Endpoint Singularities Research Articles

Generalized Gaussian quadrature rules over regions with parabolic edges

This paper presents a generalized Gaussian quadrature method for numerical integration over regions with parabolic edges. Any region represented by R 1={(x, y)| a≤x≤b, f(x)≤y≤g(x)} or R 2={(x, y)| a≤y≤b, f(y)≤x≤g(y)}, where f(x), g(x), f(y) and g(y) are quadratic functions, is a region bounded by two parabolic arcs or a triangular or a rectangular region with two parabolic edges. Using transformation of variables, a general formula for integration over the above-mentioned regions is provided. A numerical method is also illustrated to show how to apply this formula for other regions with more number of linear and parabolic sides. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear and parabolic edges. Finally, the computational efficiency of the derived formulae is demonstrated through several numerical examples.

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g?C ?[a,b], or g?C ?(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x j =a+jh and t?{x 1,?,x n?1}. These asymptotic expansions are based on some recent generalizations of the Euler---Maclaurin expansion due to the author (A. Sidi, Euler---Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which β=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler---Maclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) β , with g(x) as before and β=?1,?3,?5,?, from which suitable quadrature formulas can be obtained. We revisit the case of β=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.

A formalism for the systematic treatment of rapidity logarithms in Quantum Field Theory

Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the rele- vant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group". That is, a Collin-Soper like equation is realized as a renormalization group equation, but has a more universal applicability to observables beyond the traditional transverse momentum dependent parton distribution functions (TMDPDFs) and the Sudakov form factor. This formalism has the feature that it allows one to track the (non-standard) scheme dependence which is inherent in any scenario where one performs a resummation of rapidity divergences. We present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form factor. The formalism is then used to study observables of current interest. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. Our formalism allows one to define gauge invariant TMDPDFs which are independent of both the hard scattering amplitude and the soft function, i.e. they are uni- versal. We present details of the factorization and resummation of the jet broadening cross section including a renormalization in pT space. We furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.

Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals ∫ a b f ( x ) d x \int ^b_af(x)\,dx , where f ∈ C ∞ ( a , b ) f\in C^{\infty }(a,b) but can have arbitrary algebraic singularities at one or both endpoints. We assume that f ( x ) f(x) has asymptotic expansions of the general forms a m p ; f ( x ) ∼ K ( x − a ) − 1 + ∑ s = 0 ∞ c s ( x − a ) γ s as x → a + , a m p ; f ( x ) ∼ L ( b − x ) − 1 + ∑ s = 0 ∞ d s ( b − x ) δ s as x → b − , \begin{align*} &f(x)\sim K\,(x-a)^{-1}+\sum ^{\infty }_{s=0}c_s(x-a)^{\gamma _s} \quad \text {as}\ x\to a+,\\ &f(x)\sim L\,(b-x)^{-1}+\sum ^{\infty }_{s=0}d_s(b-x)^{\delta _s} \quad \text {as}\ x\to b-, \end{align*} where K , L K,L , and c s , d s c_s, d_s , s = 0 , 1 , … , s=0,1,\ldots , are some constants, | K | + | L | ≠ 0 , |K|+|L|\neq 0, and γ s \gamma _s and δ s \delta _s are distinct, arbitrary and, in general, complex, and different from − 1 -1 , and satisfy \[ ℜ γ 0 ≤ ℜ γ 1 ≤ ⋯ , lim s → ∞ ℜ γ s = + ∞ ; ℜ δ 0 ≤ ℜ δ 1 ≤ ⋯ , lim s → ∞ ℜ δ s = + ∞ . \Re \gamma _0\leq \Re \gamma _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \gamma _s=+\infty ;\quad \Re \delta _0\leq \Re \delta _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \delta _s=+\infty . \] Hence the integral ∫ a b f ( x ) d x \int ^b_af(x)\,dx exists in the sense of Hadamard finite part. The results we obtain in this work extend some of the results in [A. Sidi, Numer. Math. 98 (2004), pp. 371–387] that pertain to the cases in which K = L = 0. K=L=0. They are expressed in very simple terms based only on the asymptotic expansions of f ( x ) f(x) as x → a + x\to a+ and x → b − x\to b- . With h = ( b − a ) / n h=(b-a)/n , where n n is a positive integer, one of these results reads h ∑ i = 1 n − 1 f ( a + i h ) ∼ I [ f ] a m p ; + K ( C − log ⁡ h ) + ∑ s = 0 γ s ∉ { 2 , 4 , … } ∞ c s ζ ( − γ s ) h γ s + 1 a m p ; + L ( C − log ⁡ h ) + ∑ s = 0 δ s ∉ { 2 , 4 , … } ∞ d s ζ ( − δ s ) h δ s + 1 as h → 0 , \begin{align*} h\sum ^{n-1}_{i=1}f(a+ih)\sim I[f]&+K\,(C -\log h) + \sum ^{\infty }_{\substack {s=0\\ \gamma _s\not \in \{2,4,\ldots \}}}c_s \zeta (-\gamma _s)\,h^{\gamma _s+1}\\ &+L\,(C -\log h) +\sum ^{\infty }_{\substack {s=0\\ \delta _s\not \in \{2,4,\ldots \}}}d_s\zeta (-\delta _s) h^{\delta _s+1}\quad \text {as $h\to 0$}, \end{align*} where I [ f ] I[f] is the Hadamard finite part of ∫ a b f ( x ) d x \int ^b_af(x)\,dx , C C is Euler’s constant and ζ ( z ) \zeta (z) is the Riemann Zeta function. We illustrate the results with an example.

