نشریه پژوهشهای ریاضی
سال هشتم شماره 2 (پیاپی 21، تابستان 1401)

Suppose E is a Banach lattice. A net (xα) in E is said to be unbounded absolute weak convergent (uaw-convergent, for short) to x∈E provided that the net (xα-x˄u) convergences to zero, weakly, whenever u∈E+. In this note, we further investigate unbounded absolute weak convergence in E. We show that this convergence is stable under passing to and   from ideals and sublattices. Compatible with un-convergenc, we show that uaw-convergence is topological, which means that E with uaw-topology forms a topological vector space. We consider some closedness properties for this type of convergence. Some examples  are given to make the context more understandable. Finally, we introduce the notion of strongly continuous operators between Banach lattices and investigate some properties about them. Specially, we characterize Banach lattices with a strong unit in tems of this type of operators.

In this paper, we combine the order structure and the norm structure in a Banach lattice to consider the unbounded convergences in the category of all Banach lattices.

We shall show the following main results. 1. The uaw-convergence in a Banach lattice is topological. 2. In an order continuous Banach lattice, uaw-convergence is stable under passing to and from sublattices and ideals. 3. Introduce strongly continuous operators between Banach lattices and investigate some properties of them.

The following main conclusions were drawn from this research. Theorem 2. Theorem 4.  Proposition 10. Theorem 11.

‎‎ 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.

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.

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.

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 .

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.

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 multivalued mappings by applying inequalities on binomials and trinomials.

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 𝐹̃, and define it as follows: 𝐹̃ = 𝐹 ~ = {𝐹̃: 𝐹 ∈ ℱ}. Also define 𝑑̃ on 𝐹̃ with 𝑑̃(𝐹̃, 𝐺̃) = 𝐻(𝑓𝑖𝑥𝐹, 𝑓𝑖𝑥𝐺). (𝐹̃, 𝑑̃) is a metric space. In this article, by considering some conditions on the maps 𝐹 and 𝐺 in complete metric space we conclude that 𝐹̃ = 𝐺̃.

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 multivalued mappings by applying inequalities on binomials and trinomials and using different conditions.

The aim of this study, is painting of topological surfaces with the least number of colors without the distance, and the colors have a border. For this purpose, we need a color mapping. In this mapping, we have not any fixed point, and we can colorable the map with least colors.Definition: Let f∶ X → X be a graph without a fixed point. f is colorable with k colors, if there is C={C_1,…,C_K}, where all C_i do not include {(x, f(x)}. Or similarly, for every i=1,…, k, there is the equation C_i ∩ f(C_i )=∅.Also, we define some concepts such as Compression, Metric, or non-Compression of space. Also, to achieve the desired result of each space, we change the properties of the maps.

In this work, first, we define the properties and conditions of the color mapping and color number. Also, by the study of properties of each space, we choose the best of space. One of the best conditions of this space is the lowest color number and higher efficiency. Finally, we proved  that this number is finite, and we can do coloring space with some maps and conversely.

In this work, we define the properties and conditions of the color mapping and color number. We presented some theorems and Lemma in the article and proved them for coloring of any space by coloring map, the coloring number is at least 3 and at most is a n+3. Also, we proved the coloring number finite and we can  do coloring space with some maps and conversely.

The following conclusions were drawn from this research. the coloring number is at least 3. the coloring number is at most n+3. coloring number is finite and we can do coloring space with some maps. We can do the coloring of any space by the finite coloring map.

Throughout this paper, we consider monomial ideals of the polynomial ring  over a filed. We try to give some properties of the polymatroidal ideals, which are the special class of monomial ideals. Herzog and Takayama constructed explicit resolutions for all ideals with linear quotients which admit regular decomposition functions. They also shaw that this class contains all matroidal ideals. We generalize their result to the polymatroidal ideals. Therefore, we can give an explicit linear resolution for any polymatroidal ideal. We also characterize generic polymatroidal ideals. The author and Jafari [1] characterized generalized Cohen-Macaulay polymatroidal ideals. Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.

