相关论文: Generalization of Euler's Summation Formula to Pat…
We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p /(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q…
We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…
We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case…
We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…
In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp.
In this paper, we study the polynomial stability of analytical solution and convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations. Firstly, the sufficient conditions for solutions to grow…
Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on…
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…
Large optimal transport problems can be approached via domain decomposition, i.e. by iteratively solving small partial problems independently and in parallel. Convergence to the global minimizers under suitable assumptions has been shown in…
It is well known that the Euler method for a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older sample paths is estimated to be of strong…
Two formulations of quantum mechanics, inequivalent in the presence of closed timelike curves, are studied in the context of a soluable system. It illustrates how quantum field nonlinearities lead to a breakdown of unitarity, causality, and…
We generalize the property that Riemann sums of a continuous function corresponding to equidistant subdivision of an interval converge to the integral of that function, and we give some applications of this generalization.
The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on $\mathbb R$. We formulate here the analogue for functions that are just of…
We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition…
In a previous paper it was shown that the Forward Euler method applied to differential inclusions where the right-hand side is a Lipschitz continuous set-valued function with uniformly bounded, compact values, converges with rate one. The…
We present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1D models. The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented to the maximal order…
We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along…
Consider a fluid flowing through a junction between two pipes with different sections. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather…