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A new formalism to express and operate on diversity measures of qualitative variables, built in a Hilbert space, is presented. The abstract character of the Hilbert space naturally incorporates the equivalence between qualitative variables…
This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and…
The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including…
We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier…
We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the flows of completely integrable Hamiltonian systems are investigated.…
We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this operator and…
Various approaches to stochastic processes exist, noting that key properties such as measurability and continuity are not trivially satisfied. We introduce a new theory for Gaussian processes using improper linear functionals. Using a…
The purpose of this paper is to give an overview of the operator structure of frames, where the operator belongs to certain classes of linear operators and the element belongs to $H$. We discuss the size of the set of such elements. Also,…
This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…
We address classification of distributional data, where units are described by histogram or interval-valued variables. The proposed approach uses a linear discriminant function where distributions or intervals are represented by quantile…
A system of a quantum harmonic oscillator bi-linearly coupled with a Glauber amplifier is analysed considering a time-dependent Hamiltonian model. The Hilbert space of this system may be exactly subdivided into invariant finite dimensional…
We construct certain Hilbert spaces associated with a class of non-linear dynamical systems X. These are systems which arise from a generalized self-similarity, and an iterated substitution. We show that when a weight function W on X is…
Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made…
The Hilbert-Einstein equations are derived in the formalism employing the imbedding of the space-time into linear 10-dimensional space. An extra antisymmetric tensor field is needed for this task.
In an abstract Hilbert space setting, we discuss many linear phenomena of mathematical physics. The functional analytic framework presented is used to address continuous dependence of the solution operators $\mathcal{S}(\mathcal{M})$ of…
Many complex systems satisfy a set of constraints on their degrees of freedom, and at the same time, they are able to work and adapt to different conditions. Here, we describe the emergence of this ability in a simplified model in which the…
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods,…
The paper investigates dynamical systems for which the derivative of some positive-definite function along the solutions of this system depends on so-called density function. In turn, such dynamical systems are called density systems. The…
We investigate systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ where $A \in B(\mathcal{H})$ is a normal operator in a separable Hilbert space $\mathcal{H}$, $\mathcal{G}\subset \mathcal{H}$ is a countable set, and $L$ is a positive…
Time evolution of quantum systems is accelerated by the fast-forward scaling. We reformulate the method to study systems in a finite-dimensional Hilbert space. For several simple systems, we explicitly construct the acceleration potential.…