Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum decomposes, up error with superpolynomial decay, into $m$ distinct series, each associated one eigenvalues $A$. These spectral results are then applied study propagation singularit...

Starting with a W ∗-algebra M we use the inverse system obtained by cutting down M by its (central) projections to define an inverse limit of W ∗-algebras , and show that how this pro-W ∗-algebra encodes the local structure of M . For the C∗-algebras we do the same thing using their atomic enveloping W ∗-algebras . We investigate the relation of these ideas to the Akemann-Giles-Kummer non commu...

Recent projections of global sea-level rise by 2100 C.E. range from 20 cm (Meehl et al., 2007) to as much as 2 m (Vermeer and Rahmsdorf, 2009; Pfeffer et al., 2008), and sea level is projected to rise in the coming centuries even further. Without adaptation, a rise by 0.5 m would displace 3.8 million people in the most fertile part of the Nile River Delta (FitzGerald et al., 2008). A rise by 2 ...

For the weight function (1− ||x||) on the unit ball, a closed formula of the reproducing kernel is modified to include the case −1/2 < m < 0. The new formula is used to study the orthogonal projection of the weighted L space onto the space of polynomials of degree at most n, and it is proved that the uniform norm of the projection operator has the growth rate of n (d−1)/2 for m < 0, which is th...

This thesis examines the problem of reconstructing multiple discrete 2D objects, represented by a set of cells arranged in an m×n grid, from their projections. The objects being constructed are disjoint, hv-convex polyominoes, each of which has a separate colour. The main results presented here are two algorithms for unordered C-colour reconstruction that have time complexities of O(Cnm) and O(...

We assign two parameters to an M-ideal J in a Ba-nach space X, called the lower and upper grade, which are deened in terms of the size of balls contained in the set of best approximants from J. These quantities serve to measure how far J resembles an M-summand, and they enter into geometric descriptions in various ways. For instance they allow estimates of the interior of the metric complement ...

Let K be a subset of the Euclidean sphere S d−1. As seen in Lecture #1, in analyzing how well a given random projection matrix S ∈ R m×d preserves vectors in K, a central object is the random variable Z(K) = sup u∈K Su 2 2 m − 1. (2.1) Suppose that our goal is to establish that, for some δ ∈ (0, 1), we have Z(K) ≤ δ with high probability. How large must the projection dimension m be, as a funct...

Let K,L,M be orthogonal projections of a point P onto the sides of a triangle ABC. Then the locus of P such that K,L,M are collinear, is the circumcircle of ABC. This theorem has several generalizations [4], [5], [10],[6], [7], [9]. A generalization of the Simson–Wallace theorem which is by [2] ascribed to J. D. Gergonne is as follows: Let K,L,M be orthogonal projections of a point P onto the s...