Showing papers by "Lawrence Zalcman published in 2011"

Complex Proofs of Real Theorems

Abstract: Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, ""The shortest and best way between two truths of the real domain often passes through the imaginary one.'' Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics. Topics discussed include weighted approximation on the line, Muntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley-Wiener theorem, the Titchmarsh convolution theorem, the Gleason-Kahane-Zelazko theorem, and the Fatou-Julia-Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.

Normality and repelling periodic points

Abstract: Let k > 3(> 2) be an integer and F be a family of functions meromorphic in a domain D C C, all of whose poles have multiplicity at least 2 (at least 3). If in D each f ∈ F has neither repelling fixed points nor repelling periodic points of period k, then F is a normal family in D. Examples are given to show that the conditions on poles are necessary and sharp.

General relativity, geometry, and PDE

Abstract: The papers in this volume cover a wide variety of topics in differential geometry, general relativity, and partial differential equations. In addition, there are several articles dealing with various aspects of Lie groups and mathematics physics. Taken together, the articles provide the reader with a panorama of activity in general relativity and partial differential equations, drawn by a number of leading figures in the field. The companion volume (Contemporary Mathematics, Volume 553) is devoted to function theory and optimization.

Coda: Transonic airfoils and SLE

Abstract: We close by describing two rather unusual applications of complex variables. The details are beyond the scope of this book, but the ideas involved definitely deserve mention. The first area of application is fluid dynamics. It was observed already in the nineteenth century that the equations describing the incompressibility and irrotationality of fluids are just the Cauchy-Riemann equations for the velocity components in two-dimensional flow. Since low velocity flow is nearly incompressible, this made it possible to use analytic functions (more specifically, the theory of conformal mapping) to describe such flows around airfoils and to determine lift and drag. However, for high speed flows, which are compressible, this approach is not available. In high speed flows over airfoils, the flow becomes supersonic over parts of the airfoil. This leads to the formation of shock waves, an undesirable effect since shocks increase drag. Although Cathleen Morawetz proved mathematically that, in general, shock waves occur in partially supersonic flows [M1], [M2], this did not rule out the existence of special airfoils for which shockless flows are possible. In fact, Paul Garabedian and his student David Korn developed a hodograph method based on complex characteristics that enabled them to calculate supercritical wing sections free of shocks at a specified speed and angle of attack [K], [GK1]. However, the extensive trial and error involved in the selection of parameters defining the flow rendered this method impractical. After the preliminary results of [BGK], a completely satisfactory solution of the problem was obtained by Garabedian and Korn in [GK2]. They solve the partial differential equations of two-dimensional inviscid gas dynamics by analytic continuation into the domain of two independent complex characteristic coordinates. After mapping the domain of integration conformally onto the unit disk in the plane of one of these coordinates, they formulate a boundary value problem on that disk for the stream function which is well-posed even in the case of transonic flow. This enables them to give a procedure for calculating an airfoil on which the speed is prescribed as a function of arclength, leading to an exact solution of the problem in the case of subsonic flow and, in the transonic case, generally to a shockless flow which assumes the assigned subsonic values of the speed and approximates the given supersonic values. Truly a tour de force of applied complex analysis.