especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus T n = S 1...

torus so that the neighborhood of any point is more than two-dimensional. At each point of the torus there is a direction of motion that remains on the...

Lindemann–Weierstrass theorem Linear flow on the torus Schanuel's conjecture Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical...

torus/Irrational cable on a torus Knot (mathematics) Linear flow on the torus Space-filling curve Torus knot Wild knot The following topologies are a known source...

parallel lines yields a 1-dimensional foliation of the n-torus Rn/Zn associated with the linear flow on the torus. A flat bundle has not only its foliation by...

Linear density Linear dynamical system Linear elasticity Linear energy transfer Linear entropy Linear flow on the torus Linear motion Linear particle accelerator...

clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general...

such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems...

then the invariant d {\displaystyle d} -torus T d {\displaystyle {\mathcal {T}}^{d}} ( d ≥ 2 {\displaystyle d\geq 2} ) is called a KAM torus. The d = 1...

geometric analysis, the Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain partial...

compact torus. It has been shown that every principal torus bundle over a torus is of this form, see. More generally, a compact nilmanifold is a torus bundle...

potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of...

incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an Nt -torus if there are...

obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic...

cover the torus. Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by...

essential map of a torus, then M admits an essential embedding of either a torus or an annulus The JSJ decomposition, also known as the toral decomposition...

connected. In the group topology, the small open sets are single line segments on the surface of the torus and H is locally path connected. The example shows...

proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind. The problem of classifying manifolds that admit...

generated by external magnets to confine plasma in the shape of an axially symmetrical torus. The tokamak is one of several types of magnetic confinement...

angular and linear momentum, energy, and mass, with it. In the dynamics of fluid, a vortex is fluid that revolves around the line of flow. This flow of fluid...

The magnetic field lines coil loosely around a center torus. They coil outwards. Near the plasma edge, the toroidal magnetic field reverses and the field...

a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced...

of the torus), and these interpretations can also be viewed in light of the general theory of SL(2, R). Elements of PSL(2, R) are homographies on the real...

The Madison Symmetric Torus (MST) is a reversed field pinch (RFP) physics experiment with applications to both fusion energy research and astrophysical...

is a torus-shaped vortex in a fluid; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant...

dimensions other than the sphere that stay self-similar as they contract to a point under the mean-curvature flow, including the Angenent torus. A simple example...

\theta } if p {\displaystyle p} is not singular. On a flat torus the geodesic flow in a given direction has the property that it is either periodic or ergodic...

and connected, it is diffeomorphic to the N-torus T n {\displaystyle T^{n}} . There exist (local) coordinates on L c {\displaystyle L_{c}} ( θ 1 , ⋯ ,...

across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of...

the tangent bundle. Also called a vector field. Tangent space Thom space Torus Transversality – Two submanifolds M {\displaystyle M} and N {\displaystyle...