Using Hook Schur Functions to Compute Matrix Cocharacters

We develop a new integration method based on hook Schur functions instead of Schur functions to compute the cocharacters of matrices. We then use this method to compute some of the multiplicities in the cocharacter sequence of 3 × 3 matrices.     

On semi‐classical d‐orthogonal polynomials

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

The exponential matrix function and linear differential equations

Monte Carlo optimization has been shown to be useful in solving multivariate optimization problems. In most Monte Carlo simulations, to find the optimal solution of an integer programming problem, the computer gets close to the true optimal solution, but does not find the exact optimal. This can be a problem at times but it can be overcome by multi‐stage Monte Carlo integer programs that find a preliminary nearly optimal solution and use this as a focal point and springboard for finding the true optimal.     

Constructing New Braided T-Categories Over Regular Multiplier Hopf Algebras

Let A be a regular multiplier Hopf algebra, and let Aut(A) denote the set of all isomorphisms α from A to itself that are algebra maps satisfying (Δ ○ α)(a) = (α ? α) ○ Δ(a) for all a ∈ A. Let G be a certain crossed product group Aut(A) × Aut(A). The main purpose of this article is to provide a class of new braided T-categories in the sense of Turaev [\citealp9]. For this, we introduce a class of new categories _A 𝒴𝒟 ^A (α, β) of (α, β)-Yetter–Drinfel'd modules with α, β ∈Aut(A), and we show that the category ?𝒴𝒟(A) = { _A 𝒴𝒟 ^A (α, β)}_{(α, β)∈G} becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic [6 Panaite , F. , Staic , M. D. ( 2007 ). Generalized (anti) Yetter–Drinfel'd modules as components of a braided T-category . Israel J. Math. 158 : 349 – 365 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

Abstract

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.