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We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss-type results obtained earlier for specific datasets. In particular, we recover and, in several cases, improve results for sets of sparse and structured sparse vectors, low-rank matrices and tensors, and smooth manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for datasets taking the form of an infinite union of subspaces of a Hilbert space., Author(s): Sjoerd Dirksen[sup.1] Author Affiliations: (1) Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115, BonnGermany Introduction The analysis of high-dimensional data leads to various computational issues which are [...]

Known types of resultant matrices corresponding to one-parameter matrix polynomials are generalized to the multiparameter case. Based on the resultant approach suggested, methods for solving the following problems for multiparameter [...]

The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize the pointwise kth-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying pseudoblock conditions, [...]

The main result of the paper is a theorem, using which a new proof of Roth's theorem is obtained, a new solvability criterion for the matrix equation AX - YB [...]

Methods for solving the following parametric problems are considered: division of scalar and matrix polynomials and computation of their GCD and LCM; finding bases of the sum, difference, and intersection [...]

For a q-parameter polynomial m x n matrix F of rank [rho], solutions of the equation Fx = 0 at points of the spectrum of the matrix F determined by [...]

An elementary proof of a formula for the 2-norm distance from a normal matrix A to the set of matrices with a multiple zero eigenvalue is given. Earlier, the authors [...]

The paper presents new upper and lower bounds for the singular values of rectangular matrices explicitly involving the matrix sparsity pattern. These bounds are based on an upper bound for [...]

The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding k and the simple [...]

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors {[P.sub.1], [Q.sub.1]} and {[P.sub.2], [Q.sub.2]} can be verified by comparing the traces of [P.sub.1], [...]

Byline: V. Y. Pan (1) Abstract: We reexamine the Wiedemann--Coppersmith--Kaltofen--Villard algorithm for randomized computation of the determinant of an integer matrix and substantially simplify and accelerate its bottleneck stage of computing the minimum generating matrix polynomial, to make the algorithm practically promising while keeping it asymptotically fast. Bibliography: 58 titles. Author Affiliation: (1) Department of Mathematics and Computer Science, Lehman College of CUNY, USA Article History: Registration Date: 10/03/2006 Received Date: 27/11/2004 Article note: Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 163--187.

Let [S.sub.k](N)[.sup.+] be the set of primitive cusp forms of even weight k for [[GAMMA].sub.0](N) and let L(s, sy[m.sup.2]f) be the symmetric square L-function L(s, f) of a form f [...]

1. Reflection Principles for Certain Classes of CR Manifolds In this section, we consider the following general problem. Let X be a Cauchy-Riemann submanifold of [C.sup.n]; all our considerations will [...]

In previous chapters we considered the traditional situation where the reflection principle applies to the study of real analytic or algebraic structures. However, smooth structures play an important role in [...]

In this chapter, nonautonomous linear systems of the general form with Lyapunov matrices [6] are considered. By using a Lyapunov function, the stability of these systems is analyzed. The influence [...]

Byline: Ramon Carbo-Dorca (1) Keywords: space-time frames; time vectors; vector semispaces; inward matrix products; probability density functions; minkowski norms Abstract: The convenient properties of cumulative probability density functions are used to describe how sequences of infinite-dimensional time vectors can be generated from them. It is also discussed how such a description can be connected with space--time frames, while conserving a coherent connection related to the usual experimental scalar measures of time. Author Affiliation: (1) Ghent Quantum Chemistry Group, Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 (S3), B-9000 Gent, Belgium (2) Institut de Quimica Computacional/ Campus de Montilivi/, Universitat de Girona/ Girona, 17071/, (Catalonia), Spain Article History: Registration Date: 26/09/2004

