Perspex machine VII: The universal perspex machine

The perspex machine arose from the unification of projective geometry with the Turing machine. It uses a total arithmetic, called transreal arithmetic, that contains real arithmetic and allows division by zero. Transreal arithmetic is redefined here. The new arithmetic has both a positive and a negative infinity which lie at the extremes of the number line, and a number nullity that lies off the number line. We prove that nullity, 0/0, is a number. Hence a number may have one of four signs: negative, zero, positive, or nullity. It is, therefore, impossible to encode the sign of a number in one bit, as floating-, point arithmetic attempts to do, resulting in the difficulty of having both positive and negative zeros and NaNs. Transrational arithmetic is consistent with Cantor arithmetic. In an extension to real arithmetic, the product of zero, an infinity, or nullity with its reciprocal is nullity, not unity. This avoids the usual contradictions that follow from allowing division by zero. Transreal arithmetic has a fixed algebraic structure and does not admit options as IEEE, floating-point arithmetic does. Most significantly, nullity has a simple semantics that is related to zero. Zero means "no value" and nullity means "no information." We argue that nullity is as useful to a manufactured computer as zero is to a human computer. The perspex machine is intended to offer one solution to the mind-body problem by showing how the computable aspects of mind and. perhaps, the whole of mind relates to the geometrical aspects of body and, perhaps, the whole of body. We review some of Turing's writings and show that he held the view that his machine has spatial properties. In particular, that it has the property of being a 7D lattice of compact spaces. Thus, we read Turing as believing that his machine relates computation to geometrical bodies. We simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to pro-ram than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to construct a fractal of perspex machines with the cardinality of a real-numbered line or space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. We cite an experiment that has been proposed to test the physical reality of the perspex machine's model of time, but we make no claim that the physical universe works this way or that it has the cardinality of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of physical things, including manufactured computers and biological organisms, that have a cardinality no greater than the real-number line.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state.We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

Children's sociospatial (re)production of disability within primary school playgrounds

There is a contemporary shift in the institutional context of 'disabled' children's education in the United Kingdom from segregated special to mainstream schools. This change is tied to wider deinstitutionalised or reinstitutionalised geographies of disabled people, fragile globalised educational 'inclusion' agendas, and broader concerns about social cohesiveness. Although coeducating children is expected to transform negative representations of (dis)ability in future society, there are few detailed explorations of how children's everyday sociospatial practices (re)produce or transform dominant representations of (dis)ability. With this in mind, children's contextual and shifting performances of (dis)ability in two case study school playground (recreational) spaces are explored. The findings demonstrate that children with mind-body differences are variously (dis)abled, in comparison with sociospatially shifting norms of ability, which have body, learning, and emotional-social facets. The discussion therefore places an emphasis on the need to incorporate 'intellectual' and 'emotional' differences more fully into geographical studies of disability and identity. The paper has wider resonance for transformative expectations placed on colocating children with a variety of 'axes of difference' (such as gender, 'race, ethnicity, and social class) in schools.

Biases in comparative analyses of extinction risk: mind the gap

1. Comparative analyses are used to address the key question of what makes a species more prone to extinction by exploring the links between vulnerability and intrinsic species’ traits and/or extrinsic factors. This approach requires comprehensive species data but information is rarely available for all species of interest. As a result comparative analyses often rely on subsets of relatively few species that are assumed to be representative samples of the overall studied group. 2. Our study challenges this assumption and quantifies the taxonomic, spatial, and data type biases associated with the quantity of data available for 5415 mammalian species using the freely available life-history database PanTHERIA. 3. Moreover, we explore how existing biases influence results of comparative analyses of extinction risk by using subsets of data that attempt to correct for detected biases. In particular, we focus on links between four species’ traits commonly linked to vulnerability (distribution range area, adult body mass, population density and gestation length) and conduct univariate and multivariate analyses to understand how biases affect model predictions. 4. Our results show important biases in data availability with c.22% of mammals completely lacking data. Missing data, which appear to be not missing at random, occur frequently in all traits (14–99% of cases missing). Data availability is explained by intrinsic traits, with larger mammals occupying bigger range areas being the best studied. Importantly, we find that existing biases affect the results of comparative analyses by overestimating the risk of extinction and changing which traits are identified as important predictors. 5. Our results raise concerns over our ability to draw general conclusions regarding what makes a species more prone to extinction. Missing data represent a prevalent problem in comparative analyses, and unfortunately, because data are not missing at random, conventional approaches to fill data gaps, are not valid or present important challenges. These results show the importance of making appropriate inferences from comparative analyses by focusing on the subset of species for which data are available. Ultimately, addressing the data bias problem requires greater investment in data collection and dissemination, as well as the development of methodological approaches to effectively correct existing biases.

