The general linear group $\mathrm{GL}(n)$ acts on the direct sum of $m$ copies $\mathrm{Mat}(n)$ by adjoint action. action induces unitriangular subgroup $U$. We present system free generators field $U$-invariants.

In this paper, we define a [Formula: see text]-Diophantine text]-tuple to be set of text] positive integers such that the product any distinct is one less than perfect square. We study these sets in finite fields for odd prime and guarantee existence provided larger some explicit lower bound. also give formula number 3-Diophantine triples as well an asymptotic text]-tuples.

As previously known, all 3-manifolds of genus 2 can be represented by edgecoloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an elementary transformation on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of...

As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an “elementary transformation” on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifol...

For every k > 3, we give a construction of planar point sets with many collinear k-tuples and no collinear (k + 1)-tuples. We show that there are n0 = n0(k) and c = c(k) such that if n ≥ n0, then there exists a set of n points in the plane that does not contain k + 1 points on a line, but it contains at least n 2− c √ logn collinear k-tuples of points. Thus, we significantly improve the previou...

In a nested relational or complex object database, nested tuples and sets are used to represent real world objects. For various reasons, such tuples and sets can be partial or complete. In this paper, we discuss how to support them in deductive databases. In particular, we present a deductive database language RLOG II that supports partial and complete tuples and sets based on Relationlog. This...