On the Finite Basis Problem for Certain 2-limited Words

Let X＊ be a free monoid over an alphabet X and W be a finite language over X. Let S（W） be the Rees quotient X＊/I（W）, where I（W） is the ideal of X＊ consisting of all elements of X＊ that are not subwords of W. Then S（W） is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S（W） is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than using classical polynomials of degree N inside each element,in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element,using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element(AFE)basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles.The basis functions on each sub-triangle are defined,as usual,on a universal reference element,hence allowing to compute universal mass,flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only.Consequently,the construction of an efficient quadrature-free algorithm is possible,despite the unstructured nature of the computational grid.High order of accuracy in time is achieved thanks to the ADER approach,making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations.The numerical results have been checked with reference solutions available in literature and also systematically compared,in terms of computational efficiency and accuracy,with those obtained by the corresponding modal DG version of the scheme.

On the Existence for Some Special Primitive Elements in Finite Fields

Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q^n) such that α + α^(-1)is also primitive or α + α^(-1)is primitive and α is a normal element of F_(q^n) over F_q.