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Symmetries of control systems

Alexey Samokhin (1996)

Banach Center Publications

Symmetries of the control systems of the form $u_{t} = f (t, u, v)$ , $u \in ℝ^{n}$ , $v \in ℝ^{m}$ are studied. Some general results concerning point symmetries are obtained. Examples are provided.

Symmetries of the nonlinear Schrödinger equation

Benoît Grébert, Thomas Kappeler (2002)

Bulletin de la Société Mathématique de France

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\dots < λ_{k}^{-} \leq λ_{k}^{+} < λ_{k + 1}^{-} \leq \dots$ of a Zakharov-Shabat operator is symmetric,i.e. $λ_{k}^{\pm} = - λ_{- k}^{\mp}$ for all $k$ , if and only if the sequence ${(γ_{k})}_{k \in ℤ}$ of gap lengths, $γ_{k} : = λ_{k}^{+} - λ_{k}^{-}$ , is symmetric with respect to $k = 0$ .

Symmetry group analysis and invariant solutions of hydrodynamic-type systems.

Sheftel, M.B. (2004)

International Journal of Mathematics and Mathematical Sciences

Symmetry in Architecture

Kim Williams (1999)

Visual Mathematics

Symmetry operators for the Fokker-Planck-Kolmogorov equation with nonlocal quadratic nonlinearity.

Shapovalov, Alexander V., Rezaev, Roman O., Trifonov, Andrey Yu. (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions.

Bracken, Paul (2005)

International Journal of Mathematics and Mathematical Sciences

Symplectic Capacities in Manifolds

Alfred Künzle (1997)

Banach Center Publications

Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.