Logarithmic Singularity Research Articles

Wave resonances and the partition of energy in ideal compressible magnetohydrodynamic fluids

Phase mixing and resonant absorption are two processes that have been under scrutiny for some time because of their role in wave damping and in providing a mechanism for heating space and laboratory plasmas. The accumulation or absorption of energy that develops within a resonant layer is usually attributed to a logarithmic singularity, but it will be shown that this build up of energy is inextricably tied to a discontinuity in the fluid displacement at the resonant point. This change in the dynamics of the problem will be examined by establishing a partition of energy that identifies and isolates the individual resonances within the fluid. The partition is based on a variational description of the Fourier transformed equations and is guided by an electrical model of the MHD system that not only illustrates the resonant structure threading the fluid but also exposes the mechanism driving the resonant absorption process. A simplified version of this model is then constructed to help determine the approximate rate of energy absorption.

Plurisubharmonic functions and nef classes on complex manifolds

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.

Computation of integrals with oscillatory and singular integrands using Chebyshev expansions

We present a general method for computing oscillatory integrals of the form ∫−11f(x)G(x)eiωxdx, where f is sufficiently smooth on [−1,1], ω is a positive parameter and G is a product of singular factors of algebraic or logarithmic type. Based on a Chebyshev expansion of f and the properties of Chebyshev polynomials, the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor eiωx. Furthermore, due to numerically stable recurrence relations for the modified moments, the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration. Numerical examples are provided to confirm our analysis.

Unique continuation on a line for the Helmholtz equation

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.

Closed form algebra on a disk is Koszul

We prove that the algebra of closed differential forms on an (algebraic, formal, or analytic) disk with logarithmic singularities along several coordinate hyperplanes is Koszul (both nontopologically and topologically). A relation to variations of mixed Hodge-Tate structures is discussed in the introduction.

Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density

We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem \begin{eqnarray*} \rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, u(x,0)=u_{0}(x), \quad in\quad R^n, \end{eqnarray*} where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$. We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in $R^n$. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation $|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.

Power series with the von Mangoldt function

We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.

Thermodynamic singularities in the entanglement entropy at a two-dimensional quantum critical point

We study the bipartite entanglement entropy of the 2D transverse-field Ising model in the thermodynamic limit. Series expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between subsystems. Series extrapolations are used to extract power laws and logarithmic singularities as the quantum critical point is approached, giving access to new universal quantities. In 1D, we find excellent agreement with exact results as well as quantum Monte Carlo simulations. In 2D, we find compelling evidence that the entanglement at a corner is significantly different from a free boson field theory. These results demonstrate the power of the series expansion method for calculating entanglement entropy in interacting systems, a fact that will be particularly useful in searches for exotic quantum criticality in models with and without the sign problem.

Ideal focusing in the theory of electrostatic spectrographs

The problem of synthesis of field structures, in which the spectrographic principle of separating charged particle flows is implemented and ideal angular focusing is simultaneously observed in the symmetry plane for each energy value, is formulated. The search for spectrographic media is effectively carried out with the help of inverse mechanical problems. Two approaches are proposed, one of which involves the reduction to the integral Abel equation and the other is based on the application of the Goursat theorem. The algorithms for the synthesis of spectrographic media with ideal focusing of flows are described in detail, and possible versions of solutions corresponding to representatives of two structural classes (with potentials homogeneous in the Euler sense and with homogeneous potentials with a logarithmic singularity) are indicated. The prospects of developing high-transmission spectrographs for energy analysis are outlined.

Analytic functions with polar and logarithmic singularities and locally convex boundary values

We consider a family of functions which have a pole of order m and a logarithmic term at the infinity and generalize univalent convex functions defined in the exterior of the unit disc. We prove sharp estimates for the derivative and the Schwarzian and describe some geometric properties of such functions.

Singularity analysis of a reaction–diffusion equation with a solution-dependent Dirac delta source

We analyze the existence and singularity of a solution to a reaction–diffusion equation, whose reaction term is represented by a Dirac delta function which depends on the solution itself. We prove that there exists a unique analytic solution with a logarithmic singularity at the origin.

A simplified implementation of van der Waals density functionals for first-principles molecular dynamics applications

We present a simplified implementation of the non-local van der Waals correlation functional introduced by Dion et al. [Phys. Rev. Lett. 92, 246401 (2004)] and reformulated by Román-Pérez et al. [Phys. Rev. Lett. 103, 096102 (2009)]. The proposed numerical approach removes the logarithmic singularity of the kernel function. Complete expressions of the self-consistent correlation potential and of the stress tensor are given. Combined with various choices of exchange functionals, five versions of van der Waals density functionals are implemented. Applications to the computation of the interaction energy of the benzene-water complex and to the computation of the equilibrium cell parameters of the benzene crystal are presented. As an example of crystal structure calculation involving a mixture of hydrogen bonding and dispersion interactions, we compute the equilibrium structure of two polymorphs of aspirin (2-acetoxybenzoic acid, C(9)H(8)O(4)) in the P2(1)/c monoclinic structure.

