We extend the nu-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.

An abstract υ-metric was introduced by Ball and Sasane, with a view towards extending the classical υ-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an additional concrete special instance of the abstract υ-metric, by verifying all the assumptions demanded in the abstract set-up. This example links the abstract υ-metric with the one proposed by Vinnicombe as a candidate for the υ-metric for nonrational plants.

Let D-n := {z = (z(1),...,z(n)) is an element of C-n : vertical bar z(j)vertical bar < 1, j = 1,...,n}, and let <(D)over bar>(n) denote its closure in C-n. Consider the ring C-r((D) over bar (n); C) = {f : (D) over bar (n) -> C : f is continuous and f (z) = <(f<(z)over bar>)over bar> (z is an element of (D) over bar (n))} with pointwise operations, where u is used appropriately to denote both (componentwise) complex conjugation and closure. It is shown that C-r((D) over bar (n); C) is projective free, that is, finitely generated projective modules over C-r((D) over bar (n); C) are free. Let A denote the polydisc algebra, that is, the set of all continuous functions defined on (D) over bar (n) that are holomorphic in D-n. For N a positive integer, let partial derivative(-N) A denote the algebra of functions f is an element of A whose complex partial derivatives of all orders up to N belong to A. We show the projective freeness of each of the real symmetric algebras partial derivative(-N) A(r) = {f is an element of partial derivative(-N) A : f (z) = <(f<(z)over bar>)over bar> (z is an element of (D) over bar (n))}.

Let D denote the open unit disk in C centered at 0. Let H-R(infinity) denote the set of all bounded and holomorphic functions defined in D that also satisfy f(z) = )over bar> for all z is an element of D. It is shown that H-R(infinity) is a coherent ring.

Let C denote a closed convex cone in R-d with apex at 0. We denote by E'(C) the set of distributions on R-d having compact support contained in C. Then E'(C) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on (f) over cap (1), ..., (f) over cap (n) for f(n) is an element of E'(C) to generate the ring E'(C). (Here (center dot) over cap denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hormander. En route we answer an open question posed by Yutaka Yamamoto.

Let R be a commutative complex Banach algebra with the involution .* and suppose that A is an element of R-nxn, B is an element of R-nxm, C is an element of R-pxn. The question of when the Riccati equation PBB*P - PA - A*P - C*C = 0 has a solution P is an element of R-nxn is investigated. A counterexample to a previous result in the literature on this subject is given, followed by sufficient conditions on the data guaranteeing the existence of such a P. Finally, applications to spatially distributed systems are discussed.

We prove an abstract Nyquist criterion in a general set up. As applications, we recover various versions of the Nyquist criterion, some of which are new.

Let A(R) ((D) over bar) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that A(R) ((D) over bar) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set D is given for the corresponding real algebra A(R) ((D) over bar) to have Bass stable rank equal to 1. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We denote by W +(ℂ +) the set of all complex-valued functions defined in the closed right half plane ℂ + D {s ∈ ℂ | Re(s) ≥ 0} that differ from the Laplace transform of functions from L 1(0, ∞) by a constant. Equipped with pointwise operations, W +(C +) forms a ring. It is known that W +(ℂ +) is a pre-Bézout ring. The following properties are shown for W +(ℂ +): W +(ℂ +) is not a GCD domain, that is, there exist functions F 1, F 2 in W +(ℂ +) that do not possess a greatest common divisor in W +(ℂ +). W +(ℂ +) is not coherent, and in fact, we give an example of two principal ideals whose intersection is not finitely generated. We will also observe that W +(ℂ +) is a Hermite ring, by showing that the maximal ideal space of W +(ℂ +), equipped with the Gelfand topology, is contractible.

Let Omega be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H (a)(Omega) the Banach algebra of all bounded holomorphic functions on Omega, with pointwise operations and the supremum norm. We show that the topological stable rank of H (a)(Omega) is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of H (a)(D) is equal to 2, where D is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H (a"e) (a) (Omega) are 2.