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He then turned to the problems of solving, both theoretically and numerically, specific nonlinear partial differential equations arising in applications.Motivated by the work of Jean Leray and Eberhard Hopf on the Navier-Stokes equations, which describe the flow of viscous fluids such as oil and water, he generalised and modernised their methods so as to apply to a very wide range of important problems from mechanics and physics. He described many of these results in his 1969 book Quelques m?odes de r?lution des probl?s aux limites non-lin?res, which is a landmark in the modern theory of partial differential equations.His mathematical style was to identify a new, or under-developed, method having significant potential for applications in science and industry, and to make a sustained attack over a period of years on systematising it so as to be more generally useful. In doing so, he would motivate a group of collaborators and students to map out with him the area, and transfer the knowledge gained both within the academic community, and beyond it into industry.In this spirit he developed, for example, variational inequalities (the appropriate generalisation of partial differential equations when there are constraints, such as occur when one solid contacts another), homogenisation (the mathematical study of composite and heterogeneous materials) and many aspects of the optimal control of systems.The scale of this effort was enormous Lions wrote some 500 scientific papers and over 20 books. He supervised more than 50 doctoral students, from France and many other countries, a significant number of whom are now leaders of their fields, and their students in turn enlarged and internationalised his school.Lions had early on realised the profound impact that computers would have on applied mathematics.

As nonlinear analysis rapidly developed in the 1970s and 1980s, propelled by the advances of the Lions school and of other pioneers, especially in the Soviet Union and United States, it too brought about profound changes. No longer was it tenable to rely solely on physical intuition and formal calculation, disregarding rigorous analysis of the governing equations. Furthermore, the explosion in the use of computer simulations reinforced the need for fundamental understanding of the often subtle behaviour of numerical schemes, for which the analytic techniques of the Lions school were well adapted.After holding professorships in Nancy and Paris, Lions became in 1973 Professor in the Mathematical Analysis and Control of Systems at the prestigious Coll? de France. The same year, at the unusually early age of 45, he was elected a Member of the French Acad?e des Sciences.

From 1966-86 he was also Professor at the Ecole Polytechnique. At the Coll? de France he gave highly regarded lecture courses on his current research until his retirement, and held a weekly seminar, with many distinguished invited speakers, that became a Mecca of French applied mathematics.From 1980 to 1984 Lions was the first President of INRIA, the French National Institute for Research in Computer Science and Automation, a position that suited well his belief in the contribution soundly based scientific computation could make to industry. Then, from 1984 to 1992, he held the important post of President of the Centre National d'Etudes Spatiales (CNES), the French space agency. During his period of office, key decisions were made to go ahead with Ariane 5, with Topex Poseidon, an oceanography satellite built in co-operation with Nasa that has provided a breakthrough in the understanding of El Ni?nd La Ni?and the Spot family of earth observation satellites.In the same period the manned space flight programme was developed, with several French astronauts taking part in missions in co-operation with the United States and Soviet Union. Other influential posts at the interface between science and industry followed. He was President of the Scientific Committees of P?iney, Gaz de France, Electricit?e France and France Telecom, held high-level scientific advisory posts with Dassault-Aviation and Elf, and was a Director of Dassault Systems, P?iney, Saint-Gobain and Thomson Multimedia.From 1991 until 1994 Lions was President of the International Mathematical Union, the controlling body of world mathematics, which he had previously served as Secretary. As President he proposed and laid plans for World Mathematical Year 2000.

From 1996 until 1998 he was President of the Acad?e des Sciences, and through his chairmanship of its Committee 2000 co-ordinated its response to the request from the President of the Republic for a wide-ranging assessment of scientific issues facing society at the dawn of the 21st century. In 1998 President Jacques Chirac personally invested him as a Grand Officier de l'Ordre National du M?te.He received numerous prizes, including the Prix Cognacq-Jay of the Acad?e des Sciences, the 1986 John von Neumann Prize, the 1991 Japan Prize, the 1991 Harvey Prize, and the 1999 Lagrange Prize. He was a member of more than 20 foreign academies, including the Royal Society, the US and Soviet Academies of Science, and the Third World Academy of Sciences, the latter recognising his interest in and support of young mathematicians from developing countries.Jacques-Louis Lions was a man of considerable personal magnetism and charm He had clear-cut but undogmatic views on most issues. A master of the art of managing time, he made decisions quickly, tempering vision with pragmatism.He is survived by his wife Andr? whom he had met when they were both members of the French resistance, and by their son Pierre-Louis, himself a distinguished mathematician who won a Fields Medal (considered as the equivalent of the Nobel Prize for Mathematics) in 1994.John Ball. Renato Carosone, pianist and composer: born Naples 3 January 1920; married (one son); died Rome 19 May 2001 Renato Carosone, pianist and composer: born Naples 3 January 1920; married (one son); died Rome 19 May 2001. In Anthony Minghella's 1999 film The Talented Mr Ripley, Jude Law and Matt Damon leap on stage with the Italian entertainer Fiorello to intone "Tu vu?' l'americano" ("You Want to Play the American", 1957), one of the greatest hits of the musician Renato Carosone. Sophia Loren and Clark Gable duetted to the same tune in 1960 in It Started in Naples.In both cases Carosone's music is used to evoke the sunny, animated nature of Naples and its song.