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In this note we provide a natural way of defining exponential coordinates on the class of probabilities on the set Ω = [1,n] or on P = {p = (p1, ..., pn) ∈ Rn| pi > 0; Σi=1n pi = 1}. For that we have to regard P as a projective space and the exponential coordinates will be related to geodesic flows in Cn.
In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...
We first prove an extremal property
of the infinite Fibonacci*
word f: the family of the palindromic prefixes
{hn | n ≥ 6} of f
is not only a circular code but “almost” a comma-free one
(see Prop. 12 in Sect. 4).
We also extend to a more general situation
the notion of a necklace introduced
for the study of trinucleotides codes on the genetic alphabet,
and we present a hierarchy
relating two important classes of codes,
the comma-free codes and the circular ones.
In a series of papers many Boolean functions with good cryptographic properties were constructed using number-theoretic methods. We construct a large family of Boolean functions by using polynomials over finite fields, and study their cryptographic properties: maximum Fourier coefficient, nonlinearity, average sensitivity, sparsity, collision and avalanche effect.
We prove a kind of logarithmic Sobolev inequality claiming that the mutual free Fisher information dominates the microstate free entropy adapted to projections in the case of two projections.
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