A Pointwise a-priori Estimate for the d-bar Neumann Problem on Weakly Pseudoconvex Domains

We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained in case the boundary is of finite type

Regular Representations of Time-Frequency Groups

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let $G$ be a time-frequency group. More precisely, that is $G=\left\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}^{d}\right\rangle ,$ $T_{k}$, $M_{l}$ are translations and modulations operators acting in $L^{2}(\mathbb{R}^{d}),$ and $B$ is a non-singular matrix. We compute the Plancherel measure of the left regular representation of $G\$which is denoted by $L.$ The action of $G$ on $L^{2}(\mathbb{R}^{d})$ induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of $L$ into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut F\"uhr's results which are only obtained for the restricted case where $d=1$, $B=1/L,L\in\mathbb{Z}$ and $L>1.$ Even in the case where $G$ is not type I, we are able to obtain a decomposition of the left regular representation of $G$ into a direct integral decomposition of irreducible representations when $d=1$. Some interesting applications to Gabor theory are given as well. For example, when $B$ is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of $G.

The Dual Nature of Harm - In Defence of the Disjunctive View -

Much of the debate about the nature of harm regard the standard comparative views and the alternative non-comparative views. The former claim that harm always involves a subject that is made worse off. The latter acknowledges a more absolute understanding of harm, as a subject that is caused to be in a bad state. The virtues and vices of these views have been wildly debated. Even thought both notions have intuitive support, they also have serious problems. It has been suggested quite recently that the nature of harm is best understood as a combination of a comparative and a non-comparative condition - a disjunctive view. The thought is that the disjunctivist can account for many of the cases that strict views struggle with, since the two senses complement each other. However, there are some new potential problems with combining the two senses of harm. The disjunctive view has been questioned on the grounds of being ad hoc, incapable of fully solving the Non-Identity problem and contradicting Parfit's No-Difference View. This essay addresses the mentioned worries and explores possible defence strategies

Functional Equations and Fourier Analysis

By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation, on compact groups.Comment: 8 pages, to appear in CM

Axially Symmetric Cosmological Mesonic Stiff Fluid Models in Lyra's Geometry

In this paper, we obtained a new class of axially symmetric cosmological mesonic stiff fluid models in the context of Lyra's geometry. Expressions for the energy, pressure and the massless scalar field are derived by considering the time dependent displacement field. We found that the mesonic scalar field depends on only $t$ coordinate. Some physical properties of the obtained models are discussed.Comment: 13 pages, no figures, typos correcte

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.Comment: 17 page

Upper bounds for packings of spheres of several radii

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.Comment: 31 page