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.     

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.     

Evolution of Local Structural Ordering and Chemical Distribution upon Delithiation of a Rock Salt–Structured Li1.3Ta0.3Mn0.4O2 Cathode

Lithium‐rich disordered rock‐salt oxides have attracted great interest owing to their promising performance as Li‐ion battery cathodes. While experimental and theoretical efforts are critical in advancing this class of materials, a fundamental understanding of key property changes upon Li extraction is largely missing. In the present study, single‐crystal synthesis of a new disordered rock‐salt cathode material, Li_1.3Ta_0.3Mn_0.4O₂ (LTMO), and its use as a model compound to investigate Li concentration–driven evolution of local cationic ordering, charge compensation, and chemical distribution are reported. Through the combined use of 2D and 3D X‐ray nanotomography, it is shown that Li removal accompanied by oxygen oxidation is correlated with the development of morphological defects such as particle cracking. Chemical heterogeneity, quantified by subparticle level distribution of Mn valence state, is minimal during Mn redox, which drastically increases upon the formation of cracks during oxygen redox. Density functional theory and bond valence sum mismatch calculations reveal the presence of local short‐range ordering in the pristine oxide, which gradually disappears along with the extraction of Li. The study suggests that with cycling the transformation into true cation–disordered state can be expected, which likely impacts the voltage profile and obtainable energy density of the oxide cathodes.     

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.     

一种实用的多数据库量子信息检索协议

针对目前量子私有信息检索不能适用与云存储的多数据库问题,基于现在成熟的量子密钥分发方法,提出了一种适合在多数据库环境下,实用的量子私有信息检索协议。对于不同大小的数据库,协议可通过调节参数θ和k,在保证数据库安全及用户隐私的情况下,完成信息的检索。性能分析结果表明,协议的通信复杂度低,检索成功率高、易于实施。     

Potts models with magnetic field: Arithmetic,geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.     

移动机器人路径规划方法研究

移动机器人技术研究中的一个重要领域是路径规划技术，它分为基于模型的环境已知的全局路径规划和基于传感器的环境未知的局部路径规划。综述了移动机器人路径规划技术的发展现状指出了各种方法的优点与不足，最后对移动机器人路径规划技术的发展趋势进行了展望。     

小波变换在光谱特征提取方面的应用

人们在处理高光谱图像时一般要对一些典型地物进行光谱分析、特征波段的提取,以便提取出最大量的有效信息,剔除无用或冗余的信息,然后再进行分类识别.采用小波变换的分析方法,选用合适的小波进行分解,根据分解后的高频分量中包含的重要信息,利用局部相邻的正负极值点找出对应于原始光谱曲线上每个吸收带的左右边界;利用局部过零点,即可比较精确的提取出各个吸收带的中心波长.该方法比传统的光谱特征提取方法更简洁、有效,实验证明为一种比较理想的光谱特征提取方法.     

WPAN/WLAN应用环境下基于PER的链路自适应算法

本文重点研究基于包错误率测量的链路自适应算法，分析了包错误率与信道参数的关系，提出了一种有别于固定门限值方案的优化门限值方案。仿真结果表明，优化门限值方案的网络吞吐量要大于固定门限值方案和基于信干比估计的链路自适应算法，最大增幅可达12Mbit/s。在系统引入重传机制后，网络吞吐量将有所损失，但最大不超过1Mbit/s，相比较服务质量的提高是值得的。需指出的是，虽然分析和仿真均以HIPERLAN／2为例，但不失一般性，它们同样适合于WPAN/WLAN等类似系统。