Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Rescue, rehabilitation, and release of marine mammals: An analysis of current views and practices.

Stranded marine mammals have long attracted public attention. Those that wash up dead are, for all their value to science, seldom seen by the public as more than curiosities. Animals that are sick, injured, orphaned or abandoned ignite a different response. Generally, public sentiment supports any effort to rescue, treat and return them to sea. Institutions displaying marine mammals showed an early interest in live-stranded animals as a source of specimens -- in 1948, Marine Studios in St. Augustine, Florida, rescued a young short-finned pilot whale (Globicephala macrorhynchus), the first ever in captivity (Kritzler 1952). Eventually, the public as well as government agencies looked to these institutions for their recognized expertise in marine mammal care and medicine. More recently, facilities have been established for the sole purpose of rehabilitating marine mammals and preparing them for return to the wild. Four such institutions are the Marine Mammal Center (Sausalito, CA), the Research Institute for Nature Management (Pieterburen, The Netherlands), the RSPCA, Norfolk Wildlife Hospital (Norfolk, United Kingdom) and the Institute for Wildlife Biology of Christian-Albrects University (Kiel, Germany).(PDF contains 68 pages.

Smooth and Heavy Solutions to Control Problems

We introduce the concept of viability domain of a set-valued map, which we study and use for providing the existence of smooth solutions to differential inclusions. We then define and study the concept of heavy viable trajectories of a controlled system with feedbacks. Viable trajectories are trajectories satisfying at each instant given constraints on the state. The controls regulating viable trajectories evolve according a set-valued feedback map. Heavy viable trajectories are the ones which are associated to the controls in the feedback map whose velocity has at each instant the minimal norm. We construct the differential equation governing the evolution of the controls associated to heavy viable trajectories and we state their existence

Slow and Heavy Viable Trajectories of Controlled Problems. Part 1. Smooth Viability Domains

We define slow and heavy viable trajectories of differential inclusions and controlled problems. Slow trajectories minimize at each time the norm of the velocity of the state (or the control) and heavy trajectories the norm of the acceleration of the state (or the velocity of the control). Macrosystems arising in social and economic sciences or biological sciences seem to exhibit heavy trajectories. We make explicit the differential equations providing slow and heavy trajectories when the viability domain is smooth

Qualitative Equations: The Confluence Case

This paper deals with a domain of Artificial Intelligence known under the name of "qualitative simulation" or "qualitative physics", to which special volumes of "Artificial Intelligence" (1984) and or "IEEE Transactions on Systems, Man and Cybernetics" (1987) have been devoted. It defines the concept of "qualitative frame" of a set, which allows to introduce strict, large and dual confluence frames of a finite dimensional vector-space. After providing a rigorous definition of standard, lower and upper qualitative solutions in terms of confluences introduced by De Kleer, it provides a duality criterion for the existence of a strict standard solution to both linear and non linear equations. It also furnishes a dual characterization of the existence or upper and lower qualitative solutions to a linear equation. These theorems are extended to the case or "inclusions", where single-valued maps are replaced by set-valued maps. This may be useful for dealing with qualitative properties of maps which are not precisely known, or which are defined by a set of properties, a requirement which is at the heart of qualitative simulation

On the Modelisation of a Cognitive Process: A Viability Approach

A dynamical description of an abstract cognitive system which should be closer to some cognitive considerations than pure automata or networks of automata is proposed. The system operates "sensory-motor" states, whose components are the state of the environment, its variation and the cerebral motor activity. The main addition is the introduction of a "conceptual control" that is postulated in order to define a "learning process", which is a set-valued map associating conceptual controls with sensory-motor states. A learning process must be consistent with a 'recognition mechanism" which determines at each instant the set of possible metaphors, linking the perception of the environment and its variations with conceptual controls, as well as with "viability constraints" describing the consumption of the cognitive system, associating with each state of the environment the set of viable motor activities. It also has to be consistent with an 'action law", describing the evolution of the state of the environment in terms of the cerebral motor activity, and a "motor activity" law, describing the evolution of the motor activity in terms of the perception of the environment and the conceptual controls. It suggests also that the evolution of a learning process obeys an "inertia principle" which allows to select specific choices of learning procedures. This paper justifies this approach, which can used to prove mathematically the existence of a largest learning process and of specific "heavy evolutions" obeying an "inertia principle"

Study of Solutions to Differential Inclusions by the "Pipe Method"

A pipe of a differential inclusion is a set-valued map associating with each time t a subset P(t) of states which contains a trajectory of the differential inclusion for any initial state x_o belonging to P(O). As in the Liapunov method, knowledge of a pipe provides information on the behavior of the trajectory. In this paper, the characterization of pipes and non-smooth analysis of set-valued maps are used to describe several classes of pipes. This research was conducted within the framework of the Dynamics of Macrosystems study in the System and Decision Sciences Program