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The eigenvalues of Toeplitz matrices T n (f) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [ − π , π ] , are known to admit an asymptotic expansion with the form λ j (T n (f)) = f (σ j , n) + c 1 (σ j , n) h + c 2 (σ j , n) h 2 + O (h 3) , where h = 1 / (n + 1) , σ j , n = π j h , and c k are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [ − π , π ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination λ j (T n (f 0) + β n , 1 T n (f 1) + β n , 2 T n (f 2)) , where β n , 1 , β n , 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

We present an asymptotic method to compute efficiently second-order telegraph equations with high-frequency extrinsic oscillations. This method is based on asymptotic expansions in inverse powers of the oscillatory parameter. Each term of this asymptotic expansion is the sum of at most two coefficients. Each coefficient is derived either by recursion or by solving a non-oscillatory problem. This leads to method which exhibit improved performance with growing frequency of oscillation. [ABSTRACT FROM AUTHOR]

Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramér condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space–time white noise is discusses. [ABSTRACT FROM AUTHOR]

In eddy current (EC) testing applications, ECs σ E ( E : electric field, σ : conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation - which neglects the displacement current - was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O (1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants. [ABSTRACT FROM AUTHOR]

This paper is concerned with the MANOVA test when the dimensionality of the observed vector may exceed the sample size. Zhou et al. (2017) proposed the L 2 norm based testing statistic T and gave the limiting null distribution under the high-dimensional asymptotic framework that the minimum of sample sizes and the dimensionality go to infinity. In this paper we give a one-term asymptotic expansion of the null distribution for T under the asymptotic framework. We propose a correction of the critical point for Zhou et al. (2017)'s test based on this expansion. The finite sample size and the dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Zhou et al. (2017)'s test. [ABSTRACT FROM AUTHOR]

The paper is concerned with an asymptotic expansion for the oscillatory term of the solution of diffusion equation with time-like highly oscillatory forcing terms. We derive an asymptotic expansion in inverse of powers of the oscillatory parameter, for the Dirichlet case and the Neumann case. Each term of the asymptotic expansion can be computed, at a lower cost. Numerical examples are given and show that with few terms of the asymptotic expansion, we can effectively approximate the oscillatory term of the solution. [ABSTRACT FROM AUTHOR]

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and other random symbols. To specify these random symbols, it is necessary to classify the level of the effect of each term appearing in the stochastic expansion of the variable in question. To solve this problem, we consider a class L of certain sequences (I n) n ∈ N of Wiener functionals and give a systematic way of estimation of the order of (I n) n ∈ N. Based on this method, we introduce a notion of exponent of the sequence (I n) n ∈ N , and investigate the stability and contraction effect of the operators D u n and D on L , where u n is the integrand of a Skorohod integral. After constructed these machineries, we derive asymptotic expansion of the variation having anticipative weights. An application to robust volatility estimation is mentioned. [ABSTRACT FROM AUTHOR]

The flow under an Eyring-Powell description has attracted interest to model different scenarios related with non-Newtonian fluids. The goal of the present study is to provide analysis of solutions to a one-dimensional Eyring-Powell fluid in Magnetohydrodynamics (MHD) with general initial conditions. Firstly, regularity and bounds of solutions are shown as a baseline to support the construction of existence and uniqueness results. The existence analysis is based on the definition of a Hamiltonian that constitutes the underlying theory to obtain stationary profiles of solutions that are validated with a numerical approach. Afterwards, non-stationary profiles of solutions are explored based on an asymptotic approximation to a Hamilton-Jacobi type of equation. To this end, an exponential scaling is considered together with perturbation methods. Finally, a region of validity for such exponential scaling is provided. [ABSTRACT FROM AUTHOR]

We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we utilized the theory of exponents of functionals, which was introduced by the authors in Yamagishi and Yoshida (2023). [ABSTRACT FROM AUTHOR]

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re (α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [ 0 , 4 ] and are asymptotically distributed as the function g (x) = 4 sin 2 ⁡ (x / 2) on [ 0 , π ]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [ 0 , 4 ]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue. [ABSTRACT FROM AUTHOR]

