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1. Historical Remarks Convex Integration theory, first introduced
by M. Gromov [17], is one of three general methods in
immersion-theoretic topology for solving a broad range of problems
in geometry and topology. The other methods are: (i) Removal of
Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii)
the covering homotopy method which, following M. Gromov's thesis
[16], is also referred to as the method of sheaves. The covering
homotopy method is due originally to S. Smale [36] who proved a
crucial covering homotopy result in order to solve the
classification problem for immersions of spheres in Euclidean
space. These general methods are not linearly related in the sense
that succes sive methods subsumed the previous methods. Each method
has its own distinct foundation, based on an independent
geometrical or analytical insight. Conse quently, each method has a
range of applications to problems in topology that are best suited
to its particular insight. For example, a distinguishing feature of
Convex Integration theory is that it applies to solve closed
relations in jet spaces, including certain general classes of
underdetermined non-linear systems of par tial differential
equations. As a case of interest, the Nash-Kuiper Cl-isometrie
immersion theorem ean be reformulated and proved using Convex
Integration theory (cf. Gromov [18]). No such results on closed
relations in jet spaees can be proved by means of the other two
methods.
This book provides a comprehensive study of convex integration
theory in immersion-theoretic topology. Convex integration theory,
developed originally by M. Gromov, provides general topological
methods for solving the h-principle for a wide variety of problems
in differential geometry and topology, with applications also to
PDE theory and to optimal control theory. Though topological in
nature, the theory is based on a precise analytical approximation
result for higher order derivatives of functions, proved by M.
Gromov. This book is the first to present an exacting record and
exposition of all of the basic concepts and technical results of
convex integration theory in higher order jet spaces, including the
theory of iterated convex hull extensions and the theory of
relative h-principles. A second feature of the book is its detailed
presentation of applications of the general theory to topics in
symplectic topology, divergence free vector fields on 3-manifolds,
isometric immersions, totally real embeddings, underdetermined
non-linear systems of PDEs, the relaxation theorem in optimal
control theory, as well as applications to the traditional
immersion-theoretical topics such as immersions, submersions,
k-mersions and free maps. The book should prove useful to graduate
students and to researchers in topology, PDE theory and optimal
control theory who wish to understand the h-principle and how it
can be applied to solve problems in their respective disciplines.
Originally published in 1977. Professor David Spring presents
comparative histories of European landed elites in the nineteenth
century, covering English, Prussian, Russian, Spanish, and French
landed elites. European Landed Elites in the Nineteenth Century
underscores the particularities of each case and underscores the
differences between cases.
Originally published in 1963. The English Landed Estate in the
Nineteeth Century: Its Administration deals principally with the
administration of large landed estates during the years from 1830
to 1870. The book also throws new light on the work of the
Inclosure Commissioners, who, as a department of the central
government, supervised agricultural improvements made by landowners
who borrowed from the government and from land companies. Author
David Spring argues that the British government intervened in
agriculture much more than is commonly thought. In describing the
hierarchy of estate management, Spring relies, wherever possible,
on hitherto unused family papers and estate documents. Especially
important is his material on the Dukes of Bedford and on the
domestic economy and financial position of the Russell Family. The
chapter titled "The Landowner," based on the seventh Duke of
Bedford's correspondence with his agent, is a case study of a
single estate and provides insight into the workings of a great
landowner's mind. The remaining chapters, dealing with lawyers,
land agents, and the Inclosure Commissioners, include other
individual portraits. Among these are Christopher Haedy, the Duke
of Bedford's chief agent; James Loch, king of estate agents in
nineteenth-century England; Henry Morton, the Earl of Durham's land
agent; and William Blamire and James Caird, two of the Inclosure
Commissioners.
1. Historical Remarks Convex Integration theory, ?rst introduced by
M. Gromov [17], is one of three general methods in
immersion-theoretic topology for solving a broad range of problems
in geometry and topology. The other methods are: (i) Removal of
Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii)
the covering homotopy method which, following M. Gromov's thesis
[16], is also referred to as the method of sheaves. The covering
homotopy method is due originally to S. Smale [36] who proved a
crucial covering homotopy result in order to solve the
classi?cation problem for immersions of spheres in Euclidean space.
These general methods are not linearly related in the sense that
succ- sive methods subsumed the previous methods. Each method has
its own distinct foundation, based on an independent geometrical or
analytical insight. Con- quently, each method has a range of
applications to problems in topology that are best suited to its
particular insight. For example, a distinguishing feature of
ConvexIntegrationtheoryisthatitappliestosolveclosed
relationsinjetspaces, including certain general classes of
underdetermined non-linear systems of par- 1 tial di?erential
equations. As a case of interest, the Nash-Kuiper C -isometric
immersion theorem can be reformulated and proved using Convex
Integration theory (cf. Gromov [18]). No such results on closed
relations in jet spaces can be proved by means of the other two
methods. On the other hand, many classical results in
immersion-theoretic topology, such as the classi?cation of
immersions, are provable by all three methods.
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