149,943 research outputs found
A sharp centro-affine isospectral inequality of Szeg\"{o}--Weinberger type and the -Minkowski problem
We establish a sharp upper-bound for the first non-zero even eigenvalue
(corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski
operator associated to a strictly convex -smooth origin-symmetric convex
body in . Our isospectral inequality is centro-affine
invariant, attaining equality if and only if is a (centered) ellipsoid;
this is reminiscent of the (non affine invariant) classical
Szeg\"{o}--Weinberger isospectral inequality for the Neumann Laplacian. The new
upper-bound complements the conjectural lower-bound, which has been shown to be
equivalent to the log-Brunn-Minkowski inequality and is intimately related to
the uniqueness question in the even log-Minkowski problem. As applications, we
obtain new strong non-uniqueness results in the even -Minkowski problem in
the subcritical range , as well as new rigidity results for the
critical exponent and supercritical regime . In particular, we
show that any as above which is not an ellipsoid is a witness to
non-uniqueness for all and some , and that
can be taken to be arbitrarily close to .Comment: 30 page
Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming
With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given
Uniqueness of radial solutions for the fractional Laplacian
We prove general uniqueness results for radial solutions of linear and
nonlinear equations involving the fractional Laplacian with for any space dimensions . By extending a monotonicity
formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear
equation in has at most one radial and
bounded solution vanishing at infinity, provided that the potential is a
radial and non-decreasing. In particular, this result implies that all radial
eigenvalues of the corresponding fractional Schr\"odinger operator
are simple. Furthermore, by combining these findings on
linear equations with topological bounds for a related problem on the upper
half-space , we show uniqueness and nondegeneracy of ground
state solutions for the nonlinear equation in for arbitrary space dimensions and all
admissible exponents . This generalizes the nondegeneracy and
uniqueness result for dimension N=1 recently obtained by the first two authors
in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves
of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma
8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2
corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl.
Mat
The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case
We study the long-time behaviour of nonnegative solutions of the Porous
Medium Equation posed on Cartan-Hadamard manifolds having very large negative
curvature, more precisely when the sectional or Ricci curvatures diverge at
infinity more than quadratically in terms of the geodesic distance to the pole.
We find an unexpected separate-variable behaviour that reminds one of Dirichlet
problems on bounded Euclidean domains. As a crucial step, we prove existence of
solutions to a related sublinear elliptic problem, a result of independent
interest. Uniqueness of solutions vanishing at infinity is also shown, along
with comparison principles, both in the parabolic and in the elliptic case.
Our results complete previous analyses of the Porous Medium Equation flow on
negatively curved Riemannian manifolds, which were carried out first for the
hyperbolic space and then for general Cartan-Hadamard manifolds with a negative
curvature having at most quadratic growth. We point out that no similar
analysis seems to exist for the linear heat flow.
We also translate such results into some weighted Porous Medium Equations in
the Euclidean space having special weights.Comment: Slight change in the title. Final version, to appear on Math. An
Representation Learning for Clustering: A Statistical Framework
We address the problem of communicating domain knowledge from a user to the
designer of a clustering algorithm. We propose a protocol in which the user
provides a clustering of a relatively small random sample of a data set. The
algorithm designer then uses that sample to come up with a data representation
under which -means clustering results in a clustering (of the full data set)
that is aligned with the user's clustering. We provide a formal statistical
model for analyzing the sample complexity of learning a clustering
representation with this paradigm. We then introduce a notion of capacity of a
class of possible representations, in the spirit of the VC-dimension, showing
that classes of representations that have finite such dimension can be
successfully learned with sample size error bounds, and end our discussion with
an analysis of that dimension for classes of representations induced by linear
embeddings.Comment: To be published in Proceedings of UAI 201
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
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