Euler-Maclaurin Research Articles

A fast summation method for oscillatory lattice sums

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.

Thermodynamic quantities for the Klein–Gordon equation with a linear plus inverse-linear potential: Biconfluent Heun functions

We study some thermodynamics quantities for the Klein-Gordon equation with a linear plus inverse-linear, scalar potential. We obtain the energy eigenvalues with the help of the quantization rule coming from the biconfluent Heun's equation. We use a method based on the Euler-MacLaurin formula to compute the thermal functions analytically by considering only the contribution of positive part of spectrum to the partition function.

A symbolic approach to multiple zeta values at negative integers

A symbolic computation technique is used to derive closed-form expressions for an analytic continuation of the Euler–Zagier zeta function evaluated at negative integers. This continuation was recently proposed by Sadaoui. The approach presented here yields explicit contiguity identities, recurrences on the depth of the zeta values and their generating functions. Moreover, it allows to prove that the resulting multiple zeta values computed at negative integers coincide with those obtained by another analytic continuation technique that uses the Euler–MacLaurin summation formula.

Approximation Properties of q-Bernoulli Polynomials

We study the q-analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

Bicompact schemes for solving a steady-state transport equation by the quasi-diffusion method

Accurate difference schemes (of up to the fourth order of accuracy) are constructed for the numerical solutions of a neutron-transport equation and a system of quasi-diffusion equations (a low order consequences of a transport equation) used to accelerate iterations on scattering. The considered difference schemes are based on the common principles of the compact (in the context of a single cell) approximation. This enables one to make accurate resolution for contact discontinuities in the medium. The fourth order of approximation on a minimum two-point stencil is attained by widening the list of unknowns and including on it (apart from the nodal values of an unknown function) additional unknowns. As these unknowns, an integral cell-average value or a value at a half-integer node can be taken. To connect these values, Simpson quadrature formulas are used. Equations for determining additional unknowns are constructed by using the Euler–Maclaurin formulas. Computations for several onedimensional test problems have been performed; here, the high actual accuracy of the constructed difference schemes is demonstrated. The schemes are naturally generalized to the two- and three-dimensional cases. Due to their high accuracy, monotonicity, efficiency, and compactness, the proposed schemes are very attractive for engineering computations (computations of nuclear reactors, etc.).

Schrödinger equation with a non-central potential: Some statistical quantities

In this paper, we study the dependence of some statistical quantities such as the free energy, the mean energy, the entropy, and the specific heat for the Schrodinger equation on the temperature, particularly in the case of a non-central potential. The basic point is to find the partition function which is obtained by a method based on the Euler-Maclaurin formula. At first, we present the analytical results supporting them with some plots for the thermal functions for one- and three-dimensional cases to find out the effect of the angular momentum. We also study then the effect of the angle-dependent part of the non-central potential. We discuss the results briefly for a phase transition for the system. We also present our results for a three-dimesional harmonic oscillator.

Special values of partial zeta functions of real quadratic fields at nonpositive integers and the Euler-Maclaurin formula

We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field applying an asymptotic version of Euler-Maclaurin formula to the lattice cone associated to the ideal considered. The Euler-Maclaurin formula in- volved is obtained by applying the Todd series of differential opera- tors to an integral of a small perturbation of the cone. The additive property of Todd series w.r.t. the cone decomposition enables us to express the partial zeta values in terms of the continued fraction of the reduced element of the ideal. The expression obtained uses the positive continued fraction which yields a virtual decomposition of the cone. We apply the expression to some indexed families of real quadratic fields satisfying certain condition on the shape of the continued frac- tions. The families considered include those appeared in (20) and (21) as well as the Richaud-Degert types. We show that the partial zeta values at a given nonpositive integerk in the family indexed by n is a polynomial of n. Finally, we compute explicitly the polynomials producing the par- tial zeta values at s ´k for small k of some chosen families and compare these with some previously known results.

