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In this paper we present estimation formulas for the expectations of power means of large data and associate them with means of probability distribution and means of random sample. The proposed method follows from the asymptotic expansion of power means which is applicable for sufficiently large data and it is especially useful when value of such expectation is hard to obtain. We will show the accuracy of these approximations for random samples which have uniform and normal distribution and analyse their behavior for large sample volume.

Communications in mathematical physics 399(2), 1039 - 1101 (2023). doi:10.1007/s00220-022-04571-y, We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore., Published by Springer, Heidelberg

In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X (t, x) = b X(t, x) with X(0, x) = x $\in$ R d , where b is a regular Z dperiodic vector field in R d. More precisely, we provide various conditions on b to obtain a "fine" asymptotic expansion of X of the type: |X(t, x) -- x -- t $\zeta$(x)| $\le$ M < $\infty$, which is uniform with respect to t $\ge$ 0 and x $\in$ R d (or at least in a subset of R d), and where $\zeta$(x) for x $\in$ R d , are the rotation vectors induced by the flow X. On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) -- x -- t $\zeta$(x) reads as $\Phi$ X(t, x) -- $\Phi$(x), which yields immediately the desired expansion when the vector-valued function $\Phi$ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Y d and of vector-valued functions $\Phi$. On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not. It turns out that for any two-dimensional flow X associated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector $\xi$, the fine asymptotic expansion of X holds in R 2 if, and only if, $\xi$ 1 /$\xi$ 2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors $\zeta$(x) is investigated in a similar way. Finally, several examples and counterexamples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.

The Kelvin–Voigt model for a thin stratified two-dimensional visco-elastic strip is analyzed both in the quasistatic and in the dynamic cases. The Neumann boundary conditions on the upper and the lower parts of the boundary and periodicity conditions with respect to the longitudinal variable are stated. A complete asymptotic expansion of the solution is constructed in both cases, by using the dimension reduction combined with a homogenization technique. The error between the exact solution and the asymptotic one is evaluated in each case and the obtained results fully justify the asymptotic construction. The results were partially (quasistatic case) announced in the short note in C.R. Acad. Sci. Paris; the present article contains the complete proofs and generalizations in the dynamic case.

We show in this paper that complete asymptotic expansions exist for a class of holomorphic Dirichlet-Hurwitz-Lerch Eisenstein series (Theorems 1 and 2), which, together with their remainders in exact form (Theorem 3), naturally transfer to several (additive and multiplicative) character analogues of Ramanujan's formula for specific values of the Riemann zeta-function (Theorem 4 and Corollary 4.1), and further to the (quasi) modular relations for similar character analogues of the classical Eisenstein series with integer weights (Corollary 4.2). Prior to the (sketched) derivation of our main formula, we prepare several basic (but new) results on Dirichlet-Hurwitz-Lerch $L$-functions (Theorems 5, 6 and Lemmas 1-3), which play underlying roles in all aspects of the proofs; the detailed version of the proofs will appear in a forthcoming article [16].

A random polynomial is a polynomial whose coefficients are random variables. A major task in the theory of random polynomials is to examine how the real roots are distributed and correlated in situations where the degree of the polynomial is large. In this dissertation, we investigate two classes of random polynomials that have piqued the interest of researchers in probability theory and mathematical physics: elliptic polynomials and generalized Kac polynomials. Regarding elliptic polynomials, we obtain asymptotic expansions for the variances of the number of real roots on intervals whose endpoints may vary based on the polynomial's degree. Additionally, we provide sharp estimates for the cumulants of these quantities. As applications, we can determine intervals on which the number of real roots satisfies a central limit theorem and a strong law of large numbers. Our next objective is to compute the precise leading asymptotics of the variance of the number of real roots for generalized Kac polynomials whose coefficients have polynomial asymptotics. Examples of this class of random polynomials include Kac polynomials, hyperbolic polynomials, and any linear combinations of their derivatives. Before our work, such variance asymptotics had only been established for the Kac polynomials in the 1970s, thanks to Maslova's influential contribution. Our proof relies on novel asymptotic estimates for the real roots' two-point correlation function, which exposes geometric features in the distribution of real roots for these random polynomials. As a corollary, we establish asymptotic normality for the number of real roots of these random polynomials, extending and enhancing a related result of O. Nguyen and V. Vu.

