On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces. Received: 8 February 2001; in final form: 17 September 2001 / Published online: 1 February 2002     

Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ξ₂ + ξ₁ξ₂ξ₃ + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μ _k = E(ξ ^k ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Edge coloring models and reflection positivity

Solving a conjecture of M. H. Freedman, L. Lovász and A. Schrijver, we prove that a graph parameter is edge reflection positive and multiplicative if and only if it can be represented by an edge coloring model.

     

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.     

Multiple Shocks in Bricklayers' Model

In bricklayers' model, which is a generalization of the misanthrope processes, we show that a nontrivial class of product distributions is closed under the time-evolution of the process. This class also includes measures fitting to shock data of the limiting PDE. In particular, we show that shocks of this type with discontinuity of size one perform ordinary nearest neighbor random walks only interacting, in an attractive way, via their jump rates. Our results are related to those of Belitsky and Schütz⁽⁴⁾ on the simple exclusion process, although we do not use quantum formalism as they do. The structures we find are described from a fixed position. Similar ones were found in Balázs,⁽²⁾ valak as seen from the random position of the second class particle.     

Coverings of Abelian groups and vector spaces

We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many cosets we need, if we want to cover all but one element of an Abelian group. This result is a group theoretical extension of the theorem of Brouwer, Jamison and Schrijver about the blocking number of an affine space. We show that these covering problems are closely related to combinatorial problems, including the so-called additive basis conjecture, the three-flow conjecture, and a conjecture of Alon, Jaeger and Tarsi about nowhere zero vectors.     

The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n ^1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n ^1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.     

Density functional study of the conformational space of 4C1 D-glucuronic acid

The conformational space of (4)C(1) alpha- and beta-d-glucuronic acid was scanned by HF/3-21G(p) calculations followed by optimization of the 15 most stable structures for each, using the B3LYP density functional theory method in conjunction with a diffuse polarized valence triple-zeta basis set. We found a general preference of the alpha anomers in the isolated molecules in agreement with the large endo-anomeric hyperconjugation effects in these structures. From the other intramolecular interactions (exo-anomeric hyperconjugation, hydrogen-bonding, dipole-dipole, and steric interactions), the effect of the hydrogen bonding is the most pronounced and plays a major role in determining the stability order within the alpha and beta series. The most stable conformer of both alpha and beta (4)C(1) d-glucuronic acid is the structure with the maximum number (5) of intramolecular hydrogen bonds. Introduction of solvent (water) effects by the SCI-PCM model resulted in two characteristic changes of the energetic properties: the gas-phase stability order changed considerably, and the energy range of the 15 most stable conformers decreased from 30 to 15 kJ/mol. The geometrical parameters reflect well the superimposed effects of hyperconjugation and hydrogen-bonding interactions. Most characteristics are the variations of the C-O bond distances (within a range of 0.04 A) upon the combined intramolecular effects.     

Evaluation of cyclodextrin polymer as a disintegrating agent

Cyclodextrin polymer was compared to other well known disintegrants concerning the swelling properties /water uptake, moisture uptake, hydration capacity, sedimentation volume in water/. Its high disintegrating effect was proved in directly compressed tablets as well as in tablets made by wet granulation. A remarkable improvement in tablet properties was observed. Not only the disintegration of tablets and the dissolution of the drug was accelerated but also the hardness increased when CDP was used as disintegrant.