نشریه پژوهشهای ریاضی
سال هشتم شماره 1 (پیاپی 20، بهار 1401)

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.

Optimization using radial basis functions as an interpolation tool in trust-region (ORBIT), is a derivative-free framework based on fully linear models to solve unconstrained local optimization, especially when the function evaluations are computationally expensive. This algorithm stores the interpolation points and function values to using at subsequent iterations. Despite the comparatively advanced management used for interpolation points, we maintain that ORBIT ignores sorting the interpolation points based on the function values. In this paper, we propose an improved version SORT-ORBIT by sorting the interpolation points and selecting a point as the trust-region center in which the objective function reaches its minimum value. Numerical results indicate the efficiency of the improved version compared with the original version. In addition, to estimate high-accuracy solutions, we equip the ORBIT with a new gradient-free convergence test.

In this paper, random vectors following the multivariate generalized hyperbolic (GH) distribution are compared using the hessian stochastic order. This family includes the classes of symmetric and asymmetric distributions by which different behaviors of kurtosis in skewed and heavy tail data can be captured. By considering some closed convex cones and their duals, we derive some necessary and sufficient conditions for some important applied stochastic orderings. The linear convex orderings are shown to be equivalent with a certain kind of hessian orderings. Based on copulas generated by the GH distributions, it is revealed that the ordering of the GH distributions in terms of their dependence structures corresponds to some hessian stochastic orderings being satisfied. The results are shown to be relevant to some insurance and economic applications.

In this paper we present several results concerning the cofiniteness of generalized local cohomology modules.

Hagler used a diameter norm to construct a separable Banach space 𝑋 with nonseparable dual such that 𝑙1 does not embed in 𝑋. Also, Bayati Eshkaftaki considered the diameter norm on 𝑐0(𝐼) and characterized all isometries on this space. In this article, we are going to first introduce the space 𝐵0 (ℝ) and show that this space is Banakh space with supremum norm, and then we express the diameter norm on the space 𝐵0(ℝ) and examine some of the characteristics of this space. We also examine the relationship between this space and Banach space 𝑐0(ℝ) and some of the reference theorems [3] shown on space 𝑐0(ℝ)on space 𝐵0(ℝ).

 The reliability System plays an important role in the reliability of power play models. Reliability System in Stress-Strengths models is a measure of component reliability. This model was first introduced by Birnbaum in 1956. As we know, the Lindley distribution is widely used in reliability theory and many researchers are used this distribution to calculate Stress-Strengths. In 2013 Al Mutairi and Kundu work out on stress-strength reliability for Lindley distribution. In 2017 Rezaei, et al. estimate the stress-strength parameter R=P(Y<X), when the variables X and Y are from General Lindley distribution. There are different methods to estimate R=P(Y<X) such as maximum likelihood, method of moments, Bayesian, and shrinkage estimation. Recently, many researchers consider the interval shrinkage estimation to estimate parameters of statistical distribution. Here, for the first time, we estimate the R=P(X<Y) by using the interval shrinkage estimation in two parameter Lindley distribution. Ghitany et al. in 2008 showed that the behavior of this distribution in data analysis of life data and reliability is better than the exponential distribution. So, we estimate the parameters of Lindley and R=P(X<Y) by using the interval shrinkage and method of moments estimation. In 2013 Shanker, et al. considers two parameter Lindley distribution for modeling waiting and survival time data. [ Downloaded from mmr.khu.ac.ir on 2022-10-03 ] .

Material and methods:

 In statistical inference, choosing the methods of estimation is very essential in the process of it. The interval shrinkage and method of moments estimation for Lindley distribution is obtained. For interval shrinkage estimation, we consider as an interval for parameter.

Results and discussion:

 To compare the estimators, samples with different sizes and values of parameters are generated from two parameter Lindley distribution by applying R software. Consequently, comparison of the estimators, the estimation of parameters and mean square errors of them are accomplished.

