Decomposing Blaschke Products And Polynomials

The goal of this paper is to contribute to the understanding of complex polynomials and Blaschke products, two very important function classes in mathematics. For a polynomial, $f,$ of degree $n,$ we study when it is possible to write $f$ as a composition $f=g\circ h$, where $g$ and $h$ are polynomials, each of degree less than $n.$ A polynomial is defined to be \emph{decomposable }if such an $h$ and $g$ exist, and a polynomial is said to be \emph{indecomposable} if no such $h$ and $g$ exist. We apply the results of Rickards in \cite{key-2}. We show that $$C_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,(z-z_{1})(z-z_{2})...(z-z_{n})\,\mbox{is decomposable}\},$$ has measure $0$ when considered a subset of $\mathbb{R}^{2n}.$ Using this we prove the stronger result that $$D_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,\mbox{There exists\,}a\in\mathbb{C}\,\,\mbox{with}\,\,(z-z_{1})(z-z_{2})...(z-z_{n})(z-a)\,\mbox{decomposable}\},$$ also has measure zero when considered a subset of $\mathbb{R}^{2n}.$ We show that for any polynomial $p$, there exists an $a\in\mathbb{C}$ such that $p(z)(z-a)$ is indecomposable, and we also examine the case of $D_{5}$ in detail. The main work of this paper studies finite Blaschke products, analytic functions on $\overline{\mathbb{D}}$ that map $\partial\mathbb{D}$ to $\partial\mathbb{D}.$ In analogy with polynomials, we discuss when a degree $n$ Blaschke product, $B,$ can be written as a composition $C\circ D$, where $C$ and $D$ are finite Blaschke products, each of degree less than $n.$ Decomposable and indecomposable are defined analogously. Our main results are divided into two sections. First, we equate a condition on the zeros of the Blaschke product with the existence of a decomposition where the right-hand factor, $D,$ has degree $2.$ We also equate decomposability of a Blaschke product, $B,$ with the existence of a Poncelet curve, whose foci are a subset of the zeros of $B,$ such that the Poncelet curve satisfies certain tangency conditions. This result is hard to apply in general, but has a very nice geometric interpretation when we desire a composition where the right-hand factor is degree 2 or 3. Our second section of finite Blaschke product results builds off of the work of Cowen in \cite{key-3}. For a finite Blaschke product $B,$ Cowen defines the so-called monodromy group, $G_{B},$ of the finite Blaschke product. He then equates the decomposability of a finite Blaschke product, $B,$ with the existence of a nontrivial partition, $\mathcal{P},$ of the branches of $B^{-1}(z),$ such that $G_{B}$ respects $\mathcal{P}$. We present an in-depth analysis of how to calculate $G_{B}$, extending Cowen's description. These methods allow us to equate the existence of a decomposition where the left-hand factor has degree 2, with a simple condition on the critical points of the Blaschke product. In addition we are able to put a condition of the structure of $G_{B}$ for any decomposable Blaschke product satisfying certain normalization conditions. The final section of this paper discusses how one can put the results of the paper into practice to determine, if a particular Blaschke product is decomposable. We compare three major algorithms. The first is a brute force technique where one searches through the zero set of $B$ for subsets which could be the zero set of $D$, exhaustively searching for a successful decomposition $B(z)=C(D(z)).$ The second algorithm involves simply examining the cardinality of the image, under $B,$ of the set of critical points of $B.$ For a degree $n$ Blaschke product, $B,$ if this cardinality is greater than $\frac{n}{2}$, the Blaschke product is indecomposable. The final algorithm attempts to apply the geometric interpretation of decomposability given by our theorem concerning the existence of a particular Poncelet curve. The final two algorithms can be implemented easily with the use of an HTML