A monomial ideal  is said to be polymatroidal, if it is single degree and for any two elements  such that  there exists an index  with  such that. In the case that the polymatroidal ideal is squarefree, it is called matroidal. We know that the powers of a polymatroidal ideal are again polymatroidal and polymatroidal ideals have linear quotients. Therefore all powers of polymatroidal ideal have linear resolutions. Let  has linear quotients with the order  of elements of. We can associate a unique decomposition function, that is a function  which maps a monomial  to, if  is the smallest index such that , where . The decomposition function  is called regular, if  for all  and.  We show that any polymatroidal ideal has a regular decomposition function. Therefore we can give an explicit linear resolution for any polymatroidal ideal. By an example, we show that our result can not be extended to the weakly polymatroidal ideals even if they are generated in a single degree. Recall that, a monomial ideal  is called generic if two distinct minimal generators  and  have the same positive degree in some variable , there is a third generator  which  and , where  is the least common multiple of  and . In the next result, we characterize generic polymatroidal ideals. A monomial ideal  is called generalized Cohen-Macaulay, whenever  is equidimensional and monomial localization  is Cohen-Macaulay for all monomial prime ideals, where  is unique homogenous maximal ideal of . Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.

For the first result, we show that any polymatroidal ideal has a regular decomposition function. So we have an explicit linear resolution of any polymatrodal ideal. In the next, we show that ifis a fully supported polymatroidal ideal generated in degree. Then  is generic if and only if  is either a complete intersection or. Finally, we prove that if   is a fully supported monomial ideal in  and generated in degree. Then is a generalized Cohen-Macaulay polymatroidal ideal for all  if and only if where  and  for some integers  and one of the following statements holds true:  is a principal ideal.  is a Veronese ideal. is equidimensional and  for all .  is an unmixed matroidal ideal of degree 2.

The following conclusions were drawn from this research: Any polymatroidal ideal has a regular decomposition function. characterization of generic ideals. characterization of monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.

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∈𝐺 𝑑(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, FosterGreenwood 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.

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.

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.

‎‎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.

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.   

‎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. 

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.

‎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.

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.

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.

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.

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.

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 recent years, the use of the Internet has been increasing. Attractions and many educational and recreational applications, etc. have caused the emergence of a new phenomenon called Internet addiction.   Internet addiction is a new and interesting topic that arises from social changes including the cellularization of society and the family. Poor family support of their members, poor social skills, and easy access to Internet has contributed to this addiction.Today, there are Internet addiction treatment clinics around the world. In addition to the treatment, the person in these clinics receives the necessary training to use the Internet properly.The effects of Internet addiction are similar to other addictions. These effects include mental disorders, euphoria and desire to consume more, etc. Because of the great similarities between Internet addiction and other addictions, in this article we have developed the epidemic model of heroin addiction in White and Comiskey [1].

In the proposed model, we have studied the effect of education and prevention. To examine this issue, we have considered five classes: 1- Susceptible individuals, 2- Internet users without training, 3- Educated Internet users and fully aware of the harms ofthe Internet addiction, 4- Internet addicts 5- Addicts under treatment and education.We examine the dynamic behavior of the model such as backward bifurcation, local and global stability equilibrium points. We investigate the boundary and obtain the basic reproduction number of the system. We study the existence of endemic equilibrium points and, using the Chavez-Sang theorem, show that the backward bifurcation occurs. We obtain sufficient conditions for the global stability, the addiction-free equilibrium point, and the endemic equilibrium point using the Lyapunov function and the geometric stability method.

The epidemic model of heroin addiction was first introduced by White and Comiskey. In this article, we have developed this model. First, we have considered the age group of 15 to 65 years old as people who are inclined to use the Internet. Then some of these people become Internet users and a group of these people may become addicted to the Internet due to the excessive use of the Internet. At a certain rate, these addicts are treated and educated. On the one hand, Internet users may not become addicted and receive the necessary training and become professional Internet users.

In this paper, we have conducted a complete qualitative study of the model including the existence and evaluation of local and global stability of the equilibrium points of the model.
We have shown that under certain conditions the addiction-free equilibrium point is local and global asymptotical stable. Using the compound matrix, we have obtained the conditions for the global stability of the endemic equilibrium point. We have shown the occurrence of backward bifurcation. This bifurcation indicates that when, addiction remains in the society.
To control addiction, we need to get. Reduce  to less than.