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals \(I[f]=\int^{b}_{a}f(x)\,dx\), where f∈C ∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms Open image in new window where \(\widehat{P}(y),P_{s}(y)\) and \(\widehat{Q}(y),Q_{s}(y)\) are polynomials in y. The γ s and δ s are distinct, complex in general, and different from −1. They also satisfy Open image in new window The results we obtain in this work extend the results of a recent paper [A. Sidi, Numer. Math. 98:371–387, 2004], which pertain to the cases in which \(\widehat{P}(y)\equiv0\) and \(\widehat{Q}(y)\equiv0\). They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let \(D_{\omega}=\frac{d}{d\omega}\), h=(b−a)/n, where n is a positive integer, and define \(\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)\). Then with \(\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}\) and \(\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}\), one of these results reads Open image in new window where ζ(z) is the Riemann Zeta function and σ i are Stieltjes constants defined via \(\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]\), i=0,1,… .

Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

Algorithms are proposed for the numerical evaluation of Cauchy principal value integrals ⨍ − 1 1 w ( t ) f ( t ) / ( t − x ) d t , − 1 < x < 1 , with weight functions of Jacobi type singularities w ( t ) = ( 1 − t ) α ( 1 + t ) β , where α = ± 1 / 2 and β = ± 1 / 2 , for a given function f ( t ) and Hadamard finite-part integrals ⨎ − 1 1 w ( t ) f ( t ) / ( t − x ) 2 d t . The function f is interpolated by using a finite sum of Chebyshev polynomials. The present algorithms require O ( N log N ) arithmetic operations, where N is the order of the interpolation polynomial. It is shown that the present scheme gives uniform approximations, namely the errors are bounded independently of x , and is very efficient for smooth f . Further, we discuss approximations of hyper-singular integrals ∫ − 1 1 w ( t ) f ( t ) / ( t − x ) n d t , n ≥ 3 , and show their uniform convergences. Numerical examples are given to demonstrate the performance of the present schemes.

ON THE ANALYTIC-NUMERIC TREATMENT OF WEAKLY SINGULAR INTEGRALS ON ARBITRARY POLYGONAL DOMAINS

An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided.

Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities

Abstract. Let ∑∞ n=0 en[f ]Pn(x) be the Legendre expansion of a function f(x) on (−1, 1). In an earlier work [A. Sidi, Asymptot. Anal., 65 (2009), pp. 175–190], we derived asymptotic expansions as n → ∞ for en[f ], assuming that f ∈ C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x = ±1. In the present work, we extend this study to functions f(x) that are infinitely differentiable on [0, 1], except at finitely many points x1, . . . , xm in (−1, 1) and possibly at one or both of the endpoints x0 = 1 and xm+1 = −1, where they may have arbitrary algebraic singularities, including finite jump discontinuities. Specifically, we assume that, for each r, f(x) has asymptotic expansions of the form

Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R1={(x,y)∣a⩽x⩽b,g(x)⩽y⩽h(x)} and R2={(x,y)∣a⩽y⩽b,g(y)⩽x⩽h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (x,y) space to a square in (ξ,η) space, S: {(ξ,η)∣0⩽ξ⩽1,0⩽η⩽1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.

