New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Dynamic supercoiling bifurcations of growing elastic filaments

Certain bacteria form filamentous colonies when the cells fail to separate after dividing. In Bacillus subtilis, Bacillus thermus, and Cyanobacteria, the filaments can wrap into complex supercoiled structures as the cells grow. The structures may be solenoids or plectonemes, with or without branches in the latter case. Any microscopic theory of these morphological instabilities must address the nature of pattern selection in the presence of growth, for growth renders the problem nonautonomous and the bifurcations dynamic. To gain insight into these phenomena, we formulate a general theory for growing elastic filaments with bending and twisting resistance in a viscous medium, and study an illustrative model problem: a growing filament with preferred twist, closed into a loop. Growth depletes the twist, inducing a twist strain. The closure of the loop prevents the filament from unwinding back to the preferred twist; instead, twist relaxation is accomplished by the formation of supercoils. Growth also produces viscous stresses on the filament which even in the absence of twist produce buckling instabilities. Our linear stability analysis and numerical studies reveal two dynamic regimes. For small intrinsic twist the instability is akin to Euler buckling, leading to solenoidal structures, while for large twist it is like the classic writhing of a twisted filament, producing plectonemic windings. This model may apply to situations in which supercoils form only, or more readily, when axial rotation of filaments is blocked. Applications to specific biological systems are proposed.     

Structural conversions of rhodamines in solution

A study has been made of the acidic ionization, structural conversions, and prototropic tautomerism of rhodamine chromogens in methanol and dimethyl sulfoxide, using IR, ¹³ C NMR, and electronic spectroscopy. The lactone structure has been established for the colorless form of rhodamines in DMSO. For rhodamine B in mixtures of water with DMSO, the lactone/zwitterion ratio has been estimated.V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Moscow 117975. Khar'kov State University, Khar'kov 310077. Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 3, pp. 512–521, March, 1992.     

Active control of extended Van der Pol equation

We study in this paper the active control of a driven class of Van der Pol oscillator which exhibits three limit cycles. We begin by investigating the dynamics and stability analysis of the system under active control. We also analyze the effects of a time periodic perturbation included in the control process. In all these cases the domain of control gain parameters leading to a good control is obtained and verified numerically.     

Higher discriminants of binary forms

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 147–157, November, 2007.     

Nuclear Excitation by X-Ray Emission of Femtosecond Laser Plasma

The proposed model of laser plasma emission spectrum formation enables us to determine the absolute value of the laser pulse to plasma emitted radiation conversion factor, profile of the emission spectrum, and frequency distribution of the intensity and energy in the emitted spectra. This is of interest for laser plasma diagnostics and provides a means for direct calculation of the number of excited nuclei in dependent on the parameters of laser pulse. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analyzing the effect of the inventory policy on order and inventory variability with linear control theory

In this paper we apply linear control theory to study the effect of various inventory policies on order and inventory variability, which are key drivers of supply chain performance. In particular, we study a two-echelon supply chain with a stationary demand pattern under the influence of three inventory policies: an inventory-on-hand policy that bases orders on the visible inventory at an installation, an installation-stock policy that bases orders on the inventory position (on-hand plus on-order inventory) at an installation, and an echelon-stock policy that bases orders on the inventory position at that installation and all downstream installations. We prove analytically that the inventory-on-hand policy is unstable in practical settings, confirming analytically what has been observed in experimental settings and in practice. We also prove that the installation-stock and echelon-stock policies are stable and analyze their effect on order and inventory fluctuation. Specifically, we show the general superiority of the echelon-stock in our setting and demonstrate analytically the effect of forecasting parameters on order and inventory fluctuations, confirming the results in other research.