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Limit distribution theory in the econometric literature for functional coefficient cointegrating regression is incorrect in important ways, influencing rates of convergence, distributional properties, and practical work. The correct limit theory reveals that components from both bias and variance terms contribute to variability in the asymptotics. The errors in the literature arise because random variability in the bias term has been neglected in earlier research. In stationary regression this random variability is of smaller order and can be ignored in asymptotic analysis but not without consequences for finite sample performance. Implications of the findings for rate efficient estimation are discussed. Simulations in the Online Supplement provide further evidence supporting the new limit theory in nonstationary functional coefficient regressions.

Starting from a 2 × 2 discrete matrix spectral problem, we construct an integrable lattice hierarchy associated with RTL + ( α ) system which is a one-parameter perturbation of the usual Toda lattice system describing the particle motion in a lattice. Based on the obtained RTL + ( α ) hierarchy, some integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Furthermore, we establish the discrete N -fold Darboux transformation (DT) of RTL + ( α ) system. As applications of the obtained DT, multi-soliton solutions are derived on constant seed backgrounds, and discrete soliton propagation and elastic interaction structures are shown graphically and analyzed via asymptotic analysis. Numerical simulations are utilized to demonstrate the dynamical behaviors of such soliton solutions. Numerical results show that soliton evolutions are stable against a small noise. Results given in this paper might be helpful for understanding the propagation of nonlinear waves in soliton theory.

Introduction The Gross-Pitaevskii equation is a class of the nonlinear Schrodinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. Objectives We give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. Methods By constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. Results The results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. The results enrich the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

In this paper, an analytical approach based on the asymptotic analysis is proposed to develop an axisymmetric model for the radial forging process of rods. This model is able to predict the deformation inhomogeneity in the radial direction and approves that the profile of the neutral plane is not a planar cross section of the workpiece. For the first time, a relationship has been developed between the die angle, friction coefficient and back-push stress that can be used as a criterion for examining the feasibility of the radial forging process. A breakthrough method has also been proposed that can effectively find the location of the neutral plane far easier and faster than the traditional approach. Finite element simulation of the process has been performed to validate the deformation pattern obtained by the asymptotic model. Models are also verified against available experimental data. The findings of this study are especially useful for online design, analysis and control of the radial forging process.

Poisson shot noise processes are natural generalizations of compound Poisson processes that have been widely applied in insurance, neuroscience, seismology, computer science and epidemiology. In this paper we study sharp deviations, fluctuations and the stable probability approximation of Poisson shot noise processes. Our achievements extend, improve and complement existing results in the literature. We apply the theoretical results to Poisson cluster point processes, including generalized linear Hawkes processes, and risk processes with delayed claims. Many examples are discussed in detail.

Our first objective is to identify two-dimensional equations that model the displacement of a linearly elastic flexural shell subjected to the action of an external heat source. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface θ ( ω ‾ ) , where ω is a domain in R 2 and θ : ω ‾ → E 3 is a smooth enough immersion and whose thickness 2 e > 0 is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as e approaches zero, the corresponding “limit” two-dimensional variational problem. Our second objective is to identify and justify a set of two-dimensional equations that are meant to approximate the original three-dimensional model in the case where the shell under consideration is either an elliptic membrane shell or a flexural shell.

The present article describes the g-jitter effect on the rate of mass transport (Sherwood number Sh) in the electrically-conducting Newtonian fluid between two infinite horizontal plates at a distance ’d’ apart. It is assumed that concentration at the upper wall is S 0 + ▵ S and S0 is at the lower wall. A magnetic force of constant strength H0 is applied perpendicular to the XOY-plane which generates an induced magnetic field. We have used the asymptotic analysis method to solve the system of differential equations. The expression of Sherwood number depends on the amplitude of stream function which can be determined by the real cubic Landau equation. The influence of different dimensionless quantities on the convective mass transport are presented graphically. It is also observed that the Sherwood number could be tuned suitably by adjusting the amplitude and frequency of gravity modulation.

