Problems of Point Blast Theory

The first comprehensive review of the theory and practice of one of today's most powerful optimization techniques.
The explosive growth of research into and development of interior point algorithms over the past two decades has significantly improved the complexity of linear programming and yielded some of today's most sophisticated computing techniques.

by Sergei L. Semakov (Author)
The behavior of any real system is a process to a greater or lesser degree probabilistic. As a rule, it is impossible to specify exactly which external influences and internal mechanisms of interaction of the system components will be decisive in the future

This book is concerned with point estimation in Euclidean sample spaces. The first four chapters deal with exact (small-sample) theory, and their approach and organization parallel those of the companion volume, Testing Statistical Hypotheses (TSH). Optimal estimators are derived according to criteria such as unbiasedness, equivariance, and minimaxity, and the material is organized around these criteria. The principal applications are to exponential and group families, and the systematic discussion of the rich body of (relatively simple) statistical problems that fall under these headings constitutes a second major theme of the book.

One of the most important contributions to modern mathematical analysis is the theory of measure and integration developed by Henri Lebesgue during the early years of the 20th century. Lebesgue measure and integration has many advantages over ordinary Riemann integration from the point of view of application as well as theory. It is an indispensable part of the foundations of various fields as, for example, the theory of probability and statistics and Fourier series.

One of the pervasive phenomena in the history of science is the development of independent disciplines from the solution or attempted solutions of problems in other areas of science. In the Twentieth Century, the creation of specialties witqin the sciences has accelerated to the point where a large number of scientists in any major branch of science cannot understand the work of a colleague in another subdiscipline of his own science.

These notes are based on a series of lectures given in the Lefschetz Center for Dynamical Systems in the Division of Applied Mathematics at Brown University during the academic year 1978-79. The purpose of the lectures was to give an introduction to the applications of centre manifold theory to differential equations. Most of the material is presented in an informal fashion, by means of worked examples in the hope that this clarifies the use of centre manifold theory. The main application of centre manifold theory given in these notes is to dynamic bifurcation theory. Dynamic bifurcation theory is concerned with topological changes in the nature of the solutions of differential equations as para­ meters are varied. Such an example is the creation of periodic orbits from an equilibrium point as a parameter crosses a critical value. In certain circumstances, the application of centre manifold theory reduces the dimension of the system under investigation. In this respect the centre manifold theory plays the same role for dynamic problems as the Liapunov-Schmitt procedure plays for the analysis of static solutions. Our use of centre manifold theory in bifurcation problems follows that of Ruelle and Takens [57) and of Marsden and McCracken [51).

Introduction to conjugacy problems for difieomorphisms. This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G-Diff(M) such that the induced map GXM-M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeomorphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C«> or C" point of view. All manifolds maps, etc. will be differentiable (Cr, 1;;:; r ~ 00) unless stated otherwise.