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By Prof. Dr. Hans Schneeweiss, Prof. Dr. Heinrich Strecker (auth.), Prof. Dr. Hans Schneeweiss, Prof. Dr. Heinrich Strecker (eds.)

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C 8 1 [ (l-p) 8 - p8 - 1 J • i3 + (1_p)8 - 1] by (a 2 ), we have 21 - 8 - 1 8 2,3, ... Therefore, B(p) Remark 1. = P8 +1( r) 8 -p -1'8 = 2,3, ... 2 -8 -1 For a positive non-integer 8, infinite series which is convergent. (16) to include positive non-integral dimensional (1_p)8, p < 1 is equal to an In order to extend our result values of 8, we may define two- non-integral algebraic function on L2 as convergent in- finite series and then by subjecting it to the entropy properties given in (a 1 ),(a 2 ) and (a 3 ) exactly as above, we shall derive what may be called as "two-dimensional non-integral algebraic entropy" given by ( 17) Remark 2, B(p) = p8 + (1_p)8 - 1 , 8 > 0 and non-integer.

L)p. (t) p. (1) i=l l l l l n L: q2 p. (l)p. (t) i=l l l l Pi (t) :S max - - (1-) i Pi The left hand side of the mean value test is verified most conveniently by using the alternative representation I b t =-- A- e obtainable from (17); the symbols duced in b,A,e have the meaning intro- and (16). Again upon application of Schwarz' inequali- (1~) ty on the last term of the square root the appropriate lower bound will be established: L: v. (t)v. (1) l l It ~ ~---------------------[L: v 2 (1) +L: v~(t) +L: v~(1) +L: v~(t)l-L: v~(t) 2 l L:qfPi (t)p i (1) L: qf pf (1) l l l l n Pi (t) L: q2 p::' (1) i=l l l Pi(l) n L: qf pf (1) i=l p.

6) o V. 1 = diag (0, ••• , I are zero-matrices and ni Ia , 0, ••• 0) is the unit-matrix of order diag(B 1 , •. 8) a. B l , ... ,Bm <: i ( E. 1 1 ' ••• ,E. ln i )' 58 a and and introduce the vectors and Y , respectively by ordering 0 .. 1J Yij' respectively, lexicographically: (1. 10) n1 ;0 21 , ••• 0 2 n2 )' ; ••• ;0 1 ••• ,0 m rnnm Y In the quasi-normal case o. Sij = Yij - 3 = called the kurtosis of the random variable is Yij - 3 =Sij E ij o .. , while 1J is called its skewness. We denote by Diag as the matrix A A the diagonal-matrix with the same diagonal elements and by diag the diagonal matrix whose i-th a diagonal element equals the i-th component of the vector a • If A and B are two matrices of the same order then the Hadamardproduct of A Clearly, Diag and B A = A * is denoted by A * B and defined A * B = (aijb ij ).

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