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 $D$ limit of the tunneling model of Hwaking radiation proposed by Parikh and Wilczek\cite{Parikh:1999mf}. We confirm that the result $\Gamma \sim e^{\Delta S}$ is still valid for arbitrary $D>3$. The sensible large $D$ limit is given by $\sqrt{D} \ll r_0$ in order to have nonzero radiation. On the other hand, the sensible small $D$ limit is given by taking $D=3+\epsilon$ 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 $L^2$-spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of $p$-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 $L$-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 $G$ and representations $\rho$ of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,\pi,\rho)$ can be expressed as the trace of $\pi(f_{\rho,s})$ for a suitable function $f_{\rho,s}$ with non-compact support whenever $\mathrm{Re}(s) \gg 0$. Such functions can be plugged into the trace formula to study certain sums of automorphic $L$-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 $(G,\rho)$. In this article, we derive some basic properties for the basic functions $f_{\rho,s}$ 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 $F$ be a local field of characteristic not $2$. We propose a definition of stable conjugacy for all the covering groups of $\mathrm{Sp}(2n,F)$ constructed by Brylinski and Deligne, whose degree we denote by $m$. To support this notion, we follow Kaletha's approach to construct genuine epipelagic $L$-packets for such covers in the non-archimedean case with $p \nmid 2m$, or some weaker variant when $4 \mid m$; we also prove the stability of packets when $F \supset \mathbb{Q}_p$ with $p$ large. When $m=2$, the stable conjugacy reduces to that defined by J. Adams, and the epipelagic $L$-packets coincide with those obtained by $\Theta$-correspondence. This fits within Weissman's formalism of L-groups. For $n=1$ and $m$ 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 R\mathbb{R}

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 $k,$ there exists a solution to this problem for $s\in[\frac12,1)$ exhibiting exactly $k$ bumps in its $u-$component, separated from each other at a distance $O(\varepsilon^{\frac{1-2s}{4s}})$ for $s\in(\frac12,1)$ and $O(|\log\varepsilon|^{\frac12})$ for $s=\frac12$ respectively, whenever $\varepsilon$ 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 $K$-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 $K$ iterations. Simulation tests are presented to illustrate the superiority of the new bound.Comment: To appear in ICASSP 201