The automorphism group of the tetrablock

The tetrablock is a domain in 3-dimensional complex space that meets 3-dimensional Euclidean space in a regular tetrahedron. It is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation of the tetrablock by complex geodesic discs.Comment: 13 pages, 0 figure

Super-optimal approximation by meromorphic functions.

Let G be a matrix function of type m × n and suppose that G is expressible as the sum of an H∞ function and a continuous function on the unit circle. Suppose also that the (k – 1)th singular value of the Hankel operator with symbol G is greater than the kth singular value. Then there is a unique superoptimal approximant to G in : that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s∞(G – Q) or, in other words, which minimizes the sequence with respect to the lexicographic ordering, where and Sj(·) denotes the jth singular value of a matrix. This result is due to the present authors [PY1] in the case k = 0 (when the hypothesis on the Hankel singular values is vacuous) and to S. Treil[T2] in general. In this paper we give a proof of uniqueness by a diagonalization argument, a high level algorithm for the computation of the superoptimal approximant and a recursive parametrization of the set of all optimal solutions of a matrix Nehari—Takagi problem

Electrochemical detection device

A standard pH reference electrode and a platinum cathodic electrode are positioned in a container with suitable nutrient medium for microbial growth plus the sample to be tested. The two electrodes are connected to electronic circuitry including an up/down counter whicn counts up for the first 80 minutes after a test has initiated. Then the potential between the two electrodes is tracked by the electronic circuitry and after there is a change of 10 mv a signal is sent to the up/down counter to cause it to reverse its count. When there is a additional 20 mv change in the potential between the two electrodes another signal is sent to the up/down counter, signalling it to stop. The resulting count on the counter is equal to the length of time for the inoculum to begin the production of measurable amounts of H2 after inoculation

Boundary behavior of analytic functions of two variables via generalized models

We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the 2-torus. We prove the existence of a generalized model with certain properties corresponding to such a singularity and use this result to solve two function-theoretic problems. The first of these is to characterise the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carath\'eodory condition holds. The second is to obtain a representation theorem for functions in the two-variable Pick class analogous to the refined Nevanlinna representation of functions in the one-variable Pick class.Comment: 30 page

The complex geomety of a domain related to μ\mu-synthesis

We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in $\mathbb{C}^3$ given by $$\mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\}$$ where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined in connection with robust stabilization by control engineers. We demonstrate connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We show that $\mathcal{P}$ is polynomially convex and starlike.Comment: 36 pages, 2 figures. This version contains corrections of some inaccuracies and an expanded argument for Proposition 12.

Carath\'eodory extremal functions on the symmetrized bidisc

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc $$G \stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}.$$ We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath\'eodory problems for tangents of non-generic types.Comment: 24 pages, 1 figure. This version contains some minor changes. It is to appear in a volume of Operator Theory: Advamces and Applications, Birkhause

Nevanlinna representations in several variables

We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane there are representation formulae in terms of densely defined self-adjoint operators on a Hilbert space. We introduce three types of structured resolvent of a self-adjoint operator and identify four different types of representation in terms of these resolvents. We relate the types of representation that a function admits to its growth at infinity.Comment: 37 pages. In this version we have added some references and expanded the introductio

Pacific Basin Communication Study, volume 2

Users' meeting summary report, chronology of visits, economic data for forum countries, techniques used in the study, communication choices, existing resources in the Pacific Basin, and warc 79 region 3 rules and regulations were presented in volume 2

Facial behaviour of analytic functions on the bidisk

We prove that if $\phi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\phi$ satisfies Caratheodory's condition then both $\phi$ and the angular gradient $\nabla\phi$ exist and are constant on the face. Moreover, the class of all $\phi$ with prescribed $\phi(\tau)$ and $\nabla\phi(\tau)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.Comment: 18 pages. We have replaced an erroneous proof of Theorem 5.4(1) by a valid proo