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

Leiomyomatoid angiomatous neuroendocrine tumor (LANT) of the pituitary: a distinctive biphasic neoplasm with primitive secretory phenotype and smooth muscle-rich stroma

We describe a hitherto undocumented variant of dimorphic pituitary neoplasm composed of an admixture of neurosecretory cells and profuse leiomyomatous stroma around intratumoral vessels. Radiologically perceived as a macroadenoma of 3.8 cm in diameter, this pituitary mass developed in an otherwise healthy 43-year-old female. At the term of a yearlong history of amenorrhea and progressive bitemporal visual loss, subtotal resection was performed via transsphenoidal microsurgery. Discounting mild hyperprolactinemia, there was no evidence of excess hormone production. Histologically, solid sheets, nests and cords of epithelial-looking, yet cytokeratin-negative cells were seen growing in a richly vascularized stroma of spindle cells. While strong immunoreactivity for NCAM, Synaptophysin and Chromogranin-A was detected in the former, the latter showed both morphological and immunophenotypic hallmarks of smooth muscle, being positive for vimentin, muscle actin and smooth muscle actin. Architectural patterns varied from monomorphous stroma-dominant zones through biphasic neuroendocrine-leiomyomatous areas, to pseudopapillary fronds along vascular cores. Only endothelia were labeled with CD34. Staining for S100 protein and GFAP, characteristics of sustentacular cells, as well as bcl-2 and c-kit was absent. Except for alpha-subunit, anterior pituitary hormones tested negative in tumor cells, as did a panel of peripheral endocrine markers, including serotonin, somatostatin, calcitonin, parathormone and vasoactive intestinal polypeptide. Mitotic activity was absent and the MIB-1 labeling index low (1-2%). While assignment of this lesion to any established neoplastic entity is not forthcoming, we propose it is being considered as a low-grade neuroendocrine tumor possibly related to null cell adenoma.

Neonatal macrocephaly: cerebral primitive neuroectodermal tumor or neuroblastoma as an infrequent cause--a case report and review of the literature

We report a male term newborn presenting with a congenital macrocephaly 3.5 standard deviations above the median, with a wide and tense anterior fontanel, splayed calvarial sutures, and muscular hypotonia. Antenatal head circumferences were repeatedly below the median. A postnatal head ultrasound showed a large right intracerebral mass with right lateral ventricle compression, right temporal horn dilation, and right frontal horn enlargement with lateral displacement. Additional imaging by computed tomography scan and magnetic resonance imaging was performed. A decompression was performed and histology, immunohistochemistry, and molecular biology supported the diagnosis of a primitive neuroectodermal tumor. A MYCN gene amplification assay remained negative. The incidence of neonatal brain tumors is between 1.4 and 4.1/100,000 live births. Their most common presentation is macrocephaly, hydrocephalus, stillbirth, or diagnosis by pre- or postnatal imaging. Although hydrocephaly and intra- or extracranial hemorrhage are the most frequent causes of congenital macrocephaly, this should be initially investigated by head ultrasound. A suspected malignancy will be confirmed by histopathology, immunohistochemistry, and molecular biology.

Primitive ontology and quantum state in the GRW matter density theory

The paper explains in what sense the GRW matter density theory (GRWm) is a primitive ontology theory of quantum mechanics and why, thus conceived, the standard objections against the GRW formalism do not apply to GRWm. We consider the different options for conceiving the quantum state in GRWm and argue that dispositionalism is the most attractive one.

Two independent and primitive envelopes of the bilobate nucleus of comet 67P

The factors shaping cometary nuclei are still largely unknown, but could be the result of concurrent effects of evolutionary(1,2) and primordial processes(3,4). The peculiar bilobed shape of comet 67P/Churyumov-Gerasimenko may be the result of the fusion of two objects that were once separate or the result of a localized excavation by outgassing at the interface between the two lobes(5). Here we report that the comet's major lobe is enveloped by a nearly continuous set of strata, up to 650 metres thick, which are independent of an analogous stratified envelope on the minor lobe. Gravity vectors computed for the two lobes separately are closer to perpendicular to the strata than those calculated for the entire nucleus and adjacent to the neck separating the two lobes. Therefore comet 67P/Churyumov-Gerasimenko is an accreted body of two distinct objects with 'onion-like' stratification, which formed before they merged. We conclude that gentle, low-velocity collisions occurred between two fully formed kilometre-sized cometesimals in the early stages of the Solar System. The notable structural similarities between the two lobes of comet 67P/Churyumov-Gerasimenko indicate that the early-forming cometesimals experienced similar primordial stratified accretion, even though they formed independently.

Legendre Polynomials and Angular Momentum

An introduction to Legendre polynomials as precursor to studying angular momentum in quantum chemistry,

Bivariate Krawtchouk polynomials: Inversion and connection problems with the NAVIMA algorithm

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Cloning, expression, and characterization of a cDNA encoding a novel human growth factor for primitive hematopoietic progenitor cells

Multiple growth factors synergistically stimulate proliferation of primitive hematopoietic progenitor cells. A human myeloid cell line, KPB-M15, constitutively produces a novel hematopoietic cytokine, termed stem cell growth factor (SCGF), possessing species-specific proliferative activities. Here we report the molecular cloning, expression, and characterization of a cDNA encoding human SCGF using a newly developed λSHDM vector that is more efficient for differential and expression cloning. cDNA for SCGF encodes a 29-kDa polypeptide without N-linked glycosylation. SCGF transiently produced by COS-1 cells supports growth of hematopoietic progenitor cells through a short-term liquid culture of bone marrow cells and exhibits promoting activities on erythroid and granulocyte/macrophage progenitor cells in primary semisolid culture with erythropoietin and granulocyte/macrophage colony-stimulating factor, respectively. Expression of SCGF mRNA is restricted to myeloid cells and fibroblasts, suggesting that SCGF is a growth factor functioning within the hematopoietic microenvironment. SCGF could disclose some human-specific mechanisms as yet unidentified from studies on the murine hematopoietic system.

