Asymptotic Representation Research Articles

Evaluation of Effective Elastic Properties of Nitride NWs/Polymer Composite Materials Using Laser-Generated Surface Acoustic Waves

In this paper we demonstrate a high potential of transient grating method to study the behavior of surface acoustic waves in nanowires-based composite structures. The investigation of dispersion curves is done by adjusting the calculated dispersion curves to the experimental results. The wave propagation is simulated using the explicit integral and asymptotic representations for laser-generated surface acoustic waves in layered anisotropic waveguides. The analysis of the behavior permits to determine all elastic constants and effective elastic moduli of constituent materials, which is important both for technological applications of these materials and for basic scientific studies of their physical properties.

Approximations for the Likelihood Ratio Statistic for Hypothesis Testing Between Two Beta Distributions

In this paper, the likelihood ratio to test between two Beta distributions is addressed. The exact distribution of the likelihood ratio statistic, for simple hypotheses, is obtained in terms of Gamma or Generalized Integer Gamma distributions, when the first or the second of the two parameters of the Beta distributions are equal and integers. In the remaining cases addressed, near-exact or asymptotic approximations are developed for the likelihood ratio statistic. Both the exact, asymptotic or near-exact representations are obtained using a logarithm transformation of the likelihood ratio statistic and by working with the corresponding characteristic function. The numerical studies illustrate the precision of the approximations developed. Simulations are developed to analyse the power and the reproducibility probability of the tests.

Probing elastic properties of nanowire-based structures

We report the analysis of elastic properties of a composite medium consisting of GaN nanowires embedded into a dielectric matrix, which constitutes the active material of nanowire-based piezogenerators. A non-invasive technique, Transient Grating Method, is used to generate and analyze the propagation of acoustic waves in the frequency range from a few tens of MHz to several GHz. The wave propagation is simulated using the time-efficient computer model based on the explicit integral and asymptotic representations for laser-generated surface acoustic waves in layered anisotropic waveguides. Effective parameters of the multilayered model are obtained via the minimization of the discrepancy between the measured and calculated wave characteristics. A comparison with a reference composed of a dielectric matrix without nanowires shows that the mechanical behavior is strongly influenced by the presence of the nanostructures.

Covariate Distribution Balance via Propensity Scores

This paper proposes new estimators for the propensity score that aim to maximize the covariate distribution balance among different treatment groups. Heuristically, our proposed procedure attempts to estimate a propensity score model by making the underlying covariate distribution of different treatment groups as close to each other as possible. Our estimators are data-driven, do not rely on tuning parameters such as bandwidths, admit an asymptotic linear representation, and can be used to estimate different treatment effect parameters under different identifying assumptions, including unconfoundedness and local treatment effects. We derive the asymptotic properties of inverse probability weighted estimators for the average, distributional, and quantile treatment effects based on the proposed propensity score estimator and illustrate their finite sample performance via Monte Carlo simulations and two empirical applications.

A Theil-like class of inequality measures, its asymptotic normality theory and applications

In this paper, we consider a coherent theory about the asymptotic representations for a family of inequality indices called Theil-Like Inequality Measures ( TLIM ), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the element of the TLIM class in a unified approach rather than in a case by case one. The results are then applied to some UEMOA countries databases. Keywords: Inequality measure; Asymptotic behaviour; Asymptotic representations; functional empirical process AMS 2010 Mathematics Subject Classification: 62G05; 62G20; 62G07; 91B82; 62P20

Existence of a solution in the form of a moving front of a reaction-diffusion-advection problem in the case of balanced advection

We consider the initial-boundary value problem for an equation of reaction-diffusion-advection type in the case when the condition of balanced advection is satisfied. We give an algorithm for constructing an asymptotic representation of a solution which has the form of a moving front, obtain the equation of motion for the point of localization of the front, and prove the existence of that solution. The proof uses the asymptotic method of differential inequalities.

Rapidly varying solutions of a second-order differential equation with regularly and rapidly varying nonlinearities

For a second-order differential equation whose the right-hand side contains the sum of terms with regularly and rapidly varying nonlinearities, we establish the conditions of existence and asymptotic representations of rapidly varying solutions, as the argument tends to a singular point.

Fluctuations of particle systems determined by Schur generating functions

We develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems. We introduce and study the notion of the Schur generating function of a random discrete configuration. Our main result provides a Central Limit Theorem (CLT) for such a configuration given certain conditions on the Schur generating function. As applications of this approach, we prove CLT's for several probabilistic models coming from asymptotic representation theory and statistical physics, including random lozenge and domino tilings, non-intersecting random walks, decompositions of tensor products of representations of unitary groups.

Asymptotic and Spectral Analysis of a Model of the Piezoelectric Energy Harvester with the Timoshenko Beam as a Substructure

Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents Kirchhoff’s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a non-selfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order O ( 1 n ) , where n is the number of an eigenvalue. These formulas are derived for the model with variable physical parameters. The second set of the asymptotic formulas has remainder terms of the order O ( 1 n 2 ) , and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the α -branch eigenvalues and the θ -branch eigenvalues. These eigenvalues being multiplied by “i” produce the set of the vibrational modes of the system. The α -branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the θ -branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester.

A note on the asymptotic properties of least squares estimation in high dimensional constrained factor models

Constrained factor models proposed by Tsai and Tsay (2010) have wide potential applications. The existing asymptotic theory of the least squares estimator, however, falls short of asymptotic representations and limiting distributions, which limits the applicabilities. This paper fills this gap by explicitly giving the asymptotic representations and associated limiting distributions. Theoretical analysis indicates that the least square estimates are asymptotically biased. Bias-corrected estimators are therefore proposed. Monte Carlo simulations confirm our theoretical results.

Asymptotics of Natural Oscillations of Elastic Junctions with Readily Movable Elements

A one-dimensional model of harmonic oscillations of a junction of several thin elastic rods has been developed. In contrast to the classical model of a single rod, the constructed model of the junction is not purely differential but includes new algebraic unknowns and algebraic equations evoked by the so-called readily movable elements of the structure. The asymptotic representations have been found for frequencies and natural modes of the elastic body oscillations and estimates of asymptotic residues have been obtained.

Wave Motion due to a Ring Source in Two Superposed Fluids Covered by a Thin Elastic Plate

The problem of wave generation by a horizontal ring of wave sources of the same time-dependent strength present in any one layer of a two-layer fluid is investigated here. The upper fluid is of finite height above the interface and is covered by a floating thin infinite elastic plate (modeling a thin sheet of ice) while the lower fluid extends infinitely downwards. Assuming linear theory, the problem is formulated as an initial value problem and the Laplace transform in time is employed to solve it. For time-harmonic source strength, the asymptotic representations of the potential functions describing the motion in the two layers for large time and distance are derived. In these representations, the two different coefficients for each of the surface and interface wave modes have the same numerical values although it has not been possible to prove their equivalence analytically. This shows that the steady-state analysis of the potential functions produces outgoing progressive waves at the surface and at the interface. The forms of the surface and interface waves are depicted graphically for different values of the flexural rigidity of the elastic plate and the ring source being submerged in the lower or upper layer.

Computing the Fermi−Dirac Functions by Exponentially Convergent Quadratures

Highly accurate specialized quadrature formulas are constructed for directly computing the Fermi−Dirac functions of the half-integer index. It is shown that the dependence of the error on the number of nodes is not power-law but exponential. The properties of such formulas are investigated. It is demonstrated that the factor of the convergence exponent is determined by the distance to the nearest pole of the integrand. This provides a very fast convergence of the quadratures. Simple approximations of the Fermi−Dirac functions of the integer and half-integer indices with an accuracy better than 1% are constructed; these approximations are convenient for physical estimates. In passing, asymptotic representations for Bernoulli numbers are found.

