5,817 research outputs found
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
Effectively Open Real Functions
A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is
open again. Dual to this topological property, f is called OPEN iff the IMAGE
f[U] of any open set U is open again. Several classical Open Mapping Theorems
in Analysis provide a variety of sufficient conditions for openness.
By the Main Theorem of Recursive Analysis, computable real functions are
necessarily continuous. In fact they admit a well-known characterization in
terms of the mapping V+->f^{-1}[V] being EFFECTIVE: Given a list of open
rational balls exhausting V, a Turing Machine can generate a corresponding list
for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on
open real subsets to be effective.
By effectivizing classical Open Mapping Theorems as well as from application
of Tarski's Quantifier Elimination, the present work reveals several rich
classes of functions to be effectively open.Comment: added section on semi-algebraic functions; to appear in Proc.
http://cca-net.de/cca200
A Local to Global Principle for the Complexity of Riemann Mappings (Extended Abstract)
We show that the computational complexity of Riemann mappings can be bounded
by the complexity needed to compute conformal mappings locally at boundary
points. As a consequence we get first formally proven upper bounds for
Schwarz-Christoffel mappings and, more generally, Riemann mappings of domains
with piecewise analytic boundaries
A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations
AbstractWe introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions
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