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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations
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Young Measures on Topological Spaces - With Applications in Control Theory and Probability Theory (Paperback, Softcover reprint of the original 1st ed. 2004)
Loot Price: R1,570
Discovery Miles 15 700
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Young Measures on Topological Spaces - With Applications in Control Theory and Probability Theory (Paperback, Softcover reprint of the original 1st ed. 2004)
Series: Mathematics and Its Applications, 571
Expected to ship within 10 - 15 working days
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Total price: R1,590
Discovery Miles: 15 900
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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? ].
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