Logarithmic Singularity Research Articles

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of for almost every α where is ranging over a long interval and f is a 1-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for N in a set of positive upper density, the order of is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy–Koksma inequality for such f, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel–Bernstein Theorem that allows very general modularity conditions.

Pseudo and anisotropic MFS for Laplace equation and optimal sources using maximal projection method with a substitution function

The paper creates two new families of fundamental solutions for the 3D Laplace equation, presented into two parts. For the first part in terms of a planar line as a new coordinate the derived 2D like fundamental solution has a logarithmic singularity, which results in a method of pseudo fundamental solutions. We propose two methods to determine the optimal values of the offset parameter used to locate the source points. In the second part, an anisotropic distance function rg in terms of a symmetric non-negative anisotropic metric tensor is introduced, which satisfies a certain quadratic matrix equation, and then lnrg is proved to be a new fundamental solution. Using a unit orientation vector we can derive the metric tensor in closed-form, and prove that it is a singular projection operator. Given the unit orientation vector satisfying a cone condition, a method of anisotropic fundamental solutions is developed. They are distinct from the traditional 3D MFS. Owing to a weaker singularity than that of 1/r appeared in the 3D MFS, the method of pseudo fundamental solutions and the method of anisotropic fundamental solutions outperform the 3D MFS. Some numerical experiments explore the performance of these two novel methods.

The Dynamic Energy-Momentum Tensor and the Logarithmic Singularity of a Generally Accelerating Edge Dislocation

Abstract The near-field logarithmic singularities in the field quantities associated with the acceleration of an arbitrarily moving edge dislocation are calculated based on a conservation law involving the dynamic energy-momentum tensor integrated over a domain enclosed by a multi-scale contour (an annulus of inner radius ϵ02 and outer radius ϵ0). The existence of the logarithmic singularities is obtained solely from the conservation law and the leading 1/r terms in the near fields of the stress and the velocity (which are those of the steady-state motion with velocity the instantaneous velocity in the accelerating motion). From the equations of motion and the symmetry in the second partial derivatives of the displacements for y≠0 we obtain that all six logarithmic terms of the near-field expansions are independent of the angle in the polar coordinates. All logarithmic terms in the near-field expansion of the strains and velocity in an arbitrarily moving edge dislocation (subsonically) are evaluated.

Paramagnetic Singularities of the Orbital Magnetism in Graphene with a Moiré Potential.

The recent detection of the singular diamagnetism of Dirac electrons in a single graphene layer paved a new way of probing 2D quantum materials through the measurement of equilibrium orbital currents which cannot be accessed in usual transport experiments. Among the theoretical predictions is an intriguing orbital paramagnetism at saddle points of the dispersion relation. Here we present magnetization measurements in graphene monolayers aligned on hexagonal boron nitride crystals. Besides the sharp diamagnetic McClure response at the Dirac point, we detect extra diamagnetic singularities at the satellite Dirac points of the moiré lattice. Surrounding these diamagnetic satellite peaks, we also observe paramagnetic peaks located at the chemical potential of the saddle points of the graphene moiré band structure and relate them to the presence of vanHove logarithmic singularities in the density of states. These findings reveal the long ago predicted anomalous paramagnetic orbital response in 2D systems when the Fermi energy is tuned to the vicinity of saddle points.

Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity

We find two series expansions for Legendre’s second incomplete elliptic integral E ( λ , k ) E(\lambda , k) in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the ( λ , k ) (\lambda , k) plane. Partial sums of the proposed expansions form a sequence of approximations to E ( λ , k ) E(\lambda ,k) which are asymptotic when λ \lambda and/or k k tend to unity, including when both approach the logarithmic singularity λ = k = 1 \lambda =k=1 from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for E ( λ , k ) E(\lambda , k) . For the reader’s convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.

Comparisons of best approximations with Chebyshev expansions for functions with logarithmic endpoint singularities

Comparisons of best approximations with Chebyshev expansions for functions with logarithmic endpoint singularities

Renormalized Energy of a Dislocation Loop in a 3D Anisotropic Body

AbstractThe paper presents a rigorous analysis of the singularities of elastic fields near a dislocation loop in a body of arbitrary material symmetry that extends over the entire three-space. Explicit asymptotic formulas are given for the stress, strain and the incompatible distortion near the curved dislocation. These formulas are used to analyze the main object of the paper, the renormalized energy. The core-cutoff method is used to introduce that notion: first, a core in the form of a curved tube along the dislocation loop is removed; then, the energy of the complement is determined (= the core-cutoff energy). As in the case of a straight dislocation, the core-cutoff energy has a singularity that is proportional to the logarithm of the core radius. The renormalized energy is the limit, as the radius tends to 0, of the core-cutoff energy minus the singular logarithmic part. The main result of the paper are novel formulas for the coefficient of logarithmic singularity (the ‘prelogarithmic energy factor’) and for the renormalized energy.

Crack tip solution for Mode III cracks in spring interfaces

Considering an infinite linear elastic isotropic solid in antiplane shear, the Mode III crack-tip solution for a semi-infinite crack located in a straight spring interface is systematically studied for the first time. A new analytic expression for this crack-tip solution is given in the form of a double asymptotic series of the main and the so-called associated shadow terms. It is shown that the series of the shadow terms associated with a main term is infinite, and all shadow terms include logarithmic terms. Thus, although the interface tractions are bounded, the linear elastic solution at this crack-tip has a logarithmic stress singularity which is comprehensively analysed. Noteworthy, the character of this stress singularity is very different from the well-known square root singularity at the crack tip in the classical fracture mechanics. A key advantage of the present approach is its simplicity, as only elementary mathematical tools are employed, and also its easy implementation in a computer algebra software. The latter fact is very relevant because the expressions of higher-order shadow terms become increasingly complicated, so their generation by a computer code becomes crucial. The present results allow the implementation of new enriched or singular crack-tip finite elements for such cracks, and the automatic generation of analytic solutions for benchmark problems for testing the finite-element codes using these special elements. Such codes can be applied to efficient numerical modelling of interface cracks, e.g., in adhesively bonded joints with a thin adhesive layer.

High-Accuracy Numerical Schemes for Solving Plane Boundary Problem for a Polyharmonic Equation and Their Application to Problems of Hydrodynamics

Boundary value problems are considered for harmonic, biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on the plane. The problems are reduced to solving linear integral equations on boundary contours, which are assumed to be smooth. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which integral equations are expressed. The algorithm uses the periodicity of functions defined on closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed degree. Applications to solving problems of hydrodynamics, filtration and other problems of theoretical physics are considered.

Fundamental solutions of a class of degenerate elliptic equation

In this paper, for the elliptic equation with two lines of different order of degeneration, given in the first quarter of the plane , four fundamental solutions are constructed that are expressed by Appell's hypergeometric functions of two variables. By means the decomposition formula of the Appell's function in products of the Gauss hypergeometric function, it is proved that the constructed fundamental solutions possess a logarithmic singularity for .

The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background

We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the ‘cylinder’, i.e. Friedrich’s representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the Cauchy problem for the conformally invariant wave equation (Frauendiener and Hennig 2018 Class. Quantum Grav. 35 065015) showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at t = 0. From a physical point of view, any data at t = 0 are determined from the earlier history of the system and hence not exactly ‘free data’. Therefore, it is appropriate to ask what happens if we ‘go further back in time’ and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding characteristic initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders for angular modes .

Regularized limit, analytic continuation and finite-part integration

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.

Logarithmic catastrophes and Stokes’s phenomenon in waves at horizons

Waves propagating near an event horizon display interesting features including logarithmic phase singularities and caustics. We consider an acoustic horizon in a flowing Bose–Einstein condensate where the elementary excitations obey the Bogoliubov dispersion relation. In the Hamiltonian ray theory the solutions undergo a broken pitchfork bifurcation near the horizon and one might therefore expect the associated wave structure to be given by a Pearcey function, this being the universal wave function that dresses catastrophes with two control parameters. However, the wave function is in fact an Airy-type function supplemented by a logarithmic phase term, a novel type of wave catastrophe. Similar wave functions arise in aeroacoustic flows from jet engines, path integrals in radio astronomy, and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from most previous authors in that we analyze the behavior of the integral representation of the wave function using exponential coordinates. This allows for a different treatment of the branch cuts and gives rise to an analysis based purely on saddlepoint expansions. We are thereby able to resolve the multiple real and complex waves that interact at the horizon and its companion caustic. We find that the horizon is a physical manifestation of a Stokes surface, marking the place where a wave is born, and that the horizon and the caustic do not in general coincide: the finite spatial region between them delineates a broadened horizon.

