相关论文: Effectively Open Real Functions
This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we…
Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
A function $f$ on a topological space is sequentially continuous at a point $u$ if, given a sequence $(x_{n})$, $\lim x_{n}=u$ implies that $\lim f(x_{n})=f(u)$. This definition was modified by Connor and Grosse-Erdmann for real functions…
We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real…
A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that…
A function $f$ is arc-smooth if the composite $f\circ c$ with every smooth curve $c$ in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
Using the recent theory of Krein--von Neumann extensions for positive functionals we present several simple criteria to decide whether a given positive functional on the full operator algebra is normal. We also characterize those…
The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary transfinite…
Let $K_1$ and $K_2$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by $$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-\'a-vis…
For a field $\mathbb{F}$, what are all functions $f \colon \mathbb{F} \rightarrow \mathbb{F}$ that satisfy the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$? We solve…
The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been…
In mathematical biology and the theory of electric networks the firing map of an integrate-and-fire system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus…
It is proved that for every stratifiable space $Y$ and a closed subset $X\subset Y$ there exists a regular (i.e. linear positive with unit norm) extension operator $T:C(X\times X)\to C(Y\times Y)$ preserving the class of (pseudo)metrics.…
In this paper we associate with an infinite family of real extended functions defined on a locally convex space, a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems…
Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+\lambda f$ to be the curvature in the arc-length of a closed planar curve for all real $\lambda$. Several equivalent conditions are pointed out,…