Conclusion:

 Simulation and real data methods are used to compare the method of moments and interval shrinkage estimation for estimating parameters of two parameter Lindley distribution. The results show that the interval shrinkage estimator is better than the method of moment estimator. Material and methods In statistical inference, choosing the methods of estimation is very essential in the process of it. The interval shrinkage and method of moments estimation for Lindley distribution is obtained. For interval shrinkage estimation, we consider as an interval for parameter. Results and discussion To compare the estimators, samples with different sizes and values of parameters are generated from two parameter Lindley distribution by applying R software. Consequently, comparison of the estimators, the estimation of parameters and mean square errors of them are accomplished. Conclusion Simulation and real data methods are used to compare the method of moments and interval shrinkage estimation for estimating parameters of two parameter Lindley distribution. The results show that the interval shrinkage estimator is better than the method of moment estimator.

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.

In this paper we consider a fractional order model of HIV infection of  CD4+T cells and we transform this fractional order system of ordinary differential equations to a system of weakly singular integral equations. Afterwards we propose a Nystrom method for solving resulting system, convergence result and order of convergence is obtained by using conditions of existence and uniqueness of solution. Finally, we test performance of the method by numerical examples and for integer order system, we compare the obtained results with Runge-Kutta and Bessel collocation methods. Since most of the numerical methods are efficient only for intervals of small length, we also apply the introduced method for 100 days interval.

Suppose that (𝑋, 𝑑) be a compact metric space with a distinguished point𝑒 and𝐸 be a Banach space.Collection of 𝐸 −valued function 𝑓 on 𝑋 such that ℒ(𝑓) = sup ,𝑥≠𝑦 𝑥,𝑦∈𝑋 ‖𝑓(𝑥) − 𝑓(𝑦)‖ 𝑑(𝑥, 𝑦) < ∞ , 𝑓(𝑒) = 0 is called vector-valued Lipschitz space and denoted by 𝐿𝑖𝑝0 (𝑋, 𝐸). The space 𝐿𝑖𝑝0(𝑋, 𝐸) with respect to the point wise operations on functions and the norm ℒ(. ) is a Banach space that separates points of 𝑋. The subset consists of all functions such that lim 𝑑(𝑥,𝑦)→0 ‖𝑓(𝑥) − 𝑓(𝑦)‖ 𝑑(𝑥, 𝑦) = 0 is a closed subspace of 𝐿𝑖𝑝0 (𝑋, 𝐸), denoted by 𝑙𝑖𝑝0(𝑋, 𝐸) and called little vector-valued Lipschitz space. In particular when Banach space 𝐸 coincides with scaler field, 𝐿𝑖𝑝0(𝑋, 𝐸) and 𝑙𝑖𝑝0(𝑋, 𝐸) is denoted by 𝐿𝑖𝑝0(𝑋) and 𝑙𝑖𝑝0(𝑋) respectively. Definition. The space 𝑙𝑖𝑝0(𝑋) separates points of 𝑋 uniformly when there exists 𝐶 > 1 such that for each distinct pair point 𝑥, 𝑦 ∈ 𝑋 there is 𝑓 ∈ 𝑙𝑖𝑝0(𝑋, 𝐸) with 𝑓(𝑦) = 0, ‖𝑓(𝑥)‖ = 𝑑(𝑥, 𝑦), ℒ(𝑓) ≤ 𝐶. Definition.The Banach space 𝐸 has approximation property if for each ε > 0 and compact subset 𝐾 of 𝐸 there exists a finite dimensional bounded operator 𝑇: 𝐸 → 𝐸 such that sup 𝑥∈𝐾 ‖𝑇𝑥 − 𝑥‖ < 𝜀.

In this paper we deal with the uniform separation property of a metric space 𝑋 by the little vector-valued Lipschitz space, namely 𝑙𝑖𝑝0(𝑋, 𝐸).

We show that if 𝑙𝑖𝑝0 (𝑋) has the approximation property and 𝐸 be a topological dual of some Banach space, then there exists a compact metric space 𝑌 with a distinguished point and a non-expansive function 𝜋: 𝑋 → 𝑌 such that 𝑙𝑖𝑝0(𝑌, 𝐸) separates the point of 𝑌 uniformly and 𝐶𝜋, the composition operator induced by 𝜋, is a surjective linear isometry from 𝑙𝑖𝑝0(𝑌, 𝐸) to 𝑙𝑖𝑝0(𝑋, 𝐸).

