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.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

Lucas' theorem refined ∗

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a ₁,…,a _n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative p ¹, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n+ l)-gon a ₁…a _n+1 with tangent points the midpoints of then+ l sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n= 2.     

New cubature formulae and hyperinterpolation in three variables

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈n ^d /2 ^d−1 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube. Work supported by the National Science Foundation under Grant DMS-0604056, by the “ex-60%” funds of the Universities of Padova and Verona, and by the INdAM-GNCS.     

Bounds on chromatic numbers of multiple factors of a complete graph

Bounds on the sum and product of the chromatic numbers of n factors of a complete graph of order p are shown to exist. The well-known theorem of Nordhaus and Gaddum solves the problem for n = 2. Strict lower and some upper bounds for any n and strict upper bounds for n = 3 are given. In particular, the sum of the chromatic numbers of three factors is between 3p^1/3 and p + 3 and the product is between p and [(p + 3)/3]³.     

A survey of orthogonal arrays of strength two

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Error Estimates for Finite Element Approximations of a Viscous Wave Equation

We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on the energy method is developed. Optimal error estimates in both time and space are obtained. For sufficiently smooth solutions, it is demonstrated that the maximal error in the L ²-norm over a finite time interval converges optimally as O(h ^p+1 + Δt ^s ), where p denotes the polynomial degree, s = 1 or 2, h the mesh size, and Δt the time step.     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001     

Some results on (k, μ)′-almost Kenmotsu manifolds

In this paper, we study the Weyl conformal curvature tensor 𝒲 and the concircular curvature tensor 𝒞 on a (k, μ)′-almost Kenmotsu manifold M²ⁿ⁺¹ of dimension greater than 3. We obtain that if M²ⁿ⁺¹ satisfies either R · 𝒲 = 0 or 𝒞 · 𝒞 = 0, then it is locally isometric to either the hyperbolic space ?²ⁿ⁺¹ (?1) or the Riemannian product ?ⁿ⁺¹(?4) × ?ⁿ.     

Truncation error bounds for limit-periodic continued fractionsK(a n /1) with lima n =0

Summary Truncation error bounds are developed for continued fractionsK(a _n /1) where |a _n |1/4 for alln sufficiently large. The bounds are particularly suited (some are shown to be best) for the limit-periodic case when lima _n =0. Among the principal results is the following: If |a _n |/n ^p for alln sufficiently large (with constants >0,p>0), then |f–f _m |C[D/(m+2)] ^p(m+2) for allm sufficiently large (for some constantsC>0,D>0). Heref denotes the limit (assumed finite) ofK(a _n /1) andf _m denotes itsmth approximant. Applications are given for continued fraction expansions of ratios of Kummer functions₁ F ₁ and of ratios of hypergeometric functions₀ F ₁. It is shown thatp=1 for₁ F ₁ andp=2 for₀ F ₁, wherep is the parameter determining the rate of convergence. Numerical examples indicate that the error bounds are indeed sharp.Research supported in part by the National Science Foundation under Grant MCS-8202230 and DMS-8401717     

Finite linear spaces with two consecutive line degrees

Let L be a non-trivial finite linear space in which every line has n or n+1 points. We describe L completely subject to the following restrictions on n and on the number of points p: p≤n ²+n?1 and n≥3; n ²+n+2≤p≤n ²+2n?1 and n≥3; p=n ²+2n and n≥4; p=n ²+2n+2 and n≥3; p=n ²+2n+3 and n≥4.     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.     

Power closure and the Engel condition

A Liep-algebraL is calledn-power closed if, in every section ofL, any sum ofp ⁱ⁺ⁿ th powers is ap ⁱ th power (i>0). It is easy to see that ifL isp ⁿ -Engel then it isn-power closed. We establish a partial converse to this statement: ifL is residually nilpotent andn-power closed for somen≥0 thenL is (3p ^{n +2} +1)-Engel ifp>2 and (3 · 2 ⁿ⁺³+1)-Engel ifp=2. In particular, thenL is locally nilpotent by a theorem of Zel’manov. We deduce that a finitely generated pro-p group is a Lie group over thep-adic field if and only if its associated Liep-algebra isn-power closed for somen. We also deduce that any associative algebraR generated by nilpotent elements satisfies an identity of the form (x+y) ^{p n} =x ^{p n} +y ^{p n} for somen≥1 if and only ifR satisfies the Engel condition. This project was supported by the CNR in Italy and NSF-EPSCoR in Alabama during the first author’s stay at the Università di Palermo.     

Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

We study ratio asymptotics, that is, existence of the limit of P_n+1(z)/P_n(z) (P_n= monic orthogonal polynomial) and the existence of weak limits of p_n² dμ (p_n=P_n/||P_n||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z₀ with Im(z₀)≠0 implies dμ is in a Nevai class (i.e., a_n→a and b_n→b where a_n,b_n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove p_n² dμ has a weak limit if and only if lim b_n, lim a_2n, and lim a_2n+1 all exist. In both cases, we write down the limits explicitly.     

The Existence of Good Extensible Polynomial Lattice Rules

Extensible (polynomial) lattice rules have been introduced recently and they are convenient tools for quasi-Monte Carlo integration. It is shown in this paper that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomial lattice rules such that for all these infinitely many moduli and all dimensions s the quantity R ^(s) and the star discrepancy are small. The case of Korobov-type polynomial lattice rules is also considered.Received April 30, 2002; in revised form August 21, 2002 Published online April 4, 2003     

Oriented matroids with few mutations

This paper defines a “connected sum” operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ℝP ^d-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ℝP³ with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the “strong-map conjecture” of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas

The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k+1 inn-space such that the number of evaluation points isO((2n)^k)/k!),n . Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 11 but these are not presented here.