Global Analysis and Discrete Mathematics
Volume:6 Issue: 2, Winter and Spring 2021

Let (N,L) be a pair of finite dimensional nilpotent Lie algebras. If N admits a complement K in L such that dim N = n and dim K = m, then dim M(N,L) = 1/2n(n + 2m - 1) - t(N,L), where M(N,L) is the Schur multiplier of the pair (N,L) and t(N,L) is a non-negative integer. In this paper, we characterize the pair (N,L) for t(N,L)=0, 1, 2, … , 23, where N is a finite dimensional filiform Lie algebra and N,K are ideals of L such that L = N ⊕ K. Moreover, we classify the pair (N,L) for s′ (N,L) = 3, where S′ (N,L) = 1/2 (n - 1)(n - 2) + 1 + (n - 1)m – dim M(N,L), L is a finite dimensional nilpotent Lie algebra and N is a non-abelian ideal of L.

An optical model as well as an implementation model for ray casting is presented. Then it is demonstrated how this work’s proposed rendering method relates to these models. The optical model expresses how a point with in a volume is affected by the light source and density. In the implementation model a viewing plane or in other words a 2D plane comprising of pixels casts a ray for each pixel along a particular viewing direction. In order to obtain a single color and opacity value for each pixel, voxels that are aligned with a ray perform interpolation (filtering) at a constant interval. The colors and opacities are then merged via composition in front to back or back to front order.

The exit methods for producing common weight in DEA are complicated or they can’t produce full ranking for Decision Making Units (DMUs). Wang and et al introduced two methods based on regresion analysis for finding common weights. In these method they fined set of common weight so that the efficeincy which is computed by set of common weight for all units, are always smaller or and equal to obtained optimistic efficeincy from CCR model. In this methods we are trying to make the computed efficeincy by common weight closer to obtained optimistic efficeincy from CCR model, and or in other words the goal of obtaining unique hyperplane as all distance of DMUs from this hyperplane will be minimal. In this paper by using an example, we show that the obtained hyperplane from suggested methods of Wang is passing throug PPS. In other words, in introduced method of proposed hyperplane for ranking, isn’t relaying PPS, neccessarly. At the end, we introduce a new method for ranking the decision makind units that is a correspond hyperplane of relaying common weight set on PPS and finally we can use this techniqe for real data.

The Schur multiplier of a pair of groups was introduced by Ellis in 1998. In this paper, we study the Schur multiplier and the c-nilpotent multiplier of pairs of Lie algebras and give some conditions under which the Schur multiplier of a pair of Lie algebras is trivial. Moreover, we give some conditions under which the higher multiplier of a pair of Lie algebras is not trivial.

‎In this paper‎, ‎the Tau method based on shifted Legendre polynomials is proposed for solving a class of fractional stochastic integro-differential equations‎. ‎For this purpose‎, ‎shifted Legendre polynomials and their properties are introduced‎. ‎By using the operational matrices of integration and stochastic Ito-integration we transform the problem into the corresponding linear system of algebraic equations‎. ‎Finally the efficiency of the proposed method is confirmed by some examples‎. ‎The results show that this method is very accurate and efficient‎.

In this paper, the first and second kind weakly singular Volterra integral equations are approximated by using the Jacobi wavelets method. First, the operational matrices for fractional integration and product for Jacobi wavelets are computed with a new matrix approach, and then, it applied to solve numerically the first and second kind Volterra integral equations involving singularity. Illustrative numerical experiments with comparison are included to indicate the validity and practicability of the method.

Let n be a positive integer. A group G is said to be n-abelian, if (xy)n = xnyn, for any x, y ∈ G. In 1979, Fay and Waals introduced the n- potent and the n-center subgroups of a group G, as Gn = ⟨[x, yn]|x, y ∈ G⟩, Zn(G) = {x ∈ G|xyn = ynx, ∀y ∈ G}, respectively. Also, the second n-center subgroup, Zn ∈ (G), is defined by Zn 2 (G)/Zn(G) = Zn(G/Zn(G)). In this paper, we give an upper bound for the index of the second n-center subgroup of any n-abelian group G in terms of the order of n-potent subgroup Gn.

‎In this paper‎, ‎the stochastic weakly singular integro-differential equation is discussed‎. ‎The shifted Legendre Tau method is introduced for finding the unknown function‎. ‎For this purpose‎, ‎shifted Legendre polynomials and their properties are introduced‎. ‎The proposed method is based on expanding the approximate solution as the elements of shifted Legendre polynomials‎. ‎We reduce the problem to set of algebraic equations by using operational matrices‎. ‎Also the convergence analysis of shifted Legendre polynomials and error estimation for this method have been discussed‎. ‎Finally‎, ‎several numerical examples are given to demonstrate the high accuracy of the method‎.

In this paper, we will obtain analytical approximate solutions of the HIV infection model of CD4+ T-cells. This model corresponds to a class of nonlinear ordinary differential equation systems. To this end, we combine the Natural transform with the Adomian decomposition method for solving this model. The numerical results obtained by the suggested method are compared with the results obtained by other previous methods. These results indicate that this method agrees with other previous methods.

A group G is said to be n-abelian, if (xy)n=xnyn, for any x,y ∈in G and a positive integer n. In 1979, Fay and Waals introduced the n-potent and the n-center subgroups of a group G, denoted by Gn and Zn(G), respectively. In this paper, we show that the index of the n-center is bounded by an order power of the n-potent subgroup, for some classes of groups. In fact for all n-abelian groups G with finite n-potent subgroup, we prove that if G/Zn(G) is finitely generated, then [G : Zn(G)] ≤ |Gn|d(G/Zn(G)). Moreover, we conclude that [G : Zn(G)] ≤ |Gn|2log2|Gn|, for some n-capable group G.

In this study, we present the existence of solutions for Urysohn integral equations. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Petryshyn's fixed point theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a numerical approach based on Haar wavelets to solve the equation. This numerical method does not lead to a nonlinear algebraic equations system. Conducting numerical experiments confirm the theoretical results of the applied method and endorse the accuracy of the method.

The Newell-Whitehead-Segel (NWS) equation is an important model arising in fluid mechanics. Various researchers worked on approximate solutions to this model by using different methods. In this paper, the Sine-Cosine wavelets method is applied for solving numerically the NWS equation. The Sine-Cosine wavelet operational matrix of integration is obtained and used to transform the equations into a system of algebraic equations. To demonstrate the effectiveness and applicability of this method, two numerical examples are included.