1. INTRODUCTION. We present a concise and elementary derivation of the complete asymptotic expansion for the factorial function n!, which we will refer to as Stirling’s series. While there have been numerous published proofs of Stirling’s series and of its classical dominant term given by Stirling’s formula lim n→∞ n!e n n n √ 2πn = 1

Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions that involve Gamma functions. We were able to obtain some of the bounds of Γ(v) that are more accurate than some recently published inequalities.

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1/n and 1/n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

One of the main problems of fracture mechanics is to describe the behavior of media with cracks. At the same time, more and more attention is paid to the study of large systems, including periodic structures. For several cracks, the coefficient of influence is considered to be an important parameter. It is the ratio of the stress intensity factor calculated in the problem of a system of cracks to the stress intensity factor for a single crack under the same load. The paper proposes a method that allows to accurately determine the coefficients of influence for large systems of cracks, including the case of periodic structures, in which the number of cracks is infinite. The method is based on a numerical algorithm developed by the authors that uses the method of discontinuous displacements of a high order of accuracy and an asymptotic representation of stress fields at the crack tip (M. Williams). To verify the method, the results of the implemented algorithm are compared with known analytical solutions both for single cracks and for an infinite doubly periodic system of cracks.

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter ( e ) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O ( e 2 ) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number ( R e ). Furthermore, at subcritical R e with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

An analytical study on the propagation characteristics of shear wave in a spherical layered structure comprising of an isotropic sandy material medium covered by a concentric spherical transversely isotropic sandy material layer of finite thickness has been carried out in the present work. The source of excitation for wave propagation is considered in the outer layer and the analytical treatment with contour integration and Fourier-Legendre Expansion theorem is utilized to accomplish closed form of dispersion relation. Further, Debye Asymptotic Expansion analysis is used to deal the complication caused by the inclusion of Bessel's function, Hankel's function and their derivatives in the solution procedure. With help of Debye Asymptotic Expansion analysis, the obtained dispersion relation converts in the form of elementary functions and found to be in well-agreement with pre-established and classical results through the particular cases. Numerical computations are done to illustrate the influence of the affecting parameters (anisotropy, sandiness and radii ratio) on dispersion curves graphically. Moreover, to expose the impact of anisotropy, numerical data of three distinct transversely isotropic materials (Magnesium, Zinc and Beryl) are taken into the account to establish the comparative study with isotropic material which serves the major highlights of the present study.

The focus of this paper is to obtain explicit solutions to initial value problems, where numerical methods cannot provide one, and to verify the accuracy orders of the explicit solutions. One of available methods to obtain an explicit solution is the asymptotic (formal) expansion method. However, we must be sure with the accuracy order of the explicit solution. In this paper, an order verification method is proposed for truncated asymptotic formal expansion solutions to initial value problems. A least-squares fit of error data is used in the existing order verification method. The method that we propose does not involve any application of least-squares fit of error data, so is simpler, yet produces accurate expected accuracy orders of solutions of explicit truncated asymptotic formal expansions. With our proposed method, we are successful in verifying the accuracy orders of solutions of truncated asymptotic formal expansions to the linear and nonlinear initial value problems accurately.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.

