5,778 research outputs found
Local perturbations of conservative -diffeomorphisms
A number of techniques have been developed to perturb the dynamics of
-diffeomorphisms and to modify the properties of their periodic orbits.
For instance, one can locally linearize the dynamics, change the tangent
dynamics, or create local homoclinic orbits. These techniques have been crucial
for the understanding of dynamics, but their most precise forms have
mostly been shown in the dissipative setting. This work extends these results
to volume-preserving and especially symplectic systems. These tools underlie
our study of the entropy of -diffeomorphisms in (arxiv:1606.01765). We
also give an application to the approximation of transitive invariant sets
without genericity assumptions.Comment: 31 pages, companion to the paper Entropy of C1 diffeomorphisms
without a dominated splitting (arxiv:1606.01765
Lambda Models From Chern-Simons Theories
In this paper we refine and extend the results of arXiv:1701.04138, where a
connection between the superstring lambda model on
and a double Chern-Simons (CS) theory on based on the
Lie superalgebra was suggested, after introduction of
the spectral parameter . The relation between both theories mimics the
well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature.
All the statements are now valid in the strong sense, i.e. valid on the whole
phase space, making the connection between both theories precise. By
constructing a -dependent gauge field in the 2+1 Hamiltonian CS theory it is
shown that: i) by performing a symplectic reduction of the CS theory the
Maillet algebra satisfied by the extended Lax connection of the lambda model
emerges as a boundary current algebra and ii) the Poisson algebra of the
supertraces of -dependent Wilson loops in the CS theory obey some sort of
spectral parameter generalization of the Goldman bracket. The latter algebra is
interpreted as the precursor of the (ambiguous) lambda model monodromy matrix
Poisson algebra prior to the symplectic reduction. As a consequence, the
problematic non-ultralocality of lambda models is avoided (for any value of the
deformation parameter ), showing how the lambda model
classical integrable structure can be understood as a byproduct of the
symplectic reduction process of the -dependent CS theory.Comment: Published version+Erratum (of typos), 57 page
Classification of topological insulators and superconductors in three spatial dimensions
We systematically study topological phases of insulators and superconductors
(SCs) in 3D. We find that there exist 3D topologically non-trivial insulators
or SCs in 5 out of 10 symmetry classes introduced by Altland and Zirnbauer
within the context of random matrix theory. One of these is the recently
introduced Z_2 topological insulator in the symplectic symmetry class. We show
there exist precisely 4 more topological insulators. For these systems, all of
which are time-reversal (TR) invariant in 3D, the space of insulating ground
states satisfying certain discrete symmetry properties is partitioned into
topological sectors that are separated by quantum phase transitions. 3 of the
above 5 topologically non-trivial phases can be realized as TR invariant SCs,
and in these the different topological sectors are characterized by an integer
winding number defined in momentum space. When such 3D topological insulators
are terminated by a 2D surface, they support a number (which may be an
arbitrary non-vanishing even number for singlet pairing) of Dirac fermion
(Majorana fermion when spin rotation symmetry is completely broken) surface
modes which remain gapless under arbitrary perturbations that preserve the
characteristic discrete symmetries. In particular, these surface modes
completely evade Anderson localization. These topological phases can be thought
of as 3D analogues of well known paired topological phases in 2D such as the
chiral p-wave SC. In the corresponding topologically non-trivial and
topologically trivial 3D phases, the wavefunctions exhibit markedly distinct
behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap
functions are included in the dynamics, the SC phases with non-vanishing
winding number possess non-trivial topological ground state degeneracies.Comment: 20 pages. Changed title, added two table
Structure Preserving Model Reduction of Parametric Hamiltonian Systems
While reduced-order models (ROMs) have been popular for efficiently solving
large systems of differential equations, the stability of reduced models over
long-time integration is of present challenges. We present a greedy approach
for ROM generation of parametric Hamiltonian systems that captures the
symplectic structure of Hamiltonian systems to ensure stability of the reduced
model. Through the greedy selection of basis vectors, two new vectors are added
at each iteration to the linear vector space to increase the accuracy of the
reduced basis. We use the error in the Hamiltonian due to model reduction as an
error indicator to search the parameter space and identify the next best basis
vectors. Under natural assumptions on the set of all solutions of the
Hamiltonian system under variation of the parameters, we show that the greedy
algorithm converges with exponential rate. Moreover, we demonstrate that
combining the greedy basis with the discrete empirical interpolation method
also preserves the symplectic structure. This enables the reduction of the
computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy,
and stability of this model reduction technique is illustrated through
simulations of the parametric wave equation and the parametric Schrodinger
equation
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure
The Schur multiplier of finite symplectic groups
We show that the Schur multiplier of is
, when is divisible by 4.Comment: Bull. Soc. Math. France, to appea
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