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.
We introduce a degenerate–\(q\) hybrid coefficient series obtained by multiplying a normalised basic-hypergeometric coefficient array by the degenerate rising factorial defined through the Kim–Kim degenerate Gamma function. The normalisation used here deliberately omits the conventional factor \([(-1)^nq^{n(n-1)/2}]^{1+s-r}\) from the fully standard definition of \({}_r\phi_s\); the precise relationship with the standard convention is stated explicitly. Under explicit nonsingularity assumptions on the \(q\)-denominator parameters and the degenerate Gamma factors, we determine the disc of absolute convergence and give a corrected boundary analysis. The coefficient ratio \((\alpha)^*_{\lambda,n+1}/(\alpha)^*_{\lambda,n}\) is encoded as a rational Euler-transport operator, which leads coefficientwise to a hybrid Heine-type \(q\)-difference equation, numerator and denominator contiguity relations, and Jackson \(q\)-derivative identities. In the scalar case we construct two Frobenius-type local series and formulate the connection coefficients only within the corresponding nonresonant Frobenius span. For terminating series we prove genuine degenerate-Gamma moment representations; for nonterminating series the corresponding moment formalism is explicitly interpreted as a meromorphic regularisation rather than as a positive-measure integral. We also state a symmetric Beta-kernel identity with separate complex-analytic and real-positivity hypotheses and record carefully scaled limit transitions.
This paper introduces the \(Z_6\) transformation for the first time as a constructive framework for generating and solving differential equations. Its core idea is to construct a new solvable equation from two known solvable equations through a predefined transformation between the dependent and independent variables. This framework unifies two classic techniques-coordinate transformation and substitution of the dependent variable-into a single, systematic procedure. Furthermore, by iterating this construction process and combining it with the principle of common solutions proposed in this paper, an infinite number of solvable equations can be generated. Using this framework, we have obtained, for the first time, general solutions for a variety of differential equations, including general solutions for the Laplace equation in cylindrical and spherical coordinates, as well as general solutions for several classes of linear and nonlinear second-order partial differential equations with varying coefficients, among others. We also solved two boundary value problems exactly, demonstrating the practicality of this framework. The \(Z_6\) transformation, together with the \(Z_A\) method we previously proposed, provides two complementary approaches for systematically expanding the range of solvable differential equations, offering powerful tools for both theoretical analysis and practical applications.
Abstract: This paper develops a mathematics-first modular framework for completion quotients, Real-commutant parameter spaces, positive-frequency polar selection, temporal spectral triples, and conditional cone readouts for orthogonal Hilbert representations. The opening part states the ontological and operational principles used to organize the construction: generative histories are separated from completed records, observable dynamics is obtained by quotienting through completed-future tests, phase is selected only after a real orthogonal carrier has positive time orientation, and Lorentzian readout is treated as a downstream cone geometry obtained from length-delay inequalities. The theorem-bearing core concerns orthogonal Hilbert representations. For a real representation \(U:G\to O(\mathcal H_{\mathbb R})\), with complexification \(V\) on \(\mathcal H_{\mathbb C}\) and canonical conjugation \(\mathcal C\), the Real commutant
is a real \(C^*\)-algebra canonically isomorphic to the real commutant of \(U\). Symmetry-compatible orthogonal complex structures are precisely the skew-adjoint square roots of \(-\operatorname{Id}\) in this algebra, naturally under real unitary intertwiners. Abelian representations yield partial spectral phase operators, orthogonal flows yield the positive-frequency polar factorization \(A=J_A|A|=|A|J_A\) on the common domain \(\operatorname{Dom}(A)=\operatorname{Dom}(|A|)\) after removing the kernel, and compact groups yield a Peter–Weyl/Frobenius–Schur bounded-product formula in complex, real, and quaternionic multiplicity algebras. Downstream of this phase layer, the phase-balanced real tensor product is identified with the real form of the usual complex Hilbert tensor product, and a temporal shift spectral triple is built from a concrete dense \(*\)-algebra with compact resolvent, bounded generator commutators, exact diagonal Connes distance \(|m-n|/\kappa\) on the diagonal record algebra, and spectral dimension governed by an explicit counting-function hypothesis. The final module proves a conditional cone lemma: once a length budget, completion delay, and Euclidean readout are supplied, completed displacements lie in the finite cone \(\|\Delta x\|\le c_*\Theta\); the standard automorphism statement gives Lorentz similitudes, and a smooth Lorentzian representative requires additional smooth geometric input. The broader physical interpretation motivates the axioms but is not used as a hidden premise.
We study window-weighted Wijsman statistical convergence for sequences of non-empty closed sets in hyperspaces associated with idempotent bicomplex metric spaces. The ambient distance is assumed to have the form \(\rho=d_1e_1+d_2e_2,\) where \(d_1\) and \(d_2\) are ordinary metrics on the same underlying set. Thus the present paper concerns this idempotent, componentwise class of bicomplex-valued metrics, rather than arbitrary bicomplex-valued metric structures. For a closed set \(A\), the point-to-set distance is represented by the two scalar profiles \(d_1(x,A)\) and \(d_2(x,A)\). We define window-weighted Wijsman statistical convergence and the corresponding pre-Cauchy condition by requiring simultaneous control of these two profiles along weighted moving windows. The main results give exact componentwise characterizations, establish invariance under weighted-equivalent changes of the window scheme, and show that a window-weighted Wijsman pre-Cauchy sequence becomes statistically convergent whenever it has an ordinarily Wijsman convergent trace of positive lower window weight. We also prove stability under negligible perturbations and show that ordinary Wijsman convergence on a trace of full window density determines the statistical convergence of the whole sequence. Examples illustrate the dependence on the window scheme, the strict weakness of the pre-Cauchy condition, and the effect of imposing simultaneous closedness with respect to two non-equivalent component metrics.
The aim of this paper is to explore some aspects of the time-frequency analysis associated with the multidimensional Fourier–Bessel wavelet transform including the spectral analysis associated with the time-frequency operators and the scalogram analysis. The results of this paper are illustrated by some examples and figures.
