![The conference in honor of Jean Vaillant
BERNARDGAVEAU
Laboratoire Analyse et Physique Mathematique,
Universite Pierre et Marie Curie, Paris, France
Since his thesis in 1964 prepared under the direction of J. Leray and A. Lichnerowicz
the main theme of the mathematical work of Jean Vaillant has been the study of systems of
hyperbolic or holomorphic partial differential equations. The basic example of an hyperbolic
equation is the wave equation which is the mathematical description of wave propagation at
finite velocity, as, for example, the propagation of small disturbances in fluids (the sound) or
of electromagnetic waves in vacuum. Examples of hyperbolic systems include the Maxwell
system for the propagation of electromagnetic waves, the Dirac system for the propagation of
spinors and Einstein equations in general relativity. The wave equation is the fundamental
example of a strictly hyperbolic equation, for which the propagation velocities are different
and do not vanish. An approximation of the solutions of a strictly hyperbolic equation is the
high frequency approximation or geometrical optics approximation : sound or light propagates
essentially along the trajectories of the Hamilton Jacobi equation associated to the partial
differential equation. One can say, in a rather unprecise manner, that « singularities are
propagated along ^characteristics », which, a posteriori, justifies the use of geometrical
optics, the laws of reflexion and refraction. On the other hand, interference and diffraction
phenomena show that sound or light can be described as fields which can be added, rather
than particles, but again a good approximation of these phenomena is the propagation along
bicharacteristics at least in the simplest situations. Dirac system is an example of a system
with multiple characteristics with constant multiplicities. Maxwell system in a non isotropic
medium, like a crystal, is an hyperbolic system with multiple characteristics, but their
multiplicity is non constant: The velocities of propagation depends of the direction of
propagation, but for special directions, some velocities may coincide. In this situation, the
approximation of geometrical optics is no more valid : The propagation along
bicharacteristics (or rays) is not a good description or approximation of the phenomenon, and
indeed this can be checked experimentally. A light ray falling on certain crystals, is, in
general refracted along a certain direction. Nevertheless for special incidence angles,
corresponding to the geometry of the crystal, the ray is refracted, not along another ray, but on
a whole conical surface. Until the end of 19 century, this experiment was the only proof of
the electromagnetic nature of light, because all the other light propagation phenomena could
be described by a wave equation, without the use of the complete Maxwell system (see [1],
[2]).](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-22-320.jpg)
![2 Gaveau
In the beginning of the 1960's, strictly hyperbolic equations with simple
characteristics (the velocities of propagation are distinct and non zero), are well understood
(see [1], [3]). Around that time, Jean Vaillant begins a systematic study of equations or
systems of equations which are non strict. In his thesis ([4], [5]), he introduces the notion of
localization with respect to a factor of the characteristic determinant of a system with constant
coefficients and he relates this notion to the equation of propagation along the
^characteristics. This seems to be the first attempt to apply the method of localization, in
particular using invariant factors. Following the article of Garding, Kotake, Leray (Probleme
de Cauchy VI, [6]), J. Vaillant defines a new invariant associated to systems with double
characteristics and gives an application to the Goursat problem [7], and to the localization for
systems with variables coefficients and double characteristics : This is the first example of a
Levi condition in this setting. J. Vaillant relates also the difference of the subcharacteristic
polynomial, and the second coefficient of exp (-icocp) P (x, DX) exp (icocp) to the Lie
derivative of the volume form along the ^characteristics [8]. This is an important invariant,
because it is well known that the existence and the regularity of the solution depend, in
degenerate cases not only of the principal symbol but also on the lower order terms of the
operator. This result was rederived by Duistermaat and Hormander.
In 1973-74, J. Vaillant constructs the asymptotic expansion of the solution of an
hyperbolic systems with characteristics of variable multiplicities [9]. He defines the
localization of an hyperbolic system at a multiple points with application to conical refraction
[9]. This work will be extended in 1978, when he constructs the parametrix for the Cauchy
problem with multiple characteristics [10], in relation with the invariants of the system.
In [11], J. Vaillant studies the symmetrisation of localized hyperbolic systems and
defines the notion of « reduced dimension » : The property of symmetry is proved in the case
of a maximal reduced dimension. These last few years, he has continued to study the
symmetric of strongly hyperbolic systems, in particular with T. Nishitani. If the reduced
dimension of a system of rank m is not less than 2, a constant coefficient
2
systems is symmetrizable ([7], [8]). For a system with non constant coefficients, if at any
point the reduced dimension is not less than — 2, the system is also symmetrizable
with a regular symmetrizer [18]. Recently he has determined the multiple points according to
the reduced dimension [19].
In 1982, in collaboration with D. Schiltz and C. Wagschal [12], J. Vaillant has
studied the ramification of the Cauchy problem for a system in involution with triple
characteristics. This problem reduces to the question of the singularities of integrals of](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-23-320.jpg)
![Conference in honor of Jean Vaillant 3
holomorphic forms depending of parameters, on chains depending also of parameters. The
problem is to determine the singularities of these integrals with respect to the parameters. The
first systematic work in this direction was the article of J. Leray [13] in the algebraic case. J.
Vaillant studies the ramification in the general holomorphic case, using a grassmann boundle
[14]. Since 1987, J. Vaillant has started the problem of the classification of systems with
constant multiplicities : definition of invariant Levi conditions, relations to the Cauchy
problem in the C°° and Gevrey classes. For any system, he obtains systematically the Levi
conditions [15], [16].
J. Vaillant has founded a research group and a seminar, which he has maintained, for
more than thirty years, independent of fashions « mots d'ordre » and which survives in
difficult conditions. Freedom of thought, which is a necessary condition for any creative
work, is paid a very high price. Creation, scientific or artistic, cannot be judged according to
economic or social criteria, measured in monetary values. Research is not a collective activity.
The highly mysterious activity of thought can only be a personal activity. During all his
career, Jean Vaillant, following the example of Leray, has tried to defend by his attitudes and
his work, the values of scientific creation and intellectual independence.
For more than forty years, J. Vaillant has developed many collaborations with his
Japanese and Italian friends, in particular Y. Hamada, Y. Ohya, K. Kajitani, T. Nishitani and
S. Spagnolo, F. Colombini, A. Bove and E. Bernardi. He has also developed many european
collaborations and he has created a european network of belgian, french, Italian, and
Portugese universities. All his friends know that they can rely on his help and his advises.
acknowledgment: We are very grateful to Anne Durrande, Evelyne Guilloux and Maryse
Loiseau for their help during the preparation of this conference. We also thank the Maison
Europeenne des Technologies, in particular Madame Muller for her help.
B. Gaveau
Laboratoire Analyse et Physique Mathematique
14 avenue Felix Faure
75015 PARIS](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-24-320.jpg)
![Gaveau
[1] Courant-Hilbert: Methods of Mathematical Physics Vol
Interscience 1962
[2] M. Born & E. Wolf : Principles of optics
Penjamon Press 1980
[3] J. Leray : Hyperbolic differential equations
Lectures notes, Princeton 1950
[4] J. Vaillant: Sur les discontinuites du tenseur de courbure en theorie
d'Einstein-Schrodinger
CR Acad Sci Paris - 10 juillet 1961, 30 octobre 1961, 15 Janvier 1962
[5] J. Vaillant: Caracteristiques multiples et bicaracteristiques des systemes d'equations
aux derivees partielles lineaires et a coefficients constants
Annales Institut Fourier 15 (1965) et 16 (1966)
[6] L. Garding, T. Kotake, J. Leray : Uniformation et developpement asymptotique de la
solution du probleme de Cauchy lineaire a donnees holomorphes ; analogic avec la
theorie des ondes asymptotiques et approchees (Probleme de Cauchy I bis et VI)
Bull. Sci. Math. France 92 1964, 263-361.
