13,698 research outputs found
Branching laws for Verma modules and applications in parabolic geometry. I
We initiate a new study of differential operators with symmetries and combine
this with the study of branching laws for Verma modules of reductive Lie
algebras. By the criterion for discretely decomposable and multiplicity-free
restrictions of generalized Verma modules [T. Kobayashi,
http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are
brought to natural settings of parabolic geometries for which there exist
unique equivariant differential operators to submanifolds. Then we apply a new
method (F-method) relying on the Fourier transform to find singular vectors in
generalized Verma modules, which significantly simplifies and generalizes many
preceding works. In certain cases, it also determines the Jordan--H\"older
series of the restriction for singular parameters. The F-method yields an
explicit formula of such unique operators, for example, giving an intrinsic and
new proof of Juhl's conformally invariant differential operators [Juhl,
http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and its
generalizations. This article is the first in the series, and the next ones
include their extension to curved cases together with more applications of the
F-method to various settings in parabolic geometries
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Scale relativity and fractal space-time: theory and applications
In the first part of this contribution, we review the development of the
theory of scale relativity and its geometric framework constructed in terms of
a fractal and nondifferentiable continuous space-time. This theory leads (i) to
a generalization of possible physically relevant fractal laws, written as
partial differential equation acting in the space of scales, and (ii) to a new
geometric foundation of quantum mechanics and gauge field theories and their
possible generalisations. In the second part, we discuss some examples of
application of the theory to various sciences, in particular in cases when the
theoretical predictions have been validated by new or updated observational and
experimental data. This includes predictions in physics and cosmology (value of
the QCD coupling and of the cosmological constant), to astrophysics and
gravitational structure formation (distances of extrasolar planets to their
stars, of Kuiper belt objects, value of solar and solar-like star cycles), to
sciences of life (log-periodic law for species punctuated evolution, human
development and society evolution), to Earth sciences (log-periodic
deceleration of the rate of California earthquakes and of Sichuan earthquake
replicas, critical law for the arctic sea ice extent) and tentative
applications to system biology.Comment: 63 pages, 14 figures. In : First International Conference on the
Evolution and Development of the Universe,8th - 9th October 2008, Paris,
Franc
Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems
We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for
the kernels associated with power-law fundamental solutions of the polyharmonic
equation on d-dimensional Euclidean space. From these series representations we
derive Fourier expansions in certain rotationally-invariant coordinate systems
and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates.
We compare both of these expansions to generate addition theorems for the
azimuthal Fourier coefficients
Notes on conformal invariance of gauge fields
In Lagrangian gauge systems, the vector space of global reducibility
parameters forms a module under the Lie algebra of symmetries of the action.
Since the classification of global reducibility parameters is generically
easier than the classification of symmetries of the action, this fact can be
used to constrain the latter when knowing the former. We apply this strategy
and its generalization for the non-Lagrangian setting to the problem of
conformal symmetry of various free higher spin gauge fields. This scheme allows
one to show that, in terms of potentials, massless higher spin gauge fields in
Minkowski space and partially-massless fields in (A)dS space are not conformal
for spin strictly greater than one, while in terms of curvatures, maximal-depth
partially-massless fields in four dimensions are also not conformal, unlike the
closely related, but less constrained, maximal-depth Fradkin--Tseytlin fields.Comment: 38 page
Probability & incompressible Navier-Stokes equations: An overview of some recent developments
This is largely an attempt to provide probabilists some orientation to an
important class of non-linear partial differential equations in applied
mathematics, the incompressible Navier-Stokes equations. Particular focus is
given to the probabilistic framework introduced by LeJan and Sznitman [Probab.
Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al.
[Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140,
2004, in press]. In particular this is an effort to provide some foundational
facts about these equations and an overview of some recent results with an
indication of some new directions for probabilistic consideration.Comment: Published at http://dx.doi.org/10.1214/154957805100000078 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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