Search results for "Manifold"

Reduction to finite dimensions of continuous systems having only a few amplified modes

2008

In the approach of Guckenheimer and Knobloch the amplitudes of trajectories on the unstable manifold 0 are the pivotal quantities. This places a certain restriction on the applicability of this approach, as only neighbourhoods of 0 of the unstable manifold of 0 are accessible, which have a one-to-one projection into their tangent at 0, the linear space spanned by the amplified modes. This restriction may be lifted, using the arc lengths of trajectories instead.

Three physical quantum manifolds from the conformal group

1987

Critical Attractor and Universality in a Renormalization Scheme for Three Frequency Hamiltonian Systems

1998

We study an approximate renormalization-group transformation to analyze the breakup of invariant tori for three degrees of freedom Hamiltonian systems. The scheme is implemented for the spiral mean torus. We find numerically that the critical surface is the stable manifold of a critical nonperiodic attractor. We compute scaling exponents associated with this fixed set, and find that they can be expected to be universal.

Presymplectic manifolds and conservation laws

2008

In this paper we make use of a new structure called seeded fibre bundle. This allows us to combine the symplectic formalism and general relativity. A theorem of existence is obtained and some examples and properties are studied.

On the general structure of gauged Wess-Zumino-Witten terms

1998

The problem of gauging a closed form is considered. When the target manifold is a simple Lie group G, it is seen that there is no obstruction to the gauging of a subgroup H\subset G if we may construct from the form a cocycle for the relative Lie algebra cohomology (or for the equivariant cohomology), and an explicit general expression for these cocycles is given. The common geometrical structure of the gauged closed forms and the D'Hoker and Weinberg effective actions of WZW type, as well as the obstructions for their existence, is also exhibited and explained.

Modular transformations of elliptic Feynman integrals

2021

We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transf…

The Poisson Bracket Structure of the SL(2, R)/U(1) Gauged WZNW Model with Periodic Boundary Conditions

2000

The gauged SL(2, R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory. For periodic boundary conditions zero modes imply non-local Poisson brackets which, nevertheless, can be represented by canonical free fields.

Combinatorial Models in the Topological Classification of Singularities of Mappings

2018

The topological classification of finitely determined map germs \(f:(\mathbb R^n,0)\rightarrow (\mathbb R^p,0)\) is discrete (by a theorem due to R. Thom), hence we want to obtain combinatorial models which codify all the topological information of the map germ f. According to Fukuda’s work, the topology of such germs is determined by the link, which is obtained by taking the intersection of the image of f with a small enough sphere centered at the origin. If \(f^{-1}(0)=\{0\}\), then the link is a topologically stable map \(\gamma :S^{n-1}\rightarrow S^{p-1}\) (or stable if (n, p) are nice dimensions) and f is topologically equivalent to the cone of \(\gamma \). When \(f^{-1}(0)\ne \{0\}\)…

APPROXIMATE INERTIAL MANIFOLDS FOR THERMODIFFUSION EQUATIONS

2004

In this paper, we consider the two dimensional equations of thermohydraulics, i.e. the coupled system of equations of fluid and temperature in the Boussinesq approximation. We construct a family of approximate Inertial Manifolds whose order decreases exponentially fast with respect to the dimension of the manifold. We give the explicit expression of the order of the constructed manifolds.

A Remark on an Overdetermined Problem in Riemannian Geometry

2016

Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let \(\varOmega \subset M\) be a bounded domain, with \(O \in \varOmega \), and consider the problem \(\varDelta _p u = -1\ \mathrm{in}\ \varOmega \) with \(u=0\ \mathrm{on}\ \partial \varOmega \), where \(\varDelta _p\) is the p-Laplacian of g. We prove that if the normal derivative \(\partial _{\nu }u\) of u along the boundary of \(\varOmega \) is a function of d satisfying suitable conditions, then \(\varOmega \) must be a geodesic ball. In particular, our result applies to open balls of \(\mathbb {R}^n\) equipped with a rotationally symmetric metr…