In this paper we present a method for the consistent modelling and efficient ray tracing of complex natural scenes. Both plants and terrains are modelled and represented in the same way to allow mutual influences of their appearance and interdependencies of their geometry. Plants are generated together with a fractal terrain, so that they directly grow on it. This allows an accurate calculation...

Let R = Z4[u]/〈u〉 = Z4 + uZ4 + . . . + uZ4 (u = 0) where k ∈ Z satisfies k ≥ 2. For any odd positive integer n, it is known that cyclic codes over R of length n are identified with ideals of the ring R[x]/〈x − 1〉. In this paper, an explicit representation for each cyclic code over R of length n is provided and a formula to count the number of codewords in each code is given. Then a formula to c...

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast previously known examples, our proofs require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety group, for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every repr...

A graph is representable by a ring if its vertices can be labeled with distinct ring elements so the difference of the labels is a unit in the ring if and only if the vertices are adjacent. We explore representation by rings composed of the direct products of cyclic rings, including an upper bound on the representation number for a graph with a fixed number of vertices and representation number...

The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p− 3 conjecture”.

Self-similar stable mixed moving average processes can be related to nonsingular flows through their minimal representations. Self-similar stable mixed moving averages related to dissipative flows have been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is on self-similar stable mixed moving averages related to peri...

In this paper we compute the Bredon equivariant homology of representation spheres corresponding to the orientable three dimensional representations of cyclic groups and dihedral groups, as well as the symmetric group on three letters equipped with a permutation representation. These computations are greatly simplified by the introduction of a splitting theorem for the Burnside ring Mackey func...

We present some general techniques for constructing full-rank, minimal-delay, rate at least one space–time block codes (STBCs) over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras (field extensions) as well as using noncommutative division algebras of the rational field embedded in matrix rings. The first half of the paper deals with constr...

We consider an indecomposable representation of a cyclic p-group Z pr over a field of characteristic p. We show that the top degree of the corresponding ring of coinvariants is less than (r 2+3r)pr 2 . This bound also applies to the degrees of the generators for the invariant ring of the regular representation.