We compute the distribution function of single-level curvatures, P (k), for a tight binding model with site disorder, on a cubic lattice. In metals P (k) is very close to the predictions of the random-matrix theory (RMT). In insulators P (k) has a logarithmically-normal form. At the Anderson localizationdelocalization transition P (k) fits very well the proposed novel distribution P (k) ∝ (1 + k µ ) 3/µ with µ ≈ 1.58, which approaches the RMT result for large k and is non-analytical at small k. We ascribe such a non-analiticity to the spatial multifractality of the critical wave functions.PACS numbers: 71.30.+h, 72.15.Rn, 05.60.+w An important consequence of the one-parameter scaling theory [1] of the Anderson transition is the existence of three universality classes for the energy level statistics of disordered systems. In the metallic regime they are described by the strongly correlated Wigner-Dyson (WD) distribution, while in the insulating regime they follow an uncorrelated Poissonic law. This difference has its fundamental origin in the underlying nature of the corresponding eigenstates, being extended and strongly overlapping in the first case but localized in the second.The properties of the third universal statistics, describing the spectral correlations at the critical point, have been only recently the subject of intense investigation, both analytical and numerical. Using the results of numerical simulations, Shklovskii et al.[2] suggested that the spacing distribution function P (s) has the WD form P (s) ∼ s for small s and the Poissonic tail P (s) ∼ e −s for large s. Further analytical investigations [3,4] showed that the two-level correlation function R(s) in the critical region has a novel power-law asymptotic decay R(s) = −c/s 2−γ with a nontrivial exponent γ = 1 − 1/νd. Here ν is the critical exponent of the correlation/localization length ξ, which depends on the dimensionality d. Thus the two-level correlation function in the critical region resembles qualitatively the WD function which applies to the metallic phase. On the other hand, the level number variance Σ 2 (N ) = (δN ) 2 in an energy strip of width N ∆, (∆ is the mean level spacing), still contains a dominant Poissonic term, [5][6][7] linear in N , which is typical for insulators.It can be shown [3] that the Poissonic term in Σ 2 (N ) is only possible if a normalization sum rule on R(s) is violated in the thermodynamic limit (TL)

Recently it has been found that different physical systems driven by identical random noise behave exactly identical after a long time. It is also suggested that this is an outcome of finite precision in numerical experiments. Here we show that the origin of the non-chaotic behavior lies in the structural instability of the attractor of these systems which changes to a stable fixed point for strong enough drive. We see this to be true in all the systems studied in literature. Thus we affirm that in chaotic systems, synchronization can not occur only by addition of noise unless the noise destroys the strange attractor and the system is no longer chaotic.Of late there has been considerable attention drawn to the problem in which the behavior of an ensemble of initial conditions obeying the same laws of motion and driven by an identical sequence of random forces start following the same random trajectory asymptotically. This is observed in several distinct cases. Fahy and Hamann [1] observed that if a particle obeying Newton's equations (without friction) in a potential V is stopped at regular time intervals τ and all its velocity components are reset to random values, then for a given sequence of random values asymptotic trajectories become identical for all initial conditions. This a stronger statement than the observation that the statistical distribution of trajectories become independent of initial conditions. This phenonomenon occurs for the choice of a time interval τ lower than a threshold value τ c . They remark that this is common for all bounded systems.Another observation of a similar phenomenon is due to Maritan and Banavar [2]. In this letter, they show that a pair of chaotic systems driven by identical noise of sufficient strength have identical trajectories asymptotically.

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A method for the preparation of an asymmetric superbenzene is presented. The synthesis proceeds through the 12-fold cyclodehydrogenation of a polyaromatic precursor obtained by successive Diels-Alder additions of cyclopentadienones and alkynes.

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The triton angular distribution for the 12C(7Li,t)16O* reaction is measured at 20 MeV, populating discrete states of 16O. Continuum discretized coupled reaction channel  calculations are used to to extract the alpha spectroscopic properties of 16O states instead of the distorted wave born approximation theory to include the effects of breakup on the transfer process. The alpha reduced width, spectroscopic factors and the asymptotic normalization constant (ANC) of 16O states are extracted. The error in the spectroscopic factor is about 35% and in that of the ANC about 27%.

The level curvature distribution function is studied both analytically and numerically for the case of T-breaking perturbations over the orthogonal ensemble. The leading correction to the shape of the curvature distribution beyond the random matrix theory is calculated using the nonlinear supersymmetric sigma-model and compared to numerical simulations on the Anderson model.It is predicted analytically and confirmed numerically that the sign of the correction is different for T-breaking perturbations caused by a constant vector-potential equivalent to a phase twist in the boundary conditions, and those caused by a random magnetic field.In the former case it is shown using a nonperturbative approach that quasi-localized states in weakly disordered systems can cause the curvature distribution to be nonanalytic. In 2d systems the distribution function P (K) has a branching point at K = 0 that is related to the multifractality of the wave functions and thus should be a generic feature of all critical eigenstates. A relationship between the branching power and the multifractality exponent d 2 is suggested. Evidence of the branch cut singularity is found in numerical simulations in 2d systems and at the Anderson transition point in 3d systems. I. INTRODUCTION.As first suggested by Edwards and Thouless 1 , the sensitivity of the spectrum {E n } of disordered conductors to a small twist of phase φ in the boundary conditions Ψ(x = 0, ρ) = e iφ Ψ(x = L, ρ) is a powerful tool to probe the space structure of eigenfunctions and distinguish between the extended and the localized states. More precisely, the quantity K n that is now referred to as the "level curvature", was introduced in Ref.[ 1] in order to describe this sensitivity quantitatively: