We give a particularly short proof of the symmetrization of Cournot-Nash equilibrium distributions in large anonymous games with countable actions. Our proof relies on the Bollobás-Varopoulos extension of the marriage lemma.

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

Simple singularities in positive characteristic

Simple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

In recent work [5] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

In this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finite p-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finite p-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a field Fp of p elements. It was immediately clear that the parameters Fp, ℚ of the McLain group G(Fp, ℚ) could be replaced by other linearly ordered, dense sets S and by other fields F without doing much harm to the construction. If F has characteristic 0, then G(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigated G(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group Aut G(Fp, S) in his dissertation at Cambridge in 1963.

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* if

for each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

This paper may be regarded as a sequel to our earlier paper [19], where we give an elementary and self-contained proof of a very general form of the Hopf theorem on order-preserving linear operators in partially ordered vector spaces (reproduced here as Theorem 1·1).

Versions of this theorem and related ideas have been used by various authors to study both linear and nonlinear integral equations (Thompson [41], Bushell [9, 11], Potter [38, 39], Eveson [16, 17], Bushell and Okrasiriski [12, 13]); the convergence properties of nonlinear maps (Nussbaum [32, 33]); so-called DAD theorems (Borwein, Lewis and Nussbaum [8]) and in the proof of weak ergodic theorems (Fujimoto and Krause [20], Nussbaum [34]).

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.

Let ψ be a distribution function on [−1,1] from the Szegö-class, which contains in particular all Jacobi weights, and let (pn) be the monic polynomials orthogonal with respect to dψ. Let m(n)∈ℕ, n∈ℕ, be non-decreasing with limn → ∞ (n − m(n)) = ∞, l(n)∈ℕ with 0 ≤ l(n) ≤ m(n), and μj, n ∈ℝ for j = 0, …, m(n), n∈ℕ. It is shown that for each sufficiently large n, has n−l(n) simple zeros in (−1, 1)and l(n) zeros in ℂ\[−1,1] if for n ≥ n0, has m(n) − l(n) zeros in the disc |z| ≤ r < 1, l(n) zeros outside of the disc |z| ≥ R > 1 and where q > 2 max {r, 1/R}. If m(n) is constant for n ≥ n0 then the statement holds even for such polynomials (pn) orthogonal with respect to a distribution dψ satisfying the weak assumption ψ′ > 0 a.e. on [−1, 1]. For linear combinations of polynomials orthogonal on the unit circle corresponding results are derived.

Consider a knot K in a closed, oriented 3-manifold M. A genus g Heegaard decomposition of M is said to be a genus g bridge decomposition of K if it also decomposes K into trivial arcs in each handlebody. The genus g bridge number of K, denoted by bg(K), is defined as the minimal number of trivial arcs in a handlebody among all the genus g bridge decompositions of K [2, 13].

Our aim, in this paper, is to study a class of functions which occurs in pure integer programming, and to investigate conditions under which discrete subadditive functions belong to that class. The inspiration for the paper was the problem of classifying discrete metrics used in pattern recognition, while the methods of proof of the main theorem are those of pure integer programming.