Lévy-Driven Langevin Systems: Targeted Stochasticity

Langevin dynamics driven by random Wiener noise (white noise), and the resulting Fokker–Planck equation and Boltzmann equilibria are fundamental to the understanding of transport and relaxation. However, there is experimental and theoretical evidence that the use of the Gaussian Wiener noise as an underlying source of randomness in continuous time systems may not always be appropriate or justified. Rather, models incorporating general Lévy noises, should be adopted. In this work we study Langevin systems driven by general Lévy, rather than Wiener, noises. Various issues are addressed, including: (i) the evolution of the probability density function of the system's state; (ii) the system's steady state behavior; and, (iii) the attainability of equilibria of the Boltzmann type. Moreover, the issue of reverse engineering is introduced and investigated. Namely: how to design a Langevin system, subject to a given Lévy noise, that would yield a pre-specified target steady state behavior. Results are complemented with a multitude of examples of Lévy driven Langevin systems.     

A Discrete-Time American Put Option Model with Fuzziness of Stock Prices

To solve a mathematical model for American put option with uncertainty, we utilize two essentials, i.e., a λ-weighting function and a mean value of fuzzy random variables simultaneously. Estimation of randomness and fuzziness as uncertainty should be important when we deal with a reasonable and natural model extended from the original optimization/decision making. Three kinds of mean values by fuzzy measures, which are based on Possibility, Necessity and Credibility, are demonstrated particularly. We consider the optimal expected price of the American put option by dynamic programming under a reasonable assumption. A numerical example is given to illustrate our idea.     

Weak interactions involving organic fluorine: a comparative study of the crystal packing in substituted isoquinolines

Weak interactions involving fluorine have been analyzed in the structure of 6-methoxy-1,2-diphenyl-1,2,3,4-tetrahydroisoquinoline with fluorine substitution at para-, meta- and ortho- positions, respectively, on the 1-phenyl ring. The packing modes in the crystalline lattice as determined by X-ray diffraction techniques generate motifs via F?F, C-H?F and C-F?π interactions. The three structures as compared to the parent compound depict conformational changes in the saturated tetrahydroisoquinoline moiety. The salient features of the four structures in terms of weak interactions involving fluorine suggest that organic fluorine does resemble the other halogens.     

Numerical solution of singular control problems using multiple shooting techniques

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.The author would like to thank Professor Dr. R. Bulirsch, who stimulated and encouraged this work, which is part of the author's dissertation.     

Optimal design of pitched laminated wood beams

The optimal design of a pitched laminated wood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865; by the US Energy Research and Development Administration Contract No. E(04-3)-326 PA-18; and by the National Science Foundation, Grant No. DCR75-04544 at Stanford University. This work was carried out during the first author's stay at the Management Science Division of the University of British Columbia and the Systems Optimization Laboratory of Stanford University. The authors are indebted to Mr. S. Liu and Mrs. M. Ratner for their assistance in performing the computations.     

Noncentrosymmetric organic solids with very strong harmonic generation response

The molten reaction of 2-naphthol, 4-(aminomethyl)pyridine, and 4-pyridinecarboxaldehyde at about 180 degrees C yields trans-2,3-dihydro-2,3-di(4'-pyridyl)benzo[e]indole (1) which possesses two chiral centers, rather than an expected Betti-type reaction product with only one chiral carbon center. The same reactions, using 3-pyridinecarboxaldehyde, 4-cyanobenzaldehyde, or 3- cyanobenzaldehyde instead of 4-pyridinecarboxaldehyde produce the related compounds trans-2,3-dihydro-2-(4'-pyridyl)-3-(3"-pyridyl)benzo[e]indole (2), trans-2,3-dihydro-2-(4'-pyridyl)-3-(4"-cyanophenyl)benzo[e]indole (3), and trans-2,3-dihydro-2-(4'-pyridyl)-3-(3"-cyanophenyl)benzo[e]indole (4), respectively. This reaction proceeds with a high degree of stereoselectivity with a trans/cis ratio of about 98:2 at elevated temperature. Compounds 1, 2, and 4 crystallize in a noncentrosymmetric space group (Pca2(1), Pca2(1), and Cc), while compound 3 has a chiral space group (P2(1)). These successfully acentric packing arrangements are probably due to the molecule bearing both two chiral centers and potential hydrogen-bonding groups. Furthermore, the reaction of racemic 6-hydroxy-2'-methyl-2-naphthaleneacetic acid with ethyl-2-cyano-1-(4'-pyridyl)acrylic acetate in the presence of piperidine gives 1-pyridyl-2-ethoxycarbonyl-3-amino-1H-naphtho[2,1-b]pyran-2'-methylacetic acid (5), which likewise crystallizes in a chiral space group. All of compounds are second harmonic generation (SHG) active, and have a very strong SHG response approximately about 8.0, 5.0, 12.0, 6.0, and 1.4 (for 1-5 compounds) times that of urea. Ferroelectric property measurements indicate that compounds 1, 2, 4, and 5 may display ferroelectric behavior.     

Organic chlorine as a hydrogen-bridge acceptor: evidence for the existence of intramolecular O--H...Cl--C interactions in some gem-alkynols

The acceptor capabilities of "organic" halogen, CX (X=F, Cl, Br, I), with respect to hydrogen bonding are controversial, and unactivated organic chlorine is generally deemed to be a poor acceptor. Hydrogen bridges of the type O--H...Cl--C are uncommon and occur mainly in an intramolecular situation when the donor group is sterically hindered, so that the formation of intermolecular interactions is difficult. In this paper, intramolecular O--H...Cl--C interactions in a series of chloro-substituted gem-alkynols are studied. We describe various features of this interaction using crystallographic, spectroscopic and computational methods. The O--H...Cl--C interaction occurs in five of the six compounds under consideration here (CDDA, 14DDDA, 15DDDA, 18DDDA, 15MKA). Solution (1)H NMR spectroscopy shows that the interaction is intramolecular and that it is a true hydrogen bond. DFT calculations give a stabilisation energy around 4.0 kcal mol(-1). In the crystal structures of the compounds studied, the intramolecular O--H...Cl--C interactions fit into the overall scheme of cooperative interactions. These structures may be derived from that of the unsubstituted compound DDA by means of synthon exchange and the O--H...Cl--C interaction fares surprisingly well in the presence of competing stronger acceptors. The crystal structures show an unusual degree of modularity for compounds that generally form interactions that are weak and variable. It is noteworthy that the so-called "weak" acceptor, organic chlorine, is able to sustain a good intramolecular hydrogen bridge that is of an attractive and stabilizing nature and which is of potential importance in crystal engineering and supramolecular chemistry.