15,714 research outputs found
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Entanglement Entropy of the Low-Lying Excited States and Critical Properties of an Exactly Solvable Two-Leg Spin Ladder with Three-Spin Interactions
In this work, we investigate an exactly solvable two-leg spin ladder with
three-spin interactions. We obtain analytically the finite-size corrections of
the low-lying energies and determine the central charge as well as the scaling
dimensions. The model considered in this work has the same universality class
of critical behavior of the XX chain with central charge c=1. By using the
correlation matrix method, we also study the finite-size corrections of the
Renyi entropy of the ground state and of the excited states. Our results are in
agreement with the predictions of the conformal field theory.Comment: 10 pages, 6 figures, 2 table
Minimum Length from First Principles
We show that no device or gedanken experiment is capable of measuring a
distance less than the Planck length. By "measuring a distance less than the
Planck length" we mean, technically, resolve the eigenvalues of the position
operator to within that accuracy. The only assumptions in our argument are
causality, the uncertainty principle from quantum mechanics and a dynamical
criteria for gravitational collapse from classical general relativity called
the hoop conjecture. The inability of any gedanken experiment to measure a
sub-Planckian distance suggests the existence of a minimal length.Comment: 8 pages, Honorable Mention in the 2005 Gravity Research Foundation
Essay Competitio
Grand unification and enhanced quantum gravitational effects
In grand unified theories with large numbers of fields, renormalization
effects significantly modify the scale at which quantum gravity becomes strong.
This in turn can modify the boundary conditions for coupling constant
unification, if higher dimensional operators induced by gravity are taken into
consideration. We show that the generic size of these effects from gravity can
be larger than the two-loop corrections typically considered in renormalization
group analyses of unification. In some cases, gravitational effects of modest
size can render unification impossible.Comment: 4 pages, 1 figure, revtex; minor changes in v2 (version published in
Phys. Rev. Lett.
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