[7] J. Vaillant: Donnees de Cauchy portees par une caracteristique double : role des
bicaracteristiques
J. Maths Pures et Appliquees 47 (1968), 1-40
[8) J. Vaillant: Derivee de Lie de la forme element de volume le long des
bicaracteristiques et polynome sous-caracteristique de Garding-Kotake-Leray
CR Acad Sci Paris -10 mars 1969
[9] J. Vaillant: Solutions asymptotiques d'un systeme a caracteristiques de multiplicite
variable
J. Maths Pures et Appliquees 53 (1974), 71-98
[10] R. Berzin, J. Vaillant- Parametrix a caracteristiques multiples
Bull. Sci. Math 102 (1978), 287-294
[11] J. Vaillant: Symetrisation de matrices localisees
Annali della Scuola Normale Superiore di Pisa. Ser. IV, 5 (1978), 405-427
[12] D. Schiltz, J. Vaillant, C. Wagschal: Probleme de Cauchy ramifie
J. Math. Pures et appliquees (1982)
[13] J. Leray : Un complement au theoreme de N.Nilsson sur les integrales de formes
differentielles a support singulier algebrique](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-25-320.jpg)
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Title: The Architectural Review and American Builders' Journal, Aug.
1869
Author: Various
Editor: Samuel Sloan
Release date: December 22, 2019 [eBook #60997]
Most recently updated: October 17, 2024
Language: English
Credits: Produced by Paul Marshall and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK THE
ARCHITECTURAL REVIEW AND AMERICAN BUILDERS' JOURNAL,
AUG. 1869 ***](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-31-320.jpg)
![MONTHLY REVIEW.
THE LONDON BUILDER AND OURSELVES.
In a tolerantly critical notice of the Review recently published in
the Builder, we find an effort to substantiate a charge formerly made
by it, and replied to by us, on the subject of “trickery” in the
construction of the exteriors of American buildings. The Builder
reiterates the charge and points to Grace Church, New York, in proof
of the truth of it. That marble edifice, he avers, has a wooden spire,
crocketted, etc., painted in imitation of the material of which the
body of the church is constructed. Alas, we must acknowledge the
wood. And we will make a clean breast of it, and still farther
acknowledge that at the time that Grace Church was built, our land
of wooden nutmegs, and other notions, had not an architectural idea
beyond the wooden spire, and that our city and country churches,
that aspired at all, were forced to do so in the national material of
the day. That said sundry spires of wood were of necessity, painted,
is most true; and furthermore, white-lead being a great favorite with
the people generally, [when our manners, customs, and tastes were
more immaculate than in these degenerate days of many colors,]
that pigment was the ruling fashion. That the color of the marble, of
which Grace Church’s body is constructed, should be similar to that
with which said ecclesiastical edifice’s spire was coated, is
unfortunate; but, that the resemblance goes to prove any attempt at
a cheat, we most strenuously deny. Grace Church is of a by-gone
taste,—an architectural era which we now look back to in order to
see, by contrast, how far we have advanced in architectural
construction. Trinity Church, New York, was the first great effort at a
stone spire which our Architects ventured to rear. And although
hundreds have followed its lead, none in this soaring republic have](https://image.slidesharecdn.com/84501-250406051354-4e3785cf/85/Hyperbolic-Differential-Operators-and-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-34-320.jpg)
Hyperbolic Differential Operators and Related Problems 1st Edition Vincenzo Ancona (Editor)
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- 5. Hyperbolic Differential Operators and Related Problems 1st Edition Vincenzo Ancona (Editor) Digital Instant Download Author(s): Vincenzo Ancona (Editor), Jean Vaillant (Editor) ISBN(s): 9780824748227, 0824748220 Edition: 1 File Details: PDF, 13.49 MB Year: 2003 Language: english
- 7. Hyperbolic Differential Operators and Related Problems edited by Vincenzo Ancona Universita degli Studi di Firenze Florence, Italy Jean Vaillant Universite Pierre et Marie Curie, Paris VI Paris, France MARCEL DEKKER, INC. NEW YORK • BASEL
- 8. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-0963-2 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue,New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquartersaddress above. Copyright © 2003 by Marcel Dekker, Inc. AHRights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, elec- tronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA
- 9. PURE AND APPLIEDMATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Delaware New Brunswick, New Jersey Newark, Delaware EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Donald Passman JaneCronin University of Wisconsin, Rutgers University Madison Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University S. Kobayashi David L. Russell University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schempp University of California, Universitdt Siegen Santa Barbara Mark Teply W. S. Massey University of Wisconsin, Yale University Milwaukee
- 10. LECTURE NOTESIN PURE AND APPLIED MATHEMATICS 1. N. Jacobson, Exceptional Lie Algebras 2. L-A Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis 3. /. Satake, Classification Theory of Semi-Simple Algebraic Groups 4. F. Hirzebnich et a/., Differentiable Manifolds and Quadratic Forms 5. I. Chavel, Riemannian Symmetric Spaces of Rank One 6. R. B. Burckel, Characterization of C(X) Among Its Subalgebras 7. B. R. McDonald et a/., Ring Theory 8. Y.-T. Siu, Techniques of Extension on Analytic Objects 9. S. R. Caradus et a/., Calkin Algebras and Algebras of Operatorson Banach Spaces 10. E. O. Roxin et a/., Differential Games and Control Theory 11. M. Orzech and C. Small, The Brauer Group of Commutative Rings 12. S. Thornier, Topology and Its Applications 13. J. M. Lopez and K. A. Ross, Sidon Sets 14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics 15. K. McKennon andJ. M. Robertson, Locally Convex Spaces 16. M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups 17. G. B. Seligman, Rational Methods in Lie Algebras 18. D. G. de Figueiredo, Functional Analysis 19. L Cesari et a/., Nonlinear Functional Analysis and Differential Equations 20. J. J. Schaffer, Geometry of Spheres in Nomned Spaces 21. K. Yano and M. Kon, Anti-Invariant Submanifolds 22. W. V. Vasconcelos, The Rings of Dimension Two 23. R. E. Chandler, Hausdorff Compactifications 24. S. P. Franklin and B. V. S. Thomas, Topology 25. S. K. Jain, Ring Theory 26. B. R. McDonald and R. A. Morris, Ring Theory II 27. R. B. Mura and A. Rhemtulla, Orderable Groups 28. J. R. Graef, Stability of Dynamical Systems 29. H.-C.Wang, Homogeneous Branch Algebras 30. £ O. Roxin et a/., Differential Games and Control Theory II 31. R. D. Porter, Introduction to Fibre Bundles 32. M. Altman, Contractors and Contractor Directions Theory and Applications 33. J. S. Golan, Decomposition and Dimension in Module Categories 34. G. Fairweather, Finite Element Galerkin Methods for Differential Equations 35. J. D. Sally, Numbers of Generators of Ideals in Local Rings 36. S. S. Miller, Complex Analysis 37. R. Gordon, Representation Theory of Algebras 38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras 39. A. I. Arruda et a/., Mathematical Logic 40. F. Van Oystaeyen, Ring Theory 41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization 42. M. Satyanarayana, Positively Ordered Semigroups 43. D. L Russell, Mathematics of Finite-Dimensional Control Systems 44. P.-T.Liu and E. Roxin, Differential Games and Control Theory III 45. A. Geramita and J. Seberry, Orthogonal Designs 46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oystaeyen, RingTheory 52. B. Kadem, Binary Time Series 53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-ValueProblems 54. R L. Stemberg et a/., Nonlinear Partial Differential Equations in Engineering and Applied Science 55. 6. R McDonald, Ring Theory andAlgebra III 56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring 57. T. V. Narayana et a/., Combinatorics, Representation Theory and Statistical Methods in Groups 58. T.A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory
- 11. 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. O. A. Me/son, Direct Integral Theory 62. J. E. Smith et a/., Ordered Groups 63. J. Cronin, Mathematicsof Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. 7. G. McLaughlin, RegressiveSets and the Theory of Isols 67. T. L. Herdman et a/., Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra 69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Repre- sentations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Gee/, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz et a/., Algebraic Structures and Applications 75. L B/can et a/., Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C. Yang, Factorization Theory of Meromorphic Functions 79. O. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications 81. K. B. Hannsgen etal., Volterra and Functional Differential Equations 82. N. L. Johnson et a/., Finite Geometries 83. G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra 85. A. V. Fiacco, Mathematical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty et a/., Optimization 87. A. Figa Talamanca and M, A. Picardello, Harmonic Analysis on Free Groups 88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istratescu, Strict Convexity and Complex Strict Convexity 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations 91. H. L Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L. Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E.Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A. Baker and L. M. Batten, Finite Geometries 104. J. W. Brewer et a/., Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology 106. D. W. Kueke et a/., Mathematical Logic and Theoretical Computer Science 107. B.-L Lin and S. Simons, Nonlinear and ConvexAnalysis 108. S. J. Lee, Operator Methods for Optimal Control Problems 109. V. Lakshmikantham, Nonlinear Analysis and Applications 110. S. F. McCormick, Multigrid Methods 111. M. C. Tangora, Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory 113. D. V. Chudnovsky and R. D. Jenks, ComputerAlgebra 114. M. C. Tangora, Computers in Geometry and Topology 115. P. Nelson et a/., Transport Theory, Invariant Imbedding, and Integral Equations 116. P. Clement et a/., Semigroup Theory and Applications 117. J. Vinuesa, Orthogonal Polynomials and Their Applications 118. C. M. Dafermos et a/., Differential Equations 119. E. O. Roxin, Modem Optimal Control 120. J. C. Diaz, Mathematicsfor Large Scale Computing
- 12. 121. P. S. Milojevft Nonlinear Functional Analysis 122. C. Sadosky, Analysis and Partial Differential Equations 123. R. M. Shortt, GeneralTopology and Applications 124. R. Wong, Asymptotic and Computational Analysis 125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics 126. W. D. Wallis et at, Combinatorial Designs and Applications 127. S. Elaydi, DifferentialEquations 128. G. Chen et at, Distributed Parameter Control Systems 129. W. N. Everitt, Inequalities 130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ- ential Equations 131. O. Anno et a/., Mathematical Population Dynamics 132. S. Coen, Geometry and Complex Variables 133. J. A. Goldstein et a/., Differential Equations with Applications in Biology, Physics, and Engineering 134. S. J. Andima et a/., General Topology and Applications 135. P Clement et a/., Semigroup Theory and Evolution Equations 136. K. Jarosz, Function Spaces 137. J. M. Bayod et a/., p-adic Functional Analysis 138. G. A. Anastassiou, Approximation Theory 139. R. S. Rees, Graphs, Matrices, and Designs 140. G.Abrams et a/., Methods in Module Theory 141. G. L Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing 142. M. C. JoshiandA. V. Balakrishnan, Mathematical Theory of Control 143. G. Komatsu and Y. Sakane, Complex Geometry 144. /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations 145. T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections 146. L. Fuchs and R. Gdbel, Abelian Groups 147. A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum 148. G. Dore et a/., Differential Equations in Banach Spaces 149. T. West, Continuum Theory and Dynamical Systems 150. K. D. Bierstedtetal., Functional Analysis 151. K. G. Fischer et at. Computational Algebra 152. K. D. Elworthy et a/., Differential Equations, Dynamical Systems, and Control Science 153. P.-J. Cahen, et a/., Commutative RingTheory 154. S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions 155. P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics 156. M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research 157. W. O. Bray et at, Fourier Analysis 158. J. Bergen and S. Montgomery, Advances in Hopf Algebras 159. A. R. Magid, Rings, Extensions, and Cohomology 160. N. H. Pavel, Optimal Control of Differential Equations 161. M. Ikawa, Spectral and Scattering Theory 162. X. Liu and D. Siegel, Comparison Methods and Stability Theory 163. J.-P. Zolesio, Boundary Control and Variation 164. M. KHzeketat, Finite Element Methods 165. G. Da Prato and L. Tubaro, Control of Partial Differential Equations 166. E. Ballico, Projective Geometry with Applications 167. M. Costabeletal., Boundary Value Problems and Integral Equations in Nonsmooth Domains 168. G. Ferreyra, G. R. Goldstein, andF. Neubrander, Evolution Equations 169. S. Huggett, Twister Theory 170. H. Cooketal., Continue 171. D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings 172. K. Jarosz, Function Spaces 173. V.Ancona et at. Complex Analysis and Geometry 174. £ Casas, Control of Partial Differential Equations and Applications 175. N, Kalton et at, Interaction Between Functional Analysis, Harmonic Analysis, and Probability 176. Z. Deng ef at. Differential Equations and Control Theory 177. P. Marcellini et at Partial Differential Equations and Applications 178. A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type 179. M. Maruyama, Moduli of Vector Bundles 180. A, Ursini and P. Agliand, Logic and Algebra 181. X, H. Cao et at, Rings, Groups, andAlgebras 182. D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules 183. S. R Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models
- 13. 184. J. E Andersen et al., Geometry and Physics 185. P.-J. Cahen et al., Commutative Ring Theory 186. J. A. Goldstein et al., Stochastic Processes and Functional Analysis 187. A. Sorbi, Complexity, Logic, and Recursion Theory 188. G. Da Prafo and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis 189. D. D. Anderson, Factorization in Integral Domains 190. N. L Johnson, Mostly Finite Geometries 191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems 192. W. H. Schikhofet al., p-adic Functional Analysis 193. S. Sertoz, Algebraic Geometry 194. G. Caristi and E. Mitidieri, Reaction Diffusion Systems 195. A. V. Fiacco, Mathematical Programming with Data Perturbations 196. M. Kfizek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates 197. S. Caenepeeland A. Verschoren, Rings, Hopf Algebras, and Brauer Groups 198. V. Drensky et al., Methods in Ring Theory 199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions 200. P. E. Newstead, Algebraic Geometry 201. D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology 202. Z. Chen et al., Advances in Computational Mathematics 203. X. Caicedo and C. H. Montenegro,Models, Algebras, and Proofs 204. C. Y. Yildinm and S. A. Stepanov, Number Theory and Its Applications 205. D. E. Dobbs et al., Advances in Commutative Ring Theory 206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry 207. J. Kakol et al., p-adic Functional Analysis 208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory 209. S. Caenepeeland F. Van Oystaeyen, Hopf Algebras and Quantum Groups 210. F. Van Oystaeyen and M. Saon'n, Interactions Between Ring Theory and Representations of Algebras 211. R. Costa et al., Nonassociative Algebra and Its Applications 212. T.