59,165 research outputs found
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 -synthesis
We describe the basic complex geometry and function theory of the {\em
pentablock} , which is the bounded domain in given
by where denotes
the open unit ball in the space of complex matrices. We prove
several characterizations of the domain. We describe its distinguished boundary
and exhibit a -parameter group of automorphisms of . We show
that is intimately connected with the problem of -synthesis
for a certain cost function on the space of matrices defined
in connection with robust stabilization by control engineers. We demonstrate
connections between the function theories of and . We
show that 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 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 is an analytic function bounded by 1 on the bidisk
and is a point in a face of the bidisk at which satisfies
Caratheodory's condition then both and the angular gradient
exist and are constant on the face. Moreover, the class of all with
prescribed and 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
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