Let K be a field and let V be a vector space of dimension 2m over K. Let ∧V denote the exterior algebra of V and ∧kV its kth exterior power for 0[les ]k[les ]2m. Let f be a non-degenerate alternating bilinear form defined on V×V. The symplectic group Sp2m(K) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧kV. In the case that K is algebraically closed and 1[les ]k[les ]m, it is known that ∧kV contains a composition factor corresponding to the fundamental weight ωk of a root system of type Cm. We shall refer to the irreducible module for Sp2m(K) given by this composition factor as a fundamental module.

We develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn).

The paper derives a formula for the second variation of the displacement function for polynomial perturbations of Hamiltonian systems with elliptic or hyperelliptic Hamiltonians H(x, y)=½y2−U(x) in terms of the coefficients of the perturbation. As an application, the conjecture stated by C. Chicone that a specific cubic system appearing in a deformation of singularity with two zero eigenvalues has at most two limit cycles is proved.

The paper finds best constants in a class of multiplicative inequalities with one-dimensional spatial variables

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where f is a periodic function with zero mean value, and the norms in the right-hand side of the expression are the L2-norms.

We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set.

A function f defined on a domain D has an asymptotic value b∈[−∞, ∞] at a∈δD if there exists a path γ in D ending at a such that u(p) tends to b as p tends to a along γ. The set of all points on δD at which f has an asymptotic value b is denoted by A(f, b).

G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +∞) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein.

For α>0, let [Mscr ]α be the class of subharmonic functions u in IRn+1+ ≡{(x, y)[ratio ]x∈IRn, y>0} satisfying the growth condition

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for some constant C(u) depending on u. Denote by [Fscr ](u) the Fatou set of u, which consists of points on δIRn+1+ where u has finite vertical limits. For β>0, denote by Hβ the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].

Let

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be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections

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have real negative zeros only, then the series (0.1) converges in the whole complex plane C, and its sum f(z) is an entire function of order 0. Since then, formal power series with restrictions on zeros of their sections have been deeply investigated by several mathematicians. We cannot present an exhaustive bibliography here, and restrict ourselves to the references [1, 2, 3], where the reader can find detailed information.

In this paper, we propose a different kind of generalisation of Pólya's theorem. It is based on the concept of multiple positivity introduced by M. Fekete in 1912, and it has been treated in detail by S. Karlin [4].

As is well known, the classical Titchmarsh–Weyl m-function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M-matrix often turns out to be rather cumbersome. The paper interprets the m-function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m-function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.

We consider the iterates of the heat operator

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on Rn+1={(X, t); X =(x1, x2, …, xn) ∈Rn, t∈R}. Let Ω⊂Rn+1 be a domain, and let m[ges ]1 be an integer. A lower semi-continuous and locally integrable function u on Ω is called a poly-supertemperature of degree m if

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If u and −u are both poly-supertemperatures of degree m, then u is called a poly-temperature of degree m. Since H is hypoelliptic, every poly-temperature belongs to C∞(Ω), and hence

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For the case m=1, we simply call the functions the supertemperature and the temperature.

In this paper, we characterise a poly-temperature and a poly-supertemperature on a strip

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by an integral mean on a hyperplane. To state our result precisely, we define a mean A[·, ·]. This plays an essential role in our argument.

Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as

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acting on L2(RN) defines a semigroup in Lp(RN) for various values of p including p=1. The operator is defined as a quadratic form sum. That is, we put

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for f∈C∞c (all integrals are on RN and are with respect to Lebesgue measure), and note that the closure of the form is non-negative and has domain equal to the Sobolev space Wm,2. We then assume that the potential has quadratic form bound less than 1 with respect to Q0, and define

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This form is closed and is associated with a semibounded self-adjoint operator H in L2 (see [17, p. 348; 5, Theorem 4.23]). One can then ask whether the semigroup e−Ht defined on L2 for t[ges ]0 is extendable to a strongly continuous one-parameter semigroup on Lp for other values of p, and if so whether one can describe the domain and spectrum of its generator.

Functionals of higher derivative type are linear combinations of functionals of the form f→f(n)(ζ), where n[ges ]2 and 0<[mid ]ζ[mid ]<1. The paper shows that, if L is a functional of higher derivative type and f is a function in the class S of univalent functions that maximises Re{L} over S, then L(f)≠0. In addition, if the function f is a rational function, then it must be a rotation of the Koebe function k(z)=z(1−z)−2. These results are applied to establish several cases of the two-functional conjecture for functionals of higher derivative type.

