By Yuichiro Kakihara

This paintings makes a speciality of present subject matters 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 point and age of the universe; neutron celebrity and excessive density topic; statement of parts; excessive power cosmic rays; neutrino astrophysics Entropy; info assets; details channels; precise subject matters

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Then, (r, ip, U) is said to be an algebraic model for S or for (X, X, fi, S) if (r, ip) is an algebraic model for /j, by means of a one-to-one homomorphism J : T —> T(n) such that JU = Us JThen we have the following. Theorem 19. Two dynamical systems are conjugate iff they have isomorphic alge braic models. Proof. Let (Xj,Xj, ftj, Sj) (3 = 1,2) be a pair of dynamical systems with associated isometries Us1 and Us2 on L2{^{) and L2(/j,2), respectively. Suppose that Si ~ £2- Then, it is easy to see that the algebraic models ( r ^ i ) , ^ !

If (21V 03) < if (21) + if (03). if(5- 1 2l|5- 1 2J) = if(2l|2J). if(5- 1 2t)=if(2t). Proof. (1) By definition we have i(2l|2)(-) = - E U(-)logP(A|2)(-) = - E 1A(0logP(A) = /(»)(•) and hence if(2l|2) = / i(2l|2)d M = / 7(521)^ = ^(21). 3. 4 B since for 5 e 95 we have p«= £ ,ra»_ * ' ^ ' ^ and hence logP(S|av2J) = £ UlogP(p^,ff) M-a-e- Taking the expectation, we see that the desired equality holds. (3) follows from (1) and (2). (4) a < 05 implies 21 C 05 and a V 53 = 23. So (2) implies F(2t V 93|2j) = #(93|2J) = # ( a | 2 J ) + # ( 9 3 | a V 2J) > ff(a|2J).

Note that Aj,'s are disjoint and j j ^ = [ hif P(A|2J n ) < e " ' ] . Also note that A*n