The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

推广x重新标度模型的重标度参数经验公式

给出了推广x重新标度模型的重标度参数经验公式,其中建立了重标度参数与原子核的平均结合能之间的联系,由该公式可以得出A≥12的所有核的重标度参数值,利用这些参数值可以计算有关核过程并做出预言.     

An extended technicolour model for grand unification

An extended technicolour grand unification model based on the gauge groupE ₆×SU(7) extended technicolour is presented. The symmetry-breaking based on extended technicolour theory is discussed. It is shown that the existing phenomenology is well explained by the model. The strangeness changing neutral currents may not be a problem with this model.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

Locally Adaptive Wavelet Empirical Bayes Estimation of a Location Parameter

The traditional empirical Bayes (EB) model is considered with the parameter being a location parameter, in the situation when the Bayes estimator has a finite degree of smoothness and, possibly, jump discontinuities at several points. A nonlinear wavelet EB estimator based on wavelets with bounded supports is constructed, and it is shown that a finite number of jump discontinuities in the Bayes estimator do not affect the rate of convergence of the prior risk of the EB estimator to zero. It is also demonstrated that the estimator adjusts to the degree of smoothness of the Bayes estimator, locally, so that outside the neighborhoods of the points of discontinuities, the posterior risk has a high rate of convergence to zero. Hence, the technique suggested in the paper provides estimators which are significantly superior in several respects to those constructed earlier.     

模糊事件贝叶斯公式

讨论了模糊事件的概率及其基本性质 ,并通过对经典贝叶斯公式的推广 ,提出了模糊事件的贝叶斯公式 .     

Linear Relations, Differential Inclusions, and Degenerate Semigroups

A generalized linear differential equation in a Banach space is studied. The construction of a phase space and solutions with the help of the spectral theory of linear operators, ergodic theorems, and degenerate semigroups of linear operators is carried out.     

傅立叶超函数和扩充傅立叶超函数与爱米特热方程

证明了傅立叶超函数和扩充傅立叶超函数可用爱米特热方程的解来表示,且用以表示的解有很良好的性质.