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On the mobility and efficiency of mechanical systems

Gershon Wolansky — 2007

ESAIM: Control, Optimisation and Calculus of Variations

It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.


Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir;  Gershon Wolansky — 2005

Journal of the European Mathematical Society

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere S 2 , on a bounded domain Ω ⊂ ℝ 2 and on all of ℝ 2 . In some cases we also address the question of existence of minimizers.

The Lazy Travelling Salesman Problem in ℝ 2

Paz Polak;  Gershon Wolansky — 2007

ESAIM: Control, Optimisation and Calculus of Variations

We study a parameter () dependent relaxation of the Travelling Salesman Problem on  ℝ 2 . The relaxed problem is reduced to the Travelling Salesman Problem as σ → 0. For increasing it is also an ordered clustering algorithm for a set of points in ℝ 2 . A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided is large enough. ...

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