We study maximal m-rigid objects in the m-cluster category C m H associated with a finite dimensional hereditary algebra H with n nonisomorphic simple modules. We show that all maximal mrigid objects in these categories have exactly n nonisomorphic indecomposable summands, and that any almost complete m-rigid object in C m H has exactly m + 1 nonisomorphic complements. We also show that the maximal m-rigid objects and the m-cluster tilting objects in these categories coincide, and that the class of finite dimensional algebras associated with maximal m-rigid objects is closed under certain factor algebras.

We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case. The distribution of cluster-tilting objects of type D nIn this section we will show how to, given the quiver Q of a cluster-tilted algebra of type D n , explicitly find a cluster-tilting object in the cluster category of type D n inducing it. The goal is to prove the following theorem:Theorem 4.1. Given the quiver Q of a cluster-tilted algebra of type D, it will be of one of the types 1) ⋆ ⋆ 2) ⋆ ⋆ 3)

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A salient feature is a part of the scene that stands out relative to neighboring items. By that we mean that a human observer would experience a salient feature as being more prominent. It is, however, important to quantify saliency in terms of a mathematical quantity that lends itself to measurements. Different metrics have been shown to correlate with human fixations data. These include contrast, brightness and orienting gradients calculated at different image scales. In this paper, we show that these metrics can be grouped under transformations pertaining to the dihedral group D4, which is the symmetry group of the square image grid. Our results show that salient features can be defined as the image features that are most asymmetric in their surrounds.

Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.Comment: Section 3 has been removed and now is an independent article (arXiv:0912.2911v1). Section 1 and 2.2 have been modified to cope with the removal of section 3. Proof of theorem 3.5 (previously 4.5) has been improved. More details have been added to section 6 (previously 7) to clarify how section 3 (previously 4) generalizes to the infinite cas