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. • The suggested method makes it possible developing stresses in fractures interactions. • The method allows calculating infinite systems of cracks. • Numerically developed stress intensity factors are in good agreement with analytical solutions. • The interaction of periodic systems of cracks in space located in non-parallel planes was investigated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The problem of the free streamline solutions of the Falkner–Skan equation is revisited in this paper. Until now, such solutions were found for negative values of the pressure gradient parameter β only. All of them are associated with slip velocities − 1 < f ′ 0 < ∞ and emerge from the trivial solution f = η of the problem. The present paper shows, however, that in the positive range 1 < β < ∞ , just below the interval − 1 < f ′ 0 < ∞ , a further branch of free streamline solutions of slip velocities − 2 ≤ f ′ 0 < − 1 exists. These new solutions emerge from an exact solution of the Falkner–Skan equation which describes the flow in a converging channel with moving boundaries at the saddle–node bifurcation point f ′ 0 = − 2 , f ′ ′ 0 = 0. For the large- β asymptotics of this solution branch a new algorithm is presented. The occurrence of further free streamline solutions in the range β < 0 , as well as the existence of free streamlines of vanishing slip velocities, f ′ ′ 0 = f ′ 0 = 0 , both for positive and negative values of β is also addressed in the paper. The flow inside a cone is also considered shortly and the occurrence of free streamline solutions is pointed out also in this case. [ABSTRACT FROM AUTHOR]

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with n discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the n discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes G -function. • An integral representation for Fredholm determinant of the CHF kernel is obtained. • The Fredholm determinant is related to the Hamiltonian of a coupled Painlevé V system. • The aymptotics of the determinant as discontinuities tend to infinity is derived. [ABSTRACT FROM AUTHOR]

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices T n (f) generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form T n − 1 (g) T n (ℓ) with g , ℓ real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that f = ℓ / g is even and monotonic over [ 0 , π ] , matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case g ≡ 1 , combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case. Numerical experiments show a much higher accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature. [ABSTRACT FROM AUTHOR]

In this paper we study the flow an elasto-visco-plastic fluid whose deviatoric stress is made up by a "seemingly" visco-plastic part (regularized Bingham fluid) and a viscoelastic part (Rivlin–Ericksen fluid) in a "slightly" eccentric annulus. The flow is driven by an applied horizontal pressure gradient and by the motion of the inner cylinder. Using the eccentricity as perturbation parameter we show that secondary flows appear at the first order approximation. We investigate the various flow patterns depending on the different non-dimensional groups appearing in the governing equations. In particular we observe the occurrence of rotating and counter-rotating vortices depending on the material and geometrical parameters of the system. • We study the flow of an elasto-visco-plastic fluid in a "slightly" eccentric annulus. • The flow is of Couette–Poiseuille type. • We perform an asymptotic expansion around the eccentricity parameter. • We solve the zero and first order problem numerically via Spectral methods. • We show that secondary flows appear at the first order. [ABSTRACT FROM AUTHOR]

We provide a full series expansion of a generalization of the so-called u -capacity related to the Dirichlet–Laplacian in dimension three and higher, extending the results of Abatangelo et al. (2021); Abatangelo, Léna and Musolino (2022) dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem. [ABSTRACT FROM AUTHOR]

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R 3 exhibit singular behavior from the edges and corners. We present quasi-optimal h p -explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an h p -version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains. • Article initiates the study of p- and hp-versions of the boundary element method in time domain. • Quasi-optimal hp approximation for boundary traces of the solution to wave equation. • Application to time domain BEM for single layer and hypersingular integral equations. • Numerical implementation of p-method confirms theoretical results for screen and polyhedral domains. [ABSTRACT FROM AUTHOR]

In this paper, we study efficient numerical methods for the computation of highly oscillatory Bessel transform ∫ a + ∞ f (x) J ν (ω g (x)) d x , g (x) ≠ 0 , g ′ (x) ≠ 0 for all x ∈ [ a , + ∞). We first derive an asymptotic expansion formula for this integral by using integration by parts. Then we present an improved numerical steepest descent method, by applying complex integration theory to the remainder term of asymptotic expansion, which is also an oscillatory integral with Bessel kernel. In addition, the asymptotic orders about ω of the presented methods are given. The efficiency and accuracy of the proposed methods are investigated through theoretical results and numerical experiments. [ABSTRACT FROM AUTHOR]

