281,852 research outputs found
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
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems
according to their computational content. Basically, we are asking the question
which theorems can be continuously or computably transferred into each other?
For this purpose theorems are considered via their realizers which are
operations with certain input and output data. The technical tool to express
continuous or computable relations between such operations is Weihrauch
reducibility and the partially ordered degree structure induced by it. We have
identified certain choice principles which are cornerstones among Weihrauch
degrees and it turns out that certain core theorems in analysis can be
classified naturally in this structure. In particular, we study theorems such
as the Intermediate Value Theorem, the Baire Category Theorem, the Banach
Inverse Mapping Theorem and others. We also explore how existing
classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into
this picture. We compare the results of our classification with existing
classifications in constructive and reverse mathematics and we claim that in a
certain sense our classification is finer and sheds some new light on the
computational content of the respective theorems. We develop a number of
separation techniques based on a new parallelization principle, on certain
invariance properties of Weihrauch reducibility, on the Low Basis Theorem of
Jockusch and Soare and based on the Baire Category Theorem. Finally, we present
a number of metatheorems that allow to derive upper bounds for the
classification of the Weihrauch degree of many theorems and we discuss the
Brouwer Fixed Point Theorem as an example
The action of outer automorphisms on bundles of chiral blocks
On the bundles of WZW chiral blocks over the moduli space of a punctured
rational curve we construct isomorphisms that implement the action of outer
automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms
respect the Knizhnik-Zamolodchikov connection and have finite order. When all
primary fields are fixed points, the isomorphisms are endomorphisms; in this
case, the bundle of chiral blocks is typically a reducible vector bundle. A
conjecture for the trace of such endomorphisms is presented; the proposed
relation generalizes the Verlinde formula. Our results have applications to
conformal field theories based on non-simply connected groups and to the
classification of boundary conditions in such theories.Comment: 46 pages, LaTeX2e. Final version (Commun.Math.Phys., in press). We
have implemented the fact that the group of automorphisms in general acts
only projectively on the chiral blocks and corrected some typo
Interacting String Multi-verses and Holographic Instabilities of Massive Gravity
Products of large-N conformal field theories coupled by multi-trace
interactions in diverse dimensions are used to define quantum multi-gravity
(multi-string theory) on a union of (asymptotically) AdS spaces. One-loop
effects generate a small O(1/N) mass for some of the gravitons. The boundary
gauge theory and the AdS/CFT correspondence are used as guiding principles to
study and draw conclusions on some of the well known problems of massive
gravity - classical instabilities and strong coupling effects. We find examples
of stable multi-graviton theories where the usual strong coupling effects of
the scalar mode of the graviton are suppressed. Our examples require a fine
tuning of the boundary conditions in AdS. Without it, the spacetime background
backreacts in order to erase the effects of the graviton mass.Comment: 51 pages, 3 figures; v2 typos corrected, version published in NPB; v3
added appendix E on general class of fixed points in multi-trace deformation
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