A note on decomposition of solenoidal fields

The proof of the decomposition of a solenoidal field in an infinite domain is discussed.     

A note on powers in finite fields

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years, the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection, it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation that is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.     

Idempotent medialn-groupoids defined on fields

Fajtlowicz and Mycielski ([2]) showed that if we define an operationx·y=px+(1-p)x on the real numbersR for somep R, then the idempotent and medial laws form a basis for the equational theory of the groupoid (R,) if and only ifp is a transcendental number. In this paper, we generalize this ton-groupoids. Namely, if we define ann-ary operation [x ₁x₂h.x_n]=₁x₁ + ₂x₂+h.+(1-₁-h._n–1) onR for some ₁, ₂, h., _n–1 in R, then the idempotent and medial laws form a basis for the equational theory of then-groupoid (R, [ ]) if and only if ₁, ₂,..., _n–1 are algebraically independent.This is a part of Ph.D. dissertation developed under the direction of professor Trevor Evans while the author was at Emory University.     

A note on random fields forming conditional bases

We study properties of random fields that form conditional bases and their applications in spatial statistics.     

A note on character sums in function fields

We give an improvement on a character sum estimate in ${\mathbb{F}}_q[t]$ and answer a question of Shparlinski.     

A note on Pall partitions over finite fields

A Pall partition for a quadratic space V is a collection of disjoint (except for {0}) maximal totally isotropic subspaces whose union contains all of the isotropic vectors in V. In this paper it is shown that no non-degenerate quadratic space of dimension 4k+1, k?1, over a finite field of odd characteristic can have a Pall partition. The method of proof consists of assuming such a partition exists and showing by various counting arguments that this leads to the existence of an impossible array of ordered pairs.     

A note on Thue's equation over function fields

By a refinement of Mason's method we improve upon the estimate of the height of solutions of Thue's equation over function fields. We also give an application to the diophantine approximation of algebraic functions.     

A note on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+\gamma {\mathrm{Tr}}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].     

A note on Siegel's Lemma over number fields

We show that some power of the discriminant must appear in the upper bound of the formulation of Siegel's Lemma over a number field.First author partially supported by NSERC     

A note on groups definable in difference fields

We prove that a group definable in a model of is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.

     

A note on operator self-similar Gaussian vector fields

In this work, a general representation for an operator self-similar Gaussian vector field is obtained, and some properties are studied. It is also shown that such a process is the operator scaling limit of a Gaussian vector field.