761 research outputs found
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
Probability around the Quantum Gravity. Part 1: Pure Planar Gravity
In this paper we study stochastic dynamics which leaves quantum gravity
equilibrium distribution invariant. We start theoretical study of this dynamics
(earlier it was only used for Monte-Carlo simulation). Main new results concern
the existence and properties of local correlation functions in the
thermodynamic limit. The study of dynamics constitutes a third part of the
series of papers where more general class of processes were studied (but it is
self-contained), those processes have some universal significance in
probability and they cover most concrete processes, also they have many
examples in computer science and biology. At the same time the paper can serve
an introduction to quantum gravity for a probabilist: we give a rigorous
exposition of quantum gravity in the planar pure gravity case. Mostly we use
combinatorial techniques, instead of more popular in physics random matrix
models, the central point is the famous exponent.Comment: 40 pages, 11 figure
New Strings for Old Veneziano Amplitudes III. Symplectic Treatment
A d-dimensional rational polytope P is a polytope whose vertices are located
at the nodes of d-dimensional Z-lattice. Consider a number of points inside the
inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart
polynomial of P counts the number of such lattice points (nodes) inside the
inflated P and (may be) at its faces (including vertices). In Part I
(hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude
is just the Laplace transform of the generating function (considered as a
partition function in the sence of statistical mechanics) for the Ehrhart
polynomial for the regular inflated simplex obtained as a deformation retract
of the Fermat (hyper) surface living in complex projective space. This
observation is sufficient for development of new symplectic (this work) and
supersymmetric (hep-th/0411241)physical models reproducing the Veneziano (and
Veneziano-like) amplitudes. General ideas (e.g.those related to the properties
of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use
of mirror symmetry for explanation of available experimental data on pion-pion
scattering) worked out in some detail. Obtained final results are in formal
accord with those earlier obtained by Vergne [PNAS 93 (1996) 14238].Comment: 48 pages J.Geom.Phys.(in press, available on line
Mixed-integer bilevel representability
We study the representability of sets that admit extended formulations using
mixed-integer bilevel programs. We show that feasible regions modeled by
continuous bilevel constraints (with no integer variables), complementarity
constraints, and polyhedral reverse convex constraints are all finite unions of
polyhedra. Conversely, any finite union of polyhedra can be represented using
any one of these three paradigms. We then prove that the feasible region of
bilevel problems with integer constraints exclusively in the upper level is a
finite union of sets representable by mixed-integer programs and vice versa.
Further, we prove that, up to topological closures, we do not get additional
modeling power by allowing integer variables in the lower level as well. To
establish the last statement, we prove that the family of sets that are finite
unions of mixed-integer representable sets forms an algebra of sets (up to
topological closures)
Noncommutative geometry on trees and buildings
We describe the construction of theta summable and finitely summable spectral
triples associated to Mumford curves and some classes of higher dimensional
buildings. The finitely summable case is constructed by considering the
stabilization of the algebra of the dual graph of the special fiber of the
Mumford curve and a variant of the Antonescu-Christensen spectral geometries
for AF algebras. The information on the Schottky uniformization is encoded in
the spectral geometry through the Patterson-Sullivan measure on the limit set.
Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum
Toric Methods in F-theory Model Building
In this review article we discuss recent constructions of global F-theory GUT
models and explain how to make use of toric geometry to do calculations within
this framework. After introducing the basic properties of global F-theory GUTs
we give a self-contained review of toric geometry and introduce all the tools
that are necessary to construct and analyze global F-theory models. We will
explain how to systematically obtain a large class of compact Calabi-Yau
fourfolds which can support F-theory GUTs by using the software package PALP.Comment: 19 pages. Prepared for the special issue "Computational Algebraic
Geometry in String and Gauge Theory" of Advances in High Energy Physics, v2:
references added, typos correcte
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