Mathematical Physics

What is algebra? For some, it is an abstract language of x's and y’s. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. Taming the Unknown considers how these two seemingly different types of algebra evolved and how they relate. Victor Katz and Karen Parshall explore the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the authors trace the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. They show how similar problems were tackled in Alexandrian Greece, in China, and in India, then look at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. Taming the Unknown follows algebra’s remarkable growth through different epochs around the globe.

Maxwell's equations have led to many important mathematical discoveries. This text introduces mathematics students to some of their wonders.

This book presents the first comprehensive, interdisciplinary review of the rapidly developing field of air lasing. In most applications of lasers, such as cutting and engraving, the laser source is brought to the point of service where the laser beam is needed to perform its function. However, in some important applications such as remote atmospheric sensing, placing the laser at a convenient location is not an option. Current sensing schemes rely on the detection of weak backscattering of ground-based, forward-propagating optical probes, and possess limited sensitivity. The concept of air lasing (or atmospheric lasing) relies on the idea that the constituents of the air itself can be used as an active laser medium, creating a backward-propagating, impulsive, laser-like radiation emanating from a remote location in the atmosphere. This book provides important insights into the current state of development of air lasing and its applications.

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

These counterexamples deal mostly with the part of analysis known as "real variables." Covers the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, functions of 2 variables, plane sets, more. 1962 edition.