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 $p>1/2$.

This paper treats the existence of multiple positive solutions of the semi-positone Sturm–Liouville boundary value problem. \begin{eqnarray*} &\lambda (p(t)y'(t))'+g(t)f(t,y(t))=0 \qquad\mbox{almost everywhere on } [R_{0},R_{1}],\\ &\alpha z(R_{0})-\beta p(R_{0})z'(R_{0})=0,\\ &\gamma z(R_{1})+\delta p(R_{1})z'(R_{1})=0, \end{eqnarray*} where $g\in L_{+}^{\infty}[R_{0}, R_{1}]$ and $f$ is allowed to take negative values (that is, $f$ is semi-positone). When $\lambda=1$, new results on the existence of one or two nonzero positive solutions are obtained. These results generalize previous results for positone cases (that is, $f\ge 0$) to the semi-positone cases. We illustrate our results with an explicit example which has two nonzero positive solutions. These results are used to deduce results on intervals of eigenvalues for which there exist one or two nonzero positive eigenfunctions. Applications of these eigenvalue results are provided.

We obtain operator-valued analogues of Bohr's inequality involving both the absolute values of operators and their norms, that when restricted to the scalar case imply the classical Bohr inequality. In the scalar case we extend Bohr's inequality to the case where one function is majorized by another function, to the Hardy space, $H^2(\mathbb{D})$, and to bounded analytic functions on the annulus.

Let $\Gamma$ be a non-Euclidean crystallographic group. $\Gamma$ is said to be non-maximal if there exists a non-Euclidean crystallographic group $\Gamma'$ such that $\Gamma \le \Gamma'$ and the dimension of the Teichmüller space of $\Gamma$ equals the dimension of the Teichmüller space of $\Gamma'$. The full list of such pairs of groups is computed in the case when $\Gamma$ is non-normal in $\Gamma'$. The corresponding problem for Fuchsian groups was solved by Singerman.

Semigroups (with zero) and inverse semigroups (with zero) are constructed from subshifts. The semigroups and inverse semigroups that are obtained by this construction are characterized and classified.

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

We give a simple direct proof of the interpolation inequality $ \|\nabla f\|^2_{L^{2p}} \le C \|f\|_{\rm BMO} \|f\|_{W^{2,p}}$, where $ 1<p<\infty$. For $p=2$ this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo–Nirenberg inequalities where the norm of the function is taken in $L^\infty$ and other norms are in $L^q$ for appropriate $q>1$.

Let ${\mathcal P}_n$ be the collection of all polynomials of degree at most $n$ with real coefficients. A subtle Bernstein-type extremal problem is solved by establishing the inequality \[\big\|U_n^{(m)}\big\|_{L_q({\mathbb{R}})} \leq \big(c^{1+1/q}m\big)^{m/2} n^{m/2} \|U_n\|_{L_q({\mathbb{R}})}\] for all $U_n \in \widetilde{G}_n\!$, $q \in (0,\infty]$, and $m=1,2, \ldots$, where $c$ is an absolute constant and $\[ \widetilde{G}_n :=\biggl\{f: f(t) = \sum_{j=1}^N {P_{m_j}(t) e^{-(t-\lambda_j)^2}}, \; \lambda_j \in {\mathbb{R}}, \; P_{m_j}\in\, {\mathcal P}_{m_j}\!, \; \sum_{j=1}^N{(m_j+1)} \leq n \biggr\}. \]$ Some related inequalities and direct and inverse theorems about the approximation by elements of $\widetilde{G}_n$ in $L_q({\mathbb{R}})$ are also discussed.

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value $\infty$. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.

We show that the analytic rank, as defined by Murphy, of a unital graph C*-algebra is either 1 or 0, depending on whether or not the underlying graph possesses an initial loop. As a corollary, we show that the analytic rank of certain quantum spaces (including some quantum spheres, projective spaces and lens spaces) is 1.