Sinc integrals and tiny numbers

We apply a result of David and Jon Borwein to evaluate a sequence of highly-oscillatory integrals whose integrands are the products of a rapidly growing number of sinc functions. The value of each integral is given in the form $\pi(1-t)/2$, where the numbers $t$ quickly become very tiny. Using the Euler-Maclaurin summation formula, we calculate these numbers to high precision. For example, the integrand of the tenth integral in the sequence is the product of 68100152 sinc functions. The corresponding $t$ is approximately $9.6492736004286844634795531209398105309232 \cdot 10^{-554381308}$.

Euler–MacLaurin formulas via differential operators

Recently there has been a renewed interest in asymptotic Euler–MacLaurin formulas, because of their applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions on intervals, polygons, and three-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of the function in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension.

On Euler–Maclaurin formula

We use a simple approach to show that an Euler–Maclaurin like formula can be associated to any interpolatory quadrature rule. This result is obtained by successively adding correcting terms that are exact for polynomials of increasing degree. A decomposition of the coefficients of the Euler–Maclaurin formula in terms of the integral to compute and representations of powers of the nodes is pointed out. Optimal truncation error bounds are also obtained.

A simple way of approximating the canonical partition functions in statistical mechanics

We propose a simple pedagogical way of introducing the Euler–MacLaurin summation formula in an undergraduate course on statistical mechanics. The reason is that the students may feel more comfortable and confident if they are able to deduce the main equations. To this end we put forward two alternative routes: the first one is the simplest and yields the first two terms of the expansion. The second one is somewhat more elaborate and takes into account all the correction terms. We apply both to the calculation of the simplest one-particle canonical partition functions for the translational, vibrational and rotational degrees of freedom. The more elaborate, systematic calculation of the correction terms is suitable for motivating the students to explore the possibility of using available computer algebra software that enable one to avoid long and tedious manipulation of algebraic equations.

The Higher Derivation of the Hurwitz Zeta-function

In this paper, the Euler-Maclaurin Summation formula was researched, the purpose of research is to promote the application of the Hurwitz Zeta-function ; Combination method of number theory special a function and Euler-Maclaurin Summation Formula was been used ; By three derivatives of the the Euler-Maclaurin Summation formula , three formulas of Hurwitz Zeta-function were been given.

Multi-variate Hardy-type lattice point summation and Shannon-type sampling

The famous Shannon sampling theorem gives an answer to the question of how a one-dimensional time-dependent bandlimited signal can be reconstructed from discrete values in lattice points. In this work, we are concerned with multi-variate Hardy-type lattice point identities from which space-dependent Shannon-type sampling theorems can be obtained by straightforward integration over certain regular regions. An answer is given to the problem of how a signal bandlimited to a regular region in q-dimensional Euclidean space allows a reconstruction from discrete values in the lattice points of a (general) q-dimensional lattice. Weighted Hardy-type lattice point formulas are derived to allow explicit characterizations of over- and undersampling, thereby specifying not only the occurrence, but also the type of aliasing in a thorough mathematical description. An essential tool for the proof of Hardy-type identities in lattice point theory is the extension of the Euler summation formula to second order Helmholtz-type operators involving associated Green functions with respect to the “boundary condition” of periodicity. In order to circumvent convergence difficulties and/or slow convergence in multi-variate Hardy-type lattice point summation, some summability methods are necessary, namely lattice ball and Gaus–Weierstras averaging. As a consequence, multi-variate Shannon-type lattice sampling becomes available in a proposed summability context to accelerate the summation of the associated cardinal-type series. Finally, some aspects of constructive approximation in a resulting Paley–Wiener framework are indicated, such as the recovery of a finite set of lost samples, the reproducing Hilbert space context of spline interpolation.

An efficient Gaussian kernel optimization based on centered kernel polarization criterion

The success of kernel-based learning methods is heavily dependent on the choice of a kernel function and proper setting of its parameters. In this paper, we optimize the Gaussian kernel for binary-class problems by using centered kernel polarization criterion. This criterion is an extension of kernel polarization and a simplified style of centered kernel alignment. Compared with formulated kernel polarization criterion, the proposed criterion has a defined geometrical significance, and it can locate the global optimal point with less influence of threshold selection. Furthermore, the approximate criterion function can be proved to have a determined global minimum point by adopting the Euler–Maclaurin formula under weaker conditions. In addition, taking the preservation of within-class local structure into account, we present an evaluation criterion named local multiclass centered kernel polarization in multiclass classification scenario. Comparative experiments are conducted on some benchmark examples with three Gaussian kernel based learning methods and the results well demonstrate the effectiveness and efficiency of the proposed quality measures.

