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.     

Stochastic partial functional differential equations with locally monotone coefficients,locally Lipschitz non-linearity and delay

In this paper, we study the existence and uniqueness of strong solutions for stochastic partial functional differential equations with locally monotone coefficients, locally Lipschitz non-linearity, and time delay. Our results extend previous results obtained by Liu–Röckner, Caraballo et al. and Taniguchi et al. Examples are given to illustrate the wide applicability of our results.     

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

In this paper, we consider a class of split mixed vector quasivariational inequality problems in real Hilbert spaces and establish new gap functions by using the method of the nonlinear scalarization function. Further, we obtain some error bounds for the underlying split mixed vector quasivariational inequality problems in terms of regularized gap functions. Finally, we give some examples to illustrate our results. The results obtained in this paper are new.     

与 Cauchy 微分中值定理相关的几个问题

根据Cauchy微分中值定理表达式的结构引入辅助函数 F（x ，c）＝ f （x）- f （c）g（x）- g（c）（a＜ c＜ b），通过讨论其可导性，得到相关的几个不等式，由此得出Cauchy微分中值定理存在唯一“中值点”的一个条件，并给出其逆定理的一个较弱表述。     

Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

The remap phase in arbitrary Lagrangian–Eulerian (ALE) hydrodynamics involves the transfer of field quantities defined on a post‐Lagrangian mesh to some new mesh, usually generated by a mesh optimization algorithm. This problem is often posed in terms of transporting (or advecting) some state variable from the old mesh to the new mesh over a fictitious time interval. It is imperative that this remap process be monotonic, that is, not generate any new extrema in the field variables. It is well known that the only linear methods that are guaranteed to be monotonic for such problems are first‐order accurate; however, much work has been performed in developing non‐linear methods, which blend both high and low (first) order solutions to achieve monotonicity and preserve high‐order accuracy when the field is sufficiently smooth. In this paper, we present a set of methods for enforcing monotonicity targeting high‐order discontinuous Galerkin methods for advection equations in the context of high‐order curvilinear ALE hydrodynamics. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

单调性条件在Fourier级数收敛性中的最终推广:历史、发展、应用和猜想

为了对Fourier级数进行近似计算和有效应用,必须研究其收敛性,这个课题有长久的历史,形成了数学分析中吸引包括许多著名数学家在内的学者研究的一条热烈但困难的主流.其中,在三角级数(Fourier级数)一致收敛性和平均收敛性问题中人们一直关心Fourier系数的单调递减条件最终的推广.这个开始于英国Chaundy-Jollife(1916年)和Young(1913年)的工作最近出现了突破性的进展,产生了许多完善的结果.本文将对这方面的历史、发展给出综述,并重点介绍最近的应用成果,并对以后的工作给出研究思路和线索.     

Coalitional strategy-proofness and resource monotonicity for house allocation problems

We consider the problem of allocating houses to agents when monetary compensations are not allowed. We present a simple and independent proof of a result due to Ehlers and Klaus (Int J Game Theory 32:545–560, 2004) that characterizes the class of rules satisfying efficiency, strategy-proofness, resource monotonicity and nonbossiness. I am very much indebted to William Thomson for his invaluable suggestions.     

The relation between non-bossiness and monotonicity

It is a well-known fact that in some economic environments, non-bossiness and monotonicity are interrelated. In this paper, we have presented a new domain-richness condition called weak monotonic closedness, on which non-bossiness in conjunction with individual monotonicity is equivalent to monotonicity. Moreover, by applying our main result to several types of economies, we have obtained characterizations in terms of non-bossiness.     

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν _j of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν _j . We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper’s final section we present extensions of our results to more general chains. Research supported by NSF grant DMS–0406104, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.