Hopf Algebras of Linear Recurring Sequences over Rings and Modules

The module of linear recurring sequences over a commutative ring R can be considered as a Hopf algebra dual to the polynomial Hopf algebra over R. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and k-sequences.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 113–148, 2003.     

The stability of the steady rotation of a system of three equidistant vortices outside a circle

A complete non-linear analysis of the stability of the steady rotation of three point vortices, placed in a plane at the vertices of a regular triangle outside a circular domain is carried out using the results of the Kolmogorov–Arnold–Moser theory. All the resonances of up to fourth order inclusive encountered here are listed and studied. The investigations of Havelock who solved this problem in a linear formulation are thereby completed.     

Effective erosion coefficient and plasma velocity limitation in the channel of an electromagnetic rail accelerator

The concept of an effective erosion coefficient, which takes into account the capture and entrainment in motion (by accelerated plasma) of only part of the erosion mass lost by rail accelerator electrodes, is introduced to describe the plasma acceleration dynamics in the channel of an electromagnetic rail accelerator. This parameter is determined from a comparison of the experimental and calculated plasma velocities at the stage of velocity saturation. The plasma velocity is calculated using a model that takes into account the pressure force of a shock-compressed gas and the deceleration force that appears during the capture of the erosion mass by a plasma piston. The ratio of the captured mass to the mass lost by the electrodes is found to depend on the current; for copper, this ratio is 1/4–2/3. The effective erosion coefficient is 0.6–0.7 mg/C at a current of ～40 kA.     

Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system

It is well known that equilibrium in a cosymmetric system in the general position is a member of a one-parameter family. In the present paper the Lyapunov-Schmidt method and the method of the central manifold are used to analyze bifurcations of such a family of equilibria as well as internal bifurcations: transitions of the type focus-node, node-saddle, and so on during motion along the family. A series of scenarios of branching of families of equilibria and the change in the structure of their arcs, consisting of equilibria of the same type, is described. Bifurcations of stable and unstable arcs, coalescence and decomposition of families of equilibria, bifurcation of the loss of smoothness by the family of equilibria, and branching of a small equilibrium cycle from a corner point of the family of equilibria are investigated in detail. The variability of the spectrum along a family gives rise to a variety of new phenomena that are not encountered in the classical case of an isolated equilibrium or in bifurcations of families of equilibria of a system with symmetry. They include protraction with respect to the branching parameter of the family of equilibria, Lyapunov instability of a family of equilibria with the attraction properties being preserved, and the appearance and disappearance of new stable and unstable arcs on the family of equilibria. (c) 2000 American Institute of Physics.     

On Stability of Thomson’s Vortex <Emphasis Type="Italic">N</Emphasis>-gon in the Geostrophic Model of the Point Bessel Vortices

A stability analysis of the stationary rotation of a system of N identical point Bessel vortices lying uniformly on a circle of radius R is presented. The vortices have identical intensity Γ and length scale γ^?1 > 0. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for N = 2,..., 6 are studied sequentially. The case of odd N = 2?+1 ≥ 7 vortices and the case of even N = 2_n ≥ 8 vortices are considered separately. It is shown that the (2? + 1)-gon is exponentially unstable for 0 < γR<R_*(N). However, this (2? + 1)-gon is stable for γR ≥ R_*(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N = 2n ≥ 8 vortex 2n-gon is exponentially unstable for R > 0.     

Parameters of an erosion carbon plasma in the channel of a railgun

The working current dependences of the thermodynamic and electrophysical parameters of a free plasma piston moving with a near-maximal velocity in the channel of an electromagnetic rail launcher with graphite electrodes are obtained. The composition and weight of the plasma depend on the degree of electrode erosion due to discharge current passage (i = 40–80 kA). It is shown that the mean temperature of the plasma piston only slightly depends on the plasma mean pressure and plasma piston weight and increases with current by a near-power law. The measured values of the maximal velocity of the plasma piston front are compared with the calculated value of the sound velocity inside the piston. With the working current and cross-sectional area of the channel fixed, the initial gas density in the channel is found to influence the ratio of the piston maximal velocity to the sound velocity in the plasma. If the initial gas density is low (lower than some critical value), the maximal velocity of the plasma piston front exceeds the sound velocity in the plasma.     

On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

Journal of Nonlinear Science - A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a...     

On the stability of discrete tripole,quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

A two-layer quasigeostrophic model is considered in the f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.A limiting case of a homogeneous fluid is also considered.The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O _G , formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.