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We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.

Using multi-scale beyond all order methods, we investigate stationary spatially localized solutions of a Schnakenberg system, a prototype reaction–diffusion system, near the onset of a subcritical Turing bifurcation. These solutions are homoclinic orbits to a homogeneous solution and passing near a periodic solution. In bifurcation diagrams, branches of these solutions commonly show two interwining snaking curves. Here we calculate the maximal range of existence for these solutions and compare our findings with numerical computations. We derive and optimally truncate a (divergent) asymptotic series of a front solution. The remainder of the truncated series is exponentially small if and only if a specific parameter range is met. This complements work on Swift–Hohenberg equations where similar results have been obtained.

The perturbation theory is one of the tricks or tools used mathematically to find approximate solutions to fluctuating problems for which no accurate solutions can be found. In this paper we will deal with a number of basic concepts related to perturbation theory, including regular perturbation and singular perturbation, and then apply these tricks or tools theoretically to ordinary first and second order differential equations for regular perturbation using the asymptotic expansion method, which is considered one of the most important methods used to find approximate solutions of perturbation equations.

A solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter is constructed by matched asymptotic approximations. The approximation of the relic density is an asymptotic series valid when the abundance does not deviate significantly from its equilibrium value until small temperatures. Resonance and threshold effects are taken into account at leading order and found to be negligible unless the annihilation cross section is negligible at threshold. Comparisons are made to previously attempted constructions and to the freeze out approximation commonly employed in the literature. Extensions to higher order matching is outlined, and implications for solving related systems are discussed. We compare our results to a numerical determination of the relic abundance using a benchmark model and find a fantastic agreement. The method developed also serves as a solution to a wide class of problems containing an infinite order turning point., Comment: 15 pages, 6 figures

If a nonlinear function F(S) depends on a function S(x) that is represented by a factorially divergent asymptotic power series in a small parameter x, each late coefficient of the power series for F(S(x)) can be represented explicitly as an asymptotic series whose terms involve balanced combinations of the late and early coefficients of the series for S(x). The formula for the late terms was first described by R B Dingle but not published by him. Numerics for a variety of functions F(S) demonstrate this 'self-resurgence' and the accuracy of the representation.

People find that formal solutions to a type of doubly singular ordinary differential equations (systems) summable in a monomial of two variables, and that formal solutions to many singular partial differential equations were multisummable. So summability theory of formal power series in several variables is of great importance to the research of formal solutions to partial differential equations. This enriches the achievements in the research of formal solutions to differential equations. It is a application of monomial summability theory of formal power series.

To design an optimally-sized geothermal heat exchanger it is crucial to accurately forecast its thermal response over the whole lifetime of the building. Since the presence of aquifers highly affects the heat exchange between geothermal boreholes and the ground, both heat transfer mechanisms, conduction and convection, must be taken into account in the ground. In the present work the existence of large disparities in time and length scales in the problem is exploited using asymptotic expansion techniques to develop a theoretical model capable of taking into account both aforementioned heat transfer mechanisms for any Peclet number of the groundwater flow.

The efficient harnessing of geothermal energy for the heating and cooling of buildings requires extensive simulations to be carried out during the detailed design phase of a new building. These simulations, performed using semi-analytical models, serve to assess the thermal response of the geothermal heat exchanger during the building’s lifetime, that can easily excess 100 years. By using asymptotic expansion techniques, a new generation of theoretical models is being developed since 2011 that offer the flexibility and accuracy of the best performing models available nowadays, but at a fraction of their computational cost. These new models open the doors to yet untapped design and optimization possibilities.

Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that involve coalescing singularities, a more general type of exponential-over-power divergence must be used. As a model example, we study the water waves produced by flow past an obstruction such as a surface-piercing ship. In the low speed or low Froude limit, the resultant water waves are exponentially small, and their formation is attributed to the singularities in the geometry of the obstruction. However, in cases where the singularities are closely spaced, the usual asymptotic theory fails. We present both a general asymptotic framework for handling such problems of coalescing singularities, and provide numerical and asymptotic results for particular examples.

An asymptotic expansion of the solution of an inhomogeneous matrix difference equation of general form is obtained. The case when there is no bound on the differences of the arguments is considered. The effect of the roots of the characteristic equation is taken into account. An integral estimate with a submultiplicative weight is established for the remainder in terms of the submultiplicative moment of the free term of the equation. Bibliography: 14 titles.

