For all positive non-square integer multipliers, there are infinitely many triangular numbers that are multiples of other triangular numbers. With a simple change of variables, one obtains a Pell equation, whose odd solutions provide the indices of the many infinitely triangular numbers multiple of other triangular numbers. General algebraic expressions of fundamental solutions of Pell equations are found for the multiplier expressed in function of the closest integer square. Finally, recurrent relations yielding the triangular numbers and their multiples and indices are calculated for non-square multipliers.

We introduce a class of mixed general bivariational inequalities in a real Hilbert space and show that several known models, including mixed variational inequalities, bivariational inequalities, general variational inequalities, and complementarity problems, arise as special cases. An auxiliary-principle framework is then used to derive predictor–corrector type iterative schemes. A basic descent estimate is established under a g-partially relaxed monotonicity assumption, and a convergence theorem is obtained under natural continuity and uniqueness hypotheses. The presentation has been streamlined to make the algorithmic steps explicit, and a scalar example is included to illustrate the resolvent formulation.

Let \(\eta\) be a fixed positive integer. Let \(S\) be a subset of \(\mathbb{Z}\), \(\star:S\times S\to \mathbb{Z}\) be a binary function, and \(\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}\) be a function. For a simple connected graph \(G\) of order \(n\), a bijective function \(f:V(G)\to S\) (where \(|S|=n\)) is called an arithmetic cordial labeling modulo \(\eta\) under the arithmetic structure \(\langle S,\zeta_\eta,\star\rangle\) if the induced function \(f_\eta^*:E(G)\to \{0,1\}\), defined by \(f_\eta^*(ab)=1\) whenever \(\gcd(f(a)\star f(b),\eta)= 1\) and \(\zeta_\eta(f(a)\star f(b))=1\); otherwise, \(f_\eta^*(ab)=0\), satisfies the condition \(|e_{f_\eta^*}(0)-e_{f_\eta^*}(1)|\leq 1\), where \(e_{f_\eta^*}(i)\) is the number of edges with label \(i\) (\(i=0,1\)). In this paper, we explore the arithmetic cordial labeling of some graphs under conditions imposed on the function \(\zeta_\eta\). The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.

This article proposes a new extension of fixed-point theorems in the context of b-fuzzy metric spaces based on Geraghty-type inequalities. We present the concept of pair upper \((F,h)\)-class functions, which play a key role in establishing existence and uniqueness of fixed points for contractive situations. These outcomes extend and generalize some eminent fixed-point theorems in fuzzy and b-metric spaces. We support our results by proper example.

As urbanization intensifies and traffic demand continues to grow, understanding and modelling vehicular dynamics in complex transportation systems has become increasingly important. Second-order macroscopic traffic flow models provide a powerful framework for capturing the evolution of both traffic density and velocity, offering significant advantages over first-order formulations. Despite extensive developments in this field, the literature lacks a unified mathematical synthesis that systematically organizes and compares the wide range of second-order models. This paper addresses this gap by presenting a comprehensive review of second-order macroscopic traffic flow models, tracing their evolution from foundational formulations in the 1970s to recent advancements up to 2024. The review adopts a structured methodology, drawing on major academic databases to identify models based on their mathematical formulation and practical relevance. The models are classified into key families, including relaxation-based, kinetic, viscous, anisotropic, nonlocal, and multi-class formulations, providing a coherent taxonomy of the field. In addition to cataloguing model equations, this study synthesizes their fundamental mathematical properties, including hyperbolicity, stability, well-posedness, and parameter identifiability. The review further examines how successive models address limitations of earlier approaches, such as non-physical wave propagation and insufficient representation of driver behaviour. Finally, emerging trends are discussed, including the integration of connected and autonomous vehicle technologies, nonlocal interactions, and data-driven modelling approaches.

Classical orthogonal polynomials of the Askey-Wilson scheme have many different properties, e.g. they satisfy differential and recurrence equations and they have hypergeometric representations, Rodrigues formulas, generating functions, moment representations, etc. In this paper we concentrate on finding multiple hypergeometric representations for the polynomial sequences belonging to the classical continuous and classical discrete classes that are defined on a linear lattice. Currently such a database is not available. Using computer algebra, especially Zeilberger’s algorithm, it is possible to prove such identities and therefore the paper is accompanied by a Maple worksheet containing derivations or proofs of all given identities, most of which are new.

We present a unified, set–theoretic framework that extends molecular graphs to hypergraphs and superhypergraphs via iterated power sets. We define Molecular Graphs, Molecular HyperGraphs, and Molecular SuperHyperGraphs, and develop four complements over them: Weighted, Rough, Neural, and Multipolar frameworks. We prove concise inclusion results—most notably, that the Weighted Molecular SuperHyperGraph strictly contains both Weighted Molecular HyperGraphs and (unweighted) Molecular SuperHyperGraphs—while preserving alternating–path distances under canonical embeddings. Compact examples (e.g., methane, ethanol, acetic acid) illustrate how atoms, bonds, functional groups, and higher–order motifs appear as vertices, hyperedges, and superedges under rank constraints. We also provide implementation–agnostic message–passing rules for variable–arity interactions, enabling property prediction and hierarchical analysis in chemistry and chemical biology. This paper is devoted to theoretical analysis, and it is hoped that quantitative studies by domain experts will be developed in future work.