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• A new singular boundary element is constructed to model the singular stress fields. • The proposed method can obtain good results without much computational cost. • The first four amplitude coefficients can be determined without post processing. The stress singularity occurs near a V-notch. The conventional boundary element method can only approach to the exact results with gradually refined mesh. In this paper, by introducing the Williams asymptotic expansion, a novel semi-analytical element is proposed. The new element models the geometry and the known physical fields with linear interpolation, while the unknown physical fields will be simulated by the asymptotic expansion. The new element introduces six unknowns: two displacement components at the notch tip and four amplitude coefficients in the asymptotic expansion. By collocating two semi-analytical elements on both sides of the notch, the unknown displacement fields can be depicted with the asymptotic expansion, while the remaining part can be modeled by means of the conventional boundary element. The direct integral method and singular integral separation technique are adopted for the different kinds of singular integrals in the boundary integral equations. After the system of equations are solved, not only the unknown boundary physical quantities, but also the first four amplitude coefficients in the asymptotic expansion can be determined at the same time. The stress intensity factors can be determined from the amplitude coefficients in the asymptotic expansions without any post processing. [ABSTRACT FROM AUTHOR]

In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications. [ABSTRACT FROM AUTHOR]

• This paper modifies the double Heston model as a singular and regular perturbed BS model. • This paper derives explicit expressions of the asymptotic expansions for geometric Asian options with fixed and floating strikes under the modified double Heston model 3. The paper provides the convergence of the asymptotic formula. • The paper provides the Greeks of geometric Asian options prices. • The paper calibrates the modified double Heston model to real markets and examines the impacts of the two-factor volatility on geometric Asian option prices with calibrated parameters. The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices. [ABSTRACT FROM AUTHOR]

The density function of t -distribution with n degrees of freedom is given by the following formula: f n ( x ) = Γ ( n + 1 2 ) π n Γ ( n 2 ) 1 + x 2 n − ( n + 1 ) ∕ 2 , x > 0 . Ding (1988) obtained the following asymptotic formula in terms of 1 ∕ n : f n ( x ) = φ ( x ) { 1 + x 4 − 2 x 2 − 1 4 n + 3 x 8 − 28 x 6 + 30 x 4 + 12 x 2 + 3 96 n 2 + O 1 n 3 } , n → ∞ , which derives the known result lim n → ∞ f n ( x ) = φ ( x ) , where φ ( x ) = 1 2 π e − x 2 ∕ 2 is the probability density function of the standard normal distribution. In this paper, we develop Ding’s result to produce a complete asymptotic expansion. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation of limit cycles in two special near-Hamiltonian polynomial planer systems which their corresponding Hamiltonian systems have a heteroclinic loop connecting a hyperbolic saddle and a cusp of order two. In these systems, we will compute the asymptotic expansions of corresponding first order Melnikov functions near the loop and the center to analyze the number of limit cycles. Moreover, in the first system, by using the Chebychev criterion, we study the Poincaré bifurcation. [ABSTRACT FROM AUTHOR]

The Prabhakar function (namely, a three parameter Mittag–Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak–Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied. [ABSTRACT FROM AUTHOR]

In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by n th degree polynomials, n = 1 , 2 , ... , 13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n − 1 − n − 2 4 limit cycles near the non-regular homoclinic loop and n − 2 4 limit cycles near the center, and it totally has at least n − 1 limit cycles, when n ∈ { 1 , 2 , ... , 13 }. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n = 3. • A complete discussion for the maximal number of limit cycles near the non-regular homoclinic loop and the center for a quadratic reversible Hamiltonian system perturbed by n -th degree polynomials (n = 1, 2, ...,13) is given. • The maximal number of limit cycles near the non-regular homoclinic loop is at least n − 1 − n − 2 4 and near the center is n − 2 4 when n ∈ { 1 , 2 , ... , 13 }. The obtained results are novel. • The upper bound on the number of limit cycles of the perturbed quadratic reversible Hamiltonian system with n = 3 is given. This is also a new discover. [ABSTRACT FROM AUTHOR]

