necessarily positive element in a C*-algebra; Because it is not self-adjoint. Another one is this fact that there is not guarantee for existence of the maximum of two positive operators. Conclusion The following conclusions were drawn from this research. The probability inequalities are proved for acceptable random variables in Noncommutative probability spaces. They are a generalization of probability inequalities for negatively orthant dependent random variables in probability theory. We show under what situations, sequence 1ni=1nxin≥1is completely convergence, where {xn}n≥1 is a sequence of noncommutative self-adjoint random variables. We show under what situations, sequence 1nβi=1nxin≥1for every β>1 is completely convergence, where {xn}n≥1 is a sequence of noncommutative self-adjoint random variables. All the results of this paper can be used for random matrices.

In this paper, we consider the problem of two sided hypothesis testing for the parameter of coefficient of variation of an inverse Gaussian population. An approach used here is the modified signed log-likelihood ratio (MSLR) method which is the modification of traditional signed log-likelihood ratio test. Previous works show that this proposed method has third-order accuracy whereas the traditional approach has first-order one. Indeed, these methods are based on likelihood with a higher order of accuracy. For this reason, we are interested in using this method for inference about the parameter of coefficient of variation of an inverse Gaussian distribution. All necessary formulas for obtaining MSLR statistic are provided. Numerically, the performances of this method are compared with classical approaches, in terms of empirical type-I error rate and empirical test power. Simulation results show that the empirical type-I error rates of MSLR are close to nominal type-I error rate, even for small sample sizes whereas the traditional approaches are reliable only for large sample sizes. Comparing the empirical power sizes shows that the power of MSLR method is superior to other considered methods in some settings, by regarding that the competing approaches cannot perform well in controlling the type-I error probability because their empirical type-I error rates are far from the nominal type-I error rate. Finally, we illustrate the proposed methods using a real data set and then we conclude the paper.

This paper presents a numerical method for solving a fractional model of HIV infection of CD4 + T cells. Using this model, we can examine the progression and spread of HIV. This method is a collocation method based on Dikson polynomials. Chebyshev's nodes are used in this method. By performing this process, the system of fractional differential equations converted to a nonlinear equations system, and the nonlinear system is solved by one of the existing methods. To illustrate the accuracy of this method, numerical results are compared with the numerical results of several similar papers.

One of the classical fundamental motifs in differential geometry of manifolds is the notion of the almost contact structure. As a counterpart of the almost contact metric structure, the notion of the almost contact B-metric structure has been an interesting research field for many mathematicians in differential geometry of manifolds, and the geometry of such structures has been studied frequently. There is a classification for the almost contact B-metric structures, named the relevant classification, with respect to the covariant derivative of the fundamental tensor of type (1, 1). In this paper, we basically use this classification to achieve our goals. On the other hand, many of mathematicians have widely considered the concept of lifted metric on the tangent bundle and tangent sphere bundle of a Riemannian manifold (M, g). The idea of constructing a lifted metric on the tangent bundle was a strong inspiration for many of mathematicians and finally, the notion of g-natural metric as the most general type of lifted metrics on tangent bundle TM of a Riemannian manifold (M, g) was introduced in 2005. In this paper, we consider a pair of associated g-natural metrics on the unit tangent sphere bundle T1M with B-metric, and we classify this structure with respect to the relevant classification of almost contact manifold with B-metric. tensor../files/site1/files/%D9%BE%DB%8C%D8%BA%D8%A7%D9%86.pdf

In this paper, we introduce a Fréchet space and define a measure of noncompactness on it. For credit and the application to our theorems, in the application section of this paper, we present a theorem which shows the existence of solution of infinite system of nonlinear Volterra integral equations, and by using the technique of measures of noncompactness together Darbo fixed point theorem we guarantee the existence of solution in this Fréchet space . Finally, we present some examples which show the efficiency and applicability of our theorem../files/site1/files/72/5Abstract.pdf

This paper, inspired by recent advances in the application of the multilevel Monte-Carlo (MLMC) approach to Lévy driven assets, is based on the valuation of financial derivatives. First, using the weak Euler method the numerical estimate of the underlying asset, which satisfies a multi-dimensional stochastic differential equation with Lévy noise, is calculated and then applying the weak multilevel Monte-Carlo method the expected price is obtained. In this paper, as an improvement of Belomestny’s work and with a new approach in the theory, we express and prove the convergence theorems in spacefor and not only 2. We also seek to implement the weak MLMC algorithm for nonlinear equations with dependent components and . In the end, we show numerical experiments when applied to different types of processes with call options../files/site1/files/72/14Abstract.pdf

Introduction 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 Hartshorne'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 . Material and methods The main tool used in the proofs of the main results of this paper is the spectral sequences. Results and discussion 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). Conclusion 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

Introduction 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. Results and discussion 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. Conclusion 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../files/site1/files/71/5.pdf

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. ‎./files/site1/files/71/11.pdf

Introduction 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. Material and methods The main tool used in the proofs of the main results of this paper is the properties of injective modules and injective envelopes. Results and discussion We present some new properties of injective envelopes, injective modules, prime submodules and maximal submodules. Conclusion We prove the following results: 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../files/site1/files/71/4.pdf

Introduction ‎ Symmetries of an equation are closely related to conservation‎ laws‎. ‎Noether's theorem provides a method for finding conservation laws of differential‎ equations arising from a known Lagrangian and having a known Lie symmetry‎. ‎ An algorithmic method to determine conservation laws for systems of‎ Euler-Lagrange equations which their Noether symmetries are known is Noether‎ theorem‎. ‎This theorem relies on the availability of a Lagrangian and the‎ corresponding Noether symmetries which leave invariant the action integral‎. ‎ One can find the geodesic equations from the variation of the geodesic‎ Lagrangian defined by the given metric and since the Noether symmetries‎ are a subgroup of the Lie group of Lie symmetries of these equations‎, ‎one should expect‎ a relation of the Noether symmetries of this Lagrangian with the projective‎ collineations of the metric or with its degenerates‎‎. ‎ Recently Bokhari et al‎. have published many papers about the relation of the Noether symmetries‎ and Lie point symmetrys and conservation laws of some special spacetimes‎. ‎‎Tsamparlis and Paliathanasis have calculated the Lie point symmetries and the Noether symmetries explicitly‎ together with the corresponding linear and quadratic first integrals‎ for the Schwarzschild spacetime and the Friedman Robertson Walker (FRW)‎ spacetime‎. ‎More than they succeeded in establishing a connection between the Lie‎ symmetries of the geodesic equations in a Riemannian‎ space with the collineations of the metric‎. ‎ Results and discussion In this paper, Lie and Noether symmetries and conservation laws for the hyperbolic model metric of SL(2, R) geometry in a Riemannian space are obtained. Then the point symmetries of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found and then the conservation laws associated to the system of geodesic equations are calculated via Noether’s theorem. Conclusion The following conclusions were drawn from this research. The Lie point symmetry algebra of the hyperbolic model metric SL(2, R) geometry ‎has six dimensions which would include the Noether symmetries‎, have an additional basis element which is dilation in the arclength‎. The Lie point symmetry commutes with all other symmetries‎. As a more discussion in Lie algebra analysis‎, ‎one may find the Lie algebra of‎ Lie point symmetry corresponding to the metric is not semisimple or not solvable‎, because it has degenerated Killing form‎. There are five conserved flows corresponding to the Noether symmetries of given metric../files/site1/files/64/5.pdf

