arXiv preprints from January 1, 2026 through July 20, 2026 — 00:17:57 EST
This paper introduces specular differentiation, which generalizes Gâteaux and Fréchet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fréchet...
We consider abstract nonlinear equations of the form $A u = λu + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $λ\in \mathbb{R}$ is a parameter, and $u \mapsto I'(u)$ is a superlinear term of variational nature. In this abstract setting, we develop a new approach to detect prescribed norm...
We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo algebraic gauge transformations, we recast the classification problem as an explicit moduli problem...
In random matrices with independent and continuous matrix entries, the degeneracy probability of the eigenvalues is known to be zero. In this paper, random matrices including discontinuous matrix entries are analyzed in order to observe how degeneracy is generated. Using Erdös-Rényi matching probability theory of random bipartite...
We define tensor categories ${\sf Ver}_{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}_p(G)$ originating from Gelfand-Kazhdan and the higher Verlinde categories ${\sf Ver}_{p^n}$ for ${\rm SL}_2$ defined by Benson-Etingof-Ostrik....
This is an exposition of the work of O. Riemenschneider about five ''circles'' of implications relating real analysis theorems each equivalent to the Dedekind completeness of the real field. These circles cover five elements of real function theory: convergence, connectedness, differentiability, compactness and integration.
We study definite fillings of lens spaces. We classify the lens spaces $L(p,q)$ for which every smooth negative-definite filling $X$ satisfies \[ b_2(X)\ge b_2(X(p,q))-1, \] where $X(p,q)$ denotes the canonical negative-definite plumbing. The classification is given by 17 "forbidden configurations" that cannot appear as induced...
In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh-free approach based on the physics-informed neural networks (PINNs), which eliminates the need for discretization and meshing of computational domains, and is efficient in solving partial differential...
Let $T$ be a densely defined closed symmetric operator with equal deficiency indices in a separable complex Hilbert space $H$. In this paper, we prove that $T$ has a self-adjoint extension with compact resolvent if and only if the domain $D(T)$ of $T$ is compactly embedded in $H$ w.r.t. the graph norm on $D(T)$. If it is the case, we...
In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $Γ$ and a vertex-transitive subgroup $G$ of $\mathrm{Aut}(Γ)$, a $2$-factor $\mathcal{C}$ of $Γ$ is said to be {\em $G$-invariant} if the set $\mathcal{C}$ is preserved by each element of $G$....
Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gantmacher-Krein, Lusztig, and others. TP matrices form a multiplicatively closed semigroup, contained in the larger monoid of invertible totally nonnegative...
The notions of $p$-convexity and concavity are fundamental tools for studying Banach lattices, as they partition the class of Banach lattices into a scale of spaces with $L_p$-like properties. Upper and lower $p$-estimates provide a refinement of this scale, modeled by the Lorentz spaces $L_{p,\infty}$ and $L_{p,1}$, respectively. In...
We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--Bézier basis. Boundary interpolation is imposed by prescribing the boundary rows and columns of the control net, while the interior control points are selected by a Dirichlet...
Let phi(n) denote the Euler totient function. We study the analytic part associated with the summatory function of sigma_1(n) and obtain explicit bounds under the Riemann Hypothesis. In particular, we establish an upper bound of order x^{delta'} exp((log x)/(log log x)), where delta' = max(1/2, delta).
We extend our recent curve-evolution framework based on localized B-spline interpolation to present an adaptive Lagrangian framework for the geometric evolution of point-cloud data representing smooth, codimension-one surfaces in $\mathbb{R}^3$. The method constructs overlapping, localized tensor-product B-spline patches, enabling...
We develop and analyze a B-spline based arbitrary Lagrangian-Eulerian method of fundamental solutions (ALE-MFS) for curvature-driven motion of two-dimensional evolving domains. Boundary points move with the material to track the geometric flow, while interior points move within an ALE framework via a harmonic extension of the boundary...
Penalized smoothing is a standard tool in regression analysis. Classical approaches often rely on basis or kernel expansions, which constrain the estimator to a fixed span and impose smoothness assumptions that may be restrictive for discretely observed data. We introduce a class of penalized estimators that operate directly on the...
Moran-type iterated function systems (Moran-type IFS or MIFS) are defined by a sequence of iterated function systems, and their basic theoretical framework is established. We define Moran-type attractors and invariant probability measures associated with a sequence of probability weight vectors. Furthermore, separation conditions for...
Recent advances in reservoir simulation increasingly utilize hybrid approaches that couple physics-based simulators with machine-learning (ML) components. ML components offer high fidelity to training data and fast inference, enabling efficient and accurate modeling of complex multi-scale or multi-physics phenomena. Modern reservoir...
An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in several of its important subgroups. We show that not every reversible element in the Riordan group...
A novel class of Runge-Kutta discontinuous Galerkin schemes for coupled systems of conservation laws in multiple space dimensions that are separated by a fixed sharp interface is introduced. The schemes are derived from a relaxation approach and a local projection and do not require expensive solutions of nonlinear half-Riemann...
We study the symmetrization of the Novikov product. Using the embedding of a free Novikov algebra into a differential algebra over a field of characteristic zero and the Euler operators (variational derivatives), we show that the space of null Lagrangians coincides with the subspace of elements closed under the symmetrized product...
Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which preserve the ring structure in the norm in the following sense: \[...
We study the rate of convergence of two discrete processes towards the Brownian bridge: the random walk conditioned to be zero at time 2n and the empirical process which appears in the Glivencko-Cantelli theorem. Combining a functional Stein method with a Radon-Nikodym representation of the bridge, we bound the Fortet-Mourier distance...
Combined optimization problems that couple data-fidelity and regularization terms arise naturally in a wide range of inverse problems. In this paper, we study an adaptive randomized averaging block extended Bregman-Kaczmarz (aRABEBK) method for solving such problems. The proposed method incorporates iteration-wise relaxation...