27 research outputs found
Comparing computability in two topologies
Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points. We first study a purely topological version of the problem, which is to understand when two topologies are not σ-homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager w.r.t. a topology, and are computable w.r.t. another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, i.e. have a homeomorphic copy that is semicomputable but not computable
Kolmogorov superposition theorem and its applications
Hilbert’s 13th problem asked whether every continuous multivariate function can be written
as superposition of continuous functions of 2 variables. Kolmogorov and Arnold show that
every continuous multivariate function can be represented as superposition of continuous
univariate functions and addition in a universal form and thus solved the problem positively.
In Kolmogorov’s representation, only one univariate function (the outer function)
depends on and all the other univariate functions (inner functions) are independent of the
multivariate function to be represented. This greatly inspired research on representation
and superposition of functions using Kolmogorov’s superposition theorem (KST).
However, the numeric applications and theoretic development of KST is considerably
limited due to the lack of smoothness of the univariate functions in the representation.
Therefore, we investigate the properties of the outer and inner functions in detail. We show
that the outer function for a given multivariate function is not unique, does not preserve the
positivity of the multivariate function and has a largely degraded modulus of continuity.
The structure of the set of inner functions only depends on the number of variables of the
multivariate function. We show that inner functions constructed in Kolmogorov’s representation
for continuous functions of a fixed number of variables can be reused by extension
or projection to represent continuous functions of a different number of variables.
After an investigation of the functions in KST, we combine KST with Fourier transform
and write a formula regarding the change of the outer functions under different inner
functions for a given multivariate function. KST is also applied to estimate the optimal
cost between measures in high dimension by the optimal cost between measures in low
dimension. Furthermore, we apply KST to image encryption and show that the maximal
error can be obtained in the encryption schemes we suggested.Open Acces
Embeddings for -weakly differentiable functions on domains
We prove that the critical embedding
holds if and only if the -homogeneous, linear differential operator
on from to has finite
dimensional null-space. Here is a ball in and
denotes the space of maps such that the vector valued distribution
is an integrable map. The result was previously known only for
several examples of . Our result contrasts the homogeneous
embedding in full-space. Namely, Van Schaftingen proved that
if and only if is elliptic
and cancelling. We show that this condition is (strictly) implied by
having finite dimensional null-space.Comment: 23 pages, 1 tabl