We develop the asymptotic expansion theory for vector-valued sequences (F N) N ≥ 1 of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence F N. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of F N and we illustrate our results by several examples. [ABSTRACT FROM AUTHOR]

Abstract The goal of this paper is to propose new asymptotic models describing a viscous fluid flow through a pipe-like domain subjected to heating. The deformation of the pipe due to heat extension of its material is taken into account by considering a linear heat expansion law. The heat exchange between the fluid and the surrounding medium is prescribed by Newton's cooling law. An asymptotic analysis with respect to the small parameter ε (the heat expansion coefficient of the material of the pipe) is performed for the governing coupled nonlinear problem. Motivated by applications, the special case of flow through a thin (or long) pipe is also addressed leading to an asymptotic approximation of the solution in the form of an explicit formula. In both settings, we rigorously justify the usage of the proposed effective models by proving the corresponding error estimates. [ABSTRACT FROM AUTHOR]

Abstract In this study, we present a multivariate test for population bioequivalence. Recently, Chervoneva et al. (2007) proposed a multivariate generalization of the criteria for testing univariate population bioequivalence by obtaining an approximate test based on an unbiased estimator of linearized criteria. Their test involved decomposing the unbiased estimator into several terms and using the existing interval estimates of these terms. Although this test's type I error probability is often lower than the nominal significance level, its approximation accuracy is not good in case of small samples. This study intends to present a more accurate approximate test. To improve the accuracy, it is important to investigate the distribution of the unbiased estimator of the linearized bioequivalence criteria. Therefore, we derive an explicit representation of its characteristic function and asymptotic distribution and propose new tests that exploit the asymptotic results. We additionally obtain the theoretical asymptotic power function in an explicit manner and investigate the local powers of our test, proving that they are asymptotically unbiased. Further, we investigate their finite-sample performance via Monte Carlo simulations and confirm that the accuracy is better as compared to that observed in a previous test. Finally, we use our methods to analyze real data. [ABSTRACT FROM AUTHOR]

Abstract This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo simulation and other numerical integrations for estimating conditional expectations at each space–time node. Numerical examples of quadratic-growth as well as Lipschitz BSDEs suggest that the scheme works well even for large quadratic coefficients, and a fortiori for large Lipschitz constants. [ABSTRACT FROM AUTHOR]

Abstract We study the relationship between the Maxwell and eddy current (EC) models for three-dimensional configurations involving bounded regions with high conductivity σ in air and with sources placed remotely from the conducting objects, which typically occur in the numerical simulation of eddy current nondestructive testing (ECT) experiments. The underlying Maxwell transmission problem is formulated using boundary integral formulations of PMCHWT type. In this context, we derive and rigorously justify an asymptotic expansion of the Maxwell integral problem with respect to the non-dimensional parameter γ ≔ ω ε 0 ∕ σ. The EC integral problem is shown to constitute the limiting form of the Maxwell integral problem as γ → 0 , i.e. as its low-frequency and high-conductivity limit. Estimates in γ are obtained for the solution remainders (in terms of the surface currents, which are the primary unknowns of the PMCHWT problem, and the electromagnetic fields) and the impedance variation measured at the extremities of the excitating coil. In particular, the leading and remainder orders in γ of the surface currents are found to depend on the current component (electric or magnetic, charge-free or not). These theoretical results are demonstrated on three-dimensional illustrative numerical examples, where the mathematically established estimates in γ are reproduced by the numerical results. [ABSTRACT FROM AUTHOR]

