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.     

Curvature investigation in tapered fibers and its application to sensing and mode conversion

Non-adiabatic tapered fibers are basic photonic components used in a wide range of applications. Here we investigate a way to increase their utility through the controllable bending of one of their tapered sections. The experiments carried out explain, for the first time, the mechanics of this approach showing how these tapers can be used to build more sensitive sensors. Their use as highly efficient mode converters is also discussed.     

Recent advances and challenges in the recovery and purification of cellular exosomes

Exosomes are nanovesicles secreted by most cellular types that carry important biochemical compounds throughout the body with different purposes, playing a preponderant role in cellular communication. Because of their structure, physicochemical properties and stability, recent studies are focusing in their use as nanocarriers for different therapeutic compounds for the treatment of different diseases ranging from cancer to Parkinson's disease. However, current bioseparation protocols and methodologies are selected based on the final exosome application or intended use and present both advantages and disadvantages when compared among them. In this context, this review aims to present the most important technologies available for exosome isolation while discussing their advantages and disadvantages and the possibilities of being combined with other strategies. This is critical since the development of novel exosome‐based therapeutic strategies will be constrained to the effectiveness and yield of the selected downstream purification methodologies for which a thorough understanding of the available technological resources is needed.     

紫色球杆菌视紫红质光谱特性的机器学习研究

近年来，机器学习等人工智能技术被应用于蛋白质工程，其在蛋白质结构、功能预测、催化活性等研究中具有独特优势。在未知蛋白质结构的情况下，将蛋白质序列和功能特性与机器学习相结合，基于序列-活性关系（innovative sequence-activity relationship，ISAR）算法，将蛋白质氨基酸序列数字化，用快速傅里叶变换（fast four transform，FFT）进行预处理，再进行偏最小二乘回归建模，可在数据集较少情况下拟合得到最佳模型。通过机器学习对紫色球杆菌视紫红质（gloeobacter violaceus rhodopsin，GR）的突变体蛋白质氨基酸序列与光谱最大吸收波长进行建模，获得了最佳模型。用最佳索引LEVM760106建模得到的确定系数R² 为0.944，均方误差E为11.64。用小波变换进行的预处理，其R² 虽也约为0.944，但E大于11.64，不及FFT进行的预处理。方法较好地解决了蛋白质序列与功能特性之间的数学建模问题，在蛋白质工程中可为预测更优的突变体提供支持。     

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.     

具Hardy-Sobolev临界指数的奇异椭圆方程多解的存在性

运用变分方法研究了下面问题-Δpu=μupx(s)s-2u f(x,u),x∈Ω,u=0,x∈Ω,多重解的存在性,其中Ω是一个具有光滑边界的有界区域.     

Analysis of G/D/1 Queueing Systems with Inputs Satisfying Large Deviation Principle under Weak* Topology

The large deviation principle (LDP) which has been effectively used in queueing analysis is the sample path LDP, the LDP in a function space endowed with the uniform topology. Chang [5] has shown that in the discrete-time G/D/1 queueing system under the FIFO discipline, the departure process satisfies the sample path LDP if so does the arrival process. In this paper, we consider arrival processes satisfying the LDP in a space of measures endowed with the weak^* topology (Lynch and Sethuraman [12]) which holds under a weaker condition. It is shown that in the queueing system mentioned above, the departure processes still satisfies the sample path LDP. