Identification and Elimination of Interior Points for the Minimum Enclosing Ball Problem

Three series of tracer studies were performed on three constructed wetlands at the New Hanover County Landfill near Wilmington, North Carolina, USA. One vegetated free water surface wetland (FWS-R), one vegetated subsurface flow wetland (SSF-R), and one unvegetated control subsurface flow wetland (SSF-C) were studied. A conservative tracer, lithium chloride, was used to study the chemical reactor behavior of these wetlands under normal operating conditions. Results indicated that short-circuiting is quite common in SSF wetlands, while FWS wetlands are well-mixed and not as subject to short-circuiting. These results were obtained from and reinforced with tracer measurements at interior points in these wetlands, analysis of residence time distributions from two different formulations, and the construction of residence volume distributions. The short-circuiting in the SSF wetlands can be attributed to the following: (1) Vertical mixing is inhibited by a combination of physical barriers and density gradients caused by rainfall and runoff dilution of the upper layer; and (2) Leachate is drawn from the bottom of the wetland, causing it to further prefer a flow path along the bottom.

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Nonlinear Compartment Models with Time-Dependent Parameters

Mathematics ◽

10.3390/math9141657 ◽

2021 ◽

Vol 9 (14) ◽

pp. 1657

Author(s):

Jochen Merker ◽

Benjamin Kunsch ◽

Gregor Schuldt

Keyword(s):

Stable Equilibrium ◽

Compartment Model ◽

Basins Of Attraction ◽

Dynamical Behaviour ◽

Time Dependent ◽

Compartment Models ◽

Autonomous Case ◽

Hyperbolic Equilibrium ◽

Interior Points ◽

Two Parameter

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

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Markov chain approximation of one-dimensional sticky diffusions

Advances in Applied Probability ◽

10.1017/apr.2020.65 ◽

2021 ◽

Vol 53 (2) ◽

pp. 335-369

Author(s):

Christian Meier ◽

Lingfei Li ◽

Gongqiu Zhang

Keyword(s):

Markov Chain ◽

Approximate Solutions ◽

Price Model ◽

One Dimensional ◽

Markov Chain Approximation ◽

Alternative Approaches ◽

Second Order Convergence ◽

Chain Approximation ◽

Matrix Exponentials ◽

Interior Points

AbstractWe develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.

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Motion periodicity and bifurcation of a wave excited hopping ball

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0137 ◽

2019 ◽

Vol 475 (2228) ◽

pp. 20190137

Author(s):

Gaurang Ruhela ◽

Anirvan DasGupta

Keyword(s):

Wave Amplitude ◽

Harmonic Wave ◽

Inelastic Collisions ◽

Loss Of Stability ◽

Elastic Surface ◽

Minimum Frequency ◽

Surface Motion ◽

Return Map ◽

Ball Problem ◽

Subsequent Loss

We consider the problem of a hopping ball excited by a travelling harmonic wave on an elastic surface. The ball, considered as a particle, is assumed to interact with the surface through inelastic collisions. The surface motion due to the wave induces a horizontal drift in the ball. The problem is treated analytically under certain approximations. The phase space of the hopping motion is captured by constructing a phase-velocity return map. The fixed points of the return map and its compositions represent periodic hopping solutions. The linear stability of the obtained periodic solution is studied in detail. The minimum frequency for the onset of periodic hops, and the subsequent loss of stability at the bifurcation frequency, have been determined analytically. Interestingly, for small values of wave amplitude, the analytical solutions reveal striking similarities with the results of the classical bouncing ball problem.

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