Every twofold triple system can be directed

Every twofold triple system, or block design with k = 3 and λ = 2, is the underlying design of a directed triple system. Applications to the existence and enumeration of directed triple systems are described.     

Every effect algebra can be made into a total algebra

We prove that every effect algebra (E,+, 0, 1) can easily be made into a total algebra (E,⊕, ¬, 0) of type (2, 1, 0) in such a way that two elements are compatible in (E,+, 0, 1) if and only if they commute in (E,⊕, ¬, 0).     

Every integer can be written as a square plus a squarefree

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asymptotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function fully explicit.     

Every finite lattice can be embedded in a finite partition lattice

Conclusions There are many questions, which arise in connection with the theorem presented. In general, we would like to know more about the class of embeddings of a given lattice in the lattices of all equivalences over finite sets. Some of these problems are studied in [4]. In this paper, an embedding is called normal, if it preserves 0 and 1. Using regraphs, our result can be easily improved as follows: THEOREM.For every lattice L, there exists a positive integer n ₀,such that for every n≥n ₀,there is a normal embedding π: L→Eq(A), where |A|=n. Embedding satisfying special properties are shown in Lemma 3.2 and Basic Lemma 6.2. We hope that our method of regraph powers will produce other interesting results. There is also a question about the effectiveness of finding an embedding of a given lattice. In particular, the proof presented here cannot be directly used to solve the following. Problem. Can the dual of Eq(4) be embedded into Eq(2¹⁰⁰⁰)? Presented by G. Gr?tzer.     

Every weakly compact set can be uniformly embedded into a reflexive Banach space

Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.     

A quasi-Newton method can be obtained from a method of conjugate directions

In a recent paper McCormick and Ritter consider two classes of algorithms, namely methods of conjugate directions and quasi-Newton methods, for the problem of minimizing a function ofn variablesF(x). They show that the former methods possess ann-step superlinear rate of convergence while the latter are every step superlinear and therefore inherently superior. In this paper a simple and computationally inexpensive modification of a method of conjugate directions is presented. It is shown that the modified method is a quasi-Newton method and is thus every step superlinearly convergent. It is also shown that under certain assumptions on the second derivatives ofF the rate of convergence of the modified method isn-step quadratic.This work was supported by the National Research Council of Canada under Research Grant A8189.     

Every measure on the projectors of a W*-algebra can be extended to a state

Functional Analysis and Its Applications -     

Bin packing can be solved within 1 + ε in linear time

For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive ε, there exists anO(n)-time algorithmS such that, ifS(L) denotes the number of bins used byS forL, thenS(L)/L*≦1+ε for anyL providedL* is sufficiently large. The work of this author was supported by NSF Grant MCS 70-04997.     

Exceptional directions for a convex body

Let K be a convex body in Rⁿ andO be a point inside K. We examine the Grassmann manifold of k-planes passing throughO. We take as exceptional the planes intersecting K along a body having at least one (k – 1)-dimensional face such that it does not have points inside the hyperfaces of body K. We prove that in the Grassmann manifold G _k ⁿ the set of such exceptional planes is of measure zero.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 365–371, September, 1976.The author thanks V. A. Zalgaller for aid and advice on the work.     

Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators

First we show that every real Banach space satisfying a certain property, calledβ (used by Lindenstrauss and Partington) verifies the denseness of the numerical radius attaining operators. Using this result and a renorming theorem by Partington we conclude that every Banach space is isomorphic to a new one satisfying the denseness of the numerical radius attaining operators.     

A sphere can be characterized as a smooth ovaloid which contains one circle through each point

A sphere inE ³ can be characterized as a smooth ovaloid which contains one circle of an arbitrary but fixed radius through each point.Dedicated to Professor N. K. Stiphanidis on his 65th birthday     

The least number, of n-periodic points of a self-map of a solvmanifold, can be realised by a smooth map

Shub and Sullivan noticed that there is a large gap between the least number of periodic points, taken over the family of continuous maps homotopic to a given self-map f and the analogous number taken over the family of smooth maps homotopic to f. In this paper we show that in the case of self-maps of solvmanifolds the minimal numbers of n-periodic points, in both categories, are equal.     

How Hamiltonian can a finite group be?

Work supported by NSF grant DMS-8900507