Erratum to “Bayesian methods for parameter estimation in effective field theories” [Ann. Phys. 324 (2009) 682–708]

We have discovered an error in the numerical calculations for the extraction of parameters from the nucleon mass “data” presented in Sections 4 and 5 of our paper [M.R. Schindler, D.R. Phillips, Ann. Phys. 324 (2009) 682. Available from: <hep-ph/0808.3643>]. We present the corrected results and discuss the implications. We stress that the material presented in Sections 1–3 of our paper [M.R. Schindler, D.R. Phillips, Ann. Phys. 324 (2009) 682. Available from: <hep-ph/0808.3643>] is unaffected.     

Bayesian methods for parameter estimation in effective field theories

We demonstrate and explicate Bayesian methods for fitting the parameters that encode the impact of short-distance physics on observables in effective field theories (EFTs). We use Bayes’ theorem together with the principle of maximum entropy to account for the prior information that these parameters should be natural, i.e., O(1) in appropriate units. Marginalization can then be employed to integrate the resulting probability density function (pdf) over the EFT parameters that are not of specific interest in the fit. We also explore marginalization over the order of the EFT calculation, M, and over the variable, R, that encodes the inherent ambiguity in the notion that these parameters are O(1). This results in a very general formula for the pdf of the EFT parameters of interest given a data set, D. We use this formula and the simpler “augmented χ²” in a toy problem for which we generate pseudo-data. These Bayesian methods, when used in combination with the “naturalness prior”, facilitate reliable extractions of EFT parameters in cases where χ² methods are ambiguous at best. We also examine the problem of extracting the nucleon mass in the chiral limit, M₀, and the nucleon sigma term, from pseudo-data on the nucleon mass as a function of the pion mass. We find that Bayesian techniques can provide reliable information on M₀, even if some of the data points used for the extraction lie outside the region of applicability of the EFT.     

Estimating parameters in stochastic systems: A variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.     

Tau functions and the twistor theory of integrable systems

We first define τ-functions as generalized cross-ratios of four points on a finite- or infinite-dimensional Grassmannian. We show how this definition can be used to construct a natural flat connection on a determinant line bundle associated with two equivariant holomorphic vector bundles over a twistor space, provided that the action of the symmetries on the bundles has the same normal form at the fixed points for the two bundles. The determinant line bundle has a natural meromorphic section of which the logarithmic covariant derivative is the logarithmic derivative of the τ-function. We establish a natural product formula for this τ-function; we show that it vanishes at the jumping lines of one bundle and has poles at the jumping lines of the other. We also show that this definition leads to standard expressions for the τ-functions of the KdV equation, the Ernst equation, and the isomonodromic deformation equations. We describe a new twistor treatment of the isomonodromic deformation equations.     

Sum rules for zeros and intersections of Bessel functions from quantum mechanical perturbation theory

Bessel functions play an important role for quantum states in spherical and cylindrical geometries. In cases of perfect confinement, the energy of Schrödinger and massless Dirac fermions is determined by the zeros and intersections of Bessel functions, respectively. In an external electric field, standard perturbation theory therefore expresses the polarizability as a sum over these zeros or intersections. Both non-relativistic and relativistic polarizabilities can be calculated analytically, however. Hence, by equating analytical expressions to perturbation expansions, several sum rules for the zeros and intersections of Bessel functions emerge.     

Automatic phasing of MR images. Part II: voxel-wise phase estimation

Magnetic resonance images are typically displayed as the absolute value of the discrete Fourier transform of the k-space data. However, absorption-mode images, the real part of the discrete Fourier transform of the data after applying an appropriate phase correction, have significant advantages over absolute-value images. In a companion paper, the problem of estimating the phase parameters needed to produce an absorption-mode image when the phase of the complex image varies linearly as a function of position, a situation common in magnetic resonance images, was addressed. However, some magnetic resonance images have phases that can vary in a complicated, nonlinear, positionally dependent fashion. To produce an absorption-mode image from these data, one must first estimate the positionally dependent phase, and then use that phase estimate to produce an absorption-mode image. This paper addresses both of these problems by first using Bayesian probability theory to estimate the constant or zero-order phase as a function of image position, and then the calculations are illustrated by using them to generate absorption-mode images from data where the phase of the image is a nonlinear function of position.     

Approximation errors for the measurement of coupling loss factors using the ELD method

In this paper the formulae are developed to predict the approximation error resulted from the use of the energy level difference (ELD) method. The predicted approximation error can then be used to assess the feasibility of the application of the ELD method. The effects of different parameters on the approximation error are analysed and discussed. Two practical applications are presented. Both numerical and experimental results showed that the ELD method could give identical results (from experimental point of view) as other existing methods as long as α_i and β_i are small.     

Automatic phasing of MR images. Part I: linearly varying phase

In spin-echo and well shimmed gradient-echo images, the phase of the complex image often varies linearly in both the readout and phase-encode directions. Thus, in principle, it is possible to display an image in absorption mode. However, manually determining the two first-order and one zero-order phase parameters needed to display an absorption-mode image is a formidable task. In this paper, the Bayesian calculations needed to automatically determine these parameters are presented, and the calculations are illustrated using spin-echo images.     

