First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r (1¡r¡n) independent 0rst integrals which are in involution is calledpartially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings andweak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the 0rst-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging methodfor quasi-partially integrable Hamiltonian systems is brie4y reviewed. Then, basedon the averagedIt ˆo equations, a backwardKolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of 0rst-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and 0nal time conditions for the control problems of maximization of reliability andof maximization of mean 0rst-passage time are formulated. The relationship between the backwardKolmogorov equation andthe dynamical programming equation for reliability maximization, andthat between the Pontryagin equation andthe dynamical programming equation for maximization of mean 0rst-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the e9ectiveness of feedback control in reducing 0rst-passage failure.

Topographic changes in nascent and early mesoderm in amphioxus embryos studied by Dil labeling and by in situ hybridization for a Brachyury gene

In amphioxus embryos, the nascent and early mesoderm (including chorda-mesoderm) was visualized by expression of a Brachyury gene (AmBra-2). A band of mesoderm is first detected encircling the earliest (vegetal plate stage) gastrula sub-equatorially. Soon thereafter, the vegetal plate invaginates. resulting in a cap-shaped gastrula with the mesoderm localized at the blastoporal lip and completely encircling the blastopore. As the gastrula stage progresses, DiI (a vital dye) labeling demonstrates that the entire mesoderm is internalized by a slight involution of the epiblast into the hypoblast all around the perimeter of the blastopore. Subsequently. during the early neurula stage, the internalized mesoderm undergoes anterior extension mid-dorsally (as notochord) and dorsolaterally (in paraxial regions when segments will later form). By the late neurula stage, AmBra-2 is no longer transcribed throughout the mesoderm as a whole; instead. expression is detectable only in the posterior mesoderm and in the notochord, but not in par axial mesoderm where definitive somites have formed.