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.

The modern primitive and the antipodes: The visual arts and Oceania

In a book seeking to redraw the boundaries between interdisciplinary and transnational modernisms, this chapter contributes to the reorientation in modernist studies by revisiting "primitivism." While no one freely identifies as “primitive,” the spectre of primitivism was a magnet of attraction as well as of critical refusal. It resided on the knife-edge of envy and denunciation, as well as for the projection of alternate imaginative utopias and the worst forms of racial chauvinism. This chapter asserts that primitivism endures as a provocation as much as a utopian aspiration, but it also provides a different understanding of cultures on the "periphery", which is how Antipodean art history has understood itself. The spectre of primitivism not only amplifies the quandaries of modernist cultures—both alerting one to the aesthetic alternatives to modernist cultures, yet also highlighting the fate of traditional culture pitted against modernist cultures, it also suggests the quandaries of a peripheral modernity.

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.

Foxe's book of martyrs : a complete and authentic account of the lives, sufferings, and triumphant deaths of the primitive and Protestant martyrs in all parts of the world, with notes, comments and illustrations /

Includes index.

Spatial distribution of diffuse, primitive, and classic amyloid-beta deposits and blood vessels in the upper laminae of the frontal cortex in Alzheimer disease

The spatial distribution of the diffuse, primitive, and classic amyloid-beta deposits was studied in the upper laminae of the superior frontal gyrus in cases of sporadic Alzheimer disease (AD). Amyloid-beta-stained tissue was counterstained with collagen IV to determine whether the spatial distribution of the amyloid-beta deposits along the cortex was related to blood vessels. In all patients, amyloid-beta deposits and blood vessels were aggregated into distinct clusters and in many patients, the clusters were distributed with a regular periodicity along the cortex. The clusters of diffuse and primitive deposits did not coincide with the clusters of blood vessels in most patients. However, the clusters of classic amyloid-beta deposits coincided with those of the large diameter (>10 microm) blood vessels in all patients and with clusters of small-diameter (< 10 microm) blood vessels in four patients. The data suggest that, of the amyloid-beta subtypes, the clusters of classic amyloid-beta deposits appear to be the most closely related to blood vessels and especially to the larger-diameter, vertically penetrating arterioles in the upper cortical laminae.