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Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal N}\rightarrow\infty$. We prove that $[N(L) - \langle N(L)\rangle]/\sqrt{\log L}$ has a Gaussian distribution when $L\rightarrow\infty$. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function. \noindent PACS nos. 05.45.+b, 03.65.-wComment: 13 page

Inventiones mathematicae manuscript No.

We show that the tritronquée solution yt of the Painlevé equation P I that behaves algebraically for large z with arg z = π/5, is analytic in a region containing the sector z = 0, arg z ∈ − 3π 5 , π and the disk z : |z| < 37 20 . This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. The method, building on a technique developed in [4], is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than 1/100 rigorous error bounds. arXiv:1209.1009v2 [math.CA] 23 Oct 2014 24 O. COSTIN 1 , M. HUANG 2 , S. TANVEER 1 8.1. Values of intermediate constants for ρ = 3. The numerical values of these constants might be helpful to the reader who would like to double-check the estimates. These are: J M = 0.282580···, jm = 0.64374···, Y 1,M = 1.16314···, Y 1,R,M = 0.132618···, E M = 0.0490292··· z 2,R,M = 0.54226···, z 2,M = 0.91863···, Mq = 0.066702···, M L,q = 0.075708···, V M = 0.2239···, T M = 0.0385··· M 1 = 1.13838···, M 2 = 0.04303···, M 3 = 0.28346···, M 4 = 0.45227···, M 5 = 0.05430···, M 6 = 0.00231···, M 7 = 0.02018··· 9. Acknowledgments

We prove that a model atom having one bound state will be fully ionized by a time periodic potential of arbitrary strength r and frequency ω. Starting with the system in the bound state, the survival probability is for small r given by e −Γt for times of order Γ −1 ∼ r −2n , where n is the minimum number of "photons" required for ionization (with large modifications at resonances). For late times the decay is like t −3 with the power law modulated by oscillations. As r increases the time over which there is exponential decay becomes shorter and the power law behavior starts earlier. Results are for a parametrically excited one dimensional system with zero range potential but comparison with analytical works and with experiments indicate that many features are general.

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The reason to separate out the second and third terms on the right-hand side of (1.1) is that they play a special role in the asymptotic behavior of the solutions.)The functions ξ → f 0 (ξ −1 ) and (ξ, y) → g(ξ −1 , y) are taken to be analytic for small arguments. The normalization can be chosen so that f 0 (x) = O(x −2 ) for large x, and, by construction, we have g(x, y) = O(|y| 2 , x −2 y).