New expressions are derived for the Green function (GF) for diffraction and radiation of waves by a two-dimensional (2D) body in finite water depth. The finite depth GF is expressed as a sum of singularities, the infinite depth GF and smoothly-varying integrals that are convergent for all parameter values. The infinite depth component is given explicitly, making it very fast to compute. Explicit expressions are derived for the limiting cases of zero and infinite frequency, for both finite and infinite water depth. The low frequency limit of the 2D GF is inconsistent with the zero-frequency 2D GF, with the real part tending to infinity when the water depth is infinite and the imaginary part tending to infinity in finite water depth. The inconsistencies with the zero frequency GF differ between the 2D and 3D cases. These inconsistencies lead to differences between the low-frequency behaviour of the added mass and damping of an oscillating body and the values at zero frequency. It is shown that these differences can be inferred directly from the behaviour of the GF at low frequencies.

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

The generic model of a cable with small bending stiffness and anchored to flexible supports in rotation and translation is considered. An asymptotic analysis of the natural frequencies of this generic model is derived and shows that, for small bending stiffness, the first few natural frequencies can be expressed as a function of the cable axial force, the small bending stiffness and a single dimensionless group collecting the information of all other problem parameters. This formulation is used to develop an identification procedure of the cable axial force. Two formulations are proposed, one numerical and one semi-analytical based on a simple linear regression model. Both methods do not attempt at separately identifying the problem parameters since the observability analysis has revealed that only the cable axial force, the bending stiffness and the dimensionless group can be identified. In particular, the second method is very simple to implement and provides estimates of the cable axial force which account for the flexibility of the support. The proposed method can therefore be seen as an extension of usual identification techniques based on linear regressions of natural frequencies vs. mode number relations, by considering at the same time the bending stiffness and the deformability of supports. Being simple and robust as shown by means of an uncertainty quantification analysis, the proposed method can be conveniently embedded in the framework of a continuous monitoring strategy.

Stochastic semidefinite programming (SSDP) is a new class of optimization problems with a wide variety of applications. In this article, asymptotic analysis results of sample average approximation estimator for SSDP are established. Asymptotic analysis result already existing for stochastic nonlinear programming is extended to SSDP, that is, the conditions ensuring the convergence in distribution of sample average approximation estimator for SSDP to a multivariate normal are obtained and the corresponding covariance matrix is described in a closed form.

The paper obtains analytical results for the asymptotic properties of Model Selection Criteria – widely used in practice – for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical ‘i.i.d.-like’ model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is not weakly consistent. Numerical experiments support our theoretical results.

In this paper, we investigate a generalized nonlinear Schrodinger equation with two certain terms in the single-mode optical fibers. Via the binary Bell polynomials, a more general bilinear form and analytic solutions are obtained. On the basis of those solutions, we present the parametric regions for the existence of the solitons and breathers on a nonzero background. From the soliton solutions, we prove the soliton interaction to be elastic through the asymptotic analysis. The interactions can be observed between (i) two dark solitons, (ii) two anti-dark solitons, and (iii) one dark and one anti-dark solitons, and relevant parametric conditions of the three types of interactions are given. From the breather solutions, the general breathers, Akhmediev breathers and Kuznetsov–Ma breathers can be respectively derived with the different parametric conditions. Besides, we obtain some rational solutions by taking the limit of the breather solutions. Based on those rational solutions, the parametric conditions for the soliton interactions and rogue waves are given.

We study the Cauchy problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation i q t ( x , t ) + q x x ( x , t ) + 2 q 2 ( x , t ) q ¯ ( − x , t ) = 0 with a step-like initial data: q ( x , 0 ) = q 0 ( x ) , where q 0 ( x ) = o ( 1 ) as x → − ∞ and q 0 ( x ) = A + o ( 1 ) as x → ∞ , with an arbitrary positive constant A > 0 . The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane − ∞ x ∞ , t > 0 : (i) for x 0 , the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for x > 0 , the solution approaches the “modulated constant”. The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.

