Order Error Terms Research Articles

Higher moments of the pair correlation function for Sato-Tate sequences

In [BS19], Balasubramanyam and the second named author derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of [0,1] upon averaging over large families of Hecke newforms of weight k with respect to Γ0(N). The goal of this article is to study higher moments of this pair correlation function. For an integer r≥2, we present bounds for its r-th power moments. We apply these bounds to record lower order error terms in the computation of the second and third moments. As a result, one can obtain the convergence of the second and third moments of this pair correlation function for suitably small intervals, and under appropriate growth conditions for the size of the families of Hecke newforms.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^3$$\\end{document} subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2\\ln (L)$$\\end{document}, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L\\Lambda \\subset {{\\mathbb {R}}}^3$$\\end{document} as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln (L)$$\\end{document} factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}, is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the relevant trace for piecewise C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{C}^{1,\\alpha }$$\\end{document} smooth surfaces ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}.

High-order Coverage of Smoothed Bayesian Bootstrap Intervals for Population Quantiles

We characterize the high-order coverage accuracy of smoothed and unsmoothed Bayesian bootstrap confidence intervals for population quantiles. Although the original (Rubin 1981) unsmoothed intervals have the same O(n−1/2) coverage error as the standard empirical bootstrap, the smoothed Bayesian bootstrap of Banks (1988) has much smaller O(n−3/2[log(n)]3) coverage error and is exact in special cases, without requiring any smoothing parameter. It automatically removes an error term of order 1/n that other approaches need to explicitly correct for. This motivates further study of the smoothed Bayesian bootstrap in more complex settings and models.

Van der Waals–London interaction of atoms with pseudorelativistic kinetic energy

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schr\"odinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

On the Atkinson formula for the ζ function

Thanks to Littlewood (1922) and Ingham (1928), we know the first two terms of the asymptotic formula for the square mean integral value of the Riemann zeta function ζ on the critical line. Later, Atkinson (1939) presented this formula with an error term of order O(Tlog2⁡(T)), which we call the Atkinson formula. Following the latter approach and the work of Titchmarsh (1986), we present an explicit version of the Atkinson formula, improving on a recent bound by Simonič (2020). Moreover, we extend the Atkinson formula to the range ℜ(s)∈[14,34], giving an explicit bound for the square mean integral value of ζ and improving on a bound by Helfgott and the authors (2019). We use mostly classical tools, such as the approximate functional equation and the explicit convexity bounds of the zeta function given by Backlund (1918).

Strong edge coloring of circle graphs

A strong clique in a graph G is a set S of edges of G such that every two edges in S are either adjacent or joined by an edge of G. We say that G is a circle graph if vertices of G correspond to chords of a circle and G contains an edge uv if and only if chords corresponding to u and v intersect.We show that the maximum size of a strong clique in a circle graph with maximum degree Δ is at most 54Δ2. This bound is tight for even Δ and correct up to a smaller order error term for odd Δ.This result supports a conjecture of Erdős and Nešetřil from 1985, which states that the strong chromatic index of a graph with maximum degree Δ is at most 54Δ2. It also confirms a conjecture of Faudree, Gyárfás, Schelp and Tuza from 1990 for circle graphs.

Cluster Expansion for the Ising Model in the Canonical Ensemble

We show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Using the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, central limit theorem and large deviations.

The Puiseux Expansion and Numerical Integration to Nonlinear Weakly Singular Volterra Integral Equations of the Second Kind

This paper develops an efficient algorithm to solve nonlinear Volterra integral equation of the second kind with weakly singular convolution kernel. First, we show that the general Puiseux series for the solution about zero exists under smooth assumptions for the nonlinear function, and then design an algorithm to recover the finite-term truncation of the asymptotic expansion by Picard iteration. This asymptotic expansion can easily yield a more accurate Pade approximation. Second, we use trapezoidal rule to discretize the singular integral and derive the Euler–Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. By accumulating some lower order error terms to the quadrature formula, we obtain high precision evaluations to the nonlinear Volterra integral equation. Third, we prove that the scheme is convergent by extending the Gronwall inequality to be held for the scheme. Fourth, an example is provided to illustrate that the combination of the Puiseux expansion and the numerical integration can effectively increase the computational accuracy of the equation. Finally, we apply the method to solve the Lighthill integral equation, and obtain the asymptotic expansions of the solution near zero and infinity, respectively. The computation shows that the trapezoidal rule is only necessary in a small finite range of the semi-infinite interval.

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Let H_{n} = sum_{r=1}^{n} 1/r and H_{n}(x) = sum_{r=1}^{n} 1/(r+x). Let psi(x) denote the digamma function. It is shown that H_{n}(x) + psi(x+1) is approximated by frac{1}{2}log f(n+x), where f(x) = x^{2} + x + frac{1}{3}, with error term of order (n+x)^{-5}. The cases x = 0 and n = 0 equate to estimates for H_{n} - gamma and psi(x+1) itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates

We present several different codes and protocols to distillT, controlled-S, and Toffoli (orCCZ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversalT. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiencyγ→1. We also present a Reed-Muller based construction of these codes which obtains a worseγbut performs well at small sizes. Additionally, we present protocols based on checking the stabilizers ofCCZmagic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code forT-to-Toffoli distillation, punctured Reed-Muller codes forT-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a512T-gate to10Toffoli gate code with distance8as well as triorthogonal codes with parameters[[887,137,5]],[[912,112,6]],[[937,87,7]]with very low prefactors in front of the leading order error terms in those codes.

