Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times

We apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain. We show under general hypotheses that the suitably rescaled exit time converges in the zero-noise limit to an exponential random variable. Then we extend the results to an infinite-dimensional case obtained by adding a small random perturbation to a nonlinear heat equation.     

Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Multi-turn time-of-flight mass spectrometers with electrostatic sectors

The mass resolution of a time-of-flight (TOF) mass spectrometer is directly proportional to its total flight pathlength. Multi-turn or multi-passage ion optical geometries are necessary to obtain fight distances of sufficient length within reasonable size limitations. We have investigated ion optics for a multi-turn TOF mass spectrometer with electrostatic sectors. The concept of 'perfect' focusing conditions is introduced. Furthermore, a new type of multi-turn TOF mass spectrometer, the MULTUM Linear plus, was developed. It consists of four cylindrical electric sectors and 28 electric quadrupole lenses. It has a vacuum chamber 60 x 70 x 20 cm in size. Mass resolution is demonstrated to increase according to the number of ion cycles. A mass resolution of 350 000 (m/z = 28, FWHM) was achieved after 501.5 cycles. The MULTUM Linear plus analyzer is not simple, however; 28 electric quadrupole lenses are used. In order to reduce the number of ion optical parts, an improved multi-turn TOF mass spectrometer, the MULTUM II, consisting of only four toroidal electric sectors, was also developed. The possibility of tandem mass spectrometric applications using multi-turn TOF mass spectrometers is also discussed.     

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations北大核心CSCD

针对三维非稳态对流扩散反应方程,构造了一种高精度紧致有限差分格式,对空间的离散采用四阶紧致差分方法,对时间的离散采用Taylor级数展开和余项修正技术,所提格式在时间上的精度为二阶、在空间上的精度为四阶。利用Fourier稳定性分析法证明了该格式是无条件稳定的。最后给出数值算例验证了理论结果。     

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.     

On the global optimization properties of finite-difference local descent algorithms

The paper is devoted to the convergence properties of finite-difference local descent algorithms in global optimization problems with a special -convex structure. It is assumed that the objective function can be closely approximated by some smooth convex function. Stability properties of the perturbed gradient descent and coordinate descent methods are investigated. Basing on this results some global optimization properties of finite-difference local descent algorithms, in particular, coordinate descent method, are discovered. These properties are not inherent in methods using exact gradients.The paper was presented at the II. IIASA-Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Fractal behavior in trapping and reaction: A random walk study

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimensiond. We interpolate between these two limits by introducing a scaling law. An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimensiond.     

RATE OF INTRA-MOLECULAR CONTACT FORMATION IN SHORT TWO-DIMENSIONAL COMPACT POLYMERS CHAINS

It is important to know the rate of intra-molecular contact formation in proteins in order to understand how proteins fold clearly. Here we investigate the rate of intra-molecular contact formation in short two-dimensional compact polymer chains by calculating the probability distribution p(r) of end-to-end distance r using the enumeration calculation method and HP model on two-dimensional square lattice. The probability distribution of end-to-end distance p(r) of short two-dimensional compact polymers chains may consist of two parts, i.e. p(r) = p1(r) p2(r), where p1(r) and p2(r) are different for small r. The rate of contact formation decreases monotonically with the number of bonds N, and the rate approximately conforms to the scaling relation of k(N) ∝ N-α. Here the value of α increases with the contact radius a and it also depends on the percentage of H (hydrophobic) residues in the sequences of compact chains and the energy parameters of εHH, εHP and εPP . Some comparisons of theoretical predictions with experimental results are also made. This investigation may help us to understand the protein folding.     

Study on the Dynamics of Blending of Glucomannan with Carrageenan and Guaran Gum

In order to prove up the interaction between konjac glucomannan (KGM) and other amyloses, by using KGM, guaran gum and carrageenan as materials, their blending in different external conditions was modeled by means of molecular dynamics (MD) method. The result showed that 323 K was a significant turning point for the formation of hydrogen bond, and KGM and guaran gum were just juxtaposed together via intermolecular hydrogen bond; but with the addition of carrageenan, KGM, guaran gum and carrageenan were spirally twisted together, and when their blending proportion is 3:2:9 the combination was the most compact. Moreover, the research result has provided the sequent studies afterwards on amyloses with some references.     

One-dimensional ventsel-freidlin theory and its analogue for singular perturbations of degenerate diffusions

We make some remarks about deriving the large deviations estimates for the Ventsel-Freidlin perturbed system and adapt these methods to derive similar results for singular perturbations of degenerate one-dimensional diffusions where β and ω are independent Brownian motions. This corresponds to a singular perturbation of the degenerate second-order operator