In this paper, we study the dependence of the eigenvalues of one-dimensional Hamiltonian systems on indefinite weights. We first prove the local continuity of the eigenvalues with respect to weights under the weak topology. Moreover, we derive the continuous differentiability of the eigenvalues in weights under the standard norm topology and present the precise formula for the Fréchet derivatives of eigenvalues with respect to weights.
The minimal excludant of a bipartition π =(π _1, π _2) refers to the smallest integer k that does not appear simultaneously in partitions π _1 and π _2 . Let σ mex _2^o(n) denote the sum of odd minimal excludants over all bipartitions of n. In this paper, we confirm four congruences modulo small powers of 2 for σ mex _2^o(n) and establish a new refinement for the congruence that the number of bipartitions of 5n+4 is divisible by 5.
We present a functional-analytic formulation of nonlinear operator approximation on spaces of continuous functions and use it to derive structural representations underlying modern neural operator architectures. Based on compactness, equicontinuity and finite-dimensional approximation, we prove a uniform approximation on a compact set of functionals, extending classical approximation results for functionals and operators. Within this setting, continuous functionals are shown to admit uniform approximation by expressions depending on finitely many point evaluations, while continuous nonlinear operators admit finite-rank tensorized representations in which the dependence on the input function is mediated through finitely many continuous functionals. This viewpoint provides a mathematical explanation for encoder–decoder (branch–trunk) neural operator architectures and clarifies how Fourier-based models arise as basis-restricted instances of a more general approximation framework. The results thus bridge classical operator approximation theory and contemporary operator learning, offering a unified analytical perspective on finite-dimensional representations of nonlinear operators between function spaces.
A detailed study of the continuity and compactness of Hausdorff operators defined by a positive Borel measure on (0,∞ ) when they act between different weighted Banach spaces of entire functions with sup-norms is presented. Properties of the weights play a relevant role. As an application, the behavior of Hausdorff operators acting on Hörmander algebras of entire functions is investigated.
Let ℬ(ℋ) be the set of all bounded linear operators acting on a complex infinite dimensional separable Hilbert space ℋ . Let T∈ℬ(ℋ) . The local spectrum σ _1(T) is constructed by using the CFI^- operator. Using σ _1(T) , we characterize the weak property (ω ) of T and its compact perturbation. Meanwhile, we explore the relationship between compact perturbations of hypercyclic operators and the weak property (ω ) , and present an equivalent criterion for the stability of the weak property (ω ) for Toeplitz operators.
