A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a h-pseudodifferential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

On the size of the symmetry group of a perfect code

It is shown that for every nonlinear perfect code C of length n and rank r with n−log(n+1)+1≤r≤n−1, where denotes the group of symmetries of C. This bound considerably improves a bound of Malyugin.     

Semiclassical spectral estimates for Schrödinger operators at a critical energy level. Case of a degenerate minimum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on Rn. We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.     

Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on Rn. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of R and includes the singularity in t=0. For these new contributions the asymptotic expansion involves the logarithm of the parameter h. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.     

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.     

On Macdonald's formula for the volume of a compact Lie group

We give a simple proof of Macdonald's formula for the volume of a compact Lie group. Received: May 10, 1997     

Lie group symmetry analysis of transport in porous media with variable transmissivity

We determine the Lie group symmetries of the coupled partial differential equations governing a novel problem for the transient flow of a fluid containing a solidifiable gel, through a hydraulically isotropic porous medium. Assuming that the permeability (K^∗) of the porous medium is a function of the gel concentration (c^∗), we determine a number of exact solutions corresponding to the cases where the concentration-dependent permeability is either arbitrary or has a power law variation or is a constant. Each case admits a number of distinct Lie symmetries and the solutions corresponding to the optimal systems are determined. Some typical concentration and pressure profiles are illustrated and a specific moving boundary problem is solved and the concentration and pressure profiles are displayed.     

Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry

We perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a PDE with O(2) symmetry to one with SO(2) symmetry. Taking particular care of the consequences of the continuous group action, we concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. In particular, we note that although Chossat (1993) has proved that generic perturbations cause the appearance of quasiperiodic orbits, for the simplest possible flux terms this is not the case. We compare these results with numerical simulations of a Galerkin approximation of the equations.     

Discrete Lagrange problems with constraints valued in a Lie group

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group $SO(n)$.     

On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

A natural one-parameter family of Kähler quantizations of the cotangent bundle T^∗K of a compact Lie group K, taking into account the half-form correction, was studied in [C. Florentino, P. Matias, J. Mourão, J.P. Nunes, Geometric quantization, complex structures and the coherent state transform, J. Funct. Anal. 221 (2005) 303-322]. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point of view of [S. Axelrod, S. Della Pietra, E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991) 787-902]. The BKS pairing map is a composition of (unitary) coherent state transforms of K, introduced in [B.C. Hall, The Segal-Bargmann coherent state transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103-151]. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the Kähler polarizations degenerates to the vertical real polarization, leads to the unitarity of the corresponding BKS pairing map. This is in agreement with the unitarity up to scaling (with respect to a rescaled inner product) of this pairing map, established by Hall.     

Self-similar solutions of the porous medium equation in a half-space with a nonlinear boundary condition: existence and symmetry

We find existence of a nonnegative compactly supported solution of the problem Δu=u^α in , ∂u/∂ν=u on . Moreover, we prove that every nonnegative solution with finite energy is compactly supported and radially symmetric in the tangential variables.     

Reduction by stages and the Raïs-type formula for the index of a Lie algebra with an ideal

For a quite general class of Lie algebras with a nontrivial ideal we derive a formula for the index generalizing the Raïs formula for the index of semidirect products. The method of proof of our formula is based on the so-called “symplectic reduction by stages” scheme.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi-infinite stretching sheet

We investigate the steady two-dimensional flow of an incompressible water based nanofluid over a linearly semi-infinite stretching sheet in the presence of magnetic field numerically. The basic boundary layer equations for momentum and heat transfer are non-linear partial differential equations. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations for this investigation are solved numerically using Nachtsheim–Swigert shooting iteration technique together with fourth order Runge–Kutta integration scheme. Effects of the nanoparticle volume fraction ϕ, magnetic parameter M, Prandtl number Pr on the velocity and the temperature profiles are presented graphically and examined for different metallic and non-metallic nanoparticles. The skin friction coefficient and the local Nusselt number are also discussed for different nanoparticles.     

Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration,

(0.1)     

具有双不变度量的李群中超曲面的广义Gauss映照

讨论了具有双不变度量的李群中超曲面的广义Gauss映照,并且给出广义Gauss映照是相对仿射的一些条件.     

Hamiltonian systems with a stochastic force:nonlinear versus linear,and a girsanov formula

We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process     

Lie symmetry analysis,symmetry reductions with exact solutions,and conservation laws of (2+1)-dimensional Bogoyavlenskii-Schieff equation of higher order in plasma physics

In this article, similarity reductions of the (2 $+$ 1)-dimensional Bogoyavlenskii-Schieff equation of higher order have been done by Lie group method. We have determine the geometric vector field, infinitesimal generators, symmetric groups, and commutator table of Lie algebra with the help of Lie symmetry analysis. The new close form solutions and similarity solutions of the (2 $+$ 1)-dimensional Bogoyavlenskii-Schieff equation of higher order have been determined from the reduction equations. Also, the conservation laws have been derived by invoking the new conservation theorem proposed by Ibragimov.