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

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

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

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

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

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

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

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

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

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

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

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

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary. This expansion formula shows the singularity profile of solutions at the boundary. We deal with both linear and nonlinear elliptic equations, including fully nonlinear elliptic equations and equations of Monge-Ampere type.

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

The main results of this paper are related to studying the dynamics of the Van-der-Pol oscillator, which is under the influence of a driving force and a hysteretic effect, formalized by the Bouc–Wen phenomenological model. The regularizing role of the hysteretic link in the reduction of chaotic regimes is established. The main results are obtained as part of the asymptotic expansion method (small parameter method). A system of two coupled Van-der-Pol oscillators is considered under the conditions of a driving force acting on one of them: the relationship between the amplitudes and frequencies of the natural oscillations of the oscillators is established in an analytical form. The paper also describes a system of cross-related Van-der-Pol oscillators, in which the driving force acting on one of them is determined by the output of a hysteretic transducer, to the input of which the mismatch between the oscillators’ velocities is applied. It is found that the coupling parameter between the oscillators and the parameter that determines the influence of the hysteretic link have a significant effect on the synchronization in the system.

The Cauchy problem for a first-order differential equation with a small parameter multiplying the derivative in a Banach space is considered. The right-hand side of the equation contains the Fredholm operator perturbed by an additional operator term containing a small parameter. The asymptotic expansion of the solution in powers of the small parameter is constructed by the Vasil’yeva–Vishik–Lyusternik method. To calculate the components of the regular part of the expansion, the cascade decomposition method is used, which consists in the step-by-step splitting of the equation into equations in subspaces of decreasing dimensions. The conditions under which the boundary layer phenomenon occurs in the problem are determined.

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .

The asymptotic behavior of weak time-periodic solutions to the Navier–Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

We construct an asymptotic expansion in a small parameter of a boundary layer solution of the boundary value problem for a system of two ordinary differential equations, one of which is a second-order equation and the other is a first-order equation with a small parameter at the derivatives in both equations. Such a system arises in chemical kinetics when modeling the stationary process in the case of fast reactions and in the absence of diffusion of one of the reacting substances. A significant feature of the studied problem is that one of the equations of the degenerate system has a triple root. This leads to a qualitative difference in the boundary layer component of the solution compared with the case of simple (single) roots of degenerate equations. The boundary layer becomes multizonal, and the standard algorithm for constructing the boundary layer series turns out to be unsuitable and is replaced with a new algorithm.

Hodograph transformations can be used to linearize a nonlinear partial differential equation by judicious use of physical quantities (e.g. velocities or displacement gradients) as coordinate variables in the hodograph plane. This approach has been found useful for obtaining the leading order terms of eigenproblems that govern asymptotic singular crack fields in nonlinear materials. There is little work on the use of the hodograph transformation for obtaining higher order terms in the asymptotic expansion of the crack tip fields. In this paper, we develop a framework to obtain such higher order terms using the hodograph transformation. The method relies heavily on the representation of physical quantities of interest in terms of hodograph plane variables. We demonstrate the method via application to a generalized neo-Hookean material. In addition, asymptotic path-independent J-integrals are expressed in terms of either physical or hodograph variables and are used to compute the leading-order amplitude coefficients. A relationship between the asymptotic J-integrals and the energy release rate is established for a mixed crack mode. The asymptotic results are compared with numerical results from finite element computation and excellent agreement is obtained.

The topological sensitivity method is an optimization technique used in different inverse problem solutions. In this work, we adapt this method to the identification of plasma domain in a Tokamak. An asymptotic expansion of a considered shape function is established and used to solve this inverse problem. Finally, a numerical algorithm is developed and tested in different configurations.

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.

International audience; The Petrowsky type equation $y_{tt}^\eps+\eps y_{xxxx}^\eps - y_{xx}^\eps=0$, $\eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $\sqrt{\eps}$ occurring at the extremities, these boundary controls get singular as $\eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^\eps$ and derive a rigorous second order asymptotic expansion of the control of minimal weighted $L^2-$norm, with respect to the parameter $\eps$. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J-.L.Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any $\eps$ small enough. Numerical experiments support our study.

The Cauchy problem for a singularly perturbed system of equations describing a transport process with diffusion in a multiphase medium is considered. A formal asymptotic expansion of its solution is constructed in the case when exchange between the phases proceeds much more rapidly than the transport and diffusion processes. The case when the diffusion fluxes of the components have a mutual effect on each other is considered. Under the assumptions imposed on the data of the problem, the leading term of the asymptotics is described by a multidimensional generalized Burgers–Korteweg–de Vries equation. Under some additional conditions, the remainder is estimated by means of the residual.

In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain Ω : = R + × R 2 . More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the suitable initial approximation function such that the linearized equations around it admitted a partial dissipative structure when we carry out the weighted energy estimate. Meanwhile, the asymptotic expansion of Sobolev regularity solutions is given.

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u ( t , x ) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u ( t , x ) and its asymptotic approximation ν ( t , x ). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.

We study the flow past a rough sphere considering the rugosity as a parameter and its effect on the drag coefficient. The numerical implementation is carried out via a novel approach using Galerkin’s method combined with an asymptotic expansion for the stream function. The amplitude of the spatial fluctuation is used as the perturbation parameter. The numerical results for the size of the vortex ring and streamlines in the smooth case are compared with experimental data found in the literature. In addition, when the surface is rough, we compare the numerical results with the smooth case and the implications of the rugosity in the separation point and other features of the flow are discussed.

An asymptotic expansion of the solution of a nonhomogeneous 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. The asymptotic behavior of the remainder is established depending on the asymptotics of the free term of the equation.