In this paper, it is proved that for $n\geq 2$, any horizontally homothetic submersion $\varphi :\mathbb{R}^{n+1}\longrightarrow (N^{n}, h)$ is a Riemannian submersion up to a homothety. It is also shown that if $\varphi :\mathbb{S}^{n+1}\longrightarrow (N^{n}, h)$ is a horizontally homothetic submersion, then $n=2m$, $(N^{n}, h)$ is isometric to $\mathbb{C}P^{m}$ and, up to a homothety, $\varphi$ is a standard Hopf fibration $\mathbb{S}^{2m+1} \longrightarrow \mathbb{C}P^{m}$.

Let $G$ be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra $L^{1}_{\omega}(G)$ is a symmetric Banach $*$-algebra if and only if the weight function $\omega $ satisfies the GRS-condition. This condition expresses in a precise technical sense that $\omega $ grows subexponentially.

Let $A$ and $B$ be countable discrete groups, and let $\Gamma=A\ast B$ be their free product. We show that if $A$ and $B$ are uniformly embeddable into a uniformly convex Banach space, then so is $\Gamma$.

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator $T(a) \in \mathcal{L}(\ell^{p,\mu}_N)$, $1 < p < \infty$ and $\mu \in \mathbb{R}$, with a continuous matrix-valued generating function $a$. We prove that the approximation numbers of the finite sections $T_n(a) = P_n T(a) P_n$ have the $k$-splitting property, provided $T(a)$ is a Fredholm operator on $\ell^{p,\mu}_N$.

We characterize composition operators on spaces of real analytic functions which at the same time have closed image and are open onto their images. Under some mild assumptions, we also characterize composition operators with closed range and composition operators open onto their images.

By considering coverings of surfaces by annuli, we extend previous results concerning the Nielsen kernel of topologically finite Riemann surfaces to arbitrary orbifolds. Specifically, we show that the length of a boundary loop in the Nielsen kernel is strictly greater than twice the length of the corresponding boundary loop of its orbifold, and that the infinite Nielsen kernel has empty interior.

In this paper we point out how some recent developments in the theory of constant scalar curvature Kähler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces.

A unital C*-algebra $A$ is said to have cancellation of projections if the semigroup $D(A)$ of Murray–von Neumann equivalence classes of projections in matrices over $A$ is cancellative. It has long been known that stable rank one implies cancellation for any $A$, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras.

For each non-exact C*-algebra $A$ and infinite compact Hausdorff space $X$ there exists a continuous bundle ${\mathcal B}$ of C*-algebras on $X$ such that the minimal tensor product bundle $A \otimes {\mathcal B}$ is discontinuous. The bundle ${\mathcal B}$ can be chosen to be unital with constant simple fibre. When $X$ is metrizable, ${\mathcal B}$ can also be chosen to be separable. As a corollary, a C*-algebra $A$ is exact if and only if $A\otimes {\mathcal B}$ is continuous for all unital continuous C*-bundles ${\mathcal B}$ on a given infinite compact Hausdorff base space. The key to proving these results is showing that for a non-exact C*-algebra $A$ there exists a separable unital continuous C*-bundle ${\mathcal B}$ on [0,1] such that $A\otimes {\mathcal B}$ is continuous on [0,1] and discontinuous at 1, a counter-intuitive result. For a non-exact C*-algebra $A$ and separable C*-bundle ${\mathcal B}$ on [0,1], the set of points of discontinuity of $A\otimes{\mathcal B}$ in [0,1] can be of positive Lebesgue measure, and even of measure 1.

In this paper, we show that the set of all common fixed points of a one-parameter nonexpansive semigroup is that of some single nonexpansive mapping. We next compare our result with Bruck's famous fixed-point theorem. We finally prove very simple convergence theorems to a common fixed point. In our discussion, we assume neither the strict convexity of the underlying space, nor the weak compactness of the domain of a nonexpansive semigroup.