Naturally in choosing a point estimator we are interested in choosing an estimator that minimizes the risk function for all parameter space values. In practice, this is not possible due to the large number of estimators. One way to solve this problem (find optimal estimators) is to limit the range of estimators and find the best estimator in the finite range. This limitation leads us to two types of optimal estimators, namely the best equivariant estimator and the minimum risk unbiased estimator, respectively, in terms of limiting ourselves to the class of equivariant or unbiased estimators. In this paper, the role of independence in simplifying the calculation of these estimators is examined. We also deal with the stochastic independence of an invariant function and its comparison with the Basu’s theorem. To find the optimal estimators, the class of estimators can be limited. This limitation can be applied to the class of equivariant or unbiased estimators, which leads to two types of optimal estimators, namely the best equivariant estimator and the minimum risk unbiased estimator, respectively. For this purpose, in addition to the Rao-BlackwellLehmann-Scheffé theorem (Casella and Berger, 2001), two other methods have been proposed by Sathe and Varde (1969) and Eaton and Morris (1970) which can be useful to achieve this goal. In these two methods, by limiting the class of equivariant or unbiased estimators, the estimator with the minimum risk is considered as the optimal estimator. Independence can play a key role in making it easier to calculate the risk function of the best equivariant estimator and the minimum risk unbiased estimator. In fact, the role of independence is to eliminate the conditional probability in calculating the risk function of the best equivariant estimator and the minimum risk unbiased estimator, which in most cases results from Basu’s theorem and the independence of ancillary statistic from complete sufficient statistics. Similar to Basu’s theorem, it can be shown that in certain circumstances an invariant statistic and equivariant function are independent of each other, which can play a role in eliminating the conditional probability by the independence of the maximum invariant statistic from the equivariant sufficient statistic. It is noteworthy that in this case, the completeness assumption of Basu’s theorem has been replaced by equivariance assumption and the ancillarity assumption of Basu’s theorem has been replaced by invariance.

We first introduce the definitions that are needed. In the second part, by limiting the class of equivariant estimators, we create a type of optimal estimator called the best equivariant estimator and show that in groups that act as transitive on the parameter space, an invariant function is independent of the equivariant sufficient statistic. In the third part, by limiting the class of unbiased estimators, we make a type of optimal estimator called minimum risk unbiased estimator, which in a special case where the square error loss function is the same as the minimum variance unbiased estimator, which in the fourth part, in addition to Rao-Blackwell-Lehmann- Scheffé theorem (Casella and Berger, 2001), introduce two other methods proposed by Sathe and Varde (1969) and Eaton and Morris (1970) which, with the help of independence, provide a simpler method for finding a minimum variance unbiased estimator.

The following conclusions were drawn from this research.  In order to find the optimal estimators by limiting the class of estimators to the class of equivariant or unbiased estimators, the independence of complete sufficient statistics from ancillary statistics and applying Basu’s theorem can be a way to simplify calculations.  In some statistical problems with the transitive transformation group, the equivariant function can be used instead of the complete sufficient statistic. In this case, instead of using the Basu’s theorem, the independence of an invariant function and the equivariant sufficient statistic can be inferred. Hence, the assumption of completeness for establishing the Basu’s theorem is replaced by the equivariance and having a transitive transformation group.  Having a transitive group, any invariant function is also ancillary, but the incompleteness of a sufficient statistic can also result in its independence from an invariant statistic. If the group of transitive transformations and complete sufficient statistic is also equivariant, the case of Basu’s theorem concludes this view. The opposite is not always true and can be corrected in such a way that if a complete statistic is also equivariant with a transitive group, it is not necessary that each ancillary statistic be independent of it. Rather, it is possible to find an ancillary statistic that is also invariant, which is independent of the given equivariant complete sufficient statistic. By finding the condition that an ancillary statistic is also invariant, these results can be extended, in which case Basu’s theorem is the result. Of course, this open problem needs further research and it is hoped that researchers will be diligent in generalizing it.

A well-known result of Green [4] shows for any finite p-group G of order p^n, there is an integer t(G) , say corank(G), such that |M(G)|=p^(1/2n(n-1)-t(G)) . Classifying all finite p-groups in terms of their corank, is still an open problem. In this paper we classify all finite abelian p-groups by their coranks.

