We study real hypersurfaces of a complex space form Mn(c), c ≠ 0 under certain conditions of the Ricci tensor on the orthogonal distribution T0.

Elliptic units of global function fields were first studied by D. Hayes in the case that deg ∞ is assumed to be 1, and he obtained some class number formulas using elliptic units. We generalize Hayes’ results to the case that deg ∞ is arbitrary.

Pour toute courbe rectifiable du plan, nous démontrons la formule de Cauchy relative à sa longueur. La formule est donnée sous deux formes: comme intégrale de la variation totale des projections de la courbe dans les diverses directions et comme intégrale double du nombre de rencontres de la courbe avec une droite quelconque du plan.

For a well-behaved measure μ, on a locally compact totally ordered set X, with continuous part μc, we make Lp (X, μc) into a commutative Banach bimodule over the totally ordered semigroup algebra Lp (X, μ), in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from Lp (X, μ) into its dual module and in particular shows that if μc is not identically zero then Lp (X, μ) is not weakly amenable. We show that all bounded derivations from L1 (X, μ) into its dual module arise in this way and also describe all bounded derivations from Lp(X, μ) into its dual for 1 < p < ∞ the case that X is compact and μ continuous.

This note contains the classification of the finite groups G satisfying the condition NG(H)/CG(H) ≅ Aut(H) for every abelian subgroup H of G.

A space X is a D-space if, for every neighborhood assignment f there is a closed discrete set D such that f(D) = X. In this paper we give some necessary conditions and some sufficient conditions for a resolution of a topological space to be a D-space. In particular, if a space X is resolved at each x ∊ X into a D-space Yx by continuous mappings fx: X − {x} → Yx, then the resolution is a D-space if and only if {x} × Bd(Yx) is a D-space.

There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of ℓ1. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.

It is shown that every simple complex Lie algebra 𝔤 admits a 1-parameter family 𝔤q of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for Uq(𝔤)-modules; here Uq(𝔤) is the quantized enveloping algebra of 𝔤. From this it follows that the multiplication on 𝔤q is Uq(𝔤)-invariant. In the special case 𝔤 = (2), the structure constants for the deformation 𝔤 (2)q are obtained from the quantum Clebsch-Gordan formula applied to V(2)q ⊗ V(2)q; here V(2)q is the simple 3-dimensional Uq(𝔤(2))-module of highest weight q2.

This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

It is shown that if f is an entire transcendental function, l a straight line in the complex plane, and n ≥ 2, then f has infinitely many repelling periodic points of period n that do not lie on l.

For the perturbed Hill’s equation on the line,

we study the behaviour of the matrix m-function at the spectral gap endpoints. In particular, we extend the result of Hinton, Klaus and Shaw that En, a gap endpoint, is a half-bound state (HBS) if and only if (E − En)½m(E) approaches a nonzero constant as E → En, to the present case.

For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.

We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f (s, t) ≤ 4 (s + t) + 1 for all s, t.

Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other “natural” sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

This paper deals with the projective objects in the category of all Z-frames, where the latter is a common generalization of different types of frames. The main result obtained here is that a Z-frame is E-projective if and only if it is stably Z-continuous, for a naturally arising collection E of morphisms.

For positive integers p and q with (p − 2)(q − 2) > 4 there is, in the hyperbolic plane, a group [p, q] generated by reflections in the three sides of a triangle ABC with angles π/p, π/q, π/2. Hyperbolic trigonometry shows that the side AC has length ψ, where cosh ψ = c/s, c = cos π/q, s = sin π/p. For a conformal drawing inside the unit circle with centre A, we may take the sides AB and AC to run straight along radiiwhile BC appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh ψ = s/z, where z = , while its centre is at distance 1/ tanh ψ = c/z from A. In the hyperbolic triangle ABC, the altitude from AB to the right-angled vertex C is ζ, where sinh ζ = z.

If A and B are disjoint ideals on ω, there is a tower preserving σ-centered forcing which introduces a subset of ω which meets every infinite member of A in an infinite set and is almost disjoint fromeverymember of B. We can then produce a model in which all compact separable radial spaces are Fréchet, thus answering a question of P. Nyikos. The question of the existence of compact ccc radial spaces which are not Fréchet was first asked by Chertanov (see [Arh78]).

Dimension subgroups and Lie dimension subgroups are known to satisfy a ‘universal coefficient decomposition’, i.e. their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the ‘universal’ coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary N-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

In this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.