Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief,
in constructing and exploiting natural differentiable structures on abstract
probability spaces; in other words, Stochastic Calculus of Variations proceeds
from a merging of differential calculus and probability theory.
As optimization under a random environment is at the heart of mathematical
finance, and as differential calculus is of paramount importance for the
search of extrema, it is not surprising that Stochastic Calculus of Variations
appears in mathematical finance.
This book introduces a variety of problem statements in classical optimal control, in optimal estimation and filtering, and in optimal control problems with non-scalar-valued performance criteria. Many example problems are solved completely in the body of the text. All chapter-end exercises are sketched in the appendix. The theoretical part of the book is based on the calculus of variations, so the exposition is very transparent and requires little mathematical rigor.
Ebook Mathematics for physics A Guided Tour for Graduate Students. An engagingly written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics: differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables.
The mathematical modeling of microstructures in solids is a fascinating topic
that combines ideas from different fields such as analysis, numerical simulation,
and materials science. Beginning in the 80s, variational methods have
been playing a prominent rˆole in modern theories for microstructures, and
surprising developments in the calculus of variations were stimulated by questions
arising in this context.
In the quest for knowledge, it is not uncommon for researchers to push the limits
of simulation techniques to the point where they have to be adapted or totally new
techniques or approaches become necessary. True multiscale modeling techniques
are becoming increasingly necessary given the growing interest in materials and
processes on which large-scale properties are dependent or that can be tuned by their
low-scale properties. An example would be nanocomposites, where embedded nanostructures
completely change the matrix properties due to effects occurring at the
This new text, intended for the senior undergraduate finite element course in mechanical, civil and aerospace engineering departments, gives students a solid, practical understanding of the principles of the finite element method within a variety of engineering applications.
Hutton discusses basic theory of the finite element method while avoiding variational calculus, instead focusing upon the engineering mechanics and mathematical background that may be expected of senior engineering students.
Poincar´ made the ﬁrst attempt in 1896 on applying variational calculus e to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions. Little progress had been made on resolving this diﬃculty until a recent breakthrough by Chenciner and Montgomery. Afterward, variational methods were successfully applied to the N -body problem to construct new classes of solutions.
This book collects the lecture notes of two courses and one mini-course held in a
winter school in Bologna in January 2005. The aim of this school was to popularize
techniques of geometric measure theory among researchers and PhD students in
hyperbolic differential equations. Though initially developed in the context of the
calculus of variations, many of these techniques have proved to be quite powerful
for the treatment of some hyperbolic problems.
It is the space of ordinary human experience and the starting point of our
geometric intuition. Studied for two-and-a-half millenia, it has been the
object of celebrated controversies, the most famous concerning the minimum
number of properties necessary to define it completely.
We have attempted to explain the concepts which have been used and
developed to model the stochastic dynamics of natural and biological systems.
While the theory of stochastic differential equations and stochastic processes
provide an attractive framework with an intuitive appeal to many problems
with naturally induced variations, the solutions to such models are an active
area of research, which is in its infancy. Therefore, this book should provide
a large number of areas to research further.
This paper introduces to the calculus of regular expressions a replace operator and defines a set of replacement expressions that concisely encode alternate variations of the operation. Replace expressions denote regular relations, defined in terms of other regular expression operators. The basic case is unconditional obligatory replacement. We develop several versions of conditional replacement that allow the operation to be constrained by context O.