Complexity spectrum of some discrete dynamical systems

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $3 \times 3$ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the $9!$ possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil

Identification of the donor in GRS 1915+105

We report on the results of medium-resolution spectroscopy of GRS 1915+105 in the H and K band using the 8m VLT at ESO. We clearly identify absorption bandheads from CO12 and CO13. Together with other features this results in a classification of the donor as a K-M III star.Comment: 4 pages, Kluwer style, to appear in Astrophys. Space Sci. Rev, Proc. of 3rd Microquasar workshop, Granada, Sep. 2000, Eds. A.J. Castro-Tirado, J. Greiner, J.M. Paredes; all proceedings contr. available at http://www.aip.de/~jcg/granada.htm

Painleve versus Fuchs

The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit

The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q>4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeroes to be a circle. This locus, together with the density of zeroes is then shown to be sufficient to recover the singular form of the thermodynamic functions in the thermodynamic limit.Comment: 10 pages, 0 figures, LaTeX. Paper expanded and 2 references added clarifying duality relationships between discontinuities in higher cumulant

The peculiar molecular envelope around the post-AGB star IRAS 08544--4431

Circumbinary disks have been hypothesized to exist around a number of binary post-AGB stars. Although most of the circumbinary disks have been inferred through the near IR excess, a few of them are strong emitters of molecular emission. Here we present high angular resolution observations of the emission of $^{12}$CO and its isotopomer $^{13}$CO J=2--1 line from the circumstellar envelope around the binary post-AGB star IRAS 08544$-$4431, which is one of the most prominent members of this class of objects. We find that the envelope is resolved in our observations and two separate components can be identified: (a) a central extended and strong component with very narrow linewidth between 2 - 6 \kms; (b) a weak bipolar outflow with expansion velocity up to 8 \kms. The central compact component possesses low and variable $^{12}$CO/$^{13}$CO J=2--1 line ratio, indicating optically thick emission of the main isotope. We estimate a molecular gas mass of 0.0047 M$_\odot$ for this component based on the optically thinner $^{13}$CO J=2--1 line. We discuss the relation of the molecular envelope and the circumbinary disk inferred from near IR excess and compare with other known cases where the distribution of molecular gas has been imaged at high angular resolution.Comment: 14 pages, 4 figures. Accepted for publication in Astrophysical Journa

The saga of the Ising susceptibility

We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome $q$ and the modulus $k$ are compared and contrasted. The $\lambda$ generalized correlations $C(M,N;\lambda)$ are defined and explicitly computed in terms of theta functions for $M=N=0,1$.Comment: 19 pages, 1 figur