Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

The structure of 1-formyl-3-phenyl-Δ-pyrazoline in the gas phase (DFT calculations), in solution (NMR) and in the solid state (X-ray crystallography)

The molecular structure of 1-formyl-3-phenyl-Δ²-pyrazoline was determined by X-ray crystallography (triclinic, P-1). The geometry thus obtained was compared with that obtained by DFT calculations. The GIAO method was used to calculate absolute shieldings, which agree conveniently with those measured by ¹³C and ¹⁵N NMR. The title compound appears to be an essentially planar molecule.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Some identities for products and degree raising of splines

In this paper it is shown how the algebraic product of two spline functions, each represented in terms of B-splines, can again be represented as a linear combination of suitable B-splines. As a corollary to this result we obtain an explicit representation of a given B-spline function in terms of B-splines of some arbitrary higher degree. This generalizes some known results for raising the degree by one. Recurrence relations for both products and degree raising are established that may be useful for computation.Communicated by Larry L. Schumaker.     

DeWitt-Schwinger expansion for projective and Grassmann spaces

We propose a method of calculating the heat kernel expansion of coset spaces. The five first coefficients of this expansion on symmetric spaces of dimension d8 are obtained.     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

Some properties of the riesz charge associated with a δ-subharmonic function

The first property is a refinement of earlier results of Ch. de la Vallée Poussin, M. Brelot, and A. F. Grishin. Let w=u–v with u, v superharmonic on a suitable harmonic space (for example an open subset of R ⁿ ), and let [w]=[u]–[v] denote the associated Riesz charge. If w0, and if E denotes the set of those points of at which the lim inf of w in thefine topology is 0, then the restriction of [w] to E is 0. Another property states that, if e denotes a polar subset of such that the fine lim inf of |w| at each point of e is finite, then the restriction of [w] to e is 0.     

Chern forms,chern numbers and curvature conditions on a Kähler-Einstein surface

In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL ²-norm of (3C ₂ ?C ₁ ² )/C ₁ ² is sufficiently small, whereC ₁ andC ₂ are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C ₂ ?C ₁ ² = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.