Search

This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

We study the dynamics of a two-level crossing model with a polynomial modification of the linear Landau–Zener tunneling. For a cubic modification, we express the non-adiabatic transition amplitudes analytically via the bi-confluent Heun functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations in a large part of the parameter space. The analytical approximations are validated by comparison with numerical results. Finally, we discuss the application to fast high fidelity quantum gates in quantum computing. [ABSTRACT FROM AUTHOR]

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in (Ann Inst H Poincaré Anal Non Linéaire, 4(1):1–47 1987). We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work. [ABSTRACT FROM AUTHOR]

The paper investigates the interaction between the notions of expansiveness and admissibility. We consider a polynomially bounded discrete evolution family and define an admissibility notion via the solvability of an associated difference equation. Using the admissibility of weighted Lebesgue spaces of sequences, we give a characterization of discrete evolution families which are polynomially expansive and also those which are expansive in the ordinary sense. Thereafter, we apply the main results in order to infer continuous-time characterizations for the notions of expansiveness through the solvability of an associated integral equation. [ABSTRACT FROM AUTHOR]

We consider a class of power-logarithmic germs. We call them parabolic Dulac germs, as they appear as Dulac germs (first-return germs) of hyperbolic polycycles. In view of formal or analytic characterization of such a germ f by fractal properties of several of its orbits, we study the tubular ε -neighborhoods of orbits of f with initial points x 0 . We denote by A f (x 0 , ε) the length of such a tubular ε -neighborhood. We show that, even if f is an analytic germ, the function ε ↦ A f (x 0 , ε) does not have a full asymptotic expansion in ε in the scale of powers and (iterated) logarithms. Hence, this partial asymptotic expansion cannot contain necessary information for analytic classification. In order to overcome this problem, we introduce a new notion: the continuous time length of the ε -neighborhood A f c (x 0 , ε) . We show that this function has a full transasymptotic expansion in ε in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length ε ↦ A f (x 0 , ε) . Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f. [ABSTRACT FROM AUTHOR]

High accuracy analytic functions have been determined for the ground state eigenvalue of the quantum anharmonic potential x4+λx6. The procedure here used is a new application of the multipoint quasi-rational approximation method (MPQA) to quantum mechanics. Previous applications of the technique were performed in the case where the eigenvalues of the potential were already known for the particular case of λ=0. The extension of the method, where this condition is not accomplished, is presented here. As a first step, power and asymptotic expansions have been now also determined. This is followed by the search of a bridge function connecting both expansions, and later the determination of the parameters. Accuracies smaller than 0.1% have been found with only nine parameters.

For the second-order nonlinear ordinary differential equation u xx ' ' = u 5 − tu 3 − x , we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5− tu3− x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2→∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients. [ABSTRACT FROM AUTHOR]

We analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients. [ABSTRACT FROM AUTHOR]

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class of positive integer-valued distributions with power tails. [ABSTRACT FROM AUTHOR]

Linear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z). [ABSTRACT FROM AUTHOR]

The Painlevé-III equation with parameters Θ 0 = n + m and Θ ∞ = m - n + 1 has a unique rational solution u (x) = u n (x ; m) with u n (∞ ; m) = 1 whenever n ∈ Z . Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of u n (x ; m) in the limit n → + ∞ with m ∈ C held fixed. We isolate an eye-shaped domain E in the y = n - 1 x plane that asymptotically confines the poles and zeros of u n (x ; m) for all values of the second parameter m. We then show that unless m is a half-integer, the interior of E is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of E but blows up near the origin, which is the only fixed singularity of the Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ for u n (x ; m) that we compare with u n (x ; m) itself for finite values of n to illustrate their accuracy. We also consider the exceptional cases where m is a half-integer, showing that the poles and zeros of u n (x ; m) now accumulate along only one or the other of two "eyebrows," i.e., exterior boundary arcs of E. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak asymptotic in the C -plane of some wave functions resulting from the WKB-techniques applied to a Schrödinger equation with quartic oscillator and having some boundary condition. As a first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions. Especially, we investigate the properties of the Cauchy transform of the root counting measure of re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form C 2 (z) + r (z) C (z) + s (z) = 0 , where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential - p (z) d z 2 where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential. [ABSTRACT FROM AUTHOR]

