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.

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Divisibility of Trinomials by Irreducible Polynomials over F_2

Irreducible trinomials of given degree n over F_2 do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over F_2. A condition for divisibility of self-reciprocal trinomials by irreducible polynomials over F_2 is established. And we extend Welch's criterion for testing if an irreducible polynomial divides trinomials x^m + x^s + 1 to the trinomials x^am + x^bs + 1.

Low autocorrelation sequences and flat polynomials

Magdeburg, Univ., Fak. für Mathematik, Habil.-Schr., 2014

Integral inequalities for algebraic and trigonometric polynomials

Vegeu el resum a l'inici del document del fitxer adjunt.

In vivo fate mapping with SCL regulatory elements identifies progenitors for primitive and definitive hematopoiesis in mice.

One of the principal issues facing biomedical research is to elucidate developmental pathways and to establish the fate of stem and progenitor cells in vivo. Hematopoiesis, the process of blood cell formation, provides a powerful experimental system for investigating this process. Here, we employ transcriptional regulatory elements from the stem cell leukemia (SCL) gene to selectively label primitive and definitive hematopoiesis. We report that SCL-labelled cells arising in the mid to late streak embryo give rise to primitive red blood cells but fail to contribute to the vascular system of the developing embryo. Restricting SCL-marking to different stages of foetal development, we identify a second population of multilineage progenitors, proficient in contributing to adult erythroid, myeloid and lymphoid cells. The distinct lineage-restricted potential of SCL-labelled early progenitors demonstrates that primitive erythroid cell fate specification is initiated during mid gastrulation. Our data also suggest that the transition from a hemangioblastic precursors with endothelial and blood forming potential to a committed hematopoietic progenitor must have occurred prior to SCL-marking of definitive multilineage blood precursors.

Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

Let C-n(lambda)(x), n = 0, 1,..., lambda > -1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal. in (-1, 1) with respect to the weight function (1 - x(2))(lambda-1/2). Denote by X-nk(lambda), k = 1,....,n, the zeros of C-n(lambda)(x) enumerated in decreasing order. In this short note, we prove that, for any n is an element of N, the product (lambda + 1)(3/2)x(n1)(lambda) is a convex function of lambda if lambda greater than or equal to 0. The result is applied to obtain some inequalities for the largest zeros of C-n(lambda)(x). If X-nk(alpha), k = 1,...,n, are the zeros of Laguerre polynomial L-n(alpha)(x), also enumerated in decreasing order, we prove that x(n1)(lambda)/(alpha + 1) is a convex function of alpha for alpha > - 1. (C) 2002 Published by Elsevier B.V. B.V.

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

Szego polynomials: some relations to L-orthogonal and orthogonal polynomials

We consider the real Szego polynomials and obtain some relations to certain self inversive orthogonal L-polynomials defined on the unit circle and corresponding symmetric orthogonal polynomials on real intervals. We also consider the polynomials obtained when the coefficients in the recurrence relations satisfied by the self inversive orthogonal L-polynomials are rotated. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Palindromic and perturbed polynomials: zeros location

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The Missouri river and its utmost source; curtailed narration of geologic primitive and geographic distinctions descriptive of the evolution and discovery of the river and its headwaters; containing an archæological addendum ...

Mode of access: Internet.

The judgment of the Catholic Church on the necessity of believing that Our Lord Jesus Christ is very God : the primitive and apostolic tradition of the doctrine concerning the divinity of Our Saviour Jesus Christ : and brief animadversions on a treatise of Mr. Gilbert Clerke /

The first treatise is written in opposition to the views of Simon Episcopius while the second is in opposition to the Irenicum Irenicorum of Daniel Zwicker.

Buddhism, primitive and present, in Magadha and in Ceylon.

Mode of access: Internet.