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.     

A tripling construction for mutually orthogonal symmetric hamiltonian double Latin squares

We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.     

Decomposition of ideals into pseudo-irreducible ideals

A proper ideal of a commutative ring is called pseudo-irreducible if it cannot be written as a product of two comaximal proper ideals. In this paper, we give a necessary and su?cient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal pseudo-irreducible ideals. Examples of such rings include Laskerian rings, or more generally J-Noetherian rings and ZD-rings. We study when certain classes of rings satisfy this condition.     

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.     

Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.     

两类图的(d,1)-全标号

主要讨论了W_n与C_m的笛卡尔积和均衡完全r-部图K_r(n)的(d,1)-全标号,并得出了(d,1)-全数λ_d~T(W_n□C_m)和λ_d~T(K_(r(n)))的确切值.     

Toader型平均的算术与调和平均界

给出了Toader型平均T[A(a,b),G(a,b)]关于调和平均H(a,b)与算术平均A(a,b)组合的精确界.作为应用,发现了几个关于第二类完全椭圆积分的精确不等式.     

完整Coriolis力作用下带有外源强迫的非线性KdV方程

利用摄动方法,从描写既有Coriolis力垂直分量又含有水平分量的位涡方程出发,给出了近赤道非线性Rossby波所满足的具有外源强迫的非线性KdV方程,并利用Jacobi椭圆函数展开法,求解了改进后的非线性KdV方程的行波解及孤立波解.通过分析KdV方程的行波解,指出Coriolis力的水平分量和外源对Rossby波动的影响.