We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ω e in the plane R 2 . Here Ω e is Ω ∪ ω e ∪ Γ , where Ω is a fixed bounded domain with boundary Γ, ω e is a curvilinear band of width O ( e ) , and Γ = Ω ‾ ∩ ω ‾ e . The density and stiffness constants are of order e − m − t and e − t respectively in this band, while they are of order 1 in Ω; t ≥ 1 , m > 2 , and e is a small positive parameter. We address the asymptotic behavior, as e → 0 , for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Γ.

Locally weighted jackknife prediction(LW-JP) is a variant of conformal prediction, which can output interval prediction in regression problems built on traditional learning algorithms for point prediction named as underlying algorithms. Although empirical validity and efficiency of LW-JP have been reported in some works, there lacks theoretical understanding of it. This paper gives some theoretical analysis of LW-JP in the asymptotic setting, where the number of the training samples approaches infinity. Under some regularity assumptions and conditions, the asymptotic validity of LW-JP is proved in the nonlinear regression case with heteroscedastic errors. The proof is an extension of the asymptotic analysis of leave-one-out prediction intervals in linear regression with homoscedastic errors. Based on our analysis, two conformal regressors built on LW-JP are proposed and the experimental results showed that the algorithms are not only valid interval predictors, but also achieve the state-of-the-art performance of conformal regressors.

The paper develops point estimation and asymptotic theory with respect to a semiparametric model for time series with moving mean and unconditional heteroscedasticity. These two features are modelled nonparametrically, whereas autocorrelations are described by a short memory stationary parametric time series model. We first study the usual least squares estimate of the coefficient of the first-order autoregressive model based on constant but unknown mean and variance. Allowing for both the latter to vary over time in a general way we establish its probability limit and a central limit theorem for a suitably normed and centred statistic, giving explicit bias and variance formulae. As expected mean variation is the main source of inconsistency and heteroscedasticity the main source of inefficiency, though we discuss circumstances in which the estimate is consistent for, and asymptotically normal about, the autoregressive coefficient, albeit inefficient. We then consider standard implicitly-defined Whittle estimates of a more general class of short memory parametric time series model, under otherwise more restrictive conditions. When the mean is correctly assumed to be constant, estimates that ignore the heteroscedasticity are again found to be asymptotically normal but inefficient. Allowing a slowly time-varying mean we resort to trimming out of low frequencies to achieve the same outcome. Returning to finite order autoregression, nonparametric estimates of the varying mean and variance are given asymptotic justification, and forecasting formulae developed. Finite sample properties are studied by a small Monte Carlo simulation, and an empirical example is also included.

In this paper, we consider the existence of traveling wave fronts in a Belousov-Zhabotinskii system with delay. By traveling wave transformation and time scale transformation, we change the Belousov-Zhabotinskii system with delay into a singularly perturbed differential system. By applying geometric singular perturbation theory, we construct a locally invariant manifold for the associated traveling wave equation and obtain the traveling wave fronts for the equation by using the Fredholm orthogonality. Finally, we discuss the asymptotic behaviors of traveling wave solutions by applying the asymptotic theory.

We study the mean exit time from a bounded multi-dimensional domain Ω of the stochastic process governed by the overdamped Langevin dynamics. This mean exit time solves the boundary value problem ( − e 2 Δ + ∇ V ⋅ ∇ ) u e = 1 in Ω , u e = 0 on ∂ Ω , e → 0 . The function V is smooth enough and has the only minimum at the origin contained in Ω; the minimum can be degenerate. At other points of Ω, the gradient of V is non-zero and the normal derivative of V at the boundary ∂Ω does not vanish. Our main result is a complete asymptotic expansion for u e . The asymptotics for u e involves an exponentially large term, which we find in a closed form. We also construct a power in e asymptotic expansion such that this expansion and a mentioned exponentially large term approximate u e up to arbitrary power of e.

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems.

Classical portfolio selection problems that optimise expected utility can usually not be solved in closed form. It is natural to approximate the utility function, and we investigate the accuracy of this approximation when using Taylor polynomials. In the important case of a Merton market and power utility we show analytically that increasing the order of the polynomial does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. All results are illustrated by numerical examples.