Let X , X 1 , X 2 , … be a sequence of non-lattice i.i.d. random variables with E X = 0 , E X = 1 , and let S n : = X 1 + ⋯ + X n , n ≥ 1 . We refine Stone’s integro-local theorem by deriving the first term in the asymptotic expansion, as n → ∞ , for the probability P ( S n ∈ [ x , x + Δ ) ) , x ∈ R , Δ > 0 , and establishing uniform in x and Δ bounds for the remainder term, under the assumption that the distribution of X satisfies Cramér’s strong non-lattice condition and E | X | r < ∞ for some r ≥ 3 . [ABSTRACT FROM AUTHOR]

The article presents an algorithm of Student's t-statistic distribution function. The procedure provides results accurate to 11 decimal places and 8 significant digits for integer values of "n," with approximate continuation of the function through non-integer values of n. The procedure normal (x) returns the area under the standard normal frequency curve to the left of x, so that a negative argument yields the lower-tail area. The user-supplied procedure, error (n), should produce a diagnostic warning and may go to a label, terminate, or return a distinctive value (zero or -1.0) as a signal of error to the calling program. Student's series expansion of the probability integral is supplemented by a faster asymptotic approximation for large values of n and by a more precise "tail" series expansion for large values of t.

Discrete fine-scale models, in the form of either particle or lattice models, have been formulated successfully to simulate the behavior of quasi-brittle materials whose mechanical behavior is inherently connected to fracture processes occurring in the internal heterogeneous structure. These models tend to be intensive from the computational point of view as they adopt an “a priori” discretization anchored to the major material heterogeneities (e.g. grains in particulate materials and aggregate pieces in cementitious composites) and this hampers their use in the numerical simulations of large systems. In this work, this problem is addressed by formulating a general multiple scale computational framework based on classical asymptotic analysis and that (1) is applicable to any discrete model with rotational degrees of freedom; and (2) gives rise to an equivalent Cosserat continuum. The developed theory is applied to the upscaling of the Lattice Discrete Particle Model (LDPM), a recently formulated discrete model for concrete and other quasi-brittle materials, and the properties of the homogenized model are analyzed thoroughly in both the elastic and the inelastic regime. The analysis shows that the homogenized micropolar elastic properties are size-dependent, and they are functions of the RVE size and the size of the material heterogeneity. Furthermore, the analysis of the homogenized inelastic behavior highlights issues associated with the homogenization of fine-scale models featuring strain-softening and the related damage localization. Finally, nonlinear simulations of the RVE behavior subject to curvature components causing bending and torsional effects demonstrate, contrarily to typical Cosserat formulations, a significant coupling between the homogenized stress–strain and couple-curvature constitutive equations. [ABSTRACT FROM AUTHOR]

We present the formula for a certain integral with respect to multivariate Hermite polynomials. Such integrals are used for deriving higher-order local power functions of asymptotically chi-squared tests. As an example, we provide asymptotic expansion for the local power function of Rao’s score test. [ABSTRACT FROM AUTHOR]

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error. [ABSTRACT FROM AUTHOR]

It is known that the heat flux becomes infinite near the notch tip due to material or geometric discontinuities. Numerical methods such as boundary element method and finite element method cannot adequately analyze the accurate singular heat flux field since they traditionally employ piecewise polynomials. In this paper, the singularity analysis method coupled with the boundary element technique is proposed for the accurate analysis of two-dimensional singular heat flux field near the notch tip. The V-notched structure is departed into two parts, which are the near tip singular sector and far tip non-singular section. The singularity analysis is executed on the near notch tip sector for searching singularity orders and corresponding characteristic angular functions by introducing the heat flux asymptotic expansions into heat conduction governing equations. The conventional boundary element method is applied to modeling the far notch tip region because there is no heat flux singularity. The asymptotic expansions of the near tip physical field and the boundary integral equations established on the far notch tip region are combined together for solving the expansion coefficients in heat flux asymptotic expansions. Thus, the complete heat flux field near the notch vertex can be accurately determined. [ABSTRACT FROM AUTHOR]

Highlights: [•] We presented the groundwater flow equation in prolate coordinate. [•] We solved the new equation via Green’s function and perturbation methods. [•] We compared the new solution with observed data from real word. [•] We presented the error analysis. [Copyright &y& Elsevier]

Abstract: This article presents two regularization techniques for systems of state-dependent neutral delay differential equations which have a discontinuity in the derivative of the solution at the initial point. Such problems have a rich dynamics and besides classical solutions can have weak solutions in the sense of Utkin. Both of the presented techniques permit the numerical solution of such problems with the code RADAR5, which is designed to compute classical solutions of stiff and differential-algebraic (state-dependent) delay equations. [Copyright &y& Elsevier]