We investigate the existence of a weak nontrivial solution for the following problem. Our analysis is generally bathed on discussions of variational based on the Mountain Pass theorem and some recent theories one the generalized Lebesgue-Sobolev space. This paper guarantees the existence of at least one weak nontrivial solution for our problem. More precisely, by applying Ambrosetti and Rabinowitz’s mountain pass theorem and  under appropriate conditions, we show that there exists a positive number such that our problem has at least one nontrivial weak solution.

This paper presents a reliable numerical technique based on Lucas polynomials for a family of fractional differential equations and multi order fractional differential equations by means of the least square method. The fractional derivative is in the Caputo sense. A relevant feature of this approach is the analyzing of the suggested technique by Gauss quadrature method and using the theory of Lagrange multipliers to solve a constrained optimization problem. An upper error bound, the convergence, and error analysis of the scheme are investigated and the CPU time used, the values of maximum errors, the numerical convergence analysis based on the proposed technique for different values of parameters are discussed. Furthermore the results of present technique are compared with the, operational matrix of hybrid basis functions, the Jacobi orthogonal functions and pseudo-spectral scheme. In order to introduce the numerical behavior of the proposed technique in case of nonsmooth solutions, this issue is discussed. In this case, the obtained results imply an elegant superiority of our proposed technique. The numerical examples illustrate the accuracy and performance of the technique. Finally extending the proposed technique to high dimensions and system of fractional differential equations can be examined as a further works.

In this study, the least square method, the Gauss quadrature method and the theory of Lagrange multipliers are used to solve a constrained optimization problem.

Several numerical examples are examined using the proposed technique. The numerical examples illustrate the accuracy and performance of the technique. Also, the numerical results reported in the tables indicate that the accuracy improve by increasing the degree of the Lucas polynomials.

In this paper, Lucas polynomials have been successfully applied to compute the approximate solution of the fractional differential equations and multi order fractional differential equations. The results show that: • The proposed technique provides the solutions in terms of convergent series with easily computable components in a direct way, without using linearization, perturbation or restrictive assumption. • The proposed technique is very straightforward and the solution procedure can be done easily. • The numerical behavior of the proposed technique in case of non-smooth solutions, demonstrated that the obtained results imply an elegant superiority of our proposed technique.

Let n and k be two positive integers such that 𝑛 ≥ 𝑘. Let [𝑛] = {1, . . . , 𝑛} be an 𝑛-elemen set and let symbol ( [𝑛] 𝑘 ) denote the family of all k-element subsets (or k-sets) of [𝑛]. A family 𝒜 of k-sets of [𝑛] is said to be intersecting if for every two members A, B in 𝒜, we have 𝐴 ∩ 𝐵 ≠ ∅. For a fixed element 𝑖 ∈ [𝑛], if all members of 𝒜 contain i, then it is clear that 𝒜 is an intersecting family, which is called star. For each 𝑖 ∈ [𝑛], the family 𝑆𝑖 = {𝐴: |𝐴| = 𝑘, 𝐴 ⊆ [𝑛], 𝑖 ∈ 𝐴} is a maximal star. The well-known Erdős –Ko–Rado theorem is one of important results in extremal combinatorics. It has many interesting proofs and extensions. The Erdős-Ko-Rado theorem states that every intersecting family of ( [𝑛] 𝑘 ) has cardinality at most ( 𝑛 −1 𝑘 − 1 ) provided that 𝑛 ≥ 2𝑘; moreover, if 𝑛 > 2𝑘, then the only intersecting families of this cardinality are isomorphic to 𝑆𝑖 . Two families 𝒜 ⊆ ( [𝑛] 𝑘 ) and ℬ ⊆ ( [𝑛] ℓ ) are called cross intersecting if for any 𝐴 ∈ 𝒜 and 𝐵 ∈ ℬ, we have 𝐴 ∩ 𝐵 ≠ ∅. Strengthening the Erdős–Ko–Rado theorem, in 1986 Pyber showed an upper bound for |𝒜 ||ℬ| as follows. Theorem A. Let k, ℓ, n be positive integers such that k≥ℓ. Assume that 𝒜 ⊆ ( [𝑛] 𝑘 ) and ℬ ⊆ ( [𝑛] ℓ ) are cross intersecting.  If k>ℓ and 𝑛 ≥ 2𝑘 + ℓ −2, then |𝒜||ℬ| ≤ ( 𝑛 − 1 𝑘 −1 ) ( 𝑛 − 1 ℓ − 1 ).  If k=ℓ and 𝑛 ≥ 2𝑘, then |𝒜||ℬ| ≤ ( 𝑛 − 1 𝑘 − 1 ) 2 . In 1989 Matsumoto and Tokushige slightly improved Pyber's result as follows. Theorem B. Let k, ℓ, n be positive integers such that 𝑛 ≥ 2 max{𝑘, ℓ}. Assume that 𝒜 ⊆ ( [𝑛] 𝑘 ) and ℬ ⊆ ( [𝑛] ℓ ) are cross intersecting. Then |𝒜||ℬ| ≤ ( 𝑛 −1 𝑘 − 1 ) ( 𝑛 − 1 ℓ − 1 ). We say that a family 𝒜 is t-almost intersecting if for every set 𝐴 ∈ 𝒜 there are at most t elements of 𝒜 disjoint from 𝐴. In 2012 Gerbner, Lemons, Palmer, Patkós, and Szécsi proved an interesting generalization of the Erdős–Ko–Rado theorem for t-almost intersecting families. Theorem C. Let k, n. t be positive integers and 𝒜 ⊆ ( [𝑛] 𝑘 ) is a t-almost intersecting family.  If n=n(k,t) is sufficiently large, then |𝒜| ≤ ( 𝑛 − 1 𝑘 −1 ) with equality if and only if 𝒜 = 𝑆𝑖 for some 𝑖 ∈ [𝑛]  If 𝑘 ≥ 3, 𝑛 ≥ 2𝑘 +2, and 𝑡 = 1, then |𝒜| ≤ ( 𝑛 − 1 𝑘 − 1 ) with equality if and only if 𝒜 = 𝑆𝑖 for some 𝑖 ∈ [𝑛]. Main

