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" Three-layer slab waveguide with a graded-index film and a nonlinear substrate is investigated. The nonlinear substrate is considered of Kerr-type and the film index is assumed to change linearly as we move from the film-clad to the film- substrate interfaces. The solutions of the fields in the guiding film contains Airy functions which can be written as Bessel functions of order 1/3. The dispersion relation is derived in terms of three well-known normalized parameters V, a and b and the dispersion properties are plotted and studied. Many interesting features are obtained such as the nonexistence of cut-off thickness due to the nonlinear substrate and the value of the normalized guide index does not exceed unity which means they correspond to guided modes." " Three-layer slab waveguide with a graded-index film and a nonlinear substrate is investigated. The nonlinear substrate is considered of Kerr-type and the film index is assumed to change linearly as we move from the film-clad to the film- substrate interfaces. The solutions of the fields in the guiding film contains Airy functions which can be written as Bessel functions of order 1/3. The dispersion relation is derived in terms of three well-known normalized parameters V, a and b and the dispersion properties are plotted and studied. Many interesting features are obtained such as the nonexistence of cut-off thickness due to the nonlinear substrate and the value of the normalized guide index does not exceed unity which means they correspond to guided modes."

The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially two-dimensional wave propagation in the earth-ionosphere wave path using k-symbol Airy functions (KAFs) in the open unit disk. It is shown that the standard wave-mode working formula may be determined by orthogonality considerations without the use of intricate justifications of the complex plane. By taking into account the symmetry-convex depiction of the KAFs, the formula combination is derived.

The Schrödinger equation under the application of a position-dependent mass (PDM) with an exponential form is presented. Several physical models are carried out by choosing different external potential fields including the free field or a confined hard-all potential, the linear potential plus an attractive centrifugal-like term and harmonic oscillator. The eigenfunction of the first case is given by a Bessel function. The calculations of the second case are found to have the Airy function and we are able to get a general form of energy levels based on the zeros of Airy function of the first kind. The last case is found that the eigenfunctions are given by the popular associated Laguerre function. To provide a better physical insight into the solutions, some figures are plotted graphically.

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver. Financial support from Ministerio de Ciencia e Innovación, Spain, projects MTM2015-67142-P (MINECO/FEDER, UE) and PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE) is acknowledged

This paper is a contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev. The full collection is available at https://www.emis.de/journals/SIGMA/Kapaev.html In this paper we obtain large z asymptotic expansions in the complex plane forthe tau function corresponding to special function solutions of the Painlevé II differentialequation. Using the fact that these tau functions can be written as n × n Wronskiandeterminants involving classical Airy functions, we use Heine's formula to rewrite them asn-fold integrals, which can be asymptotically approximated using the classical method ofsteepest descent in the complex plane. The author acknowledges financial support from the EPSRC grant "Painlevé equations: analytical properties and numerical computation", reference EP/P026532/1, and from the project MTM2015-65888-C4-2-P from the Spanish Ministry of Economy and Competitivity. The author wishes to thank M. Fasondini, D. Huybrechs, A. Iserles, A.R. Its, A.B.J. Kuijlaars, A.F. Loureiro, C. Pechand W. Van Assche for stimulating discussions on the topic and scope of this paper, as well as the organisers of the workshop "Painlevé Equations and Applications" held at the University of Michigan, August 25-29, 2017, for their hospitality. The comments, remarks and corrections of the anonymous referees have lead to an improved version of the paper, and they are greatly appreciated. Publicado