Abstract: Higher-order asymptotic arguments for a scalar parameter of interest have been widely investigated for Bayesian inference. In this paper the theory of asymptotic expansions is discussed for a vector parameter of interest. A modified loglikelihood ratio is suggested, which can be used to derive approximate Bayesian credible sets with accurate frequentist coverage. Three examples are illustrated. [Copyright &y& Elsevier]

Abstract: A higher-order approximation to the marginal posterior distribution for a scalar parameter of interest in the presence of nuisance parameters is proposed. The approximation is obtained using a matching prior. The procedure improves the normal first-order approximation and has several advantages. It does not require the elicitation on the nuisance parameters, neither numerical integration nor Monte Carlo simulation, and it enables us to perform accurate Bayesian inference even for small sample sizes. Numerical illustrations are given for models of practical interest, such as linear non-normal models and logistic regression. Finally, it is shown how the proposed approximation can routinely be applied in practice using results from likelihood asymptotics and the R package bundle hoa. [Copyright &y& Elsevier]

Abstract: An approximation to order is obtained for the bias of the full vector of least-squares estimates obtained from a sample of size in general stable but not necessarily stationary ARX(1) models with normal disturbances. This yields generalizations, allowing for various forms of initial conditions, of Kendall’s and White’s classic results for stationary AR(1) models. The accuracy of various alternative approximations is examined and compared by simulation for particular parameterizations of AR(1) and ARX(1) models. The results show that often the second-order approximation is considerably better than its first-order counterpart and hence opens up perspectives for improved bias correction. However, order approximations are also found to be more vulnerable in the near unit root case than the much simpler order approximations. [Copyright &y& Elsevier]

Abstract: The Bartlett-type adjustment is a higher-order asymptotic method for reducing the errors of the chi-squared approximations to the null distributions of various test statistics, which ensures that the resulting test has size , where is the significance level and is the sample size. Recently, has revisited the Chandra–Mukerjee/Taniguchi adjustments in a unified way, since and originally considered the test of the simple null hypothesis, except for . This paper considers a generalization of the adjustment due to . [Copyright &y& Elsevier]

Abstract: The asymptotic expansion treatment of the homogenization problem for nonlinear purely mechanical or thermal problems exists, together with the treatment of the coupled problem in the linearized setting. In this contribution, an asymptotic expansion approach to homogenization in finite thermoelasticity is presented. The treatment naturally enforces a separation of scales, thereby inducing a first-order homogenization framework that is suitable for computational implementation. Within this framework two microscopically uncoupled cell problems, where a purely mechanical one is followed by a purely thermal one, are obtained. The results are in agreement with a recently proposed approach based on the explicit enforcement of the macroscopic temperature, thereby ensuring thermodynamic consistency across the scales. Numerical investigations additionally demonstrate the computational efficiency of the two-phase homogenization framework in characterizing deformation-induced thermal anisotropy as well as its theoretical advantages in avoiding spurious size effects. [Copyright &y& Elsevier]

Abstract: The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced at a point of the domain. In this fracture mechanics work we use the topological derivative to approximate the energy release rate field associated with a small crack at any boundary location and at any orientation. Our proposed method offers significant computational advantages over current finite element based methods since it requires a single analysis, whereas the others require a distinct analysis for each crack location–orientation combination. Moreover, the proposed method evaluates the topological derivative in the non-cracked domain which eliminates the need for tailored meshes in the crack region. [Copyright &y& Elsevier]

Abstract: We give a general matrix formula for computing the second-order skewness of maximum likelihood estimators. The formula was firstly presented in a tensorial version by . Our matrix formulation has numerical advantages, since it requires only simple operations on matrices and vectors. We apply the second-order skewness formula to a normal model with a generalized parametrization and to an ARMA model. [Copyright &y& Elsevier]

Abstract: The family of distributions proposed by  can be used to model lifetime data and it is widely applicable to model failure times of fatiguing materials. We give a simple matrix formula of order , where is the sample size, for the skewness of the distributions of the maximum likelihood estimates of the parameters in Birnbaum–Saunders nonlinear regression models, recently introduced by . The formula is quite suitable for computer implementation, since it involves only simple operations on matrices and vectors, in order to obtain closed-form skewness in a wide range of nonlinear regression models. Empirical and real applications are analyzed and discussed. [Copyright &y& Elsevier]