Introduction Border bases are a generalization of Gröbner bases for zero-dimensional ideals which have attracted the interest of many researchers recently. More precisely, border bases provide a new method to find a structurally stable monomial basis for the residue class ring of a polynomial ideal and this yields a special generating set for the ideal possessing many nice properties. Given a finite set of points, finding the set of all polynomials vanishing on it (so-called either ideal of points or vanishing ideal of the set of points) has numerous applications in several fields in Mathematics and other sciences. In 1982, Buchberger and Möller proposed an algorithm to compute a Gröbner basis for an ideal of points. This algorithm proceeds by performing Gaussian elimination on a generalized Vandermonde matrix. In 2006, Farr and Gao presented an incremental algorithm to compute a Gröbner basis for an ideal of points. The main goal of their paper is to calculate a Gröbner basis for the vanishing ideal of any finite set of points under any monomial ordering, and for points with nontrivial multiplicities they adapt their algorithm to compute the vanishing ideal via Taylor expansions. The method of border bases is a beneficial tool to obtain a set of polynomial models identified by experimental design and regression. The utilization of Gröbner bases theory in experimental design was introduced by Pistone and Wynn. However, using Gröbner bases we cannot find all possible models which form structure of an order ideal for an experiment. For example, if we consider the design {(-1,1),(1,1),(0,0),(1,0),(0,-1)}, the model {} cannot be computed by Gröbner bases method. This fact is expected this method relies on monomial orderings. Material and methods In this paper, we first present the Buchberger-Möller and Farr-Gao algorithms and then by applying these algorithms, we describe an algorithm which computes a border basis for the ideal of points corresponding to the input set of points with nontrivial multiplicity. In addition, we focus on presenting different models related to an experiment by using the concept of monomial bases for the residue class ring of a polynomial ideal. Results and discussion As we mentioned earlier, Buchberger-Möller algorithm is an efficient algorithm to compute a Gröbner basis for an ideal of points. We describe a simpler presentation of this algorithm in which we use the function NormalForm which receives as input a linear polynomial p and a Grbner basis G = { , . . . , } of linear polynomials in , . . . , and returns f and q=[, . . . , ] where f is the remainder of the division of p by G and p=+· · ·++r. Furthermore, we compare the efficiency of this algorithm with the function VanishingIdeal of Maple. Given a finite set of points, we consider the case in which some points in the set have nontrivial multiplicity. Based on the Farr-Gao algorithm, we prepare an algorithm that computes a border basis for the vanishing ideal of the finite set of points by using Taylor expansions. Suppose that n is the number of factors in an experiment. An experimental design is a finite set of points. The set of all polynomials vanishing at the design is called a design ideal. Regression analysis is a useful statistical process for the investigation of relationships between a response (or dependent) variable and one or more predictor (or independent) variables. When there is more than one predictor variable in a regression model, the model is a multiple linear regression model which we can call polynomial model. Suppose a random sample of size n is given (then we have exactly n data points are observed from (Y,X). The expession error? is the model for multiple linear regression where 's are called slopes or regression coefficients. Also, representing the merged effects of the predictor variables on the response variable is called interaction effect. By using multiple linear regression, we can analyze models containing interaction effects. For example, let us consider the following model +error. By substituting and , we have a multiple linear regression as follows +error. In addition, multiple R-squared or R2 is a statistical measure that states the square of the relationship between the predicted response value and response value. It should be noted that multiple R-squared is always any value between 0 and 1, where a value closer to 1 informing that a greater proportion of variance is computed for the model. Statistically, a high multiple R-squared shows a well-fitting regression model. Also in multiple regression, tolerance is used as an indicator of multicollinearity. Tolerance may be said to be the opposite of the coefficient of determination and is obtained as . All other things equal, researchers desire higher levels of tolerance, as low levels of tolerance are known to affect adversely the results associated with a multiple regression analysis. The smaller the tolerance of a variable, the more redundant is its contribution to the regression (i.e., it is redundant with the contribution of other independent variables). In the regression equation, if the tolerance of any of the variables is equal to zero (or very close to zero), the regression equation cannot be evaluated (the matrix is said to be ill-conditioned in this case, and it cannot be inverted). Conclusion The following conclusions were drawn from this research. We present a simpler variant of Buchberger- Möller algorithm (which seems to be easier for the implementation issue) for computing a border basis for an ideal of points. We present an algorithm that incrementally computes a border basis for the vanishing ideal of any finite set of points in which some points have multiplicity. We provide good statistical polynomial models which are more suitable for practical applications due to the stability of border bases models compared with Gröbner bases models../files/site1/files/64/6.pdf

Introduction Fractional Differential Calculus (FDC) began in the 17th century and its initial discussions were related to the works of Leibniz, Lagrange, Abel and others. In recent decades, the fractional differential equations have been considered in different fields such as fluid flow, electromagnetics, engineering, economics and finance. In the early 1970s, Black and Scholes introduced their famous model for pricing option. This model is one of the most popular models in the financial market and plays an important role in determining the price of a high-risk asset in financial modeling field. In this equation, researchers seek to obtain the option value by numerical or analytical methods or to extract new pricing models that reflect the real financial market. The Black-Scholes equation is based on some assumptions which caused constraints in the market. Some advanced models such as jump-diffusion, stochastic interest rate, and stochastic volatility models have been proposed to remove these constraints. Wang and Meng (2010) show that the distribution of stock returns have long-range dependence property which is not consistent with the classical Black -Scholes equation assumptions. This led us to use the ​​fractional modeling in this paper which is derived by applying the fractional specifications of stock market suggesting by Mandelbrot (1963). So, in this paper, we reach to the fractional Black-Scholes model by replacing fractional Brownian motion instead of standard Brownian motion in the classical Black-Scholes equation. The fractional Black-Scholes equation gives better solutions than classical Black-Scholes model to our data which their distribution of stock price depend on long-range. So, in this paper, we solve the fractional Black- Scholes equation and use the combination of radial basis functions and finite difference methods to solve the fractional Black-Scholes equation. This method is flexible because it does not depend on the position of points, and in comparison with other methods, it has a short run time in high dimensions. Material and methods In this paper, we reach to the fractional Black-Scholes equation by using the fractional underlying asset that follows the fractional Brownian motion. The model represents the price behavior of an European option. It is based on the stochastic behavior of underlying asset which is priced by fractional models. We also apply the radial basis function method to solve this model. In this method, we do not necessarily need to have points with equal distances and the convergence rate can be exponential. Therefore, this method can provide more acceptable solutions than the other numerical methods. It should be noted that the fractional derivative of the pricing function is approximated by the Caputo fractional derivative. The stability of the proposed method has also been studied. Results and discussion In this paper, we provide the numerical results for the fractional Black-Scholes equation on real data of the coin option by radial basis function method. We receive the data of working days from 95/10/01 to 96/01/06 from Tehran Stock Exchange website by Excel software and uploaded them to MATLAB software format. The parameters required by the model are also estimated by using data. We obtain option price for different α by using these parameters and present the results in figures. Figures show that if the price of the underlying asset is lower than the exercise price, the option price decreases by the increasing of the fractional order (α), which means that if the holder chooses not to exercise the contract, he will incur less loss. Increasing α plays an important role in decreasing option price, if the option price decreases, the increasing α will make the purchaser incur less loss. On the other hand, when α increases, if the underlying asset price is above the strike price, the call option price increases, and the holder will make a gain by exercising it. We also predict the price of the coin for the next 5 days by different α. The results are presented in a table and are compared with real price of the market, which show that the method is efficient and the fractional Black-Scholes equation has better performance than the classical Black-Scholes equation. Conclusion The purpose of this paper is to model the fractional Black -Scholes equation and to solve it by radial basis function method. First we modeled the fractional Black-Scholes equation by using fractional underlying asset and then we solved this equation by radial basis function method. Finally, the efficiency of this method is shown by using the real data of the coin option. The numerical results present that the method is efficient, and the fractional Black-Scholes equation performs better than the classical Black- Scholes equation .The existence of α in the fractional Black -Scholes equation has drawn the prices closer to the real price../files/site1/files/64/11.pdf

Introduction Fractional calculus has been used to model physical and chemical processes that are found to be best described by fractional differential equations. Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Fractional delay differential equations (FDDEs) are a class of fractional differential equations that the rate of change of unknown function depends not only on the values of unknown function for the same time value but also on previous time values. The solution of delay differential equations not only requires information of current state, but also requires some information about the previous state. FDDEs have received considerable recent attention and been proven to model many real life problems. For most of fractional order delay differential equations, exact solutions are not known. Therefore different numerical methods have been developed and applied for providing approximate solutions. The objective of this paper is to define the new fractional-order piecewise functions based on the Taylor polynomials for solving the FDDEs. This method is accurate and easy to implement in solving FDDEs. Material and methods In this paper, first we construct piecewise functions based on the fractional-order Taylor functions. Then, we calculate the fractional integral operational matrix for the fractional Taylor piecewise functions. This matrix and collocation method are utilized to reduce FDDEs to a system of algebraic equations which can be solved via a suitable numerical method. Results and discussion We apply mentioned paper for solving some test problems to highlight the significant features of our technique. Also, we compare our numerical results with multiquadric approximation scheme. The reported results demonstrate that there is a good agreement between approximate solution and exact solution. We plot the numerical solutions obtained by the presented method for various values of α with the exact solution. It is obvious from these Figures that, as α is close to integer value, numerical solutions converge to the exact solution. Conclusion The following conclusions were drawn from this paper. Fractional-order Taylor piecewise functions have three degrees of freedom (m, n, α) but Taylor polynomials have one degree of freedom (m). Instead of converting fractional-order Taylor piecewise functions into other functions, we have obtained the fractional-order Taylor piecewise functions operational matrix directly. The main characteristic of this method is that it reduces under study problem to a system of algebraic equations which can be easily solved by an iterative method../files/site1/files/64/9.pdf

