Related papers: Preimage Cardinalities of Continuous Functions

Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode the groupoid cardinalities of their graded components. As suggested in the work of Baez and Dolan, we use analytic continuation of…

Let $D$ be a bounded domain in $\mathbb C^n$. We study approximation of (not necessarily bounded from above) $m-$subharmonic function $D$ by continuous $m-$subharmonic ones defined on neighborhoods of $\overline{D}$. We also consider the…

Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role,…

Some physical consequences of the negation of the continuum hypothesis are considered. It is shown that quantum and classical mechanics are component parts of the multicomponent description of the set of variable infinite cardinality.…

Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where…

If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…

A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

In this paper we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. The results include approximation…

For a class C of operations on a nonempty base set A, an operation f is called a C-subfunction of an operation g, if f = g(h_1, ..., h_n), where all the inner functions h_i are members of C. Two operations are C-equivalent if they are…

A clonoid is a set of finitary functions from a set $A$ to a set $B$ that is closed under taking minors. Hence clonoids are generalizations of clones. By a classical result of Post, there are only countably many clones on a 2-element set.…

Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard…

This paper addresses the task of set prediction using deep learning. This is important because the output of many computer vision tasks, including image tagging and object detection, are naturally expressed as sets of entities rather than…

The primary goal of this document is to record the asymptotic effects that preimage constraints impose upon the sizes of the iterated images of a random function. Specifically, given a subset $\mathcal{P}\subseteq \mathbb{Z}_{\geq 0}$ and a…

The Gibbs--Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions is investigated. We prove existence of the overshoot characteristic of the phenomenon for certain cardinal…

Continual learning requires a model to adapt to ongoing changes in the data distribution, and often to the set of tasks to be performed. It is rare, however, that the data and task changes are completely unpredictable. Given a description…

In this paper some axiomatic generalization (function of open subsets) of the relative Lyusternik-Schnirelmann category is considered, incorporating the sectional category and the Schwarz genus as well. For this function and a given…

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of every…

Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets $\left\{\big\lfloor\frac{X}{n^t}\big\rfloor ~:~ 1\leq n\leq X\right\}$ have an interesting prime distribution property. We give an exact formula for the…

Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and…