Carter-Saito theorem for virtual knots

In the article we construct new non-trivial state-sum invariants of virtual knots and links using a generalization of the Carter-Saito theorem for classical knots

Two-Matrix model with ABAB interaction

Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitean matrices A and B with the action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4) -{\beta\over 2} \tr(AB)^2. This model can be mapped onto a special case of the 8-vertex model on dynamical planar graphs. The solution is parametrized in terms of elliptic functions. A phase transition is found: the critical point is a conformal field theory with central charge c=1 coupled to 2D quantum gravity.Comment: harvmac, 24 pages, 5 figures (1 color figure

On-Shell Description of Stationary Flames

The problem of non-perturbative description of stationary flames with arbitrary gas expansion is considered. On the basis of the Thomson circulation theorem an implicit integral of the flow equations is constructed. With the help of this integral, a simple explicit expression for the vortex mode of the burnt gas flow near the flame front is obtained. Furthermore, a dispersion relation for the potential mode at the flame front is written down, thus reducing the initial system of bulk equations and jump conditions for the flow variables to a set of integro-differential equations for the flame front position and the flow velocity at the front. The developed approach is applied to the case of thin flames. Finally, an asymptotic expansion of the derived equations is carried out in the case \theta\to 1 where \theta is the gas expansion coefficient, and a single equation for the front position is obtained in the second post-Sivashinsky approximation. It is demonstrated, in particular, how the well-known problem of correct normalization of the front velocity is resolved in the new approach. It is verified also that in the first post-Sivashinsky approximation, the new equation reduces to the Sivashinsky-Clavin equation corrected according to Cambray and Joulin. Analytical solutions of the derived equations are found, and compared with the results of numerical simulations.Comment: 22 pages, 4 figure

Analytical treatment of 2D steady flames anchored in high-velocity streams

The problem of burning of high-velocity gas streams in channels is revisited. Previous treatments of this issue are found to be incomplete. It is shown that despite relative smallness of the transversal gas velocity, it plays crucial role in determining flame structure. In particular, it is necessary in formulating boundary conditions near the flame anchor, and for the proper account of the flame propagation law. Using the on-shell description of steady anchored flames, a consistent solution of the problem is given. Equations for the flame front position and gas-velocity at the front are obtained. It is demonstrated that they reduce to a second-order differential equation for the front position. Numerical solutions of the derived equations are found.Comment: 15 pages, 6 figure