350 research outputs found
A priori estimates for the complex Hessian equations
We prove some a priori estimates as well as existence and
stability theorems for the weak solutions of the complex Hessian equations in
domains of and on compact K\"ahler manifolds. We also show optimal
integrability for m-subharmonic functions with compact singularities, thus
partially confirming a conjecture of Blocki. Finally we obtain a local
regularity result for solutions of the real and complex Hessian
equations under suitable regularity assumptions on the right hand side. In the
real case the method of this proof improves a result of Urbas.Comment: 18 pages, preliminary versio
On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone
We investigate the generally assumed inconsistency in light cone quantum
field theory that the restriction of a massive, real, scalar, free field to the
nullplane is independent of mass \cite{LKS}, but the
restriction of the two-point function depends on it (see, e.g., \cite{NakYam77,
Yam97}). We resolve this inconsistency by showing that the two-point function
has no canonical restriction to in the sense of distribution theory.
Only the so-called tame restriction of the two-point function exists which we
have introduced in \cite{Ull04sub}. Furthermore, we show that this tame
restriction is indeed independent of mass. Hence the inconsistency appears only
by the erroneous assumption that the two-point function would have a
(canonical) restriction to .Comment: 10 pages, 2 figure
Foliations and Chern-Heinz inequalities
We extend the Chern-Heinz inequalities about mean curvature and scalar
curvature of graphs of -functions to leaves of transversally oriented
codimension one -foliations of Riemannian manifolds. That extends
partially Salavessa's work on mean curvature of graphs and generalize results
of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and
Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of
3-dimensional Riemannian manifolds by constant mean curvature surfaces. These
Chern-Heinz inequalities for foliations can be applied to prove
Haymann-Makai-Osserman inequality (lower bounds of the fundamental tones of
bounded open subsets in terms of its inradius)
for embedded tubular neighborhoods of simple curves of .Comment: This paper is an improvment of an earlier paper titled On Chern-Heinz
Inequalities. 8 Pages, Late
Obstructions to determinantal representability
There has recently been ample interest in the question of which sets can be
represented by linear matrix inequalities (LMIs). A necessary condition is that
the set is rigidly convex, and it has been conjectured that rigid convexity is
also sufficient. To this end Helton and Vinnikov conjectured that any real zero
polynomial admits a determinantal representation with symmetric matrices. We
disprove this conjecture. By relating the question of finding LMI
representations to the problem of determining whether a polymatroid is
representable over the complex numbers, we find a real zero polynomial such
that no power of it admits a determinantal representation. The proof uses
recent results of Wagner and Wei on matroids with the half-plane property, and
the polymatroids associated to hyperbolic polynomials introduced by Gurvits.Comment: 10 pages. To appear in Advances in Mathematic
Numerical Measure of a Complex Matrix
International audienceWe introduce a natural probability measure over the numerical range of a complex matrix A â M n (C). This numerical measure ” A can be defined as the law of the random variable X â C when the vector X â C n is uniformly distributed on the unit sphere. If the matrix A is normal, we show that ” A has a piecewise polynomial density f A , which can be identified with a multivariate B-spline. In the general (nonnormal) case, we relate the Radon transform of ” A to the spectrum of a family of Hermitian matrices, and we deduce an explicit representation formula for the numerical density which is appropriate for theoretical and computational purposes. As an application, we show that the density f A is polynomial in some regions of the complex plane which can be characterized geometrically, and we recover some known results about lacunas of symmetric hyperbolic systems in 2 + 1 dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix A â M n (C) concentrates to a Dirac mass as the size n goes to infinity
Sharp Garding inequality on compact Lie groups
In this paper the sharp Garding inequality is established on compact Lie
groups. The positivity condition is expressed in the non-commutative phase
space in terms of the full symbol, which is defined using the representations
of the group. Applications are given to the and Sobolev boundedness of
pseudo-differential operators.Comment: 18 page
Polynomials with the half-plane property and matroid theory
A polynomial f is said to have the half-plane property if there is an open
half-plane H, whose boundary contains the origin, such that f is non-zero
whenever all the variables are in H. This paper answers several open questions
regarding multivariate polynomials with the half-plane property and matroid
theory.
* We prove that the support of a multivariate polynomial with the half-plane
property is a jump system. This answers an open question posed by Choe, Oxley,
Sokal and Wagner and generalizes their recent result claiming that the same is
true whenever the polynomial is also homogeneous.
* We characterize multivariate multi-affine polynomial with real coefficients
that have the half-plane property (with respect to the upper half-plane) in
terms of inequalities. This is used to answer two open questions posed by Choe
and Wagner regarding strongly Rayleigh matroids.
* We prove that the Fano matroid is not the support of a polynomial with the
half-plane property. This is the first instance of a matroid which does not
appear as the support of a polynomial with the half-plane property and answers
a question posed by Choe et al.
We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat
Fermion mixing in quasi-free states
Quantum field theoretic treatments of fermion oscillations are typically
restricted to calculations in Fock space. In this letter we extend the
oscillation formulae to include more general quasi-free states, and also
consider the case when the mixing is not unitary.Comment: 10 pages, Plain Te
Wavefront sets and polarizations on supermanifolds
In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory
- âŠ