Related papers: Preimage Cardinalities of Continuous Functions
Our objective is to determine which subsets of $\mathbb{R}^d$ arise as escaping sets of continuous functions from $\mathbb{R}^d$ to itself. We obtain partial answers to this problem, particularly in one dimension, and in the case of open…
We give full characterization of the sequences of regular cardinals that may arise as cardinal sequences of locally compact scattered spaces under GCH. The proofs are based on constructions of universal locally compact scattered spaces.
We present a novel approach for learning to predict sets using deep learning. In recent years, deep neural networks have shown remarkable results in computer vision, natural language processing and other related problems. Despite their…
We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large…
Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically.
A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be…
We study the geometry of functions from the plane to the plane. For a large special class we are able to count preimages and compute them. Both numerical and theoretical aspects are discussed. Some of the tools used are Whitney's…
We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of…
Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets?…
A pretorsion theory for the category of all categories is presented. The associated prekernels and precokernels are calculated for every functor.
Given a subset of states $S$ of a deterministic finite automaton and a word $w$, the preimage is the subset of all states mapped to a state in $S$ by the action of $w$. We study three natural problems concerning words giving certain…
We study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
On the plane, every random compact set with almost surely uncountable first projection intersects with a high probability the graph of some continuous function. Implication: every black noise over the plane fails to factorize when the plane…
A survey of recent results concerning cardinal invariants of measure and category. Submitted as a chapter of the upcoming Handbook of Set Theory.
If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these "cardinal characteristics of the continuum", it was discovered that many of the associated…
In this paper, we define a subclass of sense-preserving harmonic functions associated with a class of analytic functions satisfying a differential inequality. We then establish a close relation between both subclasses. Further, we obtain…
We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,\omega,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.
We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a…
We introduce different classical characteristics used to regularize a subharmonic function and compare them. As an application we give a complete proof of a useful characterization of the modulus of continuity of such functions in terms of…