Output price subsidies in a stochastic world

This article studies the comparative statics of output subsidies for firms, with monotonic preferences over costs and returns, that face price and production uncertainty. The modeling of deficiency payments, support-price schemes, and stochastic supply shifts in a state-space framework is discussed. It is shown how these notions can be used, via a simple application of Shephard's lemma, to analyze input-demand shifts once comparative-static results for supply are available. A range of comparative-static results for supply are then developed and discussed.

First-principles quantum dynamics in interacting Bose gases II: stochastic gauges

First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus allows first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness of the basis and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anticommuting Grassmann variables. Furthermore, because of the overcompleteness of the basis, the phase-space distribution can always be chosen positive. This has important consequences for the sign problem in fermion physics.

A diffusion approach to approximating preservation probabilities for gene duplicates

Consider a haploid population and, within its genome, a gene whose presence is vital for the survival of any individual. Each copy of this gene is subject to mutations which destroy its function. Suppose one member of the population somehow acquires a duplicate copy of the gene, where the duplicate is fully linked to the original gene's locus. Preservation is said to occur if eventually the entire population consists of individuals descended from this one which initially carried the duplicate. The system is modelled by a finite state-space Markov process which in turn is approximated by a diffusion process, whence an explicit expression for the probability of preservation is derived. The event of preservation can be compared to the fixation of a selectively neutral gene variant initially present in a single individual, the probability of which is the reciprocal of the population size. For very weak mutation, this and the probability of preservation are equal, while as mutation becomes stronger, the preservation probability tends to double this reciprocal. This is in excellent agreement with simulation studies.

Distillability of Werner states using entanglement witnesses and robust semidefinite programs

We use robust semidefinite programs and entanglement witnesses to study the distillability of Werner states. We perform exact numerical calculations that show two-undistillability in a region of the state space, which was previously conjectured to be undistillable. We also introduce bases that yield interesting expressions for the distillability witnesses and for a tensor product of Werner states with an arbitrary number of copies.

Anisotropy-based robust performance analysis of finite horizon linear discrete time varying systems

We consider a problem of robust performance analysis of linear discrete time varying systems on a bounded time interval. The system is represented in the state-space form. It is driven by a random input disturbance with imprecisely known probability distribution; this distributional uncertainty is described in terms of entropy. The worst-case performance of the system is quantified by its a-anisotropic norm. Computing the anisotropic norm is reduced to solving a set of difference Riccati and Lyapunov equations and a special form equation.

A helicopter named Dolly: Behavioural cloning for autonomous helicopter control

This paper considers the pros and cons of using behavioural cloning for the development of low-level helicopter automation modules. Over the course of this project several Behavioural cloning approaches have been investigated. The results of the most effective Behavioural cloning approach are then compared to PID modules designed for the same aircraft. The comparison takes into consideration development time, reliability, and control performance. It has been found that Behavioural cloning techniques employing local approximators and a wide state-space coverage during training can produce stabilising control modules in less time than tuning PID controllers. However, performance and reliabity deficits have been found to exist with the Behavioural Cloning, attributable largely to the time variant nature of the dynamics due to the operating environment, and the pilot actions being poor for teaching. The final conclusion drawn here is that tuning PID modules remains superior to behavioural cloning for low-level helicopter automation.

The 'moving targets' training algorithm

A simple method for training the dynamical behavior of a neural network is derived. It is applicable to any training problem in discrete-time networks with arbitrary feedback. The algorithm resembles back-propagation in that an error function is minimized using a gradient-based method, but the optimization is carried out in the hidden part of state space either instead of, or in addition to weight space. Computational results are presented for some simple dynamical training problems, one of which requires response to a signal 100 time steps in the past.

The 'moving targets' training algorithm

A simple method for training the dynamical behavior of a neural network is derived. It is applicable to any training problem in discrete-time networks with arbitrary feedback. The method resembles back-propagation in that it is a least-squares, gradient-based optimization method, but the optimization is carried out in the hidden part of state space instead of weight space. A straightforward adaptation of this method to feedforward networks offers an alternative to training by conventional back-propagation. Computational results are presented for simple dynamical training problems, with varied success. The failures appear to arise when the method converges to a chaotic attractor. A patch-up for this problem is proposed. The patch-up involves a technique for implementing inequality constraints which may be of interest in its own right.

Gaussian regression and optimal finite dimensional linear models

The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n³), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covariance operator, which is a trace-class operator. This is an infinite dimensional analogue of principal component analysis. The importance of Hilbert space methods to practical statistics is also discussed.

