Discrete Fourier Analysis on Fundamental Domain and Simplex of A d Lattice in d-Variables

A discrete Fourier analysis on the fundamental domain Ω _d of the d-dimensional lattice of type A _d is studied, where Ω₂ is the regular hexagon and Ω₃ is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of (log n) ^d . The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.     

Worst-case minimum rectilinear steiner trees in all dimensions

It is proved that the length of the longest possible minimum rectilinear Steiner tree ofn points in the unitd-cube is asymptotic toβ _dn^(d−1)/d , whereβ _d is a constant that depends on the dimensiond≥2. A method of Chung and Graham (1981) is generalized to dimensiond to show that 1≤β _d≤d4^(1−d)/d . In addition to replicating Chung and Graham's exact determination ofβ ₂=1, this generalization yields new bounds such as 1≤β ₃<1.191 and .     

Construction of Optimal Cubature Formulas Related to?Computer Tomography

We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Π _d =[0,a ₁]×⋅⋅⋅×[0,a _d ], a ₁,…,a _d >0, having a given majorant for the modulus of continuity (relative to the l ₁-metric in ℝ ^d ). An optimal cubature formula, which uses as information integrals of f along intersections of Π _d with n arbitrary (d−1)-dimensional hyperplanes in ℝ ^d (d>1) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of f along intersections of Π _d with shifts of (d−2)-dimensional coordinate subspaces of ℝ ^d (d>2).     

Product Gauss cubature over polygons based on Green’s integration formula

We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn ² nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented. AMS subject classification (2000)  65F20     

O(dlog?N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional applications. We describe the multifolding or quantics-based tensor approximation method of O(dlog N)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1,…,N}^⊗d , or briefly N-d tensors of size N ^d , and to the respective discretized differential-integral operators in ℝ ^d . As the basic approximation result, we prove that a complex exponential sampled on an equispaced grid has quantics rank 1. Moreover, a Chebyshev polynomial, sampled over a Chebyshev Gauss–Lobatto grid, has separation rank 2 in the quantics tensor format, while for the polynomial of degree m over a Chebyshev grid the respective quantics rank is at most 2m+1. For N-d tensors generated by certain analytic functions, we give a constructive proof of the O(dlog Nlog ε ⁻¹)-complexity bound for their approximation by low-rank 2-(dlog N) quantics tensors up to the accuracy ε>0. In the case ε=O(N ^−α ), α>0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/αlog ² ε ⁻¹). From numerical examples presented here, we observe that the quantics tensor method has proved its value in application to various function related tensors/matrices arising in computational quantum chemistry and in the traditional finite element method/boundary element method (FEM/BEM). The tool apparently works.     

On the existence of minimum cubature formulas for Gaussian measure on ?2 of degree t supported by [\frac{t}{4}]+1 circles

In this paper we prove that there exists no minimum cubature formula of degree 4k and 4k+2 for Gaussian measure on ℝ² supported by k+1 circles for any positive integer k, except for two formulas of degree 4.     

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.     

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e.g., all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d?2)-polytope Q on n?1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d -polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d-space that retains the complete $(\lfloor\tfrac{d}{2}\rfloor-1)We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e.g., all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope Q on n−1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d -polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d-space that retains the complete (?\tfracd2?-1)(\lfloor\tfrac{d}{2}\rfloor-1) -skeleton.     

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}ⁿ, where nearest-neighbor edges are occupied with probability p. Let p_c(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2^n/3, where λ is a small positive constant. Let ε=n(p−p_c(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2^−n/3) is of size Θ(2^2n/3), below this scaling window it is at most 2(log 2)nε⁻², and above this scaling window it is at most O(ε2ⁿ). In this paper, we prove that for the size of the largest component is at least Θ(ε2ⁿ), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.     

Kinetic Spanners in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

We present a new (1+ε)-spanner for sets of n points in ℝ ^d . Our spanner has size O(n/ε ^d−1) and maximum degree O(log  ^d n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n ²/ε ^d−1), and using a supporting data structure of size O(nlog  ^d n), we can handle events in time O(log  ^d+1 n). Moreover, the spanner can be updated in time O(log n) if the flight plan of a point changes. This is the first kinetic spanner for points in ℝ ^d whose performance does not depend on the spread of the point set.     

