Unique continuation on the boundary for Dini domains

We show that the normal derivative of a harmonic function which vanishes on an open subset of the boundary of a Dini domain cannot vanish on a subset of positive surface measure.

     

An Estimate of a Polynomial Capacity

We prove an estimate of a polynomial capacity for sets preserving Markov's inequality. Our estimate reflects some of the algebraic properties of these sets.     

Corrected sequential linear programming for sparse minimax optimization

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.Research supported partly by The Nordic Council of Ministers, The Icelandic Science Council, The University of Iceland Research Fund and The Danish Science Research Council.     

Letter to the Editor

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Librational modes and the structure of ammonium nitrate in its high temperature phases

We have measured the temperature dependence of the low frequency Raman spectrum of ammonium nitrate in its solid phases IV, II and I and also in its melt. Librational modes are observed in phases IV and II but disappear in both phase I and the melt. Our results indicate that the NO₃ group is almost freely rotating in the solid phase I.     

Linearly constrained minimax optimization

We present an algorithm for nonlinear minimax optimization subject to linear equality and inequality constraints which requires first order partial derivatives. The algorithm is based on successive linear approximations to the functions defining the problem. The resulting linear subproblems are solved in the minimax sense subject to the linear constraints. This ensures a feasible-point algorithm. Further, we introduce local bounds on the solutions of the linear subproblems, the bounds being adjusted automatically, depending on the quality of the linear approximations. It is proved that the algorithm will always converge to the set of stationary points of the problem, a stationary point being defined in terms of the generalized gradients of the minimax objective function. It is further proved that, under mild regularity conditions, the algorithm is identical to a quadratically convergent Newton iteration in its final stages. We demonstrate the performance of the algorithm by solving a number of numerical examples with up to 50 variables, 163 functions, and 25 constraints. We have also implemented a version of the algorithm which is particularly suited for the solution of restricted approximation problems.This work has been supported by the Danish Natural Science Research Council, Grant No. 511-6874.     

Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.     

Structural first-order transformation in La2/3Ba1/3MnO3: ESR study

We have studied by the electron-spin resonance (ESR) and static magnetic field techniques, the La_2/3Ba_1/3MnO₃ perovskite, which was previously shown to exhibit a martensitic phase transformation in the vicinity of T_s∼200 K [Physical Review B 68, 054109 (2003)], leading to its structural phase-segregated state. Resonant absorptions reveal that in the temperature interval from 100 K to 340 K the compound represents a mixture of two ferromagnetic phases possessing different magnetizations, in varying proportions depending on the temperature, and a small amount of a paramagnetic phase. The results agree well with the previous neutron diffraction study. Applied in the ESR experiments, magnetic fields (2–6 kOe) strongly affect the magnetization curves: even magnetic field as high as 700 Oe modifies the anomaly in the phase transformation region and removes the difference between the zero-field cooled and field-cooled magnetization curves, which implies that the difference in the magnetic susceptibility of the coexisting phases is small and the magnetic domain configuration can be easily changed.     

Cross decomposition applied to the stochastic transportation problem

In this paper we give a solution method for the stochastic transportation problem based on Cross Decomposition developed by Van Roy (1980). Solution methods to the derived sub and master problems are discussed and computational results are given for a number of large scale test problems. We also compare the efficiency of the method with other methods suggested for the stochastic transportation problem: The Frank-Wolfe algorithm and separable programming.     

Integrability of Green potentials in fractal domains

We proveL ^q -inequalities for the gradient of the Green potential (Gf) in bounded, connected NTA-domains inR ⁿ ,n≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponentq for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.