This paper presents an asymptotic method-based analysis of composite laminates having interfacial imperfections. In general, imperfect interfaces are simply modeled by introducing a linear, spring-layer model, which empirically assumes that the displacement jumps that occur at the weakened interface are proportional to the transverse shear stresses of interface positions. In this study, we propose a composite plate model derived by using asymptotic expansion that does not make any assumptions, other than the scaling of coordinate systems. Within the framework of the asymptotic analysis, the spring-layer model is introduced to describe the effect of weakened interfaces, which is realized by the separation of domains in the through-the-thickness direction, and the integration of piecewise continuous warping functions. As a result, we newly define a spring parameter that is exactly the same as the stiffness of a spring. Therefore, a set of spring elements are added to the through-the-thickness modeling of the microscopic analysis, and the plate equations derived in the macroscopic problem are the same as those of the perfectly bonded laminates. As a consequence, we also derive the proposed plate model with the mathematical rigor that the previous asymptotic models contain. We provide some numerical results verifying that the proposed method shows good agreement with the elasticity and 3D FEM solutions.

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramer–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.

Korteweg-de Vries equation governs the weakly nonlinear long wave whose phase speed reaches a simple maximum of wave with the infinite length in shallow water wave. The exponential-form variable separation solution of (2+1)-dimensional Kortweg-de Vries equation is found via the two-function method, and this solution covers many special combined solutions including sinh-cosh,sin-cos,sech-tanh,csch-coth,sec-tan and csc-cot solutions. From the exponential-form solution with choosing suitable functions, inelastic interactions between special multi-valued solitons with two loops such as anti-bell-shaped, anti-peak-shaped semifoldons and anti-foldon are graphically and analytically studied. By the asymptotic analysis, phase shift and its difference during interactions between multi-valued solitons are analytically given.

International audience; In this contribution, we recall the derivation of the anti bounce back boundary condition for the D2Q9 lattice Boltzmann scheme. We recall various elements of the state of the art for anti bounce back applied to linear heat and acoustics equations and in particular the possibility to take into account curved boundaries. We present an asymptotic analysis that allows an expansion of all the fields in the boundary cells. This analysis based on the Taylor expansion method confirms the well known behaviour of anti bounce back boundary for the heat equation. The analysis puts also in evidence a hidden differential boundary condition in the case of linear acoustics. Indeed, we observe discrepancies in the first layers near the boundary. To reduce these discrepancies, we propose a new boundary condition mixing bounce back for the oblique links and anti bounce back for the normal link. This boundary condition is able to enforce both pressure and tangential velocity on the boundary. Numerical tests for the Poiseuille flow illustrate our theoretical analysis and show improvements in the quality of the flow.

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.

The aim of this paper is to report analytical and individual-based methods forantibiotic dose selection, that are based on tools from system and control theory. A brief system analysis of standard population pharmacokinetic models proves that such models are nonnegative and stable. Then, an input-output analysis leads to an open-loop control law which yields a dosing for the "average" patient, based on the equilibrium trajectory of the system. This approach is then incorporated into a "worst-case" analysis based on the monotony of the statetrajectories with respect to the clearance (model parameter). Finally, an heuristic method of an estimated state feedback is presented. Thanks to numerical simulations, these methods were successively illustrated on a model describing the pharmacokinetic of meropenem, an intravenous antibiotic for treatment of severe sepsis., The aim of this paper is to report analytical and individual-based methods for antibiotic dose selection, that are based on tools from system and control theory. A brief system analysis of standard population pharmacokinetic models proves that such models are nonnegative and stable. Then, an input-output analysis leads to an open-loop control law which yields a dosing for the "average" patient, based on the equilibrium trajectory of the system. This approach is then incorporated into a "worst-case" analysis based on the monotony of the state trajectories with respect to the clearance (model parameter). Finally, an heuristic method of an estimated state feedback is presented. Thanks to numerical simulations, these methods were successively illustrated on a model describing the pharmacokinetic of meropenem, an intravenous antibiotic for treatment of severe sepsis.

