The main aim of this paper is to present a general method of proving Ω-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

In this paper we describe the 2-category 2Mat of 2-matrices, i.e., a strict totally coordinatized version of the 2-category 2Vect of Kapranov and Voevodsky finite dimensional 2-vector spaces, analogous to the category Mat of matrices which appears as the coordinatized version of the category Vect of finite dimensional vector spaces. In particular, explicit formulas for the composition of 1-morphisms and the two compositions between 2-morphisms are given.

In this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to an m-primary ideal I is effectively bounded by the degree of nilpotency of I. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

We establish Whitehead and Ganea characterizations for proper LS-category. We use the embedding of the proper category into the exterior category, and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given.

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

In a fundamental paper in 1984, Vaughan Jones developed his new polynomialinvariant of knots using a Markov trace on the Temperley–Liebalgebra. Subsequently, Lickorish used the associated bilinear pairing toprovided an alternative proof for the existence of the 3-manifold invariants ofWitten, Reshetinkin and Turaev. A key property of this form is thenon-degeneracy of this form except at the parameter values±2cos(π/(n+1)) [7]. Koand Smolinsky derived a recursive formula for the determinants of specificminors of Markov's form, establishing the needed non-degeneracy[6]. In this paper,we define a triangular change of basis in which the form is diagonal andexplicitly compute the diagonal entries of this matrix as products of quotientsof Chebyshev polynomials, corroborating the determinant computation of Ko andSmolinsky. The method of proof employs a recursive method for defining therequired orthogonal basis elements in the Temperley–Lieb algebra,similar in spirit to Jones' and Wenzl's recursive formulafor a family of projectors in the Temperley–Lieb algebra. We define apartial order on the non-crossing chord diagram basis and give an explicitformula for a recursive construction of an orthogonal basis, via a recursionover this partial order. Finally we relate this orthogonal basis to basesconstructed using the calculus of trivalent graphs developed by Kauffman andLins [5].

We establish some properties of Thurston's asymmetric metric L on the Teichmüller space of a surface of genus with punctures and with negative Euler characteristic. We study convergence of sequences of elements in in the sense of L, as well as sequences that tend to infinity in . We show that the topology that the asymmetric metric L induces on Teichmüller space is the same as the usual topology. Furthermore, we show that L satisfies the axioms of a (not necessarily symmetric) metric in the sense of Busemann and conclude that L is complete in the sense of Busemann.

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of “almost-geodesics”. We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is closed in the Painlevé–Kuratowski topology.

A spray is a second-order differential equation field on the slit tangent bundleof a differentiable manifold, which is homogeneous of degree 1 in the fibrecoordinates in an appropriate sense; two sprays which are projectivelyequivalent have the same base-integral curves up to reparametrization. We showhow, when the base manifold is two-dimensional, to construct from any projectiveequivalence class of sprays a conformal class of metrics on a four-dimensionalmanifold, of signature (2, 2); the Weyl conformal curvature of these metrics issimply related to the projective curvature of the sprays, and the geodesics ofthe sprays determine null geodesics of the metrics. The metrics in question havepreviously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a differentmethod involving the exploitation of a close analogy between the Cartan geometryof second-order ordinary differential equations and of three-dimensionalCauchy–Riemann structures. From this perspective the derived metricsare seen to be analoguous to those defined by Fefferman in the CR theory, andare therefore said to be of Fefferman type. Our version of the constructionreveals that the Fefferman-type metrics are derivable from the scalar tripleproduct, both directly in the flat case (which we discuss in some detail) and bya simple extension in general. There is accordingly in our formulation a verysimple expression for a representative metric of the class in suitablecoordinates.

We give estimates of the essential norms of weighted composition operators acting between Bergman spaces on strongly pseudoconvex domains. We also characterize boundedness and compactness of these operators.

A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

In analogy to the classical isomorphism between and , we show that a large class of moderate linear mappings acting between the space of Colombeau rapidly decreasing generalized functions and the space of temperate ones admits generalized integral representations, with kernels belonging to . Furthermore, this result contains the classical one.