Consider a commutative ring $R$ and a simplicial map, $X\mathop {\longrightarrow }\limits ^{\pi }K,$ of finite simplicial complexes. The simplicial cochain complex of $X$ with $R$ coefficients, $\Delta ^*X,$ then has the structure of an $(R,K)$ chain complex, in the sense of Ranicki . Therefore it has a Ranicki-dual $(R,K)$ chain complex, $T \Delta ^*X$. This (contravariant) duality functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$ was defined algebraically on the category of $(R,K)$ chain complexes and $(R,K)$ chain maps.

Our main theorem, 8.1, provides a natural $(R,K)$ chain isomorphism:

\[ T\Delta^*X\cong C(X_K) \]

$C(X_K)$

$X_K$

$X_K$

$X$

$(R,K)$

$C(X_K)$

Recent numerical studies have shown that the entire solar system is permeated with arch-like structures originating from all planets. Particles placed on such arches experience planetary close encounters after only one or few orbital revolutions. In this work, we are interested how thece arches, which we associate to encounter manifolds of Jupiter, appear in three dimensions for higher inclinations.

Our results show that about 0.5% of the observed domain [a, e, i] = [2 AU, 11.5 AU] × [0, 0.7] × [0°, 90°] is covered by the manifolds. For inclinations up to ∼5°, the arch-like structures are almost unchanged compared to those initially observed in the orbital plane of Jupiter. At higher inclinations, the number of encounter orbits rapidly decreases to narrow domains where the manifolds stretch up to inclinations of 90° (and above) in a very steep manner.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

The Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli–Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.