Quantum Fluctuations of Coulomb Potential as a Source of Flicker Noise. The Influence of External Electric Field

Fluctuations of the electromagnetic field produced by quantized matter in external electric field are investigated. A general expression for the power spectrum of fluctuations is derived within the long-range expansion. It is found that in the whole measured frequency band, the power spectrum of fluctuations exhibits an inverse frequency dependence. A general argument is given showing that for all practically relevant values of the electric field, the power spectrum of induced fluctuations is proportional to the field strength squared. As an illustration, the power spectrum is calculated explicitly using the kinetic model with the relaxation-type collision term. Finally, it is shown that the magnitude of fluctuations produced by a sample generally has a Gaussian distribution around its mean value, and its dependence on the sample geometry is determined. In particular, it is demonstrated that for geometrically similar samples, the power spectrum is inversely proportional to the sample volume. Application of the obtained results to the problem of flicker noise is discussed.Comment: 14 pages, 3 figure

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page

Ergodicity, Decisions, and Partial Information

In the simplest sequential decision problem for an ergodic stochastic process X, at each time n a decision u_n is made as a function of past observations X_0,...,X_{n-1}, and a loss l(u_n,X_n) is incurred. In this setting, it is known that one may choose (under a mild integrability assumption) a decision strategy whose pathwise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system (\Omega,B,P,T) with partial information Y\subseteq B. The existence of pathwise optimal strategies grounded in two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a general sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations \bigvee_n T^n Y. If the conditional ergodicity assumption is strengthened, the complexity assumption can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information. Our results also yield a decision-theoretic characterization of weak mixing in ergodic theory, and establish pathwise optimality of ergodic nonlinear filters.Comment: 45 page

Quantum Theory of Flicker Noise in Metal Films

Flicker (1/f^gamma) voltage noise spectrum is derived from finite-temperature quantum electromagnetic fluctuations produced by elementary charge carriers in external electric field. It is suggested that deviations of the frequency exponent \gamma from unity, observed in thin metal films, can be attributed to quantum backreaction of the conducting medium on the fluctuating field of the charge carrier. This backreaction is described phenomenologically in terms of the effective momentum space dimensionality, D. Using the dimensional continuation technique, it is shown that the combined action of the photon heat bath and external field results in a 1/f^gamma-contribution to the spectral density of the two-point correlation function of electromagnetic field. The frequency exponent is found to be equal to 1 + delta, where delta = 3 - D is a reduction of the momentum space dimensionality. This result is applied to the case of a biased conducting sample, and a general expression for the voltage power spectrum is obtained which possesses all characteristic properties of observed flicker noise spectra. The range of validity of this expression covers well the whole measured frequency band. Gauge independence of the power spectrum is proved. It is shown that the obtained results naturally resolve the problem of divergence of the total noise power. A detailed comparison with the experimental data on flicker noise measurements in metal films is given.Comment: 20 pages, 2 tables, 2 figure

Spatiotemporal complexity of the universe at subhorizon scales

This is a short note on the spatiotemporal complexity of the dynamical state(s) of the universe at subhorizon scales (up to 300 Mpc). There are reasons, based mainly on infrared radiative divergences, to believe that one can encounter a flicker noise in the time domain, while in the space domain, the scaling laws are reflected in the (multi)fractal distribution of galaxies and their clusters. There exist recent suggestions on a unifying treatment of these two aspects within the concept of spatiotemporal complexity of dynamical systems driven out of equilibrium. Spatiotemporal complexity of the subhorizon dynamical state(s) of the universe is a conceptually nice idea and may lead to progress in our understanding of the material structures at large scalesComment: references update

A discrete invitation to quantum filtering and feedback control

The engineering and control of devices at the quantum-mechanical level--such as those consisting of small numbers of atoms and photons--is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.Comment: 76 pages, 12 figures. A PDF file with high resolution figures can be found at http://minty.caltech.edu/papers.ph

Molecular epidemiology of Mycobacterium bovis in Cameroon

We describe the largest molecular epidemiological study of Bovine Tuberculosis (bTB) in a sub-Saharan African country with higher spatial resolution providing new insights into bTB. Four hundred and ninety-nine samples were collected for culture from 201 and 179 cattle with and without bTB-like lesions respectively out of 2,346 cattle slaughtered at Bamenda, Ngaoundere, Garoua and Maroua abattoirs between 2012-2013. Two hundred and fifty-five M. bovis were isolated, identified and genotyped using deletion analysis, Hain® Genotype MTBC, spoligotyping and MIRU-VNTR. African 1 was the dominant M. bovis clonal complex, with 97 unique genotypes including 19 novel spoligotypes representing the highest M. bovis genetic diversity observed in Africa to date. SB0944 and SB0953 dominated (63%) the observed spoligotypes. A third of animals with multiple lesions had multiple strain infections. Higher diversity but little evidence of recent transmission of M. bovis was more common in Adamawa compared to the North-West Region. The Adamawa was characterised by a high frequency of singletons possibly due to constant additions from an active livestock movement network compared to the North-West Region where a local expansion was more evident. The latter combined with population-based inferences suggest an unstable and stable bTB-endemic status in the North-West and Adamawa Regions respectively

Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods

In this paper we consider multidimensional mechanism design problem for selling discrete substitutable items to a group of buyers. Previous work on this problem mostly focus on stochastic description of valuations used by the seller. However, in certain applications, no prior information regarding buyers' preferences is known. To address this issue, we consider uncertain valuations and formulate the problem in a robust optimization framework: the objective is to minimize the maximum regret. For a special case of revenue-maximizing pricing problem we present a solution method based on mixed-integer linear programming formulation

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42