Unlike to traditional hierarchical and partitional clustering algorithms which always fail to deal with very large databases, a neural network architecture, Projective Adaptive Resonance Theory (PART), is developed for the high dimensional space clustering. However, the success of the PART algorithm depends on both accurate parameters and satisfied orders of input data sets. These disadvantages...

Contents Introduction 3 1. Foundations 13 1.1. A survey about the theory of tight closure 13 1.2. Solid closure and forcing algebras 23 1.3. Cohomological dimension 25 1.4. Vector bundles, locally free sheaves and projective bundles 28 2. Geometric interpretation of tight closure via bundles 30 2.1. Relation bundles 30 2.2. Affine-linear bundles arising from forcing algebras 32 2.3. Cohomology ...

We show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. Hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. As aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. This generalizes and simplifies a result of  Dung and   Smith. As another consequen...

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric i...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

In this paper, by considering the notion of extended BCK-module, we define the concepts of free extended BCK-module, free object in category of extended BCK-modules and we state and prove some related results. Specially, we define the notion of idempotent extended BCK-module and we get some important results in free extended BCK-modules. In particular, in category of idempotent extended BCK-mod...