Quantum tunnelling plays a significant role in many molecular reactions, yet it is difficult to describe for molecules with more than 10 atoms. Both commonly used approaches, conventional wavefunction methods and statistical methods, are either computationally intractable, or slowly-converging for such systems. In contrast, instanton theory can describe quantum tunnelling even in high-dimensional systems and without statistical errors. This is because instead of using the whole phase space, in instanton theory an asymptotic expansion is performed around an extremal-action trajectory called the instanton. In this work we extend instanton theory by deriving the first-order perturbative correction to the commonly used leading-order result. Furthermore, we provide mathematical background to the theory by introducing a Feynman-diagram formalism and showing equivalence of our theory to a Euclidean ϕ3 + ϕ4 theory. Physical chemists or other researchers can utilise this framework to incorporate findings from the vast field of quantum field theory (QFT) into the advancement of instanton theory. Lastly, we use the perturbatively corrected instanton theory to numerically calculate the tunnelling splitting of a proton tunnelling reaction occurring in malonaldehyde and reach good agreement with state of the art statistical methods, specifically Diffusion Monte Carlo (DMC) and Path integral molecular dynamics (PIMD). This good agreement shows promise as perturbatively corrected instanton theory requires far less computational resources than the previously mentioned methods and thus can likely be applied to high-dimensional systems, where those methods fail.

We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The novel scheme is systematically derived using the asymptotic expansion technique and a detailed analysis of the leading-order error behavior is provided. As demonstrated by a linear stability analysis, the method is stable for a very large range of Poisson’s ratios. We consider periodic problems to focus on the governing equations and rule out the influence of boundary conditions. The analytical derivations are verified by numerical experiments and convergence studies., Computer Methods in Applied Mechanics and Engineering, 404, ISSN:0045-7825, ISSN:1879-2138

Novel bounds on the left- and right-tail probability of the Binomial distribution are presented. Specifically, bounds on the left-tail probability $\text {P}\!\left [{S_{n}\leqslant an}\right]$ with $a\in (0,p)$ and the right-tail probability $\text {P}\!\left [{S_{n}\geqslant an}\right]$ with $a\in (p,1)$ , where $S_{n}$ follows the Binomial law with parameters $n\geqslant 1$ and $p\in (0,1)$ , are derived. The bounds are tighter than the best known closed-form expressions. The presented upper and lower bounds on the right-tail match an asymptotic expansion of $\log \text {P}\!\left [{S_{n}\geqslant na}\right]$ up to, and including, the $\log n$ term. In particular, the ratio between the upper and lower bounds is $1/(1-c/n)$ , where $c$ is independent of $n$ . Similar results are presented for the left-tail probability. Applications to finite blocklength source and channel coding are presented.

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

We study the existence and stability of standing pulse solutions to a singularly perturbed three-component reaction–diffusion system with one activator and two inhibitors. We apply the MAE (matched asymptotic expansion) method to construct solutions and the SLEP (singular limit eigenvalue problem) method to evaluate their stability. This approach is not just an alternative approach to geometric singular perturbation and the associated Evans function, it also yields two advantages: one is extendability to higher dimensional cases, and the other is that it allows us to obtain more precise information about the behaviors of critical eigenvalues. This implies the existence of codimension-two singularities of drift and Hopf bifurcations for the standing pulse solution. Moreover, it is numerically confirmed that stable standing and traveling breathers emerge around the singularity in a physically acceptable region.

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles.

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets.

We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the definition of the shift operator for the free QW. Then we define an anisotropic Banach space as a modified Agmon-H\"{o}rmander's $\mathcal{B}^*$ space and we derive the asymptotic behavior at infinity of generalized eigenfunctions in these spaces. The scattering matrix appears in the asymptotic expansion of generalized eigenfunctions.

Experimental results are presented of a test of the theory of local turbulent heat transfer measurements proposed by Mocikat and Herwig in 2007. A miniaturized multi-layer heat transfer sensor was developed and employed in this study. The new heat transfer sensor was designed to work in air and liquids, and this capability enabled the simultaneous investigation of different Prandtl numbers. Two basic configurations, namely the flow past a blunt plate and the flow past an inclined square cylinder, were investigated in test sections of wind and water tunnels. Convective heat transfer coefficients were obtained through conventional testing (i.e., employing thoroughly heated test objects) and using the new miniaturized sensor approach (i.e., utilizing cold test objects without heating). The main prediction of the Mocikat-Herwig theory that a specific thermal adjustment coefficient of the employed actual miniaturized heat transfer sensor should exist in the fully turbulent flow regime was proven for developed two-dimensional flow. The observed effect of the Prandtl number on this coefficient was in good agreement with the prediction of the asymptotic expansion method. The square cylinder results indicated the inherent limits of the local turbulent heat transfer measurement approach, as suggested by Mocikat and Herwig.

