相关论文: Generalization of Euler's Summation Formula to Pat…
We present and discuss a detailed derivation of a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. We develop an explicit procedure for calculating a set of…
We present a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. Using it we calculate the effective actions $S^{(p)}$ for $p\le 9$ which lead to the same continuum…
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the…
We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications…
We generalize a recently developed method for accelerated Monte Carlo calculation of path integrals to the physically relevant case of generic many-body systems. This is done by developing an analytic procedure for constructing a hierarchy…
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…
The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables $N$ included. It is proved…
Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…
We study a problem of finding good approximations to Euler's constant $\gamma=\lim_{n\to\infty}S_n,$ where $S_n=\sum_{k=1}^n\frac{1}{n}-\log(n+1),$ by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow…
The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission…
A recently developed method, introduced in Phys. Rev. Lett. 94 (2005) 180403, Phys. Rev. B 72 (2005) 064302, Phys. Lett. A 344 (2005) 84, systematically improved the convergence of generic path integrals for transition amplitudes. This was…
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration.…
This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial…
The Euler-Maclaurin formula which relates a discrete sum with an integral, is generalised to the setting of Riemann-Stieltjes sums and integrals on stochastic processes whose paths are a.s. rectifiable, namely, continuous and with bounded…
In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this "worldline Monte Carlo" approach to the standard problem of…
The use of random sampling in decision-making and control has become popular with the ease of access to graphic processing units that can generate and calculate multiple random trajectories for real-time robotic applications. In contrast to…
In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a…
This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of…