Algebraic Singularities Research Articles

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces

For the numerical solution of Fredholm integral equations on [−1,1] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods.

Deformations of Pairs of Kleinian Singularities

Abstract Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger.

Mechanism singularities and shakiness from an algebraic viewpoint

Kinematic singularities are configurations where the configuration space (c-space) V is not smooth or configurations where the rank of the constraint Jacobian drops while the c-space is locally smooth. Both can be analyzed by means of a higher-order analysis, for which the kinematic tangent cone provides a conceptual framework. It is desirable to complement such an analysis with more insight into the origin of specific singular phenomena. Such additional insight is obtained by investigating the kinematics from an algebraic viewpoint. Points of interest are then the singularities of a variety, referred to as algebraic singularities. Algebraic singularities are reflected as kinematic singularities or as shakiness. It is shown that these algebraic singularities can be attributed to properties of the ideal generated by the loop constraints — the constraint ideal. Particular situations arise depending on whether this constraint ideal is radical. If it is not radical, a singular branch is locally defined by a primary ideal. If it is radical, it must be distinguished whether real and complex dimensions of intersecting branches are different. Real intersections indicate kinematic singularities; in all other cases, the mechanism may be in a singularity or may be shaky. Various examples are presented.

Spectrally accurate numerical quadrature formulas for a class of algebraically singular periodic Hadamard finite part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integrals Hσ(t;u)=⨎0T|sinπ(x−t)T|σu(x)dx,0<t<T,σ<−1,σ⁄∈Z,u(x) being sufficiently differentiable and T-periodic on R. Thus σ=−(m+δ), m∈{1,2,3,…}, 0<δ<1. For each such σ, we regularize Hσ(t;u), and show that Hσ(t;u)=Hσ+2r(t;Uσ),r=⌊(m+1)/2⌋, where Uσ(x)=∑k=0raku(2k)(x) for some constants ak, Hσ+2r(t;Uσ) being a regular integral. We then propose to approximate Hσ(t;u) by the quadrature formula Qσ,n(t;u)≡Hσ(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qσ,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qσ,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R).We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals Hσ(t;u) with σ<−1 and σ⁄∈Z apply, with no changes, to the regular singular integrals Hσ(t;u) with σ>−1 and σ⁄∈Z.We illustrate the effectiveness of Qσ,n(t;u) with numerical examples both for σ<−1 and σ>−1.Finally, we show that the HFP or regular integral ⨎0Tf(x)dx of any T-periodic integrand f(x) that has algebraic singularities of the form |x−t+kT|σ, 0<t<T, k=0,±1,±2,…, with σ⁄∈Z, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed as Hσ(t;u), where u(x) is a T-periodic and sufficiently differentiable function of x on R that can be computed from f(x). Therefore, ⨎0Tf(x)dx can be computed efficiently using our new numerical quadrature formulas Qσ,n(t;u) on the individual Hσ(t;u). Again, only 2n function evaluations, namely, u(xn,k), k=0,1,…,2n−1, are needed for the whole process.

Fast and accurate solvers for weakly singular Volterra integral equations in weighted spaces

For the numerical solution of Volterra integral equations of the second kind whose integrands have diagonal and endpoint algebraic singularities, we investigate in this paper, a fast modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The iterated version of the standard collocation method is also defined. The proposed methods are shown to converge faster than the collocation scheme, and the Sloan iteration can be applied to the modified collocation solution to make it even more accurate. This research seems to be the first to explore superconvergent approaches for solving integral equations of this type. Some numerical tests are presented to show the effectiveness of the suggested methods.

Real tight contact structures on lens spaces and surface singularities

We give a partial classification for the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on [Formula: see text] and real lens spaces [Formula: see text]. We prove that there is a unique real tight [Formula: see text] and [Formula: see text]. We show that there is at most one real tight [Formula: see text] with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an [Formula: see text] which cannot be made convex equivariantly.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Fractional Laplacian – Quadrature Rules for Singular Double Integrals in 3D

Abstract In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.

SINGULAR HOLOMORPHIC MORSE INEQUALITIES ON NON-COMPACT MANIFOLDS

We obtain asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, q-convex manifolds and q-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.

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.

