5,823 research outputs found
Algebraic Cycles, Fundamental Group of a Punctured Curve, and Applications in Arithmetic
The results of this paper can be divided into two parts, geometric and
arithmetic. Let be a smooth projective curve over , and
be distinct points. Let be the mixed Hodge
structure of functions on given by iterated integrals
of length (as defined by Hain). In the geometric part, inspired by a
work of Darmon, Rotger, and Sols, we express the mixed Hodge extension
given by the weight filtration on
in terms of certain null-homologous algebraic cycles on
. As a corollary, we show that the extension
determines the point . The
arithmetic part of the paper gives some number-theoretic applications of the
geometric part. We assume that and , where is a subfield of and is a projective
curve over . Let be the Jacobian of . We use the extension
to associate to each a
point , which can be described analytically in terms of iterated
integrals. The proof of -rationality of uses that the algebraic cycles
constructed in the geometric part of the paper are defined over . Assuming a
certain plausible hypothesis on the Hodge filtration on
holds, we show that an algebraic cycle for which is torsion, gives
rise to relations between periods of . Interestingly,
these relations are non-trivial even when one takes to be the diagonal of
. The geometric result of the paper in case, and the fact that one
can associate to a family of points in , are
due to Darmon, Rotger, and Sols. Our contribution is in generalizing the
picture to higher weights.Comment: 65 pages. A few geometric corollaries and an application to periods
have been added (compared to the first version
On the Role of Mechanics in Chronic Lung Disease.
Progressive airflow obstruction is a classical hallmark of chronic lung disease, affecting more than one fourth of the adult population. As the disease progresses, the inner layer of the airway wall grows, folds inwards, and narrows the lumen. The critical failure conditions for airway folding have been studied intensely for idealized circular cross-sections. However, the role of airway branching during this process is unknown. Here, we show that the geometry of the bronchial tree plays a crucial role in chronic airway obstruction and that critical failure conditions vary significantly along a branching airway segment. We perform systematic parametric studies for varying airway cross-sections using a computational model for mucosal thickening based on the theory of finite growth. Our simulations indicate that smaller airways are at a higher risk of narrowing than larger airways and that regions away from a branch narrow more drastically than regions close to a branch. These results agree with clinical observations and could help explain the underlying mechanisms of progressive airway obstruction. Understanding growth-induced instabilities in constrained geometries has immediate biomedical applications beyond asthma and chronic bronchitis in the diagnostics and treatment of chronic gastritis, obstructive sleep apnea and breast cancer
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