Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators of rank and order two.

A NEW PROOF OF OKAJI'S THEOREM FOR A CLASS OF SUM OF SQUARES OPERATORS

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form ""sum of squares"", satisfying Hormander's bracket condition. Let q be a characteristic point; for P. We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji Show that P is analytic hypoelliptic at q. Hence Okaji has established the validity of Treves' conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.

COMPLETE ISOMORPHIC CLASSIFICATIONS OF SOME SPACES OF COMPACT OPERATORS

This paper is a continuation and a complement of our previous work on isomorphic classification of some spaces of compact operators. We improve the main result concerning extensions of the classical isomorphic classification of the Banach spaces of continuous functions on ordinals. As an application, fixing an ordinal a and denoting by X(xi), omega(alpha) <= xi < omega(alpha+1), the Banach space of all X-valued continuous functions defined in the interval of ordinals [0,xi] and equipped with the supremum, we provide complete isomorphic classifications of some Banach spaces K(X(xi),Y(eta)) of compact operators from X(xi) to Y(eta), eta >= omega. It is relatively consistent with ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that these results include the following cases: 1.X* contains no copy of c(0) and has the Mazur property, and Y = c(0)(J) for every set J. 2. X = c(0)(I) and Y = l(q)(J) for any infinite sets I and J and 1 <= q < infinity. 3. X = l(p)(I) and Y = l(q)(J) for any infinite sets I and J and 1 <= q < p < infinity.

ON ISOMORPHIC CLASSIFICATIONS OF SPACES OF COMPACT OPERATORS

We prove an extension of the classical isomorphic classification of Banach spaces of continuous functions on ordinals. As a consequence, we give complete isomorphic classifications of some Banach spaces K(X,Y(n)), eta >= omega, of compact operators from X to Y(eta), the space of all continuous Y-valued functions defined in the interval of ordinals [1, eta] and equipped with the supremum norm. In particular, under the Continuum Hypothesis, we extend a recent result of C. Samuel by classifying, up to isomorphism, the spaces K(X(xi), c(0)(Gamma)(eta)), where omega <= xi < omega(1,) eta >= omega, Gamma is a countable set, X contains no complemented copy of l(1), X* has the Mazur property and the density character of X** is less than or equal to N(1).

Pseudodifferential operators with C*-algebra-valued symbols: Abstract characterizations

Given a separable unital C*-algebra C with norm parallel to center dot parallel to, let E-n denote the Banach-space completion of the C-valued Schwartz space on R-n with norm parallel to f parallel to(2)=parallel to < f, f >parallel to(1/2), < f, g >=integral f(x)* g(x)dx. The assignment of the pseudodifferential operator A=a(x,D) with C-valued symbol a(x,xi) to each smooth function with bounded derivatives a is an element of B-C(R-2n) defines an injective mapping O, from B-C(R-2n) to the set H of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E-n. In this paper, we construct a left-inverse S for O and prove that S is injective if C is commutative. This generalizes Cordes' description of H in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric n x n matrix J and if C is commutative, then any A is an element of H which commutes with every pseudodifferential operator with symbol F(x+J xi), F is an element of B-C(R-n), is a pseudodifferential operator with symbol G(x - J xi), for some G is an element of B-C(R-n). That was conjectured by Rieffel.

ON SPACES OF COMPACT OPERATORS ON C(K, X) SPACES

This paper concerns the spaces of compact operators kappa(E,F), where E and F are Banach spaces C([1, xi], X) of all continuous X-valued functions defined on the interval of ordinals [1, xi] and equipped with the supremun norm. We provide sufficient conditions on X, Y, alpha, beta, xi and eta, with omega <= alpha <= beta < omega 1 for the following equivalence: (a) kappa(C([1, xi], X), C([1, alpha], Y)) is isomorphic to kappa(C([1,eta], X), C([1, beta], Y)), (b) beta < alpha(omega). In this way, we unify and extend results due to Bessaga and Pelczynski (1960) and C. Samuel (2009). Our result covers the case of the classical spaces X = l(p) and Y = l(q) with 1 < p, q < infinity.

Eigenvalues of Casimir operators for gl(m/infinity)

A full set of Casimir operators for the Lie superalgebra gl(m/infinity) is constructed and shown to be well defined in the category O-FS generated by the highest-weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from gl(m/infinity) are also determined.

Level-one highest weight representations of U-q[g1(1 vertical bar 1)] and associated vertex operators

We study the level-one irreducible highest weight representations of U-q[gl(1\1)] and associated q-vertex operators. We obtain the exchange relations satisfied by these vertex operators. The characters and supercharacters associated with these irreducible representations are calculated'. (C) 2000 Published by Elsevier Science B.V. All rights reserved.