Introduction The data obtained from observing a phenomenon over time is very common. One of the most popular models in time series and signal processing is the Autoregressive moving average model (ARMA). If the investigated time series has long memory, autoregressive fractional moving average model or in other words (ARFIMA), would be appropriate. The ARFIMA (p, d, q) model was first introduced by [1]. Classic methods for modeling, inference and estimating these processes lead to complex calculations of covariance structure and likelihood functions that make data processing difficult. Wavelet transform is one of the most powerful tools in analyzing such functions and best performs these functions from different time and location perspectives as well as high and low frequencies. Wavelet transform due to the decreasing correlation property is a very efficient method in analysis and inference for estimating long-term memory processes. The values ​​obtained from wavelet transforms for long-term memory processes, in spite of the complex covariance structure of these processes, the wavelet coefficients are almost uncorrelated and thus much easier to handle [2]. The dense covariance structure of such processes makes it difficult to accurately calculate the maximum likelihood function of data sets [3]. In these cases, the Bayesian method can be easily used to calculate wavelet coefficients. In this paper, while briefly introducing wavelet transform in section two, the ARFIMA model and covariance matrix structure of this model is investigated in section three. In section four, the Bayesian estimation of ARFIMA parameters based on wavelets are calculated. At the end section, ‎we survey the theoretical outcomes with numerical computation by using simulation to described purpose estimation. Material and methods In this scheme, first we explain wavelet transformation, ARFIMA model and covariance matrix structure of this model. By using wavelet decomposition, Bayesian estimation of ARFIMA model parameters are calculated. The performance of purpose estimation is assessed with simulated data for comparing with respect to another estimators. Results and discussion We discuss in detail wavelet transformation and autoregressive fractional moving average model with long memory. The structure of covariance matrices of wavelet coefficients and Bayesian wavelet estimation of parameters are investigated. At the end we used simulation study to examine our proposed estimation. Notice that, obtained results confirm that proposed estimation is better than another. Conclusion The following conclusions were drawn from this research. Wavelet transform due to the decreasing correlation property is very efficient method in estimating long-term memory processes. The main purpose of this paper is to provide a new wavelet estimation of ARFIMA model parameters via Bayesian method. The main characteristic of this method is that it can be easily used and therefore many calculations are reduced. The proposed method can be applied for estimating of parameters in the simulation. According to the figures, we conclude that Bayesian wavelet estimation of autoregressive process parameters is appropriate and better than with respect to other estimations. According to the table, by increasing the sample size, standard error of proposed estimator is decreased, so it was shown that the new proposed method is better with respect to others.

The main aim of this paper is to apply the Chebyshev polynomials for the solution of the linear Fredholm integro-differential-difference equation of high orders. It is usually difficult to analytically solve this equation. Our approach consists of reducing the problem to a set of linear equations by means of the matrix relations between the Chebyshev polynomials and their derivatives. The operational matrices of delay and derivative together with the Tau method are then utilized to evaluate the unknown coefficients of Chebyshev expansion of the solution. The convergence analysis is studied. Illustrative examples show the validity and applicability of the presented technique. Also, a comparison is made with existing results. Introduction The integro-difference equations arise in different applications such as biological, physical and engineering problems. In recent years, there has been a growing interest in the numerical treatment of the integro-differential-difference equations. Since the mentioned equations are usually difficult to solve analytically, numerical methods are required. Several numerical methods were used such as successive approximation method, Adomian decomposition method, the Taylor collocation method, Haar wavelet method, Legendre wavelets method, wavelet-Galerkin method, monotone iterative technique, Walsh series method, etc. In this work, we develop a framework to obtain the numerical solution of the s-order linear Fredholm integro-differential-difference equation with variable coefficients. under the mixed conditions whereand are known continuous functions. Here, the real coefficients and are given constants. Our approach consists of reducing the problem to a set of linear equations by expanding the solution in terms of Chebyshev polynomials. The operational matrices of delay and derivative are given. These matrices together with the Tau method utilized to evaluate the unknown coefficients of expansion. The Tau method has been originally proposed by Lanczos for ordinary differential equations and extended by Ortiz. The method consists of expanding the required approximate solution as the elements of a complete set of orthogonal polynomials. Recently there have been several published works in the literature on the applications of the Tau method. Conclusion This paper deals with the solution of linear Fredholm integro-differential-difference equations of high order with variable coefficients. Our approach was based on the Chebyshev Tau method which reduces a linear Fredholm integro-differential-difference equation into a set of linear algebraic equations. Numerical results show that this approach can solve the problem effectively. The approach, with some modifications, can be employed to solve differential-difference equations and Fredholm integro-differential equations../files/site1/files/64/4.pdf

Introduction Many problems in sciences and engineering can be studied by mathematical models. These models are classified as direct problems and inverse problems. In the structural vibrations, analysis and estimation of the behavior of system e.g. response of the system for an external force and natural frequencies from the known physical parameters is called a direct problem. Determination or estimation of the system physical parameters such as density, mass, stiffness and cross sectional area from the behavior of the system is called an inverse problem. A class of inverse problems which physical parameters determined from the spectral data (eigenvalues, eigenvectors, or both) is called inverse eigenvalue problem. There are many systems such as mass- spring system, vibrating Rods and Beams which are modeled as an eigenvalue problem. Free vibrations of a mass- spring system and discretization of a rod and Sturm-Liouville equations lead to Jacobi matrix eigenvalue problem. Inverse eigenvalue problem for Jacobi matrix is determination of entries using some spectral data. Different algorithms have been presented for constructing a Jacobi matrix. In this paper, we construct a Jacobi matrix and the corresponding mass-spring system using some new spectral data. Material and methods We try to construct a Jacobi matrix from two spectra and one extra data. For this purpose, using given spectral data, we find the required data of well-known Lancsoz method. Then applying Lancsoz method, we construct a positive definite Jacobi matrix. Finally, according to the relations between Jacobi matrix, mass and stiffness matrices, we obtain corresponding mass-spring system. Results and discussion Necessary and sufficient conditions on given spectral data for solvability of the inverse eigenvalue problem are presented. We find two algorithms for constructing positive definite Jacobi matrix and the corresponding mass-spring system. We solve some examples using the given algorithms. There is a good agreement between the spectral data of constructed matrix and initial given data. Conclusion The following results are obtained from this research. We find two algorithms for constructing a Jacobi matrix using two spectra and one extra data. It is observed that, for a set of spectral data, there might be exist more than one solution. It seems that, one may extend the method of this paper for matrix eigenvalue problem which arise in discretization of vibrating rod using finite element method../files/site1/files/64/16.pdf

