The study of approximation theory and the asymptotic behavior of random variables are conventionally predicated on the assumption of classical convergence. Nevertheless, the attainment of classical convergence to a unique limit is frequently impeded in various physical and stochastic processes by measurement errors or inherent system roughness. To mitigate this issue, we introduce the concept of rough asymptotically deferred weighted statistical equivalence of order α in probability. This novel structure generalizes classical asymptotic equivalence through the incorporation of a roughness degree r. We further define the notion of minimal roughness degree and scrutinize the algebraic properties of this new relation such as convexity. Moreover, we establish a rough Korovkin type approximation theorem for sequences of positive linear operators and provide an estimate regarding the rate of convergence. The manuscript concludes by presenting a numerical simulation to visualize our findings which serves to demonstrate strictly stronger generalizations of existing theories.
Identifying codes were introduced by Karpovsky et al. as dominating sets S⊆ V(G) satisfying N[u]∩ S ≠ N[v]∩ S for any distinct vertices u,v. Later, Junnila et al. introduced the concept of self-identifying codes (previously called (1,≤1)^+-identifying codes in earlier work), a dominating set S⊆ V(G) such that ⋂_c∈ N[u]∩ S N[c] = {u} for every vertex u. In this paper, we obtain bounds on the minimum size of a self-identifying code in the direct products K_m× P_n and K_m× C_n that are linear in n with coefficients depending on m, and these bounds are asymptotically tight. In particular, for K_m× P_n with m,n≥3, our bounds closely approaches the size of an identifying code in the same graph, as determined by Shinde and Waphare.
Let G be a graph with vertex V and edge set E. In this paper, we investigate the hyperbolic Sombor (HSO) index, a recently introduced degree-based topological index, defined as X HSO(G) = uv is an element of E root d2u + d2v min{du, dv}, where du denotes the degree of a vertex u is an element of V. We establish several bounds for this index in terms of fundamental graph parameters and classical topological indices. As a chemical application, we perform a correlation analysis to examine the relationship between the HSO index and the physicochemical properties of heptane and hexane isomers.
A random function on n is any function f: [n] -> [n], where [n] = {1, 2,3,. . . , n}. We study water capacity of such random functions. A bivariate generating function for water capacity of individual columns is obtained for all n and the total capacity of random functions of n is obtained.
In this paper, a new mode of convergence for sequences, called Drift-Controlled Local Cesa`ro convergence (DLC-convergence for short), is introduced. This notion requires local Cesa`ro averages taken over blocks whose lengths tend to infinity, but remain small compared to the global index, and whose drift between successive blocks is bounded. Basic properties of this convergence are established, comparisons with classical and Cesa`ro convergence are provided, counterexamples separating these notions are given, stability under Toeplitz operators is proved, and a Tauberian-type theorem is presented.
Incomplete binomial sums are sums taken over a range less than their complete coefficient range. In this paper, two incomplete analogues of the following well-known identity for the sum of the reciprocals of the binomial coefficients are deduced: n X k =0 1 n+ 1 (n ) k 2n+1 n X k =1 +1 2k k . Combining the obtained identities and some previously published results, an elegant combinatorial identity is also derived.
Let G be a simple undirected graph with vertex set V(G) = { v 0 , v1,. . . , v n-1 } . Let d i be the degree of the vertex v i in G for i = 0 , 1 , ... , n-1 . The Zagreb matrix of G is the square matrix of order n whose ( i, j)-entry is equal to d i + d j if the vertices v i and v j are adjacent, and 0 otherwise. The Zagreb spectral radius of G is the largest eigenvalue of the Zagreb matrix of G. In [P. Das, K. C. Das, S. Mondal, A. Pal, First zagreb spectral radius of unicyclic graphs and trees, J. Comb. Optim. 48 specialIntscript #5], extremal problems concerning the Zagreb spectral radius of trees were investigated. In this paper, we determine the quasi-tree graphs with the first three largest Zagreb spectral radii.
This paper investigates the categorical properties of direct limits in the context of canonical hypergroups. Fundamental results are established concerning the construction and the preservation of exactness for direct limits of directed systems of hypergroups. The main contributions are as follows: (1) a rigorous proof of the universal property of direct limits in the category of canonical hypergroups, demonstrating their existence via an explicit quotient construction; (2) a theorem showing that exact sequences of directed systems induce exact sequences at the limit level; and (3) the development of essential commutative diagram techniques for hypergroup homomorphisms in directed systems. These results extend classical algebraic constructions to the hypergroup setting, where the multivalued nature of operations requires a careful treatment of equivalence classes and compatibility conditions.
In this paper, two versions of even facial colorings of plane graphs are introduced. In the weak version, each face has at least one color that occurs an even number of times on its boundary. In the strong version, no color occurs an odd number of times on the boundary of any face. Both vertex and edge colorings are studied, and the existence of such colorings is investigated under the constraint that two colors are used.
The literature contains a wide variety of Hardy-Hilbert-type integral inequalities. In this article, two new inequalities of this kind are derived in the entire plane, characterized by their exponential-trigonometric structure. The newly obtained inequalities are formulated in a flexible manner, depending on several adjustable parameters.
In this paper, a class of phi h-s-convex functions is extended. Two new integral identities are established. By using these two integral identities and Holder's inequality, some new Hermite-Hadamard type inequalities for extended phi h-s-convex functions are derived.
