Similarity of holomorphic matrices on 1-dimensional Stein spaces

R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.Comment: This is a revised version of a part of a preprint from Sep 2016, later (Mar 2017) posted as arXiv:1703.09524. Version 2 is a mistake (the wrong file was sent). The differences of version 3 to version 1 are: a "Note added in proof" and a reference are added, typos are corrected. The difference of version 4 to version 3 is: some small corrections are carried out, rusults unchange

On holomorphic matrices on bordered Riemann surfaces

Let \D be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map A:\overline D\to \mathrm{SL}(2,\C), which is holomorphic in \D, there exist continuous maps E,F:\overline \D\to \mathfrak{sl}(2,\C), which are holomorphic in \D, such that $A=e^Ee^F$. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.Comment: 11 page

Effect of calcium phosphate and vitamin D3supplementation on bone remodelling and metabolism of calcium, phosphorus, magnesium and iron

BACKGROUND: The aim of the present study was to determine the effect of calcium phosphate and/or vitamin D(3) on bone and mineral metabolism. METHODS: Sixty omnivorous healthy subjects participated in the double-blind, placebo-controlled parallel designed study. Supplements were tricalcium phosphate (CaP) and cholecalciferol (vitamin D(3)). At the beginning of the study (baseline), all subjects documented their normal nutritional habits in a dietary record for three successive days. After baseline, subjects were allocated to three intervention groups: CaP (additional 1 g calcium/d), vitamin D(3) (additional 10 μg/d) and CaP + vitamin D(3). In the first two weeks, all groups consumed placebo bread, and afterwards, for eight weeks, the test bread according to the intervention group. In the last week of each study period (baseline, placebo, after four and eight weeks of intervention), a faecal (three days) and a urine (24 h) collection and a fasting blood sampling took place. Calcium, phosphorus, magnesium and iron were determined in faeces, urine and blood. Bone formation and resorption markers were analysed in blood and urine. RESULTS: After four and eight weeks, CaP and CaP + vitamin D(3) supplementations increased faecal excretion of calcium and phosphorus significantly compared to placebo. Due to the vitamin D(3) supplementations (vitamin D(3), CaP + vitamin D(3)), the plasma 25-(OH)D concentration significantly increased after eight weeks compared to placebo. The additional application of CaP led to a significant increase of the 25-(OH)D concentration already after four weeks. Bone resorption and bone formation markers were not influenced by any intervention. CONCLUSIONS: Supplementation with daily 10 μg vitamin D(3) significantly increases plasma 25-(OH)D concentration. The combination with daily 1 g calcium (as CaP) has a further increasing effect on the 25-(OH)D concentration. Both CaP alone and in combination with vitamin D(3) have no beneficial effect on bone remodelling markers and on the metabolism of calcium, phosphorus, magnesium and iron. TRIAL REGISTRATION: NCT0129702