3,164 research outputs found
Strictly Positive Definite Kernels on a Product of Spheres II
We present, among other things, a necessary and sufficient condition for the
strict positive definiteness of an isotropic and positive definite kernel on
the cartesian product of a circle and a higher dimensional sphere. The result
complements similar results previously obtained for strict positive
definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169]
and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016),
286-301, arXiv:1505.03695]
On bulk singularities in the random normal matrix model
We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to
study the distribution of eigenvalues close to a bulk singularity, i.e. a point
in the interior of the droplet where the density of the classical equilibrium
measure vanishes. We prove results to the effect that a certain "dominant part"
of the Taylor expansion determines the microscopic properties near a bulk
singularity. A description of the distribution is given in terms of a special
entire function, which depends on the nature of the singularity (a
Mittag-Leffler function in the case of a rotationally symmetric singularity).Comment: This version clarifies on the proof of Theorem
Positive and generalized positive real lemma for slice hyperholomorphic functions
In this paper we prove a quaternionic positive real lemma as well as its
generalized version, in case the associated kernel has negative squares for
slice hyperholomorphic functions. We consider the case of functions with
positive real part in the half space of quaternions with positive real part, as
well as the case of (generalized) Schur functions in the open unit ball
Rescaling Ward identities in the random normal matrix model
We study existence and universality of scaling limits for the eigenvalues of
a random normal matrix, in particular at points on the boundary of the
spectrum. Our approach uses Ward's equation, which is an identity satisfied by
the 1-point function.Comment: This is a substantial revision with several new results. The previous
section 7 on singular boundary points has been lifted out and developed in a
separate not
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