On the zeros of polynomials: an extension of the Enestrom-Kakeya theorem

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier Inc. All rights reserved.

Generators of supersymmetric polynomials in positive characteristic

In Kantor and Trishin (1997) [3], Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie superalgebra gl(m vertical bar n) and a related algebra A, of what they called pseudosymmetric polynomials over an algebraically closed field K of characteristic zero. The algebra A(s) was investigated earlier by Stembridge (1985) who in [9] called the elements of A(s) supersymmetric polynomials and determined generators of A(s). The case of positive characteristic p of the ground field K has been recently investigated by La Scala and Zubkov (in press) in [6]. We extend their work and give a complete description of generators of polynomial invariants of the adjoint action of the general linear supergroup GL(m vertical bar n) and generators of A(s).

Three dimensional seismic imaging and earthquake locations in a complex, segmented fault region in Southern Apennines (Italy)

The southern Apennines of Italy have been experienced several destructive earthquakes both in historic and recent times. The present day seismicity, characterized by small-to-moderate magnitude earthquakes, was used like a probe to obatin a deeper knowledge of the fault structures where the largest earthquakes occurred in the past. With the aim to infer a three dimensional seismic image both the problem of data quality and the selection of a reliable and robust tomographic inversion strategy have been faced. The data quality has been obtained to develop optimized procedures for the measurements of P- and S-wave arrival times, through the use of polarization filtering and to the application of a refined re-picking technique based on cross-correlation of waveforms. A technique of iterative tomographic inversion, linearized, damped combined with a strategy of multiscale inversion type has been adopted. The retrieved P-wave velocity model indicates the presence of a strong velocity variation along a direction orthogonal to the Apenninic chain. This variation defines two domains which are characterized by a relatively low and high velocity values. From the comparison between the inferred P-wave velocity model with a portion of a structural section available in literature, the high velocity body was correlated with the Apulia carbonatic platforms whereas the low velocity bodies was associated to the basinal deposits. The deduced Vp/Vs ratio shows that the ratio is lower than 1.8 in the shallower part of the model, while for depths ranging between 5 km and 12 km the ratio increases up to 2.1 in correspondence to the area of higher seismicity. This confirms that areas characterized by higher values are more prone to generate earthquakes as a response to the presence of fluids and higher pore-pressures.

Frobenius polynomials for Calabi-Yau equations

Sei $\pi:X\rightarrow S$ eine \"uber $\Z$ definierte Familie von Calabi-Yau Varietaten der Dimension drei. Es existiere ein unter dem Gauss-Manin Zusammenhang invarianter Untermodul $M\subset H^3_{DR}(X/S)$ von Rang vier, sodass der Picard-Fuchs Operator $P$ auf $M$ ein sogenannter {\em Calabi-Yau } Operator von Ordnung vier ist. Sei $k$ ein endlicher K\"orper der Charaktetristik $p$, und sei $\pi_0:X_0\rightarrow S_0$ die Reduktion von $\pi$ \uber $k$. F\ur die gew\ohnlichen (ordinary) Fasern $X_{t_0}$ der Familie leiten wir eine explizite Formel zur Berechnung des charakteristischen Polynoms des Frobeniusendomorphismus, des {\em Frobeniuspolynoms}, auf dem korrespondierenden Untermodul $M_{cris}\subset H^3_{cris}(X_{t_0})$ her. Sei nun $f_0(z)$ die Potenzreihenl\osung der Differentialgleichung $Pf=0$ in einer Umgebung der Null. Da eine reziproke Nullstelle des Frobeniuspolynoms in einem Teichm\uller-Punkt $t$ durch $f_0(z)/f_0(z^p)|_{z=t}$ gegeben ist, ist ein entscheidender Schritt in der Berechnung des Frobeniuspolynoms die Konstruktion einer $p-$adischen analytischen Fortsetzung des Quotienten $f_0(z)/f_0(z^p)$ auf den Rand des $p-$adischen Einheitskreises. Kann man die Koeffizienten von $f_0$ mithilfe der konstanten Terme in den Potenzen eines Laurent-Polynoms, dessen Newton-Polyeder den Ursprung als einzigen inneren Gitterpunkt enth\alt, ausdr\ucken,so beweisen wir gewisse Kongruenz-Eigenschaften unter den Koeffizienten von $f_0$. Diese sind entscheidend bei der Konstruktion der analytischen Fortsetzung. Enth\alt die Faser $X_{t_0}$ einen gew\ohnlichen Doppelpunkt, so erwarten wir im Grenz\ubergang, dass das Frobeniuspolynom in zwei Faktoren von Grad eins und einen Faktor von Grad zwei zerf\allt. Der Faktor von Grad zwei ist dabei durch einen Koeffizienten $a_p$ eindeutig bestimmt. Durchl\auft nun $p$ die Menge aller Primzahlen, so erwarten wir aufgrund des Modularit\atssatzes, dass es eine Modulform von Gewicht vier gibt, deren Koeffizienten durch die Koeffizienten $a_p$ gegeben sind. Diese Erwartung hat sich durch unsere umfangreichen Rechnungen best\atigt. Dar\uberhinaus leiten wir weitere Formeln zur Bestimmung des Frobeniuspolynoms her, in welchen auch die nicht-holomorphen L\osungen der Gleichung $Pf=0$ in einer Umgebung der Null eine Rolle spielen.

