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The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

Through Borel summation, one can often reconstruct an analytic solution of a problem from its asymptotic expansion. We view the effectiveness of Borel summation as a regularity property of the solution, and we show that the solutions of certain differential equation and integration problems are regular in this sense. By taking a geometric perspective on the Laplace and Borel transforms, we also clarify why ''Borel regular'' solutions are associated with special points on the Borel plane. The particular classes of problems we look at are level \\(1\\) ODEs and exponential period integrals over one-dimensional Lefschetz thimbles. To expand the variety of examples available in the literature, we treat various examples of these problems in detail.

Orthogonal time frequency space (OTFS) is a new two dimensional modulation technique that use the delay-Doppler domain. Despite the fact that OTFS can handle time varying channels and high Doppler scenarios, it has a high Peak-to-Average Power Ratio (PAPR). In this study, we developed an airy-based companding technique for the OTFS signal to address this issue. The airy function is a particular function that is commonly used in optics. We evaluate the proposed airy based companded OTFS with various companding techniques in order to assess its performance. Simulations signify that the proposed airy based companding outperforms the rooting companding, and  − law companding in terms of both PAPR and instantaneous to average power ratio (IAPR).

In field electron emission theory, evaluating the transmission coefficient D ET for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for D ET, by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electron's outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and 'flyover' (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler-Nordheim 1928 formula for D ET) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler-Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of D ET are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.

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.

In this paper the solutions $u_{ν} = u_{ν}(x, t)$ to fractional diffusion equations of order 0 < ν ≤ 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ν = \\frac{1}{2^{n}}, n \\geq 1$ , we show that the solutions $u_{1}/2^{n}$ correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $ν = \\frac{2}{3^{n}}, n \\geq 1$ , is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{ν}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{ν}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.

A bstract 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 AdS 4 black holes.

The Dirac equation for a particle – an electron with negative charge equal in magnitude to the elementary charge e 1 in spacetime with one spatial dimension denoted as the z axis and in the field of a point-like charge ( Ze 1 ) – the nucleus – with the one-dimensional potential φ 1 = – ( Ze 1 ) | z | (a one-dimensional hydrogen-like atom) is solved. The two-component wave function and the quantum values of the energy are expressed in terms of the Airy function and its zeros. Using the contact Lagrangian of the weak interaction with Fermi constant G 1 = g 1 ħc in such a space, where g 1 is some number, the probability of emission per unit time of a neutrino pair ( Ze 1 ) * →( Ze 1 ) + νν – by the one-dimensional hydrogen-like atom, which turns out to be proportional to the square of the mass, is found. Prospects for the realization of the considered effect and its possible significance at various stages of the evolution of the Universe are discussed.

A bstract We apply the conjecture of [ 1 ] for gravitational building blocks to the effective supergravity description of M-theory on S 7 / ℤ k . Utilizing known localization results for the holographically dual ABJM theory, we determine a complete tower of higher derivative corrections to the AdS 4 supergravity and a further set of quantum corrections. This uniquely fixes the gravitational block, leading to holographic predictions for a number of exact ABJM observables, excluding only constant and non-perturbative corrections in the gauge group rank N . The predicted S 3 partition function is an Airy function that reproduces previous results and generalizes them to include arbitrary squashing and mass deformations/R-charge assignments. The topologically twisted and superconformal indices are instead products of two different Airy functions, in agreement with direct numeric calculations in the unrefined limit of the former object. The general fixed-point formula for an arbitrary supersymmetric background is similarly given as a product of Airy functions.