Riemannian cubics are curves in a Riemannian manifold M satisfying a variational condition. They arise in computer graphics and in trajectory planning problems for rigid body motion, where M is the group SO ͑3͒ of rotations of Euclidean threespace E 3 . Riemannian cubics on a Lie group correspond to Lie quadratics in the Lie algebra. There are only a few cases where closed-form expressions are available for Lie quadratics. The present article is a qualitative analysis of null quadratics in so͑3͒, focusing on long term dynamics and internal symmetries. Conclusions are drawn for asymptotics and symmetries of null cubics in SO͑3͒.
A new method for compiling quantum algorithms is proposed and tested for a three qubit system. The proposed method is to decompose a a unitary matrix U, into a product of simpler U j via a neural network. These U j can then be decomposed into product of known quantum gates. Key to the effectiveness of this approach is the restriction of the set of training data generated to paths which approximate minimal normal subRiemannian geodesics, as this removes unnecessary redundancy and ensures the products are unique. The two neural networks are shown to work effectively, each individually returning low loss values on validation data after relatively short training periods. The two networks are able to return coefficients that are sufficiently close to the true coefficient values to validate this method as an approach for generating quantum circuits. There is scope for more work in scaling this approach for larger quantum systems.
In his paper [Takens, 1981] on strange attractors and turbulence, Floris Takens proves a theorem giving conditions under which a discrete-time dynamical system can be reconstructed from scalar-valued partial measurements of internal states. We discuss Takens' theorem in terms suitable for a general audience, and give an alternative and more detailed proof of this important result, making use of two basic facts. The first is the Whitney embedding theorem, which we use in an alternative to Takens' original argument away from periodic points of small periods. Near the periodic points we adapt a proof that typical scalar-output linear time-invariant control systems are completely observable.
To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1, 2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the formẋ(t) = (β 0 +tβ 1 )x(t), where β 0 , β 1 are skew-symmetric 3×3 matrices, and x : R → SO(3). This is done by showing that the dual of β 0 + tβ 1 is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics.
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