Multivector Functions of a Real Variable

In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key points of the development. For example, $k$-dimensional tangent spaces are replaced by representations of simple $k$-vectors supported in single points as limits of simplicial $k$-chains in a Banach space (much like Dirac monopoles). This subtle difference has powerful advantages ...

A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to handle both analytic and nonanalytic functions. This corrects a flaw in the existing methods, that is, the incorrect use of the traditional product rule. In the framework introduced, the derivatives of quaternion matrix functions can be cal...

The goal of this paper is to define the Grassmann integral in terms of a limit of a sum around a well-defined contour so that Grassmann numbers gain geometric meaning rather than symbols. The unusual rescaling properties of the integration of an exponential is due to the fact that the integral attains the known values only over a specific set of contours and not over their rescaled versions. Such contours live in infinite dimensional space and their sides are infinitesimal, a...

The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the graphically Lipschitzian mappings and thus a number of multifunctions, frequently arising in optimization and equilibrium problems. The developed theory makes use of own generalized derivatives, provides us with some calculus rules and reveals a...

The notions and certain fundamental characteristics of the proximal and limiting normal cones with respect to a set are first presented in this paper. We present the ideas of the limiting coderivative and subdifferential with respect to a set of multifunctions and singleton mappings, respectively, based on these normal cones. The necessary and sufficient conditions for the Aubin property with respect to a set of multifunctions are then described by using the limiting coderiva...

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's c...

This is a draft of my textbook on mathematical analysis and the areas of mathematics on which it is based. The idea is to fill the gaps in the existing textbooks. Any remarks from readers are welcome.

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of t...

Apart from an account of classical preliminaries, this volume contains a systematic introduction to Sobolev spaces and functions of bounded variation with selected applications. This is installment III of a four part discussion of certain aspects of Real Analysis: Functions of a Single Variable, Curves and Length, Functions of Several Variables, and Surfaces and Area.

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.