Fractional Generalization of Gradient Systems

We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems.Comment: 11 pages, LaTe

Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler-Lagrange equations are derived. Fractional equations of motion are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives.Comment: 21 pages, LaTe

Path Integral for Quantum Operations

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.Comment: 24 pages, LaTe

Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)

We consider a tensor product V(b)= \otimes_{i=1}^n\C^N(b_i) of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace $V(b)_\lambda \subset V(b)$ of weight $\lambda$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b)_\lambda$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b)^{sing}_\lambda$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b)^{sing}_\lambda$ with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map. These results we announced earlier in relation with a description of the quantum equivariant cohomology of the cotangent bundle of a partial flag variety and a description of commutative subalgebras of the group algebra of a symmetric group.Comment: Latex, 23 pages, misprints correcte

Phase-Space Metric for Non-Hamiltonian Systems

We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the time-dependent skew-symmetric phase-space metric that satisfies the Jacobi identity. The example of non-Hamiltonian systems with linear friction term is considered.Comment: 12 page