Pragmatic approach to the development of robust real-time protocols

This research is concerned with the development of distributed real-time systems, in which software is used for the control of concurrent physical processes. These distributed control systems are required to periodically coordinate the operation of several autonomous physical processes, with the property of an atomic action. The implementation of this coordination must be fault-tolerant if the integrity of the system is to be maintained in the presence of processor or communication failures. Commit protocols have been widely used to provide this type of atomicity and ensure consistency in distributed computer systems. The objective of this research is the development of a class of robust commit protocols, applicable to the coordination of distributed real-time control systems. Extended forms of the standard two phase commit protocol, that provides fault-tolerant and real-time behaviour, were developed. Petri nets are used for the design of the distributed controllers, and to embed the commit protocol models within these controller designs. This composition of controller and protocol model allows the analysis of the complete system in a unified manner. A common problem for Petri net based techniques is that of state space explosion, a modular approach to both the design and analysis would help cope with this problem. Although extensions to Petri nets that allow module construction exist, generally the modularisation is restricted to the specification, and analysis must be performed on the (flat) detailed net. The Petri net designs for the type of distributed systems considered in this research are both large and complex. The top down, bottom up and hybrid synthesis techniques that are used to model large systems in Petri nets are considered. A hybrid approach to Petri net design for a restricted class of communicating processes is developed. Designs produced using this hybrid approach are modular and allow re-use of verified modules. In order to use this form of modular analysis, it is necessary to project an equivalent but reduced behaviour on the modules used. These projections conceal events local to modules that are not essential for the purpose of analysis. To generate the external behaviour, each firing sequence of the subnet is replaced by an atomic transition internal to the module, and the firing of these transitions transforms the input and output markings of the module. Thus local events are concealed through the projection of the external behaviour of modules. This hybrid design approach preserves properties of interest, such as boundedness and liveness, while the systematic concealment of local events allows the management of state space. The approach presented in this research is particularly suited to distributed systems, as the underlying communication model is used as the basis for the interconnection of modules in the design procedure. This hybrid approach is applied to Petri net based design and analysis of distributed controllers for two industrial applications that incorporate the robust, real-time commit protocols developed. Temporal Petri nets, which combine Petri nets and temporal logic, are used to capture and verify causal and temporal aspects of the designs in a unified manner.

Self-sustained clusters and ergodicity breaking in spin models

Self-sustained spin clusters are analytically linked to ergodicity breaking in fully connected Ising and Sherrington-Kirkpatick (SK) models, relating the less understood spin space to the well understood state space. This correspondence is established through the absence of clusters in the paramagnetic phase, the presence of one dominant cluster in the Ising ferromagnet, and the formation of nontrivial clusters in SK spin glass. Yet unobserved phenomena are also revealed such as a first order phase transition in cluster sizes in the SK ferromagnet. The method could be adapted to investigate other spin models. © 2013 American Physical Society.

Transition of planar Couette flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers (Re<105) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit. © 2013 American Physical Society.

Classification of Biomedical Signals using the Dynamics of the False Nearest Neighbours (DFNN) Algorithm

* This study was supported in part by the Natural Sciences and Engineering Research Council of Canada, and by the Gastrointestinal Motility Laboratory (University of Alberta Hospitals) in Edmonton, Alberta, Canada.

An edge-based matching kernel through discrete-time quantum walks

In this paper, we propose a new edge-based matching kernel for graphs by using discrete-time quantum walks. To this end, we commence by transforming a graph into a directed line graph. The reasons of using the line graph structure are twofold. First, for a graph, its directed line graph is a dual representation and each vertex of the line graph represents a corresponding edge in the original graph. Second, we show that the discrete-time quantum walk can be seen as a walk on the line graph and the state space of the walk is the vertex set of the line graph, i.e., the state space of the walk is the edges of the original graph. As a result, the directed line graph provides an elegant way of developing new edge-based matching kernel based on discrete-time quantum walks. For a pair of graphs, we compute the h-layer depth-based representation for each vertex of their directed line graphs by computing entropic signatures (computed from discrete-time quantum walks on the line graphs) on the family of K-layer expansion subgraphs rooted at the vertex, i.e., we compute the depth-based representations for edges of the original graphs through their directed line graphs. Based on the new representations, we define an edge-based matching method for the pair of graphs by aligning the h-layer depth-based representations computed through the directed line graphs. The new edge-based matching kernel is thus computed by counting the number of matched vertices identified by the matching method on the directed line graphs. Experiments on standard graph datasets demonstrate the effectiveness of our new kernel.