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.

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.

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.

Modelling metal cutting using modern ductile fracture mechanics: Quantitative explanations for some longstanding problems

The assumption that negligible work is involved in the formation of new surfaces in the machining of ductile metals, is re-examined in the light of both current Finite Element Method (FEM) simulations of cutting and modern ductile fracture mechanics. The work associated with separation criteria in FEM models is shown to be in the kJ/m2 range rather than the few J/m2 of the surface energy (surface tension) employed by Shaw in his pioneering study of 1954 following which consideration of surface work has been omitted from analyses of metal cutting. The much greater values of surface specific work are not surprising in terms of ductile fracture mechanics where kJ/m2 values of fracture toughness are typical of the ductile metals involved in machining studies. This paper shows that when even the simple Ernst–Merchant analysis is generalised to include significant surface work, many of the experimental observations for which traditional ‘plasticity and friction only’ analyses seem to have no quantitative explanation, are now given meaning. In particular, the primary shear plane angle φ becomes material-dependent. The experimental increase of φ up to a saturated level, as the uncut chip thickness is increased, is predicted. The positive intercepts found in plots of cutting force vs. depth of cut, and in plots of force resolved along the primary shear plane vs. area of shear plane, are shown to be measures of the specific surface work. It is demonstrated that neglect of these intercepts in cutting analyses is the reason why anomalously high values of shear yield stress are derived at those very small uncut chip thicknesses at which the so-called size effect becomes evident. The material toughness/strength ratio, combined with the depth of cut to form a non-dimensional parameter, is shown to control ductile cutting mechanics. The toughness/strength ratio of a given material will change with rate, temperature, and thermomechanical treatment and the influence of such changes, together with changes in depth of cut, on the character of machining is discussed. Strength or hardness alone is insufficient to describe machining. The failure of the Ernst–Merchant theory seems less to do with problems of uniqueness and the validity of minimum work, and more to do with the problem not being properly posed. The new analysis compares favourably and consistently with the wide body of experimental results available in the literature. Why considerable progress in the understanding of metal cutting has been achieved without reference to significant surface work is also discussed.

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.

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.

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.

A method of evaluating the accuracy of human body thermoregulation models

Human Body Thermoregulation Models have been widely used in the field of human physiology or thermal comfort studies. However there are few studies on the evaluation method for these models. This paper summarises the existing evaluation methods and critically analyses the flaws. Based on that, a method for the evaluating the accuracy of the Human Body Thermoregulation models is proposed. The new evaluation method contributes to the development of Human Body Thermoregulation models and validates their accuracy both statistically and empirically. The accuracy of different models can be compared by the new method. Furthermore, the new method is not only suitable for the evaluation of Human Body Thermoregulation Models, but also can be theoretically applied to the evaluation of the accuracy of the population-based models in other research fields.

The multiple team formation problem using sociometry

The Team Formation problem (TFP) has become a well-known problem in the OR literature over the last few years. In this problem, the allocation of multiple individuals that match a required set of skills as a group must be chosen to maximise one or several social positive attributes. Speci�cally, the aim of the current research is two-fold. First, two new dimensions of the TFP are added by considering multiple projects and fractions of people's dedication. This new problem is named the Multiple Team Formation Problem (MTFP). Second, an optimization model consisting in a quadratic objective function, linear constraints and integer variables is proposed for the problem. The optimization model is solved by three algorithms: a Constraint Programming approach provided by a commercial solver, a Local Search heuristic and a Variable Neighbourhood Search metaheuristic. These three algorithms constitute the first attempt to solve the MTFP, being a variable neighbourhood local search metaheuristic the most effi�cient in almost all cases. Applications of this problem commonly appear in real-life situations, particularly with the current and ongoing development of social network analysis. Therefore, this work opens multiple paths for future research.