On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?

Direct look at the celebrated "chaotic" Mandelbrot Set in Fig..\ref{Mand2} immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific {\it forest} structure. In /hep-th/9501235 a systematic algebro-geometric approach was developed to the study of generic Mandelbrot sets, but emergency of nearly ideal circles in the special case of the family $x^2+c$ was not fully explained. In the present paper the shape of the elementary constituents of Mandelbrot Set is explicitly {\it calculated}, and difference between the shapes of {\it root} and {\it descendant} domains (cardioids and circles respectively) is explained. Such qualitative difference persists for all other Mandelbrot sets: descendant domains always have one less cusp than the root ones. Details of the phase transition between different Mandelbrot sets are explicitly demonstrated, including overlaps between elementary domains and dynamics of attraction/repulsion regions. Explicit examples of 3-dimensional sections of Universal Mandelbrot Set are given. Also a systematic small-size approximation is developed for evaluation of various Feigenbaum indices.Comment: 65 pages, 30 figure

Free-Field Representation of Group Element for Simple Quantum Group

A representation of the group element (also known as ``universal ${\cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal E}_{1/q_{i(s)}}(\chi^{(s)}T_{-i(s)})\right) q^{2\vec\phi\vec H} \left(\prod_{s=1}^{d_B}\phantom.^<\ {\cal E}_{q_{i(s)}}(\psi^{(s)} T_{+i(s)})\right)$$ where $d_B = \frac{1}{2}(d_G - r_G)$, $q_i = q^{|| \vec\alpha_i||^2/2}$ and $H_i = 2\vec H\vec\alpha_i/||\vec\alpha_i||^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi,\ \vec\phi,\ \psi$form a Heisenberg-like algebra:$\psi^{(s)}\psi^{(s')} = q^{-\vec\alpha_{i(s)} \vec\alpha_{i(s')}} \psi^{(s')}\psi^{(s)}, & \chi^{(s)}\chi^{(s')} = q^{-\vec\alpha_{i(s)}\vec\alpha_{i(s')}} \chi^{(s')}\chi^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vec\phi}\psi^{(s)} = q^{\vec h\vec\alpha_{i(s)}} \psi^{(s)}q^{\vec h\vec\phi}, & q^{\vec h\vec\phi}\chi^{(s)} = q^{\vec h \vec\alpha_{i(s)}}\chi^{(s)}q^{\vec h\vec\phi}, & \\ &\psi^{(s)} \chi^{(s')} = \chi^{(s')}\psi^{(s)} & {\rm for\ any}\ s,s'.$We argue that the$d_G$-parametric ``manifold'' which$g$spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication$g \rightarrow g'\cdot g''$. The universal${\cal R}$-matrix with the property that${\cal R} (g\otimes I)(I\otimes g) = (I\otimes g)(g\otimes I){\cal R}$is given by the usual formula$${\cal R} = q^{-\sum_{ij}^{r_G}||\vec\alpha_i||^2|| \vec\alpha_j||^2 (\vec\alpha\vec\alpha)^{-1}_{ij}H_i \otimes H_j}\prod_{ \vec\alpha > 0}^{d_B}{\cal E}_{q_{\vec\alpha}}\left(-(q_{\vec\alpha}- q_{\vec\alpha}^{-1})T_{\vec\alpha}\otimes T_{-\vec\alpha}\right).$$Comment: 68 page

Realistic interatomic potential for MD simulations

The coefficients of interatomic potential of simple form Exp-6 for neon are obtained. Repulsive part is calculated ab-initio in the Hartree-Fock approximation using the basis of atomic orbitals orthogonalized exactly on different lattice sites. Attractive part is determined empirically using single fitting parameter. The potential obtained describes well the equation of state and elastic moduli of neon crystal in wide range of interatomic distances and it is appropriate for molecular dynamic simulations of high temperature properties and phenomena in crystals and liquids.Comment: MikTex v.2.1 (AMS-TEX),11 pages, 3 EPS figure

Surface EM waves on 1D Photonic Crystals

We study surface states of 1D photonic crystals using a semiclassical coupled wave theory. Both TE and TM modes are treated. We derive analytic approximations that clarify the systematics of the dispersion relations, and the roles of the various parameters defining the crystal.Comment: 7 pages, 8 figure