The philosophical significance of Cox's theorem

Cox's theorem states that, under certain assumptions, any measure of belief is isomorphic to a probability measure. This theorem, although intended as a justification of the subjectivist interpretation of probability theory, is sometimes presented as an argument for more controversial theses. Of particular interest is the thesis that the only coherent means of representing uncertainty is via the probability calculus. In this paper I examine the logical assumptions of Cox's theorem and I show how these impinge on the philosophical conclusions thought to be supported by the theorem. I show that the more controversial thesis is not supported by Cox's theorem. (C) 2003 Elsevier Inc. All rights reserved.

A new approach to control of MIMO processes with static nonlinearities using an extended IMC framework

In this paper, a new control design method is proposed for stable processes which can be described using Hammerstein-Wiener models. The internal model control (IMC) framework is extended to accommodate multiple IMC controllers, one for each subsystem. The concept of passive systems is used to construct the IMC controllers which approximate the inverses of the subsystems to achieve dynamic control performance. The Passivity Theorem is used to ensure the closed-loop stability. (c) 2005 Elsevier Ltd. All rights reserved.

Approximate processing of massive continuous quantile queries over high-speed data streams

Quantile computation has many applications including data mining and financial data analysis. It has been shown that an is an element of-approximate summary can be maintained so that, given a quantile query d (phi, is an element of), the data item at rank [phi N] may be approximately obtained within the rank error precision is an element of N over all N data items in a data stream or in a sliding window. However, scalable online processing of massive continuous quantile queries with different phi and is an element of poses a new challenge because the summary is continuously updated with new arrivals of data items. In this paper, first we aim to dramatically reduce the number of distinct query results by grouping a set of different queries into a cluster so that they can be processed virtually as a single query while the precision requirements from users can be retained. Second, we aim to minimize the total query processing costs. Efficient algorithms are developed to minimize the total number of times for reprocessing clusters and to produce the minimum number of clusters, respectively. The techniques are extended to maintain near-optimal clustering when queries are registered and removed in an arbitrary fashion against whole data streams or sliding windows. In addition to theoretical analysis, our performance study indicates that the proposed techniques are indeed scalable with respect to the number of input queries as well as the number of items and the item arrival rate in a data stream.

Axiomatisation of an Interval Calculus for Theorem Proving

We provide an axiomatisation of the Timed Interval Calculus, a set-theoretic notation for expressing properties of time intervals. We implement the axiomatisation in the Ergo theorem prover in order to allow the machine-checked proof of laws for reasoning about predicates expressed using interval operators. These laws can be then used in the machine-assisted verification of real-time applications.

Integrating runtime assertions with dynamic types: Structuring a derivation from an incomputable specification

An inherent incomputability in the specification of a functional language extension that combines assertions with dynamic type checking is isolated in an explicit derivation from mathematical specifications. The combination of types and assertions (into "dynamic assertion-types" - DATs) is a significant issue since, because the two are congruent means for program correctness, benefit arises from their better integration in contrast to the harm resulting from their unnecessary separation. However, projecting the "set membership" view of assertion-checking into dynamic types results in some incomputable combinations. Refinement of the specification of DAT checking into an implementation by rigorous application of mathematical identities becomes feasible through the addition of a "best-approximate" pseudo-equality that isolates the incomputable component of the specification. This formal treatment leads to an improved, more maintainable outcome with further development potential.

Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Approximate, Average, Dynamic Models of Uncontrolled Rectifiers for Aircraft Applications

To carry out stability and voltage regulation studies on more electric aircraft systems in which there is a preponderance of multi-pulse, rectifier-fed motor-drive equipment, average dynamic models of the rectifier converters are required. Existing methods are difficult to apply to anything other than single converters with a low pulse number. Therefore an efficient, compact method for deriving the approximate, linear, average model of 6- and 12-pulse rectifiers, based on the assumption of a small duration of the overlap angle is presented. The models are validated against detailed simulations and laboratory prototypes.

Improvements on the Juxtaposing Theorem

A new class of binary constant weight codes is presented. We establish new lower bound and exact values on A(n1 +n2; 2(a1 +a2); n2) ≥ min {M1;M2}+1, if A(n1; 2a1; a1 +b1) = M1 and A(n2; 2b2; a2 +b2) = M2, in particular, A(30; 16; 15) = 16 and A(33; 18; 15) = 11.

On the Various Bisection Methods Derived from Vincent’s Theorem

In 2000 A. Alesina and M. Galuzzi presented Vincent’s theorem “from a modern point of view” along with two new bisection methods derived from it, B and C. Their profound understanding of Vincent’s theorem is responsible for simplicity — the characteristic property of these two methods. In this paper we compare the performance of these two new bisection methods — i.e. the time they take, as well as the number of intervals they examine in order to isolate the real roots of polynomials — against that of the well-known Vincent-Collins-Akritas method, which is the first bisection method derived from Vincent’s theorem back in 1976. Experimental results indicate that REL, the fastest implementation of the Vincent-Collins-Akritas method, is still the fastest of the three bisection methods, but the number of intervals it examines is almost the same as that of B. Therefore, further research on speeding up B while preserving its simplicity looks promising.

On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.

A Remark On S.M. Bates' Theorem

In his paper [1], Bates investigates the existence of nonlinear, but highly smooth, surjective operators between various classes of Banach spaces. Modifying his basic method, he obtains the following striking results.