Related papers: Paraunitary Matrices
Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We…
The paper presents the classification of matrix valued superpotentials corresponding to shape invariant systems of Schr\"odinger equations. All inequivalent irreducible matrix superpotentials realized by matrices of arbitrary dimension with…
Many models for chaotic systems consist of joining two integrable systems with incompatible constants of motion. The quantum counterparts of such models have a propagator which factorizes into two integrable parts. Each part can be…
The exact parameter values of mathematical models are often uncertain or even unknown. Nevertheless, we may have access to crude information about the parameters, e.g., that some of them are nonzero. Such information can be captured by…
Orthogonal wavelet transforms are a cornerstone of modern signal and image denoising because they combine multiscale representation, energy preservation, and perfect reconstruction. In this paper, we show that these advantages can be…
We describe the variety of fixed points of a unipotent operator acting on the space of matrices. We compute the determinant and the rank of a generic (symmetric, or anti-symmetric) matrix in the fixed variety, yielding information about the…
Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry…
New elliptic cylindrical wavelets are introduced, which exploit the relationship between analysing filters and Floquet's solution of Mathieu differential equations. It is shown that the transfer function of both multiresolution filters is…
In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank…
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary…
We give a number of algorithms for constructing unitary matrices and tight frames with specialized properties. These were produced at the request of researchers at the Frame Research Center (www.framerc.org) to help with their research on…
An algebraic investigation on bicomplex numbers is carried out here. Particularly matrices and linear maps defined on them are discussed. A new kind of cartesian product, referred to as an idempotent product, is introduced and studied. The…
Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing…
We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that…
A biunimodular vector of a unitary matrix $A \in U(n)$ is a vector $v \in \mathbb{T}^n\subset\bc^n$ such that $Av \in \mathbb{T}^n$ as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object…
Hyperuniform materials, characterized by their suppressed density fluctuations and vanishing structure factors as the wave number approaches zero, represent a unique state of matter that straddles the boundary between order and randomness.…
The Transfer Matrix Method is a powerful numerical tool for simulating wave propagation in layered media. It has been widely applied in many fields, although its use is typically restricted to passive media. In this paper, we develop the…
In recent years and with introduction of 5G cellular network and communication, researchers have shown great interest in Multiple Input Multiple Output (MIMO) communication, an advanced technology. Many studies have examined the problem of…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the…