New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

Boundary value problems for stochastic differential equations

A theory of two-point boundary value problems analogous to the theory of initial value problems for stochastic ordinary differential equations whose solutions form Markov processes is developed. The theory of initial value problems consists of three main parts: the proof that the solution process is markovian and diffusive; the construction of the Kolmogorov or Fokker-Planck equation of the process; and the proof that the transistion probability density of the process is a unique solution of the Fokker-Planck equation.

It is assumed here that the stochastic differential equation under consideration has, as an initial value problem, a diffusive markovian solution process. When a given boundary value problem for this stochastic equation almost surely has unique solutions, we show that the solution process of the boundary value problem is also a diffusive Markov process. Since a boundary value problem, unlike an initial value problem, has no preferred direction for the parameter set, we find that there are two Fokker-Planck equations, one for each direction. It is shown that the density of the solution process of the boundary value problem is the unique simultaneous solution of this pair of Fokker-Planck equations.

This theory is then applied to the problem of a vibrating string with stochastic density.

Effect of an annular pupil filter on differential confocal microscopy

Combining differential confocal microscopy and an annular pupil filter, we obtained the normalized axial intensity distribution curve of an optical system. We used the sharp slopes of the axial response curve of the optical system to measure the surface profile of a reflection grating. Experimental results prove that this method can extend the axial dynamic range and improve the transverse resolution of three-dimensional profilometry by sacrificing axial resolution. (C) 2000 Optical Society of America.

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

Stability of parametrically excited differential equations

Sufficient stability criteria for classes of parametrically excited differential equations are developed and applied to example problems of a dynamical nature.

Stability requirements are presented in terms of 1) the modulus of the amplitude of the parametric terms, 2) the modulus of the integral of the parametric terms and 3) the modulus of the derivative of the parametric terms.

The methods employed to show stability are Liapunov’s Direct Method and the Gronwall Lemma. The type of stability is generally referred to as asymptotic stability in the sense of Liapunov.

The results indicate that if the equation of the system with the parametric terms set equal to zero exhibits stability and possesses bounded operators, then the system will be stable under sufficiently small modulus of the parametric terms or sufficiently small modulus of the integral of the parametric terms (high frequency). On the other hand, if the equation of the system exhibits individual stability for all values that the parameter assumes in the time interval, then the actual system will be stable under sufficiently small modulus of the derivative of the parametric terms (slowly varying).

DIFFERENTIAL GEOMETRICAL METHODS IN THE STUDY OF OPTICAL-TRANSMISSION (SCALAR THEORY) .1. STATIC TRANSMISSION CASE

Static optical transmission is restudied by postulation of the optical path as the proper element in a three-dimensional Riemannian manifold (no torsion); this postulation can be applied to describe the light-medium interactive system. On the basis of the postulation, the behaviors of light transmitting through the medium with refractive index n are investigated, the investigation covering the realms of both geometrical optics and wave optics. The wave equation of light in static transmission is studied modally, the postulation being employed to derive the exact form of the optical field equation in a medium (in which the light is viewed as a single-component field). Correspondingly, the relationships concerning the conservation of optical fluid and the dynamic properties are given, and some simple applications of the theories mentioned are presented.

DIFFERENTIAL GEOMETRICAL METHODS IN THE STUDY OF OPTICAL-TRANSMISSION (SCALAR THEORY) .2. TIME-DEPENDENT TRANSMISSION THEORY

By generalization of the methods presented in Part I of the study [J. Opt. Soc. Am. A 12, 600 (1994)] to the four-dimensional (4D) Riemannian manifold case, the time-dependent behavior of light transmitting in a medium is investigated theoretically by the geodesic equation and curvature in a 4D manifold. In addition, the field equation is restudied, and the 4D conserved current of the optical fluid and its conservation equation are derived and applied to deduce the time-dependent general refractive index. On this basis the forces acting on the fluid are dynamically analyzed and the self-consistency analysis is given.

Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.

