Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Extended spectral decompositions of the Renyi map

We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map.

Fast Moving Window Algorithm for QR and Cholesky Decompositions

This paper proposes a fast moving window algorithm for QR and Cholesky decompositions by simultaneously applying data updating and downdating. The developed procedure is based on inner products and entails a similar downdating to that of the Chambers’ approach. For adding and deleting one row of data from the original matrix, a detailed analysis shows that the proposed algorithm outperforms existing ones in terms or computational efficiency, if the number of columns exceeds 7. For a large number of columns, the proposed algorithm is numerically superior compared to the traditional sequential technique.

New techniques for enhanced medial axis based decompositions in 2-D

New techniques are presented for using the medial axis to generate high quality decompositions for generating block-structured meshes with well-placed mesh singularities away from the surface boundaries. Established medial axis based meshing algorithms are highly effective for some geometries, but in general, they do not produce the most favourable decompositions, particularly when there are geometry concavities. This new approach uses both the topological and geometric information in the medial axis to establish a valid and effective arrangement of mesh singularities for any 2-D surface. It deals with concavities effectively and finds solutions that are most appropriate to the geometric shapes. Methods for directly constructing the corresponding decompositions are also put forward.

Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Custom coprocessor based matrix algorithms for image and signal processing

Matrix algorithms are important in many types of applications including image and signal processing. A close examination of the algorithms used in these, and related, applications reveals that many of the fundamental actions involve matrix algorithms such as matrix multiplication. This paper presents an investigation into the design and implementation of different matrix algorithms such as matrix operations, matrix transforms and matrix decompositions using a novel custom coprocessor system for MATrix algorithms based on Reconfigurable Computing (RCMAT). The proposed RCMAT architectures are scalable, modular and require less area and time complexity with reduced latency when compared with existing structures.