A map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I1 or I2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN, the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN.

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.

In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.

Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing are

It was proved by Mergelyan (3) that for any packing the sum of the radii diverges:

1

Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.

To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o1, o2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

Partial answers are given to two questions. When does a lattice $\Lambda $ contain a sublattice ${\Lambda }'$ of index $N$ that is geometrically similar to $\Lambda $? When is the sublattice “clean”, in the sense that the boundaries of the Voronoi cells for ${\Lambda }'$ do not intersect $\Lambda $?

Let

be a sequence of real terms with which we associate the generating power series

We consider the following definition due to Schoenberg [7, p. 362]:

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.

Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

Analogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

We consider the $s$-energy $E({{Z}_{n}};\,s)={{\Sigma }_{i\ne j}}K(\parallel {{z}_{i,n}}\,-\,{{z}_{j,n}}\parallel \,;\,s)$ for point sets $Zn\,=\,\{{{z}_{k,n}}\,:\,k\,=\,0,\,\ldots \,,\,n\} $ on certain compact sets $\Gamma $ in ${{\mathbb{R}}^{d}}$ having finite one-dimensional Hausdorff measure,where

$$K(t;\,s)\,=\,\left\{ _{-\ln \,t,\,\,\,\text{if}\,s\,=\,0,\,}^{{{t}^{-s}},\,\,\,\,\,\,\,\text{if}\,s\,>\,0,} \right\}$$

is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\,\ge \,1$, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as $n\,\to \,\infty $.

In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ${{a}^{x}}-{{b}^{y}}=c$ , where $a,\,b$ and $c$ are given nonzero integers with $a,\,b\,\ge \,2$. In particular, we obtain the sharp result that there are at most two solutions in positive integers $x$ and $y$ and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

1. Introduction. The character theory of solvable groups has undergone significant development during the last decade or so and it can now be seen to have quite a rich structure. In particular, there is an interesting interaction between characters and sets of prime numbers.

Let G be solvable and let π be a set of primes. The “π-special” characters of G are certain irreducible complex characters (defined by D. Gajendragadkar [1]) which enjoy some remarkable properties, many of which were proved in [1]. (We shall review the definition and relevant facts in Section 3 of this paper.) Actually, we need not assume solvability: that G is π-separable is sufficient, if we are willing to use the Feit-Thompson “odd order” theorem occasionally. We shall state and prove our results under this weaker hypothesis, but we stress that anything of interest in them is already interesting in the solvable case where, of course, the “odd order” theorem is irrelevant.

In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $\text{GU}\left( 1,\,s \right)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.

In the case of $\text{GU}\left( 1,\,2 \right)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

One of the many interesting conjectures proposed by S. M. Ulam in (5) can be stated as follows:

If G and H are two graphs with p points vi and ui respectively (p ⩾ 3) such that for all i, G — vi is isomorphic with H — ui then G and H are themselves isomorphic.

P. J. Kelly (3) has shown this to be true for trees. The conjecture is, of course, not true for p = 2, but Kelly has verified by exhaustion that it holds for all of the other graphs with at most six points. Harary and Palmer (2) found the same to be true of the seven-point graphs.

In (1) Harary reformulated the conjecture as a problem of reconstructing G from its subgraphs G — vi and derived several of the invariants of G from the collection G — vi.

The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum“ and “jth column-sum” of the matrix under consideration. r1 … , rm; c1, … , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = rj and σj = cj for all i and j.

(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.

The principal result of this paper is the representation of the Mathieu group M23 as a group of 11 × 11 matrices over the Galois Field GF(2). This is a new representation of M23 and in §5 an indication of how the techniques of this result might be extended to the Mathieu group M11 is given.

1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular methods 1.1(1) or at least by all such methods subject to some simple additional conditions. Questions of this kind have been treated by the author (Lorentz [2], [5]) and, from another point of view, by R. P. Agnew [2] [3] ; in this paper we wish to discuss the problem systematically.

Let 5 be a set of positive integers no three of which are in arithmetical progression, i.e., if A, B, C are distinct elements of S, A + B ≠ 2C. We call such a set a non-averaging set. Let v(n) denote the maximum number of elements not exceeding n in any non-averaging set. The problem of finding bounds for v(n) has been treated by several authors [1, 3, 5, 6, 7]. The question first arose in connection with a theorem of van der Waerden [8].

A symmetric game consists of a set of pure strategies indexed by {0, …, n} and a real payoff matrix (aij). When two players choose strategies i and j the payoffs are aij and aji to the i-player and j-player respectively. In classical game theory of Von Neumann and Morgenstern [16] the payoffs are measured in units of utility, i.e., desirability, or in units of some desirable good, e.g. money. The problem of game theory is that of a rational player who seeks to choose a strategy or mixture of strategies which will maximize his return. In evolutionary game theory of Maynard Smith and Price [13] we look at large populations of game players. Each player's opponents are selected randomly from the population, and no information about the opponent is available to the player. For each one the choice of strategy is a fixed inherited characteristic.