The Chebyshev solution of the linear matrix equationAX+YB=C

Summary In this paper we investigate the properties of the Chebyshev solutions of the linear matrix equationAX+YB=C, whereA, B andC are given matrices of dimensionsm×r, s×n andm×n, respectively, wherer ands. We separately consider two particular cases. In the first case we assumem=r+1 andn=s+1, in the second caser=s=1 andm, n are arbitrary. For these two cases, under the assumption that the matricesA andB are full rank, we formulate necessary and sufficient conditions characterizing the Chebyshev solution ofAX+YB=C and we give the formulas for the Chebyshev error. Then, we propose an algorithm which may be applied to compute the Chebyshev solution ofAX+YB=C for some particular cases. Some numerical examples are also given.     

Zweiparametrige Überrelaxation

Summary A generalization of the method of successive overrelaxation (SOR-method) toward the solution ofx=A x+b (the matrixA being of a special form) is treated. The method depends on two parameters instead of one but needs no more computational efforts. For matricesA with real and pure imaginary eigenvalues, those parameters are determined for which the method converges and those which give fastest convergence. The class of matricesA yielding convergence is enlarged in comparison to the SOR-method. The convergence is speeded up except in some cases. Three numerical examples are presented.     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

Precise domains of convergence for the block SSOR method associated withp-cyclic matrices

We apply Rouché's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.Research supported in part by NSF Grant DMS 870064.     

A divide and conquer method for unitary and orthogonal eigenproblems

Summary LetH ^{n xn} be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH ₁ andH ₂ by a rank-one modification ofH. The eigenproblems forH ₁ andH ₂ can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH ₁ andH ₂. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.WhenH ^{n xn} is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.Research supported in part by the NSF under Grant DMS-8704196 and by funds administered by the Naval Postgraduate School Research Council     

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH). We prove an extension of the theorem of Hochstadt (who proved the result in casen = N) thatn eigenvalues of anN × N Jacobi matrixH can replace the firstn matrix elements in determiningH uniquely. We completely solve the inverse problem for (δ _n , (H-z)^-1 δ _n ) in the caseN < ∞. This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.     

A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central pathH _P (XS) [PXSP ^–1 + (PXSP ^–1) ^T ]/2 = I, introduced by Zhang. At an iterate (X,S), we choose a scaling matrixP from the class of nonsingular matricesP such thatPXSP ^–1 is symmetric. This class of matrices includes the three well-known choices, namely:P = S ^1/2 andP = X ^–1/2 proposed by Monteiro, and the matrixP corresponding to the Nesterov—Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov—Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This author's research is supported in part by the National Science Foundation under grants INT-9600343 and CCR-9700448 and the Office of Naval Research under grant N00014-94-1-0340.This author's research was supported in part by DOE DE-FG02-93ER25171-A001.     

Distribution of large values of the argument of the Riemann zeta function on short intervals

An upper bound for the measure of the set of values t ∈ (T,T + H] for H = T ^27/82+ɛ for which |S(t)| ≥ λ is obtained.     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.     

Inertia-preserving secant updates

A class of rank-two, inertia-preserving updates for symmetric matricesH _c is studied. To ensure that inertia is preserved, the updates are chosen to be of the formH ₊=FH _c F ^t, whereF=I+qr ^t, withq andr selected so that the secant equation is satisfied. A characterization is given for all such updates. Using a parameterization of this family of updates, the connection between them and the Broyden class of updates is established. Also, parameter selection criteria that can be used to choose the optimally conditioned update or the update closest to the SR1 update are discussed.The work of the first author was partially supported by AFOSR Grant 84-0326. The work of the second author was partially supported by NSF Grant EAR-82-18743.     

New inequalities for the minimum eigenvalue of M-matrices

Some new inequalities for the minimum eigenvalue of M-matrices are established. These inequalities improve the results in [G. Tian and T. Huang, Inequalities for the minimum eigenvalue of M-matrices, Electr. J. Linear Algebra 20 (2010), pp. 291–302].     

Field dependence of magnetoresistance in epitaxial single-crystal La2/3Ca1/3MnO3-δ thin film

Epitaxial La_2/3Ca_1/3MnO₃ thin films were prepared on NdGaO₃(1lO) substrates by d.c. magnetron sputtering method. The measurements of magnetoresistance ρ(H) upon magnetic field at different temperatures were carried out in the field range of 0–8 T. It is found that ρ(H) obeys the following relations: when the temperature (T) is higher than the Curie temperature below and whenT is far WowT _c. It is suS8ested that the negative magnetoresistive effect is mainly due to enhancement of the magnetoconductance. Project supported by the National Natural Science Foundation of China (Grant No. 19504012) and the Chinese Academy of Sciences.     

LU decompositions of generalized diagonally dominant matrices

Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or –A is an irreducible, singularM-matrix.This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.Research sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide CorporationResearch supported in part by the National Science Foundation under Grant No. MCS 8102114Research supported in part by the U.S. Army Research Office under contract no. DAAG 29-81-k-0132     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

On the Connection Between the Projection and the Extension of a Parallelotope

This paper presents a result concerning the connection between the parallel projection P _v,H of a parallelotope P along the direction v (into a transversal hyperplane H) and the extension P + S(v), meaning the Minkowski sum of P and the segment S(v) = {λv | −1 ≤ λ ≤ 1}. A sublattice L _v of the lattice of translations of P associated to the direction v is defined. It is proved that the extension P + S(v) is a parallelotope if and only if the parallel projection P _v,H is a parallelotope with respect to the lattice of translations L _v,H , which is the projection of the lattice L _v along the direction v into the hyperplane H.     

Updating conjugate directions by the BFGS formula

Many iterative algorithms for optimization calculations form positive definite second derivative approximations,B say, automatically, butB is not stored explicitly because of the need to solve equations of the formBd--g. We consider working with matricesZ, whose columns satisfy the conjugacy conditionsZ ¹ BZ=1. Particular attention is given to updatingZ in a way that corresponds to revisingB by the BFGS formula. A procedure is proposed that seems to be much more stable than the direct use of a product formula [1]. An extension to this procedure provides some automatic rescaling of the columns ofZ, which avoids some inefficiencies due to a poor choice of the initial second derivative approximation. Our work is also relevant to active set methods for linear inequality constraints, to updating the Cholesky factorization ofB, and to explaining some properties of the BFGS algorithm. Dedicated to Martin Beale, whose achievements, advice and encouragement were of great value to my research, especially in the field of conjugate direction methods.     

On graphs containing a given graph as center

We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference ∣V(G)∣ - ∣V(H)∣ over graphs G having center isomorphic to H it is demonstrated that A(H) ≤ 4 for all H, and for 0 ≤ i ≤ 4 we characterize the class of trees T with A(T) = i. for n ≥ 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n, with chromatic number X(G) = n + X(H), and whose center is isomorphic to H. Finally, if ∣V(H)∣ ≥ 9 and k ≥ ∣V(H)∣ + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.     

On the frequency of Titchmarsh’s phenomenon for ζ(s)-VIII

For suitable functionsH = H(T) the maximum of|(ζ(σ + it)) ^z | taken overT≤t≤T + H is studied. For fixed σ(1/2≤σ≤l) and fixed complex constantsz “expected lower bounds” for the maximum are established.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.