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This monograph is a presentation of a unified approach to a certain class of semimartingale inequalities, which can be regarded as probabilistic extensions of classical estimates for conjugate harmonic functions on the unit disc. The approach, which has its roots in the seminal works of Burkholder in the 80s, enables to deduce a given inequality for semimartingales from the existence of a certain special function with some convex-type properties. Remarkably, an appropriate application of the met...

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor- mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these...

The term "weakly differentiable functions" in the title refers to those inte­ n grable functions defined on an open subset of R whose partial derivatives in the sense of distributions are either LP functions or (signed) measures with finite total variation. The former class of functions comprises what is now known as Sobolev spaces, though its origin, traceable to the early 1900s, predates the contributions by Sobolev. Both classes of functions, Sobolev spaces and the space of functions of bound...

Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun- diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theo...

In der angewandten Funktionalanalysis sind in den letzten Jahrzehnten verschie- dene relativ abgeschlossene Theorien zur Losung nichtlinearer Probleme entstanden, unter denen die Methode der monotonen Operatoren und die Variationsmethoden mit Potentialoperatoren einen hervorragenden Platz einnehmen. Die "Monotonietheorie" entstand vornehmlich im Rahmen der funktionalanaly- tischen Behandlungsweise von Randwertproblemen fur elliptische Differentialglei- chungen. In engem Zusammenhang mit diesen R...

The main topics of this book are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.The book is self-contained....

For one semester, sophomore-level courses in Vector Calculus and Multivariable Calculus. This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. The organization of the text draws strong analogies with the basic ideas of elementary calculus (derivative, integral, and fundamental theorem). Traditional in approach, it is written with an assumption that the student may have computing facilities for two- and three-dimensional gr...

Based on a streamlined presentation of the author's successful work, An Introduction to Frames and Riesz Bases, this book develops frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field. The book presents basic results in an accessible way and includes extensive exercises.

The experimental achievement of Bose-Einstein condensation (1995) and of Fermi degeneracy (1999) in ultra-cold, dilute gases has opened a new field in atomic physics and condensed matter physics. In this thesis, first we present an overview of theoretical and experimental facts on ultra-cold atomic gases. We then describe a Green's function scheme to study coherent transport by fermions through a one-dimensional array of potential wells. Within this scheme different geometries for the array like...

The material of the present book has been used for graduate-level courses at the University of Ia~i during the past ten years. It is a revised version of a book which appeared in Romanian in 1993 with the Publishing House of the Romanian Academy. The book focuses on classical boundary value problems for the principal equations of mathematical physics: second order elliptic equations (the Poisson equations), heat equations and wave equations. The existence theory of second order elliptic boundary...

Progress in mathematics is based on a thorough understanding of the mathematical objects under consideration, and yet many textbooks and monographs proceed to discuss general statements and assume that the reader can and will provide the mathematical infrastructure of examples and counterexamples. This book makes a deliberate effort to correct this situation: it is a collection of examples. The following table of contents describes its breadth and reveals the underlying motivation--differential...

This book is designed primarily for undergraduates in mathematics, engineering, and the physical sciences. Rather than concentrating on technical skills, it focuses on a deeper understanding of the subject by providing many unusual and challenging examples. The basic topics of vector geometry, differentiation and integration in several variables are explored. It also provides numerous computer illustrations and tutorials using MATLAB (R) and Maple (R), that bridge the gap between analysis and co...