The Quarterly Journal of Mathematics - recent issues

BI-ORDERINGS ON PURE BRAIDED THOMPSON'S GROUPS

Burillo, J., Gonzalez-Meneses, J. — 2008-02-21

In this paper it is proved that the pure braided Thompson's group BF admits a bi-order, analogously to the bi-order of the pure braid groups.

METAPLECTIC OPERATORS ON Cn

Feichtinger, H. G., Hazewinkel, M., Kaiblinger, N., Matusiak, E., Neuhauser, M. — 2008-02-21

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H(G), defined for a locally compact abelian group G. For G=R^d, H is the usual Heisenberg group. For the case when G is the finite cyclic group Z_n, only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on Cⁿ. We also include applications to Gabor frames.

THE DENSITY OF INTEGRAL POINTS ON COMPLETE INTERSECTIONS

Marmon, O. — 2008-02-21

In this paper, an upper bound for the number of integral points of bounded height on an affine complete intersection defined over Z is proven. The proof uses an extension to complete intersections of the method used for hypersurfaces by Heath-Brown (The density of rational points on non-singular hypersurfaces, Proc. Indian Acad. Sci. Math. Sci. 104 (1994) 13–29), the so called ‘q-analogue’ of van der Corput's AB process.

A NEW METHOD OF PRODUCING FUNCTIONAL RELATIONS AMONG MULTIPLE ZETA-FUNCTIONS

Matsumoto, K., Tsumura, H. — 2008-02-21

In this paper, we introduce a new method of producing functional relations among multiple zeta-functions. This method can be regarded as a kind of multiple analogue of Hardy's one of proving the functional equation for the Riemann zeta-function. Using this method, we give new functional relations for multiple zeta-functions. In particular, substituting positive integers into variables of them, we obtain known relation formulas for the multiple zeta-values. Furthermore, applying our method to a certain series involving hyperbolic sine functions, we can obtain certain multiple analogues of the known results given by Cauchy, Ramanujan, Berndt and so on.

ON THE STRUCTURE OF ASYMPTOTIC lp SPACES

Odell, E., Schlumprecht, Th., Zsak, A. — 2008-02-21

We prove that if X is a separable, reflexive space which is asymptotic l_p for some 1 ≤ p ≤ , then X embeds into a reflexive space Z having an asymptotic l_p finite-dimensional decomposition (FDD). This result leads to an intrinsic characterization of subspaces of spaces with an asymptotic l_p FDD. More general results of this type are also obtained. As a consequence, we prove the existence of universal spaces for certain classes of separable, reflexive and asymptotic l_p spaces.

LABELLING THE CHARACTER TABLES OF SYMMETRIC AND ALTERNATING GROUPS

Wildon, M. — 2008-02-21

Let X be a character table of the symmetric group S_n. It is shown that unless n = 4 or n = 6, there is a unique way to assign partitions of n to the rows and columns of X so that for all and , X is equal to (), the value of the irreducible character of S_n labelled by on elements of cycle type . Analogous results are proved for alternating groups, and for the Brauer character tables of symmetric and alternating groups.

GLUING TECHNIQUES IN TRIANGULAR GEOMETRY

Balmer, P., Favi, G. — 2007-11-19

We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.

FOUR-DIMENSIONAL CONFORMAL C-SPACES

Gover, A. R., Nagy, P.-A. — 2007-11-19

We investigate the structure of conformal C-spaces, a class of Riemannian manifolds which naturally arises as a conformal generalization of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension four, we obtain a full answer to this question and also investigate the incidence of the Bach condition on this class of metrics. This is related to earlier results obtained in the Einstein–Weyl context.

REAL BELYI THEORY

Kock, B., Singerman, D. — 2007-11-19

We develop a Belyi-type theory that applies to Klein surfaces, that is, (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular, we extend Belyi's famous theorem from Riemann surfaces to Klein surfaces.

AN ELEMENTARY PROOF OF THE ABRESCH ROSENBERG THEOREM ON CONSTANT MEAN CURVATURE IMMERSED SURFACES IN S2 x R AND H2 x R

Leite, M. L. — 2007-11-19

We make explicit the centers and radii of the horizontal geodesic circles on a constant mean curvature surface with null Abresch–Rosenberg differential in S² x R and in H² x R (horizontal horocycles are also determined) and prove that those centers project on to the same point, unless the complete surface is foliated by horocycles. This new visualization of the rotational and special surfaces classified by Abresch and Rosenberg is obtained in a direct way, just taking covariant derivatives of the unit normal along the flows of two global tangent fields. Moreover, this approach reveals that the special surfaces in H² x R have constant intrinsic curvature K –1+4H² (–1, 0], so they form a non-rotational family of hyperbolic examples converging to a flat one, as 4H² 1.

