Related papers: Machine learning and the Continuum Hypothesis
For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$ on $K,$ and $1 \le t < \i,$ let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of $\text{Rat}(K)$ in…
We investigate the shrinking target and recurrence set associated to non-autonomous measure-preserving systems on compact metric spaces, establishing zero-one criteria in the spirit of classical Borel-Cantelli results. Our first main…
We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass. It is a refinement of the classical notion of omega-limit sets dealing with…
Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable…
We study the connections between three seemingly different combinatorial structures - "uniform" brackets in statistics and probability theory, "containers" in online and distributed learning theory, and "combinatorial Macbeath regions", or…
We study Morse theory of the (power) distance function to a set of points in $\mathbb{R}^n$. We describe the topology of the union of the corresponding set of growing balls by a Morse poset. The Morse poset is related to the power…
We investigate Keisler measures in arbitrary theories. Our initial focus is on Borel definability. We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and…
We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations,…
Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a…
Many vision-related tasks benefit from reasoning over multiple modalities to leverage complementary views of data in an attempt to learn robust embedding spaces. Most deep learning-based methods rely on a late fusion technique whereby…
It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
It was argued [1] that there can be no extension of quantum mechanics with improved predictive power on a measurement freely chosen, independently of any event that is not in its future light cone. The assumption of measurement choice was…
Machine learning on sets towards sequential output is an important and ubiquitous task, with applications ranging from language modeling and meta-learning to multi-agent strategy games and power grid optimization. Combining elements of…
We study algorithmically random closed subsets of $2^\omega$, algorithmically random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically random Borel probability measures on $2^\omega$, especially the interplay between…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
A celebrated 1922 theorem of Kuratowski states that there are at most 14 distinct sets arising from applying the operations of complementation and closure, any number of times, in any order, to a subset of a topological space. In this paper…
We establish coupled fixed point theorems for contraction involving rational expressions in partially ordered metric spaces.
We study equivalent descriptions of the vague, weak, setwise and total-variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent…
We apply the topology of convergence on compact sets to define unpredictable functions [5, 6]. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and…