Series expansion for L^p Hardy inequalities

We consider a general class of sharp $L^p$ Hardy inequalities in $\R^N$ involving distance from a surface of general codimension $1\leq k\leq N$. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite series improvement of $L^p$ Hardy inequalities.Comment: 16 pages, to appear in the Indiana Univ. Math.

Critical Hardy--Sobolev Inequalities

We consider Hardy inequalities in $I R^n$, $n \geq 3$, with best constant that involve either distance to the boundary or distance to a surface of co-dimension $k<n$, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm.Comment: 22 page

PROPER 3C - A nucleon pion transport code

Computer programs to calculate and analyze pion and nucleon interaction within prescribed medium referred to as PROPER 3C Transport Cod

Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a byproduct of our technique we are able to answer positively to a conjecture of E.B.Davies.Comment: 40 page

Universality in Blow-Up for Nonlinear Heat Equations

We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer $k$, we construct a set of codimension $2k$ in the space of initial data giving rise to solutions that blow-up according to the given profile.Comment: 38 page

Hardy's inequality and curvature

A Hardy inequality of the form \int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, for all $f \in C_0^{\infty}({\tilde{\Omega}})$, is considered for $p\in (1,\infty)$, where ${\tilde{\Omega}}$ can be either $\Omega$ or $\mathbb{R}^n \setminus \Omega$ with $\Omega$ a domain in $\mathbb{R}^n$, $n \ge 2$, and $\delta({\bf{x}})$ is the distance from ${\bf{x}} \in {\tilde{\Omega}}$ to the boundary $\partial {\tilde{\Omega}}.$ The main emphasis is on determining the dependance of $a(\delta, \partial {\tilde{\Omega}})$ on the geometric properties of $\partial {\tilde{\Omega}}.$ A Hardy inequality is also established for any doubly connected domain $\Omega$ in $\mathbb{R}^2$ in terms of a uniformisation of $\Omega,$ that is, any conformal univalent map of $\Omega$ onto an annulus

Experiment for Testing Special Relativity Theory

An experiment aimed at testing special relativity via a comparison of the velocity of a non matter particle (annihilation photon) with the velocity of the matter particle (Compton electron) produced by the second annihilation photon from the decay Na-22(beta^+)Ne-22 is proposed.Comment: 7 pages, 1 figure, Report on the Conference of Nuclear Physics Division of Russian Academy of Science "Physics of Fundamental Interactions", ITEP, Moscow, November 26-30, 200