LARGE DIFFERENCES BETWEEN CONSECUTIVE PRIMES

Matomaki, K. — 2007-11-19

We use a sieve method together with mean and large value results for Dirichlet polynomials to prove that $$\sum _{\begin{array}{c}{p}_{n+1}-{p}_{n} > {x}^{1/2}\\ x\le {p}_{n}\le 2x\end{array}}{p}_{n+1}-{p}_{n}\ll {x}^{2/3},$$ where p_n is the nth prime number.

ON A LIMITING RELATION BETWEEN RAMANUJAN'S ENTIRE FUNCTION Aq(z) AND {theta}-FUNCTIONS

Zhang, R. — 2007-11-19

We will use a discrete analogue of the classical Laplace method to show that the main term of the asymptotic expansions of certain entire functions, including Ramanujan's entire function A_q(z), can be expressed in terms of -functions.

DISTRIBUTION OF ANGLES BETWEEN GEODESIC RAYS ASSOCIATED WITH HYPERBOLIC LATTICE POINTS

Boca, F. P. — 2007-08-30

For every two points z₀, z₁ in the upper-half plane H, consider all elements in the principal congruence group (N), acting on H by fractional linear transformations, such that the hyperbolic distance between z₁ and z₀ is at most R > 0. We study the distribution of angles between the geodesic rays [z₁, z₀] as R -> , proving that the limiting distribution exists independently of N and explicitly computing it. When z₁ = z₀, this is found to be the uniform distribution on the interval [–/2, /2].

SPECIAL SYMPLECTIC SIX-MANIFOLDS

Conti, D., Tomassini, A. — 2007-08-30

We classify nilmanifolds with an invariant symplectic half-flat structure. We study the transverse or quotient geometry of six-manifolds with an SU (3)-structure preserved by a Killing vector field, giving characterizations in the symplectic half-flat and integrable case.

REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM

Deshmukh, S. — 2007-08-30

Let M be an orientable connected and compact real hypersurface of the complex space form C^{(n + 1)/2}. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequality n²² ≤ (n² – 1) + f², where is the characteristic vector field, A is the shape operator and (n – 1) is the infimum of the Ricci curvatures of hypersurface M, then it is shown that is a constant and M is the sphere Sⁿ(²) in C^{(n + 1)/2}.

LANGLANDS DUALITY AND G2 SPECTRAL CURVES

Hitchin, N. — 2007-08-30

We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m + 1). We then show that duality for G₂ is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution.

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

Joyce, D. — 2007-08-30

An invariant of quasiprojective K-varieties X with values in a commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closed in X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We first define a unique extension ' of to finite type Artin K-stacks $$F$$, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a K-variety, G a special K-group acting on X, and [X/G] is the quotient stack. This only works if (G) is invertible in for all special K-groups G, which excludes = as (G_m) = 0. But we can extend the construction to get round this.

Then we develop the theory of stack functions on Artin stacks. These are a universal generalization of constructible functions on Artin stacks. There are several versions of the construction: the basic one $$\hbox{ }\hbox{ }SF(F)$$, and variants $$\hbox{ }\underset{\_\_}{\hbox{ }SF\phantom{\rule{-0.16666666em}{0.1em}}}\phantom{\rule{0.16666666em}{0.1em}}(F,\mathrm{{\rm Y}},\mathrm{\Lambda }),\dots $$ ‘twisted’ by motivic invariants. We associate a Q-vector space $$\hbox{ }\hbox{ }SF(F)$$ or a -module $$\hbox{ }\underset{\_\_}{\hbox{ }SF\phantom{\rule{-0.16666666em}{0.1em}}}\phantom{\rule{0.16666666em}{0.1em}}(F,\mathrm{{\rm Y}},\mathrm{\Lambda })$$ to each Artin stack $$F$$, with functorial operations of multiplication, pullbacks * and pushforwards _* under 1-morphisms $$\phi :F\to G$$;, and so on. They will be important tools in the author's series on ‘Configurations in abelian categories’.

COARSE AND UNIFORM EMBEDDINGS INTO REFLEXIVE SPACES

Kalton, N. J. — 2007-08-30

Answering an old problem in nonlinear theory, we show that c₀ cannot be coarsely or uniformly embedded into a reflexive Banach space, but that any stable metric space can be coarsely and uniformly embedded into a reflexive space. We also show that certain quasi-reflexive spaces (such as the James space) also cannot be coarsely embedded into a reflexive space and that the unit ball of these spaces cannot be uniformly embedded into a reflexive space. We give a necessary condition for a metric space to be coarsely or uniformly embeddable in a uniformly convex space.