Abstract A periodic composite microstructure consisting of permanent magnet and ferromagnetic material can be utilized to realize a controllable permanent magnet. For the microstructure design of the permanent magnet composite, its effective material properties should be estimated accurately and efficiently. For this, an asymptotic homogenization method is developed in this work. Specifically, the formulations for homogenized magnetic permeability and residual flux density are mathematically derived for both scalar and vector potential magnetostatic analysis approaches. Using the asymptotic expansion, cell problem equations are first derived, which will be solved by finite element method in microscopic coordinate. Then, the integral form formulations are derived for the calculation of homogenized effective properties. To show the design possibility of the composite, four numerical examples are investigated in this work. In each example, the accuracy of the obtained properties is numerically validated by comparing the magnetic field and energy produced by a homogeneous composite with the obtained effective properties and those produced by a heterogeneous model with actual periodic composite structures. In addition, the computation time of the homogeneous and heterogeneous models are compared to confirm the computational benefit of the developed homogenization method. [ABSTRACT FROM AUTHOR]

Highlights • A method is considered for computing oscillatory Bessel transforms. • The relation between the multiple integral and simple integral is used. • The order is discussed in two cases. • Numerical experiences show the efficiency of the proposed method. Abstract Based on the relation between the multiple integral and simple integral, we propose an asymptotic rule for computing the oscillatory Bessel transforms from position to momentum space. We begin our analysis by evaluating the generalized moments. Next, we introduce the schemes of the rule in detail. In order to demonstrate the efficiency of the rule, some numerical examples based on theoretical results are presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time τ = ω t. The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems. [ABSTRACT FROM AUTHOR]

Abstract The article shows a bridge representation for the joint density of a system of stochastic processes consisting of a Brownian motion with drift coupled with a correlated fractional Brownian motion with drift. As a result, a small time approximation of the joint density is readily obtained by substituting the conditional expectation under the bridge measure by a single path: the modal-path from the initial point to the terminal point. [ABSTRACT FROM AUTHOR]

In order to solve the advection-diffusion-reaction equation with singularity by the boundary element method, the domain integral is handled by dual reciprocity method and the semi-analytical elements are implemented to approximate the unknown physical fields. By collocating internal points in the domain and approximating the primer field with radial basis function, the integral of domain can be settled and boundary-only integral equation can be derived. In this paper, we propose two new elements containing asymptotic expressions to model the unknown temperature and singular flux fields, while the known fields are simulated by linear interpolation. The unknowns in the new elements are amplitude coefficients which can be used to evaluate the intensity of flux. Compared with conventional boundary element, the proposed method allows to avoid direct evaluation of singular values. Thus, the accuracy can be guaranteed since all the unknowns have finite values. Benefiting from the semi-analytical elements, the mesh refinement for the region with singular physical field can be avoided. Moreover, the first three amplitude coefficients as well as the whole physical fields are obtained at the same time without any post process. We demonstrate the excellent numerical results through several examples in plates with singularities. [ABSTRACT FROM AUTHOR]

In this article, two methods for evaluating highly oscillatory Bessel integrals are explored. Firstly, a polynomial is analyzed as an effective approximation of the simplex integral of a highly oscillatory Bessel function based on Laplace transform, and its error rapidly decreases as the frequency increases. Furthermore, the inner product of f and highly oscillatory Bessel function can be approximated by two other forms of inner product by which one depends on a polynomial and the higher derivatives of f , another depends on Bessel function and the interpolation polynomial of f. In addition, three issues related to highly oscillatory Bessel integrals have also been discussed: inequalities for the convergence rate of Filon-type methods, evaluation of Cauchy principal values, and simplified evaluation on infinite intervals. Through some preliminary numerical experiments, our theoretical analysis has been preliminarily confirmed, and the proposed numerical method is accurate and effective. [ABSTRACT FROM AUTHOR]

We develop series expansions in powers of q − 1 and q − 1 ∕ 2 of solutions of the equation ψ ( z ) = q , where ψ ( z ) is the Laplace exponent of a hyperexponential Lévy process. As a direct consequence we derive analytic expressions for the prices of European call and put options and their Greeks (Theta, Delta, and Gamma) and a full asymptotic expansion of the short-time Black–Scholes at-the-money implied volatility. Further we demonstrate how the speed of numerical algorithms for pricing exotic options, which are based on the Laplace transform, may be increased. [ABSTRACT FROM AUTHOR]

