New Inequalities of Gruss Type for the Stieltjes Integral and Applications

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.

math.CA↗

arXiv2026

Tensor Derivatives, Unified Tensor-Form Differential Equations, and Model Reduction via Partial Tucker Decomposition

This paper develops a unified tensor calculus for matrix-valued functions and their derivatives, and leverages this framework to construct efficient model reduction techniques for high-dimensional tensor differential equations. We first establish a systematic theory of tensor differentiation, wherein the derivative of a matrix with respect to another matrix is represented as a fourth-order tensor. Building on this calculus, we recast linear ordinary differential equations (ODEs) and partial differential equations(PDEs) into a compact tensor-matrix form $\frac{dX}{dt} = \A\ast X$. The general solution is expressed as $X = \exp(t\A)\ast C$, extending the matrix exponential to the tensor setting. Conditions under which the solution admits this exponential form are characterized in terms of the commutativity of the associated matrix slices. We introduce the partial Tucker decomposition (parTuckerD) to address the computational challenges posed by high-order tensor systems. On a synthetic electronic health record (EHR) tensor, parTuckerD achieves a relative reconstruction error of $0.0992$ with a $136.3\times$ compression ratio, matching the accuracy of the full TuckerD while preserving patient-level similarity structure. The results demonstrate that the proposed tensor calculus and parTuckerD framework provide a principle and computationally efficient approach for analyzing and solving high-dimensional tensor differential equations arising in data-intensive applications.

math.CA↗

arXiv2026

Distance preservers for Lobachevsky space

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings. These preservers admit a Lévy--Khintchine-type representation and their asymptotic characteristics are related to Krein's classification of screw lines in Lobachevsky space. Connections with complete Nevanlinna--Pick kernels and Bochner subordination are also obtained.

math.CA↗