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.     

Novel UDP-glycal derivatives as transition state analogue inhibitors of UDP-GlcNAc 2-epimerase

The "epimerisation" of UDP-GlcNAc to ManNAc, the first step in the biosynthesis of sialic acids, is catalyzed by UDP-GlcNAc 2-epimerase. In this paper we report the synthesis of transition state based inhibitors of this enzyme. To mimic the assumed first transition state of this reaction (TS 1), we designed and synthesized the novel UDP-exo-glycal derivatives 1-4. We also report herein the synthesis of 5 and 6, the first C-glycosidic derivatives of 2-acetamidoglucal, and the synthesis of the ketosides 7 and 8, which were designed as bis-substrate analogue and bis- product analogue, respectively, to mimic the second step of the reaction via the assumed second transition state TS 2.     

Thermodynamische Berechung des Systems Ge/H2O/H2 und experimentelle Untersuchung mit der Nahabstandsanordnung

In the temperature range from 950 to 1200 K K_p-values result from 5 · 10⁻⁵ to 6,9 · 10⁻³ for the heterogeneous reversible reaction Ge_(s) + H₂O_(g) ⇋ GeO_(g) + H₂O_(g), the average reaction enthalpy being 46,5 ± 05 kcal/mol. Etching rates calculated with these equilibrium constants for closed systems are 20-25% larger than the experimental etching rates for (110)-Ge in the sandwich device. The ratio of the etching rates is for (111)-, (100)- and (110)-Ge 1:1,4:1,8; the average reaction enthalpy calculated experimentally is 46 ± 1 kcal/mol for the temperature range 1000-1200 K. The dependence of transport rate on distance between source and substrate characterizes the sandwich device as a quasi-closed system with the diffusion as the rate controlling step of the material transport.     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.     

Convergence of Discretization Procedures for Problems Whose Entropy Solutions are Uniquely Characterized by Additional Relations

Weak solutions of given problems are sometimes not necessarily unique. Relevant solutions are then picked out of the set of weak solutions by so-called entropy conditions. Connections between the original and the numerical entropy condition were often discussed in the particular case of scalar conservation laws, and also a general theory was presented in the literature for general scalar problems. The entropy conditions were realized by certain inequalities not generalizable to systems of equations in a trivial way. It is a concern of this article to extend the theory in such a way that inequalities can be replaced by general relations, and this not only in an abstract way but also realized by examples.     

Über den Verschiebungsradius bei konformer Abbildung

Ohne Zusammenfassung

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Im Gedenken an Herbert Grötzsch (1902–1993)