Finite range Decomposition of Gaussian Processes

Let \D be the finite difference Laplacian associated to the lattice \bZ^{d}. For dimension $d\ge 3$, $a\ge 0$ and $L$ a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G^{a}:=(a-\D)^{-1} can be decomposed as an infinite sum of positive semi-definite functions $V_{n}$ of finite range, $V_{n} (x-y) = 0$ for $|x-y|\ge O(L)^{n}$. Equivalently, the Gaussian process on the lattice with covariance $G^{a}$ admits a decomposition into independent Gaussian processes with finite range covariances. For $a=0$, $V_{n}$ has a limiting scaling form $L^{-n(d-2)}\Gamma_{c,\ast}{\bigl (\frac{x-y}{L^{n}}\bigr)}$ as $n\to \infty$. As a corollary, such decompositions also exist for fractional powers (-\D)^{-\alpha/2}, $0<\alpha \leq 2$. The results of this paper give an alternative to the block spin renormalization group on the lattice.Comment: 26 pages, LaTeX, paper in honour of G.Jona-Lasinio.Typos corrected, corrections in section 5 and appendix

The bacterial toxin CNF1 as a tool to induce retinal degeneration reminiscent of retinitis pigmentosa.

Retinitis pigmentosa (RP) comprises a group of inherited pathologies characterized by progressive photoreceptor degeneration. In rodent models of RP, expression of defective genes and retinal degeneration usually manifest during the first weeks of postnatal life, making it difficult to distinguish consequences of primary genetic defects from abnormalities in retinal development. Moreover, mouse eyes are small and not always adequate to test pharmacological and surgical treatments. An inducible paradigm of retinal degeneration potentially extensible to large animals is therefore desirable. Starting from the serendipitous observation that intraocular injections of a Rho GTPase activator, the bacterial toxin Cytotoxic Necrotizing Factor 1 (CNF1), lead to retinal degeneration, we implemented an inducible model recapitulating most of the key features of Retinitis Pigmentosa. The model also unmasks an intrinsic vulnerability of photoreceptors to the mechanism of CNF1 action, indicating still unexplored molecular pathways potentially leading to the death of these cells in inherited forms of retinal degeneratio

Biological effects of a software-controlled voltage pulse generator (PhyBack PBK-2C) on the release of vascular endothelial growth factor (VEGF)

Electrical stimulation (ES) may induce vascular permeability and physiological angiogenesis. ES of rat muscles significantly increases the microvessel density and vascular endothelial growth factor (VEGF) protein levels. Thus, a pilot study was designed to analyze the effects of low-voltage electric impulses on VEGF levels in patients with dystrophic ulcers