- Full text PDF available (8)
1. Introduction In this paper we study non-negative smooth solutions of the conformally invariant equation in a punctured ball, B,(O) \ (0) c R " , n 2 3, with an isolated singularity at the origin. The model equation (1.1) arises in many physical contexts but its greatest interest in recent years lies in its relation to the Yamabe problem. From this… (More)
31 as a sum of scaled versions of the measure dx=x 2 on intervals 0; a], and this proves the theorem. QED Recall that the Cauchy distribution is innnitely divisible with Levy measure dx=jxj 2. This is why we call the random variables deened by C1 jxj 2 dx 0;a] `Cauchy-like'. In fact, it is easy to see that these have C 1 distribution functions. This follows… (More)
The question of symmetry in nonlinear partial differential equations has been the subject of intensive investigations over the past 25 years. The general theme is the following. Suppose the domain ft, as well as the boundary condition on dft, has some symmetry, for example radial symmetry, axial symmetry or symmetry with respect to some hyperplane. Do… (More)
Integral conditions on the traceless Ricci tensor are used to characterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for Lp-solutions of a large class of differential inequalities. Further applications of the technique are also given.
This thesis studies the statistics of natural images. It contains two chapters. Chapter 1 is on the scale invariance, a mysterious property of natural images. Chapter 2 is on the compression related to statistics of natural images. Chapter 1 explores evidence of scale invariance of natural images, and explains why natural images have this nice property.… (More)
In this thesis we restrict ourselves to stationary and discrete valued stochastic processes. A pair of stochastic processes (X, Y) is a Hidden Markov Model (HMM) if X (the state process) is a Markov process and Y (the observable process) is an incomplete observation of X. The observation can be deterministic or noisy and the observable can be a state or a… (More)