Relating geometric notions like type and cotype from local Banach space theory with some important topological invariants for Fréchet spaces (for example the density condition), we prove a version of Maurey's celebrated extension theorem for the category of Fréchet spaces, and apply it to Grothendieck's problème des topologies.

Let Ω be a fixed open cube in ℝn. For r∈[1, ∞) and α∈[0, ∞) we define

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where Q is a cube in ℝn (with sides parallel to the coordinate axes) and χQ stands for the characteristic function of the cube Q.

A well-known result of Gehring [5] states that if

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for some p∈(1, ∞) and c∈(0, ∞), then there exist q∈(p, ∞) and C=C(p, q, n, c)∈(0, ∞) such that

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for all cubes Q⊂Ω, where [mid ]Q[mid ] denotes the n-dimensional Lebesgue measure of Q. In particular, a function f∈L1(Ω) satisfying (1.1) belongs to Lq(Ω).

In [9] it was shown that Gehring's result is a particular case of a more general principle from the real method of interpolation. Roughly speaking, this principle states that if a certain reversed inequality between K-functionals holds at one point of an interpolation scale, then it holds at other nearby points of this scale. Using an extension of Holmstedt's reiteration formulae of [4] and results of [8] on weighted inequalities for monotone functions, we prove here two variants of this principle involving extrapolation spaces of an ordered pair of (quasi-) Banach spaces. As an application we prove the following Gehring-type lemmas.

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk Δ[ratio ][mid ]z[mid ]<1. Then we shall say that f(z)∈N. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that

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where h1(z), h2(z) are holomorphic in Δ and have positive real part there, while Π1(z), Π2(z) are Blaschke products, that is,

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where p is a positive integer or zero, 0<[mid ]aj[mid ]<1, c is a constant and

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We note in particular that, if c≠0, so that f(z)[nequiv ]0,

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so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in Δ such that

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If f(zj)=0 for each j and f∈N, then f(z)≡0.

N. Danikas [1] has shown that the same conclusion obtains if f(zj)→0 sufficiently rapidly as j→∞. Let εj, λj be sequences of positive numbers such that

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Danikas then defines

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and proves Theorem A.

This paper defines the n-fold central Haagerup tensor product [otimes ]ni=1[Lscr ]h[Rscr ] of a von Neumann algebra [Rscr ], and shows that the map θz[ratio ][otimes ]ni =1[Lscr ]h[Rscr ]→CB([Rscr ]n−1, [Rscr ]) given by

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is an isometry.

Let f[ratio ][Copf ]n, 0→[Copf ]p, 0 be a [Kscr ]-finite map germ, and let i=(i1, …, ik) be a Boardman symbol such that [sum ]i has codimension n in the corresponding jet space Jk(n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of [sum ]i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way: f must have rank n−i1 and, in addition, in the case p=6, f must be a singularity of type [sum ]i1,i2.

The paper gives new examples of [Ascr ]-structures on certain SO(3)-principal fibre bundles.

The paper proves several weighted imbedding theorems for domains with fractal boundaries. The weights considered are distances to the boundary to certain powers, and the domains are so-called s-John domains. The paper also proves, in the general setting, that the existence of an imbedding implies compactness of the imbedding for lower exponents. Moreover, following Maz'ya, the paper reformulates the imbedding theorems in the language of local isoperimetric and capacity estimates.

A Hilbert module over a C*-algebra B is a right B-module X, equipped with an inner product 〈·, ·〉 which is linear over B in the second factor, such that X is a Banach space with the norm ∥x∥[ratio ]=∥〈x, x〉∥1/2. (We refer to [8] for the basic theory of Hilbert modules; the basic example for us will be X=B with the inner product 〈x, y〉=x*y.) We denote by B(X) the algebra of all bounded linear operators on X, and we denote by L(X) the C*-algebra of all adjointable operators. (In the basic example X=B, L(X) is just the multiplier algebra of B.) Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule. We always assume that A is nondegenerate in the sense that [AX]=X, where [AX] denotes the closed linear span of AX.

Denote by AX the algebra of all mappings on X of the form

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where m is an integer and ai∈A, bi∈B for all i. Mappings of form (1.1) will be called elementary, and this paper is concerned with the question of which mappings on X can be approximated by elementary mappings in the point norm topology.

Let P be a polyhedron in H3 of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H3. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group.

As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H3/Γ.

In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.

Using simplicial methods developed in an earlier note, the paper constructs topological quantum field theories using an algebraic model of a homotopy n-type as initial data, generalising constructions of Yetter for n=1 and n=2.