1,383 research outputs found

    Integrable random matrix ensembles

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    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

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    The 2-point correlation form factor, K2(τ)K_2(\tau), for small values of τ\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 π/n\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 K2(0)=(n+ϵ(n))/(3(n2))K_2(0)=(n+\epsilon(n))/(3(n-2)) where ϵ(n)=0\epsilon(n)=0 for odd nn, ϵ(n)=2\epsilon(n)=2 for even nn not divisible by 3, and ϵ(n)=6\epsilon(n)=6 for even nn 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, K2(0)=13αˉ+4αˉ2K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2 where αˉ\bar{\alpha} is the fractional part of the flux through the rectangle when 0αˉ1/20\le \bar{\alpha}\le 1/2 and it is symmetric with respect to the line αˉ=1/2\bar{\alpha}=1/2 when 1/2αˉ11/2 \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 K2(0)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

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    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

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    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 kk 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 Σ2(L)χL\Sigma^2(L)\sim\chi L for large LL and the nearest-neighbor distribution decreases exponentially when ss\to \infty, P(s)exp(Λs)P(s)\sim\exp (-\Lambda s) with Λ=1/χ=kβ+1\Lambda=1/\chi=k\beta+1, where β\beta is the inverse temperature of the gas (β=\beta=1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of k=β=1k=\beta=1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. P(s)=4sexp(2s)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

    Dual methods and approximation concepts in structural synthesis

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    Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins

    ACCESS 3. Approximation concepts code for efficient structural synthesis: User's guide

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    A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of approximation concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second order primal projection method (PRIMAL2); second order Newton-type dual method (DUAL2); and first order gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero order approximation of the stress constraints are also included


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    The first 800 eigenvalues of the stadium billiard have been evaluated numerically. It is shown that the four spectra obtained (corresponding to the four types of symmetry of the wave function) exhibit the fluctuation properties of the Gaussian Orthogonal Ensemble of Random Matrices. This reinforces the belief that these fluctuation properties are characteristic of quantum chaotic systems

    Random matrix ensembles associated with Lax matrices

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    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


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    Every attempt has been made to insure that the citations appear exactly as they do in the original publication. Entries for which the original could not be obtained for verification are indicated by (-)

    Neurospora Bibliography - 1993

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    Neurospora Bibliography - 1993 - Every attempt has been made to insure that the citations appear exactly as they do in the original publication