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

In this article, we study the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise Holder continuous perturbation and investigate how the Holder constant can affect the eigenvalues. More precisely, we derive several first terms in the asymptotic expansion for the eigenvalues.

On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler-Lagrange equation at K\"ahler-Einstein structures is given by the Dolbeault Laplacian acting on $(0,1)$-forms with values in the holomorphic tangent bundle. A deformation result of Einstein ACH metrics associated with critical almost complex structures for this variational problem is given. It is also shown that the asymptotic expansion of a critical almost complex structure is determined by the induced (possibly non-integrable) CR structure on the boundary at infinity up to a certain order., Comment: 28 pages. The statement of Theorem 1.2 is modified; substantial corrections and clarifications are added throughout the text

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schr\"odinger operators on compact Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero potentials, we obtain new geometric invariants determined by the spectrum of what we call the parametric Steklov problem. In particular, for constant potentials parametric Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators., Comment: 35 pages, 2 figures

This is a pre-announcement version of the forthcoming paper [Complete asymptotic expansions for the transformed Lerch zeta-functions via the LaplaceMellin and Riemann-Liouville operators, preprint.]. For a complex variable s, and any real parameters a and λ with a > 0, the Lerch zeta-function φ(s, a, λ) is defined by the Dirichlet series Σ[∞][l=0]e(λl)(a + l)⁻[s] (Res> 1), and its meromorphic continuation over the whole s-plane, where e(λ) = e[2πiλ], and the domain of the parameter a can be extended to the whole sector |arg z| < π through the procedure in [M. Katsurada, Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 167-170.]. It is the principal aim of the present article to treat asymptotic aspects of the transformed functions obtained by applying the Laplace-Mellin and Riemann-Liouville operators (in terms of the variables), which are denoted by LM[α][z;T] and RL[α, β][z;T] respectively, to a slight modification, φ*(s, a, λ), of φ(s, a, λ). For any m ∈ ℤ, let (φ*)[(m)](s, a, λ) denote the mth derivative with respect to s if m ≥ 0, and the |m|th primitive defined with its initial point at s + ∞ if m < 0. We shall then show that complete asymptotic expansions exist, if a > 1, for .LM[α][z;T](φ*)[(m)](s+τ, a, λ) and for RL[α, β][z;T](φ*)[(m)](s+τ, a, λ) (Theorems 1-4), as well as for their severa.l iterated variants (Theorems 5-10), when the pivotal parameter z of the transforms tends to both O and oo through appropriate sectors. Most of our results include any vertical ha.If-lines in their respective regions of validity; this allows us to deduce complete asymptotic expansions for the relevant transforms through arbitrary vertical half-lines, upon taking (s, z) = (a, it) with any σ ∈ ℝ, when t → ±∞ (Corollaries 2.1, 4.1, 6.1 and 8.1).

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

An algorithm for the approximate calculation of the coefficients of the Dulac series (an asymptotic series of the monodromy transformation) in the space of vector fields with a Newton diagram containing more than one edge and a monodromic singular point is proposed. The conditions for the applicability of this algorithm are obtained. The algorithm is implemented in the MAPLE package. Examples are given for the case of a Newton diagram consisting of two edges.

