Airy Function Research Articles

Universal objects of the infinite beta random matrix theory

We develop a theory of multilevel distributions of eigenvalues which complements Dyson’s threefold \beta=1,2,4 approach corresponding to real/complex/quaternion matrices by \beta=\infty point. Our central objects are the G \infty E ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and the Airy _{\infty} line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at \beta=\infty . We develop two points of view on these objects. The probabilistic one treats them as partition functions of certain additive polymers collecting white noise. The integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of the Airy function. We also outline universal appearances of our ensembles as scaling limits.

Generation of broadband non-diffraction millimeter-wave airy beams based on high-efficiency tri-layer metasurfaces

Airy beams based on the Airy function, exhibiting a special curved parabolic trajectory, can effectively alleviate diffraction along its propagation and hold potential in many interesting applications. However, to date, the broadband and high-efficiency generation of non-diffraction Airy beams has remained largely unexplored, especially in the millimeter-wave range that is full of great application potential. In this paper, broadband 1-D and 2-D Airy beam generators utilizing high-efficiency tri-layer metasurfaces operating from 55 to 67 GHz are presented. The metasurfaces consist of elaborately tailored meta-atoms that can independently realize complete amplitude modulation from 1% to 98% and binary phase tuning of 0 and π. The Airy beam generators are engineered and verified by simulations and experiments, which demonstrate the broadband quasi-non-diffraction feature and curved trajectory of the Airy beam. Compared to the 1-D Airy beam generator, the measured quasi-non-diffraction propagation property of the 2-D Airy beam can be maintained by greater than 160 free-space wavelengths across the entire bandwidth. Self-reconstruction property of the 2-D Airy beam is also experimentally validated in the presence of metallic and dielectric obstacles. This work may benefit novel applications of Airy beams in millimeter-wave imaging and detection.

Exact and paraxial Airy propagation of relativistic electron plasma wavepackets

The different forms of propagation of relativistic electron plasma wavepackets in terms of Airy functions are studied. It is shown that exact solutions can be constructed showing accelerated propagations along coordinates transverse to the thermal speed cone coordinate. Similarly, Airy propagation is a solution for relativistic electron plasma waves in the paraxial approximation. This regime is considered in time domain, when paraxial approximation is considered for frequency, and in space domain, when paraxial approximation is considered for wavelength. In both different cases, the wavepackets remain structured in the transverse plane. Using these solutions we are able to define generalized and arbitrary Airy wavepackets for electron plasma waves, depending on arbitrary spectral functions. Examples of this construction are presented. These electron plasma Airy wavepackets are the most general solutions of this kind.

Asymptotics of the Localized Bessel Beams and Lagrangian Manifolds

The Bessel beam-type asymptotic solutions of the three-dimensional Helmholtz equation, i.e., the solutions that have maxima in the vicinity of the -axis and are described by Bessel functions in the planes normal to it, are discussed. Since the Bessel functions slowly decrease at infinity, the energy of such solutions appears unlimited. Approaches to localizing such solutions by representing them in the form of the Maslov canonical operator on proper Lagrangian manifolds with simple caustics in the form of degenerate and nondegenerate folds are described. Efficient formulas for these solutions in the form of Bessel and Airy functions of a complex argument are obtained.

Intensity of focused waves near turning points.

A wave near an isolated turning point is typically assumed to have an Airy function profile with respect to the separation distance. This description is incomplete, however, and is insufficient to describe the behavior of more realistic wave fields that are not simple plane waves. Asymptotic matching to a prescribed incoming wave field generically introduces a phase front curvature term that changes the characteristic wave behavior from the Airy function to that of the hyperbolic umbilic function. This function, which is one of the seven classic "elementary" functions from catastrophe theory along with the Airy function, can be understood intuitively as the solution for a linearly focused Gaussian beam propagating in a linearly varying density profile, as we show. The morphology of the caustic lines that govern the intensity maxima of the diffraction pattern as one alters the density length scale of the plasma, the focal length of the incident beam, and also the injection angle of the incident beam are presented in detail. This morphology includes a Goos-Hänchen shift and focal shift at oblique incidence that do not appear in a reduced ray-based description of the caustic. The enhancement of the intensity swelling factor for a focused wave compared to the typical Airy solution is highlighted, and the impact of a finite lens aperture is discussed. Collisional damping and finite beam waist are included in the model and appear as complex components to the arguments of the hyperbolic umbilic function. The observations presented here on the behavior of waves near turning points should aid the development of improved reduced wave models to be used, for example, in designing modern nuclear fusion experiments.

