Example Of Equilibrium Research Articles

Analytic MHD equilibria for rotating plasmas with X-points and elongation, triangularity and squareness control

The MHD equilibrium equation that describes plasmas rotating in the toroidal direction is solved using the Solov’ev ansatz. The equilibria have 2 X-points and the shape of the plasma can be controlled by fixing the values of the aspect ratio, elongation, triangularity, squareness and curvature at the equatorial points. The plasma current, total β and Mach number can be also specified. Examples of equilibria with positive and negative triangularity and different values of the squareness and Mach number are presented.

Extended Solov'ev type equilibria for rotating plasmas with positive and negative triangularity

The Solov'ev ansatz is employed to solve the equilibrium equation for plasmas rotating in the toroidal direction. The plasma shape can be controlled by fixing the values of the aspect ratio, elongation, triangularity, and curvature at the equatorial points. In addition, it is possible to set the values of the plasma current, total β, and Mach number. Analytic expressions for the shape coefficients and examples of equilibria with positive and negative triangularity are presented.

Quasi-analytical equilibria with toroidal rotation and shape control

Quasi-analytical equilibrium solutions for plasmas rotating in the toroidal direction are presented. The formalism provides control of the plasma shape (aspect ratio, elongation and triangularity) and allows to independently set the values of the rotation velocity (Mach number) and the plasma current and β. Explicit expressions for the shape coefficients and examples of equilibria calculated with this method are presented.

Pareto efficiency in the jungle

We revisit the jungle economy of Piccione and Rubinstein (Econ J 117(July):883–896, 2007) in which the allocation of resources is driven by coercion. In this economy the unique equilibrium satisfies lexicographic maximization. We show that relaxing the assumptions on consumption sets and preferences slightly, allocations other than lexicographic maximizers can be jungle equilibria. We attribute this result to single unilateral taking where a stronger agent takes goods from only one weaker agent. Once multiple unilateral takings are introduced, we show that jungle equilibria coincide with lexicographic maximization under weak assumptions. However, we also present examples of equilibria that are Pareto inefficient, where voluntary gift giving by stronger agents is needed to achieve Pareto efficiency and even voluntary trade has a role in the jungle.

Examples of Non-Uniqueness of the Equilibrium States for a Floating Ball

We provide a numerical algorithm for numerically approximating a centrally located floating ball. We give examples of equilibria, and we present non-unique cases for the same physical parameters when the density of the ball is either greater than the supporting liquid (heavy) or lighter than the density of the vapor above (light). We classify the non-uniqueness by analyzing a function related to the force balance. We derive the potential energy of these states, and make comparisons of the non-unique cases. In the cases of both the light and heavy floating balls, the evidence presented supports the conjecture that when there are two equilibria, the one with lower energy corresponds to the location of triple junction (between the ball, the vapor and the liquid) that is closer to the equator of the ball.

Point vortex equilibria on the sphere via Brownian ratchets

We describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, A Γ =0, where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ ∈ N is the (unit) vector of vortex strengths that must lie in the null space of A . Existence of an equilibrium is expressed by the condition det( A T A )=0, while uniqueness follows if Rank( A )= N −1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A , and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N =4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy.

Competing in Markets with Digital Convergence

The last couple of decades have witnessed widespread digitization of information and content, increasing incorporation of digital technologies into the products of diverse industries, and often, an accompanying shift to using von-Neumann-like platform-based product architec- tures. This phenomenon has been termed digital convergence, and has been observed in technology industries as diverse as handheld computing, wireline and mobile telecommunications, consumer electronics, networking equipment, residential broadband and broadcast video. Digital convergence results in substantially more valuable and flexible products and services, but also leads to increased substitutability between products in previously distinct markets. This paper analyzes the economic implications of this trade-off between increasing value and increasing substitutability. We present a new general model of imperfect competition that enables us to capture specific economic aspects that characterize digital convergence — flexible product scope, variability in consumer requirements, and non-exclusionary product choice. We derive four types of equilibrium configurations — local monopoly, kinked, competitive and non-exclusive — that emerge as outcomes of the model, and describe how each equilibrium structure characterizes a distinct stage of digital convergence. Our analysis establishes that as markets converge, prices rise initially even as competing prod- ucts become less differentiated. However, when platform scope is largely dictated by exogenous factors, prices and profits eventually fall as the stage of convergence progresses, though consumer surplus and total surplus rise. Furthermore, while digital convergence has the expected effect of shifting consumption patterns from multiple specialized products purchases to single general- purpose product purchases, we describe examples of equilibria in which consumers may buy mul- tiple general-purpose products, using each for a specialized subset of their requirements. Pricing responses to changes in variable costs and consumer functionality needs are also discussed. When firms can strategically choose platform scope, we show that in any subgame perfect equilibrium, duopoly prices are always higher than monopoly prices, and industries may sustain high levels of profitability even when their boundaries blur.

Simulation and Estimation of Hedonic Models

Making use of restrictions imposed by equilibrium, theoretical progress has been made on the nonparametric and semiparametric estimation and identification of scalar additive hedonic models (Ekeland, Heckman, and Nesheim, 2002) and scalar nonadditive hedonic models (Heckman, Matzkin, and Nesheim, 2002). However, little is known about the practical aspects of estimating such models or of the characteristics of equilibrium in such models. This paper presents computational and analytical results that fill some of these gaps. We simulate and estimate examples of equilibrium in the additive hedonic models and provide evidence on the performance of several estimation techniques. We also simulate examples of equilibria in nonadditive models and provide evidence on the performance of the nonadditive estimation techniques developed in Heckman, Matzkin, and Nesheim (2002).

Numerical calculation of compact torus equilibria

A method is presented for the calculation of a MHD equilibrium for an axisymmetric compact torus. The plasma pressure is assumed to be a scalar quantity. The plasma and magnetic fields are determined by given adiabatic quantities, entropy, poloidal and toroidal magnetic flux. The equilibrium calculation proceeds by alternating between the solution of the 2-D Grad-Shafranov equation and a 1-D flux surface average of this equation. The 2-D calculation utilizes flux surface coordinates and finite elements. The poloidal flux function ψ is computed on an approximate flux surface coordinate system (χ, λ). The points describing the χ surface are moved to coincide with the surface of constant ψ. The 1-D step computes the volume enclosed by each surface. The calculation is necessary due to the equilibrium being specified by adiabatic quantities. Three examples of equilibria computed by this method are represented, two spheromaks and one field reversed configuration (FRC).

Two examples of equilibria under price rigidities and quantity rationing

Two examples of equilibria under price rigidities and quantity rationing