Differential Neuroprotective Effects of 5 '-Deoxy-5 '-Methylthioadenosine

Background: 5'-deoxy-5'-methylthioadenosine (MTA) is an endogenous compound produced through the metabolism of polyamines. The therapeutic potential of MTA has been assayed mainly in liver diseases and, more recently, in animal models of multiple sclerosis. The aim of this study was to determine the neuroprotective effect of this molecule in vitro and to assess whether MTA can cross the blood brain barrier (BBB) in order to also analyze its potential neuroprotective efficacy in vivo. Methods: Neuroprotection was assessed in vitro using models of excitotoxicity in primary neurons, mixed astrocyte-neuron and primary oligodendrocyte cultures. The capacity of MTA to cross the BBB was measured in an artificial membrane assay and using an in vitro cell model. Finally, in vivo tests were performed in models of hypoxic brain damage, Parkinson's disease and epilepsy. Results: MTA displays a wide array of neuroprotective activities against different insults in vitro. While the data from the two complementary approaches adopted indicate that MTA is likely to cross the BBB, the in vivo data showed that MTA may provide therapeutic benefits in specific circumstances. Whereas MTA reduced the neuronal cell death in pilocarpine-induced status epilepticus and the size of the lesion in global but not focal ischemic brain damage, it was ineffective in preserving dopaminergic neurons of the substantia nigra in the 1-methyl-4-phenyl-1,2,3,6-tetrahydro-pyridine (MPTP)-mice model. However, in this model of Parkinson's disease the combined administration of MTA and an A(2A) adenosine receptor antagonist did produce significant neuroprotection in this brain region. Conclusion: MTA may potentially offer therapeutic neuroprotection.

Differential activity of GSK-3 isoforms regulates NF-kappa B and TRAIL- or TNF alpha induced apoptosis in pancreatic cancer cells

While TRAIL is a promising anticancer agent due to its ability to selectively induce apoptosis in neoplastic cells, many tumors, including pancreatic ductal adenocarcinoma (PDA), display intrinsic resistance, highlighting the need for TRAIL-sensitizing agents. Here we report that TRAIL-induced apoptosis in PDA cell lines is enhanced by pharmacological inhibition of glycogen synthase kinase-3 (GSK-3) or by shRNA-mediated depletion of either GSK-3 alpha or GSK-3 beta. In contrast, depletion of GSK-3 beta, but not GSK-3 alpha, sensitized PDA cell lines to TNF alpha-induced cell death. Further experiments demonstrated that TNF alpha-stimulated I kappa B alpha phosphorylation and degradation as well as p65 nuclear translocation were normal in GSK-3 beta-deficient MEFs. Nonetheless, inhibition of GSK-3 beta function in MEFs or PDA cell lines impaired the expression of the NF-kappa B target genes Bcl-xL and cIAP2, but not I kappa B alpha. Significantly, the expression of Bcl-xL and cIAP2 could be reestablished by expression of GSK-3 beta targeted to the nucleus but not GSK-3 beta targeted to the cytoplasm, suggesting that GSK-3 beta regulates NF-kappa B function within the nucleus. Consistent with this notion, chromatin immunoprecipitation demonstrated that GSK-3 inhibition resulted in either decreased p65 binding to the promoter of BIR3, which encodes cIAP2, or increased p50 binding as well as recruitment of SIRT1 and HDAC3 to the promoter of BCL2L1, which encodes Bcl-xL. Importantly, depletion of Bcl-xL but not cIAP2, mimicked the sensitizing effect of GSK-3 inhibition on TRAIL-induced apoptosis, whereas Bcl-xL overexpression ameliorated the sensitization by GSK-3 inhibition. These results not only suggest that GSK-3 beta overexpression and nuclear localization contribute to TNF alpha and TRAIL resistance via anti-apoptotic NF-kappa B genes such as Bcl-xL, but also provide a rationale for further exploration of GSK-3 inhibitors combined with TRAIL for the treatment of PDA.

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points t(i) is an element of [t(0), t(J)] for i = 0, 1, . . . , J of the solution. Two examples are provided.