Newtonian Lorentz Metric Spaces

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space carries a doubling measure and a weak Poincare inequality) and some restrictions on q, we show that the Lipschitz functions are dense in those spaces. Moreover, in the same setting we show that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provide a counterexample to the density result of Lipschitz functions in the Euclidean setting when q is infinite.Comment: v2: 32 pages. Formula on page 23 corrected; typos remove

Spacecraft particulate sizing spectrometer

An evaluation prototype device is described, together with conclusions and several recommendations for follow-on flight hardware. The device detects individual particles crossing an external sensing zone, and produces a histogram displaying the size distribution of particles sensed, over the nominal range of 5 to 50 microns. The output is totally independent of the particle refractive index, and is also largely unaffected by particle shape. The reported diameters are in terms of the equivalent sphere, as judged by the scattered light intercepted by the receiving channels, which develop signals whenever a particle crosses the beam of illumination in the sensing zone. Supporting evidence for the latter assertion is discussed on the basis of experimental test data for non-spherical particulates. Also included is a technical appendix which presents theoretical arguments that provide a firm foundation for this assertion

Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincar\'e inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball

Harnack's Inequality for Parabolic De Giorgi Classes in Metric Spaces

In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincare' inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale and location invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers

Deterministic and efficient minimal perfect hashing schemes

Neste trabalho apresentamos versões determinísticas para os esquemasde hashing de Botelho, Kohayakawa e Ziviani (2005) e por Botelho, Pagh e Ziviani(2007). Também respondemos a um problema deixado em aberto no primeiro dostrabalhos, relacionado à prova da corretude e à análise de complexidade do esquemapor eles proposto. As versões determinísticas desenvolvidas foram implementadase testadas sobre conjuntos de dados com até 25.000.000 de chaves, e os resultadosverificados se mostraram equivalentes aos dos algoritmos aleatorizados originais

Some isoperimetric problems in planes with density

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in $\R^2$. Finally, we give a list of related open questions.Comment: 40 pages, 19 figure

Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$