We consider continuous and discrete Schr\"odinger systems with self-adjoint matrix potentials and with additional dependence on time (i.e., dynamical Schr\"odinger systems). Transformed and explicit solutions are constructed using our generalized (GBDT) version of the B\"acklund-Darboux transformation. Asymptotic expansions of these solutions in time are of interest., Comment: In this paper, we develop further the approach to the construction of explicit solutions of dynamical systems from arXiv:1603.08709 and arXiv:1609.03451. Some results from arXiv:nlin/0311004 are actively used. In the version 2 of the paper, we consider additionally the asymptotic behaviour of the solutions in time

All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expected rates of decay of the metric near null-infinity in a free-evolution setting. We show, within the asymptotic expansion, that by carefully modifying the non-principal part of the field equations by the addition of constraints, we are able to obtain optimal decay rates even when the constraints are violated. The light-speed condition can hence be satisfied, which paves the way for the explicit numerical treatment of future null-infinity. We then study the behavior of the Trautman-Bondi mass under the decay results predicted by the asymptotic expansion. Naively the mass seems to be unbounded, but we see first that the divergent terms can be replaced with a combination of the constraints and the Einstein field equations, and second that the Bondi mass loss formula is recovered within the framework. Both of the latter results hold in the presence of small constraint violations., 18 pages, 1 figure

The task is to analyse the temperature distribution in the material under ultrashort laser irradiation. The mathematical model is built on the basis of a two-temperature model describing transition phenomena in a nonequilibrium electron gas and lattice with an ultrashort laser effect on the material. The vaporized body is considered as a thin plate and the problem is formulated as a system of one-dimensional boundary-value problems of the heat equation written for the electron and lattice components. The initial problem is reduced to solving a system of singularly perturbed boundary problems of the heat equation with nonlinear boundary conditions on moving boundaries, an approximate solution of which is obtained in the form of an asymptotic expansion of the solution in the Poincaré sense in powers of small parameters.

For a singularly perturbed parabolic - ODE system we construct the asymptotic expansion in the small parameter in the case, when the degenerate equation has a double root. Such systems, which are called partly dissipative reaction-diffusion systems, are used to model various natural processes, including the signal transmission along axons, solid combustion and the kinetics of some chemical reactions. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. The multizonal initial and boundary layers behaviour was stated.

In this paper we shall review some recent results on the asymptotic expansion of the equivariant components of an algebro geometric Szego kernel determined by the linearization of a Hamiltonian action of U(2) (with certain assumptions). We shall build on the techniques developed in [13], [1], and [11], and therefore ultimately on the microlocal description of the Szego kernel as a Fourier integral operator in [3].

This study concerns on the solution of exact Korteweg de Vries (KdV) equation in its application in generating extreme waves. The method of asymptotic expansion is employed up to the fifth order. In the previous research, the same method was applied up to the third order and fifth order but it only considered the side band solutions. Here, solutions at each order will be analyzed. The existence of resonance terms at the odd orders and side band terms are interesting to observe considering the importance of these quantities in analyzing the wave deformation which link to the phenomenon of wave’s amplitude amplification. Bichromatic signal is used as the initial wave signal as it experiences instability during its propagation which results the amplitude amplification. The amplitude amplification is presented as Maximal Temporal Amplitude (MTA) which is a quantity measuring the highest elevation at every spatial position during the observation time.

Digital photoelasticity method is used to analyzeexperimentally the complete Williams series expansion of the stress and displacement fields in the vicinity of the crack tip in isotropic linear elastic plates under Mixed Mode loading. The distribution of the isochromatic fridge patterns is employed for obtainingthe stress field near the crack tip by the use of the complete Williams asymptotic expansion for various classes of the experimental specimens (plates with two collinear cracks under tensile loading and under mixed mode loading conditions). The higher order terms of the Williams series expansion are taken into account and the coefficients of the higher order terms are experimentally obtained. The stress field equation of Williams up to fifty terms in each in mode I and mode II has been considered. The comparison of the experimental results and the calculations performed with finite element analysis has shown the importance and significant advantages of photoelastic observations for the multi-parameter description of the stress field in the neighborhood of the crack tip.

In this paper an asymptotic expansion of ergodic integrals for suspension flows over Vershik automorphisms is obtained and a limit theorem for these flows is given. Bibliography: 49 titles.

We study the long-time behaviour of solutions to the periodic problem for a non-linear Sobolev-type equation. In the case of non-small initial perturbations we get estimates for the decay as a function of time. In the case of small initial data we prove an asymptotic formula for solutions to the periodic problem for a non-linear Sobolev-type equation.