-X. He, Wavelet Analysis and Multiresolution Methods 213. H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference 214. J. Kajiwara et al.. Finite or Infinite Dimensional Complex Analysis 215. G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences 216. J. Cagnoletal., Shape Optimization and Optimal Design 217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra 218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems 219. F. AHMehmeti et al., Partial Differential Equations on Multistructures 220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra 221. A. Granja et al., Ring Theory and Algebraic Geometry 222. A. K. Katsaras et al., p-adic Functional Analysis 223. R. Salvi, The Navier-Stokes Equations 224. F. U. Coelho and H. A. Merklen, Representations of Algebras 225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory 226. G. Lyubeznik, Local Cohomology and Its Applications 227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications 228. W. A. Camiellietal., Paraconsistency 229. A. Benkirane and A. Touzani, Partial Differential Equations 230. A. Illanes et al., Continuum Theory 231. M. Fontana et al., Commutative Ring Theory and Applications 232. D. Mond and M. J. Saia, Real and Complex Singularities 233. V. Ancona and J. Vaillant, Hyperbolic Differential Operators Additional Volumes in Preparation
- 14. Preface The papers collected in this volume are concerned with hyperbolic problems, or problems the methods of which are related to hyperbolic techniques. T. NISHITANI introduces a notion of nondegenerate characteristic for systems of linear partial differential equations of general order. He shows that nondegenerate char- acteristics are stable under hyperbolic perturbations, and he proves that if the coefficients of the system are real analytic and all characteristics are nondegenerate then the Cauchy problem for the system is well posed in the class of smooth functions. K. KAJITANI studies a class of operators that generalize the linear hyperbolic op- erators, introducing the notion of time function, and proving the well-posedness of the Cauchy problem in the class of C°° functions. The Cauchyproblem is also the subject ofthe paper by A. BOVE and C. BERNARDI; they state some results for a class of linear hyperbolicoperators with double characteristics, not effectively hyperbolic. In particular they prove well-posedness in the C°° class under a geometric condition and a Levi condition, and well-posedness in the Gevrey class under more general assumptions. For a linear system whose principal part is hyperbolic and whose coefficients depend only on time, H. YAMAHARA establishes necessary and sufficient conditions for well- posedness in the Gevrey class, whatever the lower order terms are. L. MENCHERINI and S. SPAGNOLOconsider a first order hyperbolic system in two variables whose coefficients depend only on time; they define the notion of pseudosymmetry for matrix symbolsoforder zero, and determine the Gevrey class wherethe Cauchy problem is well-posed, according to the type of pseudosymmetry of the principal matrix symbol. The 2-phase Goursat problem has been solved by means of Bessel functions; here J. CARVALHO E SILVA considers the 3-phase Goursat problem, using instead some hyper- geometric functions in four variables. He also discusses the general problem, pointing out that the main difficulties are due to the lack of results on special functions. The Stricharz inequality for the classical linear wave equation has been generalized by M. REISSIG and Y.G. WANG to the case of time-dependent coefficients: the coefficient is the product of an increasing factor and an oscillatory factor. The interaction was studied by the authors; in the present paper they extend the result to a one-dimensional system describing thermoelasticity. The noncharacteristic, nonlinear Cauchy problem is the subject of the paper by M. TSUJI. The classical solution has some singularities, so that the problem arises of studying the extension of the solution beyond the singularities. The author constructs a Lagrangian solution in the cotangent bundle, getting a multivalued classical solution; then he explores how to construct a reasonable univalued solution with singularities. Y. CHOQUET considers the Einstein equations. By a suitable choice of the gauge, (for instance, an a priori hypothesis on coordinates choice) she obtains a hyperbolic system in the sense of Leray-Ohya, well-posed in the Gevrey class of index 2. She studies old and new cases where the system is strictly hyperbolic and well-posed in the C°° class. Necessary and sufficient conditions for the Cauchy-Kowalevskaya theorem on systems of linear PDEs were given by Matsumoto and Yamahara; on the other hand, Nagumo constructed a local solution, unique, for a higher order scalar Kowalevskian operator, whose coefficients are analyitic in x and continuous in t. Here W. MATSUMOTO, M. MURAI and T. NAGASE show that for a system whose coefficients are analytic in x and iii
- 15. 1V Preface C°° in t the above condition of Matsumoto and Yamahara is necessary and sufficient also when the order on d/dt is one, the order on d/dx is two, and the rank of the second order part is at most one. B.W. SCHULZE and N. TARKHANOV construct a general calculus of pseudodif- ferential operators on a smoothly stratified space, with local cone bundle neighborhood, including ellipticity and the Fredholm property. M. DREHER and I. WITT propose edge Sobolev spaces for the investigation ofweak hyperbolicity for linear and semilinear operators; propagation of singularities is stated. For the surface waves of water K. O. Friedrichs gave a systematic procedure to obtain the Airy equations from the Euler equations; a rigorous mathematical approach was more recently given by T. Kano in the analytic case. Here T. KANO and S. MIKI develop the theory in the Lagrange coordinate system. The wave equation in shallow water and the Boussineq equation, via Korteweg-de Vries, are obtained as approximate equations in Lagrangian coordinates. For certain equations of Schrodinger type, J. TAKEUCHI states necessary and suffi- cient conditions for the Cauchy problem to be well-posed in H00 ; he uses a time indepen- dent Z/2 symmetrization, with loss of regularity. D. GOURDIN studies a 2-evolution operator in the sense of Petrosky; subject to the condition that the real roots of the principal polynomial with constant coefficients have constant multiplicity. He finds sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces. Some generalizations are also given. K. KAJITANI investigates the Gevrey smoothing effects of the solution to the Cauchy problem for Schrodinger- type equations: the decay of initial data is related to the Gevrey class with respect to the space variables of the solutions. The metaplectic representation allows the construction of the solutions of the Schrodinger equation for the quadratic hamiltonians. M de GOSSON is able to obtain the result more generally for any physical hamiltonian. F. COLOMBINI and C. GRAMMATICO consider the problem of the strong unique- ness of the solution (in a neighborhood of the origin) in Rn , for particular fourth order elliptic operators flat at the origin. In the second order case, the known result in R2 is obtained in Rn . In the case of a product of some model second order operators in R2 with Gevrey coefficients strong uniqueness is obtained under a condition on the Gevrey index, related to the form of the operator. A sharp condition on the modulus of continuity was obtained by Tarama for an elliptic operator of second order to have the local uniquenessproperty. D. DEL SANTOshows that this condition is necessary by constructing a nonuniqueness example. For certain holomorphic operators with polynomial coefficients Y. HAMADA con- structs some solutions whose domain of holomorphy has an exterior point. The results are based on the earier work by Hamada, Leray, Takeuchi, as well as Darboux, Halphen and Chazy. The remaining papers contain more theoretical results.