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be the same, by utilizing the character analogue of the Euler–MacLaurin summation formula. Moreover, we extend known results on the integral of products of Bernoulli polynomials by considering the integral∫0xBn1(b1z+y1)⋯Bnr(brz+yr)dz, where bl(bl≠0) and yl(1≤l≤r) are real numbers. As a consequence of this integral we establish a connection between the reciprocity relations of sums of products of Bernoulli polynomials and of the Dedekind sums.

Structure factor of a Gaussian chain confined between two parallel plates.

We study the structure factor of a single Gaussian chain confined between two macroscopic parallel plates theoretically. The chain propagator is constructed in terms of the eigen-spectrum of the Laplace operator under the Dirichlet boundary condition enforced at the two plates, by which the confinement effect enters the treatment through size-dependent eigen-spectrum. In terms of the series expansion solution for the chain propagator, we first calculate the confinement free energy and the confinement force for an arbitrary confinement strength. It is found that the confinement force scales to the distance between the two confining surfaces with a power of -3 for strong confinements and of -2 for weak confinements. Based on the ground state dominance approximation for strong confinements and the Euler-Maclaurin formula for weak confinements, we develop approximation theories for the two limit situations, which agree with the numerical results well. We further calculate the structure factor of the confined Gaussian chain in this slit geometry. While the scattering function of the transverse chain fluctuations perpendicular to the confinement direction is still a Debye function form, the structure factor for the longitudinal fluctuations along the confinement dimension starts with the monotonic Debye function behavior for weak confinements and develops a decaying oscillation behavior with the increase of confinements. The numerical results for the structure factor are also interpreted by developing approximation theories in different confinement regimes. Finally, the orientational average of the anisotropic structure factor is performed and an analytic expression for the averaged structure factor is derived under the ground state dominance approximation for strong confinements.

Asymptotics for the Norm of Bethe Eigenstates in the Periodic Totally Asymmetric Exclusion Process

The normalization of Bethe eigenstates for the totally asymmetric simple exclusion process on a ring of \(L\) sites is studied, in the large \(L\) limit with finite density of particles, for all the eigenstates responsible for the relaxation to the stationary state on the KPZ time scale \(T\sim L^{3/2}\). In this regime, the normalization is found to be essentially equal to the exponential of the action of a scalar free field. The large \(L\) asymptotics is obtained using the Euler–Maclaurin formula for summations on segments, rectangles and triangles, with various singularities at the borders of the summation range.

Euler-Maclaurin, harmonic sums and Stirling's formula

Euler-Maclaurin, harmonic sums and Stirling's formula

Euler–Maclaurin and Gregory interpolants

Let a sufficiently smooth function $$f$$f on $$[-1,1]$$[-1,1] be sampled at $$n+1$$n+1 equispaced points, and let $$k\ge 0$$k?0 be given. An Euler---Maclaurin interpolant to the data is defined, consisting of a sum of a degree $$k$$k algebraic polynomial and a degree $$n$$n trigonometric polynomial, which deviates from $$f$$f by $$O(n^{-k})$$O(n-k) and whose integral is equal to the order $$k$$k Euler---Maclaurin approximation of the integral of $$f$$f. This interpolant makes use of the same derivatives $$f^{( j)}(\pm 1)$$f(j)(±1) as the Euler---Maclaurin formula. A variant Gregory interpolant is also defined, based on finite difference approximations to the derivatives, whose integral (for $$k$$k odd) is equal to the order $$k$$k Gregory approximation to the integral.

The Euler-Maclaurin Formula and Differential Operators of Infinite Order

The Euler-Maclaurin Formula and Differential Operators of Infinite Order