The method of least squares used to invert an orbit problem

Six parameters uniquely describe the orbit of a body about the Sun. Given these parameters, it is possible to make predictions of the body's position by solving its equation of motion. The parameters cannot be directly measured, so they must be inferred indirectly by an inversion method which uses measurements of other quantities in combination with the equation of motion. Inverse techniques are valuable tools in many applications where only noisy, incomplete, and indirect observations are available for estimating parameter values. The methodology of the approach is introduced and the Kepler problem is used as a real-world example. (C) 2003 American Association of Physics Teachers.

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.

Explaining deviations from NAV in UK property companies: rationality and sentimentality

One of the most vexing issues for analysts and managers of property companies across Europe has been the existence and persistence of deviations of Net Asset Values of property companies from their market capitalisation. The issue has clear links to similar discounts and premiums in closed-end funds. The closed end fund puzzle is regarded as an important unsolved problem in financial economics undermining theories of market efficiency and the Law of One Price. Consequently, it has generated a huge body of research. Although it can be tempting to focus on the particular inefficiencies of real estate markets in attempting to explain deviations from NAV, the closed end fund discount puzzle indicates that divergences between underlying asset values and market capitalisation are not a ‘pure’ real estate phenomenon. When examining potential explanations, two recurring factors stand out in the closed end fund literature as often undermining the economic rationale for a discount – the existence of premiums and cross-sectional and periodic fluctuations in the level of discount/premium. These need to be borne in mind when considering potential explanations for real estate markets. There are two approaches to investigating the discount to net asset value in closed-end funds: the ‘rational’ approach and the ‘noise trader’ or ‘sentiment’ approach. The ‘rational’ approach hypothesizes the discount to net asset value as being the result of company specific factors relating to such factors as management quality, tax liability and the type of stocks held by the fund. Despite the intuitive appeal of the ‘rational’ approach to closed-end fund discounts the studies have not successfully explained the variance in closed-end fund discounts or why the discount to net asset value in closed-end funds varies so much over time. The variation over time in the average sector discount is not only a feature of closed-end funds but also property companies. This paper analyses changes in the deviations from NAV for UK property companies between 2000 and 2003. The paper present a new way to study the phenomenon ‘cleaning’ the gearing effect by introducing a new way of calculating the discount itself. We call it “ungeared discount”. It is calculated by assuming that a firm issues new equity to repurchase outstanding debt without any variation on asset side. In this way discount does not depend on an accounting effect and the analysis should better explain the effect of other independent variables.

Body-centric modelling, identification, and acceleration tracking control of a quadrotor UAV

This paper presents the mathematical development of a body-centric nonlinear dynamic model of a quadrotor UAV that is suitable for the development of biologically inspired navigation strategies. Analytical approximations are used to find an initial guess of the parameters of the nonlinear model, then parameter estimation methods are used to refine the model parameters using the data obtained from onboard sensors during flight. Due to the unstable nature of the quadrotor model, the identification process is performed with the system in closed-loop control of attitude angles. The obtained model parameters are validated using real unseen experimental data. Based on the identified model, a Linear-Quadratic (LQ) optimal tracker is designed to stabilize the quadrotor and facilitate its translational control by tracking body accelerations. The LQ tracker is tested on an experimental quadrotor UAV and the obtained results are a further means to validate the quality of the estimated model. The unique formulation of the control problem in the body frame makes the controller better suited for bio-inspired navigation and guidance strategies than conventional attitude or position based control systems that can be found in the existing literature.