Urinary metabolites and antioxidant products of exogenous melatonin in the mouse

Exogenous melatonin is widely used for sleep disorders and has potential value in neuroprotection, cardioprotection and as an antioxidant. Here, a novel method is described for the determination of melatonin and six metabolites in mouse urine by use of LC-MS/MS and GC-MS. LC-MS/MS is used for the measurement of melatonin, N1-acetyl-5-methoxykynuramine (AMK), N1-acetyl-N2-formyl-5-methoxykynuramine (AFMK) and 6-hydroxymelatonin (6-HMEL), while GC/MS is used for the determination of N-[2-(5-methoxy-2-oxo-2,3-dihydro-1H-indol-3-yl)-ethyl]-acetamide (2-OMEL) and cyclic 3-hydroxymelatonin (3-HMEL) with detection limits on column of 0.02-0.5 pmol, depending on the metabolite. Following oral administration of melatonin to mice, a 0-24 hr urine collection revealed the presence of melatonin (0.2% dose), 6-HMEL (37.1%) and NAS (3.1%) comprising >90% of the total metabolites; AMK and AFMK were also detected at 0.01% each; 2-OMEL was found at 2.2% of the dose, which is >100 times more than the AMK/AFMK pathway, and comprises >5% of the melatonin-related material detected in mouse urine. 3-HMEL was largely found as a sulfate conjugate. These studies establish sensitive assays for determination of six melatonin metabolites in mouse urine and confirm the potential for antioxidant activity of melatonin through the identification in vivo of AMK and AFMK, ring-opened metabolites with a high capacity for scavenging reactive oxygen species.

Extrinsic causes of erosion. Oral hygiene products and acidic medicines

Acidic or EDTA-containing oral hygiene products and acidic medicines have the potential to soften dental hard tissues. The low pH of oral care products increases the chemical stability of some fluoride compounds, favors the incorporation of fluoride ions in the lattice of hydroxyapatite and the precipitation of calcium fluoride on the tooth surface. This layer has some protective effect against an erosive attack. However, when the pH is too low or when no fluoride is present these protecting effects are replaced by direct softening of the tooth surface. Xerostomia or oral dryness can occur as a consequence of medication such as tranquilizers, anti-histamines, anti-emetics and anti-parkinsonian medicaments or of salivary gland dysfunction e.g. due to radiotherapy of the oral cavity and the head and neck region. Above all, these patients should be aware of the potential demineralization effects of oral hygiene products with low pH and high titratable acids. Acetyl salicylic acid taken regularly in the form of multiple chewable tablets or in the form of headache powder as well chewing hydrochloric acids tablets for treatment of stomach disorders can cause erosion. There is most probably no direct association between asthmatic drugs and erosion on the population level. Consumers, patients and health professionals should be aware of the potential of tooth damage not only by oral hygiene products and salivary substitutes but also by chewable and effervescent tablets. Additionally, it can be assumed that patients suffering from xerostomia should be aware of the potential effects of oral hygiene products with low pH and high titratable acids.

Evaluation of the chemical residue monitoring in animal-derived products in Switzerland

This paper evaluates whether the Swiss monitoring programme for foreign substances in animal products fulfils basic epidemiological quality requirements, and identifies possible sources of bias in the selection of samples. The sampling was analysed over a 4-year period (2002-05). The sampling frame in 37 participating abattoirs covered 51% of all slaughtered pigs, 73% of calves, 68% of beef and 36% of cows. The analysis revealed that some sub-populations as defined by the region of origin were statistically over-represented while others were under-represented. The programme that is in accordance with European Union requirements contained some relevant bias. Patterns of under-sampled regions characterized by management type differences were identified. This could lead to an underestimate of the number of contaminated animals within the programme. Although the current sampling was stratified and partially risk-based, its efficiency could be improved by adopting a more targeted approach.

Effects of high-yield thrombocytapheresis on the quality of platelet products

BACKGROUND: The steadily increasing demands for single-donor apheresis platelet (PLT) concentrates (APCs) are a challenge to the PLT supply system. Therefore, efforts to improve plateletpheresis yield, allowing apheresis products to be split into 2 or more units, are valuable strategies. No data to demonstrate in vivo transfusion efficacy of these high-yield split-APCs are currently available, however. STUDY DESIGN AND METHODS: The transfusion efficacy of APCs produced by two apheresis methods involving different harvest and storing procedures and varying PLT yields was investigated. Efficacy measures were the 1-hour percent PLT recovery (PPR(1h)) and the 1-hour corrected count increment (CCI(1h)). In total, 400 APCs, produced with either an Amicus device (Baxter) and stored in PLT additive solution (T-Sol; Amicus method [AM], n = 107) or a Trima device (Gambro) and stored in plasma (Trima method [TM], n = 293), were transfused to 55 children (31 girls; median age, 9.5 years; range, 0.2-18.5 years) with thrombocytopenia due to chemotherapy or aplastic anemia (median, 4 APCs per child; range, 1-68). RESULTS: Transfusion efficacy was significantly lower for AM-APCs than for TM-APCs (median PPR(1h), 17 and 33%; median CCI(1h), 7.9 and 15.6, respectively; p < 0.001). Reduced transfusion efficacy correlated in a yield-dependent manner with high apheresis PLT yields (>/=6 x 10(11)) for AM-APCs (p < 0.001). CONCLUSION: Although in vitro validation of AM- and TM-APCs has been performed, only by evaluating transfusion efficacy in vivo did the AM turn out to be not suitable for high-yield thrombocytapheresis. This study recommends the implementation of in vivo transfusion efficacy studies for high-yield APC apheresis donations.