The article investigates the asymptotic behavior of the solution of a two-point boundary value problem on an interval for a linear inhomogeneous ordinary differential equation of the second order with a small parameter at the highest derivative. The essential features of the problem are the presence of a small parameter in front of the second-order derivative of the desired function, the existence of a two-dimensional boundary layer at the left end of the segment at x = 0, and the non-smoothness of the solution to the corresponding unperturbed boundary value problem. It is required to construct a uniform asymptotic expansion of the solution to a two-zone two-point boundary value problem on a unit interval, with any degree of accuracy, as the small parameter tends to zero. Due to the second and third features of the problem, it is not easy to construct an asymptotic solution expansion with respect to the small parameter using the known asymptotic methods. When solving the problem, the following methods are used: methods of integration of ordinary differential equations; the method of a small parameter; the classical method of boundary functions; and the generalized method of boundary functions and the maximum principle. The problem is solved in two stages: in the first stage, a formal expansion of the solution to the two-point boundary value problem is constructed, and in the second stage, the justification of this expansion is given, i.e. the remainder term of the expansion is estimated. In the first stage, a formal asymptotic solution is sought in the form of the sum of three solutions: a smooth outer solution on the entire segment; classical boundary layer solution in the vicinity of x = 0, which exponentially decreases outside the boundary layer; and an intermediate boundary layer solution at x = 0, which decreases in power mode outside the boundary layer. The constructed asymptotic expansion of the solution to the two-point boundary value problem is asymptotic in the sense of Erdei

A three-dimensional problem is considered for electromagnetic diffraction on a bound ideally conducting body containing a conical point. An asymptotic representation of the electromagnetic field is calculated in the vicinity of the conical point with the solution presented as the sum of singular and smooth parts.

This paper is about solutions to PDEs in which the ratio between the largest and smallest values of the coefficient is very large. Such problems are referred to as high-contrast ones. A full asympt...

A boundary value problem for a system of two ordinary differential equations, one being of the second order, and the other, of the first order, with a small parameter multiplying the derivatives in each equation is considered. The conditions are established under which the problem has a solution involving an internal transition layer near some point within which the solution jumps from a small neighborhood of one root of the corresponding degenerate system to a small neighborhood of its other root. A solution of this type is called a steplike contrast structure (SLCS). An asymptotic approximation of SLCS with respect to the small parameter is constructed and substantiated. It has certain differences from SLCS observed in other singularly perturbed problems. This is concerned primarily with the structure of the solution asymptotics in the transition layer. The constructed asymptotic expansion is substantiated using the asymptotic method of differential inequalities, whose application to the considered problem has a number of qualitative features.

In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.

Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class...

Electrovortex flows are of great interest both from the viewpoints of theoretical magnetohydrodynamics and applications. They arise when the electric current of variable density passes through a conducting medium (such as a liquid metal). The interaction between the current and self magnetic field of the current induces a nonpotential electromagnetic force that causes a vortex flow of the liquid. As a model problem, we consider a stationary flow between two parallel planes. The flow is described by the stream function for which we obtain a nonlinear fourth-order partial differential equation. For moderate values of the electrovortex flow parameter, we investigate the problem by the asymptotic series expansion in the powers of this parameter. We describe the process of successive approximations of various degrees and present the flow pattern that is given by two first terms. This asymptotic solution is shown to be rather close to the numerical solution of the problem.

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

In this paper, we consider the sine-Gordon equation with a high-frequency parametric pumping and a weak dissipative force. We examine the class of π-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable 2π-kinks, these solutions are unstable. We prove that the time of decaying of π-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametric pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a π-kink. Numerical simulations of solutions confirm the good qualitative agreement with asymptotic expansions.

In this paper, an integral transformation of the displacement is employed to determine the solution of the elastodynamics problem of two collinear Griffith cracks with constant velocity situated in a mid-plane of an infinite orthotropic strip where the boundaries are assumed to be stress-free. By use of the integral transformation of the displacement, the problem is reduced to a solution of the triple integral equations and the finite Hilbert transformation technique is used to solve it. The expressions of stresses are obtained asymptotically for large values of strip depth. An analytical expression of the stress intensity factor at the crack tip is obtained and represented graphically for two different orthotropic materials.

We consider the Helmholtz problem with source term in an anisotropic domain of R 3 . The aim of this paper is to investigate the interplay between the geometry and analysis of elliptic equations un...

We consider the dimer model on a hexagonal lattice. This model can be represented as a “pile of cubes in a box.” The energy of a configuration is given by the volume of the pile. The partition function is computed by the classical MacMahon formula or as the determinant of the Kasteleyn matrix. We use the MacMahon formula to derive the scaling behavior of free energy in the limit as the lattice spacing goes to zero and temperature goes to infinity. We consider the case of a finite hexagonal domain, the case where one side of the hexagonal box is infinite, and the case of inhomogeneous Boltzmann weights. We obtain an asymptotic expansion of free energy, which is called finite-size corrections, and discuss the universality and physical meaning of the expansion coefficients.

We study the critical periods of the Loud family of quadratic centers. We give the first terms of the asymptotic expansion of the period function and we conclude that no bifurcation occurs for certain values of the parameters. This result contributes to complete the bifurcation diagram of critical periods in the Loud family.

This paper is devoted to the proper-time method and describes a model case that reflects the subtleties of constructing the heat kernel, is easily extended to more general cases (curved space, manifold with a boundary), and contains two interrelated parts: an asymptotic expansion and a path integral representation. We discuss the significance of gauge conditions and the role of ordered exponentials in detail, derive a new nonrecursive formula for the Seeley–DeWitt coefficients on the diagonal, and show the equivalence of two main approaches using the exponential formula.