Open Journal of Mathematical Sciences (OMS)

Let \(G\) be a finite group. The product order relatively prime graph (PORP graph) \(\Gamma_{porp}(G)\) is the graph with vertex set \(V(\Gamma_{porp}(G))=G\) and two distinct vertices \(a\) and \(b\) are adjacent if \((o(a),o(ab))=1\) or \((o(b),o(ab))=1\). In this paper, we investigate the product order relatively prime graph associated with the dihedral group \(D_n\). We study the graph for two important classes of dihedral groups, namely \(D_{p^\alpha}\), where \(p\) is a prime and \(\alpha\ge1\), and \(D_{pq}\), where \(p\) and \(q\) are distinct odd primes. We determine the graph structure and establish several structural properties, including the degree sequence, total number of edges, independence number, clique number and structural decomposition. Furthermore, we investigate graph-theoretic properties such as planarity, bipartiteness, Eulericity, Hamiltonicity, and characterize the conditions under which the graph is complete.

A fractional-order predator–prey model (FOPPM) based on the Caputo derivative of order \(\alpha\) (\(0 < \alpha \le 1\)) is developed to examine the joint influence of a strong Allee effect in the prey and a nonlinear anti-predator feedback acting on the predator. The formulation couples cubic prey growth and bilinear predation with a saturating feedback loss, thereby representing threshold-dependent recovery, trophic conversion, and density-dependent behavioural regulation. The system admits three boundary equilibria, while coexistence equilibria are determined by a corrected cubic algebraic equation. Its discriminant classifies one- and three-real-root regimes, but ecological feasibility further requires \(A<X^*<K\), \(Y^*>0\), and \(kmX^*-d>0\). Nullcline geometry shows that at most two roots can be biologically feasible; the actual number is therefore established by direct root filtering for each parameter set. Local stability is characterised through the Jacobian \(\mathcal{J}(E^*)\) and the classical eigenvalue test for integer-order dynamics, while the fractional-order model requires satisfaction of Matignon’s criterion \(|\arg(\lambda_i)|>\frac{\alpha\pi}{2}, \; i=1,2.\) The coexistence equilibrium may switch between stability and instability under variations in carrying capacity \(K\), Allee threshold \(A\), interaction rate \(m\), or feedback intensity \(\varepsilon\). Because the Jacobian eigenvalues do not depend on \(\alpha\), fractional memory does not rotate them; instead, decreasing \(\alpha\) widens the admissible Matignon sector and changes the rate and shape of transient relaxation. A weighted population estimate establishes positivity, uniform boundedness, and global existence. Since the extinction equilibrium is always locally stable under a strong Allee effect, coexistence cannot be globally attractive on the entire positive quadrant. A Volterra-type logarithmic functional is therefore formulated as a basin-restricted Mittag–Leffler certificate on positively invariant sets separated from the coordinate axes. The theoretical results are tested with a fractional Adams–Bashforth–Moulton predictor–corrector (ABM–PC) scheme. A grid-refinement study gives observed orders \(1.91\), \(1.90\), and \(1.96\) for \(\alpha=0.9\), consistent with the expected order \(\min\{2,1+\alpha\}\). An independent \(\alpha=1\) comparison with a high-accuracy Runge–Kutta solution gives maximum componentwise errors below \(3.5\times10^{-4}\). Parameter sweeps across \((K,A,m,\varepsilon,\alpha)\) then quantify changes in feasibility, stability, and transient persistence. For \(\alpha=1\), a trace-zero crossing with positive determinant and transversality is the classical Hopf candidate. For \(0<\alpha<1\), crossing the Matignon boundary is instead a fractional stability-sector transition; it is not, by itself, a Hopf bifurcation. Moreover, autonomous Caputo systems of non-integer order do not possess exact nonconstant periodic solutions, so the computed oscillations are interpreted as memory-dependent transients rather than limit cycles. Overall, the FOPPM provides a coherent framework for studying memory, Allee thresholds, and behavioural feedback within one model. Corrected equilibrium algebra, explicit feasibility filtering, fractional stability theory, reproducible convergence checks, and regenerated simulations together distinguish true asymptotic conclusions from finite-time memory effects.

We introduce Lipschitz classes \(\mbox{Lip}^M(\eta,2)\) and \(\mbox{Lip}^M(\eta,\infty)\) of functions associated with the canonical Sturm–Liouville operator \(\mathscr{L}^M\); and we prove a new versions of Titchmarsh and Boas-type theorems for the canonical Sturm–Liouville transform \(\mathscr{F}^M\). An application to the canonical Sturm–Liouville multipliers is given. Titchmarsh and Boas-type theorems for the canonical Fourier–Bessel transform and the canonical Fourier–Jacobi transform are special cases of this work.