Introduction An orthonormal wavelet is a square-integrable function whose translates and dilates form an orthonormal basis for the Hilbert space . That is, given the unitary operators of translation for and dilation , we call an orthonormal wavelet if the set is an orthonormal basis for . This definition was later generalized to higher dimensions and to allow for other dilation and translation sets; let Hilbert space and an n × n expansive matrix A (i.e. a matrix with eigenvalues bigger than 1) with integer entries, then dilation operator is given by and the translation operator is given by for . A finite set is called multiwavelet if the set , is an orthogonal basis for . The concept of a multiresolution analysis, abbreviated as MRA is Central to the theory of wavelets. There is much overlaps between wavelet analysis and Fourier analysis. Indeed, wavelets can be thought of as non-trigonometric Fourier series. Thus, Fourier analysis is used as a tool to investigate properties of wavelets. Another concept is wavelet set. The term wavelet set was coined by Dai and Larson in the late 90s to describe a set W such that , the characteristic function of W, is the Fourier transform of an orthonormal wavelet on . At about the same time as the Dai and Larson paper, Fang and Wang first used the term MSF wavelet (minimally supported frequency wavelet) to describe wavelets whose Fourier transforms are supported on sets of the smallest possible measure. The importance of MSF wavelets as a source of examples and counterexamples has continued throughout wavelet history. A famous example due to Journe first showed that not all wavelets have an associated structure multiresolution analysis (MRA). The discovery of a non-MRA wavelet gave an important push to the development of more general structures such as frame multiresolution analyses (FMRAs) and generalized multiresolution analysis(GMRAs). In this paper we generalize wavelets and wavelet sets on locally compact Abelian group G with uniform lattice. Material and methods In this paper, we investigate wavelet sets on locally compact abelian groups with uniform lattice, where a uniform lattice H in LCA group G is a discrete subgroup of G such that the quotient group G/H is compact. So we review some basic facts from the theory of LCA groups and harmonic analysis. Then we define wavelet sets on these groups and characterize them by using of Fourier transform and multiresolution analysis. Results and discussion We extend theory of wavelet sets on locally compact abelian groups with uniform lattice. This is a generalization of wavelet sets on Euclidean space. We characterize wavelet sets by using of Fourier transform and multiresolution analysis. Also, we define generalized scaling sets and dimension functions on locally compact abelian groups and verify its relations with wavelet sets. Dimension functions for MSF wavelets are described by generalized scaling sets. In the setting of LCA groups, we define translation congruent and show wavelet sets are translation congruent, so we can define a map on G such that it is measurable, measure preserving and bijection. Conclusion The following conclusions were drawn from this research. Wavelet sets on locally compact groups by uniform lattice can be defined. This is a generalization of wavelet sets on Euclidean space. Characterization of wavelet sets on LCA groups can be done in different ways. A method is to use Fourier transform and translation congruent. Another way is to generaliz scaling set and dimension function. As an example, Cantor dyadic group is a non-trivial example that satisfies in the theory of wavelet sets on locally compact groups by uniform lattice. We find wavelet set and generalized scaling set for this group and show related wavelet is MRA wavelet../files/site1/files/63/7.pdf

Introduction This paper concerns an application of the modulus of continuity in characterizing geodesics. The modulus of continuity of a continuous function between metric spaces is a two variable function which assigns to each point and to each positive epsilon the greatest positive delta that satisfies the definition of continuity at that point. It is shown that if f is a function of the real numbers into a metric space, then linearity of its modulus of continuity implies that f is locally geodesic, that is, it locally preserves metric. Geodesics appear in several branches of mathematics as well as gravitational physics and general relativity. In mathematics, topics such as Riemannian geometry make extensive use of geodesics. In fact, geodesics represents the shortest paths. So for example, a mass that moves in the vacuum under the gravity passes through a geodesic. Material and Methods In Section 2 we first deal with some essential properties of the modulus of continuity. Then we mainly focus on continuous functions from the real numbers to metric spaces whose modulus of continuity is linear. In two theorems we characterize such functions and show that they are in fact locally geodesic. Results and Discussion The main result demonstrates that a continuous function with linear modulus of continuity falls into three classes: the class of functions that preserves metric with a coefficient, the class of functions that preserves metric on two rays whose intersection is a singleton, and the class of functions that preserves metric on two rays whose intersection is a closed interval of positive length. We then turn to locally convex metric spaces and show that the third class does not appear in such spaces. Finally, we apply our results to the case where both domain and range of function is the real line. Conclusion The following conclusions has been drawn in this paper: Essential properties of the modulus of continuity…….? Characterizing continuous functions of the real line into metric spaces whose modulus of continuity is linear. Dealing with continuous functions of the real line into locally convex metric spaces whose modulus of continuity is linear../files/site1/files/63/10.pdf

Introduction The kappa distribution was first introduced by Mielke (1973) and Mielke and Johnson (1973) for describing and analyzing precipitation data. This distribution is positively skewed and is widely applied when studying precipitation, wind speed and the stream flow data in hydrology. The kappa distribution has some advantages over gamma and log-normal distributions in fitting historical rainfall. Data It is because, unlike the latter two distributions, it has closed forms for the cumulative distribution function and quantile function. Due to this important feature, the kappa distribution attracts the attention of several researchers. Park et al. (2009) introduced the three-parameter kappa distribution and provided a description of the mathematical properties of the distribution and estimated the parameters by three methods. Also, they illustrated its applicability for rainfall data from Seoul, Korea. In this paper, we study the distribution and the estimation methods for the parameters considered by Park et al. (2009), and propose a new estimation method. Then, we will compare these estimation methods using a Monte Calro simulation study and a real dataset. Material and methods In this scheme, first we consider the three-parameter kappa distribution and study some of its properties and then estimate the parameters of the distribution by four methods. These methods are method of moment (MM), L-moments (LM), maximum likelihood (ML) and maximum product of spacing method (MPS). Using a Monte Carlo simulation study and a real data set, performance of these methods are compared. Results and discussion Comparing the performance of the proposed estimation methods in terms of bias and root of mean squares error (rmse), it can be concluded that the MPS method has a better performance due to its lower bias and rmse. The Kolmogorov-Smirnov test is applied for goodness-of-fit test in the three-parameter kappa distribution to the whole monthly rainfall data of Abali station in Tehran province. The results demonstrate that the MPSE method leads to better results than other mentioned methods. Conclusion The following conclusions were drawn from this research. The Monte Carlo simulation shows that the maximum product spacing method, which is proposed in this paper, is the best method for estimating the parameters of the three-parameter kappa distribution. The statistics and p-value of the Kolmogorov–Smirnov test show that the three-kappa distribution with the MPS method of estimation has better fit than the other methods.

Introduction One of the major challenges in supporting a growing human population is supplies of food. Plants play a major rule in providing human food. Hence, it is important to study plant diseases and provide appropriate models for describing the relationship between plant infection and its growth and reproduction. One of effective models that describes this relationship is mathematical model. One of the important aspects that the mathematical model can presented is the dynamic of the plant’s immune system. In this paper, a mathematical model for diffusion of infection in the host plant is introduced. The model is based on a differential equation system with two time delays. In this model, the host population of cells is divided into the classes of susceptible cells consisting of mature cells and are susceptible to infection, infected cells that spread the infection, recovered cells that are no longer infectious and are proliferating cells that become susceptible after reaching maturity. We consider two time delays, and , in equations. The proliferating cells have the average maturity time , after which they are recruited to the susceptible class. is the average time of antiviral effects. In the next sections of this paper, stability conditions of equilibrium points are investigated. In the last section, we consider a plant in two different modes, organic and non- organic. Then the solution curves are plotted with different time delays and compare solutions together. Material and methods In this scheme, first we explain the conditions of plant. Then, a mathematical model with two time delays is introduced. As follows, the dynamical behavior of the model is investigated. At the end of paper, we consider a plant with two different modes and plot the solution curves. Results and discussion We introduce a mathematical model which explain conditions of plant cells. In this model the independent variable is time, so the model is ODE with two time delays. As follows, using some theorems in dynamical systems, the dynamical behavior of the model is investigated. Using these results, we can provide good conditions for a plant that epidemic does not happen. At the end, we use of Matlab software to plot the solution curves in two different conditions. The curves explain the behavior of plant cells when they are infectious. Conclusion The following conclusions were drawn from this research. A mathematical model which is introduced in this paper is more realistic than the previous models because, the grow rate of a plant is considered to be logistic. Theorems show that how we can control the virus to prevent epidemic outbreak. We plot solution curves for two different plants (organic and non-organic). Solution curves show that how the conditions of plant cells change by changing the parameters.

Introduction Tries (from retrieval) are one of the most practical data structures with a tree construction in computer science. Tries store string data in leaves of tree. They are often used to store such data so that future retrieval can be made efficient. For example, tries are widely used in algorithms for automatically correcting words in texts. The number of nodes of the same type, which are at the same distance from the root of a rooted tree, is called profile. The analysis of the profile of a tree is of great importance. Because many of the parameters of a rooted tree can be expressed in terms of its profile. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. A generalized version of trie is called bucket trie where each leaf or bucket has a storage capacity of more than one string. Random trie is obtained by defining a random grow rule for trie. We present a detailed study of the limit behavior of the profiles in a random bucket tries. Material and methods In this paper, the methods we apply to derive recurrences satisfied by the expected profiles and to solve them asymptotically for all possible ranges of the distance from the root, are based on the use of Poissonization, Mellin transform, recurrence equations, generating functions, singularity analysis and saddle-point method. Results and discussion Here, as the number of stored strings in a random bucket trie increases, we investigate the asymptotic expectation, variance and limiting distribution of each of the two internal and bucket profiles (i.e. the number of bucket nodes or leaves, and the number of internal nodes or non-leaves which are at the same distance from the root) in random bucket tries. Both the expectation and variances of the two profiles contain periodic functions, and we show those periodic functions are not zero that this point has not been proven in the paper on the ordinary trie. Also, we examine the amount of asymptotic ratio of the expectations of the bucket and internal profiles. Conclusion In this research, we will generalize the most important part of the results for the ordinary tries and also provide a proof for an unproven point in those results. More precisely, the purpose of this article is to study the internal profile (the number of non-leaf nodes at distance from the root) and bucket profile (the number of buckets at distance from the root) in an important data structure called random bucket trie (random trie based on data and maximum capacity of data per each leaf). By some methods in complex analysis, we have shown that for every and some fixed constants and , if is in the range , then the expectations and variances of both profiles contain non-zero periodic functions (the non-zero property of these periodic functions has not been proved for the ordinary tries up to now). We also provide a graph of those periodic functions for by MAPLE. Then we study the ratio of the two profiles for Finally, by finding the asymptotic expansions of Poisson generating functions for the probability generating functions of profiles and then using the Cauchy integral formula, we obtain the asymptotic expansions for the probability generating functions, which indicate that the limiting distributions of profiles are normal. ./files/site1/files/62/3Abstract.pdf

