We study the homotopy category K(InjA) of all injective A-modules InjA and derived category D(ModA) of the category ModA of all A-modules, where A is finite dimensional algebra over an algebraically closed field. We are interested in the algebra with discrete derived category (derived discrete algebra. For a derived discrete algebra A, we get more concrete properties of K(InjA) and D(ModA). The...

We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex. For any word hyperbolic or right angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups. In hi...

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the ”truncated simple reflections” on the set of almost positive roots Φ≥−1 associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we ...

While at Ursinus Norm set a goal to become a professor and upon graduation he accepted a teaching assistantship from the Division of Mathematical Sciences at Purdue University in West Lafayette, Indiana. During his first term he met his wife Barbara (also a TA) in an Abstract Algebra class. Barbara was a secondary school mathematics teacher until their two children were born. She now teaches pa...

In our combinatorial universe, matter grows in a particular way, according to specific steps. The (n + 1) character of this growth lends itself easily to the Clifford Algebra and Clifford Spinor spaces which develop in the fashion of Mount Meru or Pascal’s Triangle. This style of development leads to Fibonacci Numbers and the Golden Section which implies that these concepts are deeply entwined ...

We construct certain subgroups of hyperbolic triangle groups which we call “congruence” subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regul...

ABSTRACT. We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor λ ∈ (0, 1). As is well known, for λ = 1/2 the invariant set, Sλ, is a fractal called the Sierpiński sieve, and for λ < 1/2 it is also a fractal. Our goal is to study Sλ for this IFS for 1/2 < λ < 2/3, i...

Let X be a weighted projective line and CX the associated cluster category. It is known that can realized as generalized category of quiver with potential. In this note, under assumption has at most three weights or tubular type, we prove if C(Q,W) Jacobi-finite non-degenerate potential (Q,W) shares 2-CY tilted algebra CX, then triangle equivalent to CX. As byproduct, determined by its provided...

Many interesting geometric structures on manifolds can be interpreted as structures locally modelled on homogeneous spaces. Given a homogeneous space (X,G) and a manifold M , there is a deformation space of structures on M locally modelled on the geometry of X invariant under G. Such a geometric structure on a manifold M determines a representation (unique up to inner automorphism) of the funda...

This paper is Part 2 of a four-part series of our research on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the problem of point-in-polygon analysis under randomness, i.e., with random measurement error (ME). It is well known that overlay is one of the most important operations in GIS, and point-i...