O objetivo principal desse trabalho foi reunir material, de forma didática, sobre a propagação horizontal das ondas de gravidade internas que existem no interior de oceanos estavelmente estratificados. Em particular, a presença de nıveis de transição onde a frequência da onda iguala, localmente, à frequência de Brunt-Vaissalla (N) é considerada usando separadamente dois modelos, tratados com o método dos modos normais verticais. A primeira simulação é numérica e supõe a existência de uma termoclina; foi baseada indiretamente em dados para o Canal de Faeroe-Shetland, um cânion submarino que se estende de (61,29 N, 4,03 E) à (61,40 N, 6,13 E). A segunda simulação considera um perfil linearmente decrescente de N² com a profundidade e é resolvido analiticamente (exceto pela determinação dos autovalores) envolvendo a aproximação clássica de WKB e o uso das funções de Airy. The main objective of this work was to amass didatical material on the horizontal propagation of internal gravity waves within stably stratified oceans. In particular, the presence of turning levels, where the wave frequency locally equals the Brunt-Vaissalla frequency (N) is examined using two distinct models, both using the vertical normal modes technique. The first simulation is numerical and concerns a thermocline in the Faeroe-Shetland Channel extending from (61.29 N, 4.03 E) to (61,40 N, 6,13 E). The second simulation uses a linearly decreasing profile of N² with respect to depth and is solved analytically (except for the eigenvalue calculation) with the classical WKB approximation and use of Airy functions.

Riesz fractional derivatives of a function, D"x^@af(x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, D"x^@a{Ai(x)Bi(x)} and D"x^@a{Ai^2(x)}, via D"x^@aAi(x) and D"x^@aGi(x). Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A"2(a,b;x)=Ai(x-a)Ai(x-b) with a,[email protected]?R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of D"x^@a{A"2(a,b;x)}. These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).

Guided-wave theory is well established, with wonderful canonical examples and classical lecture notes. Its beauty and strength comes from mathematical derivations. Guided-wave problems can be put into two groups: eigenvalue problems (i.e., the effects of the environment on the propagating waves), and the Green's function problem (the effects of the excitation on the environment). This tutorial reviews guided-wave theory in two dimensions, with characteristic examples and numerical simulations. The aim is to present alternative notes with simulation packages for both undergraduate- and graduate-level guided-wave-theory courses.

In this paper we consider fractional higher-order stochastic differential equations of the form \[ \left ( \mu + c_\alpha \frac{d^\alpha } {dt^\alpha } \right )^\beta X(t) = \mathcal{E} (t) , \quad \mu >0,\; \beta >0,\; \alpha \in (0,1) \cup \mathbb{N} \] where $\mathcal{E} (t)$ is a Gaussian white noise. We obtain explicitly the covariance functions and the spectral densities of the stochastic processes satisfying the above equations.

The double Laplace transform of the distribution function of the integral of the positive part of the Brownian bridge was determined by M. Perman and J.A. Wellner, as well as the moments of this distribution. The purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument, and the corresponding asymptotics of the moments.

Some of the work on the construction of inequalities and asymptotic approximations for the zeros λn,k(α), k = 1,2 .... ,n, of the Laguerre polynomial Lnα(x) as v = 4n + 2α + 2 → ∞, is reviewed and discussed. The cases when one or both parameters n and α unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function Jα(x), and the second is in terms of the zeros of the Airy function Ai(x). They hold for k= 1,2 .... , [qn] and for k = [pn], [pn] + 1 ..... n, respectively, where p and q are fixed numbers in the interval (0, 1 ). Numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas.

The conditional Feynman–Kac functional is used to derive the Laplace transforms of conditional maximum distributions of processes related to third- and fourth-order equations. These distributions are then obtained explicitly and are expressed in terms of stable laws and the fundamental solutions of these higher-order equations. Interestingly, it is shown that in the third-order case, a genuine non-negative real-valued probability distribution is obtained.

The polynomials Pn and Qm, having degrees n and m respectively, with Pn monic, that solve the approximation problem will be investigated for their asymptotic behaviour, in particular in connection with the distribution of their zeros. The symbol O means that the left-hand side should vanish at the origin at least to the order n+m+1. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main apprcudmants. We shall focus on the important diagonal case when n = m and the polynomials Pn and Qm, as well as the remainder En, n, (z) = Pn (z)e−z + Qn (z) can be expressed in terms of Hankel and Bessel functions. The approximate location of the zeros of Pn, Qm and En, n are given in terms of the known zeros of certain Airy functions. An application is given in which the asymptotic information on the zeros is used to obtain an estimate in an approximation of the unit block function by means of the polynomials Pn, Qn.