We say two families 𝒜 ⊆ ( [𝑛] 𝑘 ) and ℬ ⊆ ( [𝑛] ℓ ) are cross t-almost intersecting if any 𝐴 ∈ 𝒜 is disjoint from at most t elements of ℬ and any 𝐵 ∈ ℬ is disjoint from at most t elements of 𝒜. As our main result we simultaneously extend the previous results for sufficiently large n. Theorem 1. Let k, ℓ, n be positive integers such that n=n (k, ℓ ,t) is sufficiently large. Assume that 𝒜 ⊆ ( [𝑛] 𝑘 ) and ℬ ⊆ ( [𝑛] ℓ ) are cross t-almost intersecting. Then |𝒜||ℬ| ≤ ( 𝑛 −1 𝑘 − 1 ) ( 𝑛 − 1 ℓ − 1 ) with equality if and only if 𝒜 = ℬ = 𝑆𝑖 for some 𝑖 ∈ [𝑛].

Let G be a group. Neumann to answer a question of Paul Erdos proved that every infinite subset of G has two different comuting elements  if and only if G is center-by-finite. In this paper, we deal with Erdoschr('39')s question in different aspect and we show that every infinite subset X of G has two different elements x and y such that x^y=1  if and only if the exterior  center of G ihas finite index.

Through this paper all algebras and linear spaces are on the complex field ℂ. Let 𝒜 be an algebra and ℳ be an 𝒜-bimodule. The linear mapping 𝑑: 𝒜 → ℳ is called an anti-derivation if 𝑑(𝑥𝑦) = 𝑦𝑑(𝑥) + 𝑑(𝑦)𝑥 (𝑥, 𝑦 ∈ 𝒜). Also, 𝑑 is called a derivation if 𝑑(𝑥𝑦) = 𝑥𝑑(𝑦) + 𝑑(𝑥)𝑦 (𝑥, 𝑦 ∈ 𝒜). The linear mapping 𝛿: 𝒜 → ℳ is a Jordan derivation if 𝑑(𝑥 2 ) = 𝑥𝑑(𝑥) + 𝑑(𝑥)𝑥 (𝑥 ∈ 𝒜). Any anti-derivation and derivation is a Jordan derivation, but the converse is not necessarily true. Jordan in [1] has shown that every continuous Jordan derivation on C*-algebra 𝒜 into any Banach 𝒜-bimodule is a derivation. Derivations and anti-derivations are important classes of mappings on algebras which have been used to study of structure of algebras. We refer to [2] and the references there in. Bersar studied in [3] additive maps on prime ring contain a non-trivial idempotent satisfying 𝑥, 𝑦 ∈ 𝒜, 𝑥𝑦 = 0 ⟹ 𝛿(𝑥)𝑦 + 𝑥𝛿(𝑦) = 0 . Later, many studies have been done in this case and different results were obtained, for instance, see [4, 5, 6, 7, 8, 9] and the references therein. Recently [10, 11, 12, 13], the problem of characterizing continuous linear maps behaving like derivations or antiderivations at orthogonal elements for several types of orthogonality conditions on *- algebras have been studied. In this paper we study the above problems on von Neumann algebra.