Recursive moment computation in Filon methods and application to high-frequency wave scattering in two dimensions

Abstract We study the efficient approximation of highly oscillatory integrals using Filon methods. A crucial step in the implementation of these methods is the accurate and fast computation of the Filon quadrature moments. In this work we demonstrate how recurrences can be constructed for a wide class of oscillatory kernel functions, based on the observation that many physically relevant kernel functions are in the null space of a linear differential operator whose action on the Filon interpolation basis is represented by a banded (infinite) matrix. We discuss in further detail the application to two classes of particular interest: integrals with algebraic singularities and stationary points and integrals involving a Hankel function. We provide rigorous stability results for the moment computation for the first of these classes and demonstrate how the corresponding Filon method results in an accurate approximation at truly frequency-independent cost. For the Hankel kernel, we derive error estimates that describe the convergence behaviour of the method in terms of frequency and number of Filon quadrature points. Finally, we show how Filon methods with recursive moment computation can be applied to efficiently compute integrals arising in hybrid numerical-asymptotic collocation methods for high-frequency wave scattering on a screen.

Numerical evaluation and analysis of highly oscillatory singular Bessel transforms with a particular oscillator

This paper proposes and analyzes two affordable and efficient quadrature rules for the numerical approximation of the oscillatory Bessel transform ∫0bxα(b−x)βf(x)Jν(ωxγ)dx with algebraic singularities, where b,α,β,ν,ω,γ denote the given constants. Firstly, we derive the explicit formula and asymptotic estimation of the generalized moments ∫0bxα′(b−x)β′Jν(ωxγ)dx with α′,β′>−1 by means of the Meijer G function. Furthermore, we design a modified Filon-type method based on a two-endpoint Taylor interpolation polynomial. In particular, we also give a more efficient Clenshaw–Curtis–Filon-type method in view of a special Hermite interpolation polynomial at the Clenshaw–Curtis points. Moreover, this method is easily implemented by the fast Fourier transform and fast computation of the modified moments. The useful homogeneous recurrence relation of the required modified moments is derived by the Bessel equation and the properties of the Chebyshev polynomial. Importantly, the rigorous error analyses in inverse powers of ω for the proposed numerical methods are carried out in details. Some primary numerical experiments can confirm our theoretical analysis, and verify the accuracy and efficiency of the proposed numerical methods.

Singular Euler–Maclaurin expansion on multidimensional lattices

We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the expansion order for band-limited functions and that the runtime is independent of the number of particles. First, the EM summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler–Maclaurin expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of relevance in condensed matter, statistical, and quantum physics. An implementation in mathematica is provided online along with this article.

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

In a 1979 paper, Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlevé property), which is a generalisation of the Painlevé property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points.

The closures of test configurations and algebraic singularity types

Given a Kähler manifold X with an ample line bundle L, we consider the metric space of finite energy geodesic rays associated to the Chern class c1(L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems.By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge–Ampère energy of a general finite energy ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge–Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L1 rays to non-Archimedean rays.

On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

Global dynamics of the one-dimensional Euler-alignment system with weakly singular kernel

This letter studies the Euler-alignment system with weakly singular influence functions by introducing a novel technique to bound the density. Instead of resorting to a nonlinear maximum principle used in Tan (2020) to bound the interaction term ψ∗ρ by ρs with s∈(0,1) for ψ with an algebraic singularity at origin, we bound ψ∗ρ by a relaxed constant for any ψ∈L1. We thus establish the global-in-time existence results with weaker assumptions on ψ and refined solution bounds, characterized by the structure of ψ.

Comparing the Performance of Four Sinc Methods for Numerical Indefinite Integration

Aims/ Objectives: To compare the performance of four Sinc methods for the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities.&#x0D; Methodology: The first two quadrature formulas were proposed by Haber based on the sinc method, the third is Stengers Single Exponential (SE) formula and Tanaka et al.s Double Exponential (DE) sinc method completes the number. Furthermore, an application of the four quadrature formulas on numerical examples, reveals convergence to the exact solution by Tanaka et al.s DE sinc method than by the other three formulae. In addition, we compared the CPU time of the four quadrature methods which was not done in an earlier work by the same author.&#x0D; Conclusion: Haber formula A is the fastest as revealed by the CPU time.

A topological introduction to Lipschitz geometry of complex singularities

These are the notes of the three lectures I gave during the IXth Winterbraids School which took place in Reims from 4 to 7th March 2019. The aim of this course was to introduce researchers working in low-dimensional topology to Lipschitz geometry of complex singularities. In these lectures, I focussed on topological points of view on the objects, avoiding as much as possible technical material from algebraic geometry and singularity theory such as resolution of singularities, Nash transform, generic projections of curves and surfaces, etc.