Exact solutions for time-dependent complex symmetric potential well

Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.

The emergence of Airy stress function in two-dimensional disordered packings of particles

The packing of hard-core particles in contact with their neighbors is a statically determinate problem. The probability functional method allows deriving the full system of equations for the stress tensor components analytically. For an isotropic and homogeneous two-dimensional packing, we derive the classical Euler–Cauchy and Navier equations; the latter allows expressing the stress tensor of the system in terms of the Airy stress function.

ПРИБЛИЖЕННЫЕ АСИМПТОТИЧЕСКИЕ ВЫРАЖЕНИЯ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ ИСТОЧНИКОВ ВБЛИЗИ ГЛАДКОЙ ВЫПУКЛОЙ ПРОВОДЯЩЕЙ ПОВЕРХНОСТИ ВРАЩЕНИЯ

Approximate asymptotic solution is developed for the electromagnetic field due of a surface magnetic and electric currents located near an arbitrary smooth convex conducting surface of rotation of large size. The expressions of field are valid in a near-surface layer with a thickness of the order of the wavelength and have the form of a sum of a series of azimuthal harmonics and an integral over a continuous spectrum of eigenfunctions of the field. The coefficients of the series are expressed by integrals from the product of a given distribution of sources with eigenfunctions. For eigenfunctions, both general integral representations and closed expressions in terms of Airy functions are obtained, which are uniformly valid along surface of a rotation for the case of one pole by the parabolic equation method. The field expressions take into account the inseparability of the field by the sum in term of E- and H-type fields, are uniformly valid in the near-surface layer, excluding the vicinity of the poles of the rotation surface, and have no discontinuities on the caustics of surface rays. The expressions obtained were used to calculate the mutual admittances between two annular slots on a semi-infinite and smoothly truncated conical surface. Numerical results obtained by the proposed and rigorous methods for a semi-infinite cone were compared. It is shown that in the case of in-phase annular slots in a semi-infinite cone, the obtained asymptotic expression for mutual admittance coincides with the main term of the asymptotic of a strict solution based on the method of eigenfunctions.

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schrödinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schrödinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in Hloc1. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

Reconstruction of smooth shape defects in waveguides using locally resonant frequencies

This article aims to present a new method to reconstruct slowly varying width defects in 2D waveguides using locally resonant frequencies. At these frequencies, locally resonant modes propagate in the waveguide under the form of Airy functions depending on a parameter called the locally resonant point. In this particular point, the local width of the waveguide is known and its location can be recovered from boundary measurements of the wavefield. Using the same process for different frequencies, we produce a good approximation of the width in all the waveguide. Given multi-frequency measurements taken at the surface of the waveguide, we provide a -stable explicit method to reconstruct the width of the waveguide. We finally validate our method on numerical data, and we discuss its applications and limits.

An innovative and simple approach for predicting tunnelling-induced ground movements semi-analytically

ABSTRACT A semi-analytical elastic solution for predicting tunnelling-induced ground movements in clayey media is presented in this study. This study modelled the non-uniform ground movements (or non-uniform ground loss) around the tunnel cavity using the Gaussian distribution function. A new approach is proposed to model the tunnel peripheral displacements other than the simplistic trigonometric functions. It is shown that by modelling the ground loss in an elliptical fashion, (i) better prediction of ground surface settlements is obtained compared to the circular ground loss in some cases, while (ii) in other cases, circular ground loss gives better predictions. This indicates that both these approaches should be used to predict the ground surface settlements. Further, the ground displacements observed in East-West Metro, Kolkata, are discussed as a case study. The semi-analytically obtained displacements give good conformity with the observed field measurements in East-West Metro. This solution is much simpler to follow and can be easily employed in a code compared to the complex variable method or airy stress function method.

Bending effect on the energy of a particle in a semiconductor nanobelt

Semiconductor nanostructures have exhibited unique optoelectronic characteristics for a variety of applications, including flexible optoelectronic devices and systems. In this work, we analyze the effect of elastic bending on the eigenvalues (energies) of a particle in a semiconductor nanobelt in the framework of the single-band Schrödinger equation. The electronic state of the particle in the semiconductor nanobelt is described by a quantum well. Using the Airy functions, both the wave function and characteristic equation of the particle are obtained. Numerical results for an electron in a CdSe nanobelt reveal that the ground-state energy of the electron in the bent nanobelt decreases with the increase of the bending deformation and the change of the ground-state energy of the electron in the bent nanobelt decreases with the decrease of the nanobelt thickness.

