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Nature is full of spidery patterns: lightning bolts, coastlines, nerve cells, termite tunnels, bacteria cultures, root systems, forest fires, soil cracking, river deltas, galactic distributions, mountain ranges, tidal patterns, cloud shapes, sequencing of nucleotides in DNA, cauliflower, broccoli, lungs, kidneys, the scraggly nerve cells that carry signals to and from your brain, the branching arteries and veins that make up your circulatory system. These and other similar patterns in nature are called natural fractals or random fractals. This chapter contains activities that describe random fractals. There are two kinds of fractals: mathematical fractals and natural (or random) fractals. A mathematical fractal can be described by a mathematical formula. Given this formula, the resulting structure is always identically the same (though it may be colored in different ways). In contrast, natural fractals never repeat themselves; each one is unique, different from all others. This is because these processes are frequently equivalent to coin-flipping, plus a few simple rules. Nature is full of random fractals. In this book you will explore a few of the many random fractals in Nature. Branching, scraggly nerve cells are important to life (one of the patterns on the preceding pages). We cannot live without them. How do we describe a nerve cell? How do we classify different nerve cells? Each individual nerve cell is special, unique, different from every other nerve cell. And yet our eye sees that nerve cells are similar to one another.

To the Instructor The purpose of this laboratory manual is not just to help students to set up electronic circuits that function as they should. The important thing is the electronic concepts that the student learns in the process of setting up and studying these circuits. Quite often a student learns more electronics when he has to trouble shoot a circuit than when the circuit performs as it should when first built. It is unlikely that any students would be able to complete all of these experiments in one semester. The author believes that all students should have laboratory experiences with power sup- plies, amplifiers, oscillators, and integrated circuits. Additionallabomtory experiments should be de- termined by the instructor. Therefore, you can choose those that you want done. Some students are more efficient in the labomtory than others. Therefore, some would be able to complete more exper- iments in a semester than others. Also many of these experiments cannot be completed in one two- hour laboratory period. If space is available, the circuits could be left intact from one period to the next. Or you might want to select steps in an experiment that you want to delete. Neither the val- ues of the components or the magnitudes of the power supplies, as given in the instructions, are critical. Therefore you could in most cases change them if the ones recommended are not available.

viii From discussions with our colleagues, we know that they recognize the problems and worry about them, but simply do not have the time to thoroughly study the highly specialized genetic literature available. This book is an attempt to fill this void. We have made an effort to keep it as short and clear as possible and to limit it to the important and most frequent genetic abnor- malities. In particular, we have tried to take into consid- eration the difficulties of the average student in under- standing genetic logic and to eliminate the most common errors. This guide is not designed to provide more than basic information. No reader will arise from the study of this volume as an expert genetic counselor. That requires, in this as in all other sciences, knowledge of the highly spe- cialized literature as well as extensive experience. Some geneticists therefore take the position that the general practitioner (or specialist in any other field of medicine) cannot possibly give proper genetic counsel to his patients. Because he is not a genetics expert, he should, without exception, refer all such cases to the geneticist. This point of view would condemn this guide as potentially more harmful than helpful in that it might increase the cases of well-meaning error as well as encouraging those who are not competent in this field to deal with problems which are beyond their capacity. We, obviously, do not share this pessimistic standpoint.

Science students have to spend much of their time learning how to do laboratory work, even if they intend to become theoretical, rather than experimental, scientists. It is important that they understand how experiments are performed and what the results mean. In science the validity of ideas is checked by experiments. If a new idea does not work in the laboratory, it must be discarded. If it does work, it is accepted, at least tentatively. In science, therefore, laboratory experiments are the touchstones for the acceptance or rejection of results. Mathematics is different. This is not to say that experiments are not part of the subject. Numerical calculations and the examina- tion of special and simplified cases are important in leading mathematicians to make conjectures, but the acceptance of a conjecture as a theorem only comes when a proof has been constructed. In other words, proofs are to mathematics as laboratory experiments are to science. Mathematics students must, therefore, learn to know what constitute valid proofs and how to construct them. How is this done? Like everything else, by doing. Mathematics students must try to prove results and then have their work criticized by experienced mathematicians. They must critically examine proofs, both correct and incorrect ones, and develop an appreciation of good style. They must, of course, start with easy proofs and build to more complicated ones.

Animal Physiologic Surgery presents an integrated approach to the study of surgery for first-year medical students and graduate students in physiology. The primary emphasis is on the interrela- tionships between surgical techniques and physiologic phenomena observed before, during, and after surgery. All procedures described in the book are designed so that the student with a limited knowledge of surgery can successfully assume responsibility for pre- and postoperative care, as well as for the operation. Therefore, the attitude reflected in this work shows the student his obligation, and his privilege, to find the best method of treatment for the patient and to work at his highest capacity. The text begins with an introduction to operating-room proce- dures, sutures and instruments, wound healing, anesthesia, and water and electrolyte balance. The second part deals with step-by- step surgical instructions and clinical consideration in techniques, such as laparotomy, splenectomy, nephrectomy, and laminectomy. This part is followed by a section on laboratory techniques neces- sary for following and evaluating the course of the patient and on postmortem techniques. The text strikes a balance between exacting detail and discussion of basic principles; it is easily adaptable to any curriculum. I am grateful to the contributors for their close cooperation, especially Dr. William J. White for sharing much of the responsi- bilities. I am also very appreciative to Catherine Jackson and Anne Kupstas for their valuable editorial assistance, and to Joyce Greene vii Preface and Becky Robertson for their assistance in preparing the manu- script.

The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction. . . . . . . 1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation . . . . . . . . . . . . . . . 7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations. . . . . . . . . . . . . . . . . . . . . . 15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence . . . . . . . . . . . . . . . . . . . 29 The wave equation . . . . . . . . . . . . . . . . . . . . . . 32 The initial value problem for hyperbolic equations with a normal surface having multiple points . . . . . . . . . . . . . . . . . . . . 36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution . . . . . . . . . . . . . . 43 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . 45 Solution of the inhomogeneous equation with a plane wave function as right hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 The fundamental solution. . . . . . . . . . . . . . . . . . . . . .

This book is intended for &tudents, research engineers, and mathematicians interested in applications or numerical analysis. Pure analysts will also find some new problems to tackle. Most of the material can be understood by a reader with a relatively modest knowledge of differential and inte- gral equations and functional analysis. Readers interested in stochastic optimization will find a new theory of prac- tical . importance. Readers interested in problems of static and quasi-static electrodynamics, wave scattering by small bodies of arbitrary shape, and corresponding applications in geophysics, optics, and radiophysics will find explicit analytical formulas for the scattering matrix, polarizability tensor, electrical capacitance of bodies of an arbitrary shape; numerical examples showing the practical utility of these formulas; two-sided variational estimates for the pol- arizability tensor; and some open problems such as working out a standard program for calculating the capacitance and polarizability of bodies of arbitrary shape and numerical calculation of multiple integrals with weak singularities. Readers interested in nonlinear vibration theory will find a new method for qualitative study of stationary regimes in the general one-loop passive nonlinear network, including stabil- ity in the large, convergence, and an iterative process for calculation the stationary regime. No assumptions concerning the smallness of the nonlinearity or the filter property of the linear one-port are made. New results in the theory of nonlinear operator equations form the basis for the study.

The term "e;function algebra"e; usually refers to a uniformly closed algebra of complex valued continuous functions on a compact Hausdorff space. Such Banach alge- bras, which are also called "e;uniform algebras"e;, have been much studied during the past 15 or 20 years. Since the most important examples of uniform algebras consist of, or are built up from, analytic functions, it is not surprising that most of the work has been dominated by questions of analyticity in one form or another. In fact, the study of these special algebras and their generalizations accounts for the bulk of the re- search on function algebras. We are concerned here, however, with another facet of the subject based on the observation that very general algebras of continuous func- tions tend to exhibit certain properties that are strongly reminiscent of analyticity. Although there exist a variety of well-known properties of this kind that could be mentioned, in many ways the most striking is a local maximum modulus principle proved in 1960 by Hugo Rossi [RIl]. This result, one of the deepest and most elegant in the theory of function algebras, is an essential tool in the theory as we have developed it here. It holds for an arbitrary Banaeh algebra of GBPunctions defined on the spectrum (maximal ideal space) of the algebra. These are the algebras, along with appropriate generalizations to algebras defined on noncompact spaces, that we call "e;natural func- tion algebras"e;.