Let \(G = (V(G), E(G))\) be a graph with minimum degree at least \(1\). The inverse degree of \(G\), denoted \(Id(G)\), is defined as the sum of the reciprocals of degrees of all vertices in \(G\). In this note, we present inverse degree conditions for Hamiltonian and traceable graphs.

Let \(\Lambda = (\lambda_n)\) be an increasing sequence of non-negative numbers tending to \(+\infty\), with \(\lambda_0 = 0\). We denote by \(S(\Lambda, 0)\) a class of Dirichlet series \(F(s) = \sum_{n=0}^{\infty} f_n \exp\{s \lambda_n\}, \quad s = \sigma + it,\) which have an abscissa of absolute convergence \(\sigma_a = 0\). For \(\sigma < 0\), we define \( M_F(\sigma) = \sup \{|F(\sigma + it)| : t \in \mathbb{R}\}. \) The growth of the function \(F \in S(\Lambda, 0)\) is analyzed in relation to the function \( G(s) = \sum_{n=0}^{\infty} g_n \exp\{s \lambda_n\} \in S(\Lambda, 0), \) via the growth of the function \(1/|M^{-1}_G(M_F(\sigma))|\) as \(\sigma \uparrow 0\). We investigate the connection between this growth and the behavior of the coefficients \(f_n\) and \(g_n\) in terms of generalized orders.

The present paper provides a direct proof of stability of nontrivial nonnegative weak solution for fractional \(p\)-Laplacian problem under concave nonlinearity condition. The main results of this work are extend the previously known results for the fractional Laplacian problem.

In this article we provide classes of hyperbolic chains of inequalities depending on a certain parameter \(n\). New refinements as well as new results are offered. Some graphical analyses support the theoretical results.

In this article, we investigate the power convexity of two generalized forms of the invariant of the contra harmonic mean with respect to the geometric mean, and establish several inequalities involving bivariate power mean as applications. Some open problems related to the Schur power convexity and concavity are also given.

The purpose of this paper is to contribute to the development of the multidual Gamma function. For this aim, we start by defining the multidual Gamma and we propose a multidual analysis technics of in order to show a result regarding real Gamma function.

The study of innovative sequences and series is important in several fields. In this article, we examine the convergence properties of a particular product series that offers adaptability through two parameters and two functions. Based on this analysis, we extend our investigation to a related series. Our main theorems are proved in detail and include several new intermediate results that can be used for other convergence analysis purposes. This is particularly the case for a generalized version of the Riemann sum formula. Several precise examples are presented and discussed, including one related to the gamma function.

In this article, we extend the classic Banach contraction principle to a complete metric space equipped with a binary relation. We accomplish this by generalizing several key notions from metric fixed point theory, such as completeness, closedness, continuity, g-continuity, and compatibility, to the relation-theoretic setting. We then use these generalized concepts to prove results on the existence and uniqueness of coincidence points, defined by two mappings acting on a metric space with a binary relation. As a consequence of our main results, we obtain several established metrical coincidence point theorems. We further provide illustrative examples that~demonstrate~the main results.

We provide a semi-local convergence analysis of a seventh order four step method for solving nonlinear problems. Using majorizing sequences and under conditions on the first derivative, we provide sufficient convergence criteria, error bounds on the distances involved and uniqueness. Earlier convergence results have used the eighth derivative not on this method to show convergence. Hence, limiting its applicability.

An experimental study conducted by Ankit Kumar and colleagues (Kumar, Gupta, Pandey, Govil, and Patel, “Status of Arsenic Contamination in District Lakhimpur, Uttar Pradesh, India,” in Emerging Trends in Science, Social Science and Engineering, edited by Aggarwal, Pandey, Naik, Mishra, Raj, Tripathi, and Shukla, pp. 60-73, ISBN 9789358380125, Astitva Prakashan, Bilaspur, Chhattisgarh) has identified significant levels of arsenic contamination in the groundwater of Lakhimpur district, Uttar Pradesh. Their findings indicate that arsenic levels are notably higher in the shallow regions compared to the deeper India Mark II regions across eight selected study sites. Building on these findings, this paper aims to apply a dose-response Hill model to analyze and explain the observed patterns of arsenic contamination in the groundwater resources of Lakhimpur district.