14 research outputs found
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
Computability in basic quantum mechanics
The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space H. In terms of the Hilbert lattice L of closed linear subspaces of H the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch’s Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category QCB0 which is equivalent to the category AdmRep of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann’s Spectral Theorem
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page