Solution for the Dynamic Elastically Compressible Power-Law Strain Hardening Cylindrical Cavity-Expansion Problem

A closed form analytical solution for both the quasi-static or dynamic strain hardening cylindrical cavity-expansion (CCE) problems requires the plastic region to be incompressible. This is due to the fact that in both cases a hydrostatic stress state must be explicitly included in the strain hardening function, making an analytical solution intractable. Additionally, for the dynamic problem the elastic region must be compressible to avoid a logarithmic singularity at infinity. An alterative for the dynamic analytical formulation is to use the finite element method (FEM). With the FEM weak formulation, solutions for the radial stress at the cavity surface and the elastic–plastic interface interface velocity as functions of the cavity-expansion velocity can be obtained when both the strain hardening plastic and elastic regions are elastically compressible. In this study, a comparison is made using results from both analytical (strong formulations), and FE simulations for both strain hardening and perfectly plastic dynamic CCE problems. It is concluded that with increasing cavity-expansion velocities, the elastically compressible strain hardening FEM solution is less resistive than the corresponding closed form analytical solution with an incompressible plastic region. Additionally, it is shown that the elastic–plastic interface velocity asymptotes at the bulk wave speed for completely elastically compressible solutions. Furthermore, the completely elastically compressible solutions remain valid for cavity-expansion velocities beyond the critical cavity-expansion velocity associated with the closed-form analytical strain hardening solution.

Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis

In this article, we propose and analyze the Clenshaw–Curtis–Filon-type method for computing many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities at the endpoints. First, by using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. Numerical experiments verify the efficacy and correctness of the presented method.

Asymptotic existence theorem for formal solutions with singularities of nonlinear partial differential equations via multisummability

In this paper, we consider the summability of formal solutions with singularities (such as logarithmic singularities, functional power singularities, etc.) of nonlinear partial differential equations in the complex domain. The main result is as follows: when a formal solution with singularities is given, under appropriate assumptions related to the formal solution, the equation has a true solution that admits the given formal solution as an asymptotic expansion. The proof is done by constructing a new formal solution that is equivalent to the given formal solution in the asymptotic sense and is multisummable in a suitable direction. The assumptions are stated in terms of the Newton polygon associated with the given formal solution.

On the issue of a new solution of the materials resistance contact problem on compression of elastic cylinders in contact with parallel generators

As a result of logarithmic singularity in the plane classical problem of elastically deformable material mechanics, it is proved that the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic, and physically linear material is not applicable. In the special case of elastic interaction of a cylinder with a half-plane, it is established that the convergence becomes equal to infinity. This paradoxical result confirms the nadequacy of Flaman’s model of a simple radial stress state in determining displacements. Based on this model, it is possible to determine only the stresses in parallel contacting cylinders, while the calculation of displacements, in this case, is not possible. Based on apreviously developed and mathematically approximated by the authors flat design scheme of Flaman the algorithm exception of conflicts has been proposed. The algorithm is based on the integral Fredholm equation solution and can be seen as a new fundamental and applied elasticity theory problem, which is of great importance when assessing the contact of refined strength and stiffness of the cylindrical parts of the supporting structures subject to the general and local deformations (cylindrical rollers, gears, pavements with their seal steel rollers, etc.).

A new formulation of the Green function in water of finite depth at low frequencies

We consider the frequency-domain free-surface Green function (GF) of wave radiation and diffraction in water of finite depth which is important for the hydrodynamic analysis of floating bodies, in general, and some renewable devices in shallow water, in particular. Further to the sound decomposition formulation presented in Chen (1993), in which two depth-effect functions are defined, a new analysis has been performed to obtain a new formulation. It is composed of the analytical part including all singular behaviors and oscillations, and the remaining part is well suited for numerical computations with an accuracy of order 10−8. Furthermore, the formulation is developed that the remaining part is fully smoothly varying and then approximated by Chebyshev polynomials. Not only the Green function itself but also its first-order and second-order derivatives are numerically evaluated and approximated independently by polynomials. The analytical formulations plus polynomials are then used to enhance efficiency and accuracy. Unlike previous studies, the analytical part of the Green function contains explicitly the logarithmic singularity when the depth-scaled wavenumber tends to zero. Classically included in the ’smooth’ part, this singular term is then extracted and the remaining part is fully smooth and much better approximated with consistent accuracy of 10−6 and less partitions. Furthermore, the added-mass and damping coefficients around a hemisphere are given.

The twistor geometry of parabolic structures in rank two

Let X be a quasi-projective curve, compactified to (Y, D) with $$X=Y-D$$ . We construct a Deligne–Hitchin twistor space out of moduli spaces of framed $$\lambda $$ -connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the $$\lambda $$ -connection.

Logarithmic Vertex Algebras

We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic properties of quantum fields in logarithmic conformal field theory. We develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras, including in particular the Borcherds identity and Kac Existence Theorem. Several examples are investigated in detail, and they exhibit some unexpected new features that are peculiar to the logarithmic case.