In part 1 of this paper, first and second order analysis of uncertainty is applied to numerical models of groundwater flow. The models are cast in state-space form, with boundary conditions and inputs that are functionally dependent, but statistically independent, of time. Using a compact matrix calculus notation, first and second order Taylor series expansions of the model equations are derive...

A two point boundary value problem for differential inclusions with relaxed one sided Lipschitz right-hand side in Banach spaces with uniformly convex duals are studied. We use successive approximations to obtain nonemptiness and compactness of the solution set as well as its continuous dependence from the initial conditions and the right-hand side. The relaxation theorem is also proved. AMS su...

A joint experimental and computational methodology is developed and applied to investigate a vortex ring impinging normally on a wall. The method uses digital particle image velocimetry to make planar flow measurements, which are then used to initialize a second-order finite difference calculation. The experiment and the simulation are compared at later times and agree extremely well. The ring ...

We provide a new, analytic kernel for scale space filtering of dMRI data. The kernel is an approximation for the Green’s function of a hypo-elliptic diffusion on the 3D rigid body motion group SE(3), for fiber enhancement in dMRI. The enhancements are described by linear scale space PDEs in the coupled space of positions and orientations embedded in SE(3). As initial condition for the evolution...

This work addresses the problem of computing the time evolution of the probability density function (pdf) of the state in a nonlinear neuromuscular blockade (NMB) model, assuming that the source of uncertainty is the knowledge about one parameter. The NMB state is enlarged with the parameter, that verifies an equation given by its derivative being zero and has an initial condition described by ...

We derive the asymptotic traveling-wave solutions of the nonlinear 1-dimensional BalitskyKovchegov QCD equation for rapidity evolution in momentum-space, with 1-loop running coupling constant and equipped with the Balitsky-Kovchegov-Kuraev-Lipatov kernel at next-to-leading logarithmic accuracy, conveniently regularized by different resummation schemes. Traveling waves allow to define “universal...

in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy p...

The paper introduces a new notion of stability for internal autonomous system descriptions in discrete-time, referred to as ”strong stability”, which extends a parallel notion introduced in the continuous-time case. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting resp...

(1.1) i∂tψ(t) = Hψ(t) . For all initial conditions ψ(0) ∈ H, the unique solution is given by ψ(t) = e−itHψ(0), for all t ∈ R. Due to Stone’s Theorem e−itH is a strongly continuous one-parameter group of unitary operators on H, and the self-adjointness of H is the necessary and sufficient condition for the existence of a unique continuous solution for all times. An alternative description of thi...

In the piecewise affine framework, trajectories evolve among hyperrectangles in the state space. A qualitative description of the dynamics useful for models of genetic networks can be obtained by viewing each hyperrectangle as a node in a discrete system, so that trajectories follow a path in a transition graph. In this paper, a probabilistic interpretation is given for the transition between t...