Moshe Goldberg

Scheme-Independent Stability Criteria for Difference Approximations of Hyperbolic Initial-Boundary Value Problems. II.

Abstract : Convenient stability criteria are obtained for difference approximations to hyperbolic initial-boundary value problems. The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outlfow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized. (Author)

On the Numerical Radius and Its Applications

Abstract We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.

Multiplicativity factors for seminorms. II

Let S be a seminorm on an algebra A. In this paper we study multiplicativity and quadrativity factors for S, i.e., constants μ > 0 and λ > 0 for which S(xy) ⩽ μS(x)S(y) and S(x2) ⩽ λS(x)2 for all x, y ∈ A. We begin by investigating quadrativity factors in terms of the kernel of S. We then turn to the question, under what conditions does S have multiplicativity factors if it has quadrativity factors? We show that if A is commutative then quadrativity factors imply multiplicativity factors. We further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors.

Equivalence constants for lp norms of matrices

The lp norm and the lp operator-norm of an m×n complex matrix A=(α ij ) are given by respectively. The purpose of this paper is to study the equivalence relations between these norms.

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.

Operator norms, multiplicativity factors, and C-numerical radii

Abstract Let V be a normed vector space over C , let B ( V ) denote the algebra of linear bounded operators on V , and let N be an arbitrary seminorm or norm on B ( V ). In this paper we discuss multiplicativity factors for N , i.e., constants μ>0 for which N μ ≡ μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C -numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.

Multiplicativity of lp norms for matrices

Abstract The l p norm and the l p operator norm of an m × n complex matrix A = ( α ij ) are given by |A| p = ∑ i, j |α ij | p 1 p and ‖A‖ p = max {|Ax| p :xe C n , |x| p =1} , respectively. The main purpose of this paper is to investigate the multipicativity of the l p norms and their relation to the l p operator norms.

On a boundary extrapolation theorem by Kreiss

Abstract : A hardly known and very important result of Kreiss is proven explicitly: Outflow boundary extrapolation which complements stable dissipative schemes for linear hyperbolic initial value problems, maintains stability. In view of this result, the Lax-Wendroff and the Gottlieb-Turkel schemes are applied to a test problem; as expected from the rate-of-convergence theory by Gustafsson, global order to accuracy is preserved if outflow boundary computations employ extrapolation of (local) accuracy of the same order. (Author)

Mixed multiplicativity and lp norms for matrices

Abstract Let Cm×n denote the class of m×n complex matrices; and let N1, N2, and N3 be arbitrary norms on Cm×n, Cm×k, and Ck×n, respectively. In this paper we discuss the best (least) positive constant μminwhich satisfies N 1 (AB)⩽μ min N 2 (A)N 3 (B) ∀A ∈ C m×k , B ∈ C k×n . In particular, for 1 ⩽ p ⩽ ∞ let |A| p = ∑ i=1 m ∑ j=1 n |α ij | p 1 p be the lp norm of a matrix A = (αij) ∈ Cm×n. Then for arbitrary p, q, r such that 1⩽ p,q,r ⩽ ∞, we determine explicitly the best constant μmin for which |AB| p ⩽ μ min |A| q |B| r ∀A ∈ C m×k , B ∀ C k×n .

Lithium nuclear magnetic resonance measurements in halotolerant bacterium Ba1

Abstract Measurements of relaxation times T 1 and T 2 , were carried out on a high-salt- and low-salt-grown bacterial pellets of halotolerant bacterium B a 1 . In our measurements, T 1 ⪢ T 2 and both were frequency-independent. In the high-salt-grown pellet the relaxation time values were much shorter than in the case of low-salt growth medium. Intensity measurements show that only 55% of the lithium in the high-salt pellet is detected; for the low-salt pellet almost all the lithium is detected. Growth measurements were carried out on the B a 1 . It is suggested that there is some form of adaptation of the bacteria to the growth medium. The adaptation is reflected in the lithium NMR results.