Introduction General linear methods(GLM) was developed by Butcher in 1966 as an extension of the traditional Runge-Kutta and linear multistep methods [1]. The classification of GLMs is an important open and active research area. Many authors studied GLMs and developed different GLM classes for solving stiff and non-stiff problems. For example, the Diagonally-implicit multi-stage integration methods known as DIMSIM has four subclasses which is appropriate to stiff and non-stiff problems with capability of implementing in parallel or sequential forms [2]. In this paper we study the GLM representation of the predictor-corrector (PC) schemes in finite mode. The PC schemes are a class of practical schemes for solving initial value problems. The basic scheme (corrector) is an implicit scheme in which the implementation is accomplished by an iteration process with a rather good initial approximation evaluated by predictor method. The built in error estimation with local extrapolation is one of major advantages of PC schemes based on Adams schemes. In new formulation based on GLM we have enforced Milne estimation in the internal or external iterations. We present a closed form stability function for derived schemes. Numerical implementation and comparisons illustrate that in new formulation based on GLM framework we obtain more accurate solutions. This new formulation provide the opportunity of more Material and methods The methodology of this paper is based on the constructing matrix-vector formulation of the PC schemes. The matrix dimensions depend on the step numbers and the number of iterations of the PC finite mode. The derived GLM method is a scheme with r internal and s external stages. These stages are specified by the step numbers and a finite number of iterations. Results and discussion The general linear methods are a large class of schemes that include the traditional schemes such as Runge-Kutta methods. Among the PC schemes based on Adams linear multistep methods we can easily choice the methods with different step numbers and different orders. It is easy to estimate the local truncation errors. The stability function is available in the closed form. The local extrapolation method provides the application of Milne estimation in the internal and external stages of the PC iterations. In fact, we can control the error and improve the accuracy and locally increase the order of the method. Conclusion We have studied a reformulation of a class of predictor-corrector schemes in the new framework of general linear methods. The new schemes have the following properties: The Adams linear multistep methods are a large class of schemes including implicit and explicit schemes. We can choose Adams choice methods with different step numbers and orders. There is cheap error estimation for PC schemes known as Milne estimation and it is possible to use this error estimation in internal and external stages. The stability function of the resulting GLM is available for further developments and study of the stability properties. The new GLM formulation provides accurate results comparing with the original formulation in PC form../files/site1/files/62/13Abstract.pdf

Introduction In this paper, by mixed monotone operators and their fixed points, we investigate the existence and uniqueness of positive solution for the following boundary value problems via nonlinear fractional differential equations. and where is the Riemann–Liouville derivative. In recent years, fractional differential equations have been studied by many mathematicians both theoretically and practically, for example, in physics, mechanic, chemistry, engineering, biology, economy, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics and etc. First time, mixed monotone operators have been introduced by Guo and Lakshmikantham in 1987. Next, many authors studied them in Banach spaces and obtained some results not only in theory fields but also wide applications in chemistry, engineering, biology, technology and other fields. In 2009, Xu et. al studied the properties of Green's function for the nonlinear fractional differential equation boundary value problem (2). Here, the existence and uniqueness of its positive solutions are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems. Material and methods The content of this paper is organized as follows. First, we present some definitions, lemmas and basic results that will be used in the proofs of our theorems. Then, we consider the existence and uniqueness of positive solution for the operator equation, also we utilize the results obtained to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems. Results and discussion In this work, we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary-value problems. Our results guarantee the existence of a unique positive solution, and can be applied for constructing an iterative scheme for obtaining the solution. Conclusion The following conclusions were drawn from this research. Using the properties of cones and the fixed point theorem for mixed monotone operator, the existence and uniqueness of the positive solution are obtained. Our research methods are different from those in the related literature. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems. Our results extend and improve the related conclusions in the literature../files/site1/files/62/1Abstract.pdf

Introduction The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the average shadowing property. K. Sakai proved that the -interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. Asymptotic average shadowing (AASP) defined by R. Gu for continuous maps, combines to the limit shadowing property with the average shadowing property. Here we modify the notion (AASP) and define the limit average shadowing for diffeomorphisms (LASP). R. Gu presented some basic properties of the limit average shadowing for continuous maps. He proved that if a continuous map has the limit average shadowing on a compact metric space, then is chain transitive and that -hyperbolic homeomorphisms with limit average shadowing are topologically transitive. M. Kulczycki et.al., found some relations between LASP and the other notion of topological dynamics. They proved that a surjective map with specification property has the LASP. Also, they found the relation between LASP and shadowing property. They also have been shown that an expansive continuous map with shadowing property is LASP if and only if it is mixing. This paper follows the ideas of R. Gu and M. Kulczycki et.al. Here we define LASP for diffeomorphism with a slight modification of the continuous case. We give an example which shows that shadowing property and LASP are not equivalent. Also, we introduce the notion of - stably limit average shadowing for a closed -invariant subset of and show that if is -stably limit average shadowing and the minimal points of are dense there, then admits a dominated splitting. Statement of the results In this paper we give a system which has the limit average shadowing, but not the shadowing property. Also, one can give examples which have the shadowing but not the limit average shadowing property. Thus the limit average shadowing property does not imply the shadowing property. In fact, we can give a class of diffeomorphisms which have LASP, but not the shadowing property. In fact the following proposition gives a large class of diffeomorphisms satisfying the limit average shadowing. Proposition A. Let be a locally maximal -invariant set. If is the specific set for then is limit average shadowable. The main purpose of the paper is to characterize the closed -invariant set via limit average shadowing property in -open condition. So, we consider the notion of limit average shadowing property in geometric differential dynamical systems. First we show that if has the limit average shadowing property on a closed -invariant set then is chain transitive. By using chain transitivity and limit average shadowing property we can prove that is transitive. Proposition B. If is -stably limit average shadowable, then there is a neighborhood of and a neighborhood of such that contains neither almost sinks nor almost sources for any Since we have proved that if has the limit average shadowing property on a closed f-invariant set and minimal points of are dense then is transitive. It is essentially proved that under assumptions and conclusions of the Proposition B, admits a dominated splitting. Thus we get the main result of this paper. Theorem C. Let be a closed f-invariant set whose minimal points are dense there. If is -stably limit average shadowing then admits a dominated splitting../files/site1/files/62/2Abstract.pdf

Introduction Complex analysis is a comparatively active branch in mathematics which has grown significantly. A deep look at the implications of continuity, derivative and integral in complex analysis and their relation with partial differential equations determines the importance of establishing the relation between complex analysis and the theory of partial differential equations. There are three aspects of the aims of complex analysis: First it is possible to interpret the peculiarities of holomorphic functions as properties of solutions of specials systems of partial differential equations. Secondly, complex analysis becomes applicable to general classes of differential equations, not only to special ones. And thirdly, the new general complex analysis is able to construct solutions and to describe the properties of given solutions with the help of solutions of corresponding problems for holomorphic functions. The third aspect is significant, since in the case of nonlinear equations it essentially means their reduction to linear problems. In addition this third aspect shows that the general complex analysis is able to make use of results of classical function theory and of such results which originally have not been connected with partial differential equations. One of the main goals of complex analysis is its systematic application in the branch of the theory of differential equations. By using the contraction function, the Gauss-Oatrogradski integral formula and the Banach fixed point theorem of the complex space, the classical and interesting method of existence and uniqueness of the solution of differential equations with nonlinear partial derivatives are smooth, which shows the importance of complex analysis in the branch of the theory of differential equations. In recent years, complex analysis and their relationship with partial differential equations have attracted a number of researchers. In this paper, we determine the existence and uniqueness of a solution of a nonlinear partial differential equation in the form where Material and methods Firstly, we transform the nonlinear partial differential equation in a complex space to a linear differential equation of the first order and show the existence of the solution. Then, we consider the equivalence of the solution of the nonlinear partial differential equation with a singular integral equation system by using weak and strong singular integral operators and the property of the holomorphic function. Also, we consider the solution of the nonlinear partial differential equation on a bounded domain with finite area in the Holder space and Sobolev space. Results and discussion We obtain the uniqueness of the solution of the nonlinear partial differential equation by applying the Lipshitz condition in the Holder space and Sobolev space based on the contraction function and the Banach fixed point theorem. Conclusion In this paper, we discuss on the existence and uniqueness of the solution of the nonlinear partial differential equation in the continuous functions of Holder and Sobolev spaces in general case in complex space. We deduce the proposed method can be extended in other spaces../files/site1/files/62/4Abstract(1).pdf

