Massless scattering at special kinematics as Jacobi polynomials

We study the scattering equations recently proposed by Cachazo, He and Yuan in the special kinematics where their solutions can be identified with the zeros of the Jacobi polynomials. This allows for a non-trivial two-parameter family of kinematics. We present explicit and compact formulas for the n-gluon and n-graviton partial scattering amplitudes for our special kinematics in terms of Jacobi polynomials. We also provide alternative expressions in terms of gamma functions. We give an interpretation of the common reduced determinant appearing in the amplitudes as the product of the squares of the eigenfrequencies of small oscillations of a system whose equilibrium is the solutions of the scattering equations.

Métodos numéricos para encontrar zeros de funções: aplicações para o Ensino Médio

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Zero sets of bivariate Hermite polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Szegö polynomials: quadrature rules on the unit circle and on [-1, 1]

We consider some of the relations that exist between real Szegö polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again related to certain orthogonal polynomials on [-1, 1] through the transformation x = (z1/2+z1/2)/2. Using these relations we study the interpolatory quadrature rule based on the zeros of polynomials which are linear combinations of the orthogonal polynomials on [-1, 1]. In the case of any symmetric quadrature rule on [-1, 1], its associated quadrature rule on the unit circle is also given.

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

A novel universal DNA labeling and amplification system for rapid microarray-based detection of 117 antibiotic resistance genes in Gram-positive bacteria.

A rapid and simple DNA labeling system has been developed for disposable microarrays and has been validated for the detection of 117 antibiotic resistance genes abundant in Gram-positive bacteria. The DNA was fragmented and amplified using phi-29 polymerase and random primers with linkers. Labeling and further amplification were then performed by classic PCR amplification using biotinylated primers specific for the linkers. The microarray developed by Perreten et al. (Perreten, V., Vorlet-Fawer, L., Slickers, P., Ehricht, R., Kuhnert, P., Frey, J., 2005. Microarray-based detection of 90 antibiotic resistance genes of gram-positive bacteria. J.Clin.Microbiol. 43, 2291-2302.) was improved by additional oligonucleotides. A total of 244 oligonucleotides (26 to 37 nucleotide length and with similar melting temperatures) were spotted on the microarray, including genes conferring resistance to clinically important antibiotic classes like β-lactams, macrolides, aminoglycosides, glycopeptides and tetracyclines. Each antibiotic resistance gene is represented by at least 2 oligonucleotides designed from consensus sequences of gene families. The specificity of the oligonucleotides and the quality of the amplification and labeling were verified by analysis of a collection of 65 strains belonging to 24 species. Association between genotype and phenotype was verified for 6 antibiotics using 77 Staphylococcus strains belonging to different species and revealed 95% test specificity and a 93% predictive value of a positive test. The DNA labeling and amplification is independent of the species and of the target genes and could be used for different types of microarrays. This system has also the advantage to detect several genes within one bacterium at once, like in Staphylococcus aureus strain BM3318, in which up to 15 genes were detected. This new microarray-based detection system offers a large potential for applications in clinical diagnostic, basic research, food safety and surveillance programs for antimicrobial resistance.

Learning Bayesian networks from data by the incremental compilation of new network polynomials

Probabilistic graphical models are a huge research field in artificial intelligence nowadays. The scope of this work is the study of directed graphical models for the representation of discrete distributions. Two of the main research topics related to this area focus on performing inference over graphical models and on learning graphical models from data. Traditionally, the inference process and the learning process have been treated separately, but given that the learned models structure marks the inference complexity, this kind of strategies will sometimes produce very inefficient models. With the purpose of learning thinner models, in this master thesis we propose a new model for the representation of network polynomials, which we call polynomial trees. Polynomial trees are a complementary representation for Bayesian networks that allows an efficient evaluation of the inference complexity and provides a framework for exact inference. We also propose a set of methods for the incremental compilation of polynomial trees and an algorithm for learning polynomial trees from data using a greedy score+search method that includes the inference complexity as a penalization in the scoring function.

Purification and molecular cloning of an inducible gram-negative bacteria-binding protein from the silkworm, Bombyx mori.