Byline: Weigen Yan (1), Fuji Zhang (2) Keywords: outerplanar graph; adjacency matrix; skew adjacency matrix; characteristic polynomial; permanental polynomial; Cartesian product; Pfaffian orientation; nice cycle Abstract: Let G be a simple graph with adjacency matrix A(G) and [pi](G,x) the permanental polynomial of G. Let G x H denotes the Cartesian product of graphs G and H. Inspired by Klein's idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811--823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results. 1.If G is a bipartite graph containing no subgraph which is an even subdivision of K .sub.2,3, then G has an orientation G .sup.e such that [pi](G,x) = det (xI-A(G .sup.e )), where A(G .sup.e ) denotes the skew adjacency matrix of G .sup.e. 2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph [line over (term)] G with 2n+2 vertices such that [pi](G,x) is a factor of [pi] ([line over (term)] G ,x) . 3.Let T be an arbitrary tree with n vertices. Then [pi] (T K.sub.2 ,x) .sub.i = 1.sup.n (x.sup.2 + 1 + [alpha] .sub.i.sup.2 ) , where [alpha] .sub.1 , [alpha] .sub.2 , ..., [alpha] .sub.n are the eigenvalues of T. Author Affiliation: (1) Department of Mathematics, Jimei University, Xiamen, 361021, P.R. China (2) Department of Mathematics, Xiamen University, Xiamen, 361005, P.R. China Article History: Registration Date: 02/10/2004

Byline: J. Demmel (1), B. Diament (2), G. Malajovich (3) Keywords: AMS Classification. 65F35, 68Q25. Abstract: We consider the cost of estimating an error bound for the computed solution of a system of linear equations, i.e., estimating the norm of a matrix inverse. Under some technical assumptions we show that computing even a coarse error bound for the solution of a triangular system of equations costs at least as much as testing whether the product of two matrices is zero. The complexity of the latter problem is in turn conjectured to be the same as matrix multiplication, matrix inversion, etc. Since most error bounds in practical use have much lower complexity, this means they should sometimes exhibit large errors. In particular, it is shown that condition estimators that: (1) perform at least one operation on each matrix entry and (2) are asymptotically faster than any zero tester, must sometimes over or underestimate the inverse norm by a factor of at least 2.sup.[delta] k + (n) , where n is the dimension of the input matrix, k is the bitsize, and where either [delta] [greater than or equal to] 0.5 or (n) grows faster than any polynomial in n . Our results hold for the RAM model with bit complexity, as well as computations over rational and algebraic numbers, but not real or complex numbers. Our results also extend to estimating error bounds or condition numbers for other linear algebra problems such as computing eigenvectors. Author Affiliation: (1) Computer Science Division and Mathematics Department University of California Berkeley, CA 94720, USA demmel@cs.berkeley.edu , US (2) Computer Science Division University of California Berkeley, CA 94720, USA bjd@cs.berkeley.edu , US (3) Departamento de Matem&amp amp aacute&amp semi tica Aplicada Universidade Federal do Rio de Janeiro Caixa Postal 68530 CEP 21945, Rio de Janeiro, RJ, Brasil gregorio@labma.ufrj.br, BR Article note: September 10, 1999. Final version received: August 23, 2000.

Byline: H. Z. Munthe-Kaas (1), G. R. W. Quispel (2), A. Zanna (3) Keywords: Key words. Lie group, Polar decomposition, Generalized polar decompositions, Involutive automorphism, Cartan decomposition, Iwasawa decomposition. AMS Classification. 22E15, 65F25, 65Q05. Abstract: The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism [sigma] , always exist near the identity although frequently they can be extended to larger portions of the underlying group. In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion. Second, we discuss additional properties of the two factors. In particular, when [sigma] is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G . Author Affiliation: (1) Institutt for Informatikk University of Bergen Hoyteknologisenteret Thormohlensgate 55 N-5020 Bergen, Norway hans@ii.uib.no, NO (2) Department of Mathematics La Trobe University Bundoora Melbourne 3083, Australia r.quispel@latrobe.edu.au, AU (3) Institutt for Informatikk University of Bergen Hoyteknologisenteret Thormohlensgate 55 N-5020 Bergen, Norway anto@ii.uib.no, NO Article note: September 12, 2000. Final version received: April 16, 2001.

