Abstract:
A model is constructed to study the statistical properties of irregular trajectories of a log-gas whose positions are those of the complex eigenvalues of the unitary Ginibre ensemble. It is shown that statistically the trajectories form a structure that reveals the eigenvalue departure positions. It is also shown that the curvatures of the ensemble of trajectories are Cauchy distributed.

Abstract:
The (2+1)-dimension Klein-Gordon generalised equation is numerically solved through the finite differences method. Only the sine-Gordon case is focused: kink and antikink solutions are obtained in cartesian coordinates and evidence of interaction in kink-kink collision is looked for in propagation velocity. Then the change of shape in light bullet solutions is quantified during propagation and in head-on collision. Lastly, the robustness of light bullets is verified in head-on collisions with kink, antikink, standing kink and standing breather. A 30o-collision between a light bullet and a standing kink is simulated as well.

Abstract:
Complete spectroscopy (measurements of a complete sequence of consecutive levels) is often considered as a prerequisite to extract fluctuation properties of spectra. It is shown how this goal can be achieved even if only a fraction of levels are observed. The case of levels behaving as eigenvalues of random matrices, of current interest in nuclear physics, is worked out in detail.

Abstract:
The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of all eigenvalues, for medium matrix sizes, are described with a good precision by nearly normal distributions.

Abstract:
Using the simple procedure, recently introduced, of dividing Gaussian matrices by a positive random variable, a family of random matrices is generated characterized by a behavior ruled by the generalized hyperbolic distribution. The spectral density evolves from the semi-circle law to a Gaussian-like behavior while concomitantly the local fluctuations show a transition from the Wigner-Dyson to the Poisson statistics. Long range statistics such as number variance exhibit large fluctuations typical of non-ergodic ensembles.

Abstract:
By randomly removing a fraction of levels from a given spectrum a model is constructed that describes a crossover from this spectrum to a Poisson spectrum. The formalism is applied to the transitions towards Poisson from random matrix theory (RMT) spectra and picket fence spectra. It is shown that the Fredholm determinant formalism of RMT extends naturally to describe incomplete RMT spectra.

Abstract:
We analyze the form of the probability distribution function P_{n}^{(\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \times n \beta-Gaussian random matrix, \beta being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such \beta-Gaussian random matrices. We show that in the asymptotic limit n \to \infty and for arbitrary \beta the distribution P_{n}^{(\beta)}(w) converges to the Mar\v{c}enko-Pastur form, i.e., is defined as P_{n}^{(\beta)}(w) \sim \sqrt{(4 - w)/w} for w \in [0,4] and equals zero outside of the support. Furthermore, for Gaussian unitary (\beta = 2) ensembles we present exact explicit expressions for P_{n}^{(\beta=2)}(w) which are valid for arbitrary n and analyze their behavior.

Abstract:
It is shown that several effects are responsible for deviations of the intensity distributions from the Porter-Thomas law. Among these are genuine symmetry breaking, such as isospin; the nature of the transition operator; truncation of the Hilbert space in shell model calculations and missing transitions

Abstract:
A multifractal analysis is performed on the universality classes of random matrices and the transition ones.Our results indicate that the eigenvector probability distribution is a linear sum of two chi-squared distribution throughout the transition between the universality ensembles of random matrix theory and Poisson .

Abstract:
It is shown that an operator can be defined in the abstract space of random matrices ensembles whose matrix elements statistical distribution simulates the behavior of the distribution found in real physical systems. It is found that the key quantity that determines these distribution is the commutator of the operator with the Hamiltonian. Application to symmetry breaking in quantum many-body systems is discussed.