Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Let and be compact plane sets with . We define , where is analytic on . For , we define and . It is known that is a natural Banach function algebra on under the norm , where . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of to natural Banach function subalgebras of and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of .

PROPERTIES OF CERTAIN SUBALGEBRAS OF DALES-DAVIE ALGEBRAS

We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {Mn}∞n = 0 is a sequence of positive numbers such that M0 = 1 and (m + n)!/Mm+n ≤ (m!/Mm)(n!/Mn) for m, n ∈ N. Let d = lim sup(n!/Mn)1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to Xd. Let DP [DR]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of DP is $X^_d$, the polynomial convex hull of Xd, and the maximal ideal space of DR is Xd. Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

Numerical and Experimental Analysis of Monostable Mini- and Micro-Oscillators

An asymmetric micro-oscillator design based on a monostable fluidic amplifier is proposed. Experimental data with various feedback loop configurations point out that the main effect responsible for the oscillation is a capacitive and not a propagative effect. Actually, sound propagation in the feedback loop only generates a secondary oscillation which is not strong enough to provoke the jet switching. Data from a hybrid simulation using a commercial CFD code and a simple analytical model are in good agreement with the experimental data. A more compact plane design with reduced feedback loop volumes is also studied through a fully CFD simulation that confirms the previous conclusions.

Limit laws for the diameter of a random point set

Let U1,U2,… be a sequence of i.i.d. random vectors distributed uniformly in a compact plane region A of unit area. Sufficient conditions on the geometry of A are provided under which the Euclidean diameter Dn of the first n of the points converges weakly upon suitable rescaling.