Cellular Automata and Ultra-Discrete Painlev\'e Equations

Starting from integrable cellular automata we present a novel form of Painlev\'e equations. These equations are discrete in both the independent variable and the dependent one. We show that they capture the essence of the behavior of the Painlev\'e equations organizing themselves into a coalescence cascade and possessing special solutions. A necessary condition for the integrability of cellular automata is also presented.Comment: 8 pages, plainTeX, 2 figure

Conditioning bounds for traveltime tomography in layered media

This paper revisits the problem of recovering a smooth, isotropic, layered wave speed profile from surface traveltime information. While it is classic knowledge that the diving (refracted) rays classically determine the wave speed in a weakly well-posed fashion via the Abel transform, we show in this paper that traveltimes of reflected rays do not contain enough information to recover the medium in a well-posed manner, regardless of the discretization. The counterpart of the Abel transform in the case of reflected rays is a Fredholm kernel of the first kind which is shown to have singular values that decay at least root-exponentially. Kinematically equivalent media are characterized in terms of a sequence of matching moments. This severe conditioning issue comes on top of the well-known rearrangement ambiguity due to low velocity zones. Numerical experiments in an ideal scenario show that a waveform-based model inversion code fits data accurately while converging to the wrong wave speed profile

The smallest eigenvalue of Hankel matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed

Linked-cluster expansion for the Green\u27s function of the infinite-U Hubbard model

We implement a highly efficient strong-coupling expansion for the Green\u27s function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green\u27s function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green\u27s function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order

A Convergent Method for Calculating the Properties of Many Interacting Electrons

A method is presented for calculating binding energies and other properties of extended interacting systems using the projected density of transitions (PDoT) which is the probability distribution for transitions of different energies induced by a given localized operator, the operator on which the transitions are projected. It is shown that the transition contributing to the PDoT at each energy is the one which disturbs the system least, and so, by projecting on appropriate operators, the binding energies of equilibrium electronic states and the energies of their elementary excitations can be calculated. The PDoT may be expanded as a continued fraction by the recursion method, and as in other cases the continued fraction converges exponentially with the number of arithmetic operations, independent of the size of the system, in contrast to other numerical methods for which the number of operations increases with system size to maintain a given accuracy. These properties are illustrated with a calculation of the binding energies and zone-boundary spin- wave energies for an infinite spin-1/2 Heisenberg chain, which is compared with analytic results for this system and extrapolations from finite rings of spins.Comment: 30 pages, 4 figures, corrected pd

Vector Continued Fractions using a Generalised Inverse

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.Comment: Published form - minor change

Two-band random matrices

Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of spectra are directly reconstructed from the recurrence equation for orthogonal polynomials associated with a given random matrix ensemble. It is established that an eigenvalue gap does not affect the local eigenvalue correlations which follow the universal sine and the universal multicritical laws in the bulk and soft-edge scaling limits, respectively. By contrast, global smoothed eigenvalue correlations do reflect the presence of a gap, and are shown to satisfy a new universal law exhibiting a sharp dependence on the odd/even dimension of random matrices whose spectra are bounded. In the case of unbounded spectrum, the corresponding universal `density-density' correlator is conjectured to be generic for chaotic systems with a forbidden gap and broken time reversal symmetry.Comment: 12 pages (latex), references added, discussion enlarge

Generating GHZ state in 2m-qubit spin network

We consider a pure 2m-qubit initial state to evolve under a particular quantum me- chanical spin Hamiltonian, which can be written in terms of the adjacency matrix of the Johnson network J(2m;m). Then, by using some techniques such as spectral dis- tribution and stratification associated with the graphs, employed in [1, 2], a maximally entangled GHZ state is generated between the antipodes of the network. In fact, an explicit formula is given for the suitable coupling strengths of the hamiltonian, so that a maximally entangled state can be generated between antipodes of the network. By using some known multipartite entanglement measures, the amount of the entanglement of the final evolved state is calculated, and finally two examples of four qubit and six qubit states are considered in details.Comment: 22 page

Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory

The method of the quantum probability theory only requires simple structural data of graph and allows us to avoid a heavy combinational argument often necessary to obtain full description of spectrum of the adjacency matrix. In the present paper, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the quantum probability theory, we investigate quantum central limit theorem for continuous-time quantum walks on odd graphs.Comment: 19 page, 1 figure

Evaluation of effective resistances in pseudo-distance-regular resistor networks

In Refs.[1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on the stratification of the network and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effective resistances on more general networks called pseudo-distance-regular networks [21] or QD type networks \cite{obata} is investigated, where we use the stratification of these networks and show that the effective resistances between a given node such as $\alpha$ and all of the nodes $\beta$ belonging to the same stratum with respect to $\alpha$ ($R_{\alpha\beta^{(m)}}$, $\beta$ belonging to the $m$-th stratum with respect to the $\alpha$) are the same. Then, based on the spectral techniques, an analytical formula for effective resistances $R_{\alpha\beta^{(m)}}$ such that $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ (those nodes $\alpha$, $\beta$ of the network such that the network is symmetric with respect to them) is given in terms of the first and second orthogonal polynomials associated with the network, where $L^{-1}$ is the pseudo-inverse of the Laplacian of the network. From the fact that in distance-regular networks, $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ is satisfied for all nodes $\alpha,\beta$ of the network, the effective resistances $R_{\alpha\beta^{(m)}}$ for $m=1,2,...,d$ ($d$ is diameter of the network which is the same as the number of strata) are calculated directly, by using the given formula.Comment: 30 pages, 7 figure