Exotic baker and wandering domains for Ahlfors islands maps

Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere or a torus. We construct a variety of examples of analytic functions g:W->X, where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands condition". In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the omega-limit set of a Baker domain of such a function. As a corollary of our construction, we show that there are entire functions with Baker domains in which the iterates converge to infinity arbitrarily slowly. We also construct Ahlfors islands maps with wandering domains and logarithmic singularities, as well as examples where X is a compact hyperbolic surface.

Dispersion of an acoustic pulse passing through a large-grained polycrystalline film

Propagation of a short acoustic pulse through a polycrystalline film comprised of large randomly oriented elastically anisotropic grains is analyzed theoretically. For average grain size much larger than the film thickness, a short acoustic pulse launched normally into the film will traverse each grain in a time determined by the acoustic slowness in the direction normal to the film, which will depend on the local grain orientation. A typical measurement averages over a large number of grains resulting in the broadening of the composite output pulse. The resulting pulse shape is characterized by distinct features related to stationary values of the directionally dependent acoustic slowness of the crystalline material. Maxima and minima in the slowness yield discontinuities in the pulse shape, while saddle points yield logarithmic singularities. For cubic and hexagonal crystals, power law singularities result from cones of directions in which the slowness is a maximum or minimum. Numerical results, taking into account Gaussian broadening of the input pulse, are presented for thin film materials commonly encountered in picosecond ultrasonic experiments, such as copper, gold, and aluminum.

Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices

We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that P(N+,N)≈exp[-βN2ψζ(N+/N)], where β is the Dyson index of the ensemble and ψζ(κ) is a rate function that we compute explicitly in the full range 0≤κ≤1 and for any ζ. The rate function ψζ(κ) displays a quadratic behavior modulated by a logarithmic singularity close to its minimum κ⋆(ζ). This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance Δ(N) of the number of relevant components is also shown to grow universally (independent of ζ) as Δ(N)∼(βπ2)-1 lnN for large N.

Matrix form near tip solutions of interface corners

In this paper, the stresses near the tip of interface corners are expressed in matrix power function form. To make this matrix power function form valid for all possible cases of interface corners, all kinds of singularities including oscillatory and logarithmic singularity are discussed in this paper. The coefficient vector of this matrix power function form solution is then defined as a vector of stress intensity factors along the radial direction. Since the stress intensity factors are functions of radial direction, the maximum stress intensity factor of a certain radial direction may be useful for the fracture prediction of interface corners. This new definition of stress intensity factors is applicable for all possible singular orders—repeated or distinct, real or complex, and keeps the same unit for all possible interface corners. Therefore, it will be helpful for bridging the problems of cracks, corners, interface cracks and interface corners. Finally, the matrix form solutions are reduced to the scalar form solutions for two special cases. One is a special case of corner angles—cracks, and the other is a special case of anisotropic materials—isotropic materials. Through this reduction, new scalar form analytical solutions are obtained and specialized further to compare with the existing analytical solutions.

Nonlinear optical response in gapped graphene

We present a formulation for the nonlinear optical response in gapped graphene, where the low-energy single-particle spectrum is modeled by massive Dirac theory. As a representative example of the formulation presented here, we obtain a closed form formula for the third harmonic generation in gapped graphene. It turns out that the covariant form of the low-energy theory gives rise to peculiar logarithmic singularities in the nonlinear optical spectra. The universal functional dependence of the response function on dimensionless quantities indicates that the optical nonlinearity can be largely enhanced by tuning the gap to smaller values.

Five-loop Konishi in [formula omitted] SYM

We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling. It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit. It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops. The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE. Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise. We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization. At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less. This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically. We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

A clarification of the role of crack-tip conditions in linear elasticity with surface effects

We examine the role of crack-tip conditions in the reduction of stress at a crack tip in a theory of linear elasticity with surface effects. The maximum number of allowable end conditions for complete removal of a stress singularity is demonstrated for both plane and anti-plane problems. In particular, we show that the necessary and sufficient conditions for bounded stresses at a crack tip cannot be satisfied with a first-order (curvature-independent) theory of surface effects, which leads, at most, to the reduction of the classical strong square-root singularity to a weaker logarithmic singularity.

Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity

AbstractBased on potential theory, Steklov eigensolutions of elastic problems can be converted into eigenvalue problems of boundary integral equations (BIEs). The kernels of these BIEs are characterized by logarithmic and Hilbert singularities. In this article, the Nyström methods are presented for obtaining eigensolutions (λ(i),u(i)), which have to deal with the two kinds of singularities simultaneously. The solutions possess high accuracy orders O(h3) and an asymptotic error expansion with odd powers. Using h3 ‐Richardson extrapolation algorithms, we can greatly improve the accuracy orders to O(h5). Furthermore, a generalized Fourier series is constructed by the eigensolutions, and then solving the elasticity displacement and traction problems involves just calculating the coefficients of the series. A class of elasticity problems with boundary Γ is solved with high convergence rate O(h5). The efficiency is illustrated by a numerical example. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012