Asymptotic expansion based homogenization has been widely used to predict the effective macroscopic properties of periodic unit cells (PUCs). In this work, we show that the homogenization process can be done in a much more elegant manner for both continuum and discrete PUCs by taking advantage of the Cauchy–Born’s hypothesis, which is a widely used rule in the area of solid physics to relate the position of the atoms in a crystal lattice and the overall strain of the medium. It is shown that in the proposed method, the derivation process of the effective elasticity tensor is quite easy and can rely entirely on commercial CAE software (e.g., ANSYS, ABAQUS, etc.) to accomplish the homogenization task. In detail, after the discretization of the unit cell with finite elements, one only needs to apply affine boundary conditions at the exterior boundaries of the unit cell and then call the FEA solver to find the static displacement field under such affine boundary conditions. The entries of the elasticity tensor can then be expressed using the stain energy of the unit cell. After deriving the sensitivity information of the Cauchy–Born hypothesis based homogenization process, the inverse homogenization process, which attempts to find the optimal layout exhibiting pre-determined desirable material properties, can be implemented in a straightforward way as well. Some numerical examples are tested and compared with the results in the literature. It is showed that the results of both the homogenization and inverse homogenization examples obtained by our method agree very well with the ones in the literature, demonstrating the validity of the Cauchy–Born hypothesis based numerical homogenization method.

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.

We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty $. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty $. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sjöstrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.

The problem of generation of steady streaming by an oscillating circular cylinder is considered. For the case of a finite oscillation frequency, the asymptotic expansion of the solution in terms of a small oscillation amplitude is constructed. In the leading steady term the two possible regimes of secondary streaming are analyzed. The fluid pumping effect of secondary streaming is studied. Energy efficient pumping regime is determined. The limits of applicability of high-frequency asymptotic solutions are established.

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

The Bloch-Floquet theory are popular tools for the investigation of materials with periodic structures. For example, we can show that the spectrum of periodic Schrodinger operators have band structures. Here we shall extend the Bloch-Floquet theory, which is appicable to abelian groups, to the Heisenberg group. Our method is based on a combination of the representations of the discrete Heisenberg groups and of the Heisenberg Lie group. We apply this method to asymptotic problems for heat kernels and for counting prime closed geodesics. In this application, we need additional ingradients, the semi-classical analysis and the Chen's iterated integrals. As a by-product, we give another mathematical explanation of the semi-classical asymptotic expansion formula for the Hofstadter butterfly of Wilkinson, which is originally due to Helffer-Sjostrand.

A flow of Eyring–Powell type constitutes a remarkable area of analysis to model non-Newtonian processes in fluids. The associated diffusion term comes from the general kinetic theory of liquids and permits to account for a wider diffusivity, which is applicable for qualitatively low to higher shear stresses. The goal of the present article is to introduce a generalization of an Eyring–Powell fluid by the introduction of a porous reaction term (of Darcy–Forchheimer type) and a perturbation with a higher order operator. In particular, we consider that our model is an extension of a classical Eyring–Powell fluid in the same manner as introduced for other equations (see the extended Fisher–Kolmogorov model). The obtained equation is novel and requires analysis about existence, regularity and uniqueness of solutions. Stationary solutions are explored under the definition of a Hamiltonian. In addition, profiles of solutions are obtained with an exponential scaling that ends in a Hamilton–Jacobi equation. Eventually, some numerical assessments are introduced to validate the hypothesis done, and to discuss about the accuracy of the analytical approach followed. 2022-23

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O(n1−p/2), for some p≥4, is shown in both cases. From the results, it is obtained that asymptotic expansions for the Crame´r–von Mises statistics of the uniform distribution U(0,1) hold with the remainder term On1−p/2 for any p≥4. The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u(x,y). The key condition for the convergence is the nuclearity of a linear operator Tu defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables.

Designing devices composed of many small resonators is a challenging problem that can easily incur significant computational cost. Can asymptotic techniques be used to overcome this often limiting factor? Integral methods and asymptotic techniques have been used to derive concise characterisations for scattering by resonators, but can these be generalised to systems of many dispersive resonators whose material parameters have highly non-linear frequency dependence? In this paper, we study halide perovskite resonators as a demonstrative example. We extend previous work to show how a finite number of coupled resonators can be modelled concisely in the limit of small radius. We also show how these results can be used as the basis for an inverse design strategy, to design resonator systems that resonate at specific frequencies.

