Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Let $\nu=e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type $$\lambda u-L_\nu u=f,$$ where $\lambda>0$, $f\in L^2(X,\nu)$ and $L_\nu$ is the self-adjoint operator associated with the quadratic form \[(\psi,\varphi)\mapsto \int_X\left\langle\nabla_H\psi,\nabla_H\varphi\right\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(X,\nu).\

Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $\gamma$ and let $\lambda_1$ be the maximum eigenvalue of the covariance operator associated with $\gamma$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to\mathbb{R}$ and a convex set $\Omega\subseteq X$, we set $\nu:=e^{-U}\gamma$ and we consider the semigroup $(T_\Omega(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form $$(\varphi,\psi)\mapsto \int_\Omega\langle D_H\varphi,D_H\psi\rangle_Hd\nu,$$ where $\varphi,\psi$ belong to $D^{1,2}(\Omega,\nu)$, the Sobolev space defined as the domain of the closure in $L^2(\Omega,\nu)$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_\Omega(t))_{t\ge 0}$. In particular, we show that $$|D_H T_\Omega(t)f|_H^p\le e^{- p \lambda_1^{-1} t}(T_\Omega(t)|D_H f|^p_H), \qquad\, t>0,\ \nu\textrm{ -a.e. in }\Omega,$$ for any $p\in [1,+\infty)$ and $f\in D^{1,p}(\Omega ,\nu)$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincar\'e inequality in $\Omega$ for the measure $\nu$ and some improving summability properties for $(T_\Omega(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(\Omega,\nu)$ for some $p\in(1,\infty)$, then $$|D_H T_\Omega(t)f|^p_H \leq K_p t^{-\frac{p}{2}} T_\Omega(t)|f|^p,\qquad \, t>0,\ \nu\text{-a.e. in }\Omega,$$ where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_\Omega(t))_{t\geq 0}$ as $t$ goes to infinity

Creation and counting of defects in a temperature quenched Bose-Einstein Condensate

We study the spontaneous formation of defects in the order parameter of a trapped ultracold bosonic gas while crossing the critical temperature for Bose-Einstein Condensation (BEC) at different rates. The system has the shape of an elongated ellipsoid, whose transverse width can be varied to explore dimensionality effects. For slow enough temperature quenches we find a power-law scaling of the average defect number with the quench rate, as predicted by the Kibble-Zurek mechanism. A breakdown of such a scaling is found for fast quenches, leading to a saturation of the average defect number. We suggest an explanation for this saturation in terms of the mutual interactions among defects.Comment: 9 pages, 10 figure

Demonstração do Valor Adicionado como instrumento de evidenciação da riqueza distribuída à sociedade por empresas listadas na Bm

TCC (graduação) - Universidade Federal de Santa Catarina, Centro Sócio Econômico, Curso de Ciências Contábeis.Com o advento da lei 11.638/2007, a recente obrigatoriedade da publicação da Demonstração do Valor Adicionado e reconhecendo sua contribuição para verificação de desempenho encontra-se o tema deste trabalho. Tendo este por objetivo analisar distribuição do valor adicionado em sete empresas listadas na BM&F Bovespa do setor de atuação de Materiais Básicos publicadas no período de 2005 a 2009 uma vez que o ano de 2010 apesar de encerrado, ainda não teve publicação de demonstrações contábeis. Verificando ao longo dos períodos se há alguma semelhança no comportamento das distribuições de valores adicionados das empresas considerando serem estas do mesmo setor. Caracteriza-se por ser um estudo descritivo, com fonte de natureza secundária e abordagem quali-quant. Com essas delimitações visualizou-se um comportamento homogêneo na distribuição entre as empresas do setor e também ao longo do período, excetuando-se pelo ano de 2008 onde a crise econômica americana foi responsável por uma forte alteração no tipo de distribuição que em outros períodos tinha seu maior valor em Impostos, Taxas e Contribuições e devido a crise no ano de 2008 o maior percentual de distribuição foi em Remuneração de Capitais de Terceiros

Observation of Solitonic Vortices in Bose-Einstein Condensates

We observe solitonic vortices in an atomic Bose-Einstein condensate after free expansion. Clear signatures of the nature of such defects are the twisted planar density depletion around the vortex line, observed in absorption images, and the double dislocation in the interference pattern obtained through homodyne techniques. Both methods allow us to determine the sign of the quantized circulation. Experimental observations agree with numerical simulations. These solitonic vortices are the decay product of phase defects of the BEC order parameter spontaneously created after a rapid quench across the BEC transition in a cigar-shaped harmonic trap and are shown to have a very long lifetime.Comment: 7 pages, 7 figure