Integrable random matrix ensembles

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to integrability of the underlying system. Formulas for spacing distributions and level compressibility are obtained for various instances of such ensembles.Comment: 32 pages, 8 figure

Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems

The 2-point correlation form factor, $K_2(\tau)$, for small values of $\tau$ is computed analytically for typical examples of pseudo-integrable systems. This is done by explicit calculation of periodic orbit contributions in the diagonal approximation. The following cases are considered: (i) plane billiards in the form of right triangles with one angle $\pi/n$ and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that $K_2(0)=(n+\epsilon(n))/(3(n-2))$ where $\epsilon(n)=0$ for odd $n$, $\epsilon(n)=2$ for even $n$ not divisible by 3, and $\epsilon(n)=6$ for even $n$ divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, $K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2$ where $\bar{\alpha}$ is the fractional part of the flux through the rectangle when $0\le \bar{\alpha}\le 1/2$ and it is symmetric with respect to the line $\bar{\alpha}=1/2$ when $1/2 \le \bar{\alpha}\le 1$. The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of $K_2(0)$ differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems.Comment: 61 pages, 13 figures. Submitted to Communications in Mathematical Physics, 200

Distance matrices and isometric embeddings

We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-negative. Several generalizations are discussed.Comment: 17 page

Short-range plasma model for intermediate spectral statistics

We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting the logarithmic pair interaction to a finite number $k$ of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we show that the number variance has the asymptotic form $\Sigma^2(L)\sim\chi L$ for large $L$ and the nearest-neighbor distribution decreases exponentially when $s\to \infty$, $P(s)\sim\exp (-\Lambda s)$ with $\Lambda=1/\chi=k\beta+1$, where $\beta$ is the inverse temperature of the gas ($\beta=$1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of $k=\beta=1$, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. $P(s)=4s\exp(-2s)$. Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.Comment: 24 pages, 4 figure

Random matrix ensembles associated with Lax matrices

A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an integrable structure permits to calculate the joint distribution of eigenvalues for these matrices analytically. Spectral statistics of these ensembles are quite unusual and in many cases give rigorously new examples of intermediate statistics

Simple shock isolator synthesis with bilinear stiffness and variable damping

Simple shock isolator synthesis with bilinear stiffness and variable dampin

Directional emission of stadium-shaped micro-lasers

The far-field emission of two dimensional (2D) stadium-shaped dielectric cavities is investigated. Micro-lasers with such shape present a highly directional emission. We provide experimental evidence of the dependance of the emission directionality on the shape of the stadium, in good agreement with ray numerical simulations. We develop a simple geometrical optics model which permits to explain analytically main observed features. Wave numerical calculations confirm the results.Comment: 4 pages, 8 figure

Trace formula for dieletric cavities : I. General properties

The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity. The contribution of each periodic orbit is the product of the two factors. The first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary. Two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of the cavity are derived. The coefficient of the perimeter term differs from the one for closed cavities due to unusual high-energy asymptotics of the $\mathbf{S}$-matrix for the scattering on the cavity. Corrections to the leading semi-classical formula are briefly discussed. Obtained formulas agree well with numerical calculations for circular dielectric cavities.Comment: 13 pages, 10 figure

Stroke-related Effects on Maximal Dynamic Hip Flexor Fatigability and Functional Implications

Introduction: Stroke-related changes in maximal dynamic hip flexor muscle fatigability may be more relevant functionally than isometric hip flexor fatigability. Methods: Ten chronic stroke survivors performed 5 sets of 30 hip flexion maximal dynamic voluntary contractions (MDVC). A maximal isometric voluntary contraction (MIVC) was performed before and after completion of the dynamic contractions. Both the paretic and nonparetic legs were tested. Results: Reduction in hip flexion MDVC torque in the paretic leg (44.7%) was larger than the nonparetic leg (31.7%). The paretic leg had a larger reduction in rectus femoris EMG (28.9%) between the first and last set of MDVCs than the nonparetic leg (7.4%). Reduction in paretic leg MDVC torque was correlated with self-selected walking speed (r2 = 0.43), while reduction in MIVC torque was not (r2 = 0.11). Conclusions: Reductions in maximal dynamic torque of paretic hip flexors may be a better predictor of walking function than reductions in maximal isometric contractions

Insights into the economic organization of the Phoenician homeland: a multidisciplinary investigation of the later Iron Age II and Persian period Phoenician amphorae from Tell el-Burak

This paper details the results of a large-scale multi-disciplinary analysis of Iron Age pottery from a settlement in the core of the Phoenician homeland. The research presented is centred upon a large corpus of Phoenician carinated-shoulder amphorae (CSA) from the later Iron Age II and Persian period contexts at the coastal site of Tell el-Burak. Traditional typological investigations are combined with a focused archaeometric approach including a new quantitative method for the morphometric analysis of amphorae, thin-section petrography, geochemistry and organic residue analyses, aimed at gaining a more detailed understanding of the organization of the Phoenician economy. Despite gradual, but marked typological changes, very little change in the fabrics of these amphorae was noted over the 400-year Iron Age occupation of the site. The research, thus, demonstrates that the production of Iron Age amphorae from Tell el-Burak was highly organized, and was undertaken by long-lived, sustained and centralized modes. The establishment of Tell el-Burak and this new pottery industry coincides with the proliferation of the world’s first great imperial powers, the Neo-Assyrian, Neo-Babylonian and Persian empires; the outcomes of this research provide new insights into socio-economic strategies adopted in the Phoenician homeland during this pivotal time