Description of the Markov chain using the producing function

In article the way of creation of an assumed function of transformation with the help of a generating function and also a method of use of characteristic numbers and vectors for creation of a matrix which elements well describe conditions of process at any moment is described. This approach differs from preceding that, having used a concept of characteristic numbers and characteristic vectors of a matrix of the transitional probabilities, it is possible to simplify considerably calculation of the elements characterizing process. In article methods of stochastic model operation, ways of the description of a generating function, the solution of matrixes of the equations by means of characteristic numbers and vectors are used. Using properties of a generating function, made “dictionary” of z-transformations which helped to define an assumed function of transformation. The generating function of a vector was applied to a research of behavior of a vector of absolute probabilities which elements represent stationary probabilities. For definition of degree of a matrix of transition of probabilities used a concept of characteristic numbers and characteristic vectors of the transitional probabilities. Determined by such way an unlimited set of latent vectors of which made matrixes which describe a condition of a system at any moment. Reception of definition of latent vectors in more difficult examples which is that along with required coefficients of secular equations the system of auxiliary matrixes and an inverse matrix is under construction is also described.

Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

Rational cuspidal curves in a moving family of ℙ2

In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).