Paper pages (271-276) Introduction ‎Throughout this paper‎, will denote a commutative ring with‎ ‎identity and will denote the ring of integers. Let be an -module‎. A submodule of is said to be pure if for every ideal of . has the copure sum property if the sum of any two copure submodules is again copure‎. is said to be a comultiplication module if for every submodule of there exists an ideal of such that . satisfies the double annihilator conditions if for each ideal of , we have . is said to be a strong comultiplication module if is a comultiplication R-module which satisfies the double annihilator conditions. A submodule of is called fully invariant if for every endomorphism ,. In [5]‎, ‎H‎. ‎Ansari-Toroghy and F‎. ‎Farshadifar introduced the dual notion of pure submodules (that is copure submodules) and investigated the first properties of this class of modules‎. ‎A submodule of is said to be copure if for every ideal of . Material and methods We say that an -modulehas the copure intersection property if the intersection of any two copure submodules is again copure‎. In this paper, we investigate the modules with the copure intersection property and obtain some related results. Conclusion The following conclusions were drawn from this research. Every distributive -module has the copure intersection property. Every strong comultiplication -module has the copure intersection property. An -module has the copure intersection property if and only if for each ideal of and copure submodules of we have If is a , then an -module has the copure intersection property if and only if has the copure sum property. Let , where is a submodule of . If has the copure intersection property, then each has the has the copure intersection property. The converse is true if each copure submodule of is fully invariant../files/site1/files/62/12Abstract.pdf

Introduction ‎One of the most important classes of mappings is the class of‎ ‎monotone mappings due to its various applications‎. ‎For solving many‎ ‎important problems‎, ‎it is required to solve monotone inclusion‎ ‎problems‎, ‎for instance‎, ‎evolution equations‎, ‎convex optimization‎ ‎problems‎, complementarity problems and variational inequalities‎ ‎problems. The first algorithm for approximating the zero points of the‎ ‎monotone operator introduced by Martinet. ‎In the past decades‎, ‎many authors prepared various algorithms and investigated the existence and convergence of zero points for maximal monotone mappings in Hilbert spaces‎. ‎A generalization of finding zero points of nonlinear operator is to find zero points of the sum of an‎ ‎-inverse strongly monotone operator and a maximal monotone operator‎. ‎Passty introduced‎ ‎an iterative methods so called forward-backward method for finding zero points of the sum of two operators‎. ‎There are various applications of the problem of finding zero points of the sum of two operators. Recently‎, ‎some authors introduced and studied some algorithms for‎ ‎finding zero points of the sum of a -inverse strongly‎ ‎monotone operator and a maximal monotone operator under different‎ ‎conditions. In this paper‎, ‎motivated and inspired in above‎, ‎a shrinking projection algorithm is introduced for finding zero points of the sum of an inverse strongly monotone operator and a maximal monotone operator‎. ‎We prove the strong convergence theorem‎ ‎under mild restrictions imposed on the control sequences‎. Material and methods In this scheme, first we gather some ‎definitions and lemmas of geometry of Banach spaces and monotone‎ ‎operators‎, ‎which will be needed in the remaining sections‎. ‎In‎ the next section‎, ‎a shrinking projection algorithm is proposed and a‎ ‎strong convergence theorem is established for finding a zero point‎ ‎of the sum of an inverse strongly monotone operator and a maximal‎ ‎monotone operator‎. Results and discussion ‎The generated sequence by the presented algorithm converges strongly to a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces. ‎ Conclusion In this paper‎, ‎we present an iterative algorithm ‎for approximating a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces. ‎Under some mild conditions‎, ‎we show the convergence theorem of the mentioned algorithm‎. ‎Subsequently‎, ‎some corollaries and applications of those main result is provided‎. ‎This observation may lead to the future works that are to analyze and discuss the rate of convergence of these suggested algorithms‎. We obtain some applications of main theorem for solving variational inequality problems and finding fixed points of strict pseudocontractions‎../files/site1/files/62/7Abstract.pdf

Introduction Hepatitis C virus (HCV) was first identified in the year 1989. Globally, hepatitis has infected an estimated 200 million people, most of whom are chronically infected. Mathematical modeling of the spread of infectious diseases continues to provide important insights into diseases behavior and control. Over the years, it has also become an important tool in understanding the dynamics of diseases and in decision making processes regarding intervention programs for controlling these diseases in many countries. The impact of a chronic stage on the disease transmission and behavior in an exponentially growing or decaying population, is the focus of this paper. In this article a new approach for controlling hepatitis C disease for a homogeneous population based on optimal control theory and next generation matrix is proposed. In the first, we consider the Hepatitis C model using predefined parameters and variables by Yuan and Young paper [20]. The model is a SEIR epidemiological model, that the factors are Corresponds to special blocks of possible states for individuals and next calculate reproduction number ([21], [22]). In fact, we divide the population in researched area into four classes: S—susceptible, E—exposed, I—infected with acute hepatitis C, V—infected with chronic hepatitis C. The total number in time t is . In this paper, it is assumed that, after the initial infection, a host stays in a latent period before becoming infectious. We obtain the basic reproduction number of this model, which completely decides the dynamics of this model. If the disease cannot be prevalent and If the disease can be prevalent. In the following by the optimal control theory, we check the infectious states to control the outbreak, and then solve the control problem. So we get the numerical solution and we will check witch of them (variables) is the best control and compare the results. Material and methods In this scheme, first we Consider the disease model and then check it. This model is a four-dimensional SEIR model. There are a few chronic diseases that have been analyzed in a chronic fashion. We consider the population in four Blocks. (infected people (i), exposed population(e) and Chronic infected population) The costs associated with each of these strategies are also investigated by formulating the costs function problem as an optimal control problem, and we then use the Pontryagin's Maximum Principle to solve the optimal control problems. Then we create the optimal control problem, next with using Pontryagin's maximum principle we will solve the problem. Results and discussion After the modelling the hepatitis C Disease, a nonlinear ordinary differential equations system is obtained, and with simplifying the system by applying the appropriate variable change, we can optimize some of the variables that can be controlled (for reducing the prevalence) until converted to zero. Also, the numerical results reported in the tables that controlling on infected people(i), exposed population(e) and Chronic infected population. The reported results demonstrate that the disease can be have the maximum reduction by controlling on the Chronic infected population (v). These populations can be controlled by medication, using different therapies, Quarantine them to reduce the spread of the disease or with Other medical procedures. Conclusion The following conclusions were drawn from this research. The epidemic is transmitted through people's direct contacts, so It is important to reduce the incidence of the disease and we can use the Optimal control theory for some of the variables that can be controlled by reducing the prevalence until converted to zero. The results show that control over the variable sensitive(s) is weak and studying it does not lead to a desirable result. The reported results demonstrate that the disease can be the maximum reduction by controlling on the Chronic infected population (v)../files/site1/files/62/15Abstract.pdf

