Algebraic Optimization of Binary Spatially Coupled Measurement Matrices for Interval Passing

We consider binary spatially coupled (SC) low density measurement matrices for low complexity reconstruction of sparse signals via the interval passing algorithm (IPA). The IPA is known to fail due to the presence of harmful sub-structures in the Tanner graph of a binary sparse measurement matrix, so called termatiko sets. In this work we construct array-based (AB) SC sparse measurement matrices via algebraic lifts of graphs, such that the number of termatiko sets in the Tanner graph is minimized. To this end, we show for the column-weight-three case that the most critical termatiko sets can be removed by eliminating all length-12 cycles associated with the Tanner graph, via algebraic lifting. As a consequence, IPA-based reconstruction with SC measurement matrices is able to provide an almost error free reconstruction for significantly denser signal vectors compared to uncoupled AB LDPC measurement matrices.Comment: 5 pages, 2 figures, To appear in the Proceedings of 2018 IEEE Information Theory Workshop, Guangzhou, Chin

A Second-Order Stochastic Leap-Frog Algorithm for Langevin Simulation

Langevin simulation provides an effective way to study collisional effects in beams by reducing the six-dimensional Fokker-Planck equation to a group of stochastic ordinary differential equations. These resulting equations usually have multiplicative noise since the diffusion coefficients in these equations are functions of position and time. Conventional algorithms, e.g. Euler and Heun, give only first order convergence of moments in a finite time interval. In this paper, a stochastic leap-frog algorithm for the numerical integration of Langevin stochastic differential equations with multiplicative noise is proposed and tested. The algorithm has a second-order convergence of moments in a finite time interval and requires the sampling of only one uniformly distributed random variable per time step. As an example, we apply the new algorithm to the study of a mechanical oscillator with multiplicative noise.Comment: 3 pages, 4 figures, to submit to XX International LINAC conferenc

Symplectic Calculation of Lyapunov Exponents

The Lyapunov exponents of a chaotic system quantify the exponential divergence of initially nearby trajectories. For Hamiltonian systems the exponents are related to the eigenvalues of a symplectic matrix. We make use of this fact to develop a new method for the calculation of Lyapunov exponents of such systems. Our approach avoids the renormalization and reorthogonalization of usual techniques. It is also easily extendible to damped systems. We apply our method to two examples of physical interest: a model system that describes the beam halo in charged particle beams and the driven van der Pol oscillator.Comment: 10 pages, uuencoded PostScript (figures included), LA-UR-94-216

Classical Dynamics for Linear Systems: The Case of Quantum Brownian Motion

It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.Comment: 9 pages, LaTe

Lyapunov Exponents without Rescaling and Reorthogonalization

We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM_trans where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.Comment: 4 pages, no figures, uses RevTeX plus macro (included). Phys. Rev. Lett. (in press

How Wigner Functions Transform Under Symplectic Maps

It is shown that, while Wigner and Liouville functions transform in an identical way under linear symplectic maps, in general they do not transform identically for nonlinear symplectic maps. Instead there are ``quantum corrections'' whose hbar tending to zero limit may be very complicated. Examples of the behavior of Wigner functions in this limit are given in order to examine to what extent the corresponding Liouville densities are recovered.Comment: 8 pages, 6 figures [RevTeX/epsfig, macro included]. To appear in Proceedings of the Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics (Monterey, CA 1998