Byline: J. V. Burke (1), A. S. Lewis (2), M. L. Overton (3) Keywords: Key words. Eigenvalue optimization, Robust control, Stability analysis, Spectral abscissa, Nonsmooth analysis, Jordan form. AMS Classification. Primary: 15A42, 93D09, 90C30 Secondary: 65F15, 49K30. Abstract: We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the problem. Furthermore, each distinct active eigenvalue corresponds to a single Jordan block. This behavior is crucial for optimality conditions and numerical methods. Our techniques blend nonsmooth optimization and matrix analysis. Author Affiliation: (1) Department of Mathematics University of Washington Seattle, WA 98195, USA burke@math.washington.edu., US (2) Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1, Canada aslewis@math.uwaterloo.ca., CA (3) Courant Institute of Mathematical Sciences New York University New York, NY 10012, USA overton@cs.nyu.edu., US Article note: June 17, 2000. Final version received: January 4, 2001. Online publication: March 16, 2001.

Byline: H. Hornburger (1), Geerd H.F. Diercksen (2) Abstract: The expressions for Rosen--Morse Franck--Condon factors derived previously yield a double sum with alternating terms. For higher values of the quantum number the numerical calculation of the Franck--Condon factors by electronic computers using these expressions leads to numerical overflow inspite of the use of double-precision (32 digits) arithmetic. High values for the quantum number in the final ground state of the Rosen--Morse potential occur in molecular nonradiative rate calculations. Furthermore, the expressions show a lack of clearness with respect to the parameters of the potential. For out-of-plane modes exact closed form expressions and exact recurrence relations are derived. Asymptotic expressions for the matrix elements are calculated. Exact closed form expressions for matrix elements with quantum numbers which correspond to regions close to the dissociation barrier are given. Author Affiliation: (1) Max-Planck-Institut fur Quantenoptik, D-85748, Garching bei Munchen, Germany E-mail: (2) Max-Planck-Institut fur Astrophysik, D-85740, Garching bei Munchen, Germany E-mail: Article History: Registration Date: 20/10/2004

Byline: Raimund J. Ober (1), E. Sally Ward (2) Abstract: It is shown that a large class of twoadimensional NMR spectra is characterized by a matrix algebra and an invariant subspace. Both the matrix algebra and the invariant subspace are determined by the system matrices of the bilinear system which describes the NMR experiments. Author Affiliation: (1) Center for Engineering Mathematics EC35, University of Texas at Dallas, Richardson, TX, 75083, USA (2) Cancer Immunobiology Center and Department of Microbiology, University of Texas Southwestern Medical Center, 5323 Harry Hines Boulevard, Dallas, TX, 75235a8576, USA Article History: Registration Date: 11/09/2004

Byline: Emili Besalu (1), Ramon CarboaDorca (1) Abstract: An iterative method based on perturbation theory in matrix form is described as a procedure to obtain the eigenvalues and eigenvectors of square matrices. Practical vector notation and elementary schematic algorithm codes are given. The particular programming characteristics of the present computational scheme are based upon eigenvector corrections, obtained through a simple Rayleigh--Schrodinger perturbation theory algorithm. The proposed methodological processes can be used to evaluate the eigensystem of large matrices. Author Affiliation: (1) Institute of Computational Chemistry, University of Girona, 17071, Girona, Catalunya, Spain Article History: Registration Date: 23/09/2004

Sir William Rowan Hamilton discovered quaternion matrices on Oct 16, 1843. Since then, no definitive and conclusive definition of the determinant of quaternion matrices has been accepted. Many theorems, lemmas, and proofs are presented to show that quarternion matrices can have a determinant of plus or minus 1. Altogether, four axioms, five theorems and five lemmas are shown.

Byline: J.W. Barnes (1), S.P. Dokov (1), R. Acevedo (1), A. Solomon (2) Keywords: asymmetric TSP; elementary landscape; neighborhood search Abstract: Symmetric and antisymmetric distance matrices in the single agent traveling salesman problem (TSP) are not the only distance matrices to generate elementary landscapes for 'swap' and '2-opt' neighborhoods. Author Affiliation: (1) The Graduate Program in Operations Research and Industrial Engineering, The University of Texas, Austin, TX, 78712, USA (2) Faculty of Information Technology, University of Technology, Sydney, Broadway, NSW, 2007, Australia Article History: Registration Date: 09/10/2004