Let D be the closed unit disk. We study the Hurwitz numbers corresponding to the coverings of D whose only multiple critical value lies on the boundary of D and find differential equations describing the generating function of these numbers.
Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdos measure is proposed. It is shown that, for certain values of the Bernulli parameter, the algorithm gives the Hausdorff dimension of an Erdos measure to 15 decimal places.
The deﬁnition of a quantum Markov state was given by Accardi. For the classical case, this deﬁnition gives hidden Markov measures, which, generally speaking, are not Markov measures. We can use a nonnegative transfer matrix to deﬁne a Markov measure. We use a positive semideﬁnite transfer matrix and select a class of quantum Markov states (in the sense of Accardi) on the quasilocal C∗-algebras. An entangled quantum Markov state in the sense of Accardi and Fidaleo is a quantum Markov state in our sense. For the case where the transfer matrix has rank 1, we calculate the eigenvalues and the eigenvectors of the density matrices determining the quantum Markov state. The sequence of von Neumann entropies of the density matrices of this state is bounded.
A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra sl2, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. In the present paper we give a formula for the values of the sl2 weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts. Our main computational tool is the Chmutov–Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the sl2 weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S. K. Lando on the values of the weight system sl2 at the projections of chord diagrams on the space of primitive elements.
Let G be a finite Abelian group acting (linearly) on space ℝn and, therefore, on its complexification ℂn, and let W be the real part of the quotient ℂn/G (in the general case, W ≠ ℝn/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (ℝn, 0) by the subspace of forms divisible by the 1-form under consideration.