A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z ∈ : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.

We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and also show that each provides a necessary and sufficient set of conditions for the embeddability of a category in a groupoid. We show that all such conditions, and more besides, may be derived in a uniform way from a particular class of directed graphs which we call quadrangle clubs, and we prove a number of results (extending those of Mal'cev, Lambek, Bush and Krstić) on which families of quadrangle clubs provide sufficient conditions for embeddability.

In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp2n) = n + 1 for n odd.

We prove that if G is a centre-by-metabelian pro-p group of type FPm over , so every homomorphic image of G is of type FPm over .

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G2-structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G2-manifold is exactly G2 if and only if the fundamental group π1(M) is finite and neither M nor any double cover of M is homeomorphic to a cylinder.

The LMO invariant is a very strong invariant such that it is expected to classify integral homology 3-spheres. In this paper we identify the set of the degree ≤ 6 parts of the logarithm of the LMO invariant for integral homology 3-spheres. As an application, we obtain a complete set of relations which characterize the set of Ohtsuki's invariants {λi(M)} for i ≤ 6. For any simple Lie algebra , we also obtain a complete set of relations which characterize the set of perturbative P invariants {(M)} for i ≤ 3.

The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.

Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

In this paper we apply singularity theory methods to the classification of reversible-equivariant steady-state bifurcations depending on one real parameter. We assume that the group of symmetries and reversing symmetries is a compact Lie group Γ, and the equivalence is defined in order to preserve these symmetries and reversing symmetries in the normal forms and their unfoldings. When the representation of Γ is self-dual, we show that the classification can be reduced to the standard equivariant context. In this case, we establish a one-to-one association between the classification of bifurcations in the reversible-equivariant context and the classification of purely equivariant bifurcations related to them. As an application of the results, we obtain the classification of self-dual representations of Z2 ⊕ Z2 and D4 on the plane.

The notion of approximate amenability was introduced by Ghahramani and Loy, in the hope that it would yield Banach algebras without bounded approximate identity which nonetheless had a form of amenability. So far, however, all known approximately amenable Banach algebras have bounded approximate identities (b.a.i.). In this paper we define approximate amenability and contractibility of Fréchet algebras, and we prove the analogue of the result for Banach algebras that these properties are equivalent. We give examples of Fréchet algebras which are approximately contractible, but which do not have a bounded approximate identity. For a good many Fréchet algebras without b.a.i., we find either that the algebra is approximately amenable, or it is “obviously” not approximately amenable because it has continuous point derivations. So the situation for Fréchet algebras is quite close to what was hoped for Banach algebras.

A general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebra W(C,ℓ1)(d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the Hp Hardy space to Lp (or Hp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.

Assume is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + (z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

A framework is introduced for the study of general Radon shape diffusions, that is, shape diffusions induced by projections of randomly rotating shapes. This is done via a convenient representation of unoriented Radon shape diffusions in (unoriented) D.G. Kendall shape space through a Brownian motion on the hypersphere. This representation leads to a coordinate system for the generalized version of Radon diffusions since it is shown that shape can be essentially identified with unoriented shape in the projected case. A bijective correspondence between Brownian motion on real projective space and Radon shape diffusions is established. Furthermore, equations are derived for the general (unoriented) Radon diffusion of shape-and-size, and stationary measures are discussed.

Consider networks on n vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as n13/10. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order n3/2. Finally, if we require the network length to be O(n) then the mean number of hops grows as order log log n. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.

We consider classes of second order boundary value problems with a nonlinearity f(t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞ and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.

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