The purpose of this paper is the study of parabolically differentiable points of arcs in the real affine plane. In Section 2, two different definitions of convergence of a family of parabolas are given and it is observed (Theorem 1) that these are equivalent. In Section 3, tangent parabolas at a point p of an arc A are discussed and it is proved (Theorem 2) that all the non-degenerate non-tangent parabolas of A through p intersect A at p or that all of them support. In Section 4, osculating parabolas are introduced and the condition that an arc be twice parabolically differentiable at a point p is stated.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D\,>\,3705$, there is always at least one prime $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$.

For A an associative algebra with identity over a field K, [A : K] < ∞, and d an integer, we define gΛ(d) to be the number of inequivalent indecomposable Λ-modules of degree d over K. Following (6), we define Λ to be of finite representation type if . Λis said to be of bounded representation type if there exists d0 such that gΛ(d) = 0 for d ⩾ d0; Λ is of unbounded representation type if not of bounded type.

The first main result of the paper is a criterion for a partially commutative group $\mathbb{G}$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb{G}$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb{G}$ (of a coordinate group over $\mathbb{G}$) to the elementary theories of the direct factors of $\mathbb{G}$ (to the elementary theory of coordinate groups of irreducible algebraic sets).

Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb{H}$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb{H}$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb{H}$ to $\mathbb{H}\,*\,F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Let G be a group and n(≧ 2) an integer. We say that G belongs to the class of groups Pn if every product of n elements can be reordered, i.e. for all n-tuples , there exists a non-trivial element σ in the symmetric group Σn such that Let P denote the union of the classes Pn, n ≧ 2. Clearly every finite group belongs to P and each class Pn is closed with respect to forming subgroups and factor groups.

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.

A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.

In Perron integration, majorants are usually functions of points. If the domain of definition is a Euclidean space of n dimensions, we can define a finitely additive n-dimensional majorant rectangle function by taking suitable differences of the majorant point function with respect to each of the n coordinates. The way is then open to a generalization, in that we need only suppose that the majorant rectangle function is finitely superadditive. Similarly, we need only suppose that a minorant rectangle function is finitely subadditive. These kinds of rectangle functions were used by J. Mařík (5) to prove the Fubini theorem for Perron integrals in Euclidean space of m + n dimensions. He also proved that for a function that is Perron, and absolutely Perron, integrable, the majorant and minorant rectangle functions can be taken to be finitely additive. As a result he posed the following problem.

Let R be a ring with involution *, and let Z denote the center of R. In R let S = {x ∈ R|x* = x} be the set of symmetric elements of R. We shall study rings which are conditioned in the following way: given s ∈ S, then for some integer and some polynomial p(t), with integer coefficients which depend on . What can one hope to say about such rings? Certainly all rings in which every symmetric element is nilpotent fall into this class.

Introduction. In hyperbolic geometry we have three compasses, namely an ordinary compass for drawing ordinary circles with a given centre and a given radius, a hypercompass for drawing hypercycles with a given axis and a given radius, and a horocompass for drawing horocycles with a given diameter and passing through a given point.

We assume familiarity with the notation and contents of Conway [2] and Edge [3]. That the Mathieu group is a subgroup of the simple group PSU (6, 22) appears to have been first recognized by Conway and is consequent upon his identification of ·222 with PSU (6, 22). Although we know of no proof of this identification in the literature, several proofs exist in the folklore of the subject: for example, N. Patterson showed one of the authors a proof that depends on ·222 being a Fischer group, hence on consideration of order, isomorphic to PSU (6, 22). There is another proof which relies on McLaughlin's work on rank three groups.

Let R be a commutative domain with 1, Q its field of quotients, and M a torsion-free R module. By a balanced submodule of M is meant an RD-submodule N [i.e. rN = N ∩ rM for each r ∈ R] such that, for every R-submodule J of Q, every homomorphism η : J → M/N can be lifted to a homomorphism χ:J → M. This definition extends the notion of balancedness as introduced in abelian groups (see e.g. [10, p. 113]). The balanced-projective R-modules can be characterized as summands of completely decomposable R-modules (i.e. summands of direct sums of submodules of Q). If R is a valuation domain, then such summands are again completely decomposable; see [12, p. 275].

Let L be a finitely generated extension of a field K of characteristic p ≠ 0. If L/K is algebraic, then there is a unique intermediate field S such that

S is just the maximal separable extension of K in L. If L/K is not algebraic, then Dieudonne [4] showed there exist maximal separable extensions D of K in L such that L ⊆ Kp–∞⊗KD. In general, not every maximal separable extension of K in L has the property. Those which do have the property are called distinguished. Kraft [7] established that a maximal separable extension D of K in L is distinguished if and only if [L:D] is as small as possible. If the minimum of the [L:D] is pr, r is called the order of inseparability of L/K, denoted inor (L/K).

Let L1 be an intermediate field of L/K. If L/K is algebraic, then the maximal separable extension S1 of K in L1 is contained in the maximal separable extension S of K in L, and moreover S is separable over S1.

We study projections in the corona algebra of $C\left( X \right)\,\otimes \,K$, where $K$ is the ${{C}^{*}}$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X\,=\,[0,\,1],\,[0,\,\infty ),\,(-\infty ,\,\infty ),\,\text{or}\,\text{ }\!\![\!\!\text{ 0,}\,\text{1 }\!\!]\!\!\text{ / }\!\!\{\!\!\text{ 0,}\,\text{1 }\!\!\}\!\!\text{ }$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in ${{K}_{0}}$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

Let p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.

This note is concerned with a simple but rather surprising identity which emerged unexpectedly from the work of one of the authors on the characterisation of characters. Consider, for example, the seven partitions of 5. These are

(1) 5, 4 1, 3 2, 3 12, 22 1, 2 13, 15

and with each of these we can associate a product of factorials of the numbers of repetitions, respectively

(2) 1!, (1!)(1!), (1!)(1!), (1!)(2!), (2!)(1!), (1!)(3!), 5!

It is then seen that the product of all the numbers occurring in (1) coincides with that of all the numbers in (2).

Generally, for any particular natural number n, the partitions can be written in the form

1a1 2a2 3a3…,

in which ak is the frequency of repetition of the part k, and are enumerated by the distinct sets {α} = {a1, a2, …,} with ak ≧ 0 and Σkak = n.

A characterization of the Banach spaces of type $C\left( K \right)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a Namioka–Phelps compact or for some particular class of Rosenthal compacta, results which were originally proved with ad hoc methods.

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

Paul Erdôs has suggested an investigation of infinite groups from the point of view of the partition relations of set theory. In particular, he suggested that given a group G, one considers the graph T with vertex set G whose edges are the pairs ﹛g, h﹜ which do not commute.