Related papers: Nonlinear Lebesgue spaces: Dense subspaces, comple…
This paper extends the recently obtained complete and continuous map of the Lattice Isometry Space (LISP) to the practical case of dimension 3. A periodic 3-dimensional lattice is an infinite set of all integer linear combinations of basis…
The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of $n-1$ dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys…
The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…
In this review paper we study non-compact operators and embeddings between function spaces, highlighting interesting phenomena and the significance of Bernstein numbers. In particular, we demonstrate that for non-compact maps the usual…
We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric…
We look at a measure, $\lambda^\infty$, on the infinite-dimensional space, ${\mathbb R}^\infty$, for which we attempt to put forth an analogue of the Lebesgue density theorem. Although this measure allows us to find partial results, for…
It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following…
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space,…
The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about…
We consider marginal log-linear models for parameterizing distributions on multidimensional contingency tables. These models generalize ordinary log-linear and multivariate logistic models, besides several others. First, we obtain some…
Statistical mechanics has grown without bounds in space. Statistical mechanics of point particles in an unbounded perfect gas is commonly accepted as a foundation for understanding many systems, including liquids like the concentrated salt…
The Mixed Lebesgue space is a suitable tool for modelling and measuring signals living in time-space domains. And sampling in such spaces plays an important role for processing high-dimensional time-varying signals. In this paper, we first…
In this paper, we investigate the geometric properties of the variable mixed Lebesgue-sequence space $\ell^{q(\cdot)} (L^{p(\cdot)})$ as a Banach space. We show that, if $ 1<q_-,p_-,q_+,p_+<\infty $, then $\ell^{q(\cdot)} (L^{p(\cdot)})$ is…
Semantic segmentation aims to robustly predict coherent class labels for entire regions of an image. It is a scene understanding task that powers real-world applications (e.g., autonomous navigation). One important application, the use of…
We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces $L_{P}$ built from index functions $P\colon\Omega\to(0,\infty]$ on $\sigma$-finite measure spaces $(\Omega,\Sigma,\mu)$. Specifically, we prove that if…
This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory…
Understanding geometric properties of natural language processing models' latent spaces allows the manipulation of these properties for improved performance on downstream tasks. One such property is the amount of data spread in a model's…
Piecewise Translations is a class of dynamical systems which arises from some applications in computer science, machine learning, and electrical engineering. In dimension 1 it can also be viewed as a non-invertible generalization of…
Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…
Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition,…