221 research outputs found
A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets
We study a location problem that involves a weighted sum of distances to
closed convex sets. As several of the weights might be negative, traditional
solution methods of convex optimization are not applicable. After obtaining
some existence theorems, we introduce a simple, but effective, algorithm for
solving the problem. Our method is based on the Pham Dinh - Le Thi algorithm
for d.c. programming and a generalized version of the Weiszfeld algorithm,
which works well for convex location problems
Complex Multiplication Symmetry of Black Hole Attractors
We show how Moore's observation, in the context of toroidal compactifications
in type IIB string theory, concerning the complex multiplication structure of
black hole attractor varieties, can be generalized to Calabi-Yau
compactifications with finite fundamental groups. This generalization leads to
an alternative general framework in terms of motives associated to a Calabi-Yau
variety in which it is possible to address the arithmetic nature of the
attractor varieties in a universal way via Deligne's period conjecture.Comment: 28 page
Black Hole Attractor Varieties and Complex Multiplication
Black holes in string theory compactified on Calabi-Yau varieties a priori
might be expected to have moduli dependent features. For example the entropy of
the black hole might be expected to depend on the complex structure of the
manifold. This would be inconsistent with known properties of black holes.
Supersymmetric black holes appear to evade this inconsistency by having moduli
fields that flow to fixed points in the moduli space that depend only on the
charges of the black hole. Moore observed in the case of compactifications with
elliptic curve factors that these fixed points are arithmetic, corresponding to
curves with complex multiplication. The main goal of this talk is to explore
the possibility of generalizing such a characterization to Calabi-Yau varieties
with finite fundamental groups.Comment: 21 page
Tropical Principal Component Analysis and its Application to Phylogenetics
Principal component analysis is a widely-used method for the dimensionality
reduction of a given data set in a high-dimensional Euclidean space. Here we
define and analyze two analogues of principal component analysis in the setting
of tropical geometry. In one approach, we study the Stiefel tropical linear
space of fixed dimension closest to the data points in the tropical projective
torus; in the other approach, we consider the tropical polytope with a fixed
number of vertices closest to the data points. We then give approximative
algorithms for both approaches and apply them to phylogenetics, testing the
methods on simulated phylogenetic data and on an empirical dataset of
Apicomplexa genomes.Comment: 28 page
An Invitation to Generalized Minkowski Geometry
The present thesis contributes to the theory of generalized Minkowski spaces as a continuation of Minkowski geometry, i.e., the geometry of finite-dimensional normed spaces over the field of real numbers.
In a generalized Minkowski space, distance and length measurement is provided by a gauge, whose definition mimics the definition of a norm but lacks the symmetry requirement.
This seemingly minor change in the definition is deliberately chosen.
On the one hand, many techniques from Minkowski spaces can be adapted to generalized Minkowski spaces because several phenomena in Minkowski geometry simply do not depend on the symmetry of distance measurement.
On the other hand, the possible asymmetry of the distance measurement set up by gauges is nonetheless meaningful and interesting for applications, e.g., in location science.
In this spirit, the presentation of this thesis is led mainly by minimization problems from convex optimization and location science which are appealing to convex geometers, too.
In addition, we study metrically defined objects, which may receive a new interpretation when we measure distances asymmetrically.
To this end, we use a combination of methods from convex analysis and convex geometry to relate the properties of these objects to the shape of the unit ball of the generalized Minkowski space under consideration
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