George Phillip Barker

Theory of cones

Abstract This survey deals with the aspects of archimedian partially ordered finite-dimensional real vector spaces and order preserving linear maps which do not involve spectral theory. The first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations. The study of inequalities resulted in the definition of a polyhedral cone K and its face lattice F ( K ). In Section II.A the face lattice of a not necessarily polyhedral cone K in a vector space V is investigated. In particular the interplay between the lattice properties of F ( K ) and geometric properties of K is emphasized. Section II.B turns to the cones Π( K ) in the space of linear maps on V . Recall that Π( K ) is the cone of all order preserving linear maps. Of particular interest are the algebraic structure of Π( K ) as a semiring and the nature of the group Aut( K ) of nonsingular elements A ϵΠ( K ) for which A -1 ϵΠ( K ) as well. In a short final section the cone P n of n × n positive semidefinite matrices is discussed. A characterization of the set of completely positive linear maps is stated. The proofs will appear in a forthcoming paper.

Desirable performance standards for HbA1c analysis – precision, accuracy and standardisation Consensus statement of the Australasian Association of Clinical Biochemists (AACB), the Australian Diabetes Society (ADS), the Royal College of Pathologists of Australasia (RCPA), Endocrine Society of Australia (ESA), and the Australian Diabetes Educators Association (ADEA)

Abstract Background: HbA1c (glycohaemoglobin) is universally used in the ongoing monitoring of all patients with diabetes. There are many % HbA1c target control rating recommendations by national, regional and international expert bodies for diabetes patients and these are variable around the world. General patient target control ratings are currently most often recommended as either <6.5% or <7.0% HbA1c, with <6.0% HbA1c stated for individual patients where clinically possible. This necessitates very precise HbA1c assays and the same patient values, irrespective of HbA1c method or area of the world. Methods: HbA1c targets recommended by major expert groups and published HbA1c assay precision (coefficient of variation, %CV) levels have been detailed. These have been compared with published biological variation levels and with calculated HbA1c error ranges at various HbA1c levels and %CV levels. In addition, these have been compared with the analytical precision necessary to differentiate between the upper limit of the normal range for HbA1c and targets recommended by expert groups for diabetes control. Results: Intralaboratory analytical CVs of <2% are necessary and are achievable on automated HPLC analysers, and are supported on grounds of both clinical need and biological variation, as well as the need to differentiate the national, regional and international target recommendations from the upper limit of the normal range (<6.0% HbA1c level). Conclusions: Routine methods with tight long-term imprecision with CVs of <2% are recommended. International HbA1c targets essentially require that all HbA1c methods be precise, and have minimal standardisation bias and minimal methodological interferences in individual patients. Clin Chem Lab Med 2007;45:1083–97.

Algebraic Perron-Frobenius theory

Abstract Let V be a vector space over a fully ordered field F. In Sec. 2 we characterize cones K with ascending chain condition (ACC) on faces of K. In Sec.3 we show that if K has ACC on faces, then an operator A is strongly irreducible if and only if A is irreducible. In Sec. 4 we prove theorems of Perron-Frobenius type for a strongly irreducible operator A in the case that F=R, the real field, and K is a full algebraically closed cone.

The lattice of faces of a finite dimensional cone

In recent years the classical Perron-Frobenius theory has been extended to matrices which leave invariant a cone in the finite dimensional real space V. Here we shall be concerned with the geometry of those cones which are suitable for Perron-Frobenius theory. l Although there is an extensive theory dealing with the lattice of faces of a polyhedral convex set, our work is of a different spirit, especially as parts of it extend to cones which are not polyhedral.

On the completely positive and positive-semidefinite-preserving cones

Abstract The cone CP n,q of completely positive linear transformations from M n ( C )= M n to M q is shown to be isometrically isomorphic to P nq , the cone of nq by nq positive semidefinite matrices. Generalizations of scalar and matrix results to CP n, q ⊂ HP n, q ⊂ L ( M n , M q ) (where HP n,q represents the hermitian-preserving linear transformations) are discussed. Relationships among the completely positives, the set of positive semidefinite preservers π( P n ) , and its dual π( M n ) ∗ are given. Left and right facial ideals of CP are characterized. Properties of the joint angular field of values of a finite sequence of hermitian matrices H 1 ,…, H m are studied, leading to a characterization of π( P q , P n ) .

A non-commutative spectral theorem

ABSTRACT Let M,, denote the full algebra of p X matrices, and let M

Common solutions to the Lyapunov equations

) of ( pk) X pk) matrices of the form diag(B,. . ,B), where B E Mp and there are k blocks. We show that if & is an algebra of n X n matrices which is generated by a set of normal matrices, there is a unitary matrix U such that for each A E & we have U*AU=diag(B,,...,B,,O) where Bi E

The structure of cones of matrices

4); 0 is a zero matrix of some order, say r; and n = r+ Zl_ r p,k,. The result is applied to several algebras which satisfy polynomial identities. Let V be a finite dimensional inner product space over C, and let L(V) denote the algebra of linear transformations on V. The spectral theorem states that if a * -subalgebra 6? of L(V) is commutative, then V has an orthonormal basis 9, such that the matrix representation [A]% of A with respect to the basis % is diagonal for each A E @. In this paper we use the Wedderburn-Artin theorem to obtain a non-commutative spectral theorem. This latter spectral theorem immediately yields a characterization of @ if @ satisfies a polynomial identity, and so we resolve a question raised by Watters [7].’ Let

Some Observations on the Spectra of Cone Preserving Maps

Abstract Let A be an n × n matrix with complex entries. A necessary and sufficient condition is established for the existence of a Hermitian solution H to the equations AH+HA ∗ =HA+A ∗ H=I .

Automorphism groups of algebras of triangular matrices

Abstract If K is a cone in R n we let Γ( K ) denote the cone in the space M n of n X n matrices consisting of all A such that AK ⊆ K . We show first that Γ( K ) is indecomposable if and only if K is indecomposable. Next we let Γ( K ) ∗ be the dual of Γ( K ). Then we show that Γ(K)=Γ(K) ∗ if and only if K is the image of the nonnegative orthant under an orthogonal transformation.