Boolean Delay Equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular the discussion on partial BDEs is updated and enlarge

A unified analytical solution of the steady-state atmospheric diffusion equation

A unified analytical solution of the steady-state atmospheric diffusion equation for a finite and semi-infinite/infinite media was developed using the classic integral transform technique (CITT) which is based on a systematized method of separation of variable. The solution was obtained considering an arbitrary mean wind velocity depending on the vertical coordinate (z) and a generalized separable functional form for the eddy diffusivities in terms of the longitudinal (x) and vertical coordinates (z). The examples described in this article show that the well known closed-form analytical solutions, available in the literature, for both finite and semi-infinite/infinite media are special cases of the present unified analytical solution. As an example of the strength of the developed methodology, the Copenhagen and Prairie Grass experiments were simulated (finite media with the mean wind speed and the turbulent diffusion coefficient described by different functional forms). The results indicate that the present solutions are in good agreement with those obtained using other analytical procedures, previously published in the literature. It is important to note that the eigenvalue problem is associated directly to the atmospheric diffusion equation making possible the development of the unified analytical solution and also resulting in the improvement of the convergence behavior in the series of the eigenfunction-expansion.Indisponível

The geometry of sound rays in a wind

We survey the close relationship between sound and light rays and geometry. In the case where the medium is at rest, the geometry is the classical geometry of Riemann. In the case where the medium is moving, the more general geometry known as Finsler geometry is needed. We develop these geometries ab initio, with examples, and in particular show how sound rays in a stratified atmosphere with a wind can be mapped to a problem of circles and straight lines.Comment: Popular review article to appear in Contemporary Physic

Prediction and reduction of rotor broadband noise

Prediction techniques which can be or have been applied to subsonic rotors, and methods for designing helicopter rotors for reduced broadband noise generation are summarized. It is shown how detailed physical models of the noise source can be used to identify approaches to noise control

Inferring the Inclination of a Black Hole Accretion Disk from Observations of its Polarized Continuum Radiation

Spin parameters of stellar-mass black holes in X-ray binaries are currently being estimated by fitting the X-ray continuum spectra of their accretion disk emission. For this method, it is necessary to know the inclination of the X-ray-producing inner region of the disk. Since the inner disk is expected to be oriented perpendicular to the spin axis of the hole, the usual practice is to assume that the black hole spin is aligned with the orbital angular momentum vector of the binary, and to estimate the inclination of the latter from ellipsoidal modulations in the light curve of the secondary star. We show that the inclination of the disk can be inferred directly if we have both spectral and polarization information on the disk radiation. The predicted degree of polarization varies from 0% to 5% as the disk inclination changes from face-on to edge-on. With current X-ray polarimetric techniques the polarization degree of a typical bright X-ray binary could be measured to an accuracy of 0.1% by observing the source for about 10 days. Such a measurement would constrain the disk inclination to within a degree or two and would significantly improve the reliability of black hole spin estimates. In addition, it would provide new information on the tilt between the black hole spin axis and the orbital rotation axis of the binary, which would constrain any velocity kicks experienced by stellar-mass black holes during their formation.Comment: 46 pages, 8 figures, ApJ in pres

Structure and Randomness of Continuous-Time Discrete-Event Processes

Loosely speaking, the Shannon entropy rate is used to gauge a stochastic process' intrinsic randomness; the statistical complexity gives the cost of predicting the process. We calculate, for the first time, the entropy rate and statistical complexity of stochastic processes generated by finite unifilar hidden semi-Markov models---memoryful, state-dependent versions of renewal processes. Calculating these quantities requires introducing novel mathematical objects ({\epsilon}-machines of hidden semi-Markov processes) and new information-theoretic methods to stochastic processes.Comment: 10 pages, 2 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/ctdep.ht