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.
In this paper, we introduce a summability-based functional, called the sampling-induced statistical Riemann sum functional, generated by uniform partitions with a fixed sampling rule. For a function \(f:[a,b]\to\mathbb{R}\), we consider the sequence of uniformly sampled right-endpoint Riemann sums and define the functional value as the statistical limit of this sequence, whenever it exists. We show that every classically Riemann integrable function yields the same value as the classical integral, while the converse fails. Unlike the classical Riemann integral, which is tag-independent and intrinsic, the functional studied here is sampling-dependent and may change under modifications of the function on countable sets. We illustrate these phenomena with examples, discuss the relationship with the statistical derivative, and highlight the structural compatibility issues that arise when attempting a statistical Fundamental Theorem of Calculus. The paper concludes with open problems concerning sampling-scheme independence and the characterization of functions for which the functional is tag-invariant.
In this paper, we study complete minimal submanifolds of Riemannian manifolds by employing a generalized Bochner technique. First, we provide a generalization of the classical theorem of Chern, Kobayashi, and do Carmo on compact minimal submanifolds to the case of complete parabolic minimal submanifolds. Second, we investigate the rigidity of complete stable minimal parabolic hypersurfaces in Riemannian manifolds, showing that under certain curvature conditions such hypersurfaces must be totally geodesic.
The analytical solution of the Black-Scholes equation can lead to the attainment of the price of an option in an idealized fiscal market. However, this is not practically beneficial enough. This happens due to the constricting assumptions based on which the Black-Scholes model is derived. In the real financial market, one can question the constant nature of the coefficients of the Black-Scholes equation. In this paper, the solution of the Black-Scholes equation with constant parameters is reviewed. Next, the Black-Scholes equation solution taking time-dependent volatility and time-dependent risk-free interest rate into consideration is studied. Finally, a fully nonlinear model of Black-Scholes equation (Barls and Soner’s model) is considered. Because of the nonlinear nature of the model, there is not an analytical solution and numerical method is mandatory to price the derivative. Saul’yev finite difference scheme is proposed which not only is explicit, but also is unconditionally stable.
This article develops duality principles applicable to some originally non-convex primal variational formulations. More specifically, in a first step, we develop applications to a full complex Ginzburg-Landau system in superconductivity, including a magnetic field and respective magnetic potential. The results are obtained through basic tools of functional analysis, calculus of variations, duality and optimization theory in infinite dimensional spaces. It is worth emphasizing we have obtained convex dual variational formulations which may be applied to a large class of similar models in the calculus of variations. In the subsequent sections we also present a procedure for improving the convexity conditions of an originally non-convex primal formulation which is also applied to a Ginzburg-Landau type equation. Finally, in the last sections, we develop duality principles and related numerical examples for models in phase transition.
This paper investigates the existence, uniqueness, and stability of weak \(T\)-periodic solutions for nonlinear cooperative parabolic systems with Neumann boundary conditions a class of problems central to understanding recurrent phenomena in nature. By combining monotone iteration techniques with the method of sub- and super-solutions in ordered Banach spaces, we develop a robust framework that yields the existence of extremal periodic solutions between ordered barriers under general Carathéodory conditions and a cooperativity assumption on the nonlinearities. Uniqueness is further established under a mild Lipschitz condition. The power and applicability of the abstract results are demonstrated through their application to a time-periodic model of water-solute transport in porous media, offering new insights into the dynamics of solutes under environmental influence. Thus, the results of this study contribute to the theoretical advancement of periodic parabolic systems, mathematical modeling, and numerical simulation of periodic phenomena in hydrological and environmental sciences using the IMEX scheme, where seasonal phenomena such as annual recharge cycles, climatic fluctuations, and seasonal agricultural activities are particularly influenced by periodic conditions.
In this paper, we introduce the function class \(G_p^\eta(\mathbb{Z})\) alongside a more general class \(G_p(\mathbb{Z})\) to serve as parameter functions for generalized Morrey sequence spaces and generalized mixed Morrey double-sequence spaces. Using these classes, we establish necessary and sufficient conditions for the boundedness of the discrete Hardy–Littlewood maximal operator and its higher-order commutator on generalized Morrey sequence spaces and generalized mixed Morrey double-sequence spaces over \(\mathbb{Z}\). The boundedness of the maximal operator is characterized in terms of relations between the parameter functions. To obtain the sufficiency and necessity conditions, we utilize the properties of the \(G_p(\mathbb{Z})\) class. For the necessity conditions, we compute the norms of the characteristic functions of a discrete interval in generalized Morrey sequence spaces and of a rectangle in \(\mathbb{Z}\times\mathbb{Z}\) in generalized mixed Morrey double-sequence spaces. Furthermore, we characterize the boundedness of the higher-order commutator via the \(\operatorname{BMO}\) space. Specifically, the proof of sufficiency relies on the Fefferman–Stein inequality within these spaces, whereas the necessity proof adapts the techniques developed for the maximal operator itself.