Direct and Inverse Estimates For Combinations of Bernstein Polynomials with Endpoint Singularities

We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein polynomials by the $r$th Ditzian-Totik modulus of smoothness $\omega_\phi^{r}(f,t)_w$ where $\phi$ is an admissible step-weight function.

Numerical solution of system of nonlinear second-order integro-differential equations

In this article, numerical solution of a system of nonlinear second-order integro-differential equations with boundary conditions of the Fredholm and Volterra types by means of the Sinc-collocation method is considered. The method is effective for approximation in the case of the presence of end-point singularities. Properties of the Sinc-collocation method required for our subsequent development are given and utilized to reduce the computation of boundary value problems to some algebraic equations. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.

Weighted quadrature by change of variable

The standard way of dealing with singular integrals is first to write the integral in the form ∫ab: f(x)w(x)dx with a weight function w(x) > 0 and a relatively smooth function f(x). Polynomials orthogonal on [a, b] with weight function w(x) are then used to derive accurate formulas for approximating the integral. The approach developed in this paper is to use a change of variable to obtain an integral over a finite interval that has a relatively smooth integrand and no weight function. Popular formulas can be applied to this standard problem to obtain easily alternatives to all the common schemes for weighted quadrature. Moreover, the approach provides a way to apply schemes for estimating the error in the unweighted case to integrals involving weight functions. It can be used with popular codes to approximate integrals with some kinds of strong end point singularities. Implementation of the approach is quite convenient in MATLAB.

B d → π − K (*)+ and B s → π +(ρ +)K − decays with QCD factorization and flavor symmetry

The QCD factorization (QCDF) method usually contains infrared divergences which introduce large model dependence to its predictions on charmless B decays. The amplitudes of charmless B decays can be decomposed into “tree” and “penguin” parts which are conventionally defined, not from the topology of the dominant diagrams, but through their associated CKM factors V ub * V uq and V tb * V tq , respectively, with q = d, s. We find that for B d,s → π ∓ K ± decays, the “tree” amplitude can be well estimated in QCDF with small errors, as the endpoint singularities have been canceled to a large extent. With this as the only input from QCDF and combined with flavor symmetry, the branching ratio of B s → π + K − is estimated to be significantly larger than the CDF measurement. This contradiction could be solved if the form factor \( {F^{{B_s}K}} \) is smaller than the light cone sum rules estimation or the “tree” amplitude has been over estimated in QCDF. The latter possibility could happen if charming penguins are nonperturbative and not small, as argued in soft collinear effective theory. To differentiate between these two possibilities, we examine the similar B s → ρ + K − decay with the same technique. It is found that a large part of the uncertainties are canceled in the ratio \( {{\mathcal{B}\left( {{B_s} \to {\rho^{+} }{K^\user1{ - }}} \right)} \mathord{\left/{\vphantom {{\mathcal{B}\left( {{B_s} \to {\rho^{+} }{K^\user1{ - }}} \right)} {\mathcal{B}\left( {{B_s} \to {\pi^{+} }{K^\user1{ - }}} \right)}}} \right.} {\mathcal{B}\left( {{B_s} \to {\pi^{+} }{K^\user1{ - }}} \right)}} \). In QCDF, it is predicted to be 2.5 ± 0.2 which is independent on the form factor. However if charming penguins are important, this ratio could be very different from the QCDF prediction. Therefore the ratio of these two branching ratios could be an interesting indicator of the role of charming penguins in charmless B decays.

Disconnected Glass-Glass Transitions and Diffusion Anomalies in a Model with Two Repulsive Length Scales

Building on mode-coupling-theory calculations, we report a novel scenario for multiple glass transitions in a purely repulsive spherical potential: the square shoulder. The liquid-glass transition lines exhibit both melting by cooling and melting by compression as well as associated diffusion anomalies, similar to the ones observed in water. Differently from all previously investigated models, we find for small shoulder widths a glass-glass line that is disconnected from the liquid phase. Upon increasing the shoulder width such a glass-glass line merges with the liquid-glass transition lines, featuring two distinct end point singularities that give rise to logarithmic decays in the dynamics. We analytically explain these findings by considering the interplay of different repulsive length scales.

