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We consider the Poincaré's classical problem of approximation for second order linear differential equations in the class of almost periodic type functions. We obtain an explicit form for solutions of these equations by studying a Riccati equation associated with the logarithmic derivative of a solution. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions of the Riccati equation. A decomposition property of the perturbations induces a decomposition on the Riccati equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati equation.

We consider a Hankel determinant formula for generic solutions of the Painlev\'e IV equation. We show that the generating functions for the entries of the Hankel determinants are related to the asymptotic solution at infinity of the isomonodromic problem. Summability of these generating functions are also discussed., Comment: 17 pages; Corrected some errors

In this paper, we consider one-dimensional adiabatic quasi-periodic Schrodinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.

We construct generating functions for the entries of Hankel determinant formula for the Okamoto polynomials which characterize a class of rational solutions to the Painlevé IV equation. Generating functions are characterized as asymptotic expansions of log derivative of Ai and Bi, which are solutions of the Airy equation.

On s'intéresse à la formule déterminant de Hankel pour les solutions génériques de l'équation de Painlevé II. On établit une relation reliant les fonctions génératrices des coefficients des déterminants de Hankel aux solutions asymptotiques à l'infini du problème linéaire dont les déformations isomonodromiques sont décrites par cette équation de Painlevé II., In this paper we consider a Hankel determinant formula for generic solutions of the Painlevé II equation. We show that the generating functions for the entries of the Hankel determinants are related to the asymptotic solution at infinity of the linear problem of which the Painlevé II equation describes the isomonodromic deformations.

This report is devoted to a class of singularly perturbed differential equations which cannot be treated in the frame of the standard theory of singularly perturbed problems for ordinary as well as for partial differential equations. This new class of problems is called as it singularly perturbed problems in case of exchange of stabilities. The main results are concerned with the asymptotic behavior of the solutions of initial value, boundary value and initial-boundary value problems as the perturbation parameter tends to zero. The report is based essentially on papers of the authors and their colleagues published in the last six to seven years in the frame of cooperation between the Department of Mathematics of the Faculty of Physics of the Lomonosov Moscow State University and the Department of Dynamical Systems of the Weierstrass Institute for Applied Analysis and Stochastics in Berlin.

Assume that $D\subset \R^2$ is a strictly convex domain with $C^2-$smooth boundary. {\bf Theorem.} {\em If $\int_De^{ix}y^ndxdy=0$ for all sufficiently large $n$, then $D$ is a disc.}

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.Let ℜ be an expansion of the real field (ℝ, +, ·).A differentiable map F = (F1,…, F1): (a, b) → ℝi is ℜ-Pfaffian if there exists G: ℝ1+l → ℝl definable in ℜ such that F′(t) = G(t, F(t)) for all t ∈ (a, b) and each component function Gi: ℝ1+l → ℝ is independent of the last l − i variables (i = 1, …, l). If ℜ is o-minimal and F: (a, b) → ℝl is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝl is ultimately ℜ-Pfaffian if there exists r ∈ ℝ such that the restriction F ↾(r, ∞) is ℜ-Pfaffian. (In general, ultimately abbreviates “for all sufficiently large positive arguments”.)The structure ℜ is closed under asymptotic integration if for each ultimately non-zero unary (that is, ℝ → ℝ) function f definable in ℜ there is an ultimately differentiable unary function g definable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and defines ex: ℝ → ℝ (Proposition 2).Note that the above definitions make sense for expansions of arbitrary ordered fields.

The renormalization group method of Chen, Goldenfeld, and Oono offers a comprehensive approach to formally computing asymptotic expansions of the solutions to singular perturbation problems and multi-scale problems. In particular, the RG method applies to a broad array of problems customarily treated with disparate methods, such as the method of multiple scales, boundary layer theory, matched asymptotic expansions, Poincare-Lindstedt theory, the WKBJ method with and without turning points, the method of averaging, and others. For problems in which the expansions are in powers of the small parameter, it has been shown that the RG method leads to uniformly valid asymptotic expansions of the solutions. In addition, it has been shown that the RG condition constitutes an invariance condition, that the RG method is effectively a resummation technique, and that it is equivalent to normal form theory for certain broad classes of perturbed ordinary differential equations. However, there has not yet been an analysis of the validity of the RG method, i.e.,i> a demonstration that it leads to uniformly valid asymptotic expansions, for problems in which the expansions also involve logarithms of the small parameter as gauge functions. The aim of this article is to provide this justification of the RG method. In particular, we extend the approach developed in DeVille, et ali>. Physica D 237 no. 8, 1029 {-}1052] from the classes of autonomous and non-autonomous perturbations considered there to include the non-autonomous systems subject to singular perturbations for which the solutions involve logarithmic gauge functions. This framework is built upon the relationship between the RG method and normal form theory. We apply the RG method to three successively-more complex examples and also elucidate the common general features.