We study convergence of sequences of nonempty closed sets in a metric space through the power-type Musielak–Orlicz modular \(\Phi_n(t)=t^{p_n}\). We introduce Wijsman \((p_n)\)-statistical convergence, Wijsman \((p_n)\)-strong Cesàro convergence, and a lacunary block-mean statistical version associated with a lacunary sequence \(\theta=(k_r)\). In the bounded-exponent case \(1\le p_n\le p_+<\infty\), we prove that, for Wijsman bounded sequences, these three modes of convergence are equivalent whenever the lacunary moderate growth condition holds. In the power-type setting this condition is equivalent to \(h_r/k_r\to 0\). We also give comparison results and examples showing why boundedness of the exponent sequence is essential. For random closed sets, we prove a pathwise Cesàro transfer result: at a fixed base point, almost sure modular strong Cesàro control, together with almost sure boundedness of the distance evaluations and bounded exponents, yields almost sure convergence of the Cesàro averages of the distance functionals to the Wijsman limit.

This study investigates a class of number-theoretic problems related to representations of large integers as combinations of prime numbers and quartic terms, extending ideas originating from the Estermann problem. The authors focus on a refined ternary decomposition involving almost equal components, employing analytic methods from additive number theory. Central to the analysis are estimates of trigonometric sums of Weyl type and their applications to bounding error terms in asymptotic formulas. By developing sharper bounds for short exponential sums, the paper establishes a new asymptotic formula for the fourth-degree analogue under constrained conditions. To bridge theory and computation, we further introduce a numerical framework based on Gegenbauer polynomial expansions, which efficiently approximates the associated singular series and exponential integrals with spectral accuracy. The proposed algorithm leverages discrete orthogonality and Gaussian quadrature to achieve high-precision validation of the asymptotic predictions. The obtained results improve previously known bounds, provide rigorous numerical verification, and contribute to a deeper understanding of additive structures involving primes and polynomial terms. These findings have potential implications for related problems in analytic number theory and the development of spectral methods for arithmetic functions.

In this paper, we present a numerical solution to the fractional differential equation governing water pollution. The Caputo derivative will be used to model water pollution. For the numerical solution, we propose a fractional numerical scheme and use it to generate figures with mathematical software. We analyze qualitative properties of the determination of equilibrium points and examine their stability. We have analyzed the influence of the order of the fractional derivative on modeling pollution problems.

Consider a near-ring \(\Omega\) with a semigroup ideal \(\mathscr{B}\) and a Jordan ideal \(\mathscr{K}\). We analyze the structural properties of prime near-rings through a nonzero derivation \(\Lambda:\Omega \to \Omega\) by imposing suitable conditions to specific subsets. The primary objective is to establish results that lead to significant structural constraints and ensuring the existence of a nonzero derivation, induces commutativity of \(\Omega\). Our results extend prior findings in the literature by providing new insights into the structural properties of near-rings. Additionally, we include an illustrative example to validate our theoretical claims.

In this article, we investigate the boundedness and qualitative behavior of a new class of generalized Hardy-Hilbert iterated integral operators. By introducing a non-separable kernel characterized by a continuous, strictly convex homogeneous denominator \(\Phi(t,\tau)\) of degree \(\gamma > 1\) and a rational boundary profile, we establish rigorous two-sided bounds within weighted Lebesgue spaces. While typical approaches in the literature focus on one-sided inequalities under rigid symmetry assumptions, our analytical framework addresses mass dissipation near singular boundaries by restricting the operator core to a compact truncated conical sector \(\Omega_c\). We prove an upper bound criterion under explicit weight compatibility conditions and derive a non-trivial positive lower bound proportional to the local \(L_p\)-norm of the input function restricted to the active domain of \(\Omega_c\). Furthermore, we provide illustrative examples and correct previous formulations regarding metric transformations.

The Stochastic Vehicle Routing Problem (VRPSD) has been extensively studied under the assumption that all customer demands are non-negative random variables. In many real-world settings, however, demands may be signed: customers simultaneously act as sources (pickup, increasing vehicle load) and sinks (delivery, decreasing vehicle load), or excess inventory is returned to the depot. This paper introduces the Stochastic Vehicle Routing Problem with Mixed and Signed Demands (SVRP-MD), in which each customer \(i\) has a real-valued random demand \(\xi_i \in \mathbb{R}\), modelled as a net change in an abstract vehicle load-balance process under a reactive, signed detour-to-depot (DTD) recourse policy in which a violating visit triggers a depot round trip before service and the vehicle resumes at the same customer. Our contributions are: (i) a counterexample proving that the classical superadditivity of the recourse function fails when demands are signed; (ii) an incomparability result showing that the positive-part recourse neither upper- nor lower-bounds the signed recourse under the two-sided failure rule, together with a valid lower bound based on the absolute net demand that underlies our cut inequalities; (iii) a family of signed net-demand (SND) incumbent bounds, embedded in valid integer L-shaped optimality cuts in the extended \((x, \theta)\) space; (iv) a Jensen-type lower bound using a convex lower approximation of the recourse cost; and (v) a branch-and-cut sample-average bounding procedure with a Monte Carlo, confidence-based acceptance rule. Preliminary experiments suggest the SND cuts reduce the LP relaxation gap; the reported figures predate the protocol and bound corrections and are being regenerated.