In general, given the fact that every element in a ring is nilpotent, one cannot conclude that the ring itself is nilpotent. However, there are theorems which do assert that, in the presence of certain side conditions, nil implies nilpotent. We shall prove some theorems of this nature here; among them they contain or subsume many of the earlier known theorems of this sort.

Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a reflection group to be a group generated by reflections. Chevalley (1) (and also Coxeter (2) together with Shephard and Todd (4)) has shown that a reflection group G, acting on a space of n dimensions, possesses a set of n algebraically independent (polynomial) invariants which form a polynomial basis for the set of all invariants of G.

The cell growth problem is a combinatorial problem which may be stated as follows: A plane animal is made up of cells, each of which is a square of unit area. It starts as a single cell, and grows by adding cells one at a time in such a way that the new cell has at least one side in contact with a side of a cell already present in the animal. The problem is to find the number of different animals of area n, it being understood that animals which can be transformed into each other by reflections or rotations of the plane will be regarded as the same animal.

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ X

Thus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for

The space is said to be uniformly rotund if for each 0 we have .

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

Introduction. The purpose of this paper is to discuss implications of Hadamard's condition for elastic stability (2, §269) with respect to uniqueness of solutions of boundary value problems in the theory of small deformations superimposed on large. We show that a slightly refined form of his condition implies a uniqueness theorem for displacement boundary value problems. We construct a counter-example showing that his condition does not imply uniqueness of solutions for one type of stress boundary value problem.

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

In recent papers Horst Herrlich [4; 5] has demonstrated the usefulness of topological categories for applications to a large variety of special structures. A particularly striking result is his characterization of cartesian closedness for topological categories (see [5]). Spaces satisfying a separation axiom usually cannot form a topological category in Herrlich's sense however and some interesting special cases, e.g. Hausdorff C-spaces, remain excluded from his theory despite having many analogous properties. It therefore seems worthwhile to undertake a similar study in a wider setting. To this end we relax one of the axioms for a topological category and show that in the resulting initially structured categories a significant selection of results can still be proved, including the characterization of cartesian closedness.

In this paper we characterize the compactness of the commutator $\left[ b,\,T \right]$ for the singular integral operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$. More precisely, we prove that if $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$, the $\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$-closure of $C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$, then $\left[ b,\,T \right]$ is a compact operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for $1\,<\,p\,<\,\infty $ and $0\,<\,\lambda \,<\,n$. Conversely, if $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\left[ b,\,T \right]$ is a compact operator on the ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for some $p\,\left( 1\,<\,p\,<\,\infty \right)$, then $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $\left[ b,\,T \right]$ on ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges. We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n.

In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gröbner bases, theory of the resultant, Sturm’s algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

In this paper we show that for a Grothendieck category $\mathcal{A}$ and a complex $E$ in $\mathbf{C}(\mathcal{A})$ there is an associated localization endofunctor $\ell$ in $\mathbf{D}(\mathcal{A})$. This means that $\ell$ is idempotent (in a natural way) and that the objects that go to 0 by $\ell$ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of $\mathbf{D}(\mathcal{A})$ that contains $E$. As applications, we construct $\text{K}$-injective resolutions for complexes of objects of $\mathcal{A}$ and derive Brown representability for $\mathbf{D}(\mathcal{A})$ from the known result for $\mathbf{D}(R-\mathbf{mod})$, where $R$ is a ring with unit.

If S1, S2, S3, … , Sn are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that ai is in Si for (i = 1, 2, 3, … , n) and such that ai ≠ aj for i ≠ j , is that for k = 1, 2, 3, … , n, the union of any k of the sets S1, S2, … , Sn, contains at least k elements. The theorem has a number of consequences amongst which we list the following.

1. Introduction. An interesting feature of recent topological developments is the increasingly important role played by locally finite coverings. Point-finite coverings, on the other hand, even though conceptually simpler, have received very little attention. And deservedly so, since they are much less useful. Nevertheless, it sometimes happens (as it did to the author in (5)) that one is confronted by a covering which is known to be point-finite, but not necessarily locally finite.

This paper will continue certain investigations into the geometric nature of the well-known traveling salesman problem: that of determining the extreme Hamiltonian circuits (H-circuits) of a graph.

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We show that a regular Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, $\phi \,=\,\phi \left( s \right)$ , for which there are a Riemannian metric $\alpha $ and a 1-form $\beta $ on a manifold $M$ such that the scalar function $F\,=\,\alpha \phi \left( \beta /\alpha \right)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.

1. Introduction. One of the most remarkable q-extensions of the classical beta integral was recently introduced by Askey and Wilson [1]

(1.1)

where |q| < 1 and the pairwise products of {a, b, c, d} as a multiset do not belong to the set {qj, j = 0, – 1, – 2, …}. The contour C is the unit circle described in the positive direction, but with suitable deformations to separate the sequences of poles converging to zero from the sequences of poles diverging to infinity. The symbol (A; q)∞ is an infinite product defined by

(1.2)

whenever it converges.

We consider some deformations of ${{G}_{2}}$-structures on 7-manifolds. We discover a canonical way to deform a ${{G}_{2}}$-structure by a vector field in which the associated metric gets “twisted” in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy ${{G}_{2}}$. Similarly we consider analogous constructions for Spin(7)-structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the Spin(7) case.

Introduction. Les nombres entiers positifs a0, a1, … , an, … définis par la fonction génératrice

et que l'on appelle nombres exponentiels jouent, pour la sommation de certaines séries, un rôle qui rappelle le rôle sommatoire des nombres de Bernoulli. Nous avons rassemblé ici les principales propriétés des nombres an dont plusieurs sont, croyons nous, nouvelles.

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.

A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.