Transforms for minimal surfaces in the 5-sphere

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction considered in a previous paper, and illustrate this explicitly in the case of bipolar minimal surfaces.     

FFTs for the 2-Sphere-Improvements and Variations

Earlier work by Driscoll and Healy has produced an efficient algorithm for computing the Fourier transform of band-limited functions on the 2-sphere. In this article we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions. All require at most O(N log² N) operations where N is the number of sample points. We also address implementation considerations and give heuristics for allowing reliable and computationally efficient floating point implementations of slightly modified algorithms. These claims are supported by extensive numerical experiments from our implementation in C on DEC, HP, SGI and Linux Pentium platforms. These results indicate that variations of the algorithm are both reliable and efficient for a large range of useful problem sizes. Performance appears to be architecture-dependent. The article concludes with a brief discussion of a few potential applications.     

Non-Equispaced Fast Fourier Transforms with Applications to Tomography

In this article we describe a non-equispaced fast Fourier transform. It is similar to the algorithms of Dutt and Rokhlin and Beylkin but is based on an exact Fourier series representation. This results in a greatly simplified analysis and increased flexibility. The latter can be used to achieve more efficiency. Accuracy and efficiency of the resulting algorithm are illustrated by numerical examples. In the second part of the article the non-equispaced FFT is applied to the reconstruction problem in Computerized Tomography. This results in a different view of the gridding method of OSullivan and in a new ultra fast reconstruction algorithm. The new reconstruction algorithm outperforms the filtered backprojection by a speedup factor of up to 100 on standard hardware while still producing excellent reconstruction quality.     

Ratio Mercerian Theorems with Applications to Hankel and Fourier Transforms

We complete and complement our recent work on Drasin-Shea-Jordantheorems for Fourier and Hankel transforms. In improving onthe methods of our previous work, we were led to certain ratioMercerian theorems for general kernels; these yield definitiveversions of our earlier results for Fourier and Hankel transforms.1991 Mathematics Subject Classification: primary 44A15; secondary47B38.     

Legendre Transform and Applications to Finite and Infinite Optimization

We investigate convex constrained nonlinear optimization problems and optimal control with convex state constraints in the light of the so-called Legendre transform. We use this change of coordinate to propose a gradient-like algorithm for mathematical programs, which can be seen as a search method along geodesics. We also use the Legendre transform to study the value function of a state constrained Mayer problem and we show that it can be characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.     

Componentwise error analysis for FFTs with applications to fast Helmholtz solvers

We analyze the stability of the Cooley-Tukey algorithm for the Fast Fourier Transform of ordern=2 ^k and of its inverse by using componentwise error analysis.We prove that the components of the roundoff errors are linearly related to the result in exact arithmetic. We describe the structure of the error matrix and we give optimal bounds for the total error in infinity norm and inL ₂ norm.The theoretical upper bounds are based on a worst case analysis where all the rounding errors work in the same direction. We show by means of a statistical error analysis that in realistic cases the max-norm error grows asymptotically like the logarithm of the sequence length by machine precision.Finally, we use the previous results for introducing tight upper bounds on the algorithmic error for some of the classical fast Helmholtz equation solvers based on the Faster Fourier Transform and for some algorithms used in the study of turbulence.     

Legendre Pseudospectral Approximation of Boussinesq Systems and Applications to Wave Breaking

In this paper, we propose a spectral projection of a regularized Boussinesq system for wave propagation on the surface of a fluid. The spectral method is based on the use of Legendre polynomials, and is able to handle time-dependent Dirichlet boundary conditions with spectral accuracy.The algorithm is applied to the study of undular bores, and in particular to the onset of wave breaking connected with undular bores. As proposed in [2], an improved version of the breaking criterion recently introduced in [5] is used. This tightened breaking criterion together with a careful choice of the relaxation parameter yields rather accurate predictions of the onset of breaking in the leading wave of an undular bore.     

The 2-homeotoxal tilings of the plane and the 2-sphere

A complete enumeration is obtained of the finite tilings of the 2-sphere and of the infinite tilings of the plane which are normal and admit two transitivity classes of edges under topological automorphisms of the tiling. Every type is found to be representable by a tiling in which the edges form two transitivity classes under isometric symmetries of the tiling.     

The triangulations of the 3-sphere with up to 8 vertices

The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined. Using similar methods we show that one cannot always preassign the shape of a facet of a 4-polytope.     

Q-Learning Algorithms with Random Truncation Bounds and Applications to Effective Parallel Computing

Motivated by an important problem of load balancing in parallel computing, this paper examines a modified algorithm to enhance Q-learning methods, especially in asynchronous recursive procedures for self-adaptive load distribution at run-time. Unlike the existing projection method that utilizes a fixed region, our algorithm employs a sequence of growing truncation bounds to ensure the boundedness of the iterates. Convergence and rates of convergence of the proposed algorithm are established. This class of algorithms has broad applications in signal processing, learning, financial engineering, and other related fields. G. Yin’s research was supported in part by the National Science Foundation under Grants DMS-0603287 and DMS-0624849 and in part by the National Security Agency under Grant MSPF-068-029. C.Z. Xu’s research was supported in part by the National Science Foundation under Grants CCF-0611750, DMS-0624849, CNS-0702488, and CRI-0708232. L.Y. Wang’s research was supported in part by the National Science Foundation under Grants ECS-0329597 and DMS-0624849 and by the Michigan Economic Development Council.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

On the theory of fronts on the 2-sphere and the theory of space curves

We apply the singularity theory of caustics and wave fronts on the 2-sphere and the theory of Legendrian curves of the (contact) manifold ST ^{* 2} P ³ (consisting of tangent contact elements of the sphere ²) to obtain new results in the theory of curves in the Euclidean 3-space. In this way, the singularity theory of caustics and wave fronts provides a new insight to the differential geometry of curves in the Euclidean 3-space.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 7, Suzdal Conference-1, 2003.This revised version was published online in April 2005 with a corrected cover date.     

Approximation of Scattered Data by Trigonometric Polynomials on the Torus and the 2-sphere

Fast, efficient and reliable algorithms for the discrete least-square approximation of scattered points on the torus T ^d and the sphere S ² by trigonometric polynomials are presented. The algorithms are based on iterative CG-type methods in combination with fast Fourier transforms for nonequispaced data. The emphasis is on numerical aspects, in order to solve large scale problems. Numerical examples show the efficiency of the new algorithms.     

Generalized Legendre series and the fundamental solution of the Laplacian on the <Emphasis Type="Italic">n</Emphasis>-sphere

Summary Classical representations of the Legendre polynomials in terms of generating functions are nonconvergent in the usual sense for extreme values of the generating parameter. A generalized type of convergence is explored and, in conjunction with the theory of Fourier series and spherical harmonics, it is applied to the computation of the fundamental solution of the Laplace-Beltrami operator on the n-sphere.