用新型红外测温仪观察由钨丝电子发射引起的损耗

在金属电子逸出功测定实验中,未考虑由钨丝电子发射引起的损耗,存在约-1.86%的系统误差.用新型红外测温仪可以观察到因上述损耗而出现的钨丝温度的变化,为修正该系统误差提供直接的观察依据.     

On the computation of molecular auxiliary functions A n and B n

Molecular auxiliary functions A _n (p) and B _n (pt), arising in the Hartree-Fock-Roothaan (HFR) approximation for molecules, Ewald’s crystal lattice theory, electromagnetic stopping theory, and other approximate methods, are evaluated and analysed in the range of 17≤n≤60 and 25≤pt≤60.     

A theory of complex open resonators in the form of cavity chain

The previous theory ((2,6)) of microwave open resonators has been improved, which can be applicable not only to simple cavities, but also to complex cavities. The comparison between calculations and experimental results shows tha the diffraction Q value of an open resonator may be obtained more precisely from the modified theory than one from the previous theory. According to the theory suggested here, many complex cavities, e.g. a complex cavity with a fourcavity chain, have been analyzed and calculated. It is seen that a variety of field profile forms can exist in complex cavities by adjustment of their cavity geometry.     

Pair neutrino energy loss for nuclei 56^Fe at the late stages of stellar evolution

Based on the Weinberg-Salam theory, the pair neutrino energy loss rates for nuclei 56Fe are can- vassed for the wide range of density and temperature. The results of ours (QLJ) are compared with those of Beaudet G, Petrosian V and Salpeter E. E's (QBPS), and it shows that the pair neutrino energy loss rates of QBPS are always larger than QLJ .The QBPS is 12.57%, 12.86%, 14.99%, 19.80% times higher than QLJ corresponding to the temperature T9=0.385, 1.0, 5.0, 10, respectively.     

Pair neutrino energy loss for nuclei 56Fe at the late stages of stellar evolution

Based on the Weinberg-Salam theory,the pair neutrino energy loss rates for nuclei 56Fe are canvassed for the wide range of density and temperature.The results of ours (QLJ) are compared with those of Beaudet G,Petrosian V and Salpeter E.E's (QBPS),and it shows that the pair neutrino energy loss rates of QBPS are always larger than QLJ .The QBPS is 12.57%,12.86%,14.99%,19.80% times higher than QLJ corresponding to the temperature T9=0.385,1.0,5.0,10,respectively.     

Kinetic theory of time correlation functions for a dense one-component plasma in a magnetic field

The time-dependent correlations of a one-component plasma in a uniform magnetic field are studied with the help of kinetic theory. The time correlation functions of the particle density, the momentum density, and the kinetic energy density are evaluated for large time intervals. In the collision-dominated regime the results agree with those found from linearized magnetohydrodynamics.     

Renormalized theory of the time-dependent pair distribution function. I. General formulation

A classical renormalized theory of a time-dependent pair-distribution function (TDPDF), previously introduced by Oppenheim and Bloom, is presented. An equation of motion for the TDPDF is derived in which the memory function of the system appears. This is then split into a part which contains only static correlation functions and a part which describes the dynamics. The mean field approximation is discussed in some detail and contact is made witn the theory of Oppenheim and Bloom.Work supported in part by a National Research Council of Canada operating grant.     

Plasma neutrino energy loss due to the axial-vector current at the late stages of stellar evolution

Based on the Weinberg-Salam theory, the plasma neutrino energy loss rates of vector and axialvector contributions are studied.A ratable factor of the rates from the axial-vector current relative to those of the total neutrino energy loss rates is accurately calculated.The results show that the ratable factor will reach a maximum of 0.95 or even more at relatively higher temperature and lower density (such as P/μe＜ 10~7 g/cm~3).Thus the rates of the axial-vector contribution cannot be neglected.On the other hand, the rates of the axialvector contribution are on the order of～0.01% of the total vector contribution, which is in good agreement with Itoh's at relatively high density (such as p/μe＞10~7 g/cm~3) and a temperature of T≤10~(11) K.     

Model-based predictive controller design for a class of nonlinear networked systems with communication delays and data loss

This paper discusses the model-based predictive controller design of networked nonlinear systems with communication delay and data loss. Based on the analysis of the closed-loop networked predictive control systems, the model-based networked predictive control strategy can compensate for communication delay and data loss in an active way. The designed model-based predictive controller can also guarantee the stability of the closed-loop networked system. The simulation results demonstrate the feasibility and efficacy of the proposed model-based predictive controller design scheme.     

Time reversal and symmetries of time correlation functions

ABSTRACT

The time reversal invariance of classical dynamics is reconsidered in this paper with specific focus on its consequences for time correlation functions and associated properties such as transport coefficients. We show that, under fairly common assumptions on the interparticle potential, an isolated Hamiltonian system obeys more than one time reversal symmetry and that this entails non trivial consequences. Under an isotropic and homogeneous potential, in particular, eight valid time reversal operations exist. The presence of external fields that reduce the symmetry of space decreases this number, but does not necessarily impair all time reversal symmetries. Thus, analytic predictions of symmetry properties of time correlation functions and, in some cases, even of their null value are still possible. The noteworthy case of a constant external magnetic field, usually assumed to destroy time reversal symmetry, is considered in some detail. We show that, in this case too, some of the new time reversal operations hold, and that this makes it possible to derive relevant properties of correlation functions without the uninteresting inversion of the direction of the magnetic field commonly enforced in the literature.