Abstract It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair ε = ( ε 1 , ε 2 ) of positive parameters, we consider a perforated domain Ω ε obtained by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n at distance ε 1 from the boundary ∂Ω. As ε 1 → 0 , the perforation shrinks to a point and, at the same time, approaches the boundary. When ε → ( 0 , 0 ) , the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by u ε the solution of a Dirichlet problem for the Laplace equation in Ω ε . For a space dimension n ≥ 3 , we show that the function mapping ε to u ε has a real analytic continuation in a neighborhood of ( 0 , 0 ) . By contrast, for n = 2 we consider two different regimes: ε tends to ( 0 , 0 ) , and ε 1 tends to 0 with ε 2 fixed. When ε → ( 0 , 0 ) , the solution u ε has a logarithmic behavior; when only ε 1 → 0 and ε 2 is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε 1 . We also show that for n = 2 , the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials. [ABSTRACT FROM AUTHOR]

This article deals with the weak error in averaging principle for a stochastic wave equation on a bounded interval [ 0 , L ] , perturbed by an oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving on the fast time scale. Under suitable conditions, it is proved that the rate of weak convergence of the original solution to the solution of the corresponding averaged equation is of order 1 via an asymptotic expansion approach. [ABSTRACT FROM AUTHOR]

In this paper we consider old and new operators of Szász type involving Sheffer polynomials. We present an asymptotic expansion formula for operators of Ismail type. Then, in order to improve the accuracy of the approximation of a function f in a fixed point, we apply a well-known extrapolation algorithm. We also introduce some new special sequences of Appell and Sheffer polynomials and construct new generalized Szász-type operators. By using classical techniques we investigate approximation properties and rate of convergence for these operators and compare the results with other existing operators. Finally, we present numerical examples which confirm the validity of the theoretical analysis and the effectiveness of the presented operators. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast and accurate model for unsteady pressure-driven flows in circular microchannels of variable cross-section. The model is developed for channels of small diameter to length ratio, but allows for large variations in the channel’s diameter along the axis. A key feature of the model is that it puts no restriction on the time dependence of the forcing, in terms of shape and frequency. The only condition on the forcing is such that the advective component of the inertia term is small. This is a major departure from many previous expositions which assume harmonic forcing. The model is based on an extended and unsteady lubrication approximation in the aspect ratio of the channel. The resulting equations for each order are solved analytically using a finite Hankel transform, except for the implicit pressure profile, which is solved numerically with a recursive time scheme. Compared to classical CFD simulations, the reduced order semi-analytic method is two orders of magnitude faster, owing in part to the fact that the number of modes required for the convergence of these expressions is not too large. The numerical simulations reveal that the model is accurate for a large class of channels and a fairly wide range of Reynolds numbers. This, combined with the fact that it imposes no conditions on the shape and frequency of the unsteady forcing, renders the model a valuable tool for rapidly simulating large fluidic circuits (as in lab-on-a-chip, μ TAS and the human body), thereby allowing significant reduction in the design parameters space. [ABSTRACT FROM AUTHOR]

By the modified directed likelihood, higher order accurate confidence limits for a scalar parameter are obtained from the likelihood. They are conveniently described in terms of a confidence distribution, that is a sample dependent distribution function on the parameter space. In this paper we explore a different route to accurate confidence limits via tail-symmetric confidence curves, that is curves that describe equal tailed intervals at any level. Instead of modifying the directed likelihood, we consider inversion of the log-likelihood ratio when evaluated at the median of the maximum likelihood estimator. This is shown to provide equal tailed intervals, and thus an exact confidence distribution, to the third-order of approximation in regular one-dimensional models. Median bias correction also provides an alternative approximation to the modified directed likelihood which holds up to the second order in exponential families. [ABSTRACT FROM AUTHOR]

When applying Laplace’s method to approximate an integral, we require an exact mode of its integrand. In this paper, we derive a Laplace approximation using an asymptotic mode of order O p ( n − α ) ( α ≥ 1 ∕ 2 ) instead of the exact mode, and present some ways to establish asymptotic modes with a suitable order. The class of the asymptotic modes is wide and includes n α -consistent estimators ( α ≥ 1 ∕ 2 ) and a true parameter. Second-order approximations and asymptotic properties to posterior expectations of smooth functions are derived by the proposed approximation. [ABSTRACT FROM AUTHOR]

