We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z 5 p for all odd primes p and Z 4 p for all odd primes p such that p ≡ 3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.

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Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements [1]. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute Bézier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 Bézier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, i.e. write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron.

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