Abstract: In this paper, based on moment-type and location invariant Hill estimators, a new kind of location invariant moment-type extreme value index estimator is proposed. The weak and strong consistency of the estimator are discussed. The asymptotic expansion of the estimator and its distribution are also considered under second order regularly varying conditions. The asymptotic normality is employed to construct the confidence interval. Monte Carlo simulations are performed to compare the Hill estimator and the moment estimator in terms of mean squared error. [Copyright &y& Elsevier]

Abstract: A new technique concerning the evaluation of multiple stress singularity orders of a V-notch with the boundary element method is proposed. For linear elastic material, the asymptotic displacement field in a V-notch tip region is expressed as a series expansion with respect to the radial coordinate from the V-notch tip. The series expansion of the asymptotic field is then substituted into the boundary integral equation established on the V-notch. By boundary element discretization, the boundary integral equation is transformed into an eigen-equation with respect to the stress singularity orders. Hence the eigen-values, which are the singularity orders, can be obtained simultaneously by solving the eigen-equation. An evident feature of the present method is that the real and complex singularity orders together with the non-singularity ones can be obtained at the same time. [Copyright &y& Elsevier]

Abstract: Approximate closed form representations of functions are useful for mathematical manipulations. Nonlinear sequence transforms can be used to evaluate the function using a few terms of the series representation of the function and these transforms can be used for functions with complex argument as well. Moreover, if an asymptotic expansion of the function is available, an approximant for the function, valid for the entire range of the variable, can be obtained with Padé approximants as well as Levin and Weniger transforms. In addition, one can obtain an approximation for a function using quadratic Padé approximation which is also valid for the entire range of the variable. We demonstrate this for some functions frequently encountered in scientific problems. These include the error function, the Fresnel integral, the Dawson integral, the Euler integral and the elliptic integral. A comparison is made between the approximants obtained with Padé approximants and those generated by Levin and Weniger transforms. [Copyright &y& Elsevier]

Abstract: To utilize the potential of adhesive bonding, there is an increasing need for effective and accurate computational methods. The geometry and behaviour of an adhesive joint is, however, not so simple to model effectively by regular finite elements. The main reason is that the very thin adhesive layer with a low Young’s modulus must be modelled by a large number of finite elements in the thickness direction to achieve sufficiently accurate calculations. To overcome this difficulty, a material surface treatment of the adhesive and the joined parts can be attempted. This paper concerns the derivation of such a model by introducing scalings on the geometry and on the material properties in terms of a perturbation parameter. Within the framework of three-dimensional elasticity, together with an asymptotic expansion method, a family of limit models are obtained through a systematic procedure. In such a derivation no a priori assumptions on the displacements or stress fields are needed. The final result is a variational equation posed over a single reference surface. In regions near the boundary of the joint a boundary layer phenomena occurs. This indicates that the asymptotic series needs to be complemented by additional terms, in order to satisfy all boundary conditions. A structural model including shear- and peel deformation is finally proposed which improves the solution close to the boundary. [Copyright &y& Elsevier]

Abstract: In this paper, a semi-Markovian random walk process with a discrete interference of chance is constructed and the ergodicity of this process is discussed. Some exact formulas for the first- and second-order moments of the ergodic distribution of the process are obtained, when the random variable has an exponential distribution with the parameter . Here expresses the quantity of a discrete interference of chance. Based on these results, the third-order asymptotic expansions for mathematical expectation and variance of the ergodic distribution of the process are derived, when . [Copyright &y& Elsevier]

Abstract: Trapped modes and bound states occur in waveguides in a wide variety of physical situations, here we consider one class: the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. We consider the possibility of trapped modes in otherwise uniform elastic/ocean/quantum waveguides, that have a perturbation in thickness, or in curvature. These are all mathematically connected, so their study can be unified, and the same asymptotic procedure can be applied to them all. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility, or otherwise, of trapping. An asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. A mathematical explanation and physical argument for the existence, or otherwise, of trapped modes, depending on different situations, is given. The asymptotic approach leads to an ordinary differential equation eigenvalue problem and its solutions are compared both with numerics, for the full governing equations, and with a further simplification that gives highly accurate estimates for the lowest eigenvalue. [Copyright &y& Elsevier]

