Small quotients in Euclidean algorithms

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension ?? _M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of ?? _M for M???? were provided by Hensley (J. Number Theory 40:336?C358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166?C172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371?C385, 1987) and Vallée (J. Number Theory 72:183?C235, 1998) considered the case of a fixed bound M when the denominator N tends to ??. Later, Hensley (Pac. J. Math. 151(2):237?C255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M ₀; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336?C358, 1992) about the Hausdorff dimension ?? _M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531?C570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331?C386, 2005).     

Phthalocyanine-peptide conjugates via palladium-catalyzed cross-coupling reactions

Phthalocyanines (Pc) were conjugated with peptide moieties to improve their target selectivity for potential use as fluorescence and/or positron emission tomography (PET) probes in medical imaging. Three synthetic methods based on palladium-catalyzed cross-coupling reactions (Sonogashira, Buchwald-Hartwig, and Suzuki-Miyaura) were investigated. Using these methods, a series of peptides monofunctionalized with Pc at the N/C-terminal position or on a phenylalanine side chain was obtained in good yields and characterized.     

Convenient synthesis of multifunctional EDTA-based chiral metal chelates substituted with an S-mesylcysteine

We describe the synthetic route to ethylenediaminetetraacetic acid (EDTA) derivatives that can be attached to surface-exposed thiol functional groups of cysteine residues in proteins, via a methylthiosulfonate moiety that is connected in a stereochemically unique way to the C-1 carbon atom of EDTA. Such compounds can be used to align proteins in solution without the need to add liquid crystalline media, and are, therefore, of great interest for the NMR spectroscopic analysis of biomolecules. The binding constant for the paramagnetic tag to lanthanide ions was determined by measuring luminescence. For the Tb(+3)-ligand complex, a K(b) value of 6.5 x 10(17) M(-1) was obtained. This value is in excellent agreement with literature values for the related EDTA compound. In addition, it could be shown that there is no significant reduction in the luminescence intensity upon addition of a 10(4) excess of Ca2+ ions, indicating that this paramagnetic tag is compatible with buffers containing high concentrations of divalent alkaline earth ions.     

Peracetylated cellulose: end group modification and structural analysis by means of <Superscript>1</Superscript>H-NMR spectroscopy

A homologous series of partially hydrolyzed celluloses (level-off-DP cellulose) with weight-average molecular weight (DP_w) < 150 were peracetylated and characterized by ¹H-NMR spectroscopy. The results demonstrate the utility of ¹H-NMR spectroscopy to assign the chemical shifts of all end groups of the peracetylated cellulose. On the one hand, the chemical shifts of all methine and methylene protons of the non-reducing terminal end group (TEG) as well as the - and -anomer of the reducing end group (REG) could be determined by two-dimensional NMR techniques (COSY-DQF) and by selective excitation of isolated proton signals (1D-TOCSY) of these end groups. On the other hand, the spectral resolution was high enough to determine the number-average molecular weight (DP_n) of peracetylated level-off-DP cellulose (LODP cellulose acetates) as shown in comparison with the data of gel permeation chromatography (GPC). This molecular weight determination of cellulose using end group analysis by means of ¹H-NMR spectroscopy was demonstrated for the first time. Furthermore, a specific modification of hydroxyls in end groups could be exemplified in case of 1-OH-deacetylation of the REG of peracetylated LODP cellulose.     

Women's experiences of undergoing BRCA1 and BRCA2 testing: organisation of the German Hereditary Breast and Ovarian Cancer Consortium Survey and Preliminary Data from Münster