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects the density function of $\tcal$ to the Hastings-McLeod solution of the second Painlev\'e equation, we prove that as $t\rightarrow\infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly., Comment: 24 pages, 2 figures

A boundary-value problem for a second order parabolic equation on a half-line is considered. A uniform asymptotic approximation to a solution to within any power of t-1 is constructed and substantiated. © 2012 RAS(DoM) and LMS.

We study homoclinic snaking in the cubic-quintic Swift–Hohenberg equation (SHE) close to the onset of a subcritical pattern-forming instability. Application of the usual multiple-scales method produces a leading-order stationary front solution, connecting the trivial solution to the patterned state. A localized pattern may therefore be constructed by matching between two distant fronts placed back-to-back. However, the asymptotic expansion of the front is divergent, and hence should be truncated. By truncating optimally, such that the resultant remainder is exponentially small, an exponentially small parameter range is derived within which stationary fronts exist. This is shown to be a direct result of the 'locking' between the phase of the underlying pattern and its slowly varying envelope. The locking mechanism remains unobservable at any algebraic order, and can only be derived by explicitly considering beyond-all-orders effects in the tail of the asymptotic expansion, following the method of Kozyreff and Chapman as applied to the quadratic-cubic SHE (Chapman and Kozyreff 2009 Physica D 238 319–54, Kozyreff and Chapman 2006 Phys. Rev. Lett. 97 44502). Exponentially small, but exponentially growing, contributions appear in the tail of the expansion, which must be included when constructing localized patterns in order to reproduce the full snaking diagram. Implicit within the bifurcation equations is an analytical formula for the width of the snaking region. Due to the linear nature of the beyond-all-orders calculation, the bifurcation equations contain an analytically indeterminable constant, estimated in the previous work by Chapman and Kozyreff using a best fit approximation. A more accurate estimate of the equivalent constant in the cubic-quintic case is calculated from the iteration of a recurrence relation, and the subsequent analytical bifurcation diagram compared with numerical simulations, with good agreement.

Regular functions f in the half-plane Imz>0 admitting an asymptotic expansion f(z)=c{sub 1}z+c{sub 2}z{sup 2}+c{sub 3}z{sup 3}+{gamma}(z)z{sup 3}, where c{sub 1}>0, Imc{sub 2}=0 and the angular limit

This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion (and its derivatives), are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are, in fact, specific rational functions of a distinguished irrational (algebraic) function, z0(t). This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of weight of the parameter t). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions.As an intriguing application, one gains new insights into the relation between certain derivatives of the genus expansion, in a double-scaling limit, and the asymptotic expansion of the first Painleve transcendent. This provides a precise expression of the Painleve asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established in this paper. Moreover, these insights point towards a more general program relating the first Painleve hierarchy to the higher order structure of the double-scaling limit through the specific rational structure of generating functions in the genus expansion. The paper closes with a discussion of the relation of this work to recent developments in understanding the asymptotics of graphical enumeration.As a by-product, these results also yield new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights, the calculation of correlation functions for certain tied random walks on a 1D lattice, and the large time asymptotics of random matrix partition functions.

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schr\"{o}dinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N=2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended., Comment: 20 pages, journal version

This paper deals with coupled boundary layers of the primitive equations of atmosphere. It is proved that, for well-prepared initial data, the smooth solutions of the primitive equations converge to smooth solutions of quasi-geostrophic equations as the Rossby number, the vertical viscosity and the vertical heat conductivity tend to zero. The proof is based mainly on the matched asymptotic expansion techniques and the refined energy estimates. A new feature of this paper is that the velocity boundary layer and thermal layer are considered simultaneously.

We study the root of unity limit of the lens elliptic gamma function solution of the star-triangle relation, for an integrable model with continuous and discrete spin variables. This limit involves taking an elliptic nome to a primitive $rN$-th root of unity, where $r$ is an existing integer parameter of the lens elliptic gamma function, and $N$ is an additional integer parameter. This is a singular limit of the star-triangle relation, and at subleading order of an asymptotic expansion, another star-triangle relation is obtained for a model with discrete spin variables in $\mathbb{Z}_{rN}$. Some special choices of solutions of equation of motion are shown to result in well-known discrete spin solutions of the star-triangle relation. The saddle point equations themselves are identified with three-leg forms of "3D-consistent" classical discrete integrable equations, known as $Q4$ and $Q3_{(\delta=0)}$. We also comment on the implications for supersymmetric gauge theories, and in particular comment on a close parallel with the works of Nekrasov and Shatashvili., Comment: 32 pages; v2: published version

The initial value problem for a system of nonlinear ordinary differential equations with a small parameter multiplying the highest derivative is investigated. In a neighbourhood of the initial point the asymptotic behaviour of the solution has quite a complicated structure. A uniform asymptotic approximation to the solution up to an arbitrary power of the small parameter is constructed and substantiated. Bibliography: 3 titles.

