We derive Povzner–Wienholtz-type self-adjointness results for m × m matrix-valued Sturm–Liouville operatorsin L2((a, b);R dx)m, m ∈ N, for (a, b) a half-line or R.

We consider random perturbations of two-dimensional Navier–Stokes equations. Under some natural conditions on random forces, we study asymptotic properties of solutions and stationary measures.

A sequence of integers S is called Glasner if, given any ε > 0 and any infinite subset A of T = R/Z, and given y in T, we can find an integer n ∈ S such that there is an element of {nx : x ∈ A} whose distance to y is not greater than ε. In this paper we show that if a sequence of integers is uniformly distributed in the Bohr compactification of the integers, then it is also Glasner. The theorem is proved in a quantitative form.

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

There is a large catalogue of decompositions of conditioned superprocesses in terms of an ‘immortal backbone’ or ‘skeleton’ along the branches of which mass is constantly immigrated. We add to this with a study of the (infinite-variance) (1 + β)-superprocess, conditioned on survival until some fixed time T. As one would expect, we see a Poisson number of immortal trees (conditioned on there being at least one), along which mass (conditioned to die before time T) is immigrated. However, here we see a new source of immigration. Not only is mass immigrated along the branches of the immortal trees, but also there is an extra burst of immigration whenever the immortal tree branches. Moreover, the rate of immigration along the branches is no longer deterministic. In the limit as T → ∞, the immortal trees degenerate to the Evans immortal particle and the immigration (of unconditioned mass) along the particle is dictated by a stable subordinator.

Suppose that $c$ is a linear operator acting on an $n$-dimensional complex Hilbert Space $H$, and let $\tau$ denote the normalized trace on $B(H)$. Set $b_1 = (c+c^*)/2$ and $b_2 = (c-c^*)/2i$, and write $B$ for the spectral scale of $\{b_1, b_2\}$ with respect to $\tau$. We show that $B$ contains full information about $W_k(c)$, the $k$-numerical range of $c$ for each $k = 1,\dots,n$. This is in addition to the matrix pencil information that has been described in previous papers. Thus both types of information are contained in the geometry of a single 3-dimensional compact, convex set. We then use spectral scales to prove a new fact about $W_k(c)$. We show in Theorem 3.4 that the point $\lambda$ is a singular point on the boundary of $W_k(c)$ if and only if $\lambda$ is an isolated extreme point of $W_k(c)$: i.e. it is the end point of two line segments on the boundary of $W_k(c)$. In this case $\lambda = (n/k)\tau(cz)$, where $z$ is a central projection in the algebra generated by $c$ and the identity. In addition we show how the general theory of the spectral scale may be used to derive some other known properties of the $k$-numerical range.

The classical conservation of number principle is an important result in algebraic geometry. We present a version of this principle suitable for the study of topological properties of real algebraic varieties. Our self-contained topological proof does not depend on the intersection theory of algebraic cycles. Some applications are included.

We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.

The simple finite dimensional $(-1,-1)$ balanced Freudenthal Kantor triple systems over fields of characteristic zero are classified.

The tensor center of a group $G$ is the set of elements $a$ in $G$ such that $a\otimes g = 1_\otimes$ for all $g$ in $G$. It is a characteristic subgroup of $G$ contained in its center. We introduce tensor analogues of various other subgroups of a group such as centralizers and 2-Engel elements and investigate their embedding in the group as well as interrelationships between those subgroups.

We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way, establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$, discuss the existence of fixed points for these automorphisms and give sufficient conditions for $E$ to have the density property.

In this paper we study nilmanifolds which are modeled on a quotient of a free 2-step nilpotent Lie group by a 1-dimensional subgroup. In fact we obtain a very easy criterion to decide whether or not such a nilmanifold admits an Anosov diffeomorphism.

We prove that up to isomorphism $\langle a,b\,\vert\, ab=0\rangle$ is the unique principal Rees quotient of a free inverse semigroup that is not trivial or monogenic with zero, satisfying a nontrivial identity in signature with involution.

We continue the investigation of tabular algebras with trace (a certain class of associative $\mathbb{z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a knowledge of the structure constants. This problem is equivalent to understanding a certain category (the category of table data associated to a tabular algebra) which we introduce. The main result is that this category is equivalent to another category (the category of based posets associated to a tabular algebra) whose structure we describe explicitly.

We prove that for a negatively pinched ($-b^2\le\cK\le -1$) topologically tame 3-manifold $\skew5\tilde{M}/\Gamma$, all geometrically infinite ends are simply degenerate. And if the limit set of $\Gamma$ is the entire boundary sphere at infinity, then the action of $\Gamma$ on the boundary sphere is ergodic with respect to harmonic measure, and the Poincaré series diverges when the critical exponent is 2.

In this note, we give a new proof of the fact that an affine semiprime algebra $R$ of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound for the pi degree of $R$.

In this paper we introduce simple multipliers, a special subclass of multipliers on a Banach module. We show that, from a local spectral point of view, these multipliers behave like multipliers on a commutative Banach algebra. Our definition of simple multipliers relies on the notion of point multipliers. These multipliers were studied earlier. However our approach gives new insight into this topic and therefore could be of some interest by itself.

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated.

We show that locally supersoluble finitary groups over certain division rings (e.g. fields) have local systems of hypercyclic normal subgroups.

We compare the probability of generating with a given number of random elements two almost simple groups with the same socle $S$. In particular we analyse the case $S\,{=}\,{\psl}(2,p)$.