34,515 research outputs found
Fractional powers and singular perturbations of quantum differential Hamiltonians
We consider the fractional powers of singular (point-like) perturbations of
the Laplacian, and the singular perturbations of fractional powers of the
Laplacian, and we compare such two constructions focusing on their perturbative
structure for resolvents and on the local singularity structure of their
domains. In application to the linear and non-linear Schr\"{o}dinger equations
for the corresponding operators we outline a programme of relevant questions
that deserve being investigated.Comment: Published on J. Math. Phys. (2018
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
The Bianchi Ix (MIXMASTER) Cosmological Model is Not Integrable
The perturbation of an exact solution exhibits a movable transcendental
essential singularity, thus proving the nonintegrability. Then, all possible
exact particular solutions which may be written in closed form are isolated
with the perturbative Painlev\'e test; this proves the inexistence of any
vacuum solution other than the three known ones.Comment: 14 pages, no figure
On the Green function of linear evolution equations for a region with a boundary
We derive a closed-form expression for the Green function of linear evolution
equations with the Dirichlet boundary condition for an arbitrary region, based
on the singular perturbation approach to boundary problems.Comment: 9 page
Learning Mixtures of Gaussians in High Dimensions
Efficiently learning mixture of Gaussians is a fundamental problem in
statistics and learning theory. Given samples coming from a random one out of k
Gaussian distributions in Rn, the learning problem asks to estimate the means
and the covariance matrices of these Gaussians. This learning problem arises in
many areas ranging from the natural sciences to the social sciences, and has
also found many machine learning applications. Unfortunately, learning mixture
of Gaussians is an information theoretically hard problem: in order to learn
the parameters up to a reasonable accuracy, the number of samples required is
exponential in the number of Gaussian components in the worst case. In this
work, we show that provided we are in high enough dimensions, the class of
Gaussian mixtures is learnable in its most general form under a smoothed
analysis framework, where the parameters are randomly perturbed from an
adversarial starting point. In particular, given samples from a mixture of
Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give
an algorithm that learns the parameters with polynomial running time and using
polynomial number of samples. The central algorithmic ideas consist of new ways
to decompose the moment tensor of the Gaussian mixture by exploiting its
structural properties. The symmetries of this tensor are derived from the
combinatorial structure of higher order moments of Gaussian distributions
(sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop
new tools for bounding smallest singular values of structured random matrices,
which could be useful in other smoothed analysis settings
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
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