115,684 research outputs found
A Maximum Principle for Combinatorial Yamabe Flow
This article studies a discrete geometric structure on triangulated manifolds
and an associated curvature flow (combinatorial Yamabe flow). The associated
evolution of curvature appears to be like a heat equation on graphs, but it can
be shown to not satisfy the maximum principle. The notion of a parabolic-like
operator is introduced as an operator which satisfies the maximum principle,
but may not be parabolic in the usual sense of operators on graphs. A maximum
principle is derived for the curvature of combinatorial Yamabe flow under
certain assumptions on the triangulation, and hence the heat operator is shown
to be parabolic-like. The maximum principle then allows a characterization of
the curvature as well was a proof of long term existence of the flow.Comment: 20 pages, this is an almost entirely different paper. Some elements
of the old version are in the paper arxiv:math.MG/050618
On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and
of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound
in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a
(very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We
show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial
in the number of propositional variables. The proof technique for the lower bound extends the
idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle
beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal
Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. Ëť
We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles
Recombination and intrinsicality
In this paper I argue that warrant for Lewis' principle of recombination presupposes warrant for a combinatorial analysis of intrinsicality, which in turn presupposes warrant for the principle of recombination. This, I claim, leads to a vicious circularity: warrant for neither doctrine can get off the ground
Models with second order properties, V: A General principle
We present a general framework for carrying out some constructions. The
unifying factor is a combinatorial principle which we present in terms of a
game in which the first player challenges the second player to carry out
constructions which would be much easier in a generic extension of the
universe, and the second player cheats with the aid of Diamond. Section 1
contains an axiomatic framework suitable for the description of a number of
related constructions, and the statement of the main theorem in terms of this
framework. In section 2 we illustrate the use of our combinatorial principle.
The proof of the main result is then carried out in sections 3-5
Set mapping reflection
In this note we will discuss a new reflection principle which follows from
the Proper Forcing Axiom. The immediate purpose will be to prove that the
bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2
and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be
demonstrated that this reflection principle implies that combinatorial
principle Square(kappa) fails for all regular kappa > omega_1.Comment: 11 page
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