Optimal growth for linear processes with affine control

We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider $\dot x_\alpha(t) = (G + \alpha(t) F)x_\alpha(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. In the case of constant control $\alpha(t)\equiv \alpha$, we show the existence of an optimal Perron eigenvalue with respect to varying $\alpha$ under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls $\alpha(t)$. Finally we prove the existence of an eigenvalue (in the generalized sense) for the optimal control problem. The proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e] concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the relations between the three eigenvalues. Surprisingly enough, the three eigenvalues appear to be numerically the same

On monochromatic arm exponents for 2D critical percolation

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.Comment: Published in at http://dx.doi.org/10.1214/10-AOP581 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Fibonacci Knots

We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when n \not\equiv 0 \Mod 4 and $(n,j) \neq (3,3),$ the Fibonacci knot \cF_j^{(n)} is not a Lissajous knot.Comment: 7p. Sumitte

Chebyshev Knots

A Chebyshev knot is a knot which admits a parametrization of the form $x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + \phi),$ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and \phi \in \RR . Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $\phi = 0.$ We also show that every knot is a Chebyshev knot.Comment: To appear in Journal of Knot Theory and Ramification

Solving the Triangular Ising Antiferromagnet by Simple Mean Field

Few years ago, application of the mean field Bethe scheme on a given system was shown to produce a systematic change of the system intrinsic symmetry. For instance, once applied on a ferromagnet, individual spins are no more equivalent. Accordingly a new loopwise mean field theory was designed to both go beyond the one site Weiss approach and yet preserve the initial Hamitonian symmetry. This loopwise scheme is applied here to solve the Triangular Antiferromagnetic Ising model. It is found to yield Wannier's exact result of no ordering at non-zero temperature. No adjustable parameter is used. Simultaneously a non-zero critical temperature is obtained for the Triangular Ising Ferromagnet. This simple mean field scheme opens a new way to tackle random systems.Comment: 14 pages, 2 figure

Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $\eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $\epsilon$, which induces a small divergence and the time step $\delta$t tend to zero with a proportionality constraint $\epsilon$ = $\lambda$$\delta$t. Finally, when $\eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $\epsilon$ qui assure une faible divergence et le pas de temps $\delta$t tendent vers 0 avec une contrainte de proportionalit{\'e} $\epsilon$ = $\lambda$$\delta$t. Finalement nous montrons que leprob{\`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param etre de p{\'e}nalisation $\eta$ tend vers 0

Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.Comment: 8 page