On scaling laws for production functions

Summary Solutions of functional equations are used in this paper to develop laws for scaling output under proportional changes in input vectors leading to special classes of production functions, which are of significance for the question of returns to scale.I am grateful to ProfessorShephard with whom I have the great pleasure to work while writing this paper.     

Singularities on the boundaries of magnetorotational instabilities and scaling laws

In the theory of magnetorotational instability and its modern extensions such as the helical MRI, non-trivial scaling laws between the critical parameters are observed. In case of the standard MRI it is well known that the Reynolds and Hartmann numbers are scaled as Re ∼ Ha² while for the helical MRI Re ∼ Ha³ . What is less known is that the thresholds of SMRI and HMRI plotted as surfaces in the space of parameters, possess singularities that determine the scaling laws. Moreover, the two paradoxes of SMRI and HMRI in the limits of infinite and zero magnetic Prandtl number (Pm), respectively, sharply correspond to the singularities on the instability thresholds. In either case, it is the local Plücker conoid structure that explains the non-uniqueness of the critical Rossby number, and its crucial dependence on the Lundquist number. For HMRI, we have found an extension of the former Liu limit Ro_c ≃ −0.828 (valid for Lu = 0 ) to a somewhat higher value Ro ≃ −0.802 at Lu = 0.618 which is, however, still below the Kepler value. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geomagnetic plasma probe for solar wind

Magnetograms from Alibag reveal that the range Δ H of the daily variation of the horizontal component is negatively correlated with the minimum value ΔH_min. during a day. This relationship is largely unaffected by the degree of geomagnetic disturbance and holds good during all phases of the 11-year cycle of solar activity. From the nature of the relationship between ΔH and ΔH_min. it is concluded that the daily variation of the geomagnetic field at a low latitude station outside the influence of the equatorial electroject must be regarded as largely due to a weakening of the ambient field on the night side rather than an enhancement of the field on the day side due to ionospheric currents. There exists a good correlation between (ΔH)² and the kinetic energy density of the solar wind in interplanetary space measured by IMP-1 satellite. It is suggested that ΔH is largely the result of the partial ring currents related to the convective drift of the plasma from the tail of the magnetosphere. Moreover, using the relationships established during the IMP-1 period, the annual mean kinetic energy density of solar wind for geomagnetically quiet days for the past 11-year cycle is estimated, treating the earth as a plasma probe.     

Conservation laws and exact solutions for nonlinear diffusion in anisotropic media

Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.     

Modeling and simulation with operator scaling

Self-similar processes are useful models for natural systems that exhibit scaling. Operator scaling allows a different scale factor in each coordinate. This paper develops practical methods for modeling and simulation. A simulation method is developed for operator scaling Lévy processes, based on a series representation, along with a Gaussian approximation of the small jumps. Several examples are given to illustrate the range of practical applications. A complete characterization of symmetries in two dimensions is given, for any exponent and spectral measure, to inform the choice of these model parameters. The paper concludes with some extensions to general operator self-similar processes.     

Cross-country skiing motion equations,locomotive forces and mass scaling laws

This article presents differential equations for locomotive force and velocity during cross-country skiing. A muscle's work power is modelled. Thereafter, a locomotive force that depends on the skier's velocity is constructed. The external forces aerodynamic drag, friction forces and the force of gravity are incorporated in order to provide the equation of motion. Some allometric mass scaling relations are established and used to analyse the effect of a skier's mass on velocity. The model is tested by using a GPS instrument. We compare analytically and experimentally determined skiing distances and velocities as functions of time, and under different conditions. The article provides tools useful for practising athletes and coaches.     

Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion

We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms (any ??1), the flux function f(u) being mth order growth at infinity. It is shown that if ε, δ=δ(ε) tend to 0, then the sequence {uε} of the smooth solutions converges to the unique entropy solution u∈L^∞(0,T_∗;Lq(R)) to the conservation law ut+fx(u)=0 in . The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ)=λ, ?=1 and LeFloch and Natalini when ?=1.     

Optimal control laws for the model of information diffusion in a social group

The article investigates two models of information diffusion in a social group. The dynamics of the process is described by a one-dimensional controlled Riccati differential equation. Our two models differ from the original model of K. V. Izmodenova and A. P. Mikhailov in the choice of the functional being optimized. Two different choices of the optimand functional are considered. The optimal control problems are solved by the Pontryagin maximum principle. It is shown that the optimal control program is a relay function of time with at most one switching point. Conditions on the problem parameters are proposed that are easy to check and guarantee the existence of an optimal-control switching point. The theoretical analysis leads to a one-dimensional convex minimization problem to find the optimal-control switching point. The article also describes an alternative approach to the construction of the optimal solution, which does not resort to the maximum principle and instead utilizes a special representation of the optimand functional and works with reachability sets that are independent of the functional. For the two models considered in this article optimal feedback controls are derived from the programmed optimal controls.     

Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion

We concentrate on Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion describing the growth of cell populations. First, we perform a complete symmetry classification of the equation, and then we find some interesting similarity solutions by means of the symmetries and the variable coefficient heat equation. Local dynamical behaviors are analyzed via the solutions for the growing cell populations. Second, we show that the conservation law multipliers of the equation take the form Λ=Λ(t,x,u), which satisfy a linear partial differential equation, and then give the general formula of conservation laws. Finally, symmetry properties of the conservation law are investigated and used to construct conservation laws of the reduced equations.     

Singular FBSDEs and scalar conservation laws driven by diffusion processes

Motivated by earlier work on the use of fully-coupled forward–backward stochastic differential equations (henceforth FBSDEs) in the analysis of mathematical models for the CO ${}_2$ emissions markets, the present study is concerned with the analysis of these equations when the generator of the forward equation has a conservative degenerate structure and the terminal condition of the backward equation is a non-smooth function of the terminal value of the forward component. We show that a general form of existence and uniqueness result still holds. When the function giving the terminal condition is binary, we also show that the flow property of the forward component of the solution can fail at the terminal time. In particular, we prove that a Dirac point mass appears in its distribution, exactly at the location of the jump of the binary function giving the terminal condition. We provide a detailed analysis of the breakdown of the Markovian representation of the solution at the terminal time.     

A dynamic wind farm aggregate model for the simulation of power fluctuations due to wind turbulence

An important aspect related to wind energy integration into the electrical power system is the fluctuation of the generated power due to the stochastic variations of the wind speed across the area where wind turbines are installed. Simulation models are useful tools to evaluate the impact of the wind power on the power system stability and on the power quality. Aggregate models reduce the simulation time required by detailed dynamic models of multiturbine systems.In this paper, a new behavioral model representing the aggregate contribution of several variable-speed-pitch-controlled wind turbines is introduced. It is particularly suitable for the simulation of short term power fluctuations due to wind turbulence, where steady-state models are not applicable.The model relies on the output rescaling of a single turbine dynamic model. The single turbine output is divided into its steady state and dynamic components, which are then multiplied by different scaling factors. The smoothing effect due to wind incoherence at different locations inside a wind farm is taken into account by filtering the steady state power curve by means of a Gaussian filter as well as applying a proper damping on the dynamic part.The model has been developed to be one of the building-blocks of a model of a large electrical system, therefore a significant reduction of simulation time has been pursued. Comparison against a full model obtained by repeating a detailed single turbine model, shows that a proper trade-off between accuracy and computational speed has been achieved.     

Random perturbations of dynamical systems and diffusion processes with conservation laws

In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDEs. Mathematics Subject Classification (2001): 60H10; 34C29; 35B20     

Simulation of turbulent magnetic reconnection in the small-scale solar wind

Some observational examples for the possible occurrence of the turbulent magnetic reconnection in the solar wind are found by analysing Helios spacecraft's high resolution data. The phenomena of turbulent magnetic reconnections in small scale solar wind are simulated by introducing a third order accuracy upwind compact difference scheme to the compressible two_dimensional MHD flow. Numerical results verify that the turbulent magnetic reconnection process could occur in small scale interplanetary solar wind, which is a basic feature characterizing the magnetic reconnection in high_magnetic Reynolds number (RM=2 000-10 000) solar wind. The configurations of the magnetic reconnection could evolve from a single X_line to a multiple X-line reconnection, exhibiting a complex picture of the formation, merging and evolution of magnetic islands, and finally the magnetic reconnection would evolve into a low_energy state. Its life_span of evolution is about one hour order of magnitude. Various magnetic and flow signatures are recorded in the numerical test for different evolution stages and along different crossing paths, which could in principle explain and confirm the observational samples from the Helios spacecraft. These results are helpful for revealing the basic physical processes in the solar wind turbulence.     

Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor

The intermediate state of a type I superconductor is a classical example of energy‐driven pattern formation, first studied by Landau in 1937. Three of us recently derived five different rigorous upper bounds for the ground‐state energy, corresponding to different microstructural patterns, but only one of them was complemented by a lower bound with the same scaling [Choksi, Kohn, and Otto, J. Nonlinear Sci. 14 (2004), 119–171]. This paper completes the picture by providing matching lower bounds for the remaining four regimes, thereby proving that exactly those five different regimes are traversed with an increasing magnetic field. © 2007 Wiley Periodicals, Inc.