A 50-kDa hemolymph protein, having strong affinity to the cell wall of Gram(-) bacteria, was purified from the hemolymph of the silkworm, Bombyx mori. The cDNA encoding this Gram(-) bacteria-binding protein (GNBP) was isolated from an immunized silkworm fat body cDNA library and sequenced. Comparison of the deduced amino acid sequence with known sequences revealed that GNBP contained a region displaying significant homology to the putative catalytic region of a group of bacterial beta-1,3 glucanases and beta-1,3-1,4 glucanases. Silkworm GNBP was also shown to have amino acid sequence similarity to the vertebrate lipopolysaccharide receptor CD14 and was recognized specifically by a polygonal anti-CD14 antibody. Northern blot analysis showed that GNBP was constitutively expressed in fat body, as well as in cuticular epithelial cells of naive silkworms. Intense transcription was, however, rapidly induced following a cuticular or hemoceolien bacterial challenge. An mRNA that hybridized with GNBP cDNA was also found in the l(2)mbn immunocompetent Drosophila cell line. These observations suggest that GNBP is an inducible acute phase protein implicated in the immune response of the silkworm and perhaps other insects.

On decomposition of sub-linearised polynomials

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.

Extention of Apolarity and Grace Theorem

MSC 2010: 30C10

Cellular impermeability and uptake of biocides and antibiotics in Gram-positive bacteria and mycobacteria

Gram-positive bacteria possess a permeable cell wall that usually does not restrict the penetration of antimicrobials. However, resistance due to restricted penetration can occur, as illustrated by vancomycin-intermediate resistant Staphylococcus aureus strains (VISA) which produce a markedly thickened cell wall. Alterations in these strains include increased amounts of nonamidated glutamine residues in the peptidoglycan and it is suggested that the resistance mechanism involves 'affinity trapping' of vancomycin in the thickened cell wall. VISA strains have reduced doubling times, lower sensitivity to lysostaphin and reduced autolytic activity, which may reflect changes in the D-alanyl ester content of the wall and membrane teichoic acids. Mycobacterial cell walls have a high lipid content, which is assumed to act as a major barrier to the penetration of antimicrobial agents. Relatively hydrophobic antibiotics such as rifampicin and fluoroquinolones may be able to cross the cell wall by diffusion through the hydrophobic bilayer composed of long chain length mycolic acids and glycolipids. Hydrophilic antibiotics and nutrients cannot diffuse across this layer and are thought to use porin channels which have been reported in many species of mycobacteria. The occurrence of porins in a lipid bilayer supports the view that the mycobacterial wall has an outer membrane analogous to that of gram-negative bacteria. However, mycobacterial porins are much less abundant than in the gram-negative outer membrane and allow only low rates of uptake for small hydrophilic nutrients and antibiotics.

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

Model-order selection in Zernike polynomial expansion of corneal surfaces using the efficient detection criterion

Corneal-height data are typically measured with videokeratoscopes and modeled using a set of orthogonal Zernike polynomials. We address the estimation of the number of Zernike polynomials, which is formalized as a model-order selection problem in linear regression. Classical information-theoretic criteria tend to overestimate the corneal surface due to the weakness of their penalty functions, while bootstrap-based techniques tend to underestimate the surface or require extensive processing. In this paper, we propose to use the efficient detection criterion (EDC), which has the same general form of information-theoretic-based criteria, as an alternative to estimating the optimal number of Zernike polynomials. We first show, via simulations, that the EDC outperforms a large number of information-theoretic criteria and resampling-based techniques. We then illustrate that using the EDC for real corneas results in models that are in closer agreement with clinical expectations and provides means for distinguishing normal corneal surfaces from astigmatic and keratoconic surfaces.

The genetics and structure of the O-specific polysaccharide of Yersinia pseudotuberculosis serotype O:10 and its relationship with Escherichia coli O111 and Salmonella enterica O35

The O-specific polysaccharide (OPS) is a variable constituent of the lipopolysaccharide of Gram-negative bacteria. The polymorphic nature of OPSs within a species is usually first defined serologically, and the current serotyping scheme for Yersinia pseudotuberculosis consists of 21 O serotypes of which 15 have been characterized genetically and structurally. Here, we present the structure and DNA sequence of Y. pseudotuberculosis O:10 OPS. The O unit consists of one residue each of d-galactopyranose, N-acetyl-d-galactosamine (2-amino-2-deoxy-d-galactopyranose) and d-glucopyranose in the backbone, with two colitose (3,6-dideoxy-l-xylo-hexopyranose) side-branch residues. This structure is very similar to that shared by Escherichia coli O111 and Salmonella enterica O35. The gene cluster sequences of these serotypes, however, have only low levels of similarity to that of Y. pseudotuberculosis O:10, although there is significant conservation of gene order. Within Y. pseudotuberculosis, the O10 structure is most closely related to the O:6 and O:7 structures.