In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a reconstruction set, defined as a shortest pathway between the sample consistent set and the guiding set. Existence and uniqueness of the reconstruction set are inve...

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕ H1. In particular, we prove the existence of a one-parameter family of dense ...

It is shown that to every operator T in a general von Neumann factor M of type II1 and to every Borel set B in the complex plane C, one can associate a maximal, closed, T -invariant subspace, K = KT (B), affiliated with M, such that the Brown measure of T |K is concentrated on B. Moreover, K is T -hyperinvariant, and the Brown measure of PK⊥T |K⊥ is concentrated on C \ B. In particular, if T ∈ ...

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G → M → X so that M is also a complex manifold satisfying a local subelliptic estimate. In this work, we show that if G acts by holomorphic transformations in M , then the Laplacian = ∂̄∗∂̄ + ∂̄∂̄∗ on M has the following properties: The kernel of restricted to the forms Λ with q > 0 i...

A simple method of identifying plant and controller operating in closed-loop is developed by the orthogonal decomposition technique. Under the assumption that the system is open-loop as well as closed-loop stable, a subspace identification algorithm of identifying the deterministic part is derived based on two successive LQ decompositions. The idea is somewhat related to the two-stage method (V...

We give precise conditions under which the real interpolation space [Y0, X1]θ,p coincides with a closed subspace of [X0, X1]θ,p when Y0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces Hs(−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλnt) and E forms an unconditional basis in L2(−π, π). Then there...

For example, Theorem 3 says that if V is a closed subspace of f2 and if V CQp for some p < 2, then V is finite-dimensional . On the other hand, the corollary to Theorem 4 states that there exist infinite-dimensional subspaces V of f 2 none of whose nonzero elements belongs to any f p -space (p < 2) . [For L2(0, 1) the results are somewhat different: (1) if V is a closed subspace of L 2(0, 1) an...

Nagata conjectured that everyM -space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset P of size 2 such that in V P Nagata’s conjecture holds for each first countable regular space from the ground model (i.e. if a ...

Problem 1 (a) Recall that the null space and the range of a matrix A ∈ C m×n are defined as: N (A) = {x ∈ C n |Ax = 0} R(A) = {y ∈ C m |y = Ax, for some x ∈ C n } Note that N (A) is a subspace of C n since it contains 0 ∈ C n (A0 = 0) and is closed under addition and scalar multiplication: x 1 , x 2 ∈ N (A) ⇒ Ax 1 = 0 and Ax 2 = 0 ⇒ Ax 1 + Ax 2 = 0 ⇒ A(x 1 + x 2) = 0 ⇒ x 1 + x 2 ∈ N (A) x ∈ N (...