How good can polynomial interpolation on the sphere be?

This paper explores the quality of polynomial interpolation approximations over the sphere S ^r−1⊂R ^r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ _n ‖ of the interpolation operator Λ _n , considered as a map from C(S ^r−1) to C(S ^r−1), is bounded by d _n , where d _n is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d _n ^1/2(λ_avg/λ_min)^1/2, where λ_avg and λ_min are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)² and (n+1)(λ_avg/λ_min)^1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n ^1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed?

For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λ_min, the growth in ‖Λ _n ‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)² points on S ², turn out empirically to be very bad as interpolation points.

This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.

     

A lower bound for the worst-case cubature error on spheres of arbitrary dimension

This paper is concerned with numerical integration on the unit sphere S^r of dimension r≥2 in the Euclidean space ℝ^r+1. We consider the worst-case cubature error, denoted by E(Q_m;H^s(S^r)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s(S^r), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space H^s(S^r). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere S^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

Cubature over the sphere in Sobolev spaces of arbitrary order

This paper studies numerical integration (or cubature) over the unit sphere for functions in arbitrary Sobolev spaces H^s(S²), s>1. We discuss sequences of cubature rules, where (i) the rule Q_m(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤n and (ii) the sequence (Q_m(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Q_m(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space H^s(S²), s>1, the worst-case cubature error has the order of convergence O(n^-s), a result previously known only for the particular case . The crucial step in the extension to general s>1 is a novel representation of , where P_ℓ is the Legendre polynomial of degree ℓ, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Q_m(n). The order of convergence O(n^-s) is optimal for sequences (Q_m(n)) of cubature rules with properties (i) and (ii) if Q_m(n) uses m(n)=O(n²) points.     

Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.      

On extremal sort sequences

A sort sequence S_n is a sequence of all unordered pairs of indices inI _n = {1, 2, ..., n}. With a sort sequenceS_n = (s₁ ,s₂ ,...,s_{( 2ⁿ )} )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(S_n). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence S_n except that the comparisons whose outcomes can be inferred from the results of the preceding comparisons are not performed. A sort sequence is said to be extremal if it maximizes a given objective function. First we consider the extremal sort sequences with respect to the objective function ω(S_n) — the expected number of active predictions inS _n. We study ω-extremal sort sequences in terms of their prediction vectors. Then we consider the objective function Ω(S_n) — the minimum number of active predictions in S_n over all input orderings.     

Rank properties of certain semigroups

In this paper we consider certain ranks of some semigroups. These ranks are r ₁(S),r ₂(S),r ₃(S),r ₄(S) and r ₅(S) as defined below. We have r ₁≤r ₂≤r ₃≤r ₄≤r ₅. The semigroups are CL _n ,CL _m ×CL _n ,Z _n and SL _n . Here CL _n is a chain with n elements, Z _n is the zero semigroup on n elements and SL _n is the free semilattice generated by n elements and having 2 ⁿ −1 elements. We find many of the ranks for these classes of semigroups.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

有关不能全含于半球中的一些曲面的性质讨论

本文主要研究了不能全含于开半球中的一些特殊曲面.利用Lr算子的相关性质,证明了对S~(n+1)中紧致r-极小超曲面,如果第二基本形式的秩rank(h_(ij))r,则其不全含在S~(n+1)的一个开半球中.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Trisecant lemma for nonequidimensional varieties

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane , where the x _i are generic points, also meets X in a point x _r different from x ₁,..., x _r−1, then X is contained in a linear subspace L such that codim _L X ≥ r − 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℙ^r, where r ≥ n + 1. The variety of trisecant lines of Z, say V _1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V _1,3(Z) = 2n. This also implies that if dim V _1,3(Z) = 2n, then Z can be embedded in ℙ ^{n + 1}. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℙ^r, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of . We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 71–87, 2006.     

The Riemannian geometry of physical systems of curves

Summary The properties of a physical system S_k where k ≠−1, of ∞²ⁿ⁻¹ trajectories C. in a Riemannian space V_n are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system S_k, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S ₀ in a Riemannian space V_n, is obtained. Finally, contact transformations and the transformation theory of a physical system S_k where k ≠−1, are considered in detail. To Enrico Bompiani on his scientific Jubilee Kasner,Differential geometric aspecte of dynamics, The Princeton Colloquium Lectures, 1909. Published by the ? American Mathematical Society, Providence, Rhode Island, 1913, and reprinted 1934.     

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

Maximal three-independent subsets of {0, 1, 2} n

We consider a variant of the classical problem of finding the size of the largest cap in ther-dimensional projective geometry PG(r, 3) over the field IF₃ with 3 elements. We study the maximum sizef(n) of a subsetS of IF ₃ ⁿ with the property that the only solution to the equationx ₁+x₂+x₃=0 isx ₁=x₂=x₃. Letc _n=f(n)^1/n andc=sup{c ₁, c₂, ...}. We prove thatc>2.21, improving the previous lower bound of 2.1955 ...     

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

We study integrable geodesic flows on Stiefel varieties V _n,r ?=?SO(n)/SO(n?r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on V _n,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T ^* V _n,r )/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian G _n,r and on a sphere S ^n?1 in presence of Yang–Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety W _n,r ?=?U(n)/U(n?r), the matrix analogs of the double and coupled Neumann systems.     

Quadrature Rules for the Surface Integral of the Unit Sphere Based on Extremal Fundamental Systems

Quadrature rules for the surface integral of the unit Sphere S^r–1 based on an extremal fundamental system, i.e., a nodal system which provides fundamental Lagrange interpolatory polynomials with minimal uniform norm, are investigated. Such nodal systems always exist; their construction has been given in earlier work. Here the main results is that the corresponding interpolatory quadrature for the space of homogeneous polynomials of degree two is equally weighted for arbitrary r, and hence positive. For the full quadratic polynomial space we can prove positivity of the weights, only.     

Chapter 3:inertia preservers

Let Vdenote either the space of n×n hermitian matrices or the space of n×nreal symmetric matrices, Given nonnegative integers r,s,t such that r+S+t=n, let G( r,s,r) denote the set of all matrices in V with inertia (r,s,t). We consider here linear operators on V which map G(r,s,t) into itself.