- 16. Preface v P. LAUBIN presents some known results and new advances on the topology of spaces of holomorphicfunctions in an open subset of a Prechet space; he gives a projective charac- terization of holomorphic germs using seminorms whose form is similar to the one appearing in the Whitney extension theorem for C°° functions. Let Y be a smooth submanifold of a C°° manifold X; a distribution u in the com- plement of Y has the property that the closure of its wave front in the cotangent bundle of X is orthogonal to the tangent bundle of Y. M.K.V. MURTHY describes its analytic behavior in a neighborhood of Y by the notion of microlocal scaling degree along y, and obtains a result similar to the Hormander theorem for homogeneous distributions. A. DEBIARD and B. GAVEAU in their paper apply stochastic methods to determine the ground state of an atomic molecular system. The system is represented by a Hamilto- nian operator acting on a Hilbert space. A lower bound for the ground state is obtained using the Feynman-Kac formula and the special homogeneity property of the Coulomb potential. L.S. SCHULMAN raises the difficulty of conceiving that a single dynamical system could contain subsystems, in significant mutual contact, possessing opposite thermodynam- ics arrows of time. By examining possible cosmological justification for the usual arrow of time he finds that a consistent way to establish such justification is by giving symmetric boundary conditions at two (cosmologically remote) times and seeking irreversible behav- ior in between. Other boundary conditions, modeling shorter periods in the evolution of the cosmos, can be found that allow the simultaneous existence of two thermodynamic arrows, notwithstanding moderate interaction between the two systems. Vincenzo Ancona Jean Vaillant
- 18. Contents Preface Hi Contributors ix 1. The conference in honor of Jean Vaillant 1 Bernard Gaveau 2. Hyperbolic systems with nondegenerate characteristics 7 Tatsuo Nishitani 3. The Cauchy problem for hyperbolic operators dominated by the time function 31 Kunihiko Kajitani 4. A remark on the Cauchy problem for a model hyperbolic operator 41 Enrico Bernardi and Antonio Bove 5. Gevrey well-posedness of the Cauchy problem for systems 53 Hideo Yamahara 6. Gevrey well-posedness for pseudosymmetric systems with lower order terms 67 Lorenzo Mencherini and Sergio Spagnolo 7. Le role des fonctions speciales dans les problemes de Goursat pour des equations aux derives partielles a coefficients constants 83 Jaime Carvalho e Silva 8. Influence of the hyperbolic part on decay rates in 1-d thermoelasticity 89 Ya-Guang Wang and Michael Reissig 9. Integration and singularities of solutions for nonlinear second order hyperbolic equation 109 Mikio Tsuji 10. Causal evolution for Einsteinian gravitation 129 Yvonne Choquet-Bruhat 11. On the Cauchy-Kowalevskaya theorem of Nagumo type for systems 145 Waichiro Matsumoto, Minoru Murai, and Takaaki Nagase 12. Differential analysis on stratified spaces 157 B. W. Schulze and N. Tarkhanov 13. Edge Sobolev spaces, weakly hyperbolic equations, and branching of singularities 179 Michael Dreher and Ingo Witt
- 19. viii Contents 14. Sur les ondes superficieles de 1'eau et le developement de Friedrichs dans le systeme de coordonnees de Lagrange 199 Tadayoshi Kano and Sae Miki 15. Probleme de Cauchy pour certains systemes de Leray-Volevich du type de Schrodinger 233 Jiro Takenchi 16. Systemes du type de Schrodinger a raciness caracteristiques multiples 255 Daniel Gourd in 17. Smoothing effect in Gevrey classes for Schrodinger equations 269 Kunihiko Kajitani 18. Semiclassical wavefunctions and Schrodinger equation 287 Maurice de Gossan 19. Strong uniqueness in Gevrey spaces for some elliptic operators 301 F. Colombini and G. Grammatico 20. A remark on nonuniqueness in the Cauchy problem for elliptic operator having non-Lipschitz coefficients 317 Daniele Del Santo 21. Sur le prolongement analytique de la solutiondu probleme de Cauchy 321 Yusaku Harnada 22. On the projective descriptions of the space of holomorphic germs 331 P. Laubin 23. Microlocal scaling and extension of distributions 339 M. K. Venkatesha Murthy 24. A lower bound for atomic Hamiltonians and Brownian motion 349 A. Debiard and B. Gaveaii 25. A compromised arrow of time 355 L. S. Schulman
- 20. Contributors Enrico Bernard! University of Bologna, Bologna, Italy Antonio Bove Universityof Bologna, Bologna, Italy Jaime Carvalho e Silva Universidade de Coimbra, Coimbra, Portugal Yvonne Choquet-Bruhat Universite de Paris 6, Paris, France F. Colombini Universita di Pisa, Pisa, Italy Maurice de Gosson Blekinge Instituteof Technology, Karlskrona, Sweden, and University of Colorado at Boulder, Boulder, Colorado, U.S.A. A. Debiard Laboratoire Analyse et Physique Mathematique, Universite Pierre et Marie Curie, Paris, France Daniele Del Santo Universita di Trieste, Trieste, Italy Michael Dreher University of Tsukuba, Tsukuba, Japan Bernard Gaveau Laboratoire Analyse et PhysiqueMathematique, Universite Pierre et Marie Curie, Paris, France Daniel Gourdin Universite de Paris 6, Paris, France C. Grammatico Universita di Bologna, Bologna, Italy Yusaku Hamada Kyoto, Japan Kunihiko Kajitani Universityof Tsukuba, Tsukuba, Japan Tadayoshi Kano University of Osaka, Toyonaka, Japan P. Laubint Universityof Liege, Liege, Belgium Waichiro Matsumoto RyukokuUniversity,Otsu, Japan Lorenzo Mencherini Universitadi Firenze, Florence, Italy Sae Mild University of Osaka, Toyonaka, Japan Minoru Murai Ryukoku University, Otsu, Japan M. K. Venkatesha Murthy Universita di Pisa, Pisa, Italy Deceased.
- 21. Contributors Takaaki Nagase Ryukoku University,Otsu, Japan Tatsuo Nishitani Osaka University, Osaka, Japan Michael Reissig TU Bergakademie Freiberg, Freiberg, Germany L. S. Schulman Clarkson University,Potsdam, New York, U.S.A. B. W. Schulze Universitat Potsdam, Potsdam, Germany Sergio Spagnolo Universita di Pisa, Pisa, Italy Jiro Takeuchi Science University of Tokyo, Hokkaido, Japan N. Tarkhanov University of Potsdam, Potsdam, Germany Mikio Tsuji Kyoto Sangyo University,Kyoto, Japan Ya-Guang Wang Shanghai Jiao Tong University, Shanghai, P.R. China Ingo Witt University of Potsdam, Potsdam, Germany Hideo Yamahara Osaka Electro-Communication University,Osaka, Japan
- 22. The conference in honor of Jean Vaillant BERNARDGAVEAU Laboratoire Analyse et Physique Mathematique, Universite Pierre et Marie Curie, Paris, France Since his thesis in 1964 prepared under the direction of J. Leray and A. Lichnerowicz the main theme of the mathematical work of Jean Vaillant has been the study of systems of hyperbolic or holomorphic partial differential equations. The basic example of an hyperbolic equation is the wave equation which is the mathematical description of wave propagation at finite velocity, as, for example, the propagation of small disturbances in fluids (the sound) or of electromagnetic waves in vacuum. Examples of hyperbolic systems include the Maxwell system for the propagation of electromagnetic waves, the Dirac system for the propagation of spinors and Einstein equations in general relativity. The wave equation is the fundamental example of a strictly hyperbolic equation, for which the propagation velocities are different and do not vanish. An approximation of the solutions of a strictly hyperbolic equation is the high frequency approximation or geometrical optics approximation : sound or light propagates essentially along the trajectories of the Hamilton Jacobi equation associated to the partial differential equation. One can say, in a rather unprecise manner, that « singularities are propagated along ^characteristics », which, a posteriori, justifies the use of geometrical optics, the laws of reflexion and refraction. On the other hand, interference and diffraction phenomena show that sound or light can be described as fields which can be added, rather than particles, but again a good approximation of these phenomena is the propagation along bicharacteristics at least in the simplest situations. Dirac system is an example of a system with multiple characteristics with constant multiplicities. Maxwell system in a non isotropic medium, like a crystal, is an hyperbolic system with multiple characteristics, but their multiplicity is non constant: The velocities of propagation depends of the direction of propagation, but for special directions, some velocities may coincide. In this situation, the approximation of geometrical optics is no more valid : The propagation along bicharacteristics (or rays) is not a good description or approximation of the phenomenon, and indeed this can be checked experimentally. A light ray falling on certain crystals, is, in general refracted along a certain direction. Nevertheless for special incidence angles, corresponding to the geometry of the crystal, the ray is refracted, not along another ray, but on a whole conical surface. Until the end of 19 century, this experiment was the only proof of the electromagnetic nature of light, because all the other light propagation phenomena could be described by a wave equation, without the use of the complete Maxwell system (see [1], [2]).
