3,198 research outputs found
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
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
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
Using the fact that extremum of variation of generalized action can lead to
the fractional dynamics in the case of systems with long-range interaction and
long-term memory function, we consider two different applications of the action
principle: generalized Noether's theorem and Hamiltonian type equations. In the
first case, we derive conservation laws in the form of continuity equations
that consist of fractional time-space derivatives. Among applications of these
results, we consider a chain of coupled oscillators with a power-wise memory
function and power-wise interaction between oscillators. In the second case, we
consider an example of fractional differential action 1-form and find the
corresponding Hamiltonian type equations from the closed condition of the form.Comment: 30 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
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