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Note on the classical solutions of Friedmanns equation

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 Added by Domingos Soares
 Publication date 2010
  fields Physics
and research's language is English




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Graphical representations of classical Friedmanns models are often misleading when one considers the age of the universe. Most textbooks disregard conceptual differences in the representations, as far as ages are concerned. We discuss the details of the scale-factor versus time function for Friedmanns solutions in the time range that includes the ages of model universes.



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begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
This paper deals with necessary and sufficient condition for consistency of the matrix equation $AXB = C$. We will be concerned with the minimal number of free parameters in Penroses formula $X = A^(1)CB^(1) + Y - A^(1)AYBB^(1)$ for obtaining the general solution of the matrix equation and we will establish the relation between the minimal number of free parameters and the ranks of the matrices A and B. The solution is described in the terms of Rohdes general form of the {1}-inverse of the matrices A and B. We will also use Kronecker product to transform the matrix equation $AXB = C$ into the linear system $(B^T otimes A)vecX = vec C$.
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We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfeld classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to $n$. We also give examples of non-boundary solutions for the classical simple Lie algebras.
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