- 23. 2 Gaveau In the beginning of the 1960's, strictly hyperbolic equations with simple characteristics (the velocities of propagation are distinct and non zero), are well understood (see [1], [3]). Around that time, Jean Vaillant begins a systematic study of equations or systems of equations which are non strict. In his thesis ([4], [5]), he introduces the notion of localization with respect to a factor of the characteristic determinant of a system with constant coefficients and he relates this notion to the equation of propagation along the ^characteristics. This seems to be the first attempt to apply the method of localization, in particular using invariant factors. Following the article of Garding, Kotake, Leray (Probleme de Cauchy VI, [6]), J. Vaillant defines a new invariant associated to systems with double characteristics and gives an application to the Goursat problem [7], and to the localization for systems with variables coefficients and double characteristics : This is the first example of a Levi condition in this setting. J. Vaillant relates also the difference of the subcharacteristic polynomial, and the second coefficient of exp (-icocp) P (x, DX) exp (icocp) to the Lie derivative of the volume form along the ^characteristics [8]. This is an important invariant, because it is well known that the existence and the regularity of the solution depend, in degenerate cases not only of the principal symbol but also on the lower order terms of the operator. This result was rederived by Duistermaat and Hormander. In 1973-74, J. Vaillant constructs the asymptotic expansion of the solution of an hyperbolic systems with characteristics of variable multiplicities [9]. He defines the localization of an hyperbolic system at a multiple points with application to conical refraction [9]. This work will be extended in 1978, when he constructs the parametrix for the Cauchy problem with multiple characteristics [10], in relation with the invariants of the system. In [11], J. Vaillant studies the symmetrisation of localized hyperbolic systems and defines the notion of « reduced dimension » : The property of symmetry is proved in the case of a maximal reduced dimension. These last few years, he has continued to study the symmetric of strongly hyperbolic systems, in particular with T. Nishitani. If the reduced dimension of a system of rank m is not less than 2, a constant coefficient 2 systems is symmetrizable ([7], [8]). For a system with non constant coefficients, if at any point the reduced dimension is not less than — 2, the system is also symmetrizable with a regular symmetrizer [18]. Recently he has determined the multiple points according to the reduced dimension [19]. In 1982, in collaboration with D. Schiltz and C. Wagschal [12], J. Vaillant has studied the ramification of the Cauchy problem for a system in involution with triple characteristics. This problem reduces to the question of the singularities of integrals of
- 24. Conference in honor of Jean Vaillant 3 holomorphic forms depending of parameters, on chains depending also of parameters. The problem is to determine the singularities of these integrals with respect to the parameters. The first systematic work in this direction was the article of J. Leray [13] in the algebraic case. J. Vaillant studies the ramification in the general holomorphic case, using a grassmann boundle [14]. Since 1987, J. Vaillant has started the problem of the classification of systems with constant multiplicities : definition of invariant Levi conditions, relations to the Cauchy problem in the C°° and Gevrey classes. For any system, he obtains systematically the Levi conditions [15], [16]. J. Vaillant has founded a research group and a seminar, which he has maintained, for more than thirty years, independent of fashions « mots d'ordre » and which survives in difficult conditions. Freedom of thought, which is a necessary condition for any creative work, is paid a very high price. Creation, scientific or artistic, cannot be judged according to economic or social criteria, measured in monetary values. Research is not a collective activity. The highly mysterious activity of thought can only be a personal activity. During all his career, Jean Vaillant, following the example of Leray, has tried to defend by his attitudes and his work, the values of scientific creation and intellectual independence. For more than forty years, J. Vaillant has developed many collaborations with his Japanese and Italian friends, in particular Y. Hamada, Y. Ohya, K. Kajitani, T. Nishitani and S. Spagnolo, F. Colombini, A. Bove and E. Bernardi. He has also developed many european collaborations and he has created a european network of belgian, french, Italian, and Portugese universities. All his friends know that they can rely on his help and his advises. acknowledgment: We are very grateful to Anne Durrande, Evelyne Guilloux and Maryse Loiseau for their help during the preparation of this conference. We also thank the Maison Europeenne des Technologies, in particular Madame Muller for her help. B. Gaveau Laboratoire Analyse et Physique Mathematique 14 avenue Felix Faure 75015 PARIS
- 25. Gaveau [1] Courant-Hilbert: Methods of Mathematical Physics Vol Interscience 1962 [2] M. Born & E. Wolf : Principles of optics Penjamon Press 1980 [3] J. Leray : Hyperbolic differential equations Lectures notes, Princeton 1950 [4] J. Vaillant: Sur les discontinuites du tenseur de courbure en theorie d'Einstein-Schrodinger CR Acad Sci Paris - 10 juillet 1961, 30 octobre 1961, 15 Janvier 1962 [5] J. Vaillant: Caracteristiques multiples et bicaracteristiques des systemes d'equations aux derivees partielles lineaires et a coefficients constants Annales Institut Fourier 15 (1965) et 16 (1966) [6] L. Garding, T. Kotake, J. Leray : Uniformation et developpement asymptotique de la solution du probleme de Cauchy lineaire a donnees holomorphes ; analogic avec la theorie des ondes asymptotiques et approchees (Probleme de Cauchy I bis et VI) Bull. Sci. Math. France 92 1964, 263-361. [7] J. Vaillant: Donnees de Cauchy portees par une caracteristique double : role des bicaracteristiques J. Maths Pures et Appliquees 47 (1968), 1-40 [8) J. Vaillant: Derivee de Lie de la forme element de volume le long des bicaracteristiques et polynome sous-caracteristique de Garding-Kotake-Leray CR Acad Sci Paris -10 mars 1969 [9] J. Vaillant: Solutions asymptotiques d'un systeme a caracteristiques de multiplicite variable J. Maths Pures et Appliquees 53 (1974), 71-98 [10] R. Berzin, J. Vaillant- Parametrix a caracteristiques multiples Bull. Sci. Math 102 (1978), 287-294 [11] J. Vaillant: Symetrisation de matrices localisees Annali della Scuola Normale Superiore di Pisa. Ser. IV, 5 (1978), 405-427 [12] D. Schiltz, J. Vaillant, C. Wagschal: Probleme de Cauchy ramifie J. Math. Pures et appliquees (1982) [13] J. Leray : Un complement au theoreme de N.Nilsson sur les integrales de formes differentielles a support singulier algebrique
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- 30. The Project Gutenberg eBook of The Architectural Review and American Builders' Journal, Aug. 1869
- 31. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: The Architectural Review and American Builders' Journal, Aug. 1869 Author: Various Editor: Samuel Sloan Release date: December 22, 2019 [eBook #60997] Most recently updated: October 17, 2024 Language: English Credits: Produced by Paul Marshall and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK THE ARCHITECTURAL REVIEW AND AMERICAN BUILDERS' JOURNAL, AUG. 1869 ***
- 33. THE ARCHITECTURAL REVIEW AND AMERICAN BUILDERS’ JOURNAL. Vol. II.—Entered according to Act of Congress, in the year 1869, by Samuel Sloan, in the Clerk’s Office of the District Court of the United States, in and for the Eastern District of Pennsylvania.
