Related papers: Preimage Cardinalities of Continuous Functions

Let $\precsim$ be a reflexive binary relation on a topological space $(X, \tau )$. A pair $(u,v)$ of continuous real-valued functions on $(X, \tau )$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$,…

We present the class of semi-admissible subsets of an open Riemann surface on which Carleman approximation by non-critical holomorphic functions is possible. In particular we characterize closed sets with empty interior on which continuous…

Every continuous function of two or more real variables can be written as the superposition of continuous functions of one real variable along with addition.

We survey old and recent results on the problem of finding a complete set of rules describing the behavior of the power function, i.e. the function which takes a cardinal $\kappa$ to the cardinality of its power $2^\kappa$.

We consider holomorphic functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications…

These are classified by the direction of approximation (from above or below), the set family types (partition or covering) of simple functions, the coefficient signature (non-negative or signed), and cardinal number of terms of simple…

We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied…

We develop classical balayage (sweeping) measures and subharmonic functions on the ray system $S$ with a general origin on the complex plane $\mathbb C$. This allows for a subharmonic function $v$ on $\mathbb C$ to construct also a…

Let $Z\subset\mathbb{C}^N$ be an $n$-pseudoconcave subset, for $1\leq n<N$, which is locally the graph of a continuous function over a closed subset of $\mathbb{C}^n\times\mathbb{R}$. We show that $Z$ can be realised as the disjoint union…

We investigate a hierarchy of arithmetical structures obtained by a transfinite addition of a canonic universal predicate, where the canonic universal predicate for M is defined as a minimum universal predicate for M in terms of…

As a pioneering work we construct explicit real algebraic functions which may have both compact and non-compact preimages. The author has obtained explicit real algebraic functions with preimages satisfying some nice conditions. More…

We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…

In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…

We characterize continuum as the smallest cardinality of a family of compact sets needed to cover a locally compact group for which the Open Mapping Theorem does not hold.

Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.

The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for the sign change counting function is given where classical…

This paper investigates some particular limits involving nested floor functions. We'll prove some cases and then we'll show a more general result. Then we'll count the discontinuity points of those functions, and we'll prove a method to…

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers,…

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

Let $A$ and $B$ be non-constant rational functions over $\mathbb{C}$, and let $K \subset \mathbb{P}^1(\mathbb{C})$ be an infinite set. Using height functions, we prove that the inclusion $ A^{-1}(K) \subseteq B^{-1}(K) $ implies the…