Euler-Maclaurin Research Articles

Influence of various components of errors on the results of approximation using orthogonal functions

The possibility of Legendre polynomials application for approximating functions in order to avoid problems with poorly conditioned matrices for using the least squares method is considered. It is shown that the solutions based on the methods of numerical integration have a large error, which does not allow the idea of orthogonalization to be applied to functions given in a table. However, the use of Legendre polynomials as basic functions instead of algebraic ones in the least squares method can significantly improve the conditionality of matrices. In the problem of approximation of periodic functions by finite Fourier sum the numerical integration is also required. But in connection of the Euler-Maclaurin formula, the numerical integration error of periodic functions is essentially less than of polynomial integration, so the problem is solved very accurately and fast.

Analysis of Series and Products. Part 1: The Euler–Maclaurin Formula

–We explore the applications of the Euler–Maclaurin formula in analyzing functions expressed as infinite series and products. Three illustrative examples show the difficulties that may be encountered and the means by which these can be overcome.

Обобщённая формула бинома Ньютона и формулы суммирования

В основе работы лежит формула бинома Ньютона и её обобщения на последовательности многочленов биномиального типа. Даны применения к обобщённой проблеме Варинга (Хуа Ло-кен) и проблеме Гильберта – Камке (Г.И.Архипов). Доказана формула Тейлора – Маклорена для многочленов и гладких функций и даны её приложения в численном анализе (решение уравнений методом касательных Ньютона, лемма Гензеля в полныхнеархимедовских полях, приближенное вычисление значений гладких функций в точке). Даётся аналог формулы бинома Ньютона для многочленов Бернулли и доказывается формула Эйлера — Маклорена суммирования значений функции по целым точкам, выведена формула Пуассона суммирования значений функции. Рассмотрены примеры последовательностей многочленов биномиального типа (степени, нижние и верхние факториальныестепени, многочлены Абеля и Лагерра). Найдены биномиальные свойства многочленов Аппеля и Эйлера. Для многочленов и гладких функций от нескольких переменных доказана формула Тейлора, получены многомерные аналоги формул Эйлера – Маклорена и Пуассона суммирования значений функции по решётке. Рассмотрен многомерный аналог этих формул для решётки в многомерном комплексном пространстве. Доказаны ряд свойствпоследовательности многочленов биномиального типа от нескольких переменных.

Statistical properties of the two-dimensional Dirac oscillator with spin–orbit coupling

We investigate the statistical properties of a two-dimensional Dirac oscillator (2D DO) in the presence of a spin–orbit coupling, breaking its supersymmetry. The Hamiltonian of the problem is shown to admit an exact Foldy–Wouthuysen transformation, which allowed us to discard negative-energy states in carrying out our study. The main thermodynamical functions: the Helmholtz free energy, the mean energy, the entropy and the specific heat, are obtained from the canonical partition function, and their behaviours are analysed. The high temperature limit of different thermodynamical functions is probed analytically by utilizing the Euler–Maclaurin formula.

Hermitian Characteristic Scheme for a Linear Inhomogeneous Transfer Equation

In this paper, we design an interpolation-characteristic scheme for the numerical solution of the inhomogeneous transfer equation. The scheme is based on the Hermitian interpolation to reconstruct the value of an unknown function at the point of intersection of the backward characteristic with the cell faces. The Hermitian interpolation to reconstruct the function values ​​uses both the nodal values ​​of the desired function and its derivative. Unlike previous studies also based on the Hermitian interpolation, not only the differential continuation of the transfer equation but also the relationship between the integral averaged values, nodal values, ​​and derivatives according to the Euler-Maclaurin formula is used to transfer information about the derivatives to the next layer. The third-order difference scheme is shown to converge for smooth solutions. The dissipative and dispersive properties of the scheme are considered using numerical examples of solutions with decreasing smoothness.

Application of the Euler-Maclaurin Sum Formula to Obtain a Closed form Analytical Solution for Acoustical and Heat Diffusion in a Partially Bounded Region

Green’s function in the Laplace transform domain for the sound wave field in an infinite area between two parallel reflecting planes may be represented as an infinite series of images caused by the planes. Based on a comparison of the wave equation and the diffusion equation, the Green’s function representation as an infinite series for diffusion in the same region is directly obtained. A closed form expression in space time for the diffusion problem is obtained by applying the Euler-Maclaurin sum formula to a modified diffusion form of the series in the Laplace transform domain and then inverting to the time domain. Using several sets of numerical values for system parameters applicable to acoustic diffusion in the region, numerical comparisons of the infinite series vs Euler-Maclaurin closed form representation of the Green function is presented. Comparison of the infinite series vs Euler-Maclaurin transient response to an exponential and constant input are presented for cases of acoustical noise diffusion and heat diffusion respectively. Transfer function comparisons are given for the diffusion models along with the use of the closed form representation in a model-based control scheme.

A two-variable Dirichlet series and its applications

We define a two-variable Dirichlet series associated with two arithmetic functions, which is related to the Riemann zeta function, the Dirichlet L-function, the Dirichlet series associated to the harmonic numbers, and truncated multiple zeta functions. Using the periodic Euler-Maclaurin summation formula, we obtain a representation in terms of an ordinary Dirichlet series, which leads to the explicit evaluation of its values at nonpositive integers. We also find a reciprocity formula, which provides some symmetric formulas involving Bernoulli and associated numbers.

Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions Θ(m;k) associated to a piecewise quasi-polynomial function m. The family is indexed by an integer k∈Z>0, and admits an asymptotic expansion as k→∞, which generalizes the expansion obtained in the Euler–Maclaurin formula. When m is the multiplicity function arising from the quantization of a symplectic manifold, the leading term of the asymptotic expansion is the Duistermaat–Heckman measure. Our main result is that m is uniquely determined by a collection of such asymptotic expansions. We also show that the construction is compatible with pushforwards. As an application, we describe a simpler proof that formal quantization is functorial with respect to restrictions to a subgroup.

