Infinitesimal Lorentz Transformations and the States of Polarization of Free Photons

Applying the tangent dynamics technique as generated by infinitesimal Lorentz transformations, it is shown that the components of an electromagnetic field correspond to the Lie-Cartan parameters of the group SO(3,1). Thus the maximum number of non-null components of the magnetic field (i.e., the number of the states of polarization) is not decided by the number of group generators.     

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.     

On the pricing of American options

The problem of valuation for contingent claims that can be exercised at any time before or at maturity, such as American options, is discussed in the manner of Bensoussan [1]. We offer an approach which both simplifies and extends the results of existing theory on this topic.Research supported in part by the National Science Foundation under Grant No. NSF-DMS-84-16736 and by the Air Force Office of Scientific Research under Grant No. F49620-85-C-0144.     

Development and evaluation of conditions for the kinetic quantitation of acetoacetate in body fluids

This paper describes a detailed study of the kinetic behavior of reactions used in the quantitation of acetoacetate in body fluids. Reactions studied are those between acetoacetate and glycine to form a transiently stable enamine intermediate, reaction of the enamine intermediate with nitroprusside, and simultaneous reaction of acetoacetate, glycine, and nitroprusside (nitrosopentacyanoferrate(III). Variables studied include pH, temperature and concentrations of all reactants. The primary reaction between acetoacetate and glycine follows pseudo-first-order kinetics at high pH (ρ 8.6) at which the enamine is transiently stable; the other reactions deviate from first-order behavior at longer times because the reaction product is unstable. Several options involving nonlinear curve-fitting methods applied to all reaction sequences above and an initial-rate method applied to the combined reactions are evaluated. Also, the two-step and combined-reaction sequences are evaluated for quantitation of acetoacetate in urine and serum samples. For acetoacetate in the concentration range 0.5–4.25 mmol l^?1, the methods yield relative standard deviations in the range 1.0–1.8% with absolute standard deviations of 3.5–35 μmol l^?1.     

On the solution of equations with nondifferentiable and ptak error estimates

In this note we approximate solutions of equations with nondifferentiable operators and improve recent error estimates.     

On the Secant method for solving nonsmooth equations

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using the Secant method. The differentiability of the operator involved is not assumed. Using a flexible point-based approximation, we provide a local as well as a semilocal convergence analysis for the Secant method. Our results are justified by numerical examples that cannot be handled with earlier works.     

Optimal equi-partition of rectangular domains for parallel computation

We present an efficient method for the partitioning of rectangular domains into equi-area sub-domains of minimum total perimeter. For a variety of applications in parallel computation, this corresponds to a load-balanced distribution of tasks that minimize interprocessor communication. Our method is based on utilizing, to the maximum extent possible, a set of optimal shapes for sub-domains. We prove that for a large class of these problems, we can construct solutions whose relative distance from a computable lower bound converges to zero as the problem size tends to infinity. PERIX-GA, a genetic algorithm employing this approach, has successfully solved to optimality million-variable instances of the perimeter-minimization problem and for a one-billion-variable problem has generated a solution within 0.32% of the lower bound. We report on the results of an implementation on a CM-5 supercomputer and make comparisons with other existing codes.This research was partially funded by Air Force Office of Scientific Research grant F496-20-94-1-0036 and National Science Foundation grants CDA-9024618 and CCR-9306807.     

Estimation and control for linear,partially observable systems with non-gaussian initial distribution

The nonlinear filtering problem of estimating the state of a linear stochastic system from noisy observations is solved for a broad class of probability distributions of the initial state. It is shown that the conditional density of the present state, given the past observations, is a mixture of Gaussian distributions, and is parametrically determined by two sets of sufficient statistics which satisfy stochastic DEs; this result leads to a generalization of the Kalman–Bucy filter to a structure with a conditional mean vector, and additional sufficient statistics that obey nonlinear equations, and determine a generalized (random) Kalman gain. The theory is used to solve explicitly a control problem with quadratic running and terminal costs, and bounded controls.     

Coincidence set of minimal surfaces for the thin obstacle

We consider the Thin Obstacle Problem for minimal surfaces in two dimensions. The coincidence set for an analytic obstacle is proved to be a finite union of intervals. We show also that the topological structure of the coincidence set is generically identical to the above in the space of twice-continuously differentiable obstacles.