The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2], [7] and [14]. Some examples to illustrate these geometric information in practice are given. A classification of unital C*-algebras by noncommutative CW complexes and the modified Morse functions on them is the main object of this work.for some ideal I k−1 in A k−1 . In factthen are said to be homotopic if there exists a family {H t } t∈[0,1] of morphisms H t : A → B such that for each a ∈ A the map t → H t (a) is a norm continuous path in B with H 0 = α and H 1 = β. In this case we write α ∼ β. C*-algebras A and B are called of the same homotopy type if there are morphisms ϕ : A → B and ψ : B → A such that ϕoψ ∼ id B and ψoϕ ∼ id A . In this case the morphisms ϕ and ψ are called homotopy equivalent. Definition 4.2. Let A and B be unital C*-algebras. We say A is of pseudohomotopy type of B if C(P rim(A)) and B are of the same homotopy type.Remark 4.3. In the case of unital commutative C*-algebras, by the GNS construction, C(P rim(A)) = A, [10]. So the notions of pseudo-homotopy type and the same homotopy type are equivalent.

Abstract:The goal of the present article is to demonstrate a mathematical modeling for distributed applications. The present paper applies tools from topology and sheaf theory as an appropriate mathematical modeling to reflect interactions among elements of distributed applications resources. Sensors are characterized from their topological representations in distributed network system. This modeling is applied for the study of the air traffic monitoring system and discuss the model in detail.

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The increasing volume of unsolicited bulk e-mails leads to the need for reliable stochastic spam detection methods for the classification of the received sequence of e-mails. When a sequence of emails is received by a recipient during a time period, the spam filters have already classified them as spam or not spam. Due to the dynamic nature of the spam, there might be emails marked as not spam but are actually real spams and vice versa. For the sake of security, it is important to be able to detect real spam emails. This paper utilizes stochastic methods to refine the preliminary spam detection and to find maximum likelihood for spam e-mail classification. The method is based on the Bayesian theorem, hidden Markov model (HMM), and the Viterbi algorithm.

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