Since linear and multilinear algebra has many applications in different branches of sciences, the attention of many mathematicians has been attracted to it in recent decades. The determinant and the permanent are the most important functions in linear algebra and so a generalized matrix function, which is a generalization of the determinant and the permanent, becomes significant. Generalized matrix functions connect some branches of mathematics such as theory of finite groups, representation theory of groups, graph theory and combinatorics, and linear and multilinear algebra.   Let  be the symmetric group of degree , be a subgroup of , and be a function. The function given by is called the generalized matrix function associated to  and . Note that if and , the principal character of , then is the permanent and  if and , the alternating character of , then is the determinant.

  In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the determinant if and only if it preserves the product of symmetric matrices. To be precise, we show that:If and is a nonzero function, then the following are equivalent:1)  or ; 2) ; 3) ; where is the permutation matrix induced by and . Also if and is a nonzero function, then  if and only if for all symmetric matrices .

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.

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.

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.

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.

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 manifoldsLet 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, ifUgV,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.

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

Estimating the spatial hazard, or in other words, the probability of exceeding a certain boundary is one of the important issues in environmental studies that are used to control the level of pollution and prevent damage from natural disasters. Risk zoning provides useful information to decision-makers; For example, in areas where spatial hazards are high, zoning is used to design preventive policies to avoid adverse effects on the environment or harm to humans. Generally, the common spatial risk estimating methods are for stationary random fields. In addition, a parametric form is usually considered for the distribution and variogram of the random field. Whereas in practice, sometimes these assumptions are not realistic. For an example of these methods, we can point to the Indicator kriging, Disjunctive kriging, Geostatistical Markov Chain, and simple kriging.  In practice utilize the parametric spatial models caused unreliable results. In this paper, we use a nonparametric spatial model to estimate the unconditional probability or spatial risk:rcs0=PZs0⩾c.  (1) Because the conditional distribution at points close to the observations has less variability than the unconditional probability, nonparametric spatial methods will be used to estimate the unconditional probability.  

Let Z=Zs1,…,ZsnT be an observation vector from the random field {Zs;s∈D⊆Rd} which is decomposed as follows Zs=μs+εs,  (2) where μ(s) is the trend and ε(s) is the error term, that is a second-order stationary random field with zero mean and covariogram Ch=Covεs,εs+h. The local linear model for the trend is given by μHs= e1TSsTWsSs-1 SsTWsZ≡  ϕTsZ, where e1 is a vector with 1 in the first entry and all other entries 0, Ss is a matrix with ith row equal to (1, (si-s)T), Ws = diag {KHs1 – s,…,KH(sn-s)}, KHu=H-1K(H-1u), K is a triple multiplicative multivariate kernel function and H is a nonsingular symmetric d×d bandwidth matrix. In this model, the bandwidth matrix obtained from a bias corrected and estimated generalized cross-validation (CGCV). From nonparametric residuals ε(s) = Z(s) -μ(s) a local linear estimate of the variogram 2 γ(⋅)is obtained as the solution of the following least-squares problem minα.βi<jnεi-εj2-α-βT si-sj-u2 KGsi-sj-u, where G is the corresponding bandwidth matrix, that obtained from minimizing cross-validation relative squared error of semi-variogram estimate. Algorithm1: Semiparametric Bootstrap Obtain estimates of the error covariance and nonparametric residuals covariance. Generate bootstrap samples with the estimated spatial trend μHs and adding bootstrap errors generated as a spatially correlated set of errors. Compute the kriging prediction Z*s0 at each unsampled location s0 from the bootstrap sample Z*s1,…,Z*sn. Repeat steps 2 and 3 a larger number times B. Therefore, for each un-sampled location s0, B bootstrap replications Z*(1)s0.…. Z*(B)(s0) are obtained. Calculate (1) at position s_0 by calculating the relative frequency of Bootstrap repetition as follows to estimate the unconditional probability of excess of boundary c. rcs0= 1B j=1BIZ*js0≥ c

