Abstract:
The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quntization of hydrodynamics brackets is demonstrated.

Abstract:
New algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This give the topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theories are discussed.

Abstract:
We use the spectra of Dirac type operators on the sphere $S^n$ to produce sharp $L^2$ inequalities on the sphere. These operators include the Dirac operator on $S^n$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in ${mathbb R}^n$.

Abstract:
We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed.

Abstract:
Topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation are provided.

Abstract:
We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in $\mathbb{R}^{n}$.

Abstract:
Natural images in the colour space YUV have been observed to have a non-Gaussian, heavy tailed distribution (called 'sparse') when the filter G(U)(r) = U(r) - sum_{s \in N(r)} w{(Y)_{rs}} U(s), is applied to the chromacity channel U (and equivalently to V), where w is a weighting function constructed from the intensity component Y [1]. In this paper we develop Bayesian analysis of the colorization problem using the filter response as a regularization term to arrive at a non-convex optimization problem. This problem is convexified using L1 optimization which often gives the same results for sparse signals [2]. It is observed that L1 optimization, in many cases, over-performs the famous colorization algorithm by Levin et al [3].

Abstract:
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.

Abstract:
Two results are proved for $\mathrm{nul} \mathbb{P}_A$, the dimension of the kernel of the Pauli operator $\mathbb{P}_A = \bigl\{\bbf{\sigma} \cdotp \bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2 (\mathbb{R}^3)]^2$: (i) for $|\vec{B}| \in L^{3/2} (\mathbb{R}^3),$ where $\vec{B} = \mathrm{curl} \vec{A}$ is the magnetic field, $\mathrm{nul} \ \mathbb{P}_{tA} = 0$ except for a finite number of values of $t$ in any compact subset of $(0, \infty)$; (ii) $\bigl\{\vec{B}: \mathrm{nul} \mathbb{P}_{A} = 0, | \vec{B} | \in L^{3/2}(\mathbb{R}^3) \bigr\} $ contains an open dense subset of $[L^{3/2}(\mathbb{R}^3)]^3$.