Continuous approximation schemes for stochastic programs

One of the main methods for solving stochastic programs is approximation by discretizing the probability distribution. However, discretization may lose differentiability of expectational functionals. The complexity of discrete approximation schemes also increases exponentially as the dimension of the random vector increases. On the other hand, stochastic methods can solve stochastic programs with larger dimensions but their convergence is in the sense of probability one. In this paper, we study the differentiability property of stochastic two-stage programs and discuss continuous approximation methods for stochastic programs. We present several ways to calculate and estimate this derivative. We then design several continuous approximation schemes and study their convergence behavior and implementation. The methods include several types of truncation approximation, lower dimensional approximation and limited basis approximation.His work is supported by Office of Naval Research Grant N0014-86-K-0628 and the National Science Foundation under Grant ECS-8815101 and DDM-9215921.His work is supported by the Australian Research Council.     

Tuning the optical properties of fluorinated indolylfulgimides

Photochromic fluorinated indolylfulgides have been identified as potential candidates for a wide range of applications including optical switches, photoregulators of biological processes, and optical memory media. In humid environments or biological systems, hydrolytic stability is essential. In an effort to improve hydrolytic stability, a series of indolylfulgimides has been synthesized from a parent trifluoromethyl-substituted indolylfulgide. The nitrogen of the succinimide moiety is linked to either a dimethyl amino or one of seven substituted phenyl groups. The phenyl groups feature substituents with increasing electron-withdrawing ability. The spectral characteristics of each compound have been examined, revealing that the wavelength absorption maxima of each form increases with increasing electron-withdrawing ability of the substituted N-phenyl ring. The quantum yields of the photoreactions have been determined with the N-(phenyl)fulgimide showing a ring closure value of nearly 0.30 in toluene. In addition, the hydrolytic, thermal, and photochemical stabilities of each compound have been measured. The fulgimides exhibit at least a 200-fold enhancement of hydrolytic stability for the Z-form and over a 1000-fold enhancement for the C-form in comparison to the same form of the parent fulgide. The N-(2,3,5,6-tetrafluoro-4-trifluoromethylphenyl)fulgimide can undergo up to 3000 photochemical cycles (coloration followed by bleaching) before losing 20% of its initial absorbance at photostationary state.     

Electronic spectra of 12-S-CIS conformationally locked retinals

The electronic absorption spectra of the four new 12-$\text{s}$-$\text{cis}$-locked retinals (1a–1d) bearing 7-$\text{trans}$, 11-$\text{trans}$ double bond geometries are described and compared with those of analogous 7-$\text{trans}$, 11-$\text{cis}$-geometries (1e–1h) and parent retinals (2a-h).     

Homological domino effects and the first Finitistic Dimension Conjecture

Summary Counterexamples to the first Finitistic Dimension Conjecture are constructed. In fact, for any fieldk and any integerm2, there exist finite dimensionalk-algebras such that the little finitistic dimension of ism, while the big finitistic dimension ism+1. Our examples are monomial relation algebras; within this class of algebras the big and little finitistic dimensions cannot differ by more than 1. The analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras.Oblatum 5-VI-1991 & 28-X-1991This research was partially supported by a grant from the National Science Foundation. It is dedicated to Maurice Auslander on the occasion of his sixtyfifth birthday.     

Ultrabroadband beam splitter with matched group-delay dispersion

We present a general design strategy for a broadband thin-film beam splitter with matched group-delay dispersion. By taking the substrate dispersion into account in the coating design, any combination of input and output can show the same dispersion for transmission and reflection. As a specific implementation, an ultrabroadband 50:50 beam splitter from 600 to 1500 nm for femtosecond laser applications was designed, fabricated, and characterized.     

Efficient solution of two-stage stochastic linear programs using interior point methods

Solving deterministic equivalent formulations of two-stage stochastic linear programs using interior point methods may be computationally difficult due to the need to factorize quite dense search direction matrices (e.g., AA ^T ). Several methods for improving the algorithmic efficiency of interior point algorithms by reducing the density of these matrices have been proposed in the literature. Reformulating the program decreases the effort required to find a search direction, but at the expense of increased problem size. Using transpose product formulations (e.g., A ^T A) works well but is highly problem dependent. Schur complements may require solutions with potentially near singular matrices. Explicit factorizations of the search direction matrices eliminate these problems while only requiring the solution to several small, independent linear systems. These systems may be distributed across multiple processors. Computational experience with these methods suggests that substantial performance improvements are possible with each method and that, generally, explicit factorizations require the least computational effort.