Voronoi Diagrams and Convex Hulls of Random Moving Points

This report considers the expected combinatorial complexity of the Euclidean Voronoi diagram and the convex hull of sets of n independent random points moving in unit time between two positions drawn independently from the same distribution in R ^d for fixed d\ge 2 as n→∈fty . It is proved that, when the source and destination distributions are the uniform distribution on the unit d -ball, these complexities are Θ(n ^(d+1)/d ) for the Voronoi diagram and O(n ^(d-1)/(d+1) log n) for the convex hull. Additional results for the convex hull are O( log ^d n) for the uniform distribution in the unit d -cube and O( log ^(d+1)/2 n) for the d -dimensional normal distribution. Received November 23, 1998, and in revised form July 8, 1999.     

Almost tight upper bounds for lower envelopes in higher dimensions

We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(n ^d−1+∈), for any ∈>0; the constant of proportionality depends on ∈, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(n ^d-2λ _q (n)) for some constantq depending on the shape and degree of the surfaces (where λ _q (n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n ^2+∈), and give several applications of the new bounds. Work on this paper has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Degree-uniform lower bound on the weights of polynomials with given sign function

A Boolean function f: {0, 1} ⁿ → {0, 1} is called the sign function of an integer polynomial p of degree d in n variables if it is true that f(x) = 1 if and only if p(x) > 0. In this case the polynomial p is called a threshold gate of degree d for the function f. The weight of the threshold gate is the sum of the absolute values of the coefficients of p. For any n and d ≤ D ≤ $\frac{{\varepsilon n^{1/5} }} {{\log n}} $\frac{{\varepsilon n^{1/5} }} {{\log n}} we construct a function f such that there is a threshold gate of degree d for f, but any threshold gate for f of degree at most D has weight 2^{(dn)d /D4d} 2^{(\delta n)^d /D^{4d} } , where ɛ > 0 and δ > 0 are some constants. In particular, if D is constant, then any threshold gate of degree D for our function has weight 2^{W(nd )}2^{\Omega (n^d )} . Previously, functions with these properties have been known only for d = 1 (and arbitrary D) and for D = d. For constant d our functions are computable by polynomial size DNFs. The best previous lower bound on the weights of threshold gates for such functions was 2^Ω(n). Our results can also be translated to the case of functions f: {−1, 1} ⁿ → {−1, 1}.     

Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems

Due to their fundamental nature and numerous applications, sphere constrained polynomial optimization problems have received a lot of attention lately. In this paper, we consider three such problems: (i) maximizing a homogeneous polynomial over the sphere; (ii) maximizing a multilinear form over a Cartesian product of spheres; and (iii) maximizing a multiquadratic form over a Cartesian product of spheres. Since these problems are generally intractable, our focus is on designing polynomial-time approximation algorithms for them. By reducing the above problems to that of determining the L ₂-diameters of certain convex bodies, we show that they can all be approximated to within a factor of Ω((log n/n) ^d/2–1) deterministically, where n is the number of variables and d is the degree of the polynomial. This improves upon the currently best known approximation bound of Ω((1/n) ^d/2–1) in the literature. We believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Laurent Padé-Chebyshev rational approximants,A _m (z,z ⁻¹)/B _n (z, z ⁻¹), whose Laurent series expansions match that of a given functionf(z,z ⁻¹) up to as high a degree inz, z ⁻¹ as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z ⁻¹)B _n (z, z ⁻¹)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Commuting extensions and cubature formulae

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A₁,...,A_d, related to the coordinate operators x₁,...,x_d, in R^d. We prove a correspondence between cubature formulae and “commuting extensions” of A₁,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.     

On transfer inequalities in Diophantine approximation, II

Let Θ be a point in R ⁿ . We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(R ⁿ⁺¹). It turns out that our convex bodies in degree d are the dth compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables us both to split and to refine the classical Khintchine transference principle.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

The Overlay of Minimization Diagrams in a Randomized Incremental Construction

In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in ℝ ^d , for d≥2, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n ^⌊d/2⌋log n), for d odd, and O(n ^⌊d/2⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d=3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.