This paper considers the optimal multi-agent persistent monitoring problem defined for a team of agents on a set of nodes (targets) interconnected according to a fixed network topology. The aim is to control this team so as to minimize a measure of overall node state uncertainty evaluated over a finite time interval. A class of distributed threshold-based parametric controllers has been proposed in prior work to control agent dwell times at nodes and next-node destinations by enforcing thresholds on the respective node states. Under such a Threshold Control Policy (TCP), an on-line gradient technique was used to determine optimal threshold values. However, due to the non-convexity of the problem, this approach often leads to a poor local optima highly dependent on the initial thresholds used. To overcome this initialization challenge, we develop a computationally efficient off-line greedy technique based on the asymptotic analysis of the network system. This analysis is then used to generate a high-performing set of initial thresholds. Extensive numerical results show that such initial thresholds are almost immediately (locally) optimal or quickly lead to optimal values. In all cases, they perform significantly better than the locally optimal solutions known to date., Submitted to Automatica

We study stability of linear systems with fast time-varying coefficients. The classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable. However, it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis. In this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate. We transform the system to a model with time-delays of the length of the small parameter. The resulting time-delay system is a perturbation of the averaged LTI system which is assumed to be exponentially stable. The stability of the time-delay system guarantees the stability of the original one. We construct an appropriate Lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities (LMIs). The upper bound on the small parameter that preserves the exponential stability is found from LMIs. Two numerical examples (stabilization by vibrational control and by time-dependent switching) illustrate the efficiency of the method.

Optical fiber communication becomes one of the main pillars of modern communication. In this paper, we study a coupled nonlinear Schrodinger system with variable coefficients, which can describe the simultaneous propagation of the M-field components in an inhomogeneous optical fiber, where M is a positive integer. When M = 2 , the bilinear forms are constructed with the Hirota method and the N-bright-dark soliton solutions in terms of the Grammian can be obtained via the Kadomtsev-Petviashvili hierarchy reduction, where N is a positive integer. With the asymptotic analysis and graphic analysis, we find that the amplitudes of one-bright-dark solitons and the background in the dark components exhibit the periodic oscillations. Without the amplification/absorption effect, elastic interaction between the two-bright-dark solitons which keep both the amplitudes and the wave backgrounds invariant is demonstrated. Particularly, we find inelastic interaction between the two-bright-dark solitons, which possess the V-shape profiles in the zero background components and the Y-shape profiles in the periodic oscillating background components. The bound-state soliton under that inelastic condition is also shown. Besides, we present the bound-state bright-dark solitons with varying amplitudes. Furthermore, the analysis of the N-bright-dark soliton solutions are extended to those with M > 2, and as an example, inelastic interaction of the solitons with the case of M = 4 is presented.

The characterization of the stress field for notch contained structures and components at elevated temperature is a preliminary to assess their safety and integrity reasonably. In this paper, the characteristics of the sharp V-notch tip field in power-law creeping materials are presented from the asymptotic viewpoint. Through asymptotic analysis and solution, the stress exponents and the angular distribution functions for various notch angles and creep exponents are presented. The asymptotic structure of the stress field for the sharp V-notch tip field is presented by proposing the amplitude factor K(t) which is dependent on the creep time, specimen geometry, notch angle as well as loading condition. The amplitude factor K(t) can bridge the fracture parameters between notch and crack in power-law creeping solids. A reasonable and accurate estimation method for the evaluation of the K(t) factor is also proposed. With the determination of K(t), the notch tip stress fields of single edged sharp V-notch specimens in power-law creeping materials with various notch angles and different depths are presented, compared and analyzed. It has been verified that the stress components for sharp V-notch specimens can be estimated accurately and reasonably with the theory and method proposed in this paper. The differences of the asymptotic solutions for sharp V-notch tip in power-law creeping solids and power-law hardening materials are also discussed and clarified.

The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as P ( X = 2 k ) = 2 − k , k = 1 , 2 , … . The tails of X are not regularly varying and the sequence S n of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Lof, 1985; Csorgő and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that S n can be approximated by a semistable Levy process { L ( n ) , n ≥ 1 } with a.s. error O ( n ( log n ) 1 + e ) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.