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression. [ABSTRACT FROM AUTHOR]

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke L ‐function and a symmetric square L ‐function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half‐integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation. [ABSTRACT FROM AUTHOR]

In this study, a new technique is presented combining perturbation method and harmonic balance (HB) method and adopted to investigate a fourth order nonlinear Duffing equation with strong nonlinearity. The advantage of the proposed method is that the method can provide good results even the nonlinearity is strong. The result obtained by the proposed method is compared with the corresponding numerical results to verify the accuracy of the method. The results are also compared with the previous perturbation method results found in literature. Besides, the damped and undamped cases are studied, and the internal resonance is found for undamped case. [ABSTRACT FROM AUTHOR]

In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection. [ABSTRACT FROM AUTHOR]

Networked materials and micro-architectured systems gain increasingly importance in multi-scale physics and engineering sciences. Typically, computational intractable microscopic models have to be applied to capture the physical processes and numerous transmission conditions at singularities, interfaces and borders. The topology of the periodic microstructure governs the effective behaviour of such networked systems. A mathematical concept for the analysis of microscopic models on extremely large periodic networks is developed. We consider microscopic models for diffusion–advection–reaction systems in variational form on periodic manifolds. The global characteristics are identified by a homogenization approach for singularly perturbed networks with a periodic topology. We prove that the solutions of the variational models on varying networks converge to a two-scale limit function. In addition, the corresponding tangential gradients converge to a two-scale limit function for vanishing lengths of branches. We identify the variational homogenized model. Complex network models, previously considered as completely intractable, can now be solved by standard PDE-solvers in nearly no time. Furthermore, the homogenized coefficients provide an effective characterization of the global behaviour of the variational system. [ABSTRACT FROM AUTHOR]

It is well known that the one-sided stable matching problem ("stable roommates problem") does not necessarily have a solution. We had found that, for the independent, uniformly random preference lists, the expected number of solutions converges to e 1 / 2 as n, the number of members, grows, and with Rob Irving we proved that the limiting probability of solvability is below e 1 / 2 / 2 , at most. Stephan Mertens's extensive numerics compelled him to conjecture that this probability is of order n - 1 / 4 . Jimmy Tan introduced a notion of a stable cyclic partition, and proved existence of such a partition for every system of members' preferences, discovering that presence of odd cycles in a stable partition is equivalent to absence of a stable matching. In this paper we show that the expected number of stable partitions with odd cycles grows as n 1 / 4 . However the standard deviation of that number is of order n 3 / 8 ≫ n 1 / 4 , i.e. too large to conclude that the odd cycles exist with probability 1 - o (1) . Still, as a byproduct, we show that with probability 1 - o (1) the fraction of members with more than one stable "predecessor" is of order n - 1 / 2 + o (1) . Furthermore, with probability 1 - o (1) the average rank of a predecessor in every stable partition is of order n 1 / 2 . The likely size of the largest stable matching is n / 2 - O (n 1 / 4 + o (1)) , and the likely number of pairs of unmatched members blocking the optimal complete matching is O (n 3 / 4 + o (1)) . [ABSTRACT FROM AUTHOR]

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.

We derive a time scale version of the well-known result from the theory of regular variation, namely the Karamata integration theorem. We show an application of this theorem in asymptotic analysis of linear second order dynamic equations. We obtain a classification and asymptotic formulae for all (positive) solutions, which unify, extend, and improve the existing results. In addition, we utilize these results, in combination with a transformation between equations on different time scales, to study the critical double-root case in linear difference equations. This leads to solving open problems posed in the literature. [ABSTRACT FROM AUTHOR]

It is well known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemannian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. The purpose of this work is to show that, in certain circumstances, termwise integration can be used to obtain the asymptotic expansion of the heat kernel trace for Laplace operators endowed with a suitable polynomial potential on unbounded domains. This is achieved by utilizing a resummed form of the asymptotic expansion of the on-diagonal heat kernel. [ABSTRACT FROM AUTHOR]

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials Ln(α)(x), as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of Ln(α)(x) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided. [ABSTRACT FROM AUTHOR]

We investigate the large N behavior of the smallest eigenvalue, λ N , of an ( N + 1 ) × ( N + 1 ) Hankel (or moments) matrix H N , generated by the weight w ( x ) = x α ( 1 − x ) β , x ∈ [ 0 , 1 ] , α > − 1 , β > − 1 . By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials P n ( z ) , z ∈ C ∖ [ 0 , 1 ] , associated with w ( x ) , which are required in the determination of λ N . Based on this formula, we produce the expressions for λ N , for large N . Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large N . [ABSTRACT FROM AUTHOR]

In this paper, we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order 2-α and second-order approximations of the Caputo derivative by modifying the weights of the shifted Grünwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Grünwald-Letnikov approximation is obtained which allows second-order numerical solutions of fractional differential equations with arbitrary values of the solutions and their first derivatives at the initial point. [ABSTRACT FROM AUTHOR]

The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting ‘library’ of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here. [ABSTRACT FROM AUTHOR]

Considering a projectively invariant metric on a strongly convex bounded domain, we study the asymptotic expansion of the scalar curvature with respect to the distance function, and use the Fubini-Pick invariant to describe the second term in the expansion. In particular for the two-dimensional convex domain, we also show that the third term in the expansion is zero. [ABSTRACT FROM AUTHOR]

This paper formulates two 3D models using stochastic differential equations (SDEs) of two microbial populations in a chemostat competing over a single substrate. These models have two distinct noise sources. One is general noise whereas the other is dilution-rate-induced noise. Nonlinear Monod growth rates are assumed and the paper is mainly focused on the parameter values where coexistence is found in the deterministic model. Nondimensionalising the equations around the point of intersection of the two growth rates identifies the dimensionless substrate feed as a large parameter. This in turn is used to perform an asymptotic analysis leading to a reduced 2D system of equations describing the dynamics of the populations on and close to a line of steady states obtained previously from the deterministic stability analysis. That reduced system allows the formulation of a spatially 2D Fokker-Planck equation which, when solved numerically, admits results similar to those from the SDEs. Contrary to previous suggestions, one particular population becomes dominant at large times. Finally, we briefly explore the case where death rates are included. [ABSTRACT FROM AUTHOR]

We introduce the periodic Airy–Schrödinger operator and we describe its band spectrum. This is an example of solvable model with a periodic potential which is not differentiable at its extrema. We prove that there exists a sequence of explicit constants giving upper bounds of the semiclassical parameter for which explicit estimates are valid. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, we get estimates of the edges of spectral bands and thus of the widths of spectral bands and spectral gaps which are stated for values of the semiclassical parameter in fixed intervals. [ABSTRACT FROM AUTHOR]

Abstract: The present problem was first studied by Schneider, JFM 726 (2013). An asymptotic analysis was performed for small slope of the plane channel bottom and slightly supercritical, fully‐developed flow far upstream and far downstream. Turbulence modelling can be circumvented. The asymptotic analysis yields an extended Korteweg–de Vries (KdV) equation for the surface elevation. The stationary leading‐order solutions for weak dissipation have the shape of solitons in inviscid flow. Stationary solitary waves are permitted only if the surface roughness is varied along the bottom. The particular example was examined where the bottom roughness in a region is larger by a constant in comparison with the roughness in fully‐developed flow. This enlargement was considered as an eigenvalue to obtain solitary waves with strongly decaying elevations. In the present paper, the analysis is extended up to the second order to describe how the shape and the structure of the solitary waves is affected by the dissipation terms. In particular, the cases are considered when the enlargement of the bottom roughness differs from the eigenvalue. The surface elevation far downstream is then characterized by a long shallow tail that is tackled by means of matched asymptotic expansions. The required minimum enlargement of the roughness for the existence of solitary waves is given. The asymptotic solutions for the shape of the surface elevation, the height and location of the wave crest, and the eigenvalue are in good agreement with numerical results. The existence and shape of a novel kind of stationary solutions is also shown. [ABSTRACT FROM AUTHOR]