Polynomial integration over the unit sphere

The main objective of this work is the implementation of recursive formulas allowing the integration of a high order polynomial expression on the unit sphere. These formulas facilitate the evaluation of very complex computations. The proofs of the formulas are based on mathematical induction as well as the divergence theorem.     

Chaplygin ball over a fixed sphere: an explicit integration

We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel-Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.      

Rubber rolling over a sphere

“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G ₂). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S ² and base S ², that can be reduced to an almost Hamiltonian system in T*S ² with a non-closed 2-form ω_NH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I _j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω_NH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I _j are known.      

Moving triangles over a sphere

If three points on a circle are rotated around the circle in such a way that the lengths of two sides of the triangle they define do not change, then neither does the length of the third side. If instead of a circle we consider the unit sphere of a norm in a two‐dimensional normed space, then the above property characterizes the ellipses. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L-S category as its ‘once punctured submanifold’ N\{P},P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.     

Rolling of a homogeneous ball over a dynamically asymmetric sphere

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.     

Seventh degree integration rules for the sphere

A one-parameter family of rules is derived with all weights positive and all integration points within the sphere.     

Global convergence of SSM for minimizing a quadratic over a sphere

In an earlier paper [Minimizing a quadratic over a sphere, SIAM J. Optim., 12 (2001), 188-208], we presented the sequential subspace method (SSM) for minimizing a quadratic over a sphere. This method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed. We showed in this earlier paper that when the subspace contains a vector obtained by applying one step of Newton's method to the first-order optimality system, SSM is locally, quadratically convergent, even when the original problem is degenerate with multiple solutions and with a singular Jacobian in the optimality system. In this paper, we prove (nonlocal) convergence of SSM to a global minimizer whenever each SSM subspace contains the following three vectors: (i) the current iterate, (ii) the gradient of the cost function evaluated at the current iterate, and (iii) an eigenvector associated with the smallest eigenvalue of the cost function Hessian. For nondegenerate problems, the convergence rate is at least linear when vectors (i)-(iii) are included in the SSM subspace.

     

Exact point-distributions over the complex sphere

We study point-distributions over the surface of the unit sphere in unitary space that generate quadrature rules which are exact for spherical polynomials up to a certain bi-degree. In this first stage, we explore several different characterizations for this type of point sets using standard tools such as, positive definiteness, reproducing kernel techniques, linearization formulas, etc. We find bounds on the cardinality of a point-distribution, without discussing the deeper question regarding best bounds. We include examples, construction methods and explain, via isometric embeddings from real to complex spheres, the proper connections with the so-called spherical designs.     

Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

     

Minimal triangulations of sphere bundles over the circle

For integers d?2 and ε=0 or 1, let S1,d−1(ε) denote the sphere product S¹×Sd−1 if ε=0 and the twisted sphere product if ε=1. The main results of this paper are: (a) if then S1,d−1(ε) has a unique minimal triangulation using 2d+3 vertices, and (b) if then S1,d−1(ε) has minimal triangulations (not unique) using 2d+4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S1,d−1(ε) has at most one (2d+3)-vertex triangulation (one if , zero otherwise), in sharp contrast, the number of non-isomorphic (2d+4)-vertex triangulations of these d-manifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d?3, there is a unique (2d+3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a non-simply connected closed d-manifold requires at least 2d+3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

Motivic integration over Deligne-Mumford stacks

The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of Deligne-Mumford stacks.     

A note on semidefinite programming relaxations for polynomial optimization over a single sphere

We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.