Relativistic quark models and the current algebra

In this thesis we are concerned with finding representations of the algebra of SU(3) vector and axial-vector charge densities at infinite momentum (the "current algebra") to describe the mesons, idealizing the real continua of multiparticle states as a series of discrete resonances of zero width. Such representations would describe the masses and quantum numbers of the mesons, the shapes of their Regge trajectories, their electromagnetic and weak form factors, and (approximately, through the PCAC hypothesis) pion emission or absorption amplitudes.

We assume that the mesons have internal degrees of freedom equivalent to being made of two quarks (one an antiquark) and look for models in which the mass is SU(3)-independent and the current is a sum of contributions from the individual quarks. Requiring that the current algebra, as well as conditions of relativistic invariance, be satisfied turns out to be very restrictive, and, in fact, no model has been found which satisfies all requirements and gives a reasonable mass spectrum. We show that using more general mass and current operators but keeping the same internal degrees of freedom will not make the problem any more solvable. In particular, in order for any two-quark solution to exist it must be possible to solve the "factorized SU(2) problem," in which the currents are isospin currents and are carried by only one of the component quarks (as in the K meson and its excited states).

In the free-quark model the currents at infinite momentum are found using a manifestly covariant formalism and are shown to satisfy the current algebra, but the mass spectrum is unrealistic. We then consider a pair of quarks bound by a potential, finding the current as a power series in 1/m where m is the quark mass. Here it is found impossible to satisfy the algebra and relativistic invariance with the type of potential tried, because the current contributions from the two quarks do not commute with each other to order 1/m³. However, it may be possible to solve the factorized SU(2) problem with this model.

The factorized problem can be solved exactly in the case where all mesons have the same mass, using a covariant formulation in terms of an internal Lorentz group. For a more realistic, nondegenerate mass there is difficulty in covariantly solving even the factorized problem; one model is described which almost works but appears to require particles of spacelike 4-momentum, which seem unphysical.

Although the search for a completely satisfactory model has been unsuccessful, the techniques used here might eventually reveal a working model. There is also a possibility of satisfying a weaker form of the current algebra with existing models.

Distributed Optimal Control of Cyber-Physical Systems: Controller Synthesis, Architecture Design and System Identification

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and spatially distributed. Both the scale and the distributed nature of these systems has motivated the decentralization of control schemes into local sub-controllers that measure, exchange and act on locally available subsets of the globally available system information. This decentralization of control logic leads to different decision makers acting on asymmetric information sets, introduces the need for coordination between them, and perhaps not surprisingly makes the resulting optimal control problem much harder to solve. In fact, shortly after such questions were posed, it was realized that seemingly simple decentralized optimal control problems are computationally intractable to solve, with the Wistenhausen counterexample being a famous instance of this phenomenon. Spurred on by this perhaps discouraging result, a concerted 40 year effort to identify tractable classes of distributed optimal control problems culminated in the notion of quadratic invariance, which loosely states that if sub-controllers can exchange information with each other at least as quickly as the effect of their control actions propagates through the plant, then the resulting distributed optimal control problem admits a convex formulation.

The identification of quadratic invariance as an appropriate means of "convexifying" distributed optimal control problems led to a renewed enthusiasm in the controller synthesis community, resulting in a rich set of results over the past decade. The contributions of this thesis can be seen as being a part of this broader family of results, with a particular focus on closing the gap between theory and practice by relaxing or removing assumptions made in the traditional distributed optimal control framework. Our contributions are to the foundational theory of distributed optimal control, and fall under three broad categories, namely controller synthesis, architecture design and system identification.

We begin by providing two novel controller synthesis algorithms. The first is a solution to the distributed H-infinity optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an H-infinity norm bound. The second is an explicit dynamic programming solution to a two player LQR state-feedback problem with varying delays. Accommodating varying delays represents an important first step in combining distributed optimal control theory with the area of Networked Control Systems that considers lossy channels in the feedback loop. Our next set of results are concerned with controller architecture design. When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given -- indeed the task of designing this architecture is now as important as the design of the control laws themselves. To address this task, we formulate the Regularization for Design (RFD) framework, which is a unifying computationally tractable approach, based on the model matching framework and atomic norm regularization, for the simultaneous co-design of a structured optimal controller and the architecture needed to implement it. Our final result is a contribution to distributed system identification. Traditional system identification techniques such as subspace identification are not computationally scalable, and destroy rather than leverage any a priori information about the system's interconnection structure. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. To that end we propose a promising heuristic for identifying the dynamics of a subsystem that is still connected to a large system. We exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization problem. Finally, we conclude with a brief discussion of future research directions, with a particular emphasis on how to incorporate the results of this thesis, and those of optimal control theory in general, into a broader theory of dynamics, control and optimization in layered architectures.

CP and the three pion decay of the K°

The time distribution of the decays of an initially pure K° beam into π+π-π° has been analyzed to determine the complex parameter W (also known as Ƞ^+-° and (x + iy)). The K° beam was produced in a brass target by the interactions of a 2.85 GeV/c π- beam which was generated on an internal target in the Lawrence Radiation Laboratory (LRL) Bevatron. The counters and hodoscopes in the apparatus selected for events with a neutral (K°) produced in the brass target, two charged secondaries passing through a magnet spectrometer and a ɣ-ray shower in a shower hodoscope.

From the 275K apparatus triggers, 148 K → π+π-π° events were isolated. The presence of a ɣ-ray shower in the optical shower chambers and a two-prong vee in the optical spark chambers were devices used to isolate the events. The backgrounds were further reduced by reconstructing the momenta of the two charged secondaries and applying kinematic constraints.

The best fit to the final sample of 148 events distributed between .3 and 7.0 K_S lifetimes gives:

ReW = -.05 ±.17

ImW = +.39 +.35/-.37

This result is consistent with both CPT invariance (ReW = 0) and CP invariance (W = 0). Backgrounds are estimated to be less than 10% and systematic effects have also been estimated to be negligible.

An analysis of the present data on CP violation in this decay mode and other K° decay modes has estimated the phase of ɛ to be 45.3 ± 2.3 degrees. This result is consistent with the super weak theories of CP violation which predicts the phase of ɛ to be 43°. This estimate is in turn used to predict the phase of Ƞ°° to be 48.0 ± 7.9 degrees. This is a substantial improvement on presently available measurements. The largest error in this analysis comes from the present limits on W from the world average of recent experiments. The K → πuʋ mode produces the next largest error. Therefore further experimentation in these modes would be useful.

Photodisintegration of helium-3 at energies between 200 and 600 MEV

We have measured differential cross-sections for the two-body photodisintegration of Helium-3, ɣ + He³ → p + d, between incident photon energies of 200 and 600 MeV, and for center of mass frame angles between 30° and 150°. Both final state particles were detected in arrays of wire spark chambers and scintillation counters; the high momentum particle was analyzed in a magnet spectrometer. The results are interpreted in terms of amplitudes to produce the ∆(1236) resonance in an intermediate state, as well as non-resonant amplitudes. This experiment, together with an (unfinished) experiment on the inverse reaction, p + d → He³ + ɣ, will provide a reciprocity test of time reversal invariance.

Review of size- and age-dependence in batch spawning: implications for stock assessment of fish species exhibiting indeterminate fecundity

Most assessments of fish stocks use some measure of the reproductive potential of a population, such as spawning biomass. However, the correlation between spawning biomass and reproductive potential is not always strong, and it likely is weakest in the tropics and subtropics, where species tend to exhibit indeterminate fecundity and release eggs in batches over a protracted spawning season. In such cases, computing annual reproductive output requires estimates of batch fecundity and the annual number of batches—the latter subject to spawning frequency and duration of spawning season. Batch fecundity is commonly measured by age (or size), but these other variables are not. Without the relevant data, the annual number of batches is assumed to be invariant across age. We reviewed the literature and found that this default assumption lacks empirical support because both spawning duration and spawning frequency generally increase with age or size. We demonstrate effects of this assumption on measures of reproductive value and spawning potential ratio, a metric commonly used to gauge stock status. Model applications showed substantial sensitivity to age dependence in the annual number of batches. If the annual number of batches increases with age but is incorrectly assumed to be constant, stock assessment models would tend to overestimate the biological reference points used for setting harvest rates. This study underscores the need to better understand the age- or size-dependent contrast in the annual number of batches, and we conclude that, for species without evidence to support invariance, the default assumption should be replaced with one that accounts for age- or size-dependence.

Face recognition from video

In spite of over two decades of intense research, illumination and pose invariance remain prohibitively challenging aspects of face recognition for most practical applications. The objective of this work is to recognize faces using video sequences both for training and recognition input, in a realistic, unconstrained setup in which lighting, pose and user motion pattern have a wide variability and face images are of low resolution. In particular there are three areas of novelty: (i) we show how a photometric model of image formation can be combined with a statistical model of generic face appearance variation, learnt offline, to generalize in the presence of extreme illumination changes; (ii) we use the smoothness of geodesically local appearance manifold structure and a robust same-identity likelihood to achieve invariance to unseen head poses; and (iii) we introduce an accurate video sequence "reillumination" algorithm to achieve robustness to face motion patterns in video. We describe a fully automatic recognition system based on the proposed method and an extensive evaluation on 171 individuals and over 1300 video sequences with extreme illumination, pose and head motion variation. On this challenging data set our system consistently demonstrated a nearly perfect recognition rate (over 99.7%), significantly outperforming state-of-the-art commercial software and methods from the literature. © Springer-Verlag Berlin Heidelberg 2006.

Statistical image modelling using interscale phase relationships of complex wavelet coefficients

A novel method for modelling the statistics of 2D photographic images useful in image restoration is defined. The new method is based on the Dual Tree Complex Wavelet Transform (DT-CWT) but a phase rotation is applied to the coefficients to create complex coefficients whose phase is shift-invariant at multiscale edge and ridge features. This is in addition to the magnitude shift invariance achieved by the DT-CWT. The increased correlation between coefficients adjacent in space and scale provides an improved mechanism for signal estimation. © 2006 IEEE.

Biologically-inspired object recognition system with features from complex wavelets

In this paper, a novel cortex-inspired feed-forward hierarchical object recognition system based on complex wavelets is proposed and tested. Complex wavelets contain three key properties for object representation: shift invariance, which enables the extraction of stable local features; good directional selectivity, which simplifies the determination of image orientations; and limited redundancy, which allows for efficient signal analysis using the multi-resolution decomposition offered by complex wavelets. In this paper, we propose a complete cortex-inspired object recognition system based on complex wavelets. We find that the implementation of the HMAX model for object recognition in [1, 2] is rather over-complete and includes too much redundant information and processing. We have optimized the structure of the model to make it more efficient. Specifically, we have used the Caltech 5 standard dataset to compare with Serre's model in [2] (which employs Gabor filter bands). Results demonstrate that the complex wavelet model achieves a speed improvement of about 4 times over the Serre model and gives comparable recognition performance. © 2011 IEEE.

On representing chemical environments

We review some recently published methods to represent atomic neighbourhood environments, and analyse their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms, and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighbourhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbours increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether new approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps these difficulties by directly defining the similarity between any two neighbourhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.

Achieving robust face recognition from video by combining a weak photometric model and a learnt generic face invariant

In spite of over two decades of intense research, illumination and pose invariance remain prohibitively challenging aspects of face recognition for most practical applications. The objective of this work is to recognize faces using video sequences both for training and recognition input, in a realistic, unconstrained setup in which lighting, pose and user motion pattern have a wide variability and face images are of low resolution. The central contribution is an illumination invariant, which we show to be suitable for recognition from video of loosely constrained head motion. In particular there are three contributions: (i) we show how a photometric model of image formation can be combined with a statistical model of generic face appearance variation to exploit the proposed invariant and generalize in the presence of extreme illumination changes; (ii) we introduce a video sequence re-illumination algorithm to achieve fine alignment of two video sequences; and (iii) we use the smoothness of geodesically local appearance manifold structure and a robust same-identity likelihood to achieve robustness to unseen head poses. We describe a fully automatic recognition system based on the proposed method and an extensive evaluation on 323 individuals and 1474 video sequences with extreme illumination, pose and head motion variation. Our system consistently achieved a nearly perfect recognition rate (over 99.7% on all four databases). © 2012 Elsevier Ltd All rights reserved.

Direction and polarisation estimation using polarised cylindrical conformal arrays

In this study, the authors describe two-dimensional direction finding and signal polarisation estimation from a cylindrical conformal array consisting of directional and polarised antenna elements. Firstly, a simple and general transformation procedure, based on the mathematical framework of geometric algebra, is presented for arbitrary conformal arrays with polarised and directional antennas. Subsequently, the authors utilise the symmetry of cylindrical arrays to estimate signal parameters via rotational invariance techniques. The authors show how to iteratively estimate the azimuth and elevation angles of the incident signal, as well as its polarisation. To illustrate the versatility of this method, the results of simulations on a 3×4 cylindrical conformal array are shown and discussed. © 2012 The Institution of Engineering and Technology.

Periodically controlled hybrid systems verifying a controller for an autonomous vehicle

This paper introduces Periodically Controlled Hybrid Automata (PCHA) for describing a class of hybrid control systems. In a PCHA, control actions occur roughly periodically while internal and input actions may occur in the interim changing the discrete-state or the setpoint. Based on periodicity and subtangential conditions, a new sufficient condition for verifying invariance of PCHAs is presented. This technique is used in verifying safety of the planner-controller subsystem of an autonomous ground vehicle, and in deriving geometric properties of planner generated paths that can be followed safely by the controller under environmental uncertainties.

Random function priors for exchangeable arrays with applications to graphs and relational data

A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efficient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

A theory of slow feature analysis for transformation-based input signals with an application to complex cells

We develop a group-theoretical analysis of slow feature analysis for the case where the input data are generated by applying a set of continuous transformations to static templates. As an application of the theory, we analytically derive nonlinear visual receptive fields and show that their optimal stimuli, as well as the orientation and frequency tuning, are in good agreement with previous simulations of complex cells in primary visual cortex (Berkes and Wiskott, 2005). The theory suggests that side and end stopping can be interpreted as a weak breaking of translation invariance. Direction selectivity is also discussed. © 2011 Massachusetts Institute of Technology.

Regression on fixed-rank positive semidefinite matrices: A Riemannian approach

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixedrank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks. © 2011 Gilles Meyer, Silvere Bonnabel and Rodolphe Sepulchre.