In this paper, we propose a single-index panel data model with unobserved multiple interactive fixed effects. This model has the advantages of being flexible and of being able to allow for common shocks and their heterogeneous impacts on cross sections, thus making it suitable for the investigation of many economic issues. The asymptotic theories are established accordingly. Our Monte Carlo simulations show that our methodology works well for large N and T cases. In our empirical application, we illustrate our model by analysing the returns to scale of large commercial banks in the U.S.

Peristalsis of Jeffrey fluid in axisymmetric tube is studied when the fluid is subject to strong magnetic field. Mathematical analysis is made under the assumption of small Reynolds number and long wave length approximations. An alternate assumption of lubrication theory can also be applied to the peristalsis problem. Both the approaches lead to same mathematical expressions. The effects of magnetic field are investigated in the boundary layer (Hartman boundary layer) and on the boundary layer thickness. The effects of strong magnetic field, in the boundary layer, are explored using asymptotic analysis to find analytical solution. We notice that this approach facilitates to unveil the effects of strong magnetic field explicitly and determines the boundary layer thickness mathematically. The objective behind this study is: how to control the boundary layer thickness through the application of strong magnetic field. This phenomenon is central from theoretical and applied points of view. While going for the asymptotic analysis of large magnetic field; the mathematical model leads to singular perturbation problem which is an apparent diversion to most of the peristalsis problems– attempted by regular perturbation method. The boundary value problem is solved analytically using singular perturbation approach together with higher order matching technique. The important features of peristalsis like stream function, velocity, and pressure rise are calculated. In addition to the analytical solution, we explore the qualitative behavior of the flow using the theory of dynamical systems. The velocity field is determined by the phase plane analysis and the stability of the solution is found through bifurcation diagrams. Qualitative analysis of the solution has been carried out for magnetic parameter, amplitude ratio, flow rate and the Jeffrey fluid parameter. The concomitant change of stability is given through topological flow patterns. Equilibrium plots (bifurcation diagrams) give a complete description of the various flow patterns developed for the complete range of a flow parameters in contrast to most of the studies that describe the flow patterns at some particular value of a parameter. The study helps to analyze the behavior of the fluid at the critical points that represent steady solution. The mix of the analytical and qualitative approach will help to broaden the scope and understanding of peristaltic transport of fluid in channels, tubes and curved tubes (this study).

International audience; A general method to predict the steady-state regimes of a multi-degree-of-freedom unstable vibrating system (the primary system) coupled to several nonlinear energy sinks (NESs) is proposed. The method has three main steps. The first step consists in the diagonalization of the primary underline linear system using the so-called biorthogonal transformation. Within the assumption of a primary system with only one unstable mode the dynamics of the diagonalized system is reduced ignoring the stable modes and keeping only the unstable mode. The complexification method is applied in the second step with the aim of obtaining the slow-flow of the reduced system. Then, the third step is an asymptotic analysis of the slow-flow based geometric singular perturbation theory. The analysis shows that the critical manifold of the system can be reduced to a one dimensional parametric curve evolving in a multidimensional space. The shape and the stability properties of the critical manifold and the stability properties of the fixed points of the slow-flow provide an analytical tool to predict the nature of the possible steady-state regimes of the system. Finally, two examples are considered to evaluate the effectiveness and advancement of the proposed method. The method is first applied to the prediction of the mitigation limit of a breaking system subject to friction-induced vibrations coupled to two NESs, and next an airfoil model undergoing an aeroelastic instability coupled to a NESs setup (from one to four) is discussed. Theoretical results are compared, for validation purposes, to direct numerical integration of the system. The comparisons show good agreement.