Chimeric Cytokine Receptor Can Transduce Expansion Signals in Interleukin 6 Receptor α (IL-6Rα)-, IL-11Rα-, and gp130-low to -negative Primitive Hematopoietic Progenitors

We generated transgenic mice expressing chimeric receptors, which comprise extracellular domains of the human granulocyte–macrophage colony-stimulating factor (hGM-CSF) receptor and transmembrane and cytoplasmic domains of the mouse leukemia inhibitory factor receptor. In suspension cultures of lineage-negative (Lin−), 5-fluorouracil-resistant bone marrow cells of the transgenic mice, a combination of hGM-CSF and stem cell factor (SCF) induced exponential expansions of mixed colony-forming unit. The combination of hGM-CSF and SCF was effective on enriched, Lin−Sca-1+c-kit+ progenitors and increased either mixed colony-forming unit or cobblestone area–forming cells. In case of stimulation with hGM-CSF and SCF, interleukin-6 (IL-6) and SCF, or IL-11 and SCF, the most efficient expansion was achieved with hGM-CSF and SCF. When Lin−Sca-1+c-kit+CD34− further enriched progenitors were clone sorted and individually incubated in the presence of SCF, hGM-CSF stimulated a larger number of cells than did IL-6, IL-6 and soluble IL-6 receptor (IL-6R), or IL-11. These data suggest the presence of IL-6Rα-, IL-11Rα-, and gp130-low to -negative primitive hematopoietic progenitors. Such primitive progenitors are equipped with signal transduction molecules and can expand when these chimeric receptors are genetically introduced into the cells and stimulated with hGM-CSF in the presence of SCF.

In vitro and in vivo differentiation into B cells, T cells, and myeloid cells of primitive yolk sac hematopoietic precursor cells expanded >100-fold by coculture with a clonal yolk sac endothelial cell line

The yolk sac, first site of hematopoiesis during mammalian development, contains not only hematopoietic stem cells but also the earliest precursors of endothelial cells. We have previously shown that a nonadherent yolk sac cell population (WGA+, density <1.077, AA4.1+) can give rise to B cells, T cells, and myeloid cells both in vitro and in vivo. We now report on the ability of a yolk sac-derived cloned endothelial cell line (C166) to provide a suitable microenvironment for expansion of these early precursor cells. Single day 10 embryonic mouse yolk sac hematopoietic stem cells were expanded >100 fold within 8 days by coculture with irradiated C166 cells. Colony-forming ability was retained for at least three passages in vitro, with retention of the ability to differentiate into T-cell, B-cell, and myeloid lineages. Stem cell properties were maintained by a significant fraction of nonadherent cells in the third passage, although these stem cells expressed a somewhat more mature cell surface phenotype than the initial yolk sac stem cells. When reintroduced into adult allogeneic immunocompromised (scid) hosts, they were able to give rise to all of the leukocyte lineages, including T cells, B cells, and myeloid cells. We conclude that yolk sac endothelial cells can support the stable proliferation of multipotential hematopoietic stem cells, thus generating adequate numbers of cells for study of the mechanisms involved in their subsequent development and differentiation, for in vivo hematopoietic restitution, and for potential use as a vehicle for gene transfer.

Self-renewal of primitive human hematopoietic cells (long-term-culture-initiating cells) in vitro and their expansion in defined medium.

A major goal of experimental and clinical hematology is the identification of mechanisms and conditions that support the expansion of transplantable hematopoietic stem cells. In normal marrow, such cells appear to be identical to (or represent a subset of) a population referred to as long-term-culture-initiating cells (LTC-ICs) so-named because of their ability to produce colony-forming cell (CFC) progeny for > or = 5 weeks when cocultured with stromal fibroblasts. Some expansion of LTC-ICs in vitro has recently been described, but identification of the factors required and whether LTC-IC self-renewal divisions are involved have remained unresolved issues. To address these issues, we examined the maintenance and/or generation of LTC-ICs from single CD34+ CD38- cells cultured for variable periods under different culture conditions. Analysis of the progeny obtained from cultures containing a feeder layer of murine fibroblasts engineered to produce steel factor, interleukin (IL)-3, and granulocyte colony-stimulating factor showed that approximately 20% of the input LTC-ICs (representing approximately 2% of the original CD34+ CD38- cells) executed self-renewal divisions within a 6-week period. Incubation of the same CD34+ CD38- starting populations as single cells in a defined (serum free) liquid medium supplemented with Flt-3 ligand, steel factor, IL-3, IL-6, granulocyte colony-stimulating factor, and nerve growth factor resulted in the proliferation of initial cells to produce clones of from 4 to 1000 cells within 10 days, approximately 40% of which included > or = 1 LTC-IC. In contrast, in similar cultures containing methylcellulose, input LTC-ICs appeared to persist but not divide. Overall the LTC-IC expansion in the liquid cultures was 30-fold in the first 10 days and 50-fold by the end of another 1-3 weeks. Documentation of human LTC-IC self-renewal in vitro and identification of defined conditions that permit their extensive and rapid amplification should facilitate analysis of the molecular mechanisms underlying these processes and their exploitation for a variety of therapeutic applications.

A new specimen of Desmatochelys lowi Williston; a primitive cheloniid sea turtle from the Cretaceous of South Dakota / Rainer Zangerl -- and Robert E. Sloan --.

v.14:no.2(1960)

Primitive hunters of Australia / by Wilfrid D. Hambly, Assistant Curator of African Ethnology. 12 plates in photogravure and 1 map.

no.32(1936)