Let $R[\pi, \sigma, \rho]$ be a crossed group ring. An induction theorem is proved for the functor $G_{0}^{R}(R[\pi, \sigma, \rho])$ and the Swan-Gersten higher algebraic $K$-functors $K_{i}(R[\pi,\sigma,\rho])$. Using this result, a theorem on reduction is proved for the discrete normalization ring $R$ with the field of quotients $K$: I f $P$ and $Q$ are finitely generated $R[\pi, \sigma, \rho]$-projective modules and $K\bigotimes_{R}P\simeq K\bigotimes_{R}Q$ as $K[\pi, \sigma, \rho]$-modules, then $P\simeq Q.$ Under some restrictions on $n=(\pi:1)$ it is shown that finitely generated $R[\pi,\sigma,\rho]$-projective modules are decomposed into the direct sum of left ideals of the ring $R[\pi,\sigma,\rho]$. More stronger results are proved when $\sigma=id$.

The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.

In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. The structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have been studied by using decomposition method. By defining a Gray map from $F_{q}+vF_{q}+v^2F_{q}$ to $F_{q}^3$, it has been proved that the Gray image of a skew cyclic code of length $n$ over $F_{q}+vF_{q}+v^2F_{q}$ is a skew $3$-quasi cyclic code of length $3n$ over $F_{q}$. Further, it is shown that the skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ are principally generated. Finally, the idempotent generators of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have also been studied.

The main idea of this paper is to introduce the notion of a Schreier 2-category and of a crossed semimodule over categories and to prove the categorical equivalence between their categories.

In this present paper, we have generalized a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability of non-decreasing sequences to $|A, p_{n}|_{k}$ summability method by using almost increasing sequences and taking normal matrices in place of weighted mean matrices.

The aim of this paper is to discuss the uniqueness of p-adic difference monomials $f^{n}f(z+c)$. The results obtained in this paper are the p-adic analogues and supplements of the theorems given by Qi, Yang and Liu [Uniqueness and periodicity of meromorphic functions concerning the difference operator, Comput. Math. Appl. 60(2010), 1739-1746], Wang, Han and Wen [Uniqueness theorems on difference monomials of entire functions, Abstract Appl. Anal. 2012(2012), Article ID 407351], Yang and Hua [Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22(1997), 395-406].

In this paper, we modify the Cauchy additive functional equation and find all solutions of this new functional equation. Then, we study generalized Ulam-Hyers stability of such functional equation in various Banach spaces via Hyers' method.

In this current work, we acquire the general solution and Hyers-Ulam stability for a new form of Quattuorvigintic functional equation in Matrix Paranormed Space by using the direct and fixed point methods.

In the paper, we define Laguerre-based Hermite-Bernoulli polynomial with its generating function, and investigate certain properties. From this generating function, we derive summation formulas and related bilateral series associated with the newly introduced generating function. Some of whose special cases are also presented. Relevant connections of some results presented here with those involving simpler known partly unilateral and partly bilateral representations are also obtained.

A new sinc-Gauss-Jacobi collocation method for solving the fractional Volterra's population growth model in a closed system is proposed. This model is a nonlinear fractional Volterra integro-differential equation where the integral term represents the effects of toxin. The fractional derivative is considered in the Liouville-Caputo sense. In the proposed method, we first convert fractional Volterra's population model to an equivalent nonlinear fractional differential equation, and then the resulting problem is solved using collocation method. The proposed collocation technique is based on sinc functions and Gauss-Jacobi quadrature rule. In this approach, the problem is reduced to a set of algebraic equations. The obtained numerical results of the present method are compared with some well-known results in the literature to show the applicability and efficiency of the proposed method.

In this paper we present Perov type fixed point theorems for contractive mappings in Gheorghiu’s sense on spaces endowed with a family of vector valued pseudo-metrics. Applications to systems of integral equations are given to illustrate the theory. The examples also prove the advantage of using vector valued pseudo-metrics and matrices that are convergent to zero, for the study of systems of equations.

In this paper we introduce and investigate two new subclasses of the function class $\Sigma$ biunivalent functions in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-MacLaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses by using Chebyshev polynomials. Several new consequences of these results are also pointed out.