In this note we present a simple proof that 3 bugs involved in cyclic evasion, converge to an equilateral triangle configuration. The approach relies on energy type estimate which makes use of a new inequality for the triangle. The problem of the cyclic pursuit or n−bug problem is a classical one, see e.g. an article by Klamkin and Newman, " cyclic pursuit… (More)
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the trian-gulation are discussed.
Two dimensional resonators with a smooth strictly convex boundary are known to possess a whispering gallery region supporting modes concentrated near the boundary. A new class of asymmetric resonant cavities is introduced, where a whispering gallery-like region is found deep inside the resonator. The construction of such resonators is a novel application of… (More)
We consider existence and stability of dispersion-managed solitons in the two approximations of the periodic nonlinear Schrödinger (NLS) equation: (i) a dynamical system for a Gaussian pulse and (ii) an average integral NLS equation. We apply normal form transformations for finite-dimensional and infinite-dimensional Hamiltonian systems with periodic… (More)
— We study asymptotic stability properties of non-linear systems in the presence of " almost Lyapunov " functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. Nothing specific about the structure of this set is assumed besides an upper bound on its volume. We show that solutions… (More)
An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of… (More)
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy (L 2-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the… (More)
The Linear Search Problem is studied from the view point of Hamiltonian dynamics. For the specific, yet representative case of exponentially distributed position of the hidden object, it is shown that the optimal orbit follows an unstable separatrix in the associated Hamiltonian system.
We consider the nonlinear Schrodinger equation with the nonlinearity management which describes Bose-Einstein condensates under Feshbach resonance. By using an averaging theory, we derive the Hamiltonian averaged equation and compare it with other averaging methods developed for this problem. The averaged equation is used for analytical approximations of… (More)
We consider the problem of amplification of an optical signal wave with an optical pump wave when both are propagating in the fundamental mode of a single mode optical waveguide. We introduce a system of Ginzburg–Landau type and study the radiation loss due to the nonlinear interaction between the signal and the pump waves. The linear dynamics are… (More)