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A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper, we provide an analysis of the spectral envelope reduction algorithm. We described related 1- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work involved in an envelope Cholesky factorization scheme. We formulate the latter two problems as quadratic assignment problems, and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a projected quadratic assignment problem, and then show that finding a permutation matrix closest to an orthogonal matrix attaining one of the lower bounds justifies the spectral envelope reduction algorithm. The lower bound on the 2-sum is seen to be tight for reasonably 'uniform' finite element meshes. We also obtain asymptotically tight lower bounds for the envelope size for certain classes of meshes.

A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in 2. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related I- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work involved in an envelope Cholesky factorization scheme. We formulate the latter two problems as quadratic assignment problems, and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a projected quadratic assignment problem, and then show that finding a permutation matrix closest to an orthogonal matrix attaining one of the lower bounds justifies the spectral envelope reduction algorithm. The lower bound on the 2-sum is seen to be tight for reasonably 'uniform' finite element meshes. We also obtain asymptotically tight lower bounds for the envelope size for certain classes of meshes.

What I have done in this book, if I have correctly solved the great problem I have sought to investigate, is, to unite the truth perceived by the school of Smith and Ricardo to the truth perceived by the schools of Proudhon and Lasalle; to show that <i>laissez faire</i> (in its full true meaning) opens the way to a realization of the noble dreams of socialism; to identify social law with moral law, and to disprove ideas which in the minds of many cloud grand and elevating perceptions. (Summary by Henry George) <br/><br/>Audio edited by TriciaG; Proof-listened by Kimberly Krause & Larry Wilson.

2Everglades Wildguide

By Jean Craighead George

This is a United States National Parks guidebook written by a popular young people's nature writer, Jean Craighead George. It covers the Everglades in detail, from its mangrove swamps to its sawgrass prairies. (Summary by Matt Pierard)

"What I want to impress upon those who may read this paper is this: The Irish land question is not a mere local question; it is a universal question. It involves the great problem of the distribution of wealth, which is everywhere forcing itself upon attention. It can not be settled by measures which in their nature can have but a local application. It can only be settled by measures which in their nature will apply everywhere as readily as in Ireland." (Summary by Henry George)

4London Mosaic

By Walter Lionel George

An idiosyncratic view of various aspects of London, England, published in 1921. Walter Lionel George (1882 – 1926) was an English writer, chiefly known for his popular fiction. - Summary by David Wales