Neurotoxicity of glia activated by gram-positive bacterial products depends on nitric oxide production

The present study examined the mechanism by which bacterial cell walls from two gram-positive meningeal pathogens, Streptococcus pneumoniae and the group B streptococcus, induced neuronal injury in primary cultures of rat brain cells. Cell walls from both organisms produced cellular injury to similar degrees in pure astrocyte cultures but not in pure neuronal cultures. Cell walls also induced nitric oxide production in cultures of astrocytes or microglia. When neurons were cultured together with astrocytes or microglia, the cell walls of both organisms became toxic to neurons. L-NAME, a nitric oxide synthase inhibitor, protected neurons from cell wall-induced toxicity in mixed cultures with glia, as did dexamethasone. In contrast, an excitatory amino acid antagonist (MK801) had no effect. Low concentrations of cell walls from either gram-positive pathogen added together with the excitatory amino acid glutamate resulted in synergistic neurotoxicity that was inhibited by L-NAME. The induction of nitric oxide production and neurotoxicity by cell walls was independent of the presence of serum, whereas endotoxin exhibited these effects only in the presence of serum. We conclude that gram-positive cell walls can cause toxicity in neurons by inducing the production of nitric oxide in astrocytes and microglia.

Upregulation of alpha-skeletal muscle actin and myosin heavy polypeptide gene products in degenerating rotator cuff muscles

Impaired function of shoulder muscles, resulting from rotator cuff tears, is associated with abnormal deposition of fat in muscle tissue, but corresponding cellular and molecular mechanisms, likely reflected by altered gene expression profiles, are largely unknown. Here, an analysis of muscle gene expression was carried out by semiquantitative RT-PCR in total RNA extracts of supraspinatus biopsies collected from 60 patients prior to shoulder surgery. A significant increase of alpha-skeletal muscle actin (p = 0.0115) and of myosin heavy polypeptide 1 (p = 0.0147) gene transcripts was observed in parallel with progressive fat deposition in the muscle, assessed on parasagittal T1-weighted turbo-spin-echo magnetic resonance images according to Goutallier. Upregulation of alpha-skeletal muscle actin and of myosin heavy polypeptide-1 has been reported to be associated with increased muscle tissue metabolism and oxidative stress. The findings of the present study, therefore, challenge the hypothesis that increased fat deposition in rotator cuff muscle after injury reflects muscle degeneration.

Managerial perseptions on corporate social responsbility: A Transatlantic comparison between forest products companies in Europe and North America

Corporate Social Responsibility (CSR) addresses the responsibility of companies for their impacts on society. The concept of strategic CSR is becoming increasingly mainstreamed in the forest industry, but there is, however, little consensus on the definition and implementation of CSR. The objective of this research is to build knowledge on the characteristics of CSR and to provide insights on the emerging trend to increase the credibility and legitimacy of CSR through standardization. The study explores how the sustainability managers of European and North American forest companies perceive CSR and the recently released ISO 26000 guidance standard on social responsibility. The conclusions were drawn from an analysis of two data sets; multivariate survey data based on one subset of 30 European and 13 North American responses, and data obtained through in-depth interviewing of 10 sustainability managers that volunteered for an hour long phone discussion about social responsibility practices at their company. The analysis concluded that there are no major differences in the characteristics of cross-Atlantic CSR. Hence, the results were consistent with previous research that suggests that CSR is a case- and company-specific concept. Regarding the components of CSR, environmental issues and organizational governance were key priorities in both regions. Consumer issues, human rights, and financial issues were among the least addressed categories. The study reveals that there are varying perceptions on the ISO 26000 guidance standard, both positive and negative. Moreover, sustainability managers of European and North American forest companies are still uncertain regarding the applicability of the ISO 26000 guidance standard to the forest industry. This study is among the first to provide a preliminary review of the practical implications of the ISO 26000 standard in the forest sector. The results may be utilized by sustainability managers interested in the best practices on CSR, and also by a variety of forest industrial stakeholders interested in the practical outcomes of the long-lasting CSR debate.