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A Bijection Between the Adjoint and Coadjoint Orbits of a Semidirect Product
Full text linkWe prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincar ́e group, and discuss the limitations and extent of this result to other groups
Generalized point vortex dynamics on CP^2
No full textThis is the second of two companion papers.
We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space \CP ^2 interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices.
The different types of polytope depend on the values of the `vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of \CP ^2. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points.
For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense
Accurately Computing the Log-Sum-Exp and Softmax Functions
Full text linkEvaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic \new{and shifting can introduce subtractive cancellation}. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy
ABBA: Adaptive Brownian bridge-based symbolic aggregation of time series
Full text linkA new symbolic representation of time series, called ABBA, is introduced. It is based on an adaptive polygonal chain approximation of the time series into a sequence of tuples, followed by a mean-based clustering to obtain the symbolic representation. We show that the reconstruction error of this representation can be modelled as a random walk with pinned start and end points, a so-called Brownian bridge. This insight allows us to make ABBA essentially parameter-free, except for the approximation tolerance which must be chosen. Extensive comparisons with the SAX and 1d-SAX representations are included in the form of performance profiles, showing that ABBA is able to better preserve the essential shape information of time series at compression rates comparable to other algorithms. A Python implementation is provided
Non-Abelian momentum polytopes for products of CP^2
No full textThis is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope for the product of 3 copies, and numerous transition polytopes, all of which are classified here. The different polytopes depend on the weights of the symplectic form on each copy of projective space. In the second paper we use reduction techniques to study the possible dynamics of interacting point vortices.
The results are also applied to determine the inequalities satisfied by the sum of up to three 3x3 Hermitian matrices with double eigenvalues
Persistence of stationary motion under explicit symmetry breaking perturbation
Full text linkExplicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed in a way that the perturbation preserves only some symmetries of the original system. We give a geometric approach to study this phenomenon in the setting of Hamiltonian systems. We provide a method for determining the equilibria and relative equilibria that persist after a symmetry breaking perturbation. In particular a lower bound for the number of each is found, in terms of an equivariant Lyusternik-Schnirelmann category of the group orbit