A flip can be thought of as a diagram X− → X ← X+ of complex threefolds satisfying some conditions. One often thinks of a flip as being in two parts: the first part, X− → X, is the given, while the second, X ← X+, is the unknown. I calculate cohomological properties of the canonical classes, K− = KX− and so on, and in particular properties of the function

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In the case of the toric flips of Danilov [3] and Reid [7], this function can be expressed in terms of lattice points of multiplies of a polyhedron giving sharpness for the more general result.

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.

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive self- complementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congruence pm ≡ 1(mod 4), where pm is the highest power of p which divides the order of the graph.

Soient F un corps commutatif localement compact non archimédien et ψ un caractère additif non trivial de F. Soient σ une représentation du groupe de Weil–Deligne de F, et ψσˇ sa contragrédiente. Nous calculons le facteur ε(σ[otimes ]σˇ, ψ, ½). De manière analogue, nous calculons le facteur ε(π×πˇ, ψ, ½) pour toute représentation admissible irréductible π de GLn(F). En conséquence, si F est de caractéristique nulle et si σ et π se correspondent par la correspondance de Langlands construite par M. Harris, ou celle construite par les auteurs, alors les facteurs ε(σ[otimes ]σˇ, ψ, s) et ε(π×πˇ, ψ, s) sont égaux pour tout nombre complexe s.

Let F be a non-Archimedean local field and ψa non-trivial additive character of F. Let σ be a representation of the Weil–Deligne group of F and σˇ its contragredient representation. We compute ε(σ[otimes ]σˇ, ψ, ½). Analogously, we compute ε(π×πˇ, ψ, ½) for all irreducible admissible representations π of GLn(F). Consequently, if F has characteristic zero, and σ, π correspond via the Langlands correspondence established by M. Harris or the correspondence constructed by the authors, then we have ε(σ[otimes ]σˇ, ψ, s) = ε(π×πˇ, ψ, s) for all s∈[Copf ].

In iteration theory of rational functions, it is well known that any Fatou component is mapped onto another in an n-to-1 manner, and that periodic components are simply, doubly or infinitely connected.

For meromorphic functions, the situation is much more complicated. Using Ahlfors' theory of covering surfaces, we prove that Fatou components are mapped ‘nearly’ onto others, and that periodic components are again simply, doubly or infinitely connected. Instead of considering meromorphic functions with only one essential singularity, we allow countable sets of singularities and partly even sets of logarithmic capacity zero.

It remains open whether doubly connected periodic components of meromorphic functions with only one singularity are necessarily Herman rings (as holds for rational functions). However, there is a function with two singularities and a doubly connected periodic component which is not an Herman ring.

Let δ be a coaction of a locally compact group G on a C*-algebra A. We show that if I is a δ-invariant ideal in A, then 0 → I×δIG → A×δG → (A/I)×δIG → 0 for full crossed products, as Landstad et al. have done for spatial crossed products by coactions. We prove that for suitable coactions, the crossed products of C0(X)-algebras are again C0(X)-algebras, and the crossed products of continuous C*-bundles by a locally compact group are again continuous C*-bundles.

We show that all derivations from the group algebra [lscr ]1(G) of a free group into its nth dual, where n is a positive even integer, are inner. Combined with the known result for odd values of n, this shows that these algebras are permanently weakly amenable.

Let X be a real Banach space. A set K ⊆ X is called a total cone if it is closed under addition and non-negative scalar multiplication, does not contain both x and −x for any non-zero x∈X, and is such that K−K := {x − y : x, y ∈ K} is dense in X. Suppose that T is a bounded linear operator on X which leaves a closed total cone K invariant. We denote by σ(T) and r(T) the spectrum and spectral radius of T.

Krein and Rutman [5] showed that if T is compact, r(T) > 0 and K is normal (that is, inf{∥x+y∥ : x, y ∈ K, ∥x∥ = ∥y∥ = 1} > 0), then r(T) is an eigenvalue of T with an eigenvector in K. This result was later extended by Nussbaum [6] to any bounded operator T such that re(T) < r(T), where re(T) denotes the essential spectral radius of T, without the hypothesis of normality. The more general question of whether r(T) ∈ σ(T) for all bounded operators T was answered in the negative by Bonsall [1], who as well as giving counterexamples described a property of K called the bounded decomposition property, which is sufficient to guarantee that r(T) ∈ σ(T).

More recently, Toland [8] showed that if X is a separable Hilbert space and T is self-adjoint, then r(T) ∈ σ(T), without any extra hypotheses on K. In this paper we extend Toland's results to normal operators on Hilbert spaces, removing in passing the separability hypothesis.

We establish the existence of self-homeomorphisms of Rn, n [ges ] 2, which are chaotic in the sense of Devaney, preserve volume and are spatially periodic. Moreover, we show that in the space of volume-preserving homeomorphisms of the n-torus with mean rotation zero, those with chaotic lifts to Rn are dense, with respect to the uniform topology. An application is given for fixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–Franks Theorem).

If two operator algebras A and B are strongly Morita equivalent (in the sense of [5]), then their C*- envelopes C*(A) and C*(B) are strongly Morita equivalent (in the usual C*-algebraic sense due to Rieffel). Moreover, if Y is an equivalence bimodule for a (strong) Morita equivalence of A and B, then the operation, Y[otimes ]hA−, of tensoring with Y, gives a bijection between the boundary representations of C*(A) for A and the boundary representations of C*(B) for B. Thus the ‘noncommutative Choquet boundaries’ of Morita equivalent A and B are the same. Other important objects associated with an operator algebra are also shown to be preserved by Morita equivalence, such as boundary ideals, the Shilov boundary ideal, Arveson's property of admissability, and the lattice of C*-algebras generated by an operator algebra.

In this paper we consider extensions of Livsic's well-known classical results on rigidity of measurable cocycles. We consider transformations which need not be hyperbolic, and cocycles which take values in certain non-abelian non-compact groups.

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side.’ They are interested in the number S of survivors when the shooting ceases, and the surprising result, for which they give both numerical and heuristic evidence, is that when m=n,

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as m→∞ (where E denotes expectation).

It is the occurrence of this curious power of m, rather than any application to real gunfights, which makes the Williams–McIlroy process of interest. The purpose of this paper is to give an essentially elementary proof of the fact that if m and n are not too different, then S/m3/4 has a simple asymptotic distribution that leads at once to (1.1).

A theorem on gambling systems for Bernoulli sequences is extended to arbitrary sequences of random variables by using the notion of the likelihood ratio, and the martingale technique together with the tool of moment generating functions for the study of a.e. convergence is presented.

Let [Ascr ] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr ]0, and let {Tt, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr ]. The infinitesimal generator [Lscr ] of {Tt} is a bounded linear operator on the Banach space [Ascr ]. For any Hilbert space [Kscr ], denote by [Bscr ]([Kscr ]) the von Neumann algebra of all bounded operators on [Kscr ]. Christensen and Evans [3] have shown that [Lscr ] has the form

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where π is a representation of [Ascr ] in [Bscr ]([Kscr ]) for some Hilbert space [Kscr ], R: [Hscr ]0 → [Kscr ] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr ]0, X∈[Ascr ]} is total in [Kscr ], and K0 is a fixed element of [Ascr ]. The unitality of {Tt} implies that [Lscr ](1) = 0, and consequently K0 = iH−½R*R, where H is a hermitian element of [Ascr ]. Thus (1.1) can be expressed as

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We say that the quadruple ([Kscr ], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr ] of the semigroup {Tt}. It is quite possible that another set ([Kscr ]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr ]. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.