Logarithmic Singularity Research Articles

SOME MORE ON LOGARITHMIC SINGULARITY INTEGRATION IN BOUNDARY ELEMENT METOD

The accuracy of calculations of integrals with logarithmic singularities for two methods, namely the method of ignoring singularities and the method of subtraction (consisting in separating the singular part from the remaining non-singular), are presented in this paper. Only two-dimensional problems, like Dirichlet's problems, as well as acoustic problems formulated in the frequency domain are considered. Problems related to the accuracy of calculations are discussed and the influence of frequency, as well as the influence of the geometry of the analysed area on the accuracy of calculations, are indicated. When we talk about the influence of geometry, we mean not only discretization, but also the configuration of the area, such as the sharp edges of the boundary line, assuming the use of the classic, without any modifications, Boundary Element Method.

Fast Dimension Spectrum for a Potential with a Logarithmic Singularity

We regard the classic Thue–Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of the local measure decay (superpolynomial). For several scaling functions, we show that points with this behavior are abundant in the sense of full Hausdorff dimension. At the fastest possible scaling, the corresponding rates reveal several remarkable phenomena. There is a gap between level sets for dyadic rationals and non-dyadic points, and beyond dyadic rationals, non-zero accumulation points occur only within intervals of positive length. The dependence between the smallest and the largest accumulation point also manifests itself in a non-trivial joint dimension spectrum.

Critical exponents and fluctuations at BEC in a 2D harmonically trapped ideal gas

The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.

Simplicity of AdS super Yang-Mills at one loop

We perform a systematic bootstrap analysis of four-point one-loop Mellin amplitudes for super gluons in AdS5 × S3 with arbitrary Kaluza-Klein weights. The analysis produces the general expressions for these amplitudes at extremalities two and three, as well as analytic results for many other special cases. From these results we observe remarkable simplicity. We find that the Mellin amplitudes always contain only simultaneous poles in two Mellin-Mandelstam variables, extending a previous observation in the simplest case with the lowest Kaluza-Klein weights. Moreover, we discover a substantial extension of the implication of the eight-dimensional hidden conformal symmetry, which goes far beyond the Mellin poles associated with the leading logarithmic singularities. This leaves only a small finite set of poles which can be determined on a case-by-case basis from the contributions of protected operators in the OPE.

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.

Logarithmic singularity subtraction for matrix‐friendly layered medium Green's functions

AbstractA subtraction procedure is proposed to evaluate the logarithmic singularity for matrix‐friendly layered medium Green's functions. In this subtraction procedure, an integral of the Bessel function combined with the exponential function and power function, where the sum of the power of and the order of the Bessel function is −1, is designed to fit the oscillation of the tail integral. After subtracting the designed integral, the integrand of the remainder decreases rapidly. Accordingly, the remainder can be calculated without finding any surface poles or steepest descent path, while the subtracted part can be calculated analytically. Several numerical examples are provided to reveal the feasibility and the accuracy of the proposed method.

Modular differential equations with movable poles and admissible RCFT characters

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.

Computation of Neumann Localised Boundary Domain Integral Equations

Most integrals in Localised Boundary Domain Integral Equations (LBDIEs) comprise singularities. This paper aims to produce numerical solutions of the LBDIEs for the Partial Differential Equations with variable coefficients. The singularities of the boundary integrals in LBDIEs will be handled by using a semi-analytic for logarithmic singularity and a semi-quadratic analytic method for r−2 singularity. Whereas the singular domain integrals are handled by using the Duffy transformation. The LBDIEs that we consider are associated with the Neumann problem, which can be solved with a condition. If it can be solved, the solution is, however, unique up to an additive constant. We add a perturbation operator to the LBDIEs to convert the LBDIE to a uniquely solvable equation. The perturbed integral operator leads the perturbed LBDIEs to a dense matrix system that disable the use of methods in solving sparse matrix system. We solve the system of linear equations by Lower-Upper (LU) decomposition method. The numerical results indicate that high accuracy results can be attained. It gives the impression that the methods we use in this numerical experiment are reliable in handling the boundary and domain singular integrals.

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.

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.