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

Effect of segmented anodes on the performance and plume of a hall thruster

Development of alternative propellants for Hall thruster operation is an active area of research. Xenon is the current propellant of choice for Hall thrusters, but can be costly in large thrusters and for extended test periods. Condensible propellants may offer an alternative to xenon, as they will not require costly active pumping to remove from a test facility, and may be less expensive to purchase. A method has been developed which uses segmented electrodes in the discharge channel of a Hall thruster to divert discharge current to and from the main anode and thus control the anode temperature. By placing a propellant reservoir in the anode, the evaporation rate, and hence, mass flow of propellant can be controlled. Segmented electrodes for thermal control of a Hall thruster represent a unique strategy of thruster design, and thus the performance of the thruster must be measured to determine the effect the electrodes have on the thruster. Furthermore, the source of any changes in thruster performance due to the adjustment of discharge current between the shims and the main anode must be characterized. A Hall thruster was designed and constructed with segmented electrodes. It was then tested at anode voltages between 300 and 400 V and mass flows between 4 and 6 mg/s, as well as 100%, 75%, 50%, 25%, and <5% of the discharge current on the shim electrodes. The level of current on the shims was adjusted by changing the shim voltage. At each operating point, the thruster performance, plume divergence, ion energy, and multiply charged ion fraction were measured performance exhibited a small change with the level of discharge current on the shim electrodes. Thrust and specific impulse increased by as much as 6% and 7.7%, respectively, as discharge current was shifted from the main anode to the shims at constant anode voltage. Thruster efficiency did not change. Plume divergence was reduced by approximately 4 degrees of half-angle at high levels of current on the shims and at all combinations of mass flow and anode voltage. The fraction of singly charged xenon in the thruster plume varied between approximately 80% and 95% as the anode voltage and mass flow were changed, but did not show a significant change with shim current. Doubly and triply charged xenon made up the remainder of the ions detected. Ion energy exhibited a mixed behavior. The highest voltage present in the thruster largely dictated the most probable energy; either shim or anode voltage, depending on which was higher. The overall change in most probable ion energy was 20-30 eV, the majority of which took place while the shim voltage was higher than the anode voltage. The thrust, specific impulse, plume divergence, and ion energy all indicate that the thruster is capable of a higher performance output at high levels of discharge current on the shims. The lack of a change in efficiency and fraction of multiply charged ions indicate that the thruster can be operated at any level of current on the shims without detrimental effect, and thus a condensible propellant thruster can control the anode temperature without a decrease in efficiency or a change in the multiply charged ion fraction.

Segmented filamentous bacterium uses secondary and tertiary lymphoid tissues to induce gut IgA and specific T helper 17 cell responses.

Segmented filamentous bacterium (SFB) is a symbiont that drives postnatal maturation of gut adaptive immune responses. In contrast to nonpathogenic E. coli, SFB stimulated vigorous development of Peyer's patches germinal centers but paradoxically induced only a low frequency of specific immunoglobulin A (IgA)-secreting cells with delayed accumulation of somatic mutations. Moreover, blocking Peyer's patch development abolished IgA responses to E. coli, but not to SFB. Indeed, SFB stimulated the postnatal development of isolated lymphoid follicles and tertiary lymphoid tissue, which substituted for Peyer's patches as inductive sites for intestinal IgA and SFB-specific T helper 17 (Th17) cell responses. Strikingly, in mice depleted of gut organized lymphoid tissue, SFB still induced a substantial but nonspecific intestinal Th17 cell response. These results demonstrate that SFB has the remarkable capacity to induce and stimulate multiple types of intestinal lymphoid tissues that cooperate to generate potent IgA and Th17 cell responses displaying only limited target specificity.

Simultaneous multi-slice readout-segmented echo planar imaging for accelerated diffusion-weighted imaging of the breast

OBJECTIVES Readout-segmented echo planar imaging (rs-EPI) significantly reduces susceptibility artifacts in diffusion-weighted imaging (DWI) of the breast compared to single-shot EPI but is limited by longer scan times. To compensate for this, we tested a new simultaneous multi-slice (SMS) acquisition for accelerated rs-EPI. MATERIALS AND METHODS After approval by the local ethics committee, eight healthy female volunteers (age, 38.9±13.1 years) underwent breast MRI at 3T. Conventional as well as two-fold (2× SMS) and three-fold (3× SMS) slice-accelerated rs-EPI sequences were acquired at b-values of 50 and 800s/mm(2). Two independent readers analyzed the apparent diffusion coefficient (ADC) in fibroglandular breast parenchyma. The signal-to-noise ratio (SNR) was estimated based on the subtraction method. ADC and SNR were compared between sequences by using the Friedman test. RESULTS The acquisition time was 4:21min for conventional rs-EPI, 2:35min for 2× SMS rs-EPI and 1:44min for 3× SMS rs-EPI. ADC values were similar in all sequences (mean values 1.62×10(-3)mm(2)/s, p=0.99). Mean SNR was 27.7-29.6, and no significant differences were found among the sequences (p=0.83). CONCLUSION SMS rs-EPI yields similar ADC values and SNR compared to conventional rs-EPI at markedly reduced scan time. Thus, SMS excitation increases the clinical applicability of rs-EPI for DWI of the breast.

Legendre Polynomials and Angular Momentum

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

Laguerre Polynomials, an Introduction

The radial part of the Schrodinger Equation for the H-atom's electron involves Laguerre polynomials, hence this introduction.

Legendre Polynomials, Dipole Moments, Generating Functions, etc..

A standard treatment of aspects of Legendre polynomials is treated here, including the dipole moment expansion, generating functions, etc..