On the asymptotic behavior of variable exponent power–law functionals and applications

We study, via Γ-convergence, the asymptotic behavior of several classes of power–law functionals acting on fields belonging to variable exponent Lebesgue spaces and which are subject to constant rank differential constraints. Applications of the Γ-convergence results to the derivation and analysis of several models related to polycrystal plasticity arising as limiting cases of more flexible power–law models are also discussed.     

Is the statement of Murphy's law valid?

Murphy's Law is not a law in the formal sense yet popular science often compares it with the Second Law of Thermodynamics as both the statements point toward a more disorganized state with time. In this article, we first construct a mathematically equivalent statement for Murphy's Law and then disprove it using the intuitive idea that energy differences will level off along the paths of steepest descent, or along trajectories of least action. © 2015 Wiley Periodicals, Inc. Complexity 21: 374–380, 2016     

The distance-decay function of geographical gravity model: Power law or exponential law?

The distance-decay function of the geographical gravity model is originally an inverse power law, which suggests a scaling process in spatial interaction. However, the distance exponent of the model cannot be reasonably explained with the ideas from Euclidean geometry. This results in a dimension dilemma in geographical analysis. Consequently, a negative exponential function was used to replace the inverse power function to serve for a distance-decay function. But a new puzzle arose that the exponential-based gravity model goes against the first law of geography. This paper is devoted for solving these kinds of problems by mathematical reasoning and empirical analysis. New findings are as follows. First, the distance exponent of the gravity model is demonstrated to be a fractal dimension using the geometric measure relation. Second, the similarities and differences between the gravity models and spatial interaction models are revealed using allometric relations. Third, a four-parameter gravity model possesses a symmetrical expression, and we need dual gravity models to describe spatial flows. The observational data of China's cities and regions (29 elements indicative of 841 data points) in 2010 are employed to verify the theoretical inferences. A conclusion can be reached that the geographical gravity model based on power-law decay is more suitable for analyzing large, complex, and scale-free regional and urban systems. This study lends further support to the suggestion that the underlying rationale of fractal structure is entropy maximization. Moreover, it suggests that many dimensional dilemmas of spatial modeling can be solved using the concepts from fractal geometry.     

Why does the power law for stock price hold?

The aim of this paper is to explain why the power law for stock price holds. We first show that the complementary cumulative distributions of stock prices follow a power law using a large database assembled from the balance sheets and stock prices of a number of worldwide companies for the period 2004 through 2013. Secondly, we estimate company fundamentals from a simple cross-sectional regression model using three financial indicators-dividends per share, cash flow per share, and book value per share—as explanatory variables for stock price. Thirdly, we demonstrate that the complementary cumulative distributions of fundamentals follow a power law. We find that the power laws for stock prices and for fundamentals hold for the 10-year period of our study, and that the estimated values of the power law exponents are close to unity. Furthermore, we illustrate that the tail distribution of fundamentals closely matches the tail distribution of stock prices. On these grounds, we conclude that the power law for stock price is caused by the power law behavior of the fundamentals.     

Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a “direction of reflection” for each of the quadrant’s two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x→∞, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein  (Queueing Syst.: Theory Appl. 37, 259–289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

An analysis of moisture diffusion according to Fick’s law and the tensile mechanical behavior of a glass-fabric-reinforced composite

The properties of moisture diffusion parameters and their effect on the tensile mechanical behavior of a fabric composite (glass fiber/epoxy resin) in the warp and weft directions were investigated. The water up take by specimens conditioned in a humid environment under different relative humidities (0, 60, and 96% RH) at a constant temperature of 60°C was evaluated by weight gain measurements. The water absorption followed Fick’s diffusion law in the fabric composite. A comparison between the values obtained for the moisture diffusion coefficient and the equilibrium moisture content at the laboratory and those given by Loos and Springer showed that the parameters depended not only on the nature of materials, but also on environmental conditions. The effect of moisture absorption on tensile characteristics of the composite, which was tested in uniaxial tension in the warp and weft directions at constant imposed displacement rates up to failure, showed a significant reduction in the ultimate tensile strength of the specimens conditioned at 96% RH. Russian translation in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 43, pp. 479-488, May-June, 2009.     

Power series solutions of Tarski’s associativity law and of the cyclic associativity law

In the present paper we study the associativity law of Tarski and the cyclic associativity law. We characterize the power series solutions of these equations in the complex domain and we give the connection to the solutions of the ordinary associativity equation.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

Asymptotic behavior of an eigenvalue of the two-particle discrete Schr?dinger operator

We consider a two-particle discrete Schr?dinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential. We show that there is a unique eigenvalue below the bottom of the essential spectrum for all values of the coupling constant and two-particle quasimomentum. We obtain a convergent expansion for the eigenvalue.