46,526 research outputs found
Contragredient representations over local fields of positive characteristic
It is conjectured by Adams-Vogan and Prasad that under the local Langlands
correspondence, the L-parameter of the contragredient representation equals
that of the original representation composed with the Chevalley involution of
the L-group. We verify a variant of their prediction for all connected
reductive groups over local fields of positive characteristic, in terms of the
local Langlands parameterization of Genestier-Lafforgue. We deduce this from a
global result for cuspidal automorphic representations over function fields,
which is in turn based on a description of the transposes of V. Lafforgue's
excursion operators.Comment: 42 pages. Updated the reference
Doubling measures on uniform Cantor sets
We obtain a complete description for a probability measure to be doubling on
an arbitrarily given uniform Cantor set. The question of which doubling
measures on such a Cantor set can be extended to a doubling measure on [0; 1]
is also considered
The small and large D limit of Parikh-Wilczek tunneling model for Hawking radiation
In this note, we study both the small and large dimension limit of the
tunneling model of Hwaking radiation proposed by Parikh and
Wilczek\cite{Parikh:1999mf}. We confirm that the result is still valid for arbitrary . The sensible large limit is given
by in order to have nonzero radiation. On the other hand,
the sensible small limit is given by taking as a continuous
parameter. We also explicitly show the leading order correction to the thermal
radiation and discuss its connection to the two-dimensional dilaton gravity.Comment: 7 pages, updated version submitted to CQ
Zeta integrals, Schwartz spaces and local functional equations
According to Sakellaridis, many zeta integrals in the theory of automorphic
forms can be produced or explained by appropriate choices of a Schwartz space
of test functions on a spherical homogeneous space, which are in turn dictated
by the geometry of affine spherical embeddings. We pursue this perspective by
developing a local counterpart and try to explicate the functional equations.
These constructions are also related to the -spectral decomposition of
spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To
justify this viewpoint, we prove the convergence of -adic local zeta
integrals under certain premises, work out the case of prehomogeneous vector
spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore,
we explain the doubling method and show that it fits into the paradigm of
-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the
constructions of Braverman and Kazhdan (2002). In the global setting, we give
certain speculations about global zeta integrals, Poisson formulas and their
relation to period integrals.Comment: This version corrects the Lemma 7.4.5 and the proof of Theorem 7.4.7,
along with some other minor corrections for the published version in LNM 222
Basic functions and unramified local L-factors for split groups
According to a program of Braverman, Kazhdan and Ng\^o Bao Ch\^au, for a
large class of split unramified reductive groups and representations
of the dual group , the unramified local -factor
can be expressed as the trace of for a suitable function
with non-compact support whenever . Such
functions can be plugged into the trace formula to study certain sums of
automorphic -functions. It also fits into the conjectural framework of
Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term
basic functions; this is supposed to lead to a generalized
Tamagawa-Godement-Jacquet theory for . In this article, we derive
some basic properties for the basic functions and interpret them
via invariant theory. In particular, their coefficients are interpreted as
certain generalized Kostka-Foulkes polynomials defined by Panyushev. These
coefficients can be encoded into a rational generating function.Comment: 42 pages, largely revise
On The Development of Nonlinear Operator Theory
The basic results for nonlinear operators are given. These results include
nonlinear versions of classical uniform boundedness theorem and Hahn-Banach
theorem. Furthermore, the mappings from a metrizable space into another normed
space can fall in some normed spaces by defining suitable norms. The results
for the mappings on the metrizable spaces can be applied to the operators on
the space of bounded linear functionals corresponding to the Dirac's delta
function
Shear Viscosity of a Non-Relativistic Conformal Gas in Two Dimensions
The shear viscosity, eta, of a fermi gas with non-relativistic conformal
symmetry in two spatial dimensions is investigated. We find that eta/s, s being
the entropy density, diverges as a gas of free particles in this system. It is
in contrast to the eta/s=1/(4 pi) found using non-relativistic AdS/CFT
correspondence, which requires a strongly interacting CFT. It implies the
unitary fermi gas in two spatial dimensions is not likely to have a weakly
interacting gravity dual.Comment: 6 pages, 1 figure; minor corrections and references adde
Stable conjugacy and epipelagic L-packets for Brylinski-Deligne covers of Sp(2n)
Let be a local field of characteristic not . We propose a definition
of stable conjugacy for all the covering groups of
constructed by Brylinski and Deligne, whose degree we denote by . To support
this notion, we follow Kaletha's approach to construct genuine epipelagic
-packets for such covers in the non-archimedean case with , or
some weaker variant when ; we also prove the stability of packets
when with large. When , the stable conjugacy
reduces to that defined by J. Adams, and the epipelagic -packets coincide
with those obtained by -correspondence. This fits within Weissman's
formalism of L-groups. For and even, it is also compatible with the
transfer factors proposed by K. Hiraga and T. Ikeda.Comment: 92 pages, with an index. The new Section 10 is the Errata that fixes
the mistakes in Sections 7.2 and 8.3 in the published versio
Multi-bump ground states of the fractional Gierer-Meinhardt system in
In this paper we study ground-states of the fractional Gierer-Meinhardt
system on the line, namely the solutions of the problem \begin{equation*}
\left\{\begin{array}{ll} (-\Delta)^su+u-\frac{u^2}{v}=0,\quad
&\mathrm{in}~\mathbb{R},\\ (-\Delta)^sv+\varepsilon^{2s}v-u^2=0,\quad
&\mathrm{in}~\mathbb{R},\\ u,v>0,\quad
u,v\rightarrow0~&\mathrm{as}~|x|\rightarrow+\infty. \end{array}\right.
\end{equation*} We prove that given any positive integer there exists a
solution to this problem for exhibiting exactly bumps in
its component, separated from each other at a distance
for and
for respectively, whenever
is sufficiently small. These bumps resemble the shape of the
unique solution of \begin{equation*} (-\Delta)^sU+U-U^2=0,\quad
0<U(y)\rightarrow0~\mathrm{as}~|y|\rightarrow\infty. \end{equation*}Comment: 31 pages; comments welcom
Exact Sparse Signal Recovery via Orthogonal Matching Pursuit with Prior Information
The orthogonal matching pursuit (OMP) algorithm is a commonly used algorithm
for recovering -sparse signals \x\in \mathbb{R}^{n} from linear model
\y=\A\x, where \A\in \mathbb{R}^{m\times n} is a sensing matrix. A
fundamental question in the performance analysis of OMP is the characterization
of the probability that it can exactly recover \x for random matrix \A.
Although in many practical applications, in addition to the sparsity, \x
usually also has some additional property (for example, the nonzero entries of
\x independently and identically follow the Gaussian distribution), none of
existing analysis uses these properties to answer the above question. In this
paper, we first show that the prior distribution information of \x can be
used to provide an upper bound on \|\x\|_1^2/\|\x\|_2^2, and then explore the
bound to develop a better lower bound on the probability of exact recovery with
OMP in iterations. Simulation tests are presented to illustrate the
superiority of the new bound.Comment: To appear in ICASSP 201
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