The impact of universal recycling on the evolution of economic diversity.

Based on von Neumann's model of an economy characterized by processes and goods, we add to that model a component representing capital equipment. We assume that the need for capital equipment by any process is proportional to the rate at which that process is running, and therefore an increase in rate requires that capital equipment be purchased, whereas a decrease in rate allows capital equipment to be sold. We thereby construct a continuous-time dynamical model, which we use to investigate the evolution of economic diversity under two price equilibrium scenarios: the first with non-negative prices and non-positive excess demands; the second with enforced market clearing and with prices allowed to be negative. The second scenario represents an economy in which recycling is required, so that excess supply cannot be discarded. We prove that at any time during the progression of the model economy, the solution to each of the two price equilibrium problems exists, and that non-uniqueness of the solution, if any, does not affect the development of the model economy. We compare matched model economies under the two scenarios by simulating their respective evolutions. In each case, the model economy experiences a process of selection and matures to a state of balanced growth, with a higher growth rate when excess supply is discarded, but with greater economic diversity with enforced recycling. The robustness of these qualitative results is demonstrated by repeated trials of simulations on matched pairs of model economies with different randomly chosen parameters.

Echocardiographic Linear Dimensions for Assessment of Right Ventricular Chamber Volume as Demonstrated by Cardiac Magnetic Resonance.

BACKGROUND: Echocardiography-derived linear dimensions offer straightforward indices of right ventricular (RV) structure but have not been systematically compared with RV volumes on cardiac magnetic resonance (CMR). METHODS: Echocardiography and CMR were interpreted among patients with coronary artery disease imaged via prospective (90%) and retrospective (10%) registries. For echocardiography, American Society of Echocardiography-recommended RV dimensions were measured in apical four-chamber (basal RV width, mid RV width, and RV length), parasternal long-axis (proximal RV outflow tract [RVOT]), and short-axis (distal RVOT) views. For CMR, RV end-diastolic volume and RV end-systolic volume were quantified using border planimetry. RESULTS: Two hundred seventy-two patients underwent echocardiography and CMR within a narrow interval (0.4 ± 1.0 days); complete acquisition of all American Society of Echocardiography-recommended dimensions was feasible in 98%. All echocardiographic dimensions differed between patients with and those without RV dilation on CMR (P < .05). Basal RV width (r = 0.70), proximal RVOT width (r = 0.68), and RV length (r = 0.61) yielded the highest correlations with RV end-diastolic volume on CMR; end-systolic dimensions yielded similar correlations (r = 0.68, r = 0.66, and r = 0.65, respectively). In multivariate regression, basal RV width (regression coefficient = 1.96 per mm; 95% CI, 1.22-2.70; P < .001), RV length (regression coefficient = 0.97; 95% CI, 0.56-1.37; P < .001), and proximal RVOT width (regression coefficient = 2.62; 95% CI, 1.79-3.44; P < .001) were independently associated with CMR RV end-diastolic volume (r = 0.80). RV end-systolic volume was similarly associated with echocardiographic dimensions (basal RV width: 1.59 per mm [95% CI, 1.06-2.13], P < .001; RV length: 1.00 [95% CI, 0.66-1.34], P < .001; proximal RVOT width: 1.80 [95% CI, 1.22-2.39], P < .001) (r = 0.79). CONCLUSIONS: RV linear dimensions provide readily obtainable markers of RV chamber size. Proximal RVOT and basal width are independently associated with CMR volumes, supporting the use of multiple linear dimensions when assessing RV size on echocardiography.

Stochastic entrainment of a stochastic oscillator.

In this work, we consider a stochastic oscillator described by a discrete-state continuous-time Markov chain, in which the states are arranged in a circle, and there is a constant probability per unit time of jumping from one state to the next in a specified direction around the circle. At each of a sequence of equally spaced times, the oscillator has a specified probability of being reset to a particular state. The focus of this work is the entrainment of the oscillator by this periodic but stochastic stimulus. We consider a distinguished limit, in which (i) the number of states of the oscillator approaches infinity, as does the probability per unit time of jumping from one state to the next, so that the natural mean period of the oscillator remains constant, (ii) the resetting probability approaches zero, and (iii) the period of the resetting signal approaches a multiple, by a ratio of small integers, of the natural mean period of the oscillator. In this distinguished limit, we use analytic and numerical methods to study the extent to which entrainment occurs.

Computer simulation of voltage sensitive calcium ion channels in a dendritic spine.

Membrane current through voltage-sensitive calcium ion channels at the postsynaptic density of a dendritic spine is investigated. To simulate the ion channels that carry such current and the resulting temporal and spatial distribution of concentration, current, and voltage within the dendritic spine, the immersed boundary method with electrodiffusion is applied. In this simulation method a spatially continuous chemical potential barrier is used to simulate the influence of the membrane on each species of ion. The amplitudes of these barriers can be regulated to simulate channel gating. Here we introduce this methodology in a one-dimensional setting. First, we study the current-voltage relationship obtained with fixed chemical potential barriers. Next, we simulate stochastic ion-channel gating in a calcium channel with multiple subunits, and observe the diffusive wave of calcium entry within the dendritic spine that follows channel opening. This work lays the foundation for future three-dimensional studies of electrodiffusion and advection electrodiffusion in dendritic spines.

A robust numerical algorithm for studying biomolecular transport processes.

We present a numerical algorithm that is well suited for the study of biomolecular transport processes. In the algorithm a continuous Markov process is discretized as a jump process and the jump rates are derived from local solutions of the continuous system. Consequently, the algorithm has two advantages over standard numerical methods: (1) it preserves detailed balance for equilibrium processes, (2) it is able to handle discontinuous potentials. The formulation of the algorithm also allows us to calculate the effective diffusion coefficient or, equivalently, the randomness parameter. We provide several simple examples of how to implement the algorithm. All the MATLAB functions files needed to reproduce the results presented in the article are available from www.amath.unc.edu/Faculty/telston/matlab_functions.