We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmüller space of $X$ into the Fréchet space of Hölder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.

Explicit quadratic Liapunov functions that provide necessary and sufficient conditions for the asymptotic stability of the system of linear difference equations $x (t + 1) = A x(t)$ are constructed by transforming the original systems to $y (t + 1) = G y(t)$ , where $G$ is a companion matrix associated with the characteristic polynomial of $A$. A necessary and sufficient condition for all roots of the characteristic polynomial to lie in the unit circle $|z| < 1$ on the complex plane is also derived.

Let $A = (a^{ij})$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in the space of non-negative operators on $\mathbb{R}^d$ and let $b = (b^i)$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in $\mathbb{R}^d$. Let

\[Lu(t, x) = \partial_{t} u(t, x) + a^{ij}(t, x)\partial_{x_i}\partial_{x_j}u(t, x) + b^i(t, x)\partial_{x_i}u(t, x), \quad u \in C_0^\infty((0, 1)\times \mathbb{R}^d).\]

Under broad assumptions on $A$ and $b$, we construct a family $\mu = (\mu_t)_{t \in [0, 1)}$ of probability measures $\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem $L^{*}\mu = 0$ with initial condition $\mu_0 = \nu$, where $\nu$ is a probability measure on $\mathbb{R}^d$, in the following weak sense:

$$\[ \int_0^1\int_{\mathbb{R}^d} Lu(t, x)\, \mu_t(dx)\, dt=0, \quad u \in C_0^\infty((0, 1) \times \mathbb{R}^d), \] $$

and

$$\[ \lim\limits_{t \to 0} \int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx) = \int_{\mathbb{R}^d} \zeta(x)\, \nu(dx), \quad \zeta \in C_0^\infty(\mathbb{R}^d). \] $$

Such an equation is satisfied by transition probabilities of a diffusion process associated with $A$ and $b$ provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, $b$ may be locally unbounded or $A$ may be degenerate. An infinite-dimensional analogue is discussed as well. Main methods are $L^p$-analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time.

Let $A$ and $B$ be C*-algebras, with $A$ separable and $B$$\sigma$-unital and stable. It is shown that there are natural isomorphisms

$$ E(A,B) = KK(SA,Q(B)) = [SA,Q(B) \otimes K], $$

where $SA=C_0(0, 1) \otimes A$, $[\cdot, \cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B) / B$ is the generalized Calkin algebra, and $K$ denotes the C*-algebra of compact operators of an infinite-dimensional separable Hilbert space.

If a hyperbolic link has a prime alternating diagram $D$, then we show that the link complement's volume can be estimated directly from $D$. We define a very elementary invariant of the diagram $D$, its twist number $t(D)$, and show that the volume lies between $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, where $v_3$ is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology.

Let $\ast$ denote convolution and let $\delta_x$ denote the Dirac measure at a point $x$. A function in $L^2({\mathbb{R}})$ is called a difference of order $1$ if it is of the form $ g- \delta_{x} \ast g$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. Also, a difference of order $2$ is a function of the form $ g- 2^{-1}( \delta_{x} \ast g + \delta_{-x} \ast g)$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. In fact, the concept of a ‘difference of order $s$’ may be defined in a similar manner for each $s > 0$. If $\widehat{f}$ denotes the Fourier transform of $f$, it is known that a function $f$ in $L^2 ({\mathbb{R}})$ is a finite sum of differences of order $s$ if and only if $\int_{-\infty}^{\infty} |\widehat {f}(x)|^2 |x|^{-2s} dx < \infty$, and the vector space of all such functions is denoted by ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$. Every function in ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$ is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$, where ${\mathrm{int}} (t)$ denotes the integer part of $t$. Thus, every function in ${\mathcal{D}}_{1} (L^2 ({\mathbb{R}}) )$ is a sum of three first order differences, but it was proved in 1994 that there is a function in ${\mathcal{D}}_1 (L^2 ({\mathbb{R}}) )$ which is never the sum of two first order differences. This complemented, for the group ${\mathbb{R}}$, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in $L^2 ({\mathbb{R}})$ such that, for each $s \ge 1/2$, this function is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$ but it is never the sum of ${\mathrm{int}} (2s)$ differences of order $s$. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if $x_1, \ldots, x_n$ are points in the unit square,

$$\sum_{1 \le i < j \le n} |x_i - x_j|^{-2} \ge 200^{-1} n^2 \ln n.$$

Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis.