On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in Lp (p>2) by a Wiener process with an error term of order o(n1∕p). Very recently, Berkes et al. (2014) extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in Lp is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in L∞ or in L1), under which the strong approximation result holds with rate o(n1∕p). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GLd(R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L1-coupling condition is in some sense optimal.

Normal moveout coefficients for horizontally layered triclinic media

Sedimentary layers affected by vertical compaction and strong lateral tectonic stresses are often characterized by low anisotropic symmetry (e.g., tilted orthorhombic [TOR]/monoclinic or even triclinic). Considering all types of pure-mode and converted waves, we derive the normal moveout (NMO) series coefficients of near normal-incidence reflected waves in arbitrarily anisotropic horizontally layered media, for a leading error term of order six. The NMO series can be either a function of the invariant horizontal slowness (slowness domain) or the surface offset (offset domain). The NMO series coefficients, referred to also as effective parameters, are associated with the corresponding azimuthally varying NMO velocity functions. We distinguish between local (single-layer) and global (overburden multilayer) effective parameters, which are related by forward and inverse Dix-type transforms. We derive the local effective parameters for an arbitrary anisotropic (triclinic) layer, which is the main contribution of this paper. With some additional geologic constraints, the local effective parameters can then be converted into the interval elastic properties. To demonstrate the applicability of our method, we consider a synthetic layered model in which each layer is characterized with TOR symmetry. The corresponding global effective model loses the symmetries of the individual layers and is characterized by triclinic symmetry.

Pseudopotential lattice Boltzmann equation method for two-phase flow: A higher-order Chapmann-Enskog expansion.

In this paper, a higher order Chapmann-Enskog expansion technique is applied to pseudopotential lattice Boltzmann equation (LBE), and the contribution of third order error terms to pressure tensor is analyzed in detail. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to validate the LBE, and the results show that the predictions of the present LBE agree well with the analytical solution and other predictions.

On the optimality of averaging in distributed statistical learning

A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two large-sample settings: First, a classical setting where the number of parameters $p$ is fixed, and the number of samples per machine $n\to\infty$. Second, a high-dimensional regime where both $p,n\to\infty$ with $p/n \to \kappa \in (0,1)$. For both regimes and under suitable assumptions, we present asymptotically exact expressions for this estimation error. In the fixed-$p$ setting, under suitable assumptions, we prove that to leading order averaging is as accurate as the centralized solution. We also derive the second order error terms, and show that these can be non-negligible, notably for non-linear models. The high-dimensional setting, in contrast, exhibits a qualitatively different behavior: data splitting incurs a first-order accuracy loss, which to leading order increases linearly with the number of machines. The dependence of our error approximations on the number of machines traces an interesting accuracy-complexity tradeoff, allowing the practitioner an informed choice on the number of machines to deploy. Finally, we confirm our theoretical analysis with several simulations.

On the strong coupling limit of many-polaron systems in electromagnetic fields

In this paper, estimates on the ground state energy of Fröhlich N-polarons in electromagnetic fields in the strong coupling limit, α → ∞, are derived. It is shown that the ground state energy is given by α2multiplied by the minimal energy of the corresponding Pekar–Tomasevich functional for N particles, up to an error term of order α42/23 N3. The potentials A, V are suitably rescaled in α. As a corollary, binding of N-polarons for strong magnetic fields for large coupling constants is established.

Dynamics of periodic Toda chains with a large number of particles

For periodic Toda chains with a large number N of particles we consider states which are N−2-close to the equilibrium and constructed by discretizing any given C2-functions with mesh size N−1. For such states we derive asymptotic expansions of the Toda frequencies (ωnN)0<n<N and the actions (InN)0<n<N, both listed in the standard way, in powers of N−1 as N→∞. At the two edges n∼1 and N−n∼1, the expansions of the frequencies are computed up to order N−3 with an error term of higher order. Specifically, the coefficients of the expansions of ωnN and ωN−nN at order N−3 are given by a constant multiple of the nth KdV frequencies ωn− and ωn+ of two periodic potentials, q− respectively q+, constructed in terms of the states considered. The frequencies ωnN for n away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions (InN)0<n<N, asymptotics of a similar nature are derived.

On asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.

Metastable States When the Fermi Golden Rule Constant Vanishes

Resonances appearing by perturbation of embedded non-degenerate eigenvalues are studied in the case when the Fermi Golden Rule constant vanishes. Under appropriate smoothness properties for the resolvent of the unperturbed Hamiltonian, it is proved that the first order Rayleigh-Schr\"odinger expansion exists. The corresponding metastable states are constructed using this truncated expansion. We show that their exponential decay law has both the decay rate and the error term of order $\varepsilon^4$, where $\varepsilon$ is the perturbation strength.