- 34. MONTHLY REVIEW. THE LONDON BUILDER AND OURSELVES. In a tolerantly critical notice of the Review recently published in the Builder, we find an effort to substantiate a charge formerly made by it, and replied to by us, on the subject of “trickery” in the construction of the exteriors of American buildings. The Builder reiterates the charge and points to Grace Church, New York, in proof of the truth of it. That marble edifice, he avers, has a wooden spire, crocketted, etc., painted in imitation of the material of which the body of the church is constructed. Alas, we must acknowledge the wood. And we will make a clean breast of it, and still farther acknowledge that at the time that Grace Church was built, our land of wooden nutmegs, and other notions, had not an architectural idea beyond the wooden spire, and that our city and country churches, that aspired at all, were forced to do so in the national material of the day. That said sundry spires of wood were of necessity, painted, is most true; and furthermore, white-lead being a great favorite with the people generally, [when our manners, customs, and tastes were more immaculate than in these degenerate days of many colors,] that pigment was the ruling fashion. That the color of the marble, of which Grace Church’s body is constructed, should be similar to that with which said ecclesiastical edifice’s spire was coated, is unfortunate; but, that the resemblance goes to prove any attempt at a cheat, we most strenuously deny. Grace Church is of a by-gone taste,—an architectural era which we now look back to in order to see, by contrast, how far we have advanced in architectural construction. Trinity Church, New York, was the first great effort at a stone spire which our Architects ventured to rear. And although hundreds have followed its lead, none in this soaring republic have
- 35. gone so near to heaven as that yet. But the thing once effected is sure to be improved upon. We are not at all abashed then, to own to the wooden spire painted to imitate stone, which crowns the steeple of old Grace Church, New York. And the less annoyance should it give our most sensitive feelings, when we reflect that the dome of the great St. Paul’s, London, is no less a delusion and a cheat, it being of wood, coated with lead and painted on the outside, having a false dome on the inside, considerably smaller than the external diameter would naturally lead the confiding observer to expect. The body of St. Paul’s is of stone. Why, according to the requirements of the Builder, is not the dome, like that of the Pantheon at Rome, likewise of stone? Do we suppose, for an instant, that Sir Christopher Wren was guilty of a deliberate cheat in so constructing it? Certainly not. He used the material which he considered best suited to his purpose and his means. And so we should, in charity, suppose did the Architect of Grace Church, New York. The Builder, like too many of our English cousins, who do us the honor of a visit, falls into error in supposing that wood is generally used for ornamentation of exteriors. In none of our larger cities is this the case. And when that critical and usually correct authority says, “Even the Fifth avenue itself is a sham as to much of its seeming stone-work,” it displays a melancholy absence of its uniform discernment, judgment, and sense. The only other constructive material to be found on the fronts of the Fifth Avenue, New York, besides marble, brown stone, or pressed (Philadelphia) brick, is in the gutter, which is either of zinc or galvanized iron, and forms the upper portion of the cornice. Porches and Hall-door frontisces, of every style, are of marble or stone, and never of wood. Pediments and all trimmings around windows are invariably of stone. In fact we are not a little surprised at the apparent want of information on this subject by so well
- 36. posted an observer as the Builder is acknowledged to be. Some twenty years ago the taunt might lie most truthfully applied to our efforts at architectural construction, but to-day the “trick” of painted and sanded wood would be hissed down by our citizens who claim to live in residences the majority of which are greatly superior to residences of the same class in London, as far at least as material is concerned. No, no—criticism to be useful must be just; and to be just must be founded strictly on truth unbiassed by prejudice. We do not desire in these remarks to throw the slightest doubt on the good intentions of the London Builder in its monitorial check, but our wish is to correct the erroneous information which it has received, and which has led to the mistake under which it evidently labors. We as utterly despise any falsehood in construction as our honestly outspoken contemporary, and will at every opportunity disclose and denounce its adoption in this country in all cases where there is any pretension to architectural design. For a new country like this, it is at least creditable that, even in a small class of dwellings, the architect is, as a general thing, called on to design and frequently to superintend—every thing is not left to the builder as in London. Yet there is and always will be in this as in all other countries a large class of private buildings outside the pale of legitimate taste; creations ungoverned and ungovernable by rule. But such should never be taken as examples of the existing state of the constructive art of the day; they should rather prove the unfortunate exceptions to the fact of its position. Even these it will be our duty to watch over and try to set right; for we are ardent believers in the influential power of information, and look with assurance to the education of our people generally on this subject of judgment and taste in building as the infallible means of turning to good account the remarkable progress in that constructive art of the American nation, which the observant London Builder notices with the generous well-wishing of a kindly professional brother.
- 37. THE MANSARD MADNESS. Of all the intellectual qualifications which man is gifted with, there is not one as sensitive as that which enables him to discern between what is intrinsically good, and what is bad or indifferent to his eye. Yet are there none of all man’s mental attributes so frequently and so grossly outraged as is this to which we now allude, called Taste. Custom has much to say in the question of arbitrary rule which taste so imperatively claims. Persistence in any thing will, of necessity, make itself felt and recognized, no matter how odious at first may be the object put before the public eye, and ultimately that object becomes what is commonly called “fashionable.” This apparent unity of the public on one object is variable and will soon change to another, which in its turn will seem to reign by unanimous consent and so on ad infinitum. In Architecture this fickle goddess, Fashion, seems to reign as imperatively and as coquettishly as in any or all the affairs of this world of humanity. That which was at first esteemed grotesque and ridiculous, becomes in time tolerable and at last admirable. But the apathy which sameness begets cannot long be borne by the novelty worshippers, and accordingly new forms and shapes remodel the idea of the day, until it ceases to bear a vestige of its first appearance and becomes quite another thing. Of all the prominent features of architecture that which has been least changeable until late years is the “roof.” The outline of that covering has been limited to a very few ideas, some of which resolved themselves into arbitrary rules of government from which the hardiest adventurer was loath to attempt escape. Deviating from the very general style of roof which on the section presents a triangle, sometimes of one pitch, sometimes of another, but almost universally of a fourth of the span, the truncated form was to be found, but so exceedingly sombre was this peculiar roof
- 38. that it never obtained to any great extent, and indeed it presented on the exterior a very serious obstacle to its adoption by architects in the difficulty of blending it with any design in which spirit, life, or elegance, was a requisite. There are occasionally to be found in Europe, and even in America, examples of these truncated roofs, but it is very questionable whether there are to be met with any admirers of their effect. The principle on which they are constructed has, however, a very great advantage in the acquirement of head-room in the attics, giving an actual story or story and half to the height, without increasing the elevation of the walls. The architects of the middle ages took a hint from this evident advantage, and used the truncated roof on their largest constructions. Its form is that of a pyramid with the upper portion cut off (trunco, to cut off, being its derivation.) Mansart, or as he is more commonly called Mansard, an erratic but ingenious French architect, in the seventeenth century invented the curb roof, so decided an improvement on the truncated that it became known by his name. This roof adorning the palatial edifices of France soon assumed so much decorative beauty in its curb moulding and base cornice, as well as in the dormers and eyelets with which it was so judiciously pierced, that it became a source of artistic fascination in those days in France; and as Germany was indebted to French architects for her most prominent designs, the Mansard roof found its way there, and into some other parts of Europe. But, much as English architects admired, as a whole, any or all of those superb erections of the Gallic Capital, it was a century and a half before it occurred to them to imitate them even in this most desirable roof. Our architects having increased with the demand for finer houses and more showy public buildings, and having parted company with
- 39. their Greek and Roman idols to which their predecessors had been so long and so faithfully wedded, and acknowledging the necessity for novelty, ardently embraced the newly arising fashion and the Mansard roof arose at every corner in all its glory. At first the compositions which were adorned with this crowning were pleasing to the general view, if not altogether amenable to the strict rules of critical taste. But in due time (and alas that time too surely and severely came) the pseudo French style with its perverted Mansard roof palled upon the public taste for the eccentricities its capricious foster-fathers in their innate stultishness compelled it to display. Some put a Mansard roof upon an Italian building, some on a Norman, and many, oh, how many, on a Romanesque! Some put it on one story erections and made it higher than the walls that held it, in the same proportion that a high crowned hat would hold to a dwarf. Some stuck on towers at the corners of their edifices and terminated them with Mansard domes! Some had them inclined to one angle, some to another; some curved them inward, some outward, whilst others went the straight ticket. The dormers too came in for a large share of the thickening fancies and assumed every style or no style at all. The chimney shafts were not neglected. Photos of the Thuilleries were freely bought up, and bits and scraps of D’Lorme were hooked in, to make up an original idea worthy of these smoky towers. “Every dog will have his day,” is a fine old sensible remark of some long-headed lover of the canine species, and applies alike to animals, men, and things. That it particularly applies to that much abused thing called the Mansard roof is certain, as the very name is now more appropriately the absurd roof. Fashion begins to look coldly upon her recent favorite, which in truth “has been made to play such fantastic tricks before high Heaven, as make the angles weep;” and it is doomed. A few years hence, and we will all look back in amused wonder at the creations of to-day, crowned with the tortured conception of Mansard.
