It is proved that the infinitesimal generator A of a strongly continuous semigroup of linear operators on a Hilbert space also generates a strongly continuous group if and only if the resolvent of −A, (λ+A)−1, is also a bounded function on some right-hand-side half plane of complex numbers, and converges strongly to zero as the real part of λ tends to infinity. An application to a partial differential equation is given.

We study finite centralizing extensions A⊂H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H[Rarr ]spec A to coincide with the X-orbits in spec H, where X denotes the finite group of characters of H that restrict to the counit of A. In particular, all of the fibres are X-orbits if and only if the fibre over the augmentation ideal of A is an X-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group.

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann asked for a simpler proof of these results [4, Problem 2, p. 900]. We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2, p. 46], and that our method can also be applied in hyperbolic space.

For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5, p. 363].

For further information about the notions used below, we refer to Schneider's book [9]. Let [Kscr ]n be the set of all convex bodies in Euclidean space IRn, that is, the set of all compact, convex, non-empty subsets of IRn. Let Sn-1 be the unit sphere. For K∈[Kscr ]n, let NorK be the set of all support elements of K, that is, the pairs (x, u)∈IRn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), …, Θn−1(K, ·), are the unique Borel measures on IRn×Sn−1 that are concentrated on NorK and satisfy

formula here

for all integrable functions f[ratio ]IRn→IR; here λ denotes the Lebesgue measure on IRn. Equation (1), which is a consequence and a slight generalization of Theorem 4.2.1 in Schneider [9], is called the local Steiner formula. Our main result is the following.

In this note, two geometric properties of Banach spaces are proposed and discussed, related to the validity of the lemma of Riemann–Lebesgue in spaces of weakly integrable functions. The class of Banach spaces with the analytic Riemann–Lebesgue property is shown to be precisely the class of Banach spaces for which a weak form of a theorem of Ingham holds.

An example is provided of a space that does not have the homotopy type of an H-space, and yet has the same $n$ -type as a double loop space for each natural number $n$ . It is also locally homotopy equivalent to a double loop space at each prime $p$ .

In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over $\mathbb{Q}$.

We generalise to Jordan–Banach algebras some results of Aupetit on perturbation by inessential elements. This enables us to show that the kh-socle is the largest inessential ideal, and to give some spectral characterisations of it.

If the subset $E$ of ${\bb T}$ supports a synthesizable $p$ -pseudofunction $T$ , then the restriction $A_p(E)$ to $E$ of the Herz algebra on ${\bb T}$ is not Arens regular. The proof is direct, by brute force, and does not show the existence of Day points for $A_p(E)$ .

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property $O$. It is deduced both that these groups are exact and satisfy Yu's Property $A$.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.

Let [ ]P2,p (where 2 < p) be the class of all finite nilpotent groups of class 2 and of exponent p with an additional predicate for a subgroup of the centre that contains the commutator subgroup. The Fraïssé limit D of this class exists. Non-forking is described for Th(D).

Several results are proved, related to an old problem posed by G. R. MacLane, namely whether functions in the class [Ascr ] that are locally univalent can have arc tracts. In particular, a proof is given of an assertion of MacLane that if f ∈ [Ascr ] is locally univalent and has no arc tracts, then f′ ∈ [Ascr ].

Let S be an elementary operator on a C*-algebra with minimal length [lscr ](S)[ges ]1. Then S is completely positive if and only if S is n-positive for some natural number n[ges ]([lscr ](S)−1)/2.

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

An attempt is made to generalise Bloch and Kato's calculation of some of the local Tamagawa factors appearing in their conjecture, from the crystalline case to the semi-stable case. Success is attained only under severe conditions, but the result applies to the symmetric powers of an elliptic curve with multiplicative reduction.