In this paper we discuss a possibility to extend unimodular transformations to non-perfectly nested loops. The main idea behind this extension is to convert a non-perfectly nested loop into a perfectly nested one by moving code into to innermost loop and properly guarding it to avoid multiple execution. This form of the loop can be viewed as an intermediate form for the transformation. Having o...

The classical Hausdorff-Young inequality for locally compact abelian groups states that, for 1 ≤ p ≤ 2, the L-norm of a function dominates the L-norm of its Fourier transform, where 1/p + 1/q = 1. By using the theory of non-commutative L-spaces and by reinterpreting the Fourier transform, R. Kunze (1958) [resp. M. Terp (1980)] extended this inequality to unimodular [resp. non-unimodular] groups...

We find all convex lattice polygons in the plane (up to equivalence) with at most one interior lattice point. A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. In 1980, Arkinstall [1] proved that, up to lattice equivalence (defined belo...

Hermite normal form and Smith normal form are canonic forms of integral matrices with respect to congruence produced by multiplication with right unimodular matrix resp. by multiplication with left and right unimodular matrix simultaneously. This lower-diagonal resp. diagonal form has many applications in various elds. Unfortunately, the computation of them is extremely resource consuming. The ...

By an eigenvalue comparison-technique[16], the expected return probability of the delayed random walk on the finite clusters of critical Bernoulli bond percolation on the two-dimensional Euclidean lattice is estimated. The results are generalised to invariant percolations on unimodular graphs with almost surely finite clusters. A similar method has been used elsewhere[17] to derive bounds for i...