Abstract The algebraic notions of separable extension and split extension of rings, S ⊂ A, are defined dually. If compatible sections and retracts can be found, A is said to be a finite separable extension of S. Then the endomorphism ring homS(A,A) is a finite separable extension of A with a special cyclic idempotent. This construction may be iterated as in Jones theory. The Jones polynomial is...

We provide an interpretation of the homology of the loop space on the p-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If f is an idempotent in kG such that f.kG is the projective cover of the trivial module k, and e = 1− f , then we exhibit isomorphisms for n ≥ 2: Hn(ΩBG ∧ p ;...

Let A be a unital algebra and M be a unital A-bimodule. A characterization of generalized derivations and generalized Jordan derivations from A into M, through zero products or zero Jordan products, is given. Suppose that M is a unital left A-module. It is investigated when a linear mapping from A into M is a Jordan left derivation under certain conditions. It is also studied whether an algebra...

A ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean ring and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any...

in this paper, we propose a novel method for fully automatic detection and tracking of human heads and faces in video sequences. the proposed algorithm consists of two modules: a face detection module and a face tracking module. the detection module, detects the face region and approximates it with an ellipse at the first frame using a modified version of adaboost cascaded classifier. the detec...

In this paper we consider Idempotent Functional Analysis, an 'abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's corresponden...

A linear relation E acting on a Hilbert space is idempotent if E2=E. triplet of subspaces needed to characterize given idempotent: (ranE,ran(I−E),domE), or equivalently, (ker⁡(I−E),ker⁡E,mulE). The relations satisfying the inclusions E2⊆E (sub-idempotent) E⊆E2 (super-idempotent) play an important role. Lastly, adjoint and closure are studied.

A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of “category with finite products”. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of “essentially unique” and investigating its consequences. We call such 2-monads...

A category may bear manymonoidal structures, but (to within a unique isomorphism) only one structure of \category with nite products". To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical de nition of \essentially unique" and investigating its consequences. We call such 2-monads pro...