Congruences,equational theories and lattice representations

We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday     

Differential graded Lie algebras and singularities of level sets of momentum mappings

The germ of an analytic varietyX at a pointxX is said to bequadratic if it is bi-analytically isomorphic to the germ of a cone defined by a system of homogeneous quadratic equations at the origin. Arms, Marsden and Moncrief show in [2] that under certain conditions the analytic germ of a level set of a momentum mapping is quadratic. We discuss related ideas in a more algebraic context by associating to an affine Hamiltonian action a differential graded Lie algebra, which in the presence of an invariant positive complex structure, is formal in the sence of [5].Dedicated to the memory of Bruce Reinhart     

Velocity inversion: A case study in infinite-dimensional optimization

The goal of seismic velocity inversion is the estimation of seismic wave velocities inside the earth by attempting to predict, in a least-error sense, seismic waveforms measured at its surface. We present velocity inversion as a case study in the various infinite-dimensional pathologies which may afflict practically important problems of distributed parameter identification, treated as optimization problems in function spaces. These features differentiate various problem formulations far beyond the degree one would expect for finite- (small-) dimensional problems. We illustrate this differentiation by comparing the characteristics of three different least-squares formulations of velocity inversion.     

On the principal axes of diffusion in difference schemes for 2d transport problems

Via Taylor series, we associate with a difference stencil L^h approximating Lu := au_x + bu_y its modified equation:

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. By rotating axes to eliminate the 2Bu_xy term, the principle axes through which the diffusion in L^h acts is calculated. Interestingly, for many schemes proposed for 2D transport problems these axes have little to do with the “streamline” and ”crosswind” directions of the continuous problem. Several examples are considered from this point of view.     

Crystal and molecular structure of crenatine carbonate

The crystal and molecular structure of 1-ethyl-4-methoxy-9H-pyrido[3,4b]indole (crenatine) carbonate C₁₄H₁₄N₂O·H₂CO₃, (MS, m/z 226)M _R 288.3, a-carboline alkaloid, has been determined from X-ray diffraction data. The compound crystallizes in the space group Pbca with cell parameters:a=11.616(4),b=18.450(8),c=12.992(5)Å,V=2784(2)ų,Z=8,D _calc=1.375 g cm^–3, (MoK)=0.71069Å,(Mo K)=0.94 cm^–1,F(000)=1216,R/R _w =8.2/10.3% for 1099 reflections. The ring system of the-carboline nucleus is planar. The title compound shows a two center hydrogen bond between the indole N-H group and the oxygen atom of a carbonate group. The structure does not display hydrogen bonding between-carboline groups but rather a bonding network involving the carbonate group.     

A fixed-point equation for the high-temperature phase of discrete lattice spin systems

A fixed-point equation on an infinite-dimensional space is proposed as an alternative to the usual definition of the infinite-volume limit in discrete lattice spin systems in the high-temperature phase. It is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities. A theorem is then proved that says that if a certain finite-volume condition is satisfied, then this fixed-point equation has a solution and the resulting free energy is analytic in the parameters in the Hamiltonian. For particular values of the temperature this finite-volume condition may be checked with the help of a computer. The two-dimensional Ising model is considered as a test case, and it is shown that the finite-volume condition is satisfied for0.77 _critical.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550