We model the peristaltic flow of a Bingham fluid in a tube in lubrication approximation. Following the procedure developed in Fusi et al. (2015a) we derive the rigid plug equation using an integral formulation for the balance of linear momentum, modelling the unyielded domain as an evolving non-material volume. The mathematical problem is formulated for the yielded and unyielded part and appropriate boundary conditions are established at the pipe walls and at the yield surface. The zero order approximation leads to a system formed by an integral equation and an algebraic equation for the yield surface and for the plug velocity (which is uniform in space), respectively. Because of the integral approach adopted in the unyielded part of the flow, the leading order approximation does not give rise to the lubrication paradox. The problem is solved numerically and an analytical solution is found when the oscillating wall is given as a small perturbation of the uniform wall. [ABSTRACT FROM AUTHOR]

In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the n th harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujan’s asymptotic expansion for the n th harmonic number. We also give a recurrence relation for determining the coefficients a j ( r ) such that H n : = ∑ k = 1 n 1 k ∼ 1 2 ln ( 2 m ) + γ + 1 12 m ( ∑ j = 0 ∞ a j ( r ) m j ) 1 / r as n  → ∞, where m = n ( n + 1 ) / 2 is the n th triangular number and γ is the Euler–Mascheroni constant. In particular, for r = 1 , we obtain Ramanujan’s expansion for the harmonic number. [ABSTRACT FROM AUTHOR]

In this paper we provide a rigorous derivation of an asymptotic formulae for perturbations in the resonant frequencies of the two-(or three-) dimensional Laplacian operator under geometric variation of the domain. The asymptotic expansion is developed in the presence and with respect to size of the (anisotropic) imperfections of small shapes having constitutive parameters different from the background conductivity. The main feature of the method is to yield a robust procedure making it possible to recover information about the location, shape, and material properties of the anisotropic imperfections. [ABSTRACT FROM AUTHOR]

The algorithm by Al-Mohy and Higham (2011) [2] computes an approximation to e A b for given A and b , where A is an n -by- n matrix and b is, for example, a vector of dimension n . It uses a scaling together with a truncated Taylor series approximation to the exponential of the scaled matrix. In this paper, a method is developed for estimating the roundoff error of the computed solution. An asymptotic expansion of this error for small values of the unit roundoff is the basis of the method. The roundoff error is further expressed in terms of sums of rounding errors, which occur during the computation. A second approximation to e A b , which is computed with a lower precision than the first one, is used to evaluate these rounding errors in practice. The result is an upper bound on the normwise relative roundoff error. Further, an algorithm is proposed for computing the error bound. The cost for performing this algorithm depends on the type of problem and the accuracy, which is required for the error estimate. In case that all computations are performed with standard precisions, this cost can be expressed in terms of the number of computed matrix-vector products and is bounded from above by two times the cost for computing e A b . [ABSTRACT FROM AUTHOR]

We initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise for the isoparametric bilinear finite volume element scheme by employing the energy-embedded method on some non-uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Numerical examples confirm theoretical results. [ABSTRACT FROM AUTHOR]

We compute the leading mass corrections to the high-energy behavior of the massive quark vector form factor to three loops in QCD in the double-logarithmic approximation. [ABSTRACT FROM AUTHOR]

We show that the convergence rate of asymptotic expansions for solutions of SDEs is higher in the case of degenerate diffusion compared to the elliptic case, i.e. it is higher when the Brownian motion directly acts only along some directions. In the scalar case, this phenomenon was already observed in Gobet and Miri 2014 using Malliavin calculus techniques. Here, we provide a general and detailed analysis by employing the recent study of intrinsic functional spaces related to hypoelliptic Kolmogorov operators in Pagliarani et al. 2016. Applications to finance are discussed, in the study of path-dependent derivatives (e.g. Asian options) and in models incorporating dependence on past information. [ABSTRACT FROM AUTHOR]