The steady flow in a rotating sphere is investigated by asymptotic analysis in the limit of strong precession. The whole spherical body is divided into three regions in terms of the flow characteristics: the critical band, which is the close vicinity surrounding the great circle perpendicular to the precession axis, the boundary layer, which is attached to the whole sphere surface and the inviscid region that occupies the majority of the sphere. The analytic expressions, in the leading order of the asymptotic expansion, of the velocity field are obtained in the former two, whereas partial differential equations for the velocity field are derived in the latter, which are solved numerically. This steady flow structure is confirmed by the corresponding direct numerical simulation.

We consider a problem on a half-line with a Neumann-type boundary condition. We prove the global existence of solutions and find the leading term of their large-time asymptotic expansion. The case of large initial data and critical non-linearity is considered in the case when the non-linear term of the equation decays in time as rapidly as the linear terms.

We use the method of regularization to construct two kinds of regularized asymptotic expansions (in a complex parameter) for a fundamental system of solutions of the Bessel equation. Expansions of the first kind are defined in the closed complex plane of the independent variable except for singular points of the spectral functions of the initial operator. We determine the domains of uniform and non-uniform convergence of the series involved. We study the resulting formulae on the positive real axis and prove that they yield Debye's familiar asymptotic expansions for Bessel functions on the interval (0,1), which lies in the domain of non-uniform convergence. The second kind of regularized uniform asymptotic expansions is constructed near a regular singular point in another domain of values of the parameter in the equations. Using these results, we get uniform asymptotic expansions of solutions of a boundary-value problem for the non-homogenous and homogeneous Bessel equations.

The asymptotic behaviour of solutions of a second-order semilinear parabolic equation is analyzed in a cylindrical domain that is bounded in the space variables. The dominant term of the asymptotic expansion of the solution as is found. It is shown that the solution of this problem is asymptotically equivalent to the solution of a certain non-linear ordinary differential equation. Bibliography: 8 titles.

We consider the Dirichlet Laplacian in a domain formed by two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them. It is also shown that the eigenvalues emerge from the threshold of the essential spectrum as the window passes through certain critical shapes. We prove the necessary condition for the window to be of critical shape. Under an additional assumption we show that this condition is sufficient and obtain the asymptotic expansion for the emerging eigenvalue and for the associated eigenfunction.

The asymptotic expansion of the solution u to the equation ∇ (α∇u) = −∇ β + γ in , with respect to the size of an inclusion (at a point x0 of the domain Ω) in which the parameters α, β and γ are changed, is studied. Writing the difference with the unperturbed solution as u − u = nz, it is shown that the sequence z converges weakly, for all p 0, to a function z in Lp(Ω)∩H1(Ω Bρ(x0)), where Bρ(x0) is the ball of radius ρ around x0. This allows for the calculation of asymptotic expansions of cost functions of the form J() = ∫ΩF(u) dx, for example, rendering it useful for many applications. It also extends available estimates which hold uniformly on the boundary of Ω. In addition, a link is provided with the adjoint method of calculating topological expansions of cost functions. A specific application to lubricated devices is illustrated.

The case of small intercentre distance in the D-dimensional two Coulomb centres problem (Z1eZ2)D (D ≥ 2) is studied by solving the wave equations using the separations of variables. Asymptotic expansions for the electronic terms and the quantum defect are obtained. Results obtained are compared with previous asymptotic and numerical treatments. Correspondence between energy terms of the three-dimensional system (Z1eZ2)3 and the D-dimensional system (Z1eZ2)D is found.

We investigate holography for asymptotically Schrodinger spacetimes, using a frame formalism. Our dictionary is based on the anisotropic scaling symmetry. We consider z, Comment: 34 pages, no figures. v2: corrected typos, improved explanation on some technical points, to be published in CQG

We consider the dynamics and thermodynamics of a pair of magnetic dipoles interacting via their magnetic fields. We consider only the ‘spin’ degrees of freedom; the dipoles are fixed in space. With this restriction it is possible to provide the general solution of the equations of motion in analytical form. Thermodynamic quantities, such as the specific heat and the zero field susceptibility are calculated analytically or by combining low temperature asymptotic series and a complete high temperature expansion. The thermal expectation value of the autocorrelation function is determined for the low temperature regime and short times including terms linear in T. Furthermore, we have performed Monte Carlo simulations for the system under consideration and compared our analytical results with these.