Large N partition functions of the ABJM theory

We study the large N limit of some supersymmetric partition functions of the U(N)k × U(N)−k ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.

Airy stress function for proposed thermoelastic triangular elements

Airy stress function for proposed thermoelastic triangular elements

Airy function, mass spectra and decay constants of pseudo-scalar heavy-flavor mesons at N3LO level in a perturbative approach

In this paper, we incorporate three-loop contributions in the strong coupling constant and study their effects in the wave functions, masses and decay constants of heavy–light pseudo-scalar mesons (PSM) [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in V-scheme. In this work, we use the standard linear plus inverse distance QCD potential in association with Dalgarno’s method of perturbation theory for linear parent option. Detailed comparison is done with the results from Lattice QCD, QCD sum rules, light-front quark model (LFQM), relativistic harmonic confinement model (RHCM) and recent PDG data.

A developed hybrid experimental–analytical method for thermal stress analysis of a deep U-notched plate

A general framework of hybridizing experimentally-recorded load-induced thermal information by means of thermoelastic stress analysis (TSA) of a loaded structure with the Michell solution of Airy stress function in isotropic linear elasticity is proposed for in-plane stresses determination. The capability of the proposed hybrid method in separating the TSA signals into individual stresses is demonstrated by stress-analyzing a deep U-notched aluminum plate without neither knowing the entire geometry and distant loading/boundary conditions nor using supplementary experimental information. Even though no experimental data were processed at, and adjacent to, the edges of the U-notched plate, the individual stresses are determined throughout the surface of the plate. Prior processing actual experimental TSA data, the accuracy and the stability of the proposed hybrid method through adding artificial noise scatters in the processed simulated data with the number of retained terms in the generalized Airy stress expressions are firstly investigated. The effects of the superimposed noise scatters as well as the number of employed data in the evaluated stresses are also assessed. Finally, the reliability of the determined hybrid-experimental stresses is supported through a comparison with finite-element predictions and strain gauge measurements. The ease of implementation, the accuracy and the precision of the evaluated stress, and the versatility to the type of employed data make the proposed hybrid method more attractive and superior than other hybrid methods.

Extensive novel waves evolution of three-dimensional Yu–Toda–Sasa–Fukuyama equation compatible with plasma and electromagnetic applications

There is no doubt that the investigation of the interaction and propagation of plasma and electromagnetic waves play an important role in understanding these phenomena. The three-dimensional Yu–Toda–Sasa–Fukuyama equation (YTSFE) is a competent mathematical model of waves in plasma, electromagnetics, or fluids. An optimal system of infinitesimal symmetries is constructed to discover extensive and astonishingly exact solutions to the YTSFE. The outstanding solutions include periodic, polynomials, fractional, logarithmic, exponential, hyperbolic, exponential integral, Airy and complex functions. These solutions are significant because they help understand how plasma and electromagnetic applications work at different boundary or initial conditions.

The effects of the electric-magnetic barrier on the spin transmission in the multibarrier semiconductor heterostructures

In the present paper, we study spin transmission in the multibarrier semiconductor heterostructures based on single particle effective mass approximation. These structures are double-barrier and triple-barrier semiconductor hetero-structures that a metallic ferromagnetic is deposited on them. Using Airy function and magnetic barriers approximated by delta function, we calculate transmission coefficient of tunneling electrons and spin polarization. Our results have shown that the parameters as the height and width of the electrical potential barrier, wave vector parallel to the barrier, applied bias voltage and magnetic field are effective parameters in determination of the transmission coefficient.

Numerical solution of problem of equilibrium of parallelepiped in stresses

Usually, planar and spatial problems of the theory of elasticity in stresses are solved using the Airy or Maxwell stress function, respectively. Solving a plane problem usually comes down to solving a biharmonic equation. In the spatial case, it can be solved by the Castigliano variational method. This work is devoted to the mathematical and numerical solution of the spatial problem of elasticity theory directly concerning stresses. In this case, the boundary value problem consists of three equilibrium equations and three Beltrami-Michell equations with the corresponding boundary conditions. At the same time, equilibrium equations are considered as missing boundary conditions. Symmetric finite-difference equations are constructed, and the problems of equilibrium of a parallelepiped under the action of domed and uniformly distributed loads are solved numerically. The reliability of the results obtained is substantiated by comparing the numerical results with the wellknown Filonenko-Borodich solution and with the results of solving the boundary value problem of the equilibrium of the parallelepiped with respect to displacements.