We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice ${\mathbb Z}^2$ is equal to $1/2$. The lower bound was proved by Harris, who showed in 1960 that percolation does not occur at $p=1/2$. The other, more difficult, bound was proved by Kesten, who showed in 1980 that percolation does occur for any $p>1/2$.

For each finitely presented group $Q$ , a short exact sequence $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ is produced, such that $G$ is residually finite and $N$ is finitely generated.

Transformation formulas are obtained for generalized Dedekind eta functions; these are simpler to apply than Schoeneberg's formulas. As an application, a list is given of the generators of all the function fields associated with torsion-free genus zero congruence subgroups of $\hbox{PSL}_2(\mathbb R)$.

We prove that certain universality properties of the partial sums force a power series to have Ostrowski-gaps. This has interesting consequences for some classes of universal functions.

Given a graph of $\delta$-hyperbolic spaces, this paper gives sufficient conditions that ensure that the graph itself is $\delta$-hyperbolic. As an application, a simple proof is given to show that limit groups are relatively hyperbolic.

A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group $F$, weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l1. It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t)=Au(t)+ϕ(t) (t[ges ]0). Suppose that u has uniformly convergent means, σ(A)∩iR is countable, and ϕ is asymptotically almost periodic. Then u is asymptotically almost periodic. Related results have been obtained by Ruess and Vũ, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Räbiger, and applications to Volterra equations are discussed.

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz– Minkowski space L3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L3, involving the local geometry of the surface and its hyperbolic image. As an application of this, a new proof of the Calabi–Bernstein theorem is provided.

Barbasch and Vogan showed that the Kazhdan–Lusztig cells of a finite Weyl group are compatible with parabolic subgroups. Their proof uses the known bridge between the theory of cells and the theory of primitive ideals. In this paper, an elementary, self-contained proof of this result is provided, which works for arbitrary Coxeter groups and Lusztig's general definition of cells (involving Iwahori–Hecke algebras with unequal parameters). The argument is based on a recent paper by Howlett and Yin.

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality

∫Mf(x)dHα(x) [les ]C∫[mid ]f(x)[mid ]dHα(x), 0<α<n, (1)

where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in ℝn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of ϕ[ges ]0 with respect to a set function C is defined by

formula here

Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (ℝn), see [1, p. 114].

The purpose of this note is to provide a self-contained, direct proof of a result more general than (1).

Using properties of the Steinberg character, we obtain a congruence modulo p for the number of ways in which a p-regular element may be expressed as a commutator in a finite simple group G of Lie type of characteristic p. This congruence shows that such an element is a commutator in G. We also show that if K and L are conjugacy classes of G consisting of elements whose centralizers have order relatively prime to p, then any p-regular element of G is expressible as the product of an element of K and an element of L.

Using Morse theory and the truncation technique, a proof is given of the existence of at least three nontrivial solutions for a class of $p$-Laplacian equations. When $p=2$, the existence of four nontrivial solutions is also considered.

In this paper we are going to study the solutions of a complex T-periodic Riccati equation

formula here

and we are going to determine all the possible dynamics of this type of equation. Also, we can say that generically there are two different T-periodic solutions.

Let $\phi\,{:}\,G \to G$ be a group endomorphism where G is a finitely generated group of exponential growth, and denote by $R(\phi)$ the number of twisted ϕ-conjugacy classes. Fel'shtyn and Hill (K-theory 8 (1994) 367–393) conjectured that if ϕ is injective, then R(ϕ) is infinite. This paper shows that this conjecture does not hold in general. In fact, R(ϕ) can be finite for some automorphism ϕ. Furthermore, for a certain family of polycyclic groups, there is no injective endomorphism ϕ with $R({\phi}^n)\,{<}\,\infty$ for all n.

There exists a set $\cal U$ in the plane, such that elements of $\cal U$ correspond to minimal stable solutions of a two-parameter elliptic system. For points on the boundary $\Upsilon$ of $\cal U$, there exist weak solutions to the elliptic system. The paper studies the properties of the curve $\Upsilon$.

In the representation theory of finite groups, it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups [Sfr ]n, this amounts to determining which Specht modules are irreducible over a field of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifying the conjecture in [3, p. 97].

Recall that a partition is 2-regular if all of its non-zero parts are distinct; otherwise the partition is 2-singular. The irreducible Specht modules Sλ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If Sλ is irreducible, then Sλ′ is irreducible, since Sλ′ is isomorphic to the dual of Sλ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then Sλ is irreducible only if λ = (2, 2).

In order to state our theorem, we let l(k), for an integer k, be the least non-negative integer such that k<2l(k).

Let [Mscr ]n denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of [Mscr ]n, namely the set of matrices in [Mscr ]n whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper.

Let φ be an analytic function from [ ] to the symmetrized bidisc

(formula here)

We show that if φ(0) = (0,0) and φ(λ) = (s, p) in the interior of Γ, then

(formula here)

Moreover, the inequality is sharp: we give an explicit formula for a suitable φ in the event that the inequality holds with equality. We show further that the inverse hyperbolic tangent of the left-hand side of the inequality is equal to both the Caratheodory distance and the Kobayashi distance from (0,0) to (s, p) in int Γ

It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broué's abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.