Open Journal of Mathematical Sciences (OMS)

This article considers a second-order difference equation with constant coefficients in its standard form and two different classes of two-point homogeneous boundary conditions. First, we construct the corresponding Green functions and derive some important properties for further analysis. Next, we propose adequate conditions for the existence of solutions to the considered boundary value problems. Finally, we offer two examples to show the applicability of the main results.

In this paper, we establish several new arctangent- and logarithmic-Hardy-Hilbert integral inequalities. The approach combines fundamental principles with refined techniques from the theory of integral inequalities, leading to a range of original results. Complete proofs are presented together with a discussion of their sharpness and potential applications.

This note introduces a \(1\)-parameter of cubic curves naturally associated to the sphere \(S^4\) considered in the unique \(5\)-dimensional irreducible representation space of \(SO(3)\). Eight examples are discussed with the last two being elliptic curves. Also, two conics are defined naturally in our setting by a special basis of the Lie algebra \(sl(3, \mathbb{R})\).

This paper introduces a unified framework for fixed point theorems involving asymptotically regular mappings in \(b\)-metric spaces through the concept of contractive families. We establish a general fixed point theorem that encompasses various existing results, including those of Kannan-type and generalized contractive conditions, as special cases. In particular, we demonstrate that the recent results of Nagac and Tas [1] emerge naturally as special cases of our main theorem through appropriate parameter choices. The main result employs coefficient functions and a general auxiliary function with strengthened continuity conditions, providing flexibility that allows the derivation of numerous particular cases. Several corollaries with complete proofs are presented to demonstrate that our results properly generalize and extend well-known theorems in the literature.

This paper presents a thorough investigation of Laplace transforms for a wide range of probability distributions, including both classical and generalized forms. Analytical derivations are complemented with extensive numerical examples and graphical simulations to validate the theoretical results. Applications in reliability theory and system availability are discussed, highlighting the practical significance of the findings for modeling and performance evaluation of complex systems.

Pre-exposure prophylaxis (PrEP) is an effective strategy for HIV prevention, offering individual protection and broader public health benefits. Enrollment in PrEP programs not only provides access to HIV prevention but also serves as a strategic entry point for the diagnosis of other sexually transmitted infections (STIs). Because participation requires regular HIV testing and routine STI screening (e.g., for syphilis), PrEP implementation facilitates early detection and treatment of coexisting infections, strengthening integrated sexual health surveillance and control efforts. We developed a mathematical model capturing syphilis dynamics, incorporating PrEP as a mechanism for diagnosis and treatment engagement. The model considers coinfection via high-risk sexual contact, partial protection of PrEP (HIV but not syphilis), and diagnostic pathways linked to PrEP program entry. Independent analysis of syphilis (without PrEP) established population persistence, basic reproduction numbers, and stability of disease-free and endemic equilibria. Integrating PrEP, we derived conditions under which PrEP-related parameters—particularly diagnostic access—positively influence syphilis transmission dynamics. Sensitivity analysis showed that higher PrEP adherence reduces reproduction numbers for syphilis and coinfection. Computational simulations using literature-based parameters confirmed these findings: increased PrEP use and lower discontinuation rates decreased new infections and improved treatment outcomes. These results highlight the role of PrEP in improving the detection and treatment of syphilis and HIV–syphilis coinfection.

In recent decades, a wide range of Hardy-Hilbert-type integral inequalities have been established. This article focuses on a one-parameter result introduced by Waadallah Tawfeeq Sulaiman in 2010, which has a unique structure: the double integral involves a power-sum of the variables, as well as a technical power-minimum. The sharp constant factor is also elegantly expressed in terms of the beta function. However, the parameter involved is subject to restrictions on its values. In this article, we refine the inequality by removing this restriction and addressing a theoretical gap in the original proof to yield a sharper result. We provide a thorough, step-by-step proof and demonstrate how this new result can be used to derive additional variants and extensions.

We study the Abel-type family \(y’=C\,y^r(1-y)^s\) under a parity-driven mapping of \((r,s)\), which yields symmetric dynamics for odd \(k\) and asymmetric, potentially stiff dynamics for even \(k\). We correct the normalization by peaking at the true maximizer \(y^\star=r/(r+s)\) and provide the analytic Jacobian \(g'(y)\) for implicit solvers. A matched-accuracy benchmarking protocol sweeps rtol/atol and reports global errors against ultra-tight references (separable/explicit for odd \(k\), Radau for even \(k\)), alongside wall time, \(nfev\), \(njev\), linear-solve counts, rejected steps, and step-size histories. Stiffness is quantified through the proxy \(\tau(t)=1/\lvert g'(y(t))\rvert\) and correlated with step-size adaptation; trajectories are constrained to \(y\in[0,1]\) via terminal events. Across tolerances, DOP853 and LSODA are strong all-rounders in non-stiff regimes, while Radau/BDF dominate when asymmetry and proximity to multiple roots induce stiffness; observed orders align with nominal ones under matched error. The study clarifies how parity and nonlinearity govern solver efficiency for polynomial nonlinearities and provides full environment details and code for reproducibility.

Let \(G\) be a graph of order \(n\) and size \(m\), with adjacency matrix eigenvalues \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n\). The energy of \(G\), denoted by \(\mathcal{E}(G)\), is defined as the sum of the absolute values of its eigenvalues. A classical upper bound on the energy, originally established by McClelland [1], states that \(\mathcal{E}(G) \leq \sqrt{2mn}\,.\) In this paper, we refine the spectral analysis of graph energy by deriving an exact analytical expression relating \(\mathcal{E}(G)\) to the variance of the vector of absolute eigenvalues \(x = (|\lambda_1|, |\lambda_2|, \dots, |\lambda_n|)\,.\) Specifically, we prove that \(\mathcal{E}(G) = \sqrt{2mn – n^2 \operatorname{Var}(x)},\) providing a more precise and quantitative spectral characterization of graph energy. As an application, this identity allows us to derive improved lower bounds for \(\mathcal{E}(G)\), thereby strengthening and generalizing previously known inequalities. Furthermore we conjecture that for any non-singular graph \(G\) of order \(n\), \(\mathcal{E}(G) \geq 2 \sqrt{\langle d \rangle (n-1)},\) where \(\langle d \rangle = 2m/n\) is the average vertex degree of \(G\). Equality holds if and only if \(G \cong K_n\).

First, this paper provides some approximation and estimation type results for some moments of the Gauss function, motivated by the fact that the moments of even orders \(n=2l,\ l\in \mathbb{N}\mathrm{=}\mathrm{\{}0,1,\dots \}\) of the function \(exp\left(-t^2\right)\) on bounded intervals . Second, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subseteq \left[0,{1}/{\sqrt{2}}\right]\) is studied and solved. Here the point is using Jensen inequality. Third, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subset \left[0,+\infty \right)\) is deduced, via elements of complex analysis (Vitali’s theorem). The convergence holds uniformly on compact subsets of the complex plane. Fourth, the asymptotic behavior of the sequence of all moments on \(\left[0,1\right],\ \)as \(n\to \infty ,\) for an arbitrary function \(f\in C\left(\left[0,1\right]\right)\) is determined precisely, by means of Korovkin’s approximation theorem. Consequently, a similar result for complex analytic functions is deduced, using Vitali’s theorem. This is the fifth aim of the paper.