This study presents a novel multi-scale computational method to analyze the dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations. The heterogeneities of heterogeneous shell structures are taken into account by periodic layouts of unit cells on the microscale in orthogonal curvilinear coordinate system. The new second-order two-scale approximate solutions for these multi-scale problems are constructed based on the multi-scale asymptotic analysis. Furthermore, the error estimates for the second-order two-scale (SOTS) solutions are obtained under some hypotheses. And then, a novel SOTS numerical algorithm based on finite element method (FEM), finite difference method (FDM) and decoupling method is brought forward in detail. Finally, some numerical examples are presented to verify the feasibility and validity of our multi-scale computational method. They also demonstrate that our multi-scale computational method can accurately capture the micro-scale dynamic thermo-mechanical responses in heterogeneous block structure, plate, cylindrical and doubly-curved shallow shells. In this paper, a unified multi-scale computational framework is established for dynamic thermo-mechanical problems of heterogeneous materials and structures with orthogonal periodic configurations. The asymptotic homogenization theory in Cartesian coordinate system and cylindrical coordinate system can be directly obtained based on the results in this paper.

We investigate the spatial dependence between commercial and residential mortgage defaults. A new class of observation-driven frailty factor models is introduced to do so. The idea of dynamic parameters embedded in the class of GAS models is utilized to estimate dynamic models of default risk with potentially multiple factors which are driven by stratified grouping of large panels of mortgage loan records. The score dynamics in the models is driven by so-called generalized residuals, and have therefore a fairly intuitive interpretation of ARMA-like dynamics. The asymptotic analysis recognizes the fact that we deal with both cross-sectional and time series data features. The proposed models are computationally easy to implement and therefore attractive in big data applications, something that gives them a considerable advantage in comparison to the typical latent factor frailty models proposed in the literature. Our empirical analysis demonstrates strong spatial dependence between commercial default and residential defaults.

A novel comprehensive analysis based on the matched asymptotic expansions method is curried out for a problem of flame propagation in narrow channels with heat-losses using the activation energy as a large parameter. The novelty consists in presentation of the results, namely the dependence of the flame velocity versus the heat-loss intensity, in a parametric form. Comparison with the numerical calculations reveals that the asymptotic results becomes in close agreement at relatively low values of the expansion parameter in such a manner extending a practical implementation of the method. The analysis may be of utility to flame-sheet model applications increasing its accuracy in general combustion studies.

We consider a family of linearly elastic shells all sharing the same middle surface and in unilateral contact with a rigid foundation on the lower face. The shells are elliptic and their lateral face is clamped. Under these conditions, when the thickness tends to zero, the solution of the three-dimensional problem converges to the solution of a two-dimensional obstacle problem for an elastic membrane shell. In this paper we provide error estimates for this convergence. The proof uses a corrector method.

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 $\varepsilon$ (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.

A novel three-scale asymptotic expansion is proposed in this work to investigate ageing linear viscoelastic problems of composites with multiple periodic configurations. Here, the composite structures are established by periodical distribution of local cells on microscale and mesoscale. The new three-scale asymptotic expansion formulas based on classical homogenized methods in time domain are constructed at first, and the microscale and mesoscale functions are also derived in detail. Further, two distinct homogenized parameters defined on microscale and mesoscale domains are obtained by upscaling methods, and the newly developed homogenized equations are given on whole structure in time domain. Then, the three-scale strain and stress solutions are constructed by assembling the unit cell solutions and homogenized solutions. Also, the efficient finite element algorithms based on the three-scale asymptotic expansion and homogenized method are brought forward. Finally, some representative numerical examples are evaluated to verify the proposed methods. They show that the three-scale asymptotic expansions developed in this work are effective and valid for predicting the ageing linear viscoelastic properties of the composite materials with multiple configurations.

We study approximation properties of centred additive random fields Y d , d ∈ N . The average case approximation complexity n Y d ( e ) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Y d , with relative 2-average error not exceeding a given threshold e ∈ ( 0 , 1 ) . We investigate the growth of n Y d ( e ) for arbitrary fixed e ∈ ( 0 , 1 ) and d → ∞ . Under natural assumptions we obtain general results concerning asymptotics of n Y d ( e ) . We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.

In this paper, we use the matched asymptotic analysis to derive a reduced model for domain wall dynamics within thin films of Neel-wall type in soft ferromagnets. It is found that the front dynamics is driven by mean curvature in the absence of external fields, and always relaxes to a planar interface where the one-dimensional ansatz is valid. The analysis predicts a linear dependence of front migration velocity on the field strength if the external magnetic field is weak, which is consistent with the result of Walker’s ansatz. Moreover, the derivation also identifies a regime where such an analysis fails, which corresponds to the Walker’s breakdown.

We consider a family of linearly viscoelastic shells of thickness 2 e , all with the same middle surface and fixed on the lateral boundary. By using asymptotic analysis, we find that for external forces of a particular order of e , a two-dimensional viscoelastic flexural shell model is an accurate approximation of the three-dimensional quasistatic problem. Most noticeable is that the limit problem includes a long-term memory that takes into account the previous history of deformations. We provide convergence results which justify our asymptotic approach.

Asynchronous iterative methods for solving linear systems are gaining renewed interest due to the high cost of synchronization points in massively parallel codes. Historically, theory on asynchronous iterative methods has focused on asymptotic behavior, while the transient behavior remains poorly understood. In this paper, we study a model of the asynchronous Jacobi method without communication delays, which we call simplified asynchronous Jacobi. Simplified asynchronous Jacobi can be used to model asynchronous Jacobi implemented in shared memory or distributed memory with fast communication networks. Our analysis uses the idea of a propagation matrix, which is similar in concept to an iteration matrix. We show that simplified asynchronous Jacobi can continue to reduce the residual when some processes are slower than other processes. We also show that simplified asynchronous Jacobi can converge when synchronous Jacobi does not. We verify our analysis of simplified asynchronous Jacobi using results from asynchronous Jacobi implemented in shared and distributed memory.

We provide a complete asymptotic distribution theory for clustered data with a large number of independent groups, generalizing the classic laws of large numbers, uniform laws, central limit theory, and clustered covariance matrix estimation. Our theory allows for clustered observations with heterogeneous and unbounded cluster sizes. Our conditions cleanly nest the classical results for i.n.i.d. observations, in the sense that our conditions specialize to the classical conditions under independent sampling. We use this theory to develop a full asymptotic distribution theory for estimation based on linear least-squares, 2SLS, nonlinear MLE, and nonlinear GMM.

Asymptotic analyses and numerical computations are reported for surges of viscoplastic fluid down an incline with low inertia. The asymptotic theory applies for relatively shallow gravity currents. The computations use the volume-of-fluid method for tracking the interface; the constitutive law is dealt with by the augmented-Lagrangian method. The anatomy of the surge consists of an upstream region that converges to a uniform sheet flow, and over which a truly rigid plug sheaths the surge. The plug breaks further downstream due to the build up of the extensional stress acting upon it, leaving instead a weakly yielded superficial layer, or pseudo-plug. Finally, the surge ends in a steep flow front that lies beyond the validity of shallow asymptotics.

We study the sensitivity of optimal consumption streams with respect to perturbations of the random endowment. At the leading order, the consumption adjustment does not matter: any choice that matches the budget constraint simply shifts the original utility by the marginal value of the perturbation. Nontrivial results can be obtained by considering the next-to-leading order. Here, one first solves the problem for a deterministic perturbation, which leads to a “prognosis measure”. The desired consumption adjustment for a general endowment perturbation is in turn given by the conditional expectation of the latter, computed under this measure and appropriately weighted with the conditional expectations of the remaining risk-tolerance.

The most challenging problem for simulating conjugate mass transfer problem is the variable concentration jump described by the double film theory at the phase interface. In the present paper, the physical mechanism of Henry’s law is analyzed, and it is found that Henry’s law can be expressed as a relation between two constant weight coefficients in lattice Boltzmann method (LBM). As a consequence, a unified multi-relaxation-time (MRT) lattice Boltzmann equation (LBE) for conjugate mass transfer is proposed. The asymptotic analysis proves that the interface conditions, including the concentration jump determined by the double film theory and mass flux continuity, can be ensured with second-order accuracy. Five test cases containing both transient and steady conjugate mass transfer problems with straight or curved interfaces are calculated to validate the present method. The results show that the proposed MRT LBE has both simplicity and good accuracy. It is capable of simulating conjugate mass transfer at both steady and transient periods, with a stationary or moving interface. Especially, it has a good numerical stability at large distribution coefficient and diffusivity coefficient ratio.