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 IM ⫅ N. 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.
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.
Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces wpΣα(X), wpQα(X) and wDα(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.
Authors:S. Hassi, Z. Sebestyén, H. De Snoo, and F. Szafraniec
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
Weak atomic decompositions of B-valued martingales with two-parameters in weak Hardy spaces wpΣα and wpHα 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.
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.
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.
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.
Authors:P. Cristofori, M. Mulazzani, and A. Vesnin
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.