Interpolation between and

We show that if is a rearrangement invariant space on that is an interpolation space between and and for which we have only a one-sided estimate of the Boyd index 1/p, 1 < p < \infty$">, then is an interpolation space between and . This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type and weak type .

     

Effect of helium/argon gas ratio in a He-Ar-Cu<Superscript>+</Superscript> IR hollow-cathode discharge laser: modeling study and comparison with experiments

The He-Ar-Cu⁺ IR laser operates in a hollow-cathode discharge, typically in a mixture of helium with a few-% Ar. The population inversion of the Cu⁺ ion levels, responsible for laser action, is attributed to asymmetric charge transfer between He⁺ ions and sputtered Cu atoms. The Ar gas is added to promote sputtering of the Cu cathode. In this paper, a hybrid modeling network consisting of several different models for the various plasma species present in a He-Ar-Cu hollow-cathode discharge is applied to investigate the effect of Ar concentration in the gas mixture on the discharge behavior, and to find the optimum He/Ar gas ratio for laser operation. It is found that the densities of electrons, Ar⁺ ions, Ar_m ^* metastable atoms, sputtered Cu atoms and Cu⁺ ions increase upon the addition of more Ar gas, whereas the densities of He⁺ ions, He₂ ⁺ ions and He_m ^* metastable atoms drop considerably. The product of the calculated Cu atom and He⁺ ion densities, which determines the production rate of the upper laser levels, and hence probably also the laser output power, is found to reach a maximum around 1–5 % Ar addition. This calculation result is compared to experimental measurements, and reasonable agreement has been reached. Received: 14 October 2002 / Revised version: 28 November 2002 / Published online: 19 March 2003 RID="*" ID="*"Corresponding author. Fax: +32-3/820-23-76, E-mail: annemie.bogaerts@ua.ac.be     

Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

Free and Sulfated Sterols of Two Far-East Leptasterias Starfish

NMR spectroscopy, capillary GLC, and GLC-MS are used to study the composition of free and sulfated sterols from the far-east starfishLeptasterias alaskensis asiatica(Fischer) andL. Fisheri(Djakonov). The total free sterols of both species are shown to have similar qualitative and quantitative compositions and contain mainly⁷ -sterols. Sterol sulfate fractions contain cholesterol sulfate as the main component but differ in the ratios of⁵ :⁰ :⁷ -sterol derivatives. Possible reasons for these differences are discussed. A new steroid, 3-hydroxycholest-5-en-7-one sulfate, was isolated.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

On harmonic typically real mappings

In this paper, we prove that a univalent orientation-preserving harmonic mapping defined on the unit disk U with the normalization f(0)=0, , is a typically real mapping, if f(U) is a starlike domain with respect to the origin or f(U) is convex in one direction.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.