It was shown in Groeneboom (1983) that the least concave majorant of one-sided Brownian motion without drift can be characterized by a jump process with independent increments, which is the inverse of the process of slopes of the least concave majorant. This result can be used to prove the result in Sparre Andersen (1954) that the number of vertices of the smallest concave majorant of the empirical distribution function of a sample of size $n$ from the uniform distribution on $[0,1]$ is asymptotically normal, with an asymptotic expectation and variance which are both of order $\log(n)$. A similar (Markovian) inverse jump process was introduced in Groeneboom (1989), in an analysis of the least concave majorant of two-sided Brownian motion with a parabolic drift. This process is quite different from the process for one-sided Brownian motion without drift: the number of vertices in a (corresponding slopes) interval has an expectation proportional to the length of the interval and the variance of the number of vertices in such an interval is about half the size of the expectation, if the length of the interval tends to infinity. We prove an asymptotic normality result for the number of vertices in an increasing interval, which translates into a corresponding result for the least concave majorant of an empirical distribution function of a sample of size $n$, generated by a strictly concave distribution function. In this case the number of vertices is of order cube root $n$ and the variance is again about half the size of the asymptotic expectation. As a side result we obtain some interesting relations between the first moments of the number of vertices, the square of the location of the maximum of Brownian motion minus a parabola, the value of the maximum itself, the squared slope of the least concave majorant at zero, and the value of the least concave majorant at zero. An erratum is available in EJP volume 18 paper 46 .

In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{N}, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes \bm{S}_n^{2\beta} whose random time is represented by the inverse \mathpzc{L}^\nu (t), t>0, of the superposition of independent positively-skewed stable processes, \mathpzc{H}^\nu (t) = H_1^{2\nu} (t) + (2\lambda \r^{\frac{1}{\nu}} H_2^\nu (t), t>0, (H_1^{2\nu}, H_2^\nu, independent stable subordinators). As special cases for n=1, \nu = 1/2 and \beta = 1 we examine the telegraph process T at Brownian time B (Orsingher and Beghin) and establish the equality in distribution B (c^2 \mathpzc{L}^{1/2} (t)) \stackrel{\textrm{law}}{=} T (|B(t)|), t>0. Furthermore the iterated Brownian motion (Allouba and Zheng) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stable-distributed time., Comment: 34 pages

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_m$ related to the higher-order heat-type equation with positively skewed stable r.v.'s $T^j_{1/3}$, $j=1,2, ..., n$ we obtain genuine r.v.'s whose explicit distribution is given for $n=3$ in terms of Cauchy asymmetric laws. We also prove that $X_3(T^1_{1/3}(...(T^n_{(1/3)}(t))...))$ has a stable asymmetric law.

We analyze the tail behavior of the maximum N of Brownian motion minus a parabola and give an asymptotic expansion for P(N>x) as x tends to infinity. This extends a first order result on the tail behavior, which can be deduced from Huesler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-L��f (2010)., 12 pages, submitted to the Electronic Communications in Probability

It was shown in Groeneboom (1983) that the least concave majorant of one-sided Brownian motion without drift can be characterized by a jump process with independent increments, which is the inverse of the process of slopes of the least concave majorant. This result can be used to prove the result of Sparre Andersen (1954) that the number of vertices of the smallest concave majorant of the empirical distribution function of a sample of size n from the uniform distribution on [0,1] is asymptotically normal, with an asymptotic expectation and variance which are both of order log n. A similar (Markovian) inverse jump process was introduced in Groeneboom (1989), in an analysis of the least concave majorant of two-sided Brownian motion with a parabolic drift. This process is quite different from the process for one-sided Brownian motion without drift: the number of vertices in a (corresponding slopes) interval has an expectation proportional to the length of the interval and the variance of the number of vertices in such an interval is about half the size of the expectation, if the length of the interval tends to infinity. We prove an asymptotic normality result for the number of vertices in an increasing interval, which translates into a corresponding result for the least concave majorant of an empirical distribution function of a sample of size n, generated by a strictly concave distribution function. In this case the number of vertices is of order cube root n, and the variance is again about half the size of the asymptotic expectation. As a side result we obtain some interesting relations between the first moments of the number of vertices, the square of the location of the maximum of Brownian motion minus a parabola, the value of the maximum itself, the squared slope of the least concave majorant at zero, and the value of the least concave majorant at zero., Comment: 17 pages, 4 figures, to appear in the Electronic Journal of Probability