Risk management for scuba diving operators on Australia's Great Barrier Reef

Australia's Great Barrier Reef is one of the world's most popular scuba diving destinations. Unfortunately, a series of recent diving injuries and deaths has tarnished the region's safety record. In particular, media attention surrounding the disappearance of American divers Thomas and Eileen Lonergan has focused attention on dive operators' legal responsibilities and the consequences of failing to discharge their duty of care to customers. This paper briefly examines the relevant Australian law for recreational diving operations, and reviews risk management strategies that may reduce or prevent the occurrence of future problems. (C) 2000 Elsevier Science Ltd. All rights reserved.

U-q[(sl(2)over-cap vertical bar 1)] vertex operators, screen currents, and correlation functions at an arbitrary level

Bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra U-q[sl((2) over cap\1)] are constructed for arbitrary level k=alpha, where alpha not equal 0,-1 is a complex parameter appearing in the four-dimensional evaluation representations. They are intertwiners among the level-alpha highest weight Fock-Wakimoto modules. Screen currents which commute with the action of U-q[sl((2) over cap/1)] up to total differences are presented. Integral formulas for N-point functions of type I and type II q-vertex operators are proposed. (C) 2000 American Institute of Physics. [S0022-2488(00)00608-3].

Ladder operators for integrable one-dimensional lattice models

A generalised ladder operator is used to construct the conserved operators for any one-dimensional lattice model derived from the Yang-Baxter equation. As an example, the low order conserved operators for the XYh model are calculated explicitly.

On the loss of regularity for a class of weakly hyperbolic operators

In this paper we determine bounds for the optimal loss of regularity in the Sobolev scale for a class of weakly hyperbolic operators. (C) 2009 Elsevier Inc. All rights reserved.

Bispectral monitoring in dogs subjected to ovariohysterectomy and anesthetized with halothane, isoflurane or sevoflurane

Objective To assess the effect of halothane (H), isoflurane (I) or sevoflurane (S) on the bispectral index (BIS), and the effect of the addition of meperidine in dogs subjected to ovariohysterectomy. Study design Prospective, randomized, blinded, clinical trial. Animals Forty-eight female mixed-breed dogs, with weights varying from 10 to 25 kg. Methods All dogs were premedicated with acepromazine (A) (0.1 mg kg(-1) IM) or A and meperidine (M) (3 mg kg(-1) IM) and they were divided into six groups of eight animals (AH, AMH, AI, AMI, AS, and AMS). Fifteen minutes after premedication they were anesthetized with propofol (5 mg kg(-1) IV) and then orotracheally intubated. Anesthesia was maintained with halothane, isoflurane or sevoflurane, respectively. The BIS, E`(anest) variables were recorded at 15 minutes after administering pre-anesthetic medication (T0); 10 minutes of anesthesia maintenance (T1); right ovarian pedicle ligation (T2); muscle suturing (T3); skin suture (T4) and 10 minutes after terminating the inhalant anesthetic (T5), respectively. Results BIS values were decreased at all times when compared to the baseline values in all groups (p < 0.05). In the comparative assessment between groups, the values obtained at T0 and T1 were similar for all groups. At T2, the values in AMH were lower than those obtained in AI, AMI and AS (p < 0.05). At the same time significantly higher values were found for AI when compared to AMS (p < 0.01). There was a correlation between the bispectral index and the expired anesthetic fraction in all groups. Conclusions and clinical relevance Within groups given the same inhalant anesthetic the bispectral index was a good indicator for the degree of hypnosis in dogs, indicating a good correlation with the amount of anesthetic and the nociceptive stimulation. BIS was a less reliable indicator of relative anesthetic depth when comparing equipotent end-tidal concentrations between the three inhalants.

Bispectral index in dogs at three intravenous infusion rates of propofol

Objective To establish the correlation between the bispectral index (BIS) and different rates of infusion of propofol in dogs. Study design Prospective experimental trial. Animals Eight adult dogs weighing 6-20 kg. Methods Eight animals underwent three treatments at intervals of 20 days. Propofol was used for induction of anesthesia (10 mg kg(-1) IV), followed by a continuous rate infusion (CRI) at 0.2 mg kg(-1) minute(-1) (P2), 0.4 mg kg(-1) minute(-1) (P4) or 0.8 mg kg(-1) minute(-1) (P8) for 55 minutes. The BIS values were measured at 10, 20, 30, 40, and 50 minutes (T10, T20, T30, T40, and T50, respectively) after the CRI of propofol was started. Numeric data were submitted to analysis of variance followed by Tukey test (p < 0.05). Results The BIS differed significantly among groups at T40, when P8 was lower than P2 and P4. At T50, P8 was lower than P2. The electromyographic activity (EMG) in P2 and P4 was higher than P8 at T40 and T50. Conclusion An increase in propofol infusion rates decreases the BIS values and EMG.