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Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IMN. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

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A pair of linear bounded commuting operators T1, T2 in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T1)(I-T2) = Ker (I-T1) + Ker (I-T2). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Rvsz that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included.

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The aim of this paper is to give a decomposition of a weaker form of continuity, namely gα**-continuity by providing the concepts of K-sets and K-continuity.

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Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces w pΣα(X), w p Q α(X) and wD α(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the operators f*, S (p)(f) and σ(p)(f) are obtained.

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An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear relation.

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Weak atomic decompositions of B-valued martingales with two-parameters in weak Hardy spaces w pΣα and w p H α are established and the boundedness of sublinear operators on these spaces are proved. By using them, some characterizations of the smoothness of Banach spaces are obtained.

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The following two decomposition theorems are obtained. (1) A function f is α-continuous if and only if f is pre-continuous and αα-continuous, (2) A function f is semi-continuous if and only if f is spr-continuous and αLC-continuous.

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A topological space (X, π) is said to be nearly Lindelf if every regularly open cover of (X, π) has a countable subcover. In this paper we study the effect of mappings and some decompositions of continuity on nearly Lindelf spaces. The main result is that a δ-continuous image of a nearly Lindelf space is nearly Lindelf.

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The main aim of the paper is to prove still another version of the Lvy--Khintchine decomposition of conditionally positive definite functions on a general locally compact Abelian group. The exposition is based on the two-cones theorem proved by N. Drumm in 1976. Application of the main result to the Euclidean group shows the novelty of the approach.

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Strongly-cyclic branched coverings of knots are studied by using their (g, 1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations.

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