Euler–Maclaurin expansions without analytic derivatives

The Euler–Maclaurin (EM) formulae relate sums and integrals. Discovered nearly 300 years ago, they have lost none of their importance over the years, and are nowadays routinely taught in scientific computing and numerical analysis courses. The two common versions can be viewed as providing error expansions for the trapezoidal rule and for the midpoint rule, respectively. More importantly, they provide a means for evaluating many infinite sums to high levels of accuracy. However, in all but the simplest cases, calculating very high-order derivatives analytically becomes prohibitively complicated. When approximating such derivatives with finite differences (FD), the choice of step size typically requires a severe trade-off between errors due to truncation and to rounding. We show here that, in the special case of EM expansions, FD approximations can provide excellent accuracy without the step size having to go to zero. While FD approximations of low-order derivatives to high orders of accuracy have many applications for solving ODEs and PDEs, the present context is unusual in that it relies on FD approximations to derivatives of very high orders. The application to infinite sums ensures that one can use centred FD formulae (which are not subject to the Runge phenomenon).

Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain

We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σz-basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.

Contour integrals of analytic functions given on a grid in the complex plane

Abstract Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a $5\times 5$ ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from $2$nd to $26$th order.

Asymptotic formulas for Vasyunin cotangent sums

Abstract We study the Vasyunin-type cotangent sum c 0 ( h / k ) = − ∑ m = 1 k − 1 ( m / k ) cot ( π h m / k ) , where h and k are positive coprime integers. This sum is related to Estermann zeta function. By applying the Euler-Maclaurin summation formula to a suitable function, we improve a previous large-k asymptotic approximation of c0(h/k). We also provide a procedure to compute an arbitrary number of terms of the approximation.

Chemical Langevin equation: A path-integral view of Gillespie's derivation.

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path-integral description of the CME and show how applying Gillespie's two conditions to it directly leads to a path-integral equivalent to the CLE. We compare this approach to the path-integral equivalent of a large system size derivation and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems and are readily generalizable.

Exact and approximate energy sums in potential wells

Sums of the N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regime N ≫ 1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levels N + 1, N + 2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10−7. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on how N and the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits.

Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald

Let V be a real vector space of dimension n and let $$M\subset V$$ be a lattice. Let $$P\subset V$$ be an n-dimensional polytope with vertices in M, and let $$\varphi :V\rightarrow {\mathbb {C}}$$ be a homogeneous polynomial function of degree d. For $$q\in {\mathbb {Z}}_{>0}$$ and any face F of P, let $$D_{\varphi ,F} (q)$$ be the sum of $$\varphi $$ over the lattice points in the dilate qF. We define a generating function $$G_{\varphi }(q,y) \in {\mathbb {Q}}[q] [y]$$ packaging together the various $$D_{\varphi ,F} (q)$$ , and show that it satisfies a functional equation that simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how $$G_{\varphi }$$ can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.

Fermionic Model with a Non-Hermitian Hamiltonian

This paper deals with the mathematical spectral analysis and physical interpretation of a fermionic system described by a non-Hermitian Hamiltonian possessing real eigenvalues. A statistical thermodynamical description of such a system is considered. Approximate expressions for the energy expectation value and the number operator expectation value, in terms of the absolute temperature $T$ and of the chemical potential $\mu$, are obtained, based on the Euler-Maclaurin formula.

On a New Extended Hardy–Hilbert’s Inequality with Parameters

In this paper, by introducing parameters and weight functions, with the help of the Euler–Maclaurin summation formula, we establish the extension of Hardy–Hilbert’s inequality and its equivalent forms. The equivalent statements of the best possible constant factor related to several parameters are provided. The operator expressions and some particular cases are also discussed.

On an extended Hardy-Littlewood-Polya’s inequality

By utilization of the weight coefficients, the idea of introducing parameters and Euler-Maclaurin summation formula, an extended Hardy-Littlewood-Polya's inequality and its equivalent form are established. The equivalent statements of the best possible constant factor involving several parameters, and some particular cases are provided. The operator expressions of the obtained results are also considered.

Thermodynamic properties of neutral Dirac particles in the presence of an electromagnetic field

In this paper, we investigate the thermodynamic properties of a set of neutral Dirac particles in the presence of an electromagnetic field in contact with a heat bath for the relativistic and nonrelativistic cases. In order to perform the calculations, the high temperatures regime is considered and the Euler–MacLaurin formula is taking into account. Next, we explicitly determine the behavior of the main thermodynamic functions of the canonical ensemble: the Helmholtz free energy, the mean energy, the entropy, and the heat capacity. As a result, we verify that the mean energy and the heat capacity for the relativistic case are twice the values of the nonrelativistic case, thus, satisfying the so-called Dulong–Petit law. In addition, we also verify that the Helmholtz free energy increases and the entropy decreases as functions of the electric field. Finally, we note that there exists no influence on the thermodynamic functions due to the magnetic field.

Thermal properties of three dimensional Morse potential for some diatomic molecules via Euler-Maclaurin approximation

The purpose of this study is to develop a method of calculating the vibration partition function of diatomic molecules for the Morse potential energy. After a brief introduction about the eigensolutions obtained for the problem in question, Via the Euler-Maclaurin formula, we have determined the thermal properties for four diatomics such as \text{H}_{2}, HCl, LiH, and CO. Different situation has been exposed and explained by the appropriate curves of the thermal properties for these diatomics molecules in consideration. In addition, we have shown that our method exposed to calculate these thermal properties can be used to determine these thermodynamic quantities.