Asymptotics of large eigenvalues for a class of band matrices

We investigate the asymptotic behaviour of large eigenvalues for a class of finite difference self-adjoint operators with compact resolvent in $l^2$

Legendrian Distributions with Applications to Poincar\'e Series

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |\Lambda, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|\Lambda, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $\Lambda$, and a sequence itself has a symbol which is a half-form, $\sigma$, on $\Lambda$. We prove estimates, as $k\to\infty$, of the norm squares $\langle \Lambda, k|\Lambda, k\rangle$ in terms of $\int_\Lambda \sigma\overline{\sigma}$. More generally, we show that if $\Lambda_1$ and $\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langle\Lambda_1, k|\Lambda_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $\Lambda_1\cap\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincar\'e series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe

Molecular portrait of chronic joint diseases: Defining endotypes toward personalized medicine.

Joint diseases affect hundreds of millions of people worldwide, and their prevalence is constantly increasing. To date, despite recent advances in the development of therapeutic options for most rheumatic conditions, a significant proportion of patients still lack efficient disease management, considerably impacting their quality of life. Through the spectrum of rheumatoid arthritis (RA), psoriatic arthritis (PsA), and osteoarthritis (OA) as quintessential and common rheumatic diseases, this review first provides an overview of their epidemiological and clinical features before exploring how the better definition of clinical phenotypes has helped their clinical management. It then discusses the recent progress in understanding the diversity of endotypes underlying disease phenotypes. Finally, this review highlights the current challenges of implementing molecular endotypes towards the personalized management of RA, PsA and OA patients in the future

The DAG1 transcription factor negatively regulates the seed-to-seedling transition in Arabidopsis acting on ABA and GA levels

BACKGROUND: In seeds, the transition from dormancy to germination is regulated by abscisic acid (ABA) and gibberellins (GAs), and involves chromatin remodelling. Particularly, the repressive mark H3K27 trimethylation (H3K27me3) has been shown to target many master regulators of this transition. DAG1 (DOF AFFECTING GERMINATION1), is a negative regulator of seed germination in Arabidopsis, and directly represses the GA biosynthetic gene GA3ox1 (gibberellin 3-β-dioxygenase 1). We set to investigate the role of DAG1 in seed dormancy and maturation with respect to epigenetic and hormonal control. RESULTS: We show that DAG1 expression is controlled at the epigenetic level through the H3K27me3 mark during the seed-to-seedling transition, and that DAG1 directly represses also the ABA catabolic gene CYP707A2; consistently, the ABA level is lower while the GA level is higher in dag1 mutant seeds. Furthermore, both DAG1 expression and protein stability are controlled by GAs. CONCLUSIONS: Our results point to DAG1 as a key player in the control of the developmental switch between seed dormancy and germination

Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.Comment: 40 pages, the constants in Theorems 1.1-1.8 have been modified by a multiplicative constant 1/2 ; v.2 is a final updat

Toeplitz Quantization of K\"ahler Manifolds and gl(N)gl(N) N→∞N\to\infty

For general compact K\"ahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.Comment: 17 pages, AmsTeX 2.1, Sept. 93 (rev: only typos are corrected

Scattering Theory for Jacobi Operators with Steplike Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.Comment: 14 page

Toeplitz operators on symplectic manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page

The density of stationary points in a high-dimensional random energy landscape and the onset of glassy behaviour

We calculate the density of stationary points and minima of a $N\gg 1$ dimensional Gaussian energy landscape. We use it to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of size $L=R\sqrt{N}$ corresponds to the onset of exponential in $N$ growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures we construct a simple variational upper bound on the true free energy of the $R=\infty$ version of the problem and show that this approximation is able to recover the position of the whole de-Almeida-Thouless line.Comment: a revised and shortened version with a few typos corrected and references added. To appear in JETP Letter

On perturbations of Dirac operators with variable magnetic field of constant direction

We carry out the spectral analysis of matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.Comment: 12 page