Abstract: A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip plane under the action of an external force in the direction of a locked dislocation in that plane is considered. As there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away. [Copyright &y& Elsevier]

Abstract: This paper is concerned with asymptotic behavior of sample skewness and kurtosis. The asymptotic expansions under non-normality are obtained for the sample measures of multivariate skewness and kurtosis defined by Srivastava [1984. A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statist. Probab. Lett. 2, 263–267.]. [Copyright &y& Elsevier]

Abstract: Trapped modes within elastic waveguides have been shown to exist for bent guides, rods and thickened plates (in plane stress). Here we concentrate upon the complementary problem of thickened or thinned elastic guides in plane strain. We develop an asymptotic procedure that encapsulates the essential physics within a single ordinary differential equation. It is shown that this reduced model is functionally the same as that for a bent plate and hence the two problems are mathematically identical. Two-humped, or doubly bent, waveguides are also considered and it is shown that trapping can occur localised at the geometric variation whether it be at the humps or bends or in-between them. Physical arguments for the trapping are advanced and numerical simulations of the full elasticity equations are performed to both demonstrate the accuracy of the asymptotic model and to validate it. [Copyright &y& Elsevier]

In this paper, we derive a simple matrix formula for second-order covariances of maximum likelihood estimates in generalized linear models. The formula covers many important and commonly used models and is also simple enough to be used algebraically to obtain closed-form expressions in special models. The practical use of this formula is illustrated in a simulation study. [Copyright &y& Elsevier]

Asymptotic expansions of densities of the normalized sums of random vectors with at least finite third moment have been studied extensively (Normal Approximation and Asymptotic expansions. Wiley, New York.). In this note, we obtain the asymptotic expansions of densities of the normalized sums of i.i.d. random vectors with regularly varying density with index between 4 and 5, which implies that third moment is infinite. [Copyright &y& Elsevier]

• Planar near-Hamiltonian systems are considered. • Bifurcation of limit cycles near heteroclinic loops is studied. • More coefficients of the expension of Melnikov functions are computed. • With more coefficients, more limit cycles could be determined. In this paper, we study the bifurcation of limit cycles near a heterocilinc loop with hyperbolic saddles in a perturbed planar Hamiltonian system. We present a method for computing the coefficients in the corresponding expansion of the first order Melnikov function. With more those coefficients, more limit cycles could be determined around the heteroclinic loop. An example of studying limit cycles produced from a heteroclinic loop with 2 saddles is investigated to illustrate our method. [ABSTRACT FROM AUTHOR]

• This paper studies the vulnerable option under the multiscale stochastic volatility model. • An analytic pricing formula of option is derived using the asymptotic expansion method. • The Greek Delta for the hedge is derived. • Numerical examples are provided to illustrate the properties of the option under the proposed model. In this study, we examine the pricing of vulnerable options under a stochastic volatility model based on the partial differential equation approach. Specifically, we consider a multiscale stochastic volatility model that is assumed to be driven by two diffusions (fast-scale and slow-scale) and use an asymptotic expansion approach to drive the approximate pricing formulas of vulnerable options, which allows the counterparty credit risk at maturity. Furthermore, we provide the Greek Delta of vulnerable options for the dynamic hedge and present the numerical results to examine the effect of the multiscale stochastic volatility model and to show the accuracy of our formula. [ABSTRACT FROM AUTHOR]

Meta-analyses of diagnostic test accuracy (DTA) studies have been gathering attention in research in clinical epidemiology and health technology development, and bivariate random-effects model is becoming a standard tool. However, standard inference methods usually underestimate statistical errors and possibly provide highly overconfident results under realistic situations since they ignore the variability in the estimation of variance parameters. To overcome the difficulty, a new improved inference method, namely, an accurate confidence region for the meta-analysis of DTA, by asymptotically expanding the coverage probability of the standard confidence region. The advantage of the proposed confidence region is that it holds a relatively simple expression and does not require any repeated calculations such as Bootstrap or Monte Carlo methods to compute the region, thereby the proposed method can be easily carried out in practical applications. The effectiveness of the proposed method is demonstrated through simulation studies and an application to meta-analysis of screening test accuracy for alcohol problems. [ABSTRACT FROM AUTHOR]