BACKGROUND: In order to promote safe and effective testing for BRCA1 and BRCA2 mutations in clinical practice, a network of expert centres in hereditary breast and ovarian cancer ('Verbundprojekt famili?rer Brust- und Eierstockkrebs'; German Consortium on Hereditary Breast and Ovarian Cancer) has been established by the Deutsche Krebshilfe (German Cancer Aid). To improve practice, evaluations based upon the views of patients who have undergone testing concerning the impact of the genetic diagnosis and the quality of the services they receive and require are undertaken. METHODS: Herein, we first describe the protocols for interdisciplinary pre- and post-test counselling and for molecular diagnostics, then the feedback from the patients undergoing testing. Women and men who had obtained their test results at least 6 months earlier were interviewed using a questionnaire including open and standardised questions at 11 participating centres in 2002-2003. Reported here are the survey protocol and preliminary data from interviews with women conducted by the centre based at the University of Münster's Medical School in 2002 (n = 46). RESULTS: Compared to international guidelines, the Consortium's protocols provide more specifically outlined indications based on the mutation frequencies observed in the German population for families that should be offered interdisciplinary counselling and genetic testing. The data from the Münster survey suggest that there is little regret regarding the decision to undergo testing and that the vast majority of women would undergo the test again. However, women with positive findings experienced more problems than women with negative results. They were less prone to recommend the test and to communicate the tests results in their family compared to women with negative test results. Communication of test results within the family was characterised by preferential information of female family members. CONCLUSION: BRCA1/2 testing should be provided within a framework that ensures harmonisation and standardisation of services and that protects users. Women and men with positive test results may need special support from counsellors on how to handle test results in their families. Strategies need to be developed and evaluated on how to help stimulate and facilitate the dissemination of information within families without potentially 'overstraining' the 'messenger patient', at least when a mutation has been identified.     

On the Convergence of Cone Splitting Algorithms with ω-Subdivisions

We present a convergence proof of the Tuy cone splitting algorithm with a pure -subdivision strategy for the minimization of a concave function over a polytope. The key idea of the convergence proof is to associate with the current hyperplane a new hyperplane that supports the whole polytope instead of only the portion of it contained in the current cone. A branch-and-bound variant of the algorithm is also discussed.     

Sensitive routine liquid chromatography–tandem mass spectrometry method for serum estradiol and estrone without derivatization

The need for a routinely applicable assay to measure low estradiol levels in adult men, postmenopausal women, and young adolescents was recently discussed in an Endocrine Society position statement. Our aim was to develop a sensitive liquid chromatography–tandem mass spectrometry method for estradiol and estrone in human serum without the need for derivatization or extended extraction protocols. After protein precipitation of serum with a mixture of methanol/acetonitrile (85/15) (v/v) containing isotopic internal standards (17β-estradiol-16,16,17-d ₃ and estrone-2,3,4-¹³C), we quantified estradiol and estrone by two-dimensional liquid chromatography–tandem mass spectrometry with electrospray ionization in the negative mode monitoring 5?×?271.20→145.00 (17β-estradiol) and 269.20→145.00 (estrone). Sensitivity was increased by using fluoride and summation of 5 identical transitions for estradiol. Our method was analytically validated, compared against direct immunoassays using serum of 25 adult men, and clinically tested by measuring samples of 3 men at baseline and after chemical castration, 30 postmenopausal women and 15 patients receiving aromatase inhibitors. Total imprecision was below 20 % for the low quality controls. Limit of quantification was 1.3 ng/L (4.8 pmol/L) for estradiol and 1.2 ng/L (4.4 pmol/L) for estrone. Estradiol in Certified Reference Material BCR-576 was within specified uncertainty limits. No significant ion suppression or interference was observed. Our method showed modest correlation with direct immunoassay for estradiol (r ²?=?0.64) but no correlation for estrone (r ²?=?0.12). Patient sample results were within expected ranges. In conclusion, we developed a routinely applicable liquid chromatography–tandem mass spectrometry method for estradiol and estrone measurement which is sensitive enough for use in men, postmenopausal women, and young adolescents.

Bio‐based Biodegradable and Biocompatible Hyperbranched Polyurethane: A Scaffold for Tissue Engineering

Hyperbranched polyurethanes are synthesized using TDI, PCL diol, butanediol, and pentaerythritol (1–5 wt%) as the B₄ reactant with and without the monoglyceride of sunflower oil. The biodegradation, physico‐mechanical, and thermal properties are found to be tailored by varying the percentage weight of the branching unit. An MTT/hemolytic assay and subcutaneous implantation in Wistar rats followed by cytokine/ALP assay and histopathology studies confirm a better biocompatibility of HBPU with MG than without MG. HBPU supports the proliferation of dermatocytes with no toxic effect in major organs, in addition the in vitro degraded products are non‐toxic. Cell adherence and proliferation endorse the bio‐based HBPU as a prospective scaffold material in the niche of tissue engineering.