44 research outputs found
On symmetries of KdV-like evolution equations
The -dependence of the symmetries of (1+1)-dimensional scalar
translationally invariant evolution equations is described. The sufficient
condition of (quasi)polynomiality in time of the symmetries of evolution
equations with constant separant is found. The general form of time dependence
of the symmetries of KdV-like non-linearizable evolution equations is
presented.Comment: LaTeX, 8 pages, no figures, very minor change
Continuous rotation invariant valuations on convex sets
The famous Hadwiger theorem classifies all rigid motion invariant continuous
valuations on convex sets as linear conbinations of quermassintegrals. We prove
much more general result. We classify continuous valuations which are invariant
with respect to the orthogonal (or special orthogonal) group. Some applications
to integral geometry are given.Comment: 29 pages, published version, abstract added in migratio
Harish-Chandra's volume formula via Weyl's Law and Euler-Maclaurin formula
Harish-Chandra's volume formula shows that the volume of a flag manifold
, where the measure is induced by an invariant inner product on the Lie
algebra of , is determined up to a scalar by the algebraic properties of
. This article explains how to deduce Harish-Chandra's formula from Weyl's
law by utilizing the Euler-Maclaurin formula. This approach leads to a mystery
that lies under the Atiyah-Singer index theorem
On boundary confinements for the Coulomb gas
We introduce a family of boundary confinements for Coulomb gas ensembles, and
study them in the two-dimensional determinantal case of random normal matrices.
The family interpolates between the free boundary and hard edge cases, which
have been well studied in various random matrix theories. The confinement can
also be relaxed beyond the free boundary to produce ensembles with more fuzzy
boundaries, i.e., where the particles are more and more likely to be found
outside of the boundary. The resulting ensembles are investigated with respect
to scaling limits and distribution of the maximum modulus. In particular, we
prove existence of a new point field - a limit of scaling limits to the
ultraweak point when the droplet ceases to be well defined
Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH
We provide counter-examples to Mulmuley's strong saturation conjecture
(strong SH) for the Kronecker coefficients. This conjecture was proposed in the
setting of Geometric Complexity Theory to show that deciding whether or not a
Kronecker coefficient is zero can be done in polynomial time. We also provide a
short proof of the #P-hardness of computing the Kronecker coefficients. Both
results rely on the connections between the Kronecker coefficients and another
family of structural constants in the representation theory of the symmetric
groups: Murnaghan's reduced Kronecker coefficients.
An appendix by Mulmuley introduces a relaxed form of the saturation
hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.Comment: 25 pages. With an appendix by Ketan Mulmuley. To appear in
Computational Complexity. See also
http://emmanuel.jean.briand.free.fr/publications
Local Euler-Maclaurin formula for polytopes
We give a local Euler-Maclaurin formula for rational convex polytopes in a
rational euclidean space . For every affine rational polyhedral cone C in a
rational euclidean space W, we construct a differential operator of infinite
order D(C) on W with constant rational coefficients, which is unchanged when C
is translated by an integral vector. Then for every convex rational polytope P
in a rational euclidean space V and every polynomial function f (x) on V, the
sum of the values of f(x) at the integral points of P is equal to the sum, for
all faces F of P, of the integral over F of the function D(N(F)).f, where we
denote by N(F) the normal cone to P along F.Comment: Revised version (July 2006) has some changes of notation and
references adde
Local formulas for Ehrhart coefficients from lattice tiles
The coefficients of the Ehrhart polynomial of a lattice polytope can be written as a weighted sum of facial volumes. The weights in such a 'local formula' depend only on the outer normal cones of faces, but are far from being unique. In this thesis, we present local formulas μ based on choices of fundamental domains that, which allows a geometric interpretation of the values. Additionally, we generalize the results to Ehrhart quasipolynomials, prove new results about the symmetric behavior and introduce a variation well-suited for implementations.Die Koeffizienten der Ehrhart-Polynome eines Gitterpolytops können als eine gewichtete Summe über die Volumen der Seiten dargestellt werden. Die Gewichte einer solchen 'lokalen Formel' hängen nur von den Normalenkegeln der Seiten ab, sind aber nicht eindeutig. Wir präsentieren hier lokale Formeln μ. Die Konstruktion basiert auf Fundamentalzellen und erlaubt so eine geometrische Interpretation der Werte. Zudem verallgemeinern wir μ auf Ehrhart Quasipolynome, beweisen neue Symmetrieeigenschaften und zeigen Implementierungen