364 research outputs found
Bipodal structure in oversaturated random graphs
We study the asymptotics of large simple graphs constrained by the limiting
density of edges and the limiting subgraph density of an arbitrary fixed graph
. We prove that, for all but finitely many values of the edge density, if
the density of is constrained to be slightly higher than that for the
corresponding Erd\H{o}s-R\'enyi graph, the typical large graph is bipodal with
parameters varying analytically with the densities. Asymptotically, the
parameters depend only on the degree sequence of
Kodiak: An Implementation Framework for Branch and Bound Algorithms
Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements
Toric partial density functions and stability of toric varieties
Let denote a polarized toric K\"ahler manifold. Fix a
toric submanifold and denote by the
partial density function corresponding to the partial Bergman kernel projecting
smooth sections of onto holomorphic sections of that vanish to
order at least along , for fixed such that . We
prove the existence of a distributional expansion of as , including the identification of the coefficient of as a
distribution on . This expansion is used to give a direct proof that if
has constant scalar curvature, then must be slope semi-stable
with respect to . Similar results are also obtained for more general partial
density functions. These results have analogous applications to the study of
toric K-stability of toric varieties.Comment: Accepted by Mathematische Annalen on 13 September 201
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