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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Calculus of variations
The first part of the book is devoted to the transport equation for
a given vector field, exploiting the lagrangian structure of
solutions. It also treats the regularity of solutions of some
degenerate elliptic equations, which appear in the eulerian
counterpart of some transport models with congestion. The second
part of the book deals with the lagrangian structure of solutions
of the Vlasov-Poisson system, which describes the evolution of a
system of particles under the self-induced
gravitational/electrostatic field, and the existence of solutions
of the semigeostrophic system, used in meteorology to describe the
motion of large-scale oceanic/atmospheric flows.
Proceedings of the Conference on Control Theory for Distributed
Parameter Systems, Held at the Chorherrenstift Vorau, Styria, July
11-17, 1982
A linear optimization problem is the task of minimizing a linear
real-valued function of finitely many variables subject to linear
con straints; in general there may be infinitely many constraints.
This book is devoted to such problems. Their mathematical
properties are investi gated and algorithms for their computational
solution are presented. Applications are discussed in detail.
Linear optimization problems are encountered in many areas of appli
cations. They have therefore been subject to mathematical analysis
for a long time. We mention here only two classical topics from
this area: the so-called uniform approximation of functions which
was used as a mathematical tool by Chebyshev in 1853 when he set
out to design a crane, and the theory of systems of linear
inequalities which has already been studied by Fourier in 1823. We
will not treat the historical development of the theory of linear
optimization in detail. However, we point out that the decisive
break through occurred in the middle of this century. It was urged
on by the need to solve complicated decision problems where the
optimal deployment of military and civilian resources had to be
determined. The availability of electronic computers also played an
important role. The principal computational scheme for the solution
of linear optimization problems, the simplex algorithm, was
established by Dantzig about 1950. In addi tion, the fundamental
theorems on such problems were rapidly developed, based on earlier
published results on the properties of systems of linear
inequalities."
This book is based on a seminar given at the University of
California at Los Angeles in the Spring of 1975. The choice of
topics reflects my interests at the time and the needs of the
students taking the course. Initially the lectures were written up
for publication in the Lecture Notes series. How ever, when I
accepted Professor A. V. Balakrishnan's invitation to publish them
in the Springer series on Applications of Mathematics it became
necessary to alter the informal and often abridged style of the
notes and to rewrite or expand much of the original manuscript so
as to make the book as self-contained as possible. Even so, no
attempt has been made to write a comprehensive treatise on
filtering theory, and the book still follows the original plan of
the lectures. While this book was in preparation, the two-volume
English translation of the work by R. S. Liptser and A. N. Shiryaev
has appeared in this series. The first volume and the present book
have the same approach to the sub ject, viz. that of martingale
theory. Liptser and Shiryaev go into greater detail in the
discussion of statistical applications and also consider inter
polation and extrapolation as well as filtering."
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