Abstract. We consider the following conjecture (from Huang, et al): Let ∆ + denote the upper half disc in C and let γ = (−1, 1) (viewed as an interval in the real axis in C). Assume that F is a holomorphic function on ∆ + with continuous extension up to γ such that F maps γ into {| Im z| ≤ C| Re z|}, for some positive C. If F vanishes to infinite order at 0 then F vanishes identically.We show that given the conditions of the conjecture, either F ≡ 0 or there is a sequence in ∆ + , converging to 0, along which Im F/ Re F (defined where Re F = 0) is unbounded.
Background:In scoliosis, kypholordos and wedge properties of the vertebrae should be involved in determining how stress is distributed in the vertebral column. The impact is logically expected to be maximal at the apex.Aim:To introduce an algorithm for constructing artificial geometric models of the vertebral column from DICOM stacks, with the ultimate aim to obtain a formalized way to create simplistic models, which enhance and focus on wedge properties and relative tilting.Material/Methods:
Our procedure requires parameter extraction from DICOM image-stacks (with PACS,IDS-7), mechanical FEM-modelling (with Matlab and Comsol). As a test implementation, models were constructed for five patients with thoracal idiopathic scoliosis with varying apex rotation. For a selection of load states, we calculated a response variable which is based upon distortion energy.Results:For the test implementation, pairwise t-tests show that our response variable is non-trivial and that it is chiefly sensitive to the transversal stresses (transversal stresses where of main interest to us, as opposed to the case of additional shear stresses, due to the lack of explicit surrounding tissue and ligaments in our model). Also, a pairwise t-test did not show a difference (n = 25, p-value≈0.084) between the cases of isotropic and orthotropic material modeling.Conclusion: A step-by-step description is given for a procedure of constructing artificial geometric models from chest CT DICOM-stacks, such that the models are appropriate for semi-global stress-analysis, where the focus is on the wedge properties and relative tilting. The method is inappropriate for analyses where the local roughness and irregularities of surfaces are wanted features. A test application hints that one particular load state possibly has a high correlation to a certain response variable (based upon distortion energy distribution on a surface of the apex), however, the number of patients is too small to draw any statistical conclusions.
We generalize to infinite dimensions the concept of polyanalytic functions.These are our main results: (1) A characterization based upon restriction, which generalizes a known characterization of holomorphic functions on Banach spaces. (2) A property of special local uniform limits which yields an approximation result. (3) We introduce meta-analytic functions on Hilbert manifolds together with a characterization compatible with (1). We also point out several corollaries to our characterizations.
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