Exterior billiards plakhov alex ander
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Conference topics reflected the diversity of operations research applications in the natural sciences, to wit: analysis of micro array data or next generation sequencing; applications of modeling and optimization in physics, biology, chemistry, and medicine; billiard theory and applications; biomedical engineering; design optimization; data visualization for optimal decisions; image processing and inverse problems; infinite and semi-infinite optimization with applications; multi-criteria optimization with applications; optimal control applied to biological models; optimization in bioinformatics and computational biology; shape optimization; and statistical and probabilistic modeling. The book will appeal to mathematicians working in dynamical systems and calculus of variations. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text. We show that failed unlearning results in total memory breakdown. This set is the disjoint union of infinitely many domains with piecewise smooth boundary. By measure we mean the natural measure in the space of straight lines in R2 associated with initial parts of billiard trajectories.

A self-consistent system of equations of the spectral dynamics of a synaptic matrix is obtained at the thermodynamic limit. It is assumed that each particle is reflected no more than once no multiple reflections are allowed ; then the resistance of the graph to the flow is expressed as {equation presented}. This estimate is obtained with the help of a result related to a Monge problem of mass transportation. Of course, one can pose new questions about our solutions: are they sensitive to the thermal motion of the particles or to rotational motion of the body? We consider the problem of mirror invisibility for plane sets. The notions of rough body and law of scattering on a body are discussed.

The particles make elastic reflections when colliding with the body surface and do not interact with each other. The body's center of mass moves with constant velocity and, in addition, the body performs slow rotational motion. We shall see that the method of solution is quite conventional as compared with the original problem. Moreover, there is, in a sense, a huge amount of such families. The book will appeal to mathematicians working in dynamical systems and calculus of variations.

It begins with an overview of the mathematical notations used throughout the book and a brief review of the main results. Specialists working in the areas of applications discussed will also find it useful. This problem has been considered for various classes of admissible bodies. The theory is illustrated by an example from mathematical epidemiology. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1. In chapters 4 and 5, scattering of billiards by nonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied.

The maximum is attained on a sequence of bodies with a very complicated boundary. The function ϕ may have one or several zeros; the random values ξ t are independent and identically distributed, with zero mean and finite variance. The numerical study was made for somewhat more restricted classes of bodies. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text. The center of mass of the body moves with a constant speed, and, moreover, the body executes slow rotational motions. Chapters 2 and 3 are focused on problems of minimal resistance and Newton's problem in media with positive temperature. We define the notion of a rough body and give a characterization of scattering by rough bodies.

The body performs both translational and slow rotational motion. However, the authors kept the initial assumption that the body must have a fixed length and width, that is, can be inscribed in a fixed right circular cylinder. The fourth object - notched angle - is a new one; a proof of its retroreflectivity is given. The authors propose a solvable iterative algorithm of unlearning type for self-correction of Hebbian connectivity. This problem is as follows: find , where , , , and is the set of one-to-one maps of onto itself preserving the measure. In chapters 4 and 5, scattering of billiards by nonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied. All collisions of particles with the disk are perfectly elastic; there may happen multiple collisions in the cavities.

Optimization is an important operation in many domains of science and technology. This book distinguishes itself from existing literature by presenting billiard dynamics outside bounded domains, including scattering, resistance, invisibility and retro-reflection. In particular, the particles have no thermal motion, that is, the medium has temperature zero, and the body is not allowed to oscillate or rotate. Synopsis A billiard is a dynamical system in which a point particle alternates between free motion and specular reflections from the boundary of a domain. The first four cases are realized for any distribution of particles over velocities, and the case e is only realized for some distributions. A specific 2-D parameter space has been introduced to provide respective classification.

A family of sets with similar properties is also constructed in the three-dimensional case. We show that the nonzero transversal component of the force generally appears, resulting in deflection of the disk trajectory. Pressure force exerted on a rough disk spinning in a flow of noninteracting particles is determined by considering that a flow of point particles impinges on a body spinning around a fixed point. But its interest does not lie solely in the maximization of Newtonian resistance; on regarding its characteristics, other areas of application are seen to begin to appear which are thought to be capable of having great utility. Cases a and b are realized when the velocity of the body in the medium exceeds a critical value and when it is smaller than this value, respectively. Then we define the resistance of a rough body; it can be interpreted as the aerodynamic resistance of the somersaulting body moving through a rarefied medium.