Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

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.     

On the support of midconvex operators

Summary LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D Y is midconvex,g: D Y {– } is midconcave andg f onD, then there exists a Jensen mappingh: D Y {– } such thatg h f onD. Using this theorem we show that a mappingf: D Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

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.     

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.     

Functional inequality characterizing nonnegative concave functions in (0, ∞) k

Summary In the present note we prove that every functionf: (0, ) [0, ) satisfying the inequalityaf(s) + bf(t) f(as + bt), s, t > 0, for somea andb such that 0 <a < 1 <a + b must be of the formf(t) = f(1)t, (t > 0). This improves our recent result in [2], where the inequality is assumed to hold for alls, t 0, and gives a positive answer to the question raised there.An analogue for functions of several real variables of the above result characterizes concave functions. Conjugate functions and some relations to Hölder's and Minkowski's inequalities are mentioned.     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

Countable Fréchetα 1-spaces may be first countable

It is shown to be consistent that countable, Fréchet, ₁-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of length ₂.The research of both authors is partially supported by NSERC     

Wavelength-modulated,continuum-source excited atomic fluorescence spectrometric system for wear metals in jet engine lubricating oils using electrothermal atomization

A system for measuring atomic fluorescence of atoms produced via an electrically-heated graphite filament in a flame (acetylene/air or acetylene/nitrous oxide) and excited with a 300-W Eimac xenon are lamp is described. The experimental system also included wavelength modulation for background emission/fluorescence/scatter correction and an optically-triggered electronic integrator for efficient monitoring of the analyte fluorescence signal. Copper, aluminum and molybdenum were determined in jet engine lubricating oil samples (1 μl) with no pretreatment. The determinations are evaluated with respect to the accuracy and repeatability criteria of the U.S. Joint Oil Analysis Program.     

On bounds for the range of ordered variates II

Letx ₁, ,x _n be real numbers with ₁ ⁿ x _j =0, |x ₁ ||x ₂ ||x _n |, and ₁ ⁿ f(|x _i |)=A>0, wheref is a continuous, strictly increasing function on [0, ) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x _m | isf ^–1 (A/(n–m+1)). If (n–m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n–m+1) is odd provided either (i)f is strictly convex on [0, ), or (ii)f is strictly concave on [0, ). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x ^p forp>1 and 0<p<1 respectively.