The Hamilton--Jacobi Theory and the Analogy between Classical and Quantum Mechanics

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.Comment: 42 pages, LaTeX with AIMS clas

The archaeal ATPase PINA interacts with the helicase Hjm via its carboxyl terminal KH domain remodeling and processing replication fork and Holliday junction.

PINA is a novel ATPase and DNA helicase highly conserved in Archaea, the third domain of life. The PINA from Sulfolobus islandicus (SisPINA) forms a hexameric ring in crystal and solution. The protein is able to promote Holliday junction (HJ) migration and physically and functionally interacts with Hjc, the HJ specific endonuclease. Here, we show that SisPINA has direct physical interaction with Hjm (Hel308a), a helicase presumably targeting replication forks. In vitro biochemical analysis revealed that Hjm, Hjc, and SisPINA are able to coordinate HJ migration and cleavage in a concerted way. Deletion of the carboxyl 13 amino acid residues impaired the interaction between SisPINA and Hjm. Crystal structure analysis showed that the carboxyl 70 amino acid residues fold into a type II KH domain which, in other proteins, functions in binding RNA or ssDNA. The KH domain not only mediates the interactions of PINA with Hjm and Hjc but also regulates the hexameric assembly of PINA. Our results collectively suggest that SisPINA, Hjm and Hjc work together to function in replication fork regression, HJ formation and HJ cleavage

Bohmian trajectories and the Path Integral Paradigm. Complexified Lagrangian Mechanics

David Bohm shown that the Schr{\"o}dinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions - action and probability density. The first equation is the Hamilton-Jacobi (HJ) equation, a "visiting card" of classical mechanics, to be modified by the Bohmian quantum potential. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed to two Bohmian quantum correctors. The first corrector modifies kinetic energy term of the HJ equation, and the second one modifies potential energy term. Unification of the quantum HJ equation and the entropy balance equation gives complexified HJ equation containing complex kinetic and potential terms. Imaginary parts of these terms have order of smallness about the Planck constant. The Bohmian quantum corrector is indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to imaginary sector.Comment: 14 pages, 3 figures, 46 references, 48 equation

Stochastic Discount Factor Bounds with Conditioning Information

Hansen and Jagannathan (HJ, 1991) describe restrictions on the volatility of stochastic discount factors (SDFs) that price a given set of asset returns. This paper compares the sampling properties of different versions of HJ bounds that use conditioning information in the form of a given set of lagged instruments. HJ describe one way to use conditioning information. Their approach is to multiply the original returns by the lagged variables, and much of the asset pricing literature to date has followed this ihmultiplicativel. approach. We also study two versions of optimized HJ bounds with conditioning information. One is from Gallant, Hansen and Tauchen (1990) and the second is based on the unconditionally-efficient portfolios derived in Ferson and Siegel (2000). We document finite-sample biases in the HJ bounds, where the biased bounds reject asset-pricing models too often. We provide useful correction factors for the bias. We also evaluate the asymptotic standard errors for the HJ bounds, from Hansen, Heaton and Luttmer (1995).

First Order Actions: a New View

We analyse systems described by first order actions using the Hamilton-Jacobi (HJ) formalism for singular systems. In this study we verify that generalized brackets appear in a natural way in HJ approach, showing us the existence of a symplectic structure in the phase spaces of this formalism

Fast Reachable Set Approximations via State Decoupling Disturbances

With the recent surge of interest in using robotics and automation for civil purposes, providing safety and performance guarantees has become extremely important. In the past, differential games have been successfully used for the analysis of safety-critical systems. In particular, the Hamilton-Jacobi (HJ) formulation of differential games provides a flexible way to compute the reachable set, which can characterize the set of states which lead to either desirable or undesirable configurations, depending on the application. While HJ reachability is applicable to many small practical systems, the curse of dimensionality prevents the direct application of HJ reachability to many larger systems. To address computation complexity issues, various efficient computation methods in the literature have been developed for approximating or exactly computing the solution to HJ partial differential equations, but only when the system dynamics are of specific forms. In this paper, we propose a flexible method to trade off optimality with computation complexity in HJ reachability analysis. To achieve this, we propose to simplify system dynamics by treating state variables as disturbances. We prove that the resulting approximation is conservative in the desired direction, and demonstrate our method using a four-dimensional plane model.Comment: in Proceedings of the IEE Conference on Decision and Control, 201