Recent zbMATH articles in MSC 41A60https://www.zbmath.org/atom/cc/41A602022-05-16T20:40:13.078697ZWerkzeugAsymptotics of compound meanshttps://www.zbmath.org/1483.260262022-05-16T20:40:13.078697Z"Hilberdink, Titus"https://www.zbmath.org/authors/?q=ai:hilberdink.titus-wIn this paper, the author studies bivariate means \(m\) and \(M\) which can be formed from sequences \(a_n, b_n\) defined recursively by \(a_{n+1} = m(a_n, b_n)\), \(b_{n+1} = M(a_n, b_n)\) with \(a_0, b_0 > 0\). He investigates under mild conditions when these means will converge to a new mean \(\mathcal{M}(a_0,b_0)\), called a compound mean. The special case \(m\) and \(M\) are homogeneous, it is shown that \(\mathcal{M}\) is also homogeneous and satisfies a functional equation. Further, the author studies the asymptotic behaviour of \(\mathcal{M}(1, x)\) as \(x \to \infty\) given that of \(m\) and \(M\), and obtains the main term up to a possible oscillatory function. The oscillatory behaviour is also investigated if \(m\) and \(M\) are coming from some well-known classes of means. Some numerical computations are also reported which show that the oscillations are generic.
Reviewer: James Adedayo Oguntuase (Abeokuta)Design of a mode converter using thin resonant ligamentshttps://www.zbmath.org/1483.350142022-05-16T20:40:13.078697Z"Chesnel, Lucas"https://www.zbmath.org/authors/?q=ai:chesnel.lucas"Heleine, Jérémy"https://www.zbmath.org/authors/?q=ai:heleine.jeremy"Nazarov, Sergei A."https://www.zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: The goal of this work is to design an acoustic mode converter. The wave number is fixed so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory.Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-timehttps://www.zbmath.org/1483.351752022-05-16T20:40:13.078697Z"Cheng, Qiaoyuan"https://www.zbmath.org/authors/?q=ai:cheng.qiaoyuan"Fan, Engui"https://www.zbmath.org/authors/?q=ai:fan.enguiSummary: We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation
\[ u_{x t} + \alpha \beta^2 u - 2 i \alpha \beta u_x - \alpha u_{x x} - i \alpha \beta^2 | u |^2 u_x = 0\]
with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at \(k = 0\). (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone
\[ \mathcal{C} ( x_1 , x_2 , v_1 , v_2 ) = \{ ( x , t ) \in \mathbb{R}^2 | x = x_0 + v t , x_0 \in [ x_1 , x_2 ] , v \in [ v_1 , v_2 ] \},\]
the long-time asymptotic behavior of the solution \(u(x, t)\) for the focusing FL equation can be characterized with an \(N(\mathcal{I})\)-soliton on discrete spectrums and a leading order term \(\mathcal{O}( t^{- 1 / 2})\) on continuous spectrum up to a residual error order \(\mathcal{O}( t^{- 3 / 4})\). The main tool is a \(\overline{\partial} \)-generalization of the Deift-Zhou nonlinear steepest descent method.On Shallit's minimization problemhttps://www.zbmath.org/1483.371132022-05-16T20:40:13.078697Z"Sadov, S. Yu."https://www.zbmath.org/authors/?q=ai:sadov.sergey-yuSummary: In Shallit's problem [\textit{PJ. Shallit}, SIAM Rev., 36, No. 3, 490--491 (1994)], it was proposed to justify a two-term asymptotics of the minimum of a rational function of \(n\) variables defined as the sum of a special form whose number of terms is of order \(n^2\) as \(n\to\infty \). Of particular interest is the second term of this asymptotics (``Shallit's constant''). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of \(n\)-sums. In addition, a sharp exponential estimate of the remainder in Shallit's asymptotic formula is obtained.Asymptotic relations for the products of elements of some positive sequenceshttps://www.zbmath.org/1483.410112022-05-16T20:40:13.078697Z"Chmielowska, Agata"https://www.zbmath.org/authors/?q=ai:chmielowska.agata"Różański, Michał"https://www.zbmath.org/authors/?q=ai:rozanski.michal"Smoleń, Barbara"https://www.zbmath.org/authors/?q=ai:smolen.barbara"Sobstyl, Ireneusz"https://www.zbmath.org/authors/?q=ai:sobstyl.ireneusz"Wituła, Roman"https://www.zbmath.org/authors/?q=ai:witula.romanSummary: The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first \(k\) primes.Lagrangian manifolds and efficient short-wave asymptotics in a neighborhood of a caustic cusphttps://www.zbmath.org/1483.530942022-05-16T20:40:13.078697Z"Dobrokhotov, S. Yu."https://www.zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nazaikinskii, V. E."https://www.zbmath.org/authors/?q=ai:nazaikinskii.vladimir-eSummary: We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type \(A_3\) (standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.CCF approach for asymptotic option pricing under the CEV diffusionhttps://www.zbmath.org/1483.912562022-05-16T20:40:13.078697Z"Muroi, Yoshifumi"https://www.zbmath.org/authors/?q=ai:muroi.yoshifumiSummary: In the last two decades, the asymptotic expansion approach has become popular in mathematical finance because it enables us to obtain closed-form approximation formulae for many kinds of options within various kinds of financial models, such as local and stochastic volatility models. In this study, we propose an asymptotic expansion formula for the option price in a constant elasticity of variance model using the asymptotic expansion technique and Fourier analysis. This approach enables us to derive the higher order terms using only algebraic computation. Furthermore, this method enables us to derive not only the price of European options but also the price of options with an early exercise feature, such as Bermudan options and American options.