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

Let $A$ be a non-commutative normed algebra with centre $Z(A)$. A simple application of the triangle inequality shows that if $a \in A$ is close to $Z(A)$ then $a$ almosts commutes with elements of the unit ball of $A$. The extent to which the converse holds is a fundamental question in the study of non-commutativity and has been previously investigated for various classes of Banach algebras, including von Neumann algebras and other C*-algebras. In this context, $K(A)$ is defined to be the smallest number in $[0, \infty]$ such that $$ {\rm distance}(a, Z(A)) \leq K(A) \Vert {\rm ad}(a) \Vert $$ for all $a \in A$, where ${\rm ad}(a)$ is the inner derivation $x \to xa-ax$ for $(x \in A)$. For a C*-algebra $A$, the constant $K_s(A)$ is defined similarly, restricting to self-adjoint $a \in A$.

For a unital C*-algebra $A$, it is already known that the only possible values for $K_s(A)$ are $ \frac{1}{2}, 1, \frac{3}{2}, \ldots, \infty $, that $K(A) \in \{\frac{1}{2}, \frac{1}{\sqrt{3}}\} \cup [1, \infty]$ and that these cases are distinguished by purely topological conditions on the primitive ideal space ${\rm Prim}(A)$. In this paper, further progress is made on the question of possible values for $K(A)$ in $[1, \infty]$. It is shown that $ K(A) \leq \frac{2}{\sqrt{3}}K_s(A)$ and that either $ K(A) \leq \frac{4}{\sqrt{15}} \sim 1.033$ or $ K(A) \geq \frac{1}{2} + \frac{1}{\sqrt{3}} \sim 1.077$. The values $\frac{4}{\sqrt{15}}$ and $\frac{1}{2} + \frac{1}{\sqrt{3}}$ are attained and each of them arises as the maximal bounding radius of a constrained subset of the plane which is obtained from the structure of ${\rm Prim}(A)$.

It is also shown that if $A$ has trivial centre then $K(A)$ is further restricted: $$ K(A) \in \{\frac{1}{2}, 1, \frac{1}{2} + \frac{1}{\sqrt{3}}\} \cup [\frac{3}{2}, \infty].$$ Again, the various cases are distinguished by purely topological conditions on ${\rm Prim}(A)$, some of which are expressed in terms of primal ideals.

The main purpose of this paper is to study the validity of the Hausdorff–Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.

For a dense $G_\delta$-set of parameters, the irrational rotation algebra is shown to contain infinitely many C*-subalgebras satisfying the following properties. Each subalgebra is isomorphic to a direct sum of two matrix algebras of the same (perfect square) dimension; the Fourier transform maps each summand onto the other; the corresponding unit projection is approximately central; the compressions of the canonical generators of the irrational rotation algebra are approximately contained in the subalgebra.

The set N of all null geodesics of a globally hyperbolic $(d + 1)$-dimensional spacetime $(M, g)$ is naturally a smooth $(2d - 1)$-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to $S^{(d - 1)}$. It was conjectured by Low that for $d = 2$ two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for $d = 3$ smooth linking should be replaced with Legendrian linking.

Let $H$ be the discrete Schrödinger operator

$$Hu(n):=u(n-1)+u(n+1)+v(n)u(n),\quad u(0)=0$$

acting on $l^2 (\mathbb{Z}^+)$, where the potential $v$ is real-valued and $v(n) \to 0$ as $n \to \infty$. Let <formula form="inline" disc="math" id="frm007"><formtex notation="AMSTeX">$P$ be the orthogonal projection onto a closed linear subspace $\Lambda \subset l^2 (\mathbb{Z}^+)$. In a recent paper E. B. Davies defines the second order spectrum ${\rm Spec}_2(H, \Lambda)$ of $H$ relative to $\Lambda$ as the set of $z \in \mathbb{C}$ such that the restriction to $\Lambda$ of the operator $P(H - z)^2P$ is not invertible within the space $\Lambda$. The purpose of this article is to investigate properties of ${\rm Spec}_2(H, \Lambda)$ when $\Lambda$ is large but finite dimensional. We explore in particular the connection between this set and the spectrum of $H$. Our main result provides sharp bounds in terms of the potential $v$ for the asymptotic behaviour of ${\rm Spec}_2(H, \Lambda)$ as $\Lambda$ increases towards $l^2(\mathbb{Z}^+)$.