This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p>2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized étale cohomology theory called étale BP2.

We study the distribution of a family of generalized Euler constants arising from integers sieved by finite sets of primes . For , the set of the first r primes, as r → ∞. Calculations suggest that is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of and the Riemann hypothesis.

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

Let (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

For a locally compact group G, let B(G) denote its Fourier–Stieltjes algebra. Any continuous, piecewise affine map α: Y ⊂ H → G induces a completely bounded algebra homomorphism jα: B(G) → B(H) [14, 15] and we prove that jα is w* – w* continuous if and only if α is an open map. This extends one of the main results in [3], due to M.B. Bekka, E. Kaniuth, A.T. Lau and G. Schlichting. Several classical theorems regarding isomorphisms and extensions of homomorphisms on group algebras of abelian groups are extended to the setting of Fourier–Stieltjes algebras of amenable groups. As applications, when G is amenable we provide complete characterizations of those maps between Fourier–Stieltjes algebras that are either associated to a piecewise affine mapping, or are completely bounded and w* – w* continuous.

Let B ⊆ A be an inclusion of C*-algebras. Then B is said to norm A if, for each X ∈ (A),In this paper we introduce and study the conesThese are shown to coincide with the standard positive cones precisely when B norms A, and we apply this to obtain automatic complete positivity of certain positive maps between C*-algebras.

A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. We describe Wirtinger presentations realizable by virtual link groups.

We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).

Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimension n+2 to a Riemannian manifold of dimension 2, which can be factorised as an n-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

We study properties of finitely determined corank 2 quasihomogeneous map germs f:(, 0) → (, 0). Examples and counter examples of such map germs are presented.

Let ϵ > 0 and be a family of finite subsets of the Cantor set . Following D.H. Fremlin, we say that is ϵ-filling over if is hereditary and for every F ⊆ finite there exists G ⊆ F such that G ϵ and . We show that if is ϵ-filling over and C-measurable in , then for every P ⊆ perfect there exists Q ⊆ P perfect with . A similar result for weaker versions of density is also obtained.

We present an approach to compute the Hausdorff dimension of a class of sets of real numbers that are defined in terms of nonlinear relations between frequencies of digits in some integer base m. We consider the model case of quadratic perturbations, for which the computations are already rather involved. We show that the Hausdorff dimension is analytic in the parameter determining the perturbation. Our approach also allows to estimate the asymptotic behavior of the Taylor coefficients of the dimension in terms of m.

We consider a system of “forms” defined for ẕ = (zij) on a subset of bywhere d = d1 + ⋅ ⋅ ⋅ + dl and for each pair of integers (i,j) with 1 ≤ i ≤ l, 1 ≤ j ≤ di we denote by a strictly increasing sequence of natural numbers. Let = {z ∈ : |z| < 1} and let where for each pair (i, j) we have Xij = . We study the distribution of the sequence on the l-polydisc defined by the coordinatewise polar fractional parts of the sequence Xk(ẕ) = (L1(ẕ)(k),. . ., Ll(ẕ)(k)) for typical ẕ in More precisely for arcs I1, . . ., I2l in , let B = I1 × ⋅ ⋅ ⋅ × I2l be a box in and for each N ≥ 1 define a pair correlation function byand a discrepancy by ΔN = {VN(B) − N(N−1)leb(B)}, where the supremum is over all boxes in . We show, subject to a non-resonance condition on , that given ε > 0 we have ΔN = o(N(log log N)1+ε) for almost every . Similar results on extremal discrepancy are also proved. Our results complement those of I. Berkes, W. Philipp, M. Pollicott, Z. Rudnick, P. Sarnak, R Tichy and the author in the real setting.

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-DimsE = DimsE for all integers s > 0 and all analytic sets E ⊆ RN.

The aim of this paper is to prove that P-DimsE = DimsE for all real numbers s > 0 and analytic sets E ⊆ RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

We consider certain sets of non-normal continued fractions for which the asymptotic frequencies of digit strings oscillate in one or other ways. The Hausdorff dimensions of these sets are shown to be the same value 1/2 as long as they are non-empty. An interesting example among them is the set of “extremely non-normal continued fractions” which was previously conjectured to be of Hausdorff dimension 0.

We present an example of an entire function with a Baker domain such that the complement of the set of singular values of the function contains annuli of arbitrarily large width. This is the first known example of a function with this property.

An asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.