To analysis the practical behavior of the proposed methods a simulation study is conducted under different scenarios. For N=150 samples and n=16×16 were generated on a regular grid in the unit square following model (2), with mean function μs=2.5 + sin( 2π x1) + 4x2 - 0.5 2, and random errors normally distributed with zero mean and isotropic exponential covariogram  Ch= 0.04 + 2.01 1- exp-3 ∥ h∥0.5,   h∈ R2. For comparing nonparametric spatial methods for estimate unconditional risk, conditional risk, and Indicator kriging, we considered 7 missing observations in certain situations. Empirical spatial risk and its estimates are presented in Table 1. The Indicator kriging is overestimating and estimate spatial risk larger than 1. Generally, an estimated risk with unconditional and conditional methods is near value to empirical value. Table 1. Empirical spatial risk and its estimates Locations (0.13, 0) (0.87, 0.87) (0.80, 0.20) (0.94, 0.27) (0, 0.47) (074, .60) (0.34, 0.60) Methods 0.999 0.300 0.069 0.317 0.504 0.011 0.989 Empirical 0.998 0.351 0.054 0.347 0.494 0.057 0.954 Conditional 1.002 0.230 0.091 0.091 0.652 0.006 0.996 Indicator 1.000 0.388 0.418 0.481 0.602 0.024 0.994 Unconditional The spatial risk mapping for the maximum temperature means of Iran in 364 stations in March 2018 is obtained. By applying Algorithm 1 final trend and semi-variogram estimates are smoother than the pilot version.  The conditional and unconditional spatial risk with 150 bootstrap replicates for two values of threshold 25 and 31 on a 75×75 grid are estimated. The unconditional risk estimate is smoother than the conditional risk estimate. Because of this in the unconditional version, biased residual unused directly in the spatial prediction but in the conditional risk estimating, original residuals and simple kriging used.

The spatial risk estimated with the nonparametric spatial method. For the trend and variability of the random field, modeling applied a local linear nonparametric model. In the simulation study, this method better results than Indicator kriging. Because the flexibility of the nonparametric spatial method could apply for the construction of confidence or prediction intervals and hypothesis testing.

  In Throughout all groups are abelian. Suppose that G is a group and n is a positive integer. For a ∈ G, if we consider the solution of the equation nx = a in G, two subsets of G are proposed. One of them is {a ∈ G | ∃x ∈ G,nx = a} and the other is {x ∈ G | nx = a} for given a ∈ G. The first is nG, which is clearly a subgroup of G, but the second does not have to be a subgroup. However, if we replace the equation nx = a with nx ∈< a > then we come to the equation nx = 0 in the group, whose solutions determine a subgroup of   (hence of G). In this regard, we state something about divisibility from [2]. Let a is an element in a group G. The element a is called divisible whenever for every n ≥ 1 there exists x ∈ G such that nx = a. Also a is called torsion whenever there exists positive integer m such that a is a solution of the equation mx = 0. The group G is then called divisible (resp. torsion) if every element in G is divisible (resp. torsion). Furthermore, G is called reduced (resp. torsionfree) if it has no non-zero divisible (resp. torsion) subgroup. Therefore, G is divisible if and only if nG = G for every n ≥ 1. As canonical examples, we can mention the additive group Q and. Here, is the subgroup of Q/Z generated by {1/pi + }. Also, and all proper subgroup of Q are reduced; [1] and [2] are excellent references on the subject. Suppose that n ≥ 1. It is easy to verify that nG = G if and only if pG = G for every prime number p | n. This follows that G is divisible if and only if pG = G for every prime number p. Thus G is non − divisible if there exists a prime number q such that qG ≠ G. Based on the above, we may define the divisibility (non-divisibility) with respect to a number. Definition 1.1. Let n ≥ 1 a group G is called:(a) n-divisible if nG = G. (b)   Fully non-divisible if pG ≠ G for every prime number p. (c)   Absolutely non-divisible if pH ≠ H for every prime number p and non-zero subgroup H of G. Thus, we deal with three class of groups as blow:{Absolutely non-divisible groups} ⊆ {Fully non-divisible} ∩ {Reduced groups}. Examples are presented to show that these three classes are mutually distinct. main

