Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, f, s) = p2/2+ cos q - 1 + I2/2 + h(q, f, s; e) — proving that for any small periodic perturbation of the form h(q, f, s; e) = e cos q (a00 + a10 cosf + a01 cos s) (a10a01 ¿ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ p/2µ, µ = a10/a01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any µ). The bifurcations of the scattering map are also studied as a function of µ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.Peer ReviewedPostprint (published version

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems

International study on inter-reader variability for circulating tumor cells in breast cancer

Introduction: Circulating tumor cells (CTCs) have been studied in breast cancer with the CellSearch® system. Given the low CTC counts in non-metastatic breast cancer, it is important to evaluate the inter-reader agreement.Methods: CellSearch® images (N = 272) of either CTCs or white blood cells or artifacts from 109 non-metastatic (M0) and 22 metastatic (M1) breast cancer patients from reported studies were sent to 22 readers from 15 academic laboratories and 8 readers from two Veridex laboratories. Each image was scored as No CTC vs CTC HER2- vs CTC HER2+. The 8 Veridex readers were summarized to a Veridex Consensus (VC) to compare each academic reader using % agreement and kappa (κ) statistics. Agreement was compared according to disease stage and CTC counts using the Wilcoxon signed rank test.Results: For CTC definition (No CTC vs CTC), the median agreement between academic readers and VC was 92% (range 69 to 97%) with a median κ of 0.83 (range 0.37 to 0.93). Lower agreement was observed in images from M0 (median 91%, range 70 to 96%) compared to M1 (median 98%, range 64 to 100%) patients (P < 0.001) and from M0 and <3CTCs (median 87%, range 66 to 95%) compared to M0 and ≥3CTCs samples (median 95%, range 77 to 99%), (P < 0.001). For CTC HER2 expression (HER2- vs HER2+), the median agreement was 87% (range 51 to 95%) with a median κ of 0.74 (range 0.25 to 0.90).Conclusions: The inter-reader agreement for CTC definition was high. Reduced agreement was observed in M0 patients with low CTC counts. Continuous training and independent image review are required

New insights into the genetic etiology of Alzheimer's disease and related dementias

Characterization of the genetic landscape of Alzheimer's disease (AD) and related dementias (ADD) provides a unique opportunity for a better understanding of the associated pathophysiological processes. We performed a two-stage genome-wide association study totaling 111,326 clinically diagnosed/'proxy' AD cases and 677,663 controls. We found 75 risk loci, of which 42 were new at the time of analysis. Pathway enrichment analyses confirmed the involvement of amyloid/tau pathways and highlighted microglia implication. Gene prioritization in the new loci identified 31 genes that were suggestive of new genetically associated processes, including the tumor necrosis factor alpha pathway through the linear ubiquitin chain assembly complex. We also built a new genetic risk score associated with the risk of future AD/dementia or progression from mild cognitive impairment to AD/dementia. The improvement in prediction led to a 1.6- to 1.9-fold increase in AD risk from the lowest to the highest decile, in addition to effects of age and the APOE ε4 allele

Slope-changing solutions of elliptic problems on Rn\R^n.

We consider the problem $-\Delta u+F_u(x,u)=0$ on $\R^n$, where $F$ is a smooth function periodic of period 1 in all its variables. We are going to find a nondegeneracy condition on $F$ for which the following holds. If we are given a sequence of positive integers $\{ \tilde N_i \}_{i\in\Z}$ and a sequence \{ \a_i \}_{i\in\Z} of real numbers (the slopes), then we shall find an increasing sequence $\{ Q_i\}$ of integers and a solution $u$ which is entire, periodic in $(x_2,\dots,x_n)$ and which is close to the plane \a_1(x_1-Q_i)+u(Q_i,0,\dots,0) for $x_1\in [Q_i,Q_i+\tilde N_i]$

A cost on paths of measures which induces the Fokker-Planck equation.

In [12], Feng and Nguyen (2012) define a cost on curves of measures which is finite exactly on the curves which solve a Fokker–Planck equation with L2 drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen (2012)

The Aubry set for a version of the Vlasov equation.

We check that several properties of the Aubry set, first proven for finite-dimensional Lagrangians by Mather and Fathi, continue to hold in the ase of the infinitely many interacting particles of the Vlasov equation