Product Description
This work celebrates the work of Eberhard Hopf, a founding father of ergodic theory, a mathematician who produced many beautiful, elegantly written, and now classical, results in integral equations and partial differential equations. Hopf's results remain at the core of these fields, and the title includes Hopf's original mathematical papers, still notable for their elegance and clarity of the writing, with accompanying summaries and commentary by well-known mathematicians. Today, ergodic theory and P.D.E. continue to be active, important areas of mathematics. In this volume the reader will find the roots of many ergodic theory concepts and theorems. Hopf authored fundamental results for P.D.E., such as the maximum principle of elliptic equations and the complete solution of Burger's equation. The familiar properties of elliptic equations were proved for the first time in his earliest work and are included here. His bifurcation theorem, still used over and over again, is a particular gem. The proof of the Wiener-Hopf Theorem is a stunning application of deep analysis. The volume is presented in two main parts. The first section is dedicated to classical papers in analysis and fluid dynamics, and the second to ergodic theory. These works and all the others in the Selected Works carry commentaries by a stellar group of mathematicians who write of the origin of the problems, the important results that followed. Many a mathematical researcher and graduate student will find these collected works to be an excellent resource. ... Read more

Product Description
MATHEMATICAL PROBLEMS OF RADIATIVE EQUILIBRIUM BY EBERHAUD HOPF, PH. D. - Ix ixtunf Proftt ot of J tt iHHiti t it the J riHsm7cM. s ttff Institute of eclntoloi if CAMBRIDGE AT THE UNIVERSITY PRESS 1934 CONTENTS Introduction qxige vii CHAPTER I FUNDAMENTAL PRINCIPLES AND PROBLEMS 1. Transmission of radiation through absorbing and scattering material 1 2. Absorption and scattering of the mass element. The equation of transfer 5 3. Radiative equilibrium. Radiation and temperature 7 4. The main problem JO 5. Schwarzschilds models. Material stratified in parallel planes 1 6. First consequences. The radiation from great depth I tf-7. The integral equations of the problems 21 8. Some properties of the functions E n x 2f CHAPTER II SOLUTION OF PROBLEMS I AND TI 9. The solution of Problem la 28 10. Uniqueness of the solution 31 11. A differentiation formula 33 12. Asymptotically linear solutions of A 35 13. An auxiliary theorem with application to Pro blem II 38 14. Other applications. The boundary temperature 41 15. Proof that q r increases 47 16. Other properties of q r 51 17. Problem 16 for parallel incident radiation 55 18. The emergent light 58 19. Problem 16 for arbitrary incident radiation 59 vi CONTENTS CHAPTER III DISCUSSION OF PROBLEMS III AND IV 20. Solution of Problem III jxige 61 21. Discussion of Problem IV in the case 65 63 22. Narrow limits for the solution in Schwarzschilds case, y l 66 23. The net flux in Schwarzschilds problem 69 CHAPTER IV EXPLICIT SOLUTION OF CERTAIN INTEGRAL EQUATIONS 24. The characteristic equation 72 25. The integral formulae for the solution of 226 75 26. Preliminaries 78 27. Proof of Theorem XVIII 79 28. Proof of the asymptotic form of the solutions 82 29. New pi oof of Theorem X 83 CHAPTER V OTHER PROBLEMS OF RADIATIVE EQUILIBRIUM 30. Pure absorption. Non-gray material 86 31. Discussion of the non-linear integral equation 88 32. Remarks on the case of infinite optical depth 90 33. Absorption and scattering. Schwarzschilds integral equation 91 34. Milnes model of a planetary nebula in radiative equilibrium 94 Bibliography and supplementary remarks 100 List of some formulae of physical interest 105 INTRODUCTION K. Schwarzschilds classical work on absorption and diffusion in the suns atmosphere, and the continuation of these investiga tions through A. 8. Eddington, J. H. Jeans, R. A. Milne and others, have rendered the theory of radiative equilibrium a definite chapter of mathematical astrophysics. The problems connected with this theory are of the following type. iaseous material filling a certain part of space is subject to given incident radia tion. What is the distribution of light and of temperature in the medium, when in radiative equilibrium In the outer layers of a star, where the curvature can be neglected, i. e. where the material can be considered as stratified in parallel planes, those problems appear in their simplest mathematical form. The astrophysicists mostly contented themselves with an approxi mate solution of these problems. Owing to a certain inherent beauty, however, they aroused also the interest of the rigorous mathematician. It is the purpose of this tract to attempt a coherent representation of all that has been achieved in the direction of a rigorous treatment of those standard problems. The rigorous solution of a physical problem naturally pre supposes an exact formulation of the physical assumptions im plied in the problem. Has then the rigorous solution more than merely mathematical significance, when these assumptions prove right only with a limited degree of approximation The answer to this question is not always in the negative. The Schuster-Schwarzschild model of a purely scattering atmosphere, or the Schwarzschild model of a purely absorbing gray atmosphere, both in radiative equilibrium, represent for instance typical standard models which play a similar role in the theory of the stellar atmospheres as the intermediate orbits in celestial mechanics... ... Read more