On function and operator modules

Let be a unital Banach algebra. We give a characterization of the left Banach -modules for which there exists a commutative unital -algebra , a linear isometry , and a contractive unital homomorphism such that for any . We then deduce a ``commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a -version of the latter characterization, which generalizes some previous work of Effros and Ruan.

     

Sums of unit fractions

Erdos and Straus conjectured that for any positive integer the equation has a solution in positive integers , and . Let k \geq 3$"> and

We show that parametric solutions can be used to find upper bounds on where the number of parameters increases exponentially with . This enables us to prove

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This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization' of the work of Vaughan (1970), who considered the case .

     

Diffractively produced charm final states in 800-GeV/c pp collisions

We report the first observation of diffractively produced open charm in 800-GeV/c pp collisions of the type pp-->pD*X. We measure cross sections of sigma(diff)(D*+) = (0.185+/-0.044+/-0.054) (mu)b and sigma(diff)(D(*-)) = (0.174+/-0.034+/-0.029) (mu)b. Our measurements are based on 4.3x10(9) events recorded by FNAL E690 in the fixed-target run of 1991. We compare our results with previous fixed-target charm experiments.     

Tamely Ramified Towers and Discriminant Bounds for Number Fields

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R _2m be the minimal root discriminant for totally complex number fields of degree 2m, and put ₀ = lim inf _m R _2m . One knows that ₀ 4e 22.3, and, assuming the Generalized Riemann Hypothesis, ₀ 8e 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that ₀ < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate ₀ < 83.9.     

Local Power Properties of Kernel Based Goodness of Fit Tests

If (X_i, i ) is a strictly stationary process with marginal density function f, we are interested in testing the hypothesis H₀: {f=f₀}, where f₀ is given. We consider different test statistics based on integrated quadratic forms measuring the proximity between f_n, a kernel estimator of f, and f₀, or between f_n and its expected value computed under H₀. We study the asymptotic local power properties of the testing procedures under local alternatives. This study generalizes to the multidimensional case in a context of dependence the corresponding one made by P. J. Bickel and M. Rosenblatt in 1973 (Ann. Statist.1, 1071–1095).     

Computational methods for the structural alignment of molecules

In drug design, often enough, no structural information on a particular receptor protein is available. However, frequently a considerable number of different ligands is known together with their measured binding affinities towards a receptor under consideration. In such a situation, a set of plausible relative superpositions of different ligands, hopefully approximating their putative binding geometry, is usually the method of choice for preparing data for the subsequent application of 3D methods that analyze the similarity or diversity of the ligands. Examples are 3D-QSAR studies, pharmacophore elucidation, and receptor modeling. An aggravating fact is that ligands are usually quite flexible and a rigorous analysis has to incorporate molecular flexibility. We review the past six years of scientific publishing on molecular superposition. Our focus lies on automatic procedures to be performed on arbitrary molecular structures. Methodical aspects are our main concern here. Accordingly, plain application studies with few methodical elements are omitted in this presentation. While this review cannot mention every contribution to this actively developing field, we intend to provide pointers to the recent literature providing important contributions to computational methods for the structural alignment of molecules. Finally we provide a perspective on how superposition methods can effectively be used for the purpose of virtual database screening. In our opinion it is the ultimate goal to detect analogues in structure databases of nontrivial size in order to narrow down the search space for subsequent experiments.     

Embedded surfaces and almost complex structures

In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4-manifold to be pseudoholomorphic with respect to an almost complex structure on . In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.

     

On the Classification of q-Algebras

The problem is the classification of the ideals of free differential algebras, or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac–Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac–Moody algebras and of the (super) Kac–Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points q=1 and q=–1 are the most singular points in parameter space. In this Letter, one of two major hurdles in this classification program has been overcome. Fix a set of integers n ₁,...,n _k, and consider the space of homogeneous polynomials of degree n ₁ in the generator e ₁, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.