Euler-Maclaurin Research Articles

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Thermodynamic properties of the noncommutative quantum Hall effect with anomalous magnetic moment

In this paper, we study the thermodynamic properties of the noncommutative quantum Hall effect (NCQHE) with anomalous magnetic moment (AMM) for both relativistic and nonrelativistic cases at high temperatures, where the thermodynamic properties are: the Helmholtz free energy, the entropy, the mean energy, and the heat capacity. We also work with the Euler-MacLaurin formula to construct the partition function. Next, we plotted the graphs of properties as a function of temperature for different values of the magnetic field and of the NC parameters. We note that the Helmholtz free energy decreases with the temperature, increases with the NC parameters, and can decrease or increase with the magnetic field, while the entropy increases with the temperature, decreases with the NC parameters, and can decrease or increase with the magnetic field. Besides, the mean energy increases linearly with the temperature and the heat capacity satisfies the Dulong-Petit law, which also verified that the AMM does not influence the thermodynamic properties.

Монотонизация модифицированной схемы с эрмитовой интерполяцией для численного решения неоднородного уравнения переноса с поглощением

A hybrid scheme for the numerical solving of a transport equation is implemented. The high order scheme is a modified CIP (Cubic Interpolation Polynomial) scheme with Hermitian interpolation. The third order of approximation of the CIP scheme is achieved by including in the list of unknowns not only the nodal values of the function, but also the nodal values of its derivatives. In the modification under consideration the Euler–Maclaurin formula is used to calculate derivatives. The low order scheme is the characteristic scheme of the first order of approximation. Local, layerwise and global monotonization are considered, the hybridization is performed, respectively, after a cell, a time layer or the whole grid processing. It is shown that the best results are obtained by a scheme with local monotonization. The convergence orders of the hybrid scheme on tests of different smoothness of exact solution do not differ significantly from the convergence orders of the CIP scheme. It is proposed to calculate an integral along a characteristic using the Simpson formula for the Stieltjes integral in the case of the large optical thickness; it provides a significant reduction in the numerical solution errors.

Особенности реализации модифицированной схемы с эрмитовой интерполяцией для численного решения уравнения переноса с переменным коэффициентом поглощения

The grid-characteristic scheme based on Hermitian interpolation is used for solving the transport equation with a varying absorption coefficient. Hermitian interpolation is built on the previous time layer using the nodal values of the function itself and its spatial derivatives. To calculate the nodal values of the space derivative on the new time layer the integral averages and the Euler–Maclaurin formula relating nodal values, integral average and derivative values at nodes are used. Convergence tests for various smoothness functions were carried out. It is shown that at the points of the absorption coefficient discontinuity it is most accurate to use the absorption coefficient in the cell where backward characteristic lies.

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities.

Seaweed algebras and the index statistic for partitions

In 2018 Coll, Mayers, and Mayers conjectured that the q-series (q,−q3;q4)∞−1 is the generating function for a certain parity statistic related to the index of seaweed algebras. We prove this conjecture. Thanks to earlier work by Seo and Yee, the conjecture would follow from the non-negativity of the coefficients of this infinite product. Using a variant of the circle method along with Euler–Maclaurin summation, we establish this non-negativity, thereby confirming the Coll–Mayers–Mayers Conjecture.

Euler–MacLaurin Summation Formula on Polytopes and Expansions in Multivariate Bernoulli Polynomials

We provide a multidimensional weighted Euler–MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if chi _{tau {mathcal {P}}} denotes the characteristic function of a dilated integer convex polytope {mathcal {P}} and q is a function with suitable regularity, we prove that the periodization of qchi _{tau {mathcal {P}}} admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are related to the Lerch Zeta function. In order to prove our results we need to carefully study the asymptotic expansion of widehat{qchi _{tau {mathcal {P}}}}, the Fourier transform of qchi _{tau {mathcal {P}}}.

