Daskaloyannis, C. Tanoudis, Y.
The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.
Capel, J. J.; Kress, J. M.
In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Stäckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a seven-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Gröbner basis calculations using the computer algebra software packages Maple and Singular.
Rajaratnam, Krishan McLenaghan, Raymond G.
We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.
Kalnins, L. M.; Valentine, A. P.; Trampert, J.
One relatively obvious surface reflection of certain types of tectonic and mantle processes is volcanic activity. Ocean covers two thirds of our planet, so naturally much of this evidence will be marine, yet the evidence of volcanic activity in the oceans remains very incompletely mapped. Many seamounts, the products of 'excess' volcanism, have been identified (10,000--20,000 over 1 km in height, depending on the study), but it is estimated that up to 60% of seamounts in this height range remain unmapped. Given the scale of the task, identification of probable seamounts is a process that clearly needs to be automated, but identifying naturally occurring features such as these is difficult because of the degree of inherent variation. A very promising avenue for these questions lies in the use of learning algorithms, such as neural networks, designed to have complex pattern recognition capabilities. Building on the work of Valentine et al. (2013), we present preliminary results of a new global seamount study based on neural network methods. Advantages of this approach include an intrinsic measure of confidence in the seamount identification and full automation, allowing easy re-picking to suit the requirements of different types of studies. Here, we examine the resulting spatial and temporal distribution of marine volcanism and consider what insights this offers into the shifting patterns of plate tectonics and mantle activity. We also consider the size distribution of the seamounts and explore possible classes based on shape and their distributions, potentially reflecting both differing formational processes and later erosional processes. Valentine, A. P., L. M. Kalnins, and J. Trampert (2013), Discovery and analysis of topographic features using learning algorithms: A seamount case study, Geophysical Research Letters, 40(12), p. 3048--3054.
Szereszewski, A.; Sym, A.
are binary metrics. In this paper we solve the following problem: to classify all conformally flat (of arbitrary signature) 4-dimensional binary metrics. Among them there are 1) those that are separable in the sense of SRE metrics Kalnins-Miller (1978 Trans. Am. Math. Soc. 244 241-61 1982 J. Phys. A: Math. Gen. 15 2699-709 1984 Adv. Math. 51 91-106 1983 SIAM J. Math. Anal. 14 126-37) and 2) new examples of non-Stäckel R-separability in 4 dimensions.
Valentine, Andrew; Kalnins, Lara; van Dinther, Chantal; Trampert, Jeannot
global seamount census from altimetry-derived gravity data, Geophysical Journal International, 186, pp.615-631. Valentine, A., Kalnins, L. & Trampert, J., 2012. Hunting for seamounts using neural networks: learning algorithms for geomorphic studies, EGU General Assembly, Abstract EGU2012-4560. Valentine, A. & Trampert, J., 2012. Data-space reduction, quality assessment and searching of seismograms: Autoencoder networks for waveform data. Geophysical Journal International, 189, pp.1183-1201.