Introduction In 1976, A. Lambert characterized subnormal weighted shifts. Then he studied hyponormal weighted composition operators on in 1986 and in 1988 subnormal composition operators studied again by him. Recently, A. Lambert, et al., have published an interesting paper: Separation partial normality classes with composition operators (2005). In 1978, R. Whitley showed that a composition operator is normal if and only if essentially. Normal and quasinormal weighted composition operators were worked by J.T. Campbell, et al. in 1991. In 1993, J.T. Campbell, et al. worked also seminormal composition operators. Burnap C. and Jung I.B. studied composition operators with weak hyponormality in 2008. Material and methods Let be a complete -finite measure space and be a complete -finite measure space where is a subalgebra of . For any non-negative -measurable functions as well as for any , by the Radon-Nikodym theorem, there exists a unique -measurable function such that for all As an operator, is a contractive orthogonal projection which is called the conditional expectation operator with respect For a non-singular transformation again by the Radon-Nikodym theorem, there exists a non-negative unique function such that The function is called Radon-Nikodym derivative of with respect . These are two most useful tools which play important roles in this review. For a non-negative finite-valued - measurable function and a non-singular transformation the weighted composition operator on induced by and is given by , where is called the composition operator on . is bounded on for if and only if Results and discussion In this paper, we review some known classes of composition operators, weighted composition operators, their adjoints and Aluthge transformations on such as normal, subnormal, normaloid, hyponormal, -hyponormal, -quasihyponormal, -paranormal, and weakly hyponormal, Furthermore, miscellaneous examples are given to illustrate that weighted composition operators lie between these classes. We discuss from the point of view of measure theory and all results depend strongly to the Radon-Nikodym derivative and the conditional expectation operator with their various types. Hence we study their fundamental properties in sections 1 and 2. Then, we review some results by A. Lambert, D.J. Harringston, R. Whitley, J.T. Campbell and W.E. Hornor. Conclusion According to the given miscellaneous examples in the final section, we can conclude that composition and weighted composition operators lie between these classes../files/site1/files/62/10Abstract.pdf

Introduction One of ‎the ‎intersecting aspects of the reliability theory and economics is investigating non-negative data generally skewed‎ ‎(lifetimes and income)‎, ‎extrapolation of a suitable model for the data and finding their characteristics‎. The concepts of inequality measures play an important role in economic, social sciences and the other areas. In recent years, various inequality curves have been developed or investigated as the descriptors of income inequality. The Lorenz curve is a strong tool extensively used in economics to consider the inequality of income distributions and wealth. It is the graphical display of wealth distribution created by the American economist, Max Lorenz, in 1905. It plots the percentage of the total income obtained by various portions of the population when the population is ordered by the size of its income. Several alternative inequality curves have been proposed in the literature. The alternative inequality curves are considered as competitors of the classical Lorenz curve as descriptors of income inequality. The Bonferroni curve and the Zenga-2007 curve appear to be essentially the function of the Lorenz curve. Reliability is a broad concept. It is applied whenever we expect something to behave in a certain way. It is one of the metrics that are used to measure quality. The notion of reliability, in the statistical sense, is the probability that an equipment or unit will perform the required function under the conditions specified for its operations for a given period of time. The primary concern in reliability theory is to understand the patterns in which failures occur, for different mechanisms and under varying operating environments, as a function of its age. This is accomplished by identifying the probability distribution of the lifetime represented by a non-negative random variable. Accordingly, several concepts have been developed that help in evaluating the effect of age, based on the distribution function of the lifetime random variable and the residual life X. Concepts of aging describe how a component or a system improves or deteriorates with age; and they are very serious in the reliability analysis. In reliability, several aging classes of life distributions have been presented to explain the various forms of aging. Different order relations have been developed using measures in connection with many fields such as reliability, economics, queuing theory, survival analysis, insurance, operations research, etc. Material and methods In this paper, we provide a brief review of the widely used income inequality measures, their inter relationships and their properties and we discuss on basic reliability concepts such as hazard rate, mean residual life, reversed hazard rate and reversed mean residual life in distribution function frame work. The main aim of this paper is finding the relationship between the indices of economic inequality and reliability theory‎. For example, Zenga curve shall be interpreted as the difference in average age of components which has survived beyond age X from those which has failed before attaining age X, expressed in terms of average age of components exceeding age X. Therefore, we present some of the important results about the relationship between aging concepts and several important theorems have been proved in this subject that actually are novelty of the paper. We also present some characterizations of the reliability concept by using the Lorenz curve. ‎‎In this article‎, ‎first‎, ‎the relationship between inequality indices and reliability indices which have a very close relationship are discussed‎.‎ In fact, proofing of several propositions to express these connections is the main novelty of this article. In addition,‎ some of the ageing concepts can be expressed through inequality indices‎. ‎Finally‎, ‎to gain better understanding of the basic idea the Iranian income and expenditure data between the years 1388 to 1393 are numerically studied. Results and discussion In the present work, we have examined the connection between the other existing inequality measures, the relationship of the concepts of inequality indices with certain reliability concepts are exploited to obtain characterization results for probability distributions. Further some results on a stochastic order using inequality curves are also established. Finally some numerical results were given in order to indicate the usefulness. There are many reasons to study the relationship between the reliability concepts and the inequality measures. For example, the study of the relationship between inequality indices and measurement standards of reliability makes it possible to use each of these two concepts for studying the other one‎. We are able to set some other criteria for exponentiation based on the Lorenz curve and Gini index. Also, we can obtain some new properties for the Lorenz curve, the Gini index, and the other inequality indices. Moreover, we are able to indicate that the Lorenz curve can be expended to create a variant explanation of lifetime data and vice versa and, as well, to determine the bound of the class of lifetime distributions in terms of its Lorenz curve and the other index inequalities. Conclusion The following conclusions were drawn from this research. The bound of the class of lifetime distributions are determined in terms of its Lorenz curve and the other index inequalities. Some interesting relationships that exist between commonly used notions in reliability theory and economic theory and reliability. Some new properties for income inequality are obtained‎./files/site1/files/61/3.pdf

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1) y]=[[x ,_( n) y] , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y) ,_( n(x,y)) y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group and G' is a p-group, 2) if r=0 or r is not a power of a prime, then G is abelian. In this paper, also, we define some generalized Engel conditions on groups, then we present a result about unit group of group algebras which satisfies this kind of generalized Engel conditions. ./files/site1/files/61/6.pdf

Introduction The graph is a mathematical model for a discrete set whose members are interlinked in some way. The members of this collection can be the different parts of the earth and the connections between them are bridges that tie them together (like the Konigsberg problem). Graph theory is one of the important issues in discrete mathematics, which studies graphs and modeling issues by them. In 1736, Leonard Euler established the graph theory for solving the Konigsberg Bridge problem. But James Joseph Sylvester was the first to use the word "graph" in 1878 to name these mathematical models. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. The convention of using colors originates from coloring the countries of a map, where each face is literally colored. Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still a very active field of research. This paper is concerned with a specific coloring, say the distinguishing coloring which is originated from a classic elementary problem, Frank Rubin's key problem, which Stan Wagon circulated in the Macalester College problem column: Professor X, who is blind, keeps keys on a circular key ring. Suppose there are a variety of handle shapes available that can be distinguished by touch. Assume that all keys are symmetrical so that a rotation of the key ring about an axis in its plane is undetectable from an examination of a single key. How many shapes does Professor X need to use in order to keep n keys on the ring and still be able to select the proper key by feel? The surprise is that if six or more keys are on the ring, there need only be 2 different handle shapes; but if there are three, four, or five keys on the ring, there must be 3 different handle shapes to distinguish them. The answer to the key problem depends on the shape of the key ring. A labeling of a graph G, ɸ: V(G) → {1,2, …, r}, is said to be r-distinguishing if no automorphism of G preserves all of the vertex labels. The point of the labels on the vertices is to destroy the symmetries of the graph, that is, to make the automorphism group of the labeled graph trivial. Consequently, we define the distinguishing number of a graph G by D(G)=min {r| G has a labeling that is r-distinguishing}. Similarly, the distinguishing index D′(G) of a graph G is the least integer d such that G has an edge labeling with d labels that is preserved only by a trivial‎ automorphism‎. Let G be a connected graph of order n ≥ 3 and let c: E(G) → {1, 2, …, k} be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v with respect to c is the k-tuple c(v) = (a1, a2, …, ak), where ai is the number of edges incident with v that are colored i (1 ≤ i ≤ k). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. Material and methods We use cycle rank parameter and first consider the case and prove that and then using spanning trees we obtain an upper bound for distinguishing number. Results and discussion In this paper, we consider the relationship between the distinguishing number and index with the detection number of a graph. In particular, we show that the distinguishing index of a connected graph is at most equal with the detection number, i.e., D'(G) ≤ det(G). Conclusion The following conclusions were drawn from this research. An upper bound for D(G) by the detection number of its spanning trees. The upper and lower bounds for the distinguishing number by the detection number of a graph. Every detectable coloring is a distinguishing labeling of the edges of a graph. The upper bounds for the distinguishing index by the detection number of a graph../files/site1/files/61/11.pdf