Definition 2.1. For every prime number p, let radp(G) = ∩n≥1pnG and Tp(G), the sum of all p-divisible subgroups of G. Dn be the class of all n-divisible groups and Fp be the class of all groups G with Tp(G) = {0}. Let D={G|G=H⩽GH such that H ∈ ∪n≥1Dn} and Cp be the class of all groups G with radp(G) = {0}. Theorem 2.2. Let p be a prime number. For every group homomorphism f: G1 → G2 we have f (radp(G1)) ⊆ radp(G2). Furthermore radp(G) is a fully invariant subgroup of G. For every H⩽G we have radp(H) ⊆ radp(G). Also if G = H⊕K then radp(H) = H ∩ radp(G). radp(⊕i∈IGi)=⊕i∈Iradp(Gi). radp(i∈IGi)=i∈Iradp(Gi). pG = G if and only if radp(G) = G if and only if  HomZ(G,Zp)={0}. For every H⩽G we haveradp(G)+HH⊆radp(GH). Also, if H⊆radp(G) thenradp(G)H=radp(GH). FurthermoreGradp(G)={0} . radp(G)=Rej(G,Cp). radp(G)=Rej(G,{Zpi}i≥1). Theorem 2.3. Let p be a prime number. The class of p-divisible groups is closed under direct sum and homomorphic image. For every group G, Tp(G) is p-divisible and we have Tp(G) ⊆ radp(G). Furthermore Tp(radp(G)) = radp(Tp(G)) = Tp(G). If G is a p-torsionfree group, then radp(G) is a p-divisible subgroup and radp(G) = Tp(G). Let G be a p-torsionfree group and H ≤ G. H ⊆ radp(G) if and only if radp(G) = radp(H). Furthermore radp(radp(G)) = radp(G). If  Tp(G) = {0}, then p divide the order of every torsion element in G. Let p and q be two different prime numbers. If Tp(G) = Tq(G) = {0}, then radp(G) = radq(G) = {0}. Tp(GTp(G))={0}. Theorem 2.4. For every prime number p, (Dp,Fp) is a torsion theory. Theorem 2.5. Every absolutely non-divisible group G is torsion free and so G is isomorphic to a subgroup ofQΛ. Theorem 2.6. The following statements are equivalent for every group G. G is absolutely non-divisible, for every prime number p, radq(G) = {0}, HomZ(D,G)={0}. Theorem 2.7. The class of absolutely non-divisible is closed under direct product and subgroup. Theorem 2.8. If H and G/H are absolutely non-divisible groups then G is absolutely non-divisible group. Theorem 2.9. For every group G the following statements hold. . (b) G is an absolutely non-divisible group if and only if for every prime number p there exists a natural number n such that  is absolutely non-divisible. For H⩽Q and prime number p, letBp(H)={t∈N|∃mn∈H,(m,n)=1,pt|n}, and bp(H) = |Bp(H)|. Theorem 2.10. Let {0} ≠ G ≤ Q G is absolutely non-divisible if and only if for every prime number p, bp(G)<∞.

Regression analysis is a common method for modeling relationships between variables. Usually Ordinary Least Squares method is applied to estimate regression model parameters. These estimators are unbiased and appropriate when design matrix is nonsingular.In presence of multicollinearity, design matrix is singular and Ordinary Least Squares estimates cannot be obtained. In this situation, other methods‎, ‎such as Lasso‎, ‎Ridge‎ and ‎Lars may be considered.Other hand, in many fields such as medicine, number of variables is greater than the number of observations‎ and usual methods such as Ordinary Least Squares are not proper and shrinkage methods‎, ‎such as Lasso‎, ‎Ridge‎ ‎and‎ ... ‎have a better performance to estimate regression model coefficients‎. ‎In the shrinkage methods‎, ‎tuning parameter plays an essential role in selecting variables and estimating parameters‎. ‎Bridge shrinkage estimators is an estimator that can be obtained by changing its tuning parameter‎. In other words, Bridge method is the extension of Ridge and Lasso regression methods. Selecting the appropriate amount of tuning parameter is important. There are many studies ‎on each of these methods under the assumed conditions. In this paper‎, performance of Bridge shrinkage estimators‎, ‎such as Lasso and Ridge‎ ‎are compared with Lars and Ordinary Least Squares estimators in a simulation study.

A simulation study is applied to compare performance of the regression methods Ridge, Lasso, Lars and Ordinary Least Squares. MSE criterion is applied to evaluate the method performance. Statistical software R is applied for simulation and comparing the regression methods.

In the presence of collinearity, Bridge regression estimators will result in appropriate estimators. These estimators are biased but their performance is better than unbiased estimators such as Ordinary Least Squares. Indeed, Bridge estimators have the best performance in the class of biased estimators.

In this article, Ridge and Lasso estimators as special cases of Bridge estimators are compared with Lasso and Ordinary Least Squares in a simulation study. This study shows that under the supposed conditions, Ridge, Lasso and Lars have better action than Ordinary Least Squares method. Lars method has the best performance and Ridge estimators is better than Lasso Regression.

  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.

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.

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.

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.

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, foundsome 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 multiplicationa,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 multiplicationa1,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.

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.