Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C∗correspondences over N 0 , generalising the classical result for a doubly commuting pair of isometries due to M. S lociński. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are pr...

The classical (m, k)-Landen transform Fm,k is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m’th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P, we prove that Fm,0 induces a dominant rational self-map Rd,m,0 of P 2d+1 of algebraic degree m, and for 1 ≤ k < m, the t...

We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the N\'{e}ron-Severi groups. The discrete logarithm problem is reduci...

Active commuting could contribute to increasing physical activity. The objective of this study was to characterise patterns of active commuting to and from schools in children and adolescents in Chile. A total of 453 Chilean children and adolescents aged between 10 and 18 years were included in this study. Data regarding modes of commuting and commuting distance was collected using a validated ...

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+\omega(y), y)$ $T_\alpha y)=(x+\alpha, y)$, where $\omega(y)$ is non-degenerate $\alpha$ Diophantine. Using KAM iterative scheme for group action we show are simultaneously $C^\infty$-linearizable if has intersection p...

Here the solid arrows represent maps that are given, and the problem is to find a map h commuting in the diagram. Usually the map i is an inclusion, and p is some kind of “bundle map” such as a local product. Note the special cases: (i) if i is an inclusion, E = A, and f = 1A, then the extension problem asks whether A is a retract of X; and (ii) if p is surjective, X=B, and g = 1B, the lifting ...

Let Fλ(S) be the space of tensor densities of degree (or weight) λ on the circle S. The spaceDk λ,μ(S1) of k-th order linear differential operators from Fλ(S) to Fμ(S) is a natural module over Diff(S), the diffeomorphism group of S. We determine the algebra of symmetries of the modules Dk λ,μ(S1), i.e., the linear maps on Dk λ,μ(S1) commuting with the Diff(S)-action. We also solve the same prob...

Let f(z) = ez + z, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we als...

In a previous paper, the authors obtained a model for a bi-isometry, that is, a pair of commuting isometries on complex Hilbert space. This representation is based on the canonical model of Sz. Nagy and the third author. One approach to describing the invariant subspaces for such a bi-isometry using this model is to consider isometric intertwining maps from another such model to the given one. ...