David E. Amundsen

Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Networks of interacting signaling pathways are formulated with systems of reaction-diffusion (RD) equations. We show that weak interactions between signaling pathways have negligible effects on formation of spatial patterns of signaling molecules. In particular, a weak interaction between Retinoic Acid (RA) and Notch signaling pathways does not change dynamics of Notch activity in the spatial domain. Conversely, large interactions of signaling pathways can influence effects of each signaling pathway. When the RD system is largely perturbed by RA-Notch interactions, new spatial patterns of Notch activity are obtained. Moreover, analysis of the perturbed Homogeneous System (HS) indicates that the system admits bifurcating periodic orbits near a Hopf bifurcation point. Starting from a neighborhood of the Hopf bifurcation, oscillatory standing waves of Notch activity are numerically observed. This is of particular interest since recent laboratory experiments confirm oscillatory dynamics of Notch activity.

Study and simulation of reaction-diffusion systems affected by interacting signaling pathways.

Possible effects of interaction (cross-talk) between signaling pathways is studied in a system of Reaction–Diffusion (RD) equations. Furthermore, the relevance of spontaneous neurite symmetry breaking and Turing instability has been examined through numerical simulations. The interaction between Retinoic Acid (RA) and Notch signaling pathways is considered as a perturbation to RD system of axon-forming potential for N2a neuroblastoma cells. The present work suggests that large increases to the level of RA–Notch interaction can possibly have substantial impacts on neurite outgrowth and on the process of axon formation. This can be observed by the numerical study of the homogeneous system showing that in the absence of RA–Notch interaction the unperturbed homogeneous system may exhibit different saddle-node bifurcations that are robust under small perturbations by low levels of RA–Notch interactions, while large increases in the level of RA–Notch interaction result in a number of transitions of saddle-node bifurcations into Hopf bifurcations. It is speculated that near a Hopf bifurcation, the regulations between the positive and negative feedbacks change in such a way that spontaneous symmetry breaking takes place only when transport of activated Notch protein takes place at a faster rate.

Resonant oscillations of an inhomogeneous gas between concentric spheres

We investigate the effects of nonlinearity, geometry and stratification on the resonant motion of a gas contained between two concentric spheres. The emphasis is on whether the motion is continuous, and on how the inhomogeneity, geometry or nonlinearity can move the motion to a shocked state. Linear undamped theory yields a standing wave of arbitrary amplitude and an eigenvalue equation in which the higher eigenvalues are not integer multiples of the fundamental; the system is said to be dissonant. Higher modes, generated by the nonlinearity, are not resonant and consequently shocks may not form. When the output is shockless, the amplitude is two orders of magnitude greater than that of the input. When the eigenvalues for a homogeneous gas are not sufficiently dissonant and shocks form, the introduction of a stratification in the gas can restore dissonance and allow a continuous output. Similarly, the introduction of an inhomogeneity can change a continuous motion to a shocked one, as can an increase in the input Mach number, or an increase in the geometrical parameter. Various limits of the eigenvalue equation are considered and previous results for simpler geometries are recovered; e.g. a full sphere, a cone and a straight tube.

A Time Series Modeling Approach in Risk Appraisal of Violent and Sexual Recidivism

For over half a century, various clinical and actuarial methods have been employed to assess the likelihood of violent recidivism. Yet there is a need for new methods that can improve the accuracy of recidivism predictions. This study proposes a new time series modeling approach that generates high levels of predictive accuracy over short and long periods of time. The proposed approach outperformed two widely used actuarial instruments (i.e., the Violence Risk Appraisal Guide and the Sex Offender Risk Appraisal Guide). Furthermore, analysis of temporal risk variations based on specific time series models can add valuable information into risk assessment and management of violent offenders.

Resonances in Dispersive Wave Systems

Weakly nonlinear wave interactions under the assumption of a continuous, as opposed to discrete, spectrum of modes is studied. In particular, a general class of one-dimensional (1-D) dispersive systems containing weak quadratic nonlinearity is investigated. It is known that such systems can possess three-wave resonances, provided certain conditions on the wavenumber and frequency of the constituent modes are met. In the case of a continuous spectrum, it has been shown that an additional condition on the group velocities is required for a resonance to occur. Nonetheless, such so-called double resonances occur in a variety of physical regimes. A direct multiple scale analysis of a general model system is conducted. This leads to a system of three-wave equations analogous to those for the discrete case. Key distinctions include an asymmetry between the temporal evolution of the modes and a longer time scale of O(∈√t) as opposed to O(∈t). Extensions to additional dimensions and higher-order nonlinearities are then made. Numerical simulations are conducted for a variety of dispersions and nonlinearities providing qualitative and quantitative agreement.

Asymptotic solutions for shocked resonant acoustic oscillations between concentric spheres and coaxial cylinders

For resonant oscillations of a gas in a straight tube with a closed end, shocks form and all harmonics are generated, see Chester [“Resonant oscillations in a closed tube,” J. Fluid Mech. 18, 44 (1964)]10.1017/S0022112064000040. When the gas is confined between two concentric spheres or coaxial cylinders, the radially symmetric resonant oscillations may be continuous or shocked. For a fixed small Mach number of the input, the flow is continuous for sufficiently small L, defined as the ratio of the inner radius to the difference of the radii, see Seymour et al. [“Resonant oscillations of an inhomogeneous gas between concentric spheres,” Proc. R. Soc. London, Ser. A 467, 2149 (2011)]10.1098/rspa.2010.0576. However, shocks appear in the resonant flow for either larger values of L or larger input Mach number. A nonlinear geometric acoustics approximation is used to analyse the shocked motion of the gas when L ≫ 1. This approximation and the exact numerical solution are compared for the shocked wave profiles a...

Unifying the steady state resonant solutions of the periodically forced KdVB, mKdVB, and eKdVB equations

The periodically forced extended KdVB (eKdVB) equation, which contains both KdVB and modified KdVB (mKdVB) equations as special cases, is known to possess a rich array of resonant steady solutions. We present an analytic methodology based on singular perturbation and asymptotic matching in order to illustrate and approximate these solutions in the limit that the dispersive effects are small relative to the nonlinear and forcing terms. Weak Burgers damping is also included at the same order as dispersion. Solutions across the resonant band may be constructed and show good agreement with solutions of the full equation, showing clearly the role of the various physical effects. In this way, direct comparisons and connections are made between the various classes of KdVB equations, illustrating, in particular, the underlying mathematical connections between the KdVB and mKdVB equations.

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

Abstract In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.

Time Stepping Via One-Dimensional Padé Approximation

The numerical solution of time-dependent ordinary and partial differential equations presents a number of well known difficulties—including, possibly, severe restrictions on time-step sizes for stability in explicit procedures, as well as need for solution of challenging, generally nonlinear systems of equations in implicit schemes. In this note we introduce a novel class of explicit methods based on use of one-dimensional Padé approximation. These schemes, which are as simple and inexpensive per time-step as other explicit algorithms, possess, in many cases, properties of stability similar to those offered by implicit approaches. We demonstrate the character of our schemes through application to notoriously stiff systems of ODEs and PDEs. In a number of important cases, use of these algorithms has resulted in orders-of-magnitude reductions in computing times over those required by leading approaches.