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.     

Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Properties of bounded stochastic processes employed in biophysics

Abstract

Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated within a solid mathematical ground. Four families are object of investigation: the Sine-Wiener (SW), the Doering–Cai–Lin (DCL), the Tsallis–Stariolo–Borland (TSB), and the Kessler–Sørensen (KS) families. We address mathematical questions on existence and uniqueness of the processes defined through Stochastic Differential Equations, which often conceal non-obvious behavior, and we explore the behavior of the solutions near the boundaries of the state space. The expression of the time-dependent probability density of the Sine-Wiener noise is provided in closed form, and a close connection with the Doering–Cai–Lin noise is shown. Further relationships among the different families are explored, pathwise and in distribution. Finally, we illustrate an analogy between the Kessler–Sørensen family and Bessel processes, which allows to relate the respective local times at the boundaries.     

公理化局部有限空间中映射连续性的扩展

用公理化方法研究了局部有限空间中的连续映射及其扩张问题。给出了局部有限空间的公理化定义方法;利用邻近关系研究了局部有限空间中的连续映射、同胚和局部同胚等问题;通过对局部有限空间变形的研究,定义了局部有限空间的一种特殊收缩核,有效地解决了局部有限空间中连续映射的扩张问题。     

高静态动作压力下粉尘爆炸泄放标准的可靠性

粉尘爆炸泄放设计标准的可靠性对能否实现爆炸防护尤为关键.以中径70μm石松子粉尘为介质,采用20 L球形粉尘爆炸装置进行了4种泄放口径、静态动作压力在0.12~0.6 MPa的爆炸泄放实验,对比了实验值与NFPA 68和EN 14491计算值.结果表明,NFPA 68在预测高静态动作压力下的泄放面积时,随着泄放口径的增大,预测结果由保守变为危险,但预测结果精度较好且数值稳定;EN 14491在预测高静态动作压力下的泄放面积时非常保守,预测精度较差,数据不可靠.依据两种标准推荐的经验函数关系,分析了NFPA 68和EN 14491在预测高静态动作压力时产生这种差异的原因.     

矩形型钢弯曲回弹曲率半径r值的确定

文章在分析矩形型钢弯曲变形过程及应力与应变等方面的关系的基础上，推导出工件曲率半径与工件回弹后曲率半径之间的关系，从而确定了凸模半径。     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

一阶最优性条件研究

本对由Botsko的关于多变量函数取极值的一阶导数检验条件定理^[1]进行了分析研究，给出了更实用而简捷的差别条件。最后，举出若干例子予以说明。     

Surfaces of Constant Mean Curvature Bounded by Two Planar Curves

In this paper we study constant mean curvature compact surfaces with two Jordan curves in parallel planes as boundary and we investigate the point at which the surface inherits the symmetries of its boundary.