The past twenty years have seen an explosion in the realm of infrared detection technology fueled by improvements in III-V semiconductor technology and by new semiconductor growth methods. One of the fastest growing areas of this research involves the use of bandgap engineering in order to create artificial quantum wells for use in Quantum Well Infrared Photodetectors (QWIPs). QWIPs have an advantage over other infrared detectors such as Mercury Cadmium Telluride (MCT) because they have larger bandgaps and are therefore stronger and cheaper to manufacture. This thesis introduces one method of "multi-color" detection through the use of an asymmetric quantum well structure in which all energy transitions are possible. The QWIP structure in this thesis was designed to detect a laser wavelength of 1.06 æm and a wavelength in the 8-10 æm atmospheric window. The relevance of a detector that is tuned to these wavelengths is that it can be used on military aircraft as a laser spot tracker and an infrared imager providing much greater accuracy and dependability than older systems. http://archive.org/details/twocolorphotodet109455884 Ensign, United States Navy Approved for public release; distribution is unlimited.

We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with $A^+ = \int_0^1 B^+ (t) dt$ and $A_0^+ = \int_0^1 U^+ (t) dt$, we use excursion theory to show that the Laplace transforms $\Psi^+ (s) = E \exp (-sA^+)$ and $\Psi_0^+ (s) = E \exp (-sA_0^+)$ of $A^+$ of $A_0^+$ satisfy $$\int_0^{\infty} e^{-\lambda s \Psi +} (\sqrt{2} s^{3/2}) ds = \frac{\lambda^{-1/2} Ai(\lambda) + (1/3 - \int_0^{\lambda} Ai(t) dt)}{\sqrt{\lambda} Ai(\lambda) - Ai (\lambda)}.$$ and $$\int_0^{\infty} \frac{e^{-\lambda s}{\sqrt{s}} \Psi_0^+ (\sqrt{2} s^{3/2}) ds = 2 \sqrt{\pi} \frac{Ai(\lambda)}{\sqrt{\lambda} Ai'(\lambda) - Ai(\lambda)}.$$ where Ai is Airy's function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takács for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the "positive part" areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, $\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)$.

We consider the integral equation p(t) = ∫ 0 t K(t−τ)p(τ)dτ+r(t) where both K(t) and r(t) behave as exp(αt3) as t → ∞ (α > 0). So straightforward application of the Laplace transform technique is not possible. By introducing a complex parameter the equation is solved in the complex domain. Analytic continuation with respect to this parameter yields the desired solution. For a particular example (which arose in a statistical problem on estimating monotone densities) we describe the construction of the explicit solution of the equation.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:24:39Z No. of bitstreams: 0 Made available in DSpace on 2014-05-27T11:24:39Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-03-01 The nonrelativistic problem of a particle immersed in a triangular potential well, set forth by N. A. Rao and B. A. Kagali, is revised. It is shown that these researchers misunderstood the full meaning of the potential and obtained a wrong quantization condition. By exploring the space inversion symmetry, this work presents the correct solution to this problem with potential applications in electronics in a simple and transparent way. © Electronic Journal of Theoretical Physics. All rights reserved. UNESP-Campus de Guaratinguetá Departamento de Física e Química, Ratinguetá SP UNESP-Campus de Guaratinguetá Departamento de Física e Química, Ratinguetá SP