Ethynyl terminated ester oligomers and polymers therefrom

A new class of ethynyl-terminated oligomers and the process for preparing same are disclosed. Upon the application of heat, with or without a catalyst, the ethynyl groups react to provide crosslinking and chain extension to increase the polymer use temperature and improve the polymer solvent resistance. These improved polyesters are potentially useful in packaging, magnetic tapes, capacitors, industrial belting, protective coatings, structural adhesives and composite matrices

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study the connection between eigenvalue statistics on microscopic energy scales $\eta\ll1$ and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order $\eta \sim\log N/N$. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized in the sense that their $\ell^p$-norms are comparable with $N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound $N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.Comment: Published in at http://dx.doi.org/10.1214/08-AOP421 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cartan matrices and presentations of the exceptional simple Elduque Lie superalgebra

Recently Alberto Elduque listed all simple and graded modulo 2 finite dimensional Lie algebras and superalgebras whose odd component is the spinor representation of the orthogonal Lie algebra equal to the even component, and discovered one exceptional such Lie superalgebra in characteristic 5. For this Lie superalgebra all inequivalent Cartan matrices (in other words, inequivalent systems of simple roots) are listed together with defining relations between analogs of its Chevalley generators.Comment: 5 pages, 1 figure, LaTeX2

Factoring nonnegative matrices with linear programs

This paper describes a new approach, based on linear programming, for computing nonnegative matrix factorizations (NMFs). The key idea is a data-driven model for the factorization where the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C such that X approximately equals CX and some linear constraints. The constraints are chosen to ensure that the matrix C selects features; these features can then be used to find a low-rank NMF of X. A theoretical analysis demonstrates that this approach has guarantees similar to those of the recent NMF algorithm of Arora et al. (2012). In contrast with this earlier work, the proposed method extends to more general noise models and leads to efficient, scalable algorithms. Experiments with synthetic and real datasets provide evidence that the new approach is also superior in practice. An optimized C++ implementation can factor a multigigabyte matrix in a matter of minutes.Comment: 17 pages, 10 figures. Modified theorem statement for robust recovery conditions. Revised proof techniques to make arguments more elementary. Results on robustness when rows are duplicated have been superseded by arxiv.org/1211.668

Approximating orthogonal matrices by permutation matrices

Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by ``non-commutative convex combinations'' A of permutation matrices of the type A=sum A_sigma sigma, where sigma are permutation matrices and A_sigma are positive semidefinite nxn matrices summing up to the identity matrix. We prove that for every nxn orthogonal matrix U there is a non-commutative convex combination A of permutation matrices which approximates U entry-wise within an error of c n^{-1/2}ln n and in the Frobenius norm within an error of c ln n. The proof uses a certain procedure of randomized rounding of an orthogonal matrix to a permutation matrix.Comment: 18 page

Conditions for separability in generalized Laplacian matrices and nonnegative matrices as density matrices

Recently, Laplacian matrices of graphs are studied as density matrices in quantum mechanics. We continue this study and give conditions for separability of generalized Laplacian matrices of weighted graphs with unit trace. In particular, we show that the Peres-Horodecki positive partial transpose separability condition is necessary and sufficient for separability in ${\mathbb C}^2\otimes {\mathbb C}^q$. In addition, we present a sufficient condition for separability of generalized Laplacian matrices and diagonally dominant nonnegative matrices.Comment: 10 pages, 1 figur