FFTs on the Rotation Group

We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B ⁴) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B ³)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B ⁶) algorithm derived from a basic quadrature rule on O(B ³) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B ³log ² B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well. First author was supported by NSF ITR award; second author was supported by NSF Grant 0219717 and the Santa Fe Institute.     

Monotone and Convex C 1 Hermite Interpolants Generated by a Subdivision Scheme

   Abstract. We propose C ¹ Hermite interpolants generated by the general subdivision scheme introduced by Merrien [17] and satisfying monotonicity or convexity constraints. For arbitrary values and slopes of a given function f at the end-points of a bounded interval, which are compatible with the contraints, the given algorithms construct shape-preserving interpolants. Moreover, these algorithms are quite simple and fast as well as adapted to CAGD. We also give error estimates in the case of interpolation of smooth functions.     

Phase Spaces and Deformation Theory

In the papers (Laudal in Contemporary Mathematics, vol. 391, [2005]; Geometry of time-spaces, Report No. 03, [2006/2007]), we introduced the notion of (non-commutative) phase algebras (spaces) Ph ⁿ (A), n=0,1,…,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to study this construction in some more detail. This seems to give us a possible framework for the study of non-commutative partial differential equations. We refer to the paper (Laudal in Phase spaces and deformation theory, Report No. 09, [2006/2007]), for the applications to non-commutative deformation theory, Massey products and for the construction of the versal family of families of modules. See also (Laudal in Homology, Homotopy, Appl. 4:357–396, [2002]; Proceedings of NATO Advanced Research Workshop, Computational Commutative and Non-Commutative Algebraic Geometry, [2004]).      

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

An FIO calculus for marine seismic imaging, II: Sobolev estimates

We establish sharp L ²-Sobolev estimates for classes of pseudodifferential operators with singular symbols [Guillemin and Uhlmann (Duke Math J 48:251–267, 1981), Melrose and Uhlmann (Commun Pure Appl Math 32:483–519, 1979)] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in \mathbb R²{\mathbb R^2} with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics [Felea (Comm PDE 30:1717–1740, 2005), Felea and Greenleaf (Comm PDE 33:45–77, 2008), Nolan (SIAM J Appl Math 61:659–672, 2000)].     

Monotone and Convex C 1 Hermite Interpolants Generated by a Subdivision Scheme

Abstract. We propose C ¹ Hermite interpolants generated by the general subdivision scheme introduced by Merrien [17] and satisfying monotonicity or convexity constraints. For arbitrary values and slopes of a given function f at the end-points of a bounded interval, which are compatible with the contraints, the given algorithms construct shape-preserving interpolants. Moreover, these algorithms are quite simple and fast as well as adapted to CAGD. We also give error estimates in the case of interpolation of smooth functions.     

On some new general theorems for the explicit evaluations of Ramanujan’s remarkable product of theta-functions

On pages 338 and 339 in his first notebook (Notebooks (2 volumes), [1957]), Ramanujan records eighteen values for a certain product of theta-functions. All these have been proved by B.C. Berndt, H.H. Chan and L.-C. Zhang (Proc. Edinburg Math. Soc. 40:583–612, [1997]). The main purpose of this paper is to establish some new general theorems for the explicit evaluations of Ramanujan’s product of theta-functions and some new explicit evaluations there from.      

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.     

A <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Multivariate Clough–Tocher Interpolant

We show that in dimensions four and higher, to insure a smooth interpolant, additional geometric constraints must be imposed on the generalized Clough–Tocher split introduced in Worsey and Farin (Constr. Approx. 3:99–110, [1987]).      

Symmetries and First Integrals of Differential Equations

It is known that an n-dimensional system of ordinary differential equations with Lie symmetry which involves a divergence-free Liouville vector field possesses n−1 independent first integrals (i.e., it is algebraically integrable) (ünal in Phys. Lett. A 260:352–359, [1999]). In the present paper, we show that if an n-dimensional system of ordinary differential equations admits a C ^∞-symmetry vector field which satisfies some special conditions, then it also possesses n−1 independent first integrals. Several examples are given to illustrate our result. Y. Li’s research was partially supported by NSFC Grants 10531050, 10225107, SRFDP Grant 20040183030, and the 985 program of Jilin University.     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

Approximate Isomorphisms in <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C ^*-algebras:

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]). Research supported by a grant from CONACYT (Mexico).     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

Solution Methods for Pseudomonotone Variational Inequalities

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. This research was supported in part by a grant from the National Sun Yat-Sen University, Kaohsiung, Taiwan. It has been carried out under the agreement between the National Sun Yat-Sen University, Kaohsiung, Taiwan and the University of Pisa, Pisa, Italy. The authors thank the anonymous referee for useful comments and suggestions.     

On m-regular systems on ℋ(5,q 2)

The notion of m-regular system on the Hermitian variety ℋ(n,q ²) was introduced by B. Segre (Ann. Math. Pura Appl. 70:1–201, 1965). Here, three infinite families of hemisystems on ℋ(5,q ²), q odd, are constructed.     

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), López and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.      

On the Convergence of the Pshenichnyi-Pironneau-Polak Minimax Algorithm with an Active Set Strategy

The purpose of this technical note is to present a proof of convergence of the Pshenichnyi-Pironneau-Polak (PPP) minimax algorithm (see Algorithm 2.4.1 in Polak, Optimization: Algorithms and Consistent Approximations, Springer, [1997]), modified to use an active set strategy. This active set strategy was formally derived in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]) from those used in the methods of feasible directions developed by Zoutendijk (Methods of Feasible Directions, Elsevier, [1960]) and Polak (Computational Methods in Optimization: A Unified Approach, Academic, [1971]). The resulting ε-Active PPP algorithm was presented as Algorithm 2.4.34, in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]), without any proofs.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.