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A lot of economic problems can formulated as constrained
optimizations and equilibration of their solutions. Various
mathematical theories have been supplying economists with
indispensable machineries for these problems arising in economic
theory. Conversely, mathematicians have been stimulated by various
mathematical difficulties raised by economic theories. The series
is designed to bring together those mathematicians who were
seriously interested in getting new challenging stimuli from
economic theories with those economists who are seeking for
effective mathematical tools for their researchers. Members of the
editorial board of this series consists of following prominent
economists and mathematicians: Managing Editors: S. Kusuoka (Univ.
Tokyo), T. Maruyama (Keio Univ.) Editors: R. Anderson
(U.C.Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke
(Univ. Lyon I), G. Debreu (U.C. Berkeleyer), E. Dierker (Univ.
Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T.
Fujimoto (Okayama Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano
(Yokohama National Univ.), L. Hurwicz (Univ. of Minnesota), T.
Ichiishi (Ohio State Univ.), A. Ioffe (Israel Institute of
Technology), S. Iwamoto (Kyushu Univ.), K. Kamiya (Univ. Tokyo), K.
Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ.
Tokyo), K. Nishimura (Kyoto Univ.), M. K. Richter (Univ.
Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ.
Montpellier II), M. Yano (Keio Univ).
Classicalexamples of moreand more oscillatingreal-valued functions
on a domain N ?of R are the functions u (x)=sin(nx)with x=(x ,...,x
) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[,u (x)=r
(x) = sgn(sin(2 ?x))(seelater3.1.4). n n They may appear as the
gradients?v of minimizing sequences (v ) in some n n n?N
variationalproblems. Intheseexamples,thefunctionu
convergesinsomesenseto n ameasure on ? xR, called Young measure. In
Functional Analysis formulation, this is the narrow convergence to
of the image of the Lebesgue measure on ? by ? ? (?,u (?)). In the
disintegrated form ( ) ,the parametrized measure n ? ??? ? captures
the possible scattering of the u around ?. n Curiously if (X ) is a
sequence of random variables deriving from indep- n n?N dent ones,
the n-th one may appear more and more far from the k ?rst ones as 2
if it was oscillating (think of orthonormal vectors in L which
converge weakly to 0). More precisely when the laws L(X ) narrowly
converge to some probability n measure , it often happens that for
any k and any A in the algebra generated by X ,...,X , the
conditional law L(X|A) still converges to (see Chapter 9) 1 k n
which means 1 ??? C (R) ?(X (?))dP(?)?? ?d b n P(A) A R or
equivalently, ? denoting the image of P by ? ? (?,X (?)), n X n (1l
??)d? ?? (1l ??)d[P? ].
Classicalexamples of moreand more oscillatingreal-valued functions
on a domain N ?of R are the functions u (x)=sin(nx)with x=(x ,...,x
) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[,u (x)=r
(x) = sgn(sin(2 ?x))(seelater3.1.4). n n They may appear as the
gradients?v of minimizing sequences (v ) in some n n n?N
variationalproblems. Intheseexamples,thefunctionu
convergesinsomesenseto n ameasure on ? xR, called Young measure. In
Functional Analysis formulation, this is the narrow convergence to
of the image of the Lebesgue measure on ? by ? ? (?,u (?)). In the
disintegrated form ( ) ,the parametrized measure n ? ??? ? captures
the possible scattering of the u around ?. n Curiously if (X ) is a
sequence of random variables deriving from indep- n n?N dent ones,
the n-th one may appear more and more far from the k ?rst ones as 2
if it was oscillating (think of orthonormal vectors in L which
converge weakly to 0). More precisely when the laws L(X ) narrowly
converge to some probability n measure , it often happens that for
any k and any A in the algebra generated by X ,...,X , the
conditional law L(X|A) still converges to (see Chapter 9) 1 k n
which means 1 ??? C (R) ?(X (?))dP(?)?? ?d b n P(A) A R or
equivalently, ? denoting the image of P by ? ? (?,X (?)), n X n (1l
??)d? ?? (1l ??)d[P? ].
A lot of economic problems can formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories. The series is designed to bring together those mathematicians who were seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking for effective mathematical tools for their researchers. Members of the editorial board of this series consists of following prominent economists and mathematicians: Managing Editors: S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.) Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeleyer), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama National Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio State Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), K. Kamiya (Univ. Tokyo), K. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), K. Nishimura (Kyoto Univ.), M. K. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).
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