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On the stability of Hamiltonian relative equilibria with non-trivial isotropy
Get PDFWe consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extended to apply to systems whose momentum map is not coadjoint equivariant
An Exploration of the Relationship between Mathematics and Music
Get PDFAn exploration of the relationship between math and music: a readable and scholarly double project, drawing from many sources, and beautifully illustrated
Adaptive Solvers for Elliptic and Parabolic Partial Differential Equations
Get PDFIn this thesis our primary interest is in developing adaptive solution methods for parabolic and elliptic partial differential equations. The convection-diffusion equation is used as a representative test problem.
Investigations are made into adaptive temporal solvers implementing only a few changes to existing software. This includes a comparison of commercial code against some more academic releases. A novel way to select step sizes for an adaptive BDF2 code is introduced.
A chapter is included introducing some functional analysis that is required to understand aspects of the finite element method and error estimation. Two error estimators are derived and proofs of their error bounds are covered.
A new finite element package is written, implementing a rather interesting error estimator in one dimension to drive a rather standard refinement/coarsening type of adaptivity. This is compared to a commercially available partial differential equation solver and an investigation into the properties of the two inspires the development of a new method designed to very quickly and directly equidistribute the errors between elements. This new method is not really a refinement technique but doesn�t quite fit the traditional description of a moving mesh either. We show that this method is far more effective at equidistribution of errors than a simple moving mesh method and the original simple adaptive method. A simple extension of the new method is proposed that would be a mesh reconstruction method.
Finally the new code is extended to solve steady-state problems in two dimensions. The mesh refinement method from one dimension does not offer a simple extension, so the error estimator is used to supply an impression of the local topology of the error on each element. This in turn allows us to develop a new anisotropic refinement algorithm, which is more in tune with the nature of the error on the parent element. Whilst the benefits observed in one dimension are not directly transferred into the two-dimensional case, the obtained meshes seem to better capture the topology of the solution
Adaptive time-stepping for incompressible flow Part II: Navier-Stokes Equations
No full textWe outline a new class of robust and efficient methods for solving the Navier-Stokes equations. We describe a general solution strategy that has two basic building blocks; a fully implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach
A novel Ocular Anaesthetic Scoring System, OASS, tool to measure both motor and sensory function following local anaesthesia
Get PDFBackground/Aims: To devise and evaluate a novel Ocular Anaesthetic Scoring System (OASS) for non-topical local anaesthesia.
Methods: In OASS a score of between 0 (poor) and 14 (excellent) was devised measuring motor (ocular motility, levator and orbicularis function) and sensory functions (digital spear pressure at limbus and topical anaesthetic sting). 40 patients were studied prospectively to analyse inter-observer consistency in OASS. A further 100 patients were collected into four groups receiving either sub-Tenon's or peribulbar block with 150 or 300 units of hyaluronidase. Patient satisfaction was determined using Visual Analogue Pain Scale (VAPS) and Iowa Satisfaction with Anaesthesia Scale (ISAS).
Results: There was no significant difference in OASS scores between two independent observers (p=0.8910). The sub-Tenon's approach achieved significantly better OASS scores than the peribulbar approach (p sub-Tenon's with 150 units of hyaluronidase > peribulbar with 300 units of hyaluronidase > peribulbar with 150 units of hyaluronidase.
Conclusion: OASS is a simple and robust system for assessing and comparing non-topical local anaesthetic techniques. Of the techniques evaluated a sub-Tenon's block with 300 units of hyaluronidase gives best anaesthesia, analgesia and patient satisfaction results
Bar-like S-cone stimuli reveal the importance of an intermediate temporal filter
Get PDFThe relative involvement of different temporal frequency-selective filters underlying detection of chromatic stimuli were studied. Diverse spectral stimuli were used, namely flashed blue and yellow light spots, wide bars and narrow bars. The stimuli were temporally modulated in luminance having constant wavelength. Although stimulus elongation apparently reduced the sensitivity at short and long wavelengths, the cone-opponent mechanism still remained responsible for the actual stimulus detection at different temporal frequencies. Stimulus elongation increased sensitivity for temporal frequencies around 3-6 Hz, revealing involvement of the intermediate temporal frequency-selective filters to detection, the so-called first transient-1 filter. A probability summation model for the method of adjustment was developed that assumes that detection depends on the properties of the temporal filters underlying the temporal frequency-sensitivity curve. The model supports the notion that at least two temporal frequency-selective filters are necessary to account for the shape of the sensitivity curves obtained for blue stimuli
Mercator's projection, logarithm and seafaring
Get PDFThe principal mathematical content of this paper amounts to a classical proposition:
"the cylinder is the uniformising surface for the
logarithm".
We believe that this interpretation of the Mercator projection as the logarithm in the complex domain should be part of every course of complex analysis -- but, unfortunately, we failed to find this simple observation in modern undergraduate textbooks
A Comparison of Multivariable & Decentralized Control Strategies for Robust Humanoid Walking
Get PDFBipedal walking is one of the most interesting control problems in humanoids research. Walking is modelled as a hybrid system in the sense that it involves various phases such as single support phase, impacts with the ground (i.e. a state reset) and the double support phase. The control system has to provide good dynamic performance in these different modes to achieve fast walking speeds while guaranteeing its safe and robust operation. Most humanoids use local joint PID loops (decentralized) control systems while the robot is a multivariable system and walking involves significant interactions between the robot links. Hence in this paper a centralized LQR multivariable controller is designed for the robot and analyzed for its stability, robustness to noise and disturbances and dynamic performance. Then, an LQR based iterative algorithm is used to tune the local PID servos. A comparison between the two schemes is done, where it is shown that the multivariable LQR has better robustness and energy efficiency. Finally, both controllers are simulated using the linearized model of a 10 degree of freedom robot called “C-Cub”
THE KO*-RINGS OF BTm, THE DAVIS-JANUSZKIEWICZ SPACES AND CERTAIN TORIC MANIFOLDS
Get PDFThis paper contains an explicit computation of the KO*-ring structure of an
m-fold product of CP1, the Davis-Januszkiewicz spaces and of toric manifolds which have
trivial Sq2-homology. A key ingredient is the stable splitting of the Davis-Januszkiewicz
spaces given in [6]
Computing a Nearest Correlation Matrix with Factor Structure
Get PDFAn correlation matrix has factor structure if its off-diagonal agrees with that of a rank matrix. Such correlation matrices arise, for example, in factor models of collateralized debt obligations (CDOs) and multivariate time series. We analyze the properties of these matrices and, in particular, obtain an explicit formula for the rank in the one factor case. Our main focus is on the nearness problem of finding the nearest factor correlation matrix C(X) = \diag(I-XX^T) + XX^T to a given symmetric matrix, subject to natural nonlinear constraints on the elements of the matrix , where distance is measured in the Frobenius norm. For a special one parameter case we obtain an explicit solution. For the general factor case we obtain the gradient and Hessian of the objective function and derive an instructive result on the positive definiteness of the Hessian when . We investigate several numerical methods for solving the nearness problem: the alternating directions method; a principal factors method used by Anderson, Sidenius, and Basu in the CDO application, which we show is equivalent to the alternating projections method and lacks convergence results; the spectral projected gradient method of Birgin, Mart{\'\i}nez, and Raydan; and Newton and sequential quadratic programming methods. The methods differ in whether or not they can take account of the nonlinear constraints and in their convergence properties. Our numerical experiments show that the performance of the methods depends strongly on the problem, but that the spectral projected gradient method is the clear winner