- 40. HYDRAULIC CEMENT. The rapid hardening under water of the cement which from that property derives its name of “Hydraulic Cement,” has been, and indeed is still, a subject of discussion as to the true theory of such action. We find in the June number of the Chemical News a paragraph which must prove very interesting to manufacturers as well as to all who use and take an interest in that most useful of building materials to which the Architect and the Engineer are so deeply indebted. “In order to test the truth of the different hypotheses made concerning this subject, A. Schulatschenko, seeing the impossibility of separating, from a mixture of silicates, each special combination thereof, repeated Fuch’s experiment, by separating the silica from 100 parts of pure soluble silicate of potassa, and, after mixing it with fifty parts of lime, and placing the mass under water, when it hardened rapidly. A similar mixture was submitted to a very high temperature, and in this case, also, a cement was made. As a third experiment, a similar mixture was heated till it was fused; after having been cooled and pulverized, the fused mass did not harden any more under water. Hence it follows that hardening does take place in cement made by the wet as well as dry process, and that the so-called over-burned cement is inactive, in consequence of its particles having suffered a physical change.”
- 41. IRON STORE-FRONTS, No. V. By Wm. J. Fryer, Jr., with Messrs. J. J. Jackson & Bros., New York. NATIVE COLORED MARBLES.
- 42. In the preceding number we have spoken in general terms of this beautiful acquisition to our art materials, and indeed we feel that we cannot esteem this new American discovery too highly; for even in Europe such stone is extremely scarce at the present day, and it is fortunate that the location in which the quarries exist is open to the Old World to freely supply the wants of its artists, as well as our own. The beautiful Lake Champlain affords excellent commercial facilities, the Chambly Canal and Sorel River improvements opening a free navigation both with the great chain of lakes, and the Atlantic Ocean. The Champlain Canal connecting it with the Erie Canal and Hudson River, giving it uninterrupted communication with New York State and its Empire City, from the latter end of March to the middle of December. The quarry is situated in a great lode projecting up in the bosom or bay of Lake Champlain, forming an island of several acres outcropping on each shore, and giving evidence that the deposit extends and really forms, at this point, the bed of the lake, its supply being thought to be inexhaustible. The marble occurs in beds and strata varying in thickness from one to six feet, and will split across the bed or grain; blocks of any required size being readily obtained. Its closeness of texture and hardness render it susceptible of a very high polish, and it will resist in a remarkable degree all atmospheric changes. It is hard to deface with acids or scratches, and this one fact should attach to it much additional value. Its variegation in color, as shown by the specimens taken from its outcroppings, give promise of a much richer development as the bed of the quarry is approached; and must equal in beauty and durability the highly prized oriental marble of ancient and modern times. The facilities, already alluded to, of its transportation to all the markets for such material in the country and to the seaboard, whence it can be shipped to any part of the world, must tend to bring it into general use here and elsewhere, that colored marbles are required for building and ornamental purposes.
- 43. We are much indebted to a gentleman of Philadelphia, whose taste and liberal enterprise have so opportunely brought to our knowledge this most remarkable deposit of one of Nature’s most beautiful hidden treasures, which must, at no distant day, add vastly and more cheaply to the art material of our country. The palace in course of construction at Ismalia, for the reception of the Empress Eugenie during her stay in Egypt, will be 180 feet wide and 120 deep. The estimate cost is 700,000fr. According to the contract it is to be finished by the 1st of October, for every day’s delay the architect will be subject to a fine of 300fr per day, and if finished before he will receive a bonus of 300fr per day. The building will be square; in the centre there is to be a dome covered with Persian blinds. On the ground floor there will be the ball, reception, and refreshment rooms. An idea can be formed of the importance of this structure and of the work necessary to complete it within the required time, as it will contain no less than 17,400 cubic feet of masonry. To Remove Writing Ink—To remove writing ink from paper, without scratching—apply with a camel’s hair brush pencil a solution of two drachms of muriate of tin in four drachms of water; after the writing has disappeared, pass the paper through the water and dry.
- 44. DESCRIPTIONS. IRON STORE FRONTS, No. 5. By W. J. Fryer, Jr., New York. The elevation, shown in the accompanying page illustration, shows an iron front of five stories, having a pedimented centre frontispiece of three stories in alto relievo. The style, though not in strict accordance with rule, is showy, without being objectionably so, and goes far to prove the capabilities of iron as a desirable material in commercial Architecture, where strength, display, and economy may be very well combined. Such an elevation as this, now under consideration, could not be executed in cut stone, so as to produce the same appearance, without incurring a much greater expense, and in the event of a continuous block of such fronts, the balance of economy would be wonderfully in favor of the iron, for the moulds could be duplicated and triplicated with ease, whilst the same composition executed to a like extent in stone would not be a cent cheaper in proportion. Every capital and every truss, and every fillet, should be cut in stone independently of each other, no matter how many were called for. It may be very well to say that stone is the proper material, according to the long-accepted notion of art judgment, and that iron has to be painted to give it even the semblance of that material, being, therefore, but a base imitation at best. All very true. But, nevertheless, iron, even as a painted substitute, possesses advantages over the original material of which it is a copy, rendering it a very acceptable medium in the constructive line, and one which will be sought after by a large class of the community who desire to have this cheap yet practical material, even though it be not that
- 45. which it represents. As a representative it is in most respects the peer of stone though not it identically. SUBURBAN RESIDENCE IN THE FRENCH STYLE. BY CARL PFEIFFER, ESQ., ARCHITECT, N. Y. This design is of one of those homes of moderate luxury wherein the prosperous man of business may enjoy in reason the fruits of his energetic toil. There is nothing about it to indicate presumptuous display, but rather the contented elegance of a mind at ease, surrounded with unostentatious comfort. Fig. 1.
- 47. SUBURBAN RESIDENCE IN THE FRENCH STYLE. Carl Pfeiffer, Esq., Architect, New York. On the westerly slope of the Palisades, and two miles to the west of the Hudson, this residence was built by one of New York’s retired merchants. It is sixteen miles from Jersey City, in a town of but a few years growth, named “Terrafly,” in Bergen county, and stands on a hill commanding some of the most charming pieces of pastoral scenery, occupying about thirty acres laid out in lawns, walks, gardens, etc., and tastefully ornamented with shrubbery, having a fountain on the lawn in front of the house (as shown.) The approach is from the public road, by a drive through a grove of about ten acres of stately trees, passing by the side of a pretty pond formed by the contributions of several streams and making a considerable sheet of water. About the middle of this pond the sides approach so near to each other as to be spanned by an artistic little stone arched bridge which leads to the garden.
- 48. From the house one looks on a lovely panorama of inland scenery. The Palisades towards the east, the Ramapo mountains to the northwest; and looking in a southerly direction the numerous suburban villages and elegant villas near New York may be seen. The house is constructed of best Philadelphia pressed brick with water-table, quoins, and general trimmings of native brown stone neatly cut. It stands high on a basement of native quarry building stone and has for its foundation a permanent bed of concrete which likewise forms the basement floors, as well as a durable bedding for the blue flagging of Kitchen and Laundry hearths. Fig. 2.
- 49. The arrangement of plan is admirably calculated to conduce to the comfort of the family. It is as follows: Fig. 1 shows the plan of the basement. A, steps and passage leading from Yard. B, Servant’s Dining Room. C, C, C, Coal Cellar and Passages. D, Kitchen. E, Pantry. F. Laundry. G, G, Cellars. H, Water Closet. I, Wash tubs in Laundry. J, Dumb waiter. K, Wash-tray. L, Sink. M, Back stairs.
- 50. Fig. 2 shows the plan of the principal story. A, Dining Room. B, Drawing Room. C, and D, Parlors connected by sliding doors with the Drawing Room through the hall. E, Principal staircase. F, Back Hall. G, Butler’s Pantry with dumb waiter, plate closet, wash-trays, etc. H, Back stairs. J, Conservatory. K, Steps leading down to Yard. L, L, L, Verandahs. M, M, Piscinæ. Fig. 3.
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