Introduction Optimal control problems (OCPs) appear in a wide class of applications. In the classical control problems, the state-space equations are expressed as differential equations. Many physical systems, technology, biology, viscoelastic, electrochemical, economic, and generally the systems that have a memory effect cannot properly be described as ordinary differential equations. Hence, the equation of these systems expresses as integral equations, integro-differential equations, fractional differential equations and fractional integro-differential equations. Almost every system of controlled ordinary differential equations or controlled integro-differential equations can be modeled by a class of systems of controlled Volterra integral equations. There are many methods for solving optimal control problems with the state space of the system in the form of ordinary, fractional, and integral equations; can be mentioned the Euler-Lagrange method, the method of using Pontryagin’s maximum principle, the numerical methods based on finite difference, finite element methods, conjugate gradient method, spectral methods, the methods of continuous orthogonal functions, the operational matrices of integrals and embedding method. The method which we used in this paper is based on using the operational matrix of cubic B-spline scaling functions and wavelets with collocation method to reduce the optimal control problem governed by the nonlinear integral equation and integro-differential equation system with quadratic performance index to a nonlinear programming. The semi-orthogonal B-spline scaling functions and wavelets and their dual functions used in this paper have compact support, vanishing moments. These properties make many of the operational matrix elements be very small compared with the largest ones. These scaling functions and wavelets can be represented in a closed form so working with them is easy. The convergence of control and state functions and the performance index of the optimal approximation of the proposed method and also the upper bound of the error are given. Material and methods In this paper, a numerical method based on cubic B-spline scaling functions and wavelets for solving optimal control problems with the dynamical system of the integral equation or the differential-integral equation is discussed. The Operational matrices of derivative and integration of the product of two cubic B-spline wavelet vectors, collocation method and Gauss-Legendre integration rule for the discretization of the continuous optimal control problem and its transformation into a problem of non-linear programming is used. Results and discussion We solve two examples of optimal control with the dynamical system of integral equation and two examples with the dynamical system of the integro-differential equation by using present method to demonstrate validity, applicability and the simplicity of the new technique, then compare the present method with hybrid pseudo-spectral and Legendre wavelets method. These results illustrate that the accuracy of our numerical solutions are a few better than the numerical solutions obtained in the other method and there is a good agreement between the approximate solution and exact solution. Also, the numerical results reported in the tables and convergence analysis demonstrate that the accuracy improve by increasing the. Therefore, to get more accurate results, using the larger is recommended. Conclusion The following conclusions were drawn from this research. The operation matrix can be simply obtained for any basis of the approximation space and it is always available, therefore it can be applied to obtain the numerical solution of various kind of optimal control problems. Numerical results and convergence analysis indicates that the approximation solution fairly matches with the exact solution and the upper bound of error exponentially decreases by growing of approximation space. Due to the characteristics of the B-spline wavelet and dual of them, a nonlinear objective function can be obtained without calculating the integral. The semi-orthogonal B-spline scaling functions and wavelets used in the present paper have the properties of compact support, vanishing moments, smoothness function and the representation by a closed-form expression. With these assumptions, time is reduced, computer memory is less occupied and the operation matrix is always available../files/site1/files/61/12.pdf

Introduction Despite wide applications of constant order fractional derivatives, some systems require the use of derivatives whose order changes with respect to other parameters. Samko and Ross produced an extension of the classical fractional calculus with a continuously varying order for differential and integral operators. Variable-order fractional (V-OF) calculus has applications in optimal control, processing of geographical data, diffusion processes, description of anomalous diffusion, heat-transfer problems, etc. Due to the V-OF operators which are non-local with singular kernels, finding the exact solutions of V-OF problems is difficult. Therefore, efficient numerical techniques are necessary to be developed. The numerical solution of V-OF differential equation has been considered in some papers. Recently, discrete orthogonal polynomials have been considered as basis functions instead of continuous orthogonal polynomials. Discrete orthogonal polynomials are orthogonal with respect to a weighted discrete inner product. These polynomials have important applications in chemical engineering, theory of random matrices, queuing theory and image coding. In this paper, we focus on a special class of discrete polynomials, called Hahn polynomials. In this work, first, a new operational matrix is obtained for V-OF integral of Hahn polynomials. Then, we use a spectral collocation technique combined with the associated operational matrices of V-OF integral for solving weakly singular fractional integro-differential equations. Material and methods In this scheme, the operational matrix of fractional integration of Hahn polynomials is calculated. This method converts the weakly singular fractional integro-differential equations into an algebraic system which can be solved by a technique of linear algebra. Results and discussion In this paper, some numerical examples are provided to show the accuracy and efficiency of the presented method. By using a small number of Hahn polynomials, significant results are achieved which are compared to other methods. A comparison to the numerical solutions by CAS and Haar wavelets and Adomain decomposition method, shows that this technique is accurate enough to be known as a powerful device. Conclusion The following results are obtained from this research. The operational matrix of fractional integration of Hahn polynomials is presented for the first time. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy at interval [0,N], where N is the number of bases. Furthermore, for estimating the coefficients of the expansion of approximate solution, we only have to compute a summation which is calculated exactly. Using Hahn polynomials, the numerical results achieved only by a small number of bases, are accurate in a larger interval and significant results are achieved../files/site1/files/61/7.pdf

Paper pages (47-56) Introduction The Voronoi game is a simple geometric model for competitive facility location problem which is played by two players, White and Black, in a continuous space (one-dimensional or two-dimensional). In the one-round game, White places all his n points. After that, Black places the same number of points on the game space. The players Cannot change or reuse a point that was placed before. At the end of the game, the Voronoi diagram of all 2n points is constructed and a player who obtains the larger total area is the winner. Ahn et al. considered the Voronoi game on a unit circle and a line. They presented a winning strategy for the second player and showed that the first player can preserve the winning margin as small as possible. Cheong et al. presented a winning strategy for the second player when on a square. Fekete et al. considered the two-dimensional version of Voronoi game on a rectangular area of aspect ratio ρ. They showed that there is a winning strategy for the second player where with and for with and the first player wins in the other situations. Rashid et al. introduced a new version of Voronoi game called neighbor Voronoi and presented the winning strategies based on the neighborhood of points. Bandyapadhyay et al. studied Voronoi game on a weighted graph and showed that the problem of finding a winning strategy for the second player is NP-Complete. In the previous studies, the researchers considered the Voronoi game in which the facilities are the same. In the real world, the customers usually consider both the preferences and the distances for shopping which is modeled by the multiplicity weighted Voronoi diagram. Accordingly, we introduce a new version of the Voronoi game for modeling this problem, called multiplicity weighted Voronoi diagram. Material and methods In this paper, the one-round weighted Voronoi game is studied in both one-dimensional and two-dimensional cases. In the multiplicity weighted Voronoi game, the customer can consider not only the distance of the facility but also its. Therefore, in the one-dimensional, two different models of the facility are considered and it is showed that the second player has a winning strategy in the one-round Voronoi game. In the weighted Voronoi diagram, the points have the assigned weights according to their performance. Firstly, we consider the topology of the cells of weighted Voronoi diagram whose edges are the parts of Apollonius circles and it is too complicated to calculate the areas of the weighted Voronoi cells. So Black just tries to earn a little more than White without calculating the exact amount of the winning area. Also, we assume that the sum of all weights of black points equals with the sum of the weights of all whites points, i.e., . Results and discussion In the one-dimensional model in which the area is a line segment, we study both the same-weight points and the different-weight points separately. We show that Black always has a winning strategy when the points have the same weight. In the other case, Black has a winning strategy when the variance of the distribution of the black points is less than or equal to the variance of the distribution of the white points. In the two-dimensional model, we study the same-weight points and show that Black has a winning strategy. Conclusion In this study, we introduced the weighted Voronoi game to consider the performance of the facilities as well as the distance from customers. We showed that the second player has a winning strategy. In this model, customers can choose their service based on performances of the facilities such as parking, price, variety, quality of products, etc. Therefore this model is closer to the real world. In the one-dimensional model, we showed that the second player can always win the game when the points have the same weight. Also, the second player can win the game with the different-weight points when the variance of the distribution of the black points is less than or equals the variance of the distribution of the white points. In the two-dimensional model, we showed that the black player can always win the one-round game../files/site1/files/61/5.pdf

Introduction In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. ‎An abstract cone is analogous to a real vector space‎, ‎except that we take as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset of a real Banach space X, is norm dense in and the set of support points of is dense in the boundary of ". Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf