We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate co...

This article consists of a detailed geometric study of the one-dimensional vorticity model equation ωt + uωx + 2ωux = 0, ω = Hux, t ∈ R, x ∈ S 1 , which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on Diff(S) when the latter is endowed with the rightinvariant homogeneous Ḣ–metric. In this article we prove th...

We consider the filtering problem for a Markov chain {Xk, Yk}k≥0 with state X and observation Y . The state process {Xk}k≥0 is an homogeneous Markov chain taking value in a measurable set X equipped with a σ-algebra B(X). We let Q be the transition kernel of the chain. The observations {Yk}k≥0 takes values in a measurable set Y (BY is the associated σ-algebra). For i ≤ j, denote Yi:j , (Yi, Yi+...

In the present work, we study autocatalytic pathways which contain reactions that need the use of one of their own productions. These pathways are common in biology; one of the simplest and widely studied autocatalytic pathways is Glycolysis. This pathway produces energy by breaking down Glucose. It is shown that this pathway can be simplified as a network of three biochemical reactions. We fir...

We examine the stability properties of a class of LTV di"erence equations on an in&nite-dimensional state space that arise in backstepping designs for parabolic PDEs. The nominal system matrix of the di"erence equation has a special structure: all of its powers have entries that are −1, 0, or 1, and all of the eigenvalues of the matrix are on the unit circle. The di"erence equation is driven by...

We propose an extended quantum theory, in which the number K of parameters necessary to characterize a quantum state behaves as fourth power of the number N of distinguishable states. As the simplex of classical N–point probability distributions can be embedded inside a higher dimensional convex body M Q N of mixed quantum states, one can further increase the dimensionality constructing the set...

It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Korteweg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (B...

Scale spaces allow us to organize, compare and analyse differently sized structures of an object. The linear scale space of a monochromatic image is the solution of the heat equation using that image as an initial condition. Alternatively, this linear scale space can also be obtained applying Gaussian filters of increasing variances to the original image. In this work, we compare (by looking at...

Our goal is to develop model reduction processes for linear dynamical systems with non-zero initial conditions. Standard model reduction schemes optimize system input/output characteristics but frequently destroy initial condition state space information. Additionally, existing model reduction schemes for initial conditions inappropriately assign weight to initial condition information. We prop...

As the first constrained step in the investigation of the Fokker-Planck equation as a source of the probability density function of the position (or velocity) of a Brownian particle, we studied the connection between the inverse problem for the diffusion equation and the kernel methods developed in the area of statistical learning theory and its applications. The initial condition (an initial p...