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By Yuichiro Kakihara

This paintings makes a speciality of present issues in astronomy, astrophysics and nuclear astrophysics. The parts coated are: foundation of the universe and nucleosynthesis; chemical and dynamical evolution of galaxies; nova/supernova and evolution of stars; astrophysical nuclear response; constitution of nuclei with risky nuclear beams; beginning of the heavy point and age of the universe; neutron big name and excessive density subject; statement of parts; excessive strength cosmic rays; neutrino astrophysics Entropy; details resources; info channels; distinctive subject matters

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1) By Lemma 1, {H) —> H). Suppose first that H < v. For n > 1 let /n„ = ^ l ^ and vn = i/|3)„, the restrictions of n and v to 2)„, respectively. Then /i 1. If we let / „ = ^ and / = g^, then it follows from Theorem 2 that for n > 1 # S > „ ( M I " ) = / /nlog/„aV, Hv(n\v)= j flogfdv. e. e. 6. Relative entropy and Kullback-Leibler tier information in = liminf HVn(fi\u) 53 < H

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). (5) is obtained from (1) and (4).

Let Ci = {7 G T : ip(-y) = 1}. (7l) = ^(72), 7172 -1 € G\. Let f = r / C i , the quotient group and define (p on T by £(7) = P(7). 7 e r> where 7 is the equivalence class containing 7. Then it is verified that (p is positive definite on f such that £(7) = 1 iff 7 = 1, so that (f,

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