In this article, the subsequent conditions on a continuous linear map 𝛿: 𝒜 → 𝒜 where 𝒜 is a *-algebra has been considered: 𝑥𝑦 ∗ = 0 ⟹ 𝑥𝛿(𝑦) ∗ + 𝛿(𝑥)𝑦 ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥𝑦 ∗ = 0 ⟹ 𝑥 ∗𝛿(𝑦) + 𝑥𝛿(𝑦) ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜). We consider following conditions on continuous linear map on von Neumann algebras: 𝑥𝑦 = 0 ⟹ 𝑦𝛿(𝑥) + 𝛿(𝑦)𝑥 = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥𝑦 ∗ = 0 ⟹ 𝑦 ∗𝛿(𝑥) + 𝛿(𝑦) ∗𝑥 = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥 ∗𝑦 = 0 ⟹ 𝑦𝛿(𝑥) ∗ + 𝛿(𝑦)𝑥 ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜). Over methods are based on structure of von Neumann algebras and the fact that every derivation on von Neumann algebras is inner. Main Results The followings are the main results of our paper. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝑦 𝛿(𝑥) + 𝛿(𝑦)𝑥 = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥𝑦 = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝑥 𝜇 − 𝜈𝑥, where 𝜇 − 𝜈 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + 2[𝑥, 𝑦](𝜇 − 𝜈) = 0 for all 𝑥 , 𝑦 ∈ 𝒜. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝑦 ∗𝛿(𝑥) + 𝛿(𝑦) ∗𝑥 = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥𝑦 ∗ = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝜈𝑥 − 𝜇𝑥, where Re𝜇 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + (𝜈 − 𝜇) ∗ [𝑥, 𝑦] + [𝑥, 𝑦](𝜈 − 𝜇) = 0, for all 𝑥 , 𝑦 ∈ 𝒜. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝛿(y)𝑥 ∗ + 𝑦𝛿(𝑥) ∗ = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥 ∗y = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝑥𝜇 − 𝜈𝑥, where Re𝜇 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + [𝑥, 𝑦](𝜇 − 𝜈) ∗ + (𝜇 − 𝜈)[𝑥, 𝑦] = 0, for all 𝑥 , 𝑦 ∈ 𝒜.

Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 be a continuous linear map. Let 𝛿 be anti-derivation at orthogonal elements. We characterized the structure of 𝛿 according to the )generalized) inner derivation. We guess that the results obtained can also be proved on standard operator algebras.

Let  be a simple graph with vertex set  and edges set . A set  is a dominating set if every vertex in  is adjacent to at least one vertex  in . An eternal 1-secure set of a graph G is defined as a dominating set  such that for any positive integer k and any sequence  of vertices, there exists a sequence of guards   with  and either  or  and  is a dominating set. If we take a guard on every vertex in an eternal 1-secure set, then for any sequence of attacks to vertices of the graph only by moving one guard during one of the edges adjacent with the vertex, the result set still remains secure. Now let for every sequence of attacks to vertices, all guards could move during one of the edges adjacent with the vertex and the result set still remains secure. This set is called eternal -  secure set. The eternal -  security number  is defined as the minimum number of an eternal - secure set. secure set in G. An edge  is subdivided if the edge  is deleted and a new vertex  is added, along with two new edges and . The eternal - security subdivision number  of a graph  is the minimum cardinality of a set of edges that must be subdivided (where each edge in  can be subdivided at most once) in order to increase the eternal - security number of  to increase the eternal m- security number of G. In this paper, we show that the eternal - security subdivision number is at most 3 for any nontrivial graph .