Outer billiard around a curvilinear triangle with a fixed diameter

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes

On the limit cycles of polynomial vector field

In this paper we study the limit cycles which can bifurcate from the periodic orbits of the center located at the origin of the quadratic polynomial differential system x˙ = −y(1+x), y˙ = x(1+x), and of the cubic polynomial differential system x˙ = −y(1−x2 −y 2 ), y˙ = x(1 − x2 − y 2 ), when we perturb them in the class of all polynomial vector fields with quadratic and cubic homogenous nonlinearities, respectively. For doing this study we use the averaging theory

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness

Weakened Markus-Yamabe Conditions for 2Dimensional Global Asymptotic Stability, Nonlinear Analalysis

Abstract. For a general 2–dimensional autonomous system ˙x = f(x), it is difficult to find easily verifiable sufficient conditions guaranteeing global asymptotic stability of an equilibrium point. This paper considers three conditions which imply global asymptotic stability for a large class of systems, weakening the so–called Markus–Yamabe condition. The new conditions are: (1) the system admits a unique equilibrium point, (2) it is locally asymptotically stable, and (3) the trace of the Jacobian matrix of f is negative everywhere. We prove that under these three conditions global asymptotic stability is obtained when the components of f are polynomials of degree two or represent a Liénard system. However, we provide examples that global asymptotic stability is not obtained under these conditions for other classes of planar differential systems

Outer billiard around a curvilinear triangle with a fixed diameter

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes