Algebraic Singularities Research Articles

Could MT2 be a singularity variable?

The algebraic singularity method is a framework for analyzing collider events with missing energy. It provides a way to draw out a set of singularity variables that can catch singular features originating from the projection of full phase space onto the observable phase space of measured particle momenta. It is a promising approach applicable to various physics processes with missing energy but still requires more studies for use in practice. Meanwhile, in the double-sided decay topology with an invisible particle on each side, the MT2 variable has been known to be a useful collider observable for measuring particle masses from missing energy events or setting signal regions of collider searches. We investigate the relation between the two different types of kinematic variables in double-sided decay topology. We find that the singularity variables contain the MT2 variable in many cases, although the former is not a strict superset of the latter.

Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given.

Numerical study of non-uniqueness for 2D compressible isentropic Euler equations

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

Algebraic singularities of scattering amplitudes from tropical geometry

We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar mathcal{N} = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

Evaluation of the 4-D singular and near singular potential integrals via the Stokes' theorem

A closed-form solution for singular and near-singular double surface integrals arising in boundary integral equations of electrostatics is given for the case of arbitrary coplanar polygonal surfaces. To date, a limited number of closed-form solutions for these integrals were published only for the case of coincident triangular surfaces. The second result of the paper is the evaluation method itself, which is extendable to non-coplanar and non-polygonal surfaces. The main idea of the evaluation method is to construct exact differential forms to perform integration via the Stokes' theorem. Free of coordinates, the differential forms approach has several important advantages over the traditional algebraic singularity subtraction and singularity cancellation methods. Numerical tests on triangles show that the proposed method maintains accuracy even as the aspect ratio of a triangle tends to infinity while the existing methods fail in that limit. Thus, the results of this paper are expected to greatly improve the accuracy and efficiency of the computational electrostatics codes. We also show that the proposed method is extendable to other kernels, e.g., Helmholtz, and therefore has the potential to greatly speed up the matrix build of the computational electromagnetics and acoustics codes.

The radiated acoustic pressure and time scales of a spherical bubble

Numerical simulations of violent bubble dynamics are often associated with numerical instabilities at the end of collapse, when a shock wave is emitted. Based on the Keller–Miksis equation, we show that this is caused by two time scales associated with the phenomenon. Nonsingular equations are thus formed based on asymptotic expansion theory and the time derivatives of the bubble radius are shown to have algebraic singularities in the Mach number. The period of oscillation is shown to divide into two asymptotic layers: a long and short time scale. The short time scale, on which significant acoustic radiation is emitted from the bubble, has been determined to be , where c is the speed of sound in the liquid, the maximum bubble radius, ρ the liquid density, the hydrostatic pressure of the liquid, the vapour pressure of the liquid and κ the polytropic index of the bubble gas. Using the scalings for this short time scale, the radiated acoustic pressure scale has been deduced to be , where is the radial distance from the bubble centre to the point of measurement. The results are validated by comparison with experimental results.

A mixed collocation scheme for solving second kind Fredholm integral equations in [-1,1

In this paper we propose a suitable combination of two collocation methods based on the zeros of Jacobi polynomials in order to approximate the solution of Fredholm integral equations on [−1,1]. One of the main interesting aspects of this procedure is that our approach is cheaper than the usual collocation method based on standard Lagrange interpolation using Jacobi zeros. Moreover, we can successfully manage functions with algebraic singularities at the endpoints. The error of the method is comparable with the error of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method is proved, and some numerical tests, which confirm the theoretical estimates, are provided.

Asymptotics and numerical approximation of highly oscillatory Hilbert transforms

In this paper, we study asymptotics and fast computation of highly oscillatory Hilbert transforms ⨍0+∞f(x)x−τeiωxHν(1)(ωx)dx, where τ > 0 and f is analytic in the complex plane ℜ(z) ≥ 0, which may have an algebraic singularity at the point x=0. By analytic continuation, we first transform Hilbert transforms into the integrals on [0,+∞) with the integrands that don’t oscillate and decay exponentially, and then present the asymptotic behaviour about ω based on asymptotic expansion. For the computation of highly oscillatory Hilbert transforms, we present efficient numerical methods according to the position of τ. To be precise, we construct a Gaussian quadrature rule for it if τ=O(1) or τ ≥ 1. If 0 < τ ≪ 1, we rewrite it as a sum of three integrals, which can be efficiently evaluated by using Chebyshev approximation, Gauss–Laguerre quadrature rule and Meijer G–function. The effectiveness of the proposed methods are demonstrated by several numerical experiments.

Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes

The paper deals with de la Vallée Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments.

On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results.

Asymptotics of Bivariate Analytic Functions with Algebraic Singularities

In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.

Levin methods for highly oscillatory integrals with singularities

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.

Multiplicity of singularities is not a bi-Lipschitz invariant

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture, constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.

The series expansions and Gauss-Legendre rule for computing arbitrary derivatives of the Beta-type functions

The beta-type functions play an important role in many applied sciences. The partial derivatives of the beta function and the incomplete beta function are integrals involving algebraic and logarithmic endpoint singularities. In this paper, some series expansions for these beta-type functions are found, which are easily used to evaluate these functions with prescribed precision. On the other hand, an accurate Gauss-Legendre quadrature formula is designed to compute these beta-type functions and their partial derivatives based on the Puiseux series for the integrands at their singularities. Some numerical examples confirm the high accuracy and high efficiency of the two algorithms, and also show that the algorithms can be used to effectively evaluate the generalized beta-type functions.

Correlation functions of one-dimensional strongly interacting two-component gases

We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitute the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution $n(k)$ at zero temperature does not present algebraic singularities but the tails obey the universal decay $\lim_{k\rightarrow\pm\infty}n(k)\sim C/k^4$ with the amplitude $C$ given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.

On Generalised Abundance, II

In our previous work, we introduced the Generalised Nonvanishing Conjecture, which generalises several central conjectures in algebraic geometry. In this paper, we derive some remarkable nonvanishing results for pluricanonical bundles which were not predicted by the Minimal Model Program, by making progress towards the Generalised Nonvanishing Conjecture in every dimension. The main step is to establish that a somewhat stronger version of the Generalised Nonvanishing Conjecture holds almost always in the presence of metrics with generalised algebraic singularities, assuming the Minimal Model Program in lower dimensions.

Numerical methods for two classes of singularly oscillatory Bessel transforms and their error analysis

This work introduces and analyzes both the complex integration method and the modified complex integration method for two classes of highly oscillatory Bessel transforms with algebraic or algebraic-logarithmic singularities. We first transform the original integrals into the infinite integrals on [1,+∞) by the change of variable t=x−β, where β>0. With the help of the integral representation of the Bessel function, the integral representation and the asymptotic series of the Whittaker function, we then use the Cauchy’s residue theorem and convert the above infinite integrals into the infinite integrals on [0,+∞). The integrands of the resulting infinite integrals are not oscillatory and decay exponentially fast. Fortunately, they can be efficiently evaluated by using the Gauss-Laguerre quadrature rule. Furthermore, we conduct their error analysis in inverse powers of the frequency ω. Finally, numerical experiments are provided to confirm our theoretical analysis. We can see from theoretical analysis and numerical experiments that the accuracy of the quadrature increases both for the case of growing the number of nodes n and fixed ω, and for the case of an increasing ω and fixed the number of nodes n. The comparison of the proposed two methods and the Filon-type method presented in Kang and Ma (2017), is briefly discussed as well.

A mixed scheme of product integration rules in (−1,1)

The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

Quadratic Vector Equations On Complex Upper Half-Plane

We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb{H}$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb{R}$. In [AEK17a] we qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16b], we present a complete stability analysis of the equation for any $z\in \mathbb{H}$, including the vicinity of the singularities.

Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation

In this paper we study the continuous coagulation and multiple fragmentation equation for the mean-field description of a system of particles taking into account the combined effect of the coagulation and the fragmentation processes in which a system of particles growing by successive mergers to form a bigger one and a larger particle splits into a finite number of smaller pieces. We demonstrate the global existence of mass-conserving weak solutions for a wide class of coagulation rate, selection rate and breakage function. Here, both the breakage function and the coagulation rate may have algebraic singularity on both the coordinate axes. The proof of the existence result is based on a weak $ L^1 $ compactness method for two different suitable approximations to the original problem, namely, the conservative and non-conservative approximations. Moreover, the mass-conservation property of solutions is established for both approximations.