Value sets of bivariate Chebyshev maps over finite fields

We determine the cardinality of the value sets of bivariate Chebyshev maps over finite fields. We achieve this using the dynamical properties of these maps and the algebraic expressions of their fixed points in terms of roots of unity.Comment: 11 pages, 2 figure

Bivariate polynomial mappings associated with simple complex Lie algebras

There are three families of bivariate polynomial maps associated with the rank-$2$ simple complex Lie algebras $A_2, B_2 \cong C_2$ and $G_2$. It is known that the bivariate polynomial map associated with $A_2$ induces a permutation of $\mathbf{F}_q^2$ if and only if $\gcd(k,q^s-1)=1$ for $s=1, 2, 3$. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.Comment: 16 pages, 6 figure

On the Arithmetic Exceptionality of Polynomial Mappings

In this note we prove that certain polynomial mappings $P_\mathfrak{g}^k(\mathbf{x}) \in \mathbf{Z}[\mathbf{x}]$ in $n$-variables obtained from simple complex Lie algebras $\mathfrak{g}$ of arbitrary rank $n \ge 1$, are exceptional.Comment: 5 page

Hamiltonian Formulation of Jackiw--Pi 3-Dimensional Gauge Theories

A 3-dimensional non-abelian gauge theory was proposed by Jackiw and Pi to create mass for the gauge fields. However, the quadratic action obtained by switching off the non-abelian interactions possesses more gauge symmetries than the original one, causing some difficulties in quantization. Jackiw and Pi proposed another action by introducing new fields, whose gauge symmetries are consistent with the quadratic part. It is shown that all of these theories have the same number of physical degrees of freedom in the hamiltonian framework. Hence, as far as the physical states are considered there is no inconsistency. Nevertheless, perturbation expansion is still problematic. To cure this we propose to modify one of the constraints of the non-abelian theory without altering neither its canonical hamiltonian nor the number of physical states.Comment: A new system is proposed and some comments and references added. Latex, 10 page

Genome sequences of five Lactobacillus sp. isolates from traditional Turkish sourdough

A high level of variation in microflora can be observed in profiles of lactic acid bacteria (LAB) from sourdoughs. Here, we present draft genome sequences of Lactobacillus reuteri E81, L. reuteri LR5A, L. rhamnosus LR2, L. plantarum PFC-311, and the novel Lactobacillus sp. strain PFC-70, isolated from traditional Turkish backslopped wheat sourdoughs. © 2018 Dertli et al