We describe a flexible algorithm that can create many different kinds of aesthetic fractal patterns. The main idea of the algorithm is to randomly place progressively smaller copies of a motif within a region, thus producing a fractal pattern. This has been found to work for a wide variety of shapes for both the motif and the enclosing region. We show a several examples. If the region is a fund...

A Selberg zeta function is attached to the three-dimensional BTZ black hole, and a trace formula is developed for a general class of test functions. The trace formula differs from those of more standard use in physics in that the black hole has a fundamental domain of infinite hyperbolic volume. Various thermodynamic quantities associated with the black hole are conveniently expressed in terms ...

A fundamental domain D for T2 is a connected closed subset D c H behaving like the quotient space T2 \ H, at least up to boundary identifications. Thus, for every z g H, there is a y G T2 such that yz g D. Moreover, if z and w lie in the interior of D and z = yw for y G T2, then y = ±1, where / is the identity matrix. It is easily seen (cf. Terras [17]) that the region (1.1) F2= (zg//| -i< Rez<...

We use conference matrices to define an action of the complex numbers on the real Euclidean vector space R. In certain cases, the lattice D n becomes a module over a ring of quadratic integers. We can then obtain new unimodular lattices, essentially by multiplying the lattice D n by a non-principal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the ...

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. Let Γ ⊂ PSL2(R) be a Fuchsian group, a discrete group of orientationpreserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental domain for Γ is a closed domain D ⊂ H such th...

Morgenstern ([Mor95]) claimed to have constructed fundamental domains for congruence subgroups of the lattice group Γ = PGL2(Fq[t]), and subgraphs providing the first known examples of linear families of bounded concentrators. His method was to construct the fundamental domain for a congruence subgroup as a ‘ramified covering’ of the fundamental domain for Γ on the Bruhat-Tits tree X = Xq+1 of ...

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. Let Γ ⊂ PSL2(R) be a Fuchsian group, a discrete group of orientationpreserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental domain for Γ is a closed domain D ⊂ H such th...

We exploit symmetry to reduce the memory storage requirements of linear explicit model predictive controllers. In the first part of the paper we define controller symmetry. We describe how symmetry can be used to compress the explicit controller and discuss the implementation of the resulting compressed controller. In the second part we develop a method for computing the reduced-memory controll...

We generalize the grid–projection method for the construction of quasiperiodic tilings. A rather general fundamental domain of the associated higher dimensional lattice is used for the construction of the acceptance region. The arbitrariness of the fundamental domain allows for a choice which obeys all the symmetries of the lattice, which is important for the construction of tilings with a give...

The adult Drosophila midgut is established during the larval/pupal transition from undifferentiated cells known as adult midgut precursors (AMPs). Four fundamental cell types are found in the adult midgut epithelium: undifferentiated intestinal stem cells (ISCs) and their committed daughter cells, enteroblasts (EBs), plus enterocytes (ECs) and enteroendocrine cells (EEs). Using the Drosophila p...