Terry D. Lenker

On the minimum semidefinite rank of a simple graph

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.

On Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 2 × 2 matrices and real n × n tridiagonal matrices. Additional results describing the class of n × n complex Schur D-stable matrices are given.

Local and global Lipschitz constants

Abstract Let X be a closed subset of I = [−1, 1], and let B n ( f ) be the best uniform approximation to f ϵ C [ X ] from the set of polynomials of degree at most n . An extended global Lipschitz constant is defined for f, and it is shown that this constant is asymptotically equivalent to the strong unicity constant. Estimates of the size of the local Lipschitz constant for f are given when the cardinality of the set of extremal points of f − B n ( f )is n + 2. Examples which illustrate that the local and extended global Lipschitz constants may have very different asymptotic behavior are constructed.

Classes of Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 3×3 matrices. Also, principally nilpotent n×n complex matrices are shown to be perfectly Schur D-stable, and additional characterizations of these matrices are given.

Local Lipschitz constants

Abstract Let X be a closed subset of I= [− 1, 1], For f ϵ C[X], the local Lipschitz constant is defined to be λ nδ (f) = sup { ∥B n (f) − B n (g)∥ ∥f − g∥: 0 , where Bn(g) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as δ → 0 of the local Lipschitz constant of f, and an explicit expression is given for this common value. The, possibly very different, characterizations of local and global Lipschitz constants are also considered.

Linear Convergence and the Bisection Algorithm

(1986). Linear Convergence and the Bisection Algorithm. The American Mathematical Monthly: Vol. 93, No. 1, pp. 48-51.

Limit cycles for successive projections onto hyperplanes in Rn

Abstract In this paper we consider successive orthogonal projections onto m hyperplanes in R n, where m ⩾ 2 and n ⩾ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in R 2, are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in R n.

Near orderings of topological spaces

Let X be a topological space with an equivalance relation . Solutions to the following problems are of fundmental importance in economics (utility theory) [2, 6, 9, 10], psychology (foundations of measurement) [8, 11] and current models of thermodynamics [3, 5, 7]. I. Under what conditions does there exist an essentially unique near order < on X that agrees with the topology and ; that is, for all x, y, z, w ~ X :

Caratheodory's concept of temperature

EXAMPLE: Let S{ = S2 = S3 = R and let F^Si, s?) = sin [tt(S? s?)] for i, j = 1, 2, 3. Assume there exist continuous functions t?: S,?>R, ? = 1, 2, 3 such that t\(s\) = t2(s2) = t3(s3) if and only if Fi2(si, s2) = F13(si, s3) = F23(s2, s3) = 0. Thus t\(s\) = t2(s2) = t3(s3) if and only if Si = s2 mod 1 = s3 mod 1. Consider the triples (0, 2, 3) and (1, 2, 3) belonging to Si x S2 x S3. This implies i,(0) = f,(l) and ^,(5) * tx(0) for s i (0,1). Thus t\ is not continuous.

Local Lipschitz and strong unicity constants for certain nonlinear families

Abstract Let X be a compact metric space, and let V = { F ( a , x ): a ϵ A } where A is an open subset of R n , and F ( a , x ) and ∂F ∂a i , 1 ⩽ i ⩽ n , are continuous on A × X . Suppose f ϵ C ( X ) is weakly normal; that is (i) f has a best approximation F(a ∗ , ·) = B v (f) such that N = dim W(a ∗ ) ≡ dim span {( ∂F ∂a i )(a ∗ , ·): 1 ⩽ i