This paper introduces a generalization of asymptotic notation using filters, a topological structure. We present key properties of this generalization, including reflexivity, symmetry, and transitivity, supported by illustrative examples. Our research demonstrates that classical asymptotic notations imply their filter-generalized counterparts, but we provide examples showing the converse is not universally true. We also propose a characterization of traditional asymptotic notations using filters, offering a new perspective on these fundamental concepts. Furthermore, we establish relationships between bounded or vanishing sequences and filter-based asymptotic notations, enabling the determination of properties central to this study. This generalization provides a more nuanced framework for analyzing algorithmic complexity, potentially capturing behaviors overlooked by classical notations and opening new avenues for theoretical computer science research.

We give a construction of a new family of Deza graphs of girth at least 5 that possess an arc-transitive group of automorphisms isomorphic to a Suzuki simple group \(Sz(q)\). To study their combinatorical properties, we elaborate some group-theoretic arguments involving classical results on the groups of given type.

This paper discusses the exact controllability of linear and nonlinear impulsive Caputo fractional systems. The exact controllability of a linear impulsive system is studied using the concept of generators and functional analysis. In contrast, the controllability of a nonlinear system is discussed using nonlinear functional analysis. An example is provided in the paper to support the results.

In this article, we introduce and rigorously analyze the concept of difference \(\lambda\)-weak convergence for sequences defined by an Orlicz function. This notion generalizes the classical weak convergence by incorporating a \(\lambda\)-density framework and an Orlicz function, providing a more flexible tool for analyzing convergence behavior in sequence spaces. We systematically investigate the algebraic and topological properties of these newly defined sequence spaces, establishing that they form linear and normed spaces under suitable conditions. Our results include demonstrating the convexity of these spaces and identifying several important inclusion relationships among them, such as strict inclusions between spaces involving different orders of difference operators and Orlicz functions satisfying the \(\Delta_{2}\)-condition.

In this paper, we establish a result on the Hyers–Ulam–Rassias stability of the Euler–Lagrange functional equation. The work presented here is in the framework of modular spaces. We obtain our results by applying a fixed point theorem. Moreover, we do not use the \(\Delta_\alpha\)-condition of modular spaces in the proofs of our theorems, which introduces additional complications in establishing stability. We also provide some corollaries and an illustrative example. Apart from its main objective of obtaining a stability result, the present paper also demonstrates how fixed point methods are applicable in modular spaces.

Perfect Italian Domination is a type of vertex domination which can also be viewed as a graph labelling problem. The vertices of a graph \(G\) are labelled by 0, 1 or 2 in such a way that a vertex labelled 0 should have a neighbourhood with exactly two vertices in it labelled 1 each or with exactly one vertex labelled 2. The remaining vertices in the neighbourhood of the vertex labelled 0 should be all 0's. The minimum sum of all labels of the graph G satisfying these conditions is called its Perfect Italian domination number. We study the behaviour of graph complements and how the Perfect Italian Domination number varies between a graph and its complement. The Nordhaus–Gaddum type inequalities in the Perfect Italian Domination number are also discussed.

A graph \(G\) is splittable if its set of vertices can be represented as the union of a clique and a coclique. We will call a graph \(H\) a splittable ancestor of a graph \(G\) if the graph \(G\) is reducible to the graph \(H\) using some sequential lifting rotations of edges and \(H\) is a splittable graph. A splittable \(r\)-ancestor of \(G\) we will call its splittable ancestor whose Durfey rank is \(r\). Let us set \(s = ({1}/{2}) (\mathrm{sum}\,\mathrm{tl}(\lambda) - \mathrm{sum}\,\mathrm{hd}(\lambda))\), where \(\mathrm{hd}(\lambda)\) and \(\mathrm{tl}(\lambda)\) are the head and the tail of a partition \(\lambda\). The main goal of this work is to prove that any graph \(G\) of Durfey rank \(r\) is reducible by \(s\) successive lifting rotations of edges to a splittable \(r\)-ancestor \(H\) and \(s\) is the smallest non-negative integer with this property. Note that the degree partition \(\mathrm{dpt}(G)\) of the graph \(G\) can be obtained from the degree partition \(\mathrm{dpt}(H)\) of the splittable \(r\)-ancestor \(H\) using a sequence of \(s\) elementary transformations of the first type. The obtained results provide new opportunities for investigating the set of all realizations of a given graphical partition using splittable graphs.

For a finite poset (partially ordered set) \(U\) and a natural number \(n\), let \(S(U,n)\) denote the largest number of pairwise unrelated copies of \(U\) in the powerset lattice (AKA subset lattice) of an \(n\)-element set. If \(U\) is the singleton poset, then \(S(U,n)\) was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A.P. Dove, J.R. Griggs, G.O.H. Katona, D.J. Stahl, and W.T.Jr. Trotter. We determine \(S(U,n)\) for all finite posets with 0 and 1, and we give reasonable estimates for the "\(V\)-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice \(L\), let \(G_{\min}L\) denote the minimum size of generating sets of \(L\). We prove that if \(U\) is the poset of the join-irreducible elements of a finite distributive lattice \(D\), then the function \(k\mapsto G_{\min}{D^k}\) is the left adjoint of the function \(n\mapsto S(U,n)\). This allows us to determine \(G_{\min}{D^k}\) in many cases. E.g., for a 5-element distributive lattice \(D\), \(G_{\min}{D^{2023}}=18\) if \(D\) is a chain and \(G_{\min}{D^{2023}}=15\) otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.

Here we consider a certain transfert operator \(\mathrm{M}_{(c,\omega)}=I_{\mathcal{P}}-c \, \tau_{\omega},\) \(\omega\neq0,\) \({c \in \mathbb{R}-\{0,1\},}\) and we prove the following statement: up to an affine transformation, the only orthogonal sequence that remains orthogonal after application of this transfert operator is the Meixner polynomials of the first kind.

We investigate abelian (in the sense of Godsil and Hensel) distance-regular covers of complete graphs with the following property: there is a vertex-transitive group of automorphisms of the cover which possesses at most two orbits in the induced action on its arc set. We focus on covers whose parameters belong to some known infinite series of feasible parameters. We also complete the classification of arc-transitive covers with a non-solvable automorphism group and show that the automorphism group of any unknown edge-transitive cover induces a one-dimensional affine permutation group on the set of its antipodal classes.