Asymptotic mean value properties for fractional anisotropic operators

We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain–Brezis–Mironescu type limit formula for a related class of anisotropic nonlocal norms.

Existence and Asymptotic Representation of the Autowave Solution of a System of Equations

A singularly perturbed parabolic system of nonlinear reaction–diffusion equations is studied. Systems of this class are used to simulate autowave processes in chemical kinetics, biophysics, and ecology. A detailed algorithm for constructing an asymptotic approximation of a travelling front solution is proposed. In addition, methods for constructing an upper and a lower solution based on the asymptotics are described. According to the method of differential inequalities, the existence of an upper and a lower solution guarantees the existence of a solution to the problem under consideration. These methods can be used for asymptotic analysis of model systems in applications. The results can also be used to develop and justify difference schemes for solving problems with moving fronts.

Stability of the Reconstruction Discontinuous Sturm-Liouville Problem

In this work, we study stability of the inverse spectral problem for the Sturm-Liouville operator -D²+q with discontinuity boundary conditions inside a finite closed interval. We use the method which is given by Ryabushko for regular Sturm-Liouville operator in ryb to obtain stability results. These results give a bound for the difference between the spectral functions of associated problems. In addition, we give asymptotic representation of the eigenvalues and a formula for the representation of the norming constants by two spectra.

Asymptotic representation of solutions for second-order impulsive differential equations

We obtain sufficient conditions which guarantee the existence of a solution of a class of second order nonlinear impulsive differential equations with fixed moments of impulses possessing a prescribed asymptotic behavior at infinity in terms of principal and nonprincipal solutions. An example is given to illustrate the relevance of the results.

Asymptotic representations of solutions of nonlinear two term difference equations

In this paper we establish the necessary and sufficient conditions for existence of solutions of the nonlinear two term difference equation where and are regularly varying functions, Also we obtain their asymptotic representations as Since and could be more complicated than power functions and for proving the main result we formulate and prove an additional statement about existence of vanishing solutions of quasilinear system of difference equations.

Extension of the effective relativistic coupling by asymptotic representation (ERCAR) approach to multi-dimensional potential energy surfaces: 3D model for CH3I

The Effective Relativistic Coupling by Asymptotic Representation (ERCAR) approach is a new method developed by us over the past few years that allows for the accurate diabatic representation of a molecular Coulomb and spin-orbit Hamiltonian and yields an analytic potential energy surface (PES) model for use in quantum dynamics simulations. So far, we focused on the single one dissociation coordinate defining the asymptote for diabatic representation and corresponding to removing a single, strongly relativistic atom from the remaining fragment. In the present study, we extend this approach to multiple dimensions for the first time. To this end, a 3D PES model is developed for the methyl iodide (CH3I) system accounting for all totally symmetric coordinates (C–I stretch, CH3 umbrella, and CH3 breathing modes). The model parameters are fitted with respect to high-level ab initio reference data for the spin space (“spin-free”) states which are reproduced with very good accuracy. The ERCAR method also yields the fine structure states and energies which are not computed ab initio. This is particularly important for the 1Q1 and 3Q0 fine structure states of CH3I which form an intersection that is considered key for the photodissociation dynamics of the system. Our new model shows that this intersection is considerably curved in the 2D subspace of the C–I stretch and CH3 umbrella coordinate. This will certainly affect the complicated nonadiabatic photodissociation dynamics of CH3I. The construction of a full 9D diabatic PES model is currently in progress.

Asymptotic Behavior of the Solutions of Ordinary Second-Order Differential Equations with Rapidly Varying Nonlinearities

We establish conditions for the existence of one class of solutions of two-term nonautonomous second-order differential equations with rapidly varying nonlinearities and obtain the asymptotic representations for these solutions and their first-order derivatives as t ↑ ω(ω ≤ + ∞).