Error estimates in the determination of rigid body displacements in protein crystals

We report the estimation of random errors in the refinement of the rigid body displacements of the -helices of the enzyme 6-phosphogluconate dehydrogenase. Least-squares refinement of the TLS parameters of the helices has been carried out using X-ray reflection data of 2.1 Å resolution, resulting in anR-factor of 19.5%. Standard deviations were estimated from the normal matrix. The results show that the translational mean-square displacements of nearly all the helices are significant at this resolution. However the libration parameters are only significant when the helices have at least four turns. Screw-rotation tensor values cannot be determined at this resolution.     

OR Utilization in Nigeria: A Sample Survey

The study presents the results of a mail survey on the utilization of Operational Research (OR) techniques in tackling complex operational problems in the private sector in Nigeria. The survey was mailed to OR practitioners in nearly 1000 companies and was aimed at determining the familiarity with, and usage of, OR techniques, background of OR personnel, degree of computer usage, application areas, implementation and contributions of OR, problems attendant in using OR techniques, and future prospects for OR usage.     

Minimax risk overl p -balls forl p -error

Summary Consider estimating the mean vector from dataN _n (, ² I) withl _q norm loss,q1, when is known to lie in ann-dimensionall _p ball,p(0, ). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

Moment closure hierarchies for kinetic theories

This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a 14-moment closure, which also recovers the behavior of the Grad 13-moment system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.     

Positivity of entropy production in nonequilibrium statistical mechanics

We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee() in a state . When is steady state describing the long term behavior of a system we show thate()0, and sometimes we can provee()>0.     

Cohomology of the complement of a free divisor

We prove that if is a ``strongly quasihomogeneous" free divisor in the Stein manifold , and is its complement, then the de Rham cohomology of can be computed as the cohomology of the complex of meromorphic differential forms on with logarithmic poles along , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

     

Exact controllability and stabilizability of the Korteweg-de Vries equation

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation

 

on the interval , with periodic boundary conditions

 

where the distributed control is restricted so that the ``volume' of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control is allowed to act on the whole spatial domain , it is shown that the system is globally exactly controllable, i.e., for given and functions , with the same ``volume', one can alway find a control so that the system (i)--(ii) has a solution satisfying

If the control is allowed to act on only a small subset of the domain , then the same result still holds if the initial and terminal states, and , have small ``amplitude' in a certain sense. In the case of closed loop control, the distributed control is assumed to be generated by a linear feedback law conserving the ``volume' while monotonically reducing . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

     

Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where satisfies a ``maximal' estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is

(with analytic , naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [4] and Himonas in [8] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [1], Hanges and Himonas [6], and Christ [3]) were globally analytic hypoelliptic on products of tori.

     

Chemistry of 1-Alkoxy-1-glycosyl Radicals: The Manno- and Rhamnopyranosyl Series. Inversion of alpha- to beta-Pyranosides and the Fragmentation of Anomeric Radicals

The formation and stereoselective quenching of 1-mannopyranosyl radicals by a tributyltin hydride-mediated intramolecular 1,5-hydrogen abstraction sequence is described. A competing process is 1,4-hydrogen atom abstraction leading principally to glucopyran-2-ulosides. Fragmentation of the anomeric radical resulting in the formation of ring opened products is a problem in certain series. The chemistry is dictated to a considerable extent by the nature of the protecting groups employed with the 4,6-benzylidene series and, for rhamnose, the Ley 3,4-dispiroketal, being particularly susceptible to the 1,4-hydrogen atom abstraction but less to the fragmentation. Photochemical conditions are described, in which these side reactions are practically eliminated, and applied to the inversion of an alpha- to a beta-mannoside in a disaccharide.