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.

Let be a very ample vector bundle of rank 2 on with c1() = 4 and c2() = 6. Then it is proved that is the cokernel of a bundle monomorphism , where is the tangent bundle of . This gives a new example of a threefold containing a Bordiga surface as a hyperplane section.

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.

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 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.

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras and are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

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.

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).

A classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu = 4e2u near isolated singularities is generalized to solutions of the Gaussian curvature equation Δu = −κ(z)e2u where κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

Under certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

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

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.

We study the existence of finite absolutely continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

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.

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.

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.

The asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

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.