Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using tridiagonal matrix models, in the limit as β→∞. We prove that the fluctuations are described by multivariate Gaussians of covariance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. The covariance matrix itself is expressed as combinations of Hermite or Laguerre polynomials respectively.We show that the approximations are of real value even for small β; we can use them to approximate the true functions even for the traditional β=1,2,4 values.     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

On characterizing collections arising from N-gons in the plane

Consider a labeled convex n-gon in the plane and a point x in general position in the interior. We associate with this a collection (x) of 3-element sets, namely those 3-element subsets of the vertices which contain x in their interior. We investigate the problem of characterizing those collections of subsets which can be represented in this matter. We give a complete characterization of this and a related problem.School of Mathematics, University of Minnesota, Minneapolis, MN 55455. This work was supported in part by NSF grant DMS-8700995.Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England. Work done while Hill Visiting Professor at the University of Minnesota.     

Enantioselective, chirally templated sol-gel thin films

Enantioselective surfactant-templated thin films were fabricated through the sol-gel (SG) process. The enantioselectivity is general in the sense that it discriminates between pairs of enantiomers not used for the imprinting process. The chiral cationic surfactant (-)-N-dodecyl-N-methylephedrinium bromide (1) was used as the surfactant template, and after its extraction chiral domains were created. The chiral discriminative feature of these films was examined by challenging with pure enantiomer solutions for rebinding. Selective adsorption was shown using (R)- and (S)-propranolol, (R)-2 and (S)-2, respectively, and (R)- and (S)-2,2,2-trifluoro-1-(9-anthryl)ethanol, (R)-3 and (S)-3, respectively, as the chiral probes. The selective adsorption was measured by fluorescence analysis, and the chiral selectivity factors were found to be 1.6 for 2 and 2.25 for 3. In both cases, (R)-enantiomer was adsorbed preferably. The resulting material was characterized by transmission electron microscopy, by diffraction, and by surface area measurements, and was found to be semicrystalline with short-range ordered domains (50 A) of hexagonal symmetry.     

Free arrangements and rhombic tilings

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:

1. When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
2. When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?

Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.     

Raman spectroscopy of tris-(hydroxymethyl)aminomethane as a model system for the studies of α-chymotrypsin activation by crown ether in organic solvents

Raman spectroscopy is employed to study tris-(hydroxymethyl)aminomethane and its complexes with 18-crown-6. The results obtained are used to interpret the known effect of α-chymotrypsin activation by crown ether in organic solvents. Raman spectra of the samples lyophilized from aqueous solutions at various pH values are measured in solid state, acetonitrile and cyclohexane.     

Si-E (E = N, O, F) bonding in a hexacoordinated silicon complex: new facts from experimental and theoretical charge density studies

The concept of hypervalency in molecules, which hold more than eight valence electrons at the central atom, still is a topic of constant debate. There is general interest in silicon compounds with more than four substituents at the central silicon atom. The dispute, whether this silicon is hypervalent or highly coordinated, is enlightened by the first experimental charge density determination and subsequent topological analysis of three different highly polar Si-E (E = N, O, F) bonds in a hexacoordinated compound. The experiment reveals predominantly ionic bonding and much less covalent contribution than commonly anticipated. For comparison gas-phase ab initio calculations were performed on this compound. The results of the theoretical calculations underline the findings of the experiment.