By Yuichiro Kakihara

ISBN-10: 9810237111

ISBN-13: 9789810237110

This paintings makes a speciality of present issues in astronomy, astrophysics and nuclear astrophysics. The components lined are: starting place of the universe and nucleosynthesis; chemical and dynamical evolution of galaxies; nova/supernova and evolution of stars; astrophysical nuclear response; constitution of nuclei with volatile nuclear beams; beginning of the heavy aspect and age of the universe; neutron superstar and excessive density topic; remark of components; excessive strength cosmic rays; neutrino astrophysics Entropy; details assets; info channels; targeted subject matters

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1) H(IJI\V) > 0; H{p\v) = 0&fi = v. u, relative entropy has an integral form as is shown in the following. Theorem 2. Let /J,,V G P{X). 2) then H(n\v) = oo. Proof. Since (2) is obvious we prove (1). °g kg&\v{A)J (-TK I| --TTT A v /" log fdv (-TK log n r J f fdv)< fdA < --FT: f/ Jfflog )-v{A)J \V(A)JAA ) &\v(A)JAAJ )-v{A)JAAJ by Jensen's Inequality (cf. 2 (12)). 1) »<*>*/, (£*£)* *<*>*/,(£*£)*• To show the reverse inequality, let . 6. Relative entropy and Kullback-Leibler information 51 Then {An>k : 0 < k < n2} € P(X) for each n > 1.

3) if( m y o 1 5 , -J'2t,S' m ) = mff(2l,5) for m > 1. (4) if (21,5) < if (23, S) + H(21|** 1 we see that by definition ff(Vs-»'a,H -H(ny\smrk(mylS-m)\ = Um = lim ra- — jyf V S- fc 2l) n-voo mn fc=o / n-voo mn V V fc=o = mH(%,S). **

Since (2) is obvious we prove (1). °g kg&\v{A)J (-TK I| --TTT A v /" log fdv (-TK log n r J f fdv)< fdA < --FT: f/ Jfflog )-v{A)J \V(A)JAA ) &\v(A)JAAJ )-v{A)JAAJ by Jensen's Inequality (cf. 2 (12)). 1) »<*>*/, (£*£)* *<*>*/,(£*£)*• To show the reverse inequality, let . 6. Relative entropy and Kullback-Leibler information 51 Then {An>k : 0 < k < n2} € P(X) for each n > 1. Moreover, by the definition of A»,fc's / lAn,k^dv< J ' du Jxx ' du -v{An,k)< n n n v(A>,k) n for 0 < k < n2 — 1 and n > 1. fc) < fc + 1 v{An

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