31,331 research outputs found
Approximation systems for functions in topological and in metric spaces
A notable feature of the TTE approach to computability is the representation
of the argument values and the corresponding function values by means of
infinitistic names. Two ways to eliminate the using of such names in certain
cases are indicated in the paper. The first one is intended for the case of
topological spaces with selected indexed denumerable bases. Suppose a partial
function is given from one such space into another one whose selected base has
a recursively enumerable index set, and suppose that the intersection of base
open sets in the first space is computable in the sense of Weihrauch-Grubba.
Then the ordinary TTE computability of the function is characterized by the
existence of an appropriate recursively enumerable relation between indices of
base sets containing the argument value and indices of base sets containing the
corresponding function value.This result can be regarded as an improvement of a
result of Korovina and Kudinov. The second way is applicable to metric spaces
with selected indexed denumerable dense subsets. If a partial function is given
from one such space into another one, then, under a semi-computability
assumption concerning these spaces, the ordinary TTE computability of the
function is characterized by the existence of an appropriate recursively
enumerable set of quadruples. Any of them consists of an index of element from
the selected dense subset in the first space, a natural number encoding a
rational bound for the distance between this element and the argument value, an
index of element from the selected dense subset in the second space and a
natural number encoding a rational bound for the distance between this element
and the function value. One of the examples in the paper indicates that the
computability of real functions can be characterized in a simple way by using
the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc
Twistor Approach to String Compactifications: a Review
We review a progress in obtaining the complete non-perturbative effective
action of type II string theory compactified on a Calabi-Yau manifold. This
problem is equivalent to understanding quantum corrections to the metric on the
hypermultiplet moduli space. We show how all these corrections, which include
D-brane and NS5-brane instantons, are incorporated in the framework of the
twistor approach, which provides a powerful mathematical description of
hyperkahler and quaternion-Kahler manifolds. We also present new insights on
S-duality, quantum mirror symmetry, connections to integrable models and
topological strings.Comment: 99 pages; minor corrections; journal versio
Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
The hypermultiplet moduli space M_H in type II string theories compactified
on a Calabi-Yau threefold X is largely constrained by supersymmetry (which
demands quaternion-K\"ahlerity), S-duality (which requires an isometric action
of SL(2, Z)) and regularity. Mathematically, M_H ought to encode all
generalized Donaldson-Thomas invariants on X consistently with wall-crossing,
modularity and homological mirror symmetry. We review recent progress towards
computing the exact metric on M_H, or rather the exact complex contact
structure on its twistor space.Comment: 31 pages; Contribution to the Proceedings of String Math 2012; v2:
references added, misprints corrected, published versio
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