The Blasius function, denoted by f, is the solution to a simple nonlinear boundary layer problem, a third order ordinary differential equation on $x \in [0, \infty]$. In this work, we calculate several numerical constants, such as the second derivative of f at the origin and the two parameters of the linear asymptotic approximation to f, to at least eleven digits. Although the Blasius function is unbounded, we nevertheless derive an expansion in rational Chebyshev functions $\TL_{j}$ which converges exponentially fast with the truncation, and tabulate enough coefficients to compute f and its derivatives to about nine decimal places for all positive real x. The power series of f has a finite radius of convergence, ut the Euler-accelerated expansion is apparently convergent for all real x. We show that the singularities, which are first order poles to lowest order, have an infinite series of cosine-of-a-logarithm corrections. Lastly, we chart the behavior of f in the complex plane and conjecture that all singularities lie within three narrow sectors.

We consider Markovian many-server systems with admission control operating in a QED regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold., Comment: 36 pages, 7 figures

We study the existence of solutions of a class of singularly perturbed BVPs $\varepsilon y'' +2y' +f(y) =0, ~y(0)=0, ~y(A)=0$ for some $A>0$ and $f>0$. Given an $f$ satisfying certain conditions, we will show that for each $\varepsilon>0$, there exists $A(\varepsilon)$ such that, if $0 < A < A(\varepsilon)$, then the problem has exactly two solutions. As an example we consider $f(y)=e^y$, where we show that $A(\varepsilon)>1$ for $\varepsilon>0$ sufficiently small. Both solutions have the same outer solution on $(0,1]$, but have different boundary behavior near $x=0$. One of them is bounded, the other unbounded as $\varepsilon \to 0$. A uniform expansion for the smaller solution has been obtained from matched asymptotic expansions. We prove rigorously that these asymptotics are correct up to $O(\varepsilon).$ We also prove that for a class of functions, $\underset{\varepsilon \to 0}{\lim} A(\varepsilon) = 2\int_0^{\infty} dy/f(y)$.

We consider a hybrid diffusion process that is a combination of two Ornstein-Uhlenbeck processes with different restraining forces. This process serves as the heavy-traffic approximation to the Markovian many-server queue with abandonments in the critical Halfin-Whitt regime. We obtain an expression for the Laplace transform of the time-dependent probability distribution, from which the spectral gap is explicitly characterized. The spectral gap gives the exponential rate of convergence to equilibrium. We further give various asymptotic results for the spectral gap, in the limits of small and large abandonment effects. It turns out that convergence to equilibrium becomes extremely slow for overloaded systems with small abandonment effects., 48 pages

A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions ($\Lambda$-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size $n$ tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen-Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen--Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that $n\ge 4$.

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and higher-order scalar equations where growth behavior is expressed in terms of elementary functions. We then recast our equations in mollified form - thereby obtaining stability., Comment: 12 pages

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.

We study the time decay of scattering solutions to one-dimensional Schrodinger equations and prove a weighted dispersive estimate with stronger time decay than the case of unweighted estimates for the non-resonant state. Furthermore asymptotic expansions in time of scattering solutions are given. The key of the proof is the study of the Fourier properties of the Jost functions. We improve the Fourier properties of the Jost functions obtained by D'Ancona and Fanelli [2].

We present a sufficient condition of existence of asymptotic expansion in negative power series for a function defined by Taylor series and unitary formulas for coefficients of this expansion. An example of computing scheme for arctangent function is represented., 25 pages, no figures. Some new results added

We study a nonlocal parabolic problem modeling the temperature in a thin region during linear friction welding for a hard material. We derive the structures of steady states of this nonlocal problem and its associated approximated problems. Moreover, some remarks on the parabolic problem are given.

An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n +\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape and in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and analysis of algorithms, and based on the psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed and several attractive phenomena are discovered.