Direct observation of a self-assembled monolayer of heptyl xanthate at the germanium/water interface: a polarized FTIR study

The adsorption of heptyl xanthate on germanium has been studied by the attenuated total reflection (ATR) technique. Polarized infrared light was used in situ to determine the average orientation of the alkyl chain in heptyl xanthate adsorbed at the germanium/solution interface. Spectra reveal the formation of closely packed xanthate ions with the alkyl chains in the all-trans conformation. The average tilt angle of the alkyl chains of heptyl xanthate was approximately 47 degrees from the surface normal.     

On the Value of a Random Minimum Weight Steiner Tree

Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1+o(1))(k-1)(log n-log k)/n when k =o(n) and n.* Research supported in part by NSF grant DSM9971788 Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Part of this research was done while visiting IBM T. J. Watson Research Center.     

The Number of Edge Colorings with no Monochromatic Cliques

Let F(n,r,k) denote the maximum possible number of distinctedge-colorings of a simple graph on n vertices with r colorswhich contain no monochromatic copy of K_k. It is shown thatfor every fixed k and all n>n₀(k), and , where t_k–1(n)is the maximum possible number of edges of a graph on n verticeswith no K_k (determined by Turán's theorem). The caser=2 settles an old conjecture of Erds and Rothschild, whichwas also independently raised later by Yuster. On the otherhand, for every fixed r>3 and k>2, the function F(n,r,k)is exponentially bigger than . The proofs are based on Szemerédi's regularity lemmatogether with some additional tools in extremal graph theory,and provide one of the rare examples of a precise result provedby applying this lemma.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

Optimal closing of a pair trade with a model containing jumps

A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely used investment strategy in the financial industry. Recently, Ekström, Lindberg, and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In the present work the model is generalized to also include jumps. More precisely, we assume that the difference between the assets is an Ornstein-Uhlenbeck type process, driven by a Lévy process of finite activity. We prove a necessary condition for optimality (a so-called verification theorem), which takes the form of a free boundary problem for an integro-differential equation. We analyze a finite element method for this problem and prove rigorous error estimates, which are used to draw conclusions from numerical simulations. In particular, we present strong evidence for the existence and uniqueness of an optimal solution.     

A STUDY OF FAT OXIDATION IN A MICROEMULSION SYSTEM

The effect of ascorbic acid and α-tocopherol on fat oxidation has been studied in a microemulsion consisting of soybean oil, sunflower oil monoglycerides and water. It was observed that the presence of ascorbic acid in the water aggregates gave a pronounced reduction in the oxidation rate, and even after storage at 40°C for 100 days only a minor oxidation of the oil was observed, whereas the control, without ascorbic acid, was highly oxidized.     

Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non-linear heat equation

Error estimates are shown for some spatially discrete Galerkin finite element methods for a non-linear heat equation. The approximation schemes studied are based on the introduction of the enthalpy as a new dependent variable, and also on the application of the Kirchhoff transformation and on interpolation of the non-linear coefficients into standard Lagrangian finite element spaces.     

Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.