Related papers: Differentiation in Topological Vector Spaces
This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and is closely related to metric and geometric properties of the set such as…
We find an expression for Gateaux derivative of the $C^*$-algebra norm. This gives us alternative proofs or generalizations of various known results on the closely related notions of subdifferential sets, smooth points and Birkhoff-James…
Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a…
Periodically driven (Floquet) systems have been under active theoretical and experimental investigations. This paper aims at a systematic study in the following aspects of Floquet systems: (i) A systematic formulation of topological…
We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $\delta > 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for…
This is an experiential study of investigating a consistent method for deriving the correlation between sentence vector and semantic meaning of a sentence. We first used three state-of-the-art word/sentence embedding methods including…
In this paper, we present a novel workflow consisting of algebraic algorithms and data structures for fast and topologically accurate conversion of vector data models such as Boundary Representations into voxels (topological voxelization);…
We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization…
We define a Fr\'echet topology on the space $C^\infty(X)[[\hbar]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^\infty(X)[[\hbar]]$ making it a…
A common approach for sequence tagging tasks based on contextual word representations is to train a machine learning classifier directly on these embedding vectors. This approach has two shortcomings. First, such methods consider single…
Persistent homology is a cornerstone of topological data analysis, offering a multiscale summary of topology with robustness to nuisance transformations, such as rotations and small deformations. Persistent homology has seen broad use…
We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of…
A system of differential forms will establish a topology and a topological structure on a domain of independent variables such that is possible to determine which maps or processes acting on the system are continuous. Perhaps the most…
We formalize the notion of vector semi-inner products and introduce a class of vector seminorms which are built from these maps. The classical Pythagorean theorem and parallelogram law are then generalized to vector seminorms that have a…
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and scalar multiplication are continuous. We prove that, if an isomorphism between the lattice of topologies of two…
We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be…
A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact…
In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved.…