Effects of the turnover rate on the size distribution of firms: An application of the kinetic exchange models

We address the issue of the distribution of firm size. To this end we propose a model of firms in a closed, conserved economy populated with zero-intelligence agents who continuously move from one firm to another. We then analyze the size distribution and related statistics obtained from the model. There are three well known statistical features obtained from the panel study of the firms i.e., the power law in size (in terms of income and/or employment), the Laplace distribution in the growth rates and the slowly declining standard deviation of the growth rates conditional on the firm size. First, we show that the model generalizes the usual kinetic exchange models with binary interaction to interactions between an arbitrary number of agents. When the number of interacting agents is in the order of the system itself, it is possible to decouple the model. We provide exact results on the distributions which are not known yet for binary interactions. Our model easily reproduces the power law for the size distribution of firms (Zipf’s law). The fluctuations in the growth rate falls with increasing size following a power law (though the exponent does not match with the data). However, the distribution of the difference of the firm size in this model has Laplace distribution whereas the real data suggests that the difference of the log of sizes has the same distribution.     

Surface optical Raman modes in GaN nanoribbons

Raman scattering studies were performed in GaN nanoribbons grown along [1 0 0]. These samples were prepared inside Na‐4 mica nanochannels by the ion‐exchange technique and subsequent annealing in NH₃ ambient. Detailed morphological and structural studies including the crystalline orientation were performed by analyzing the vibrational properties in these GaN nanoribbons. Pressure in the embedded structure was calculated from the blue shift of the E₂(high) phonon mode of GaN. Possible red shift of optical phonon modes due to the quantum confinement is also discussed. In addition to the optical phonons allowed by symmetry, two additional Raman peaks were also observed at ∼633 and 678 cm⁻¹ for these nanoribbons. Calculations for the wavenumbers of the surface optical (SO) phonon modes in GaN in Na‐4 mica yielded values close to those of the new Raman modes. The SO phonon modes were calculated in the slab (applicable to belt‐like nanoribbon) mode, as the wavenumber and intensity of these modes depend on the size and the shape of the nanostructures. The effect of surface‐modulation‐assisted electron–SO phonon scattering is suggested to be responsible for the pronounced appearance of SO phonon modes. A scaling factor is also estimated for the interacting surface potential influencing the observed SO Raman scattering intensities. Copyright © 2010 John Wiley & Sons, Ltd.     

Intersectingp — p′ branes inpp-wave background

Several supergravity solutions corresponding to bothDp, as well asDp—Dp′ systems, inNS-NS andR-R pp-wave background originating fromAdS ₃ xS ³ xR ⁴ are presented. The supersymmetry properties of these solutions are analysed along with a brief outline of the world sheet construction for thep — p′ branes.     

Partially Observable Semi-Markov Games with Discounted Payoff

Abstract

We study partially observable semi-Markov game with discounted payoff on a Borel state space. We study both zero sum and nonzero sum games. We establish saddle point equilibrium and Nash equilibrium for zero sum and nonzero sum cases, respectively.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.