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The concept of statistical order convergence of sequences in Riesz spaces was introduced and studied. In the present paper, we define the statistical order limit points of a sequence (xn ) as a vector x that is the order limit of a subsequence (xk ) k∈K of (xn ) such that the set K does not have density zero. Moreover, we introduce the statistical order cluster points of sequences in Riesz spaces, and also, we give some relations between them.

This paper explores the concepts of J-lacunary statistical limit points, J-lacunary statistical cluster points, and J-lacunary statistical Cauchy multiset sequences. Building upon previous work in the field, we investigate the relationships between J-lacunary statistical convergence and J*-lacunary statistical convergence in multiset sequences. The findings contribute to a deeper understanding of the convergence behaviour of multiset sequences and provide new insights into the application of ideal convergence in this context.

The main object of this paper is to study the concept of weak $I^K$-convergence, a generalization of weak $I^*$-convergence of sequences in a normed space, introducing the idea of weak* $I^K$-convergence of sequences of functionals where $I,K$ are two ideals on $\\mathbb{N}$, the set of all positive integers. Also we have studied the ideas of weak $I^K$ and weak* $I^K$-limit points to investigate the properties in the same space.

The principal aim of this paper is to present a new proof of Weyl's criterion in which it is shown that the natural framework for the associated Sturm-Liouville operators is W 2,1 ∩ L 2 -i.e.- the intersection of a particular Sobolev space and of the L2 space. Indeed, we will deal with the special case of the radial operator − d 2 dx 2 + q ( x ) on a real line segment (either bounded or unbounded) that often occurs in the study of quantum systems in central potentials. We also derive from first principles the functional behaviour of the coefficients for a general second-order Sturm-Liouville operator by using some extensions of a milestone Carathéodory existence theorem.

This paper is concerned with the defect index of singular symmetric linear difference equations of order 2n2n with real coefficients and one singular endpoint. We show that their defect index dd satisfies the inequalities n≤d≤2nn\\leq d \\leq 2n and that all values of dd in this range are realized. This parallels the well known result of Glazman for differential equations established about 1950. In addition, several criteria of the limit point and strong limit point cases are established.

In this study, we investigate the topological positions of points relative to sets in various topologies induced by raw binary operations. The fact that the raw binary operation is weaker than both partial and multivalued ones allows for a relatively wider variety of topologies induced by it. This gives us the ability to determine the topological positions of a point in a raw binary structure relative to a set by considering the lo-topology, ro-topology, and o-topology induced by the raw binary operation. Through the analysis of topological positions such as interior points, limit points, and closure points, we gain a deeper understanding of the nature of raw binary operations from a topological perspective. The potential benefit of this work is to expand the conceptual framework in terms of topological evaluability of problems in algebraic structures. In this study, we introduce the concepts of raw-binary interior point, raw-binary limit point, raw-binary closure point and give some useful facts about finding raw-binary interior, raw-binary limit and raw-binary closure points of a given set in lo-topology, ro-topology and o-topology. The potential benefit of this study is to provide topological evaluability of problems in algebraic structures by extending the conceptual framework.

The main idea of this research is to define a new neutrosophic crisp points in neutrosophic crisp topological space namely (NncPN), the concept of Nnc limit point was defined using (NncPN), with some of its properties, the separation axioms (Nnceτi-space (i = 0,1, 2) were constructed in neutrosophic crisp topological space using (NncPN) and examine the relationship between them in details.

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results.

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.