Using the so-called direct scheme method, an asymptotic expansion of n-th order to the solution of a class of singularly perturbed linear-quadratic optimal problem with integral constraint on control is constructed. The expansion contains three type functions. Two of them are boundary layer functions in the vicinities of two fixed end-points, and the remain is regular function. A numerical example is represented to illustrate the obtained results.

We consider the undivided boundary value problem for singularly perturbed linear integro-differential equations of th order with Fredholm’s integral terms have an initial jump of the phenomenon th order on the left end of the segment. Defined regular and boundary layer members of the asymptotic expansions of the solutions. Regular members of the asymptotics constructed in the form of integro-differential equations, which different from the usual unperturbed equations by the presence of additional term called initial jump of integral terms. The values of the initial jumps are defined. Boundary conditions for the regular members of the asymptotics also contain additional term, called an initial jump of the derivatives th order. Thus, for the determination of the regular members of the asymptotics obtained boundary value problems for linear integro-differential with additional parameter. To determine the boundary layer of the asymptotic expansion of solution we obtained initial problems for homogeneous an inhomogeneous ordinary differential equations with constant coefficients. Exponential estimates for boundary layer terms of the asymptotics are obtained. A theorem of existence and uniqueness and the asymptotic representations of solution with an estimate of the remainder term of the asymptotics. It is found that the constructed asymptotics approximation to the solution of the original singularly perturbed integro-differential boundary value problem is uniformly throughout the considered interval.

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $$\mathbb {R}^d$$ with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $$|x|^\beta $$ with $$\beta

In the present paper, we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of orders $$\alpha ,1+\alpha ,2+\alpha ,3+\alpha $$ and $$4+\alpha $$ . The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.

We study the uniqueness and the asymptotic expansion of the solution of the mixed Dirichlet–Neumann problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of this problem has a finite Dirichlet integral with power-law weight with exponent $$a\in \mathbb {R}$$ . Depending on the value of the parameter $$a$$ , uniqueness (nonuniqueness) theorems are proved and exact formulas are found for the dimension of the linear solution space of the mixed Dirichlet–Neumann problem.

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to $\pm \infty $ . In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

Let s ∈ IC be variable, α, λ ∈ R parameters with α > 0, φ(s, α, λ) the Lerch zeta-function defined below, and φ*(s, α, λ) its slight modification obtained by extracting the only singularity at s = 1 (if λ ∈ Z) of the Hurwitz zeta-function ζ(s, α). We denote by (φ*)(m)(s, α, λ) for any m ∈ Z the m-th derivative (with respect to s) if m ≥ 0, while the lml-th primitive (with its initial point at s + ∞) if m ≤ 0. It is shown in the present article that complete asymptotic expansions exist for the multiple LaplaceMellin transform (with respect to s) of (φ*)(m)(s, α, λ) for any m ∈ Z if a > 1, when the multivariate pivotal parameter z = (z1 , ... , zd) ∈ C[d] (of the transforms) becomes both small and large through an appropriate poly-sector (Theorems 5 and 6), which extends our previous results on one dimensional case (Theorems 1 and 2). Further consideration on the excluded singular part of ζ(s, α) at s = 1 is supplemented to establish complete asymptotic expansions for the Lapalce-Mellin transform of ζ(m)(s, α) (Theorems 7 and 8). A topic on complete asymptotic expansions for certain mean values of multiple zetafunctions, which is positioned on a sligntly different direction of research, is discussed in the final section (Theorems 9-13). Several open problems, relevant to the present study, are to be posed along with the presentation of our results (Problems 1-5).

In this paper, we construct the utility-based optimal hedging strategy for a European-type option in the Almgren-Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first order terms in the associated HJB equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.