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type ( x − y ) ln | x − y | and ( x − y ) 2 ln | x − y | . In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected. Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.

On an interesting variant of the IMT quadrature

We indicate an interesting variant of the IMT quadrature. The IMT quadrature is known to converge exponentially in the presence of an end-point integrable singularity. The variant of IMT quadrature we indicate here retains this property and in addition it has exponential convergence for an integrand which has poles lying just below or above the mid-point of the integration interval.

On Threshold Eigenvalues and Resonances for the Linearized NLS Equation

We prove the instability of threshold resonances and eigenvalues of the linearized NLS operator. We compute the asymptotic approximations of the eigenvalues appearing from the endpoint singularities in terms of the perturbations applied to the original NLS equation. Our method involves such techniques as the Birman-Schwinger principle and the Feshbach map.

Two-pseudoscalar-meson decay ofχcJwith twist-3 corrections

The decays of ${\ensuremath{\chi}}_{cJ}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${K}^{+}{K}^{\ensuremath{-}}$ ($J=0,2$) are discussed within the standard and modified hard-scattering approach when including the contributions from twist-3 distribution amplitudes and wave functions of the light pseudoscalar meson. A model for twist-2 and twist-3 distribution amplitudes and wave functions of the pion and kaon with BHL prescription are proposed as the solution to the end-point singularities. The results show that the contributions from twist-3 parts are actually not power suppressed comparing with the leading-twist contribution. After including the effects from the transverse momentum of light meson valence-quark state and Sudakov factors, the decay widths of the ${\ensuremath{\chi}}_{cJ}$ into pions or kaons are comparable with the their experimental data.

Anatomy of the perturbative QCD approach to the baryonic decaysΛb→pπ,pK

We calculate the $CP$-averaged branching ratios and $CP$-violating asymmetries for the two-body charmless hadronic decays ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}p\ensuremath{\pi}$, $pK$ in the perturbative QCD (pQCD) approach to lowest order in ${\ensuremath{\alpha}}_{s}$. The baryon distribution amplitudes involved in the factorization formulas are considered to the leading twist accuracy, and the distribution amplitudes of the proton are expanded to the next-to-leading conformal spin (i.e., ``P'' waves), the moments of which are determined from QCD sum rules. Our work shows that the contributions from the factorizable diagrams in ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}p\ensuremath{\pi}$, $pK$ decays are much smaller compared to the nonfactorizable diagrams in the conventional pQCD approach. We argue that this reflects the estimates of the ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}p$ transition form factors in the ${k}_{T}$ factorization approach, which are found to be typically an order of magnitude smaller than those estimated in the light-cone sum rules and in the nonrelativistic quark model. As an alternative, we adopt a hybrid pQCD approach, in which we compute the factorizable contributions with the ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}p$ form factors taken from the light-cone QCD sum rules. The nonfactorizable diagrams are evaluated utilizing the conventional pQCD formalism, which is free from the endpoint singularities. The predictions worked out here are confronted with the recently available data from the CDF Collaboration on the branching ratios and the direct $CP$ asymmetries for the decays ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}p\ensuremath{\pi}$ and ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}pK$. The asymmetry parameter $\ensuremath{\alpha}$ relevant for the anisotropic angular distribution of the emitted proton in the polarized ${\ensuremath{\Lambda}}_{b}$ baryon decays is also calculated for the two decay modes.

Generalized Bernstein polynomials with Pollaczek weight

Bernstein polynomials are a useful tool for approximating functions. In this paper, we extend the applicability of this operator to a certain class of locally continuous functions. To do so, we consider the Pollaczek weight w ( x ) ≔ exp ( − 1 x ( 1 − x ) ) , 0 < x < 1 , which is rapidly decaying at the endpoints of the interval considered. In order to establish convergence theorems and error estimates, we need to introduce corresponding moduli of smoothness and K -functionals. Because of the unusual nature of this weight, we have to overcome a number of technical difficulties, but the equivalence of the moduli and K -functionals is a benefit interesting in itself. Similar investigations have been made in [B. Della Vecchia, G. Mastroianni, J. Szabados, Weighted approximation of functions with endpoint or inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004) 19–41] in connection with Jacobi weights.