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.
In this paper, we study the extension of the Riemann zeta function to finite-dimensional commutative algebras and present examples of such extensions for several hypercomplex systems. We also investigate the zeros of the quaternionic zeta function and prove that its nontrivial zeros are spherical.
We develop fixed point and periodic point results for self-mappings on \(n\)-G-metric spaces. The definition is stated in a form compatible with the classical case \(n=3\), while the paper specializes to the genuinely higher-order case \(n\geq4\). The first result proves a sharp bound: a mapping satisfying an \(n\)-tuple contraction on pairwise distinct points has at most \(n-1\) periodic points, and hence at most \(n-1\) fixed points. This formulation makes clear the role of short periodic orbits, which are the natural obstruction to applying a distinct-tuple contraction along Picard blocks. Under \(d_G\)-completeness, orbital continuity and an \(n\)-orbit-separation condition, every Picard orbit either reaches a fixed point or converges to one. We also state direct \(n\)-ary Kannan- and Reich-type analogues and separate them from the standard consequences obtained from the associated metric. Finally, the total pairwise polygonal-length functional on normed spaces provides a concrete geometric model of the theory. To make the model computationally explicit, we add a Picard algorithm in \(n\)-G-metric spaces, pseudocode, convergence and error tables for several values of \(n\), \(\lambda\) and \(\rho\), and both linear and nonlinear discrete polygonal systems illustrating the comparison between the associated metric \(d_G\) and the pairwise polygonal-length dynamics.
We investigate an inverse source problem for a general linear parabolic equation with space-time dependent coefficients. While the forward operator \(S: f \mapsto u\) is stable and characterized by a smoothing mechanism, its inversion is severely ill-posed. To stabilize the reconstruction of \(f\), we develop a framework based on orthogonal Hilbert-space expansions. By decomposing the evolution equation into an infinite system of ordinary differential equations (ODEs), we explicitly reveal how high-frequency noise is exponentially amplified during the recovery process. A spectral filtering regularization method is proposed, and we establish optimal convergence rates under suitable source conditions.
Given a permutation \(\pi\) of size \(n\), we consider the associated grid graph \(G_{\pi}\) whose \(i\)th column has height \(\pi_i\), with vertical edges between consecutive levels and horizontal edges between equal levels in adjacent columns. We investigate global degree statistics of these graphs when \(\pi\) ranges over the Catalan avoidance class \(\mathrm{Av}_n(213)\) (and, by reversal, also over \(\mathrm{Av}_n(312)\)). We obtain explicit closed formulas for two accumulated quantities over \(\mathrm{Av}_n(213)\): the total number of horizontal edges and, for each \(r\in\{1,2,3,4\}\), the total number of degree-\(r\) vertices. The proofs rely on the Catalan decomposition induced by the position of the minimum entry, which leads to gluing recurrences and algebraic ordinary generating functions, and are completed using universal identities for the total number of vertices and the sum of degrees. As an application, we derive asymptotic degree proportions for a uniformly random permutation in \(\mathrm{Av}_n(213)\): the expected proportion of degree-4 vertices, equivalently the global degree-4 proportion over the class, tends to \(1\), with a deficit of order \(n^{-1/2}\), contrasting with the unrestricted case.