A reverse extended Hardy–Hilbert’s inequality with parameters

In this paper, by virtue of the symmetry principle, applying the techniques of real analysis and Euler–Maclaurin summation formula, we construct proper weight coefficients and use them to establish a reverse extended Hardy–Hilbert’s inequality with multi-parameters. Then, we obtain the equivalent forms and some equivalent statements of the best possible constant factor related to several parameters. Finally, we illustrate how the obtained results can generate some new reverse Hardy–Hilbert-type inequalities.

Extrapolation quadrature from equispaced samples of functions with jumps

Based on the Euler–Maclaurin formula, the Romberg quadrature method extrapolates trapezoidal values to improve their accuracy when computing the integral of smooth functions from equispaced samples. It has been known at least since an article of Lyness in 1971 that the Euler–Maclaurin formula may be extended to accommodate functions with jumps. In the present work, we develop an extrapolation method, based on this extended formula, for the quadrature of such discontinuous functions. We illustrate the method with numerical examples, using one as well as several sample vectors.

New Integrals Involving a Function Associated With Euler-Maclaurin Summation Formula

New Integrals Involving a Function Associated With Euler-Maclaurin Summation Formula

Evolution of the Euler-Maclaurin sum formula

The correspondence between the discrete and the continuous is a fascinating theme in mathematics. The Euler-Maclaurin sum formula, discovered independently and almost contemporaneously by Leonhard Euler (1707–1783) and Colin Maclaurin (1698–1746), in the early 1730s, relates the sum of the values of a function at the integers in the interval [a, b] with its integral over [a, b]. It thus equates a discrete sum with a continuous sum (integral) of a related function.

Systematic derivation of angular-averaged Ewald potential

In this work we provide a step by step derivation of an angular-averaged Ewald potential suitable for numerical simulations of disordered Coulomb systems. The potential was first introduced by E Yakub and C Ronchi without a clear derivation. Two methods are used to find the coefficients of the series expansion of the potential: based on the Euler–Maclaurin and Poisson summation formulas. The expressions for each coefficient is represented as a finite series containing derivatives of Jacobi theta functions. We also demonstrate the formal equivalence of the Poisson and Euler–Maclaurin summation formulas in the three-dimensional case. The effectiveness of the angular-averaged Ewald potential is shown by the example of calculating the Madelung constant for a number of crystal lattices.

An algebraic construction of sum-integral interpolators

This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-level intersections in Chow rings, that first entered the theory of polytopes from algebraic geometry.

A Reverse Hardy–Hilbert’s Inequality Containing Multiple Parameters and One Partial Sum

In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy–Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we characterize the equivalent conditions of the best possible constant factor associated with several parameters. At the end of the paper, we illustrate that more new inequalities can be generated from the special cases of the reverse Hardy–Hilbert’s inequality.

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.

Approximating sums by integrals only: multiple sums and sums over lattice polytopes

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was presented by the author, which approximates the sum by a linear combination of integrals only, without using derivatives of $f$. It was shown that the Alt formula will in most cases outperform the EM formula. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.

Revisiting the Formula for the Ramanujan Constant of a Series

The main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the Euler–Boole summation formula. Such an expression is not present in the literature. We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods for divergent series. Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler–Maclaurin summation formula. Finally, we present several examples of the Ramanujan constant of divergent series.

A Reverse Hardy-Hilbert’s Inequality Involving One Partial Sum as the Terms of Double Series

In this paper, by constructing proper weight coefficients and utilizing the Euler-Maclaurin summation formula and the Abel partial summation formula, we establish reverse Hardy-Hilbert’s inequality involving one partial sum as the terms of double series. On the basis of the obtained inequality, the equivalent conditions of the best possible constant factor associated with several parameters are discussed. Finally, we illustrate that more reverse inequalities of Hardy-Hilbert type can be generated from the special cases of the present results.

More accurate and strengthened forms of half-discrete Hilbert inequality

In this work, by introducing a new half-discrete kernel function with parameters, and using the method of weight coefficient, Euler-Maclaurin summation formula and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality as well as its strengthened form are established. Additionally, some equivalent inequalities in Hardy's form are also considered. Furthermore, it is proved that the constant factors in the newly obtained inequalities are the best possible. At last, by taking some special values of parameters and using the numerical calculation function of Matlab, some special cases are presented at the end of the paper.