We consider the direct and inverse obstacle scattering problem for the Laplace–Beltrami operator with complex-valued electromagnetic potentials where outside of some large ball the operator is simply the negative Laplacian. It is assumed there are an arbitrary number of impenetrable obstacles with various boundary conditions (including mixed boundary conditions, i.e., partially coated obstacles). We prove that given the scattering amplitude a(θ, ω, k) for all θ, ω Sn−1 and any fixed k > 0, the location of the obstacles and their respective boundary conditions are uniquely determined. In addition, we give a new proof of the asymptotic expansion of the Green's function in terms of distorted plane waves and, as an immediate corollary, prove the determination of the Dirichlet-to-Neumann operator from the scattering data.

The (3+1)-dimensional Jimbo–Miwa (JM) equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new (3+1)-dimensional integrable models, which are conformal invariant and possess Painleve property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.

The present work is based on Bordag M et al 2005 (J. Phys. A: Math. Gen. 38 11027) where the spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out. The solutions to Maxwell equations with the appropriate matching conditions at the plasma layer are derived and the spectrum of electromagnetic oscillations is determined. The spectral zeta function and the integrated heat kernel are constructed for different branches of the spectrum in an explicit form. The asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived. The local heat kernels are considered also.

We derive expansions with exact asymptotic expressions for the remainders for solutions of multi-dimensional renewal equations. The effect of the roots of the characteristic equation on the asymptotic representation of solutions is taken into account. The resulting formulae are used to investigate the asymptotic behaviour of the average number of particles in age-dependent branching processes having several types of particles.

A time-asymptotic analytical expression of the wave function transmitted by a double-barrier potential is presented. This analytical expression is validated by comparing it with the complete numerical solution of the Schroedinger equation. In the resonant regime, where the asymptotic expression is not valid, an approximation of the wave function, valid for all times, is given.

We consider the heat kernel for a massless fermion of spin 1/2 interacting with all types of non-Abelian boson fields, i.e. scalar, pseudo-scalar, vector, axial-vector and antisymmetric tensor fields, in a four-dimensional Riemannian space. The couplings of the fermion with the boson fields contain irreducible matrices of the product of the γ-matrices. In this model, the components of the first and second asymptotic expansion coefficients of the heat kernel with respect to the irreducible matrices are explicitly presented. The form of the second coefficients is useful for evaluation of some fermionic anomalies in four-dimensional curved space, and the concrete forms of the chiral U(1) and the trace anomalies are presented.

Entropic forces in models of Motzkin paths in a wedge geometry are considered as models of forces in polymers in confined geometries. A Motzkin path in the square lattice is a path from the origin to a point in the line Y = X while it never visits sites below this line, and it is constrained to give unit length steps only in the north and east directions and steps of length in the north-east direction. Motzkin path models may be generalized to ensembles of NE-oriented paths above the line Y = rX, where r > 0 is an irrational number. These are paths giving east, north and north-east steps from the origin in the square lattice, and confined to the r-wedge formed by the Y-axis and the line Y = rX. The generating function gr of these paths is not known, but if r > 1, then I determine its radius of convergence to be and if r (0, 1), then tr = 1/3. The entropic force the path exerts on the line Y = rX may be computed from this. An asymptotic expression for the force for large values of r is given by In terms of the vertex angle α of the r-wedge, the moment of the force about the origin has leading terms as α → 0+ and F(α) = 0 if α [π/4, π/2]. Moreover, numerical integration of the force shows that the total work done by closing the wedge is 1.085 07... lattice units. An alternative ensemble of NE-oriented paths may be defined by slightly changing the generating function gr. In this model, it is possible to determine closed-form expressions for the limiting free energy and the force. The leading term in an asymptotic expansions for this force agrees with the leading term in the asymptotic expansion of the above model, and the subleading term only differs by a factor of 2.

We study the asymptotics of the heat trace $\Tr\{fPe^{-tP^2}\}$ where $P$ is an operator of Dirac type, where $f$ is an auxiliary smooth smearing function which is used to localize the problem, and where we impose spectral boundary conditions. Using functorial techniques and special case calculations, the boundary part of the leading coefficients in the asymptotic expansion is found., Comment: 19 pages, LaTeX, extended Introduction

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation., 17 pages. 1 figure

It is proved that the asymptotic series for the leading term K∞(a) of the Keesom integral K(a) obtained by us in a previous paper (2004 J. Phys. A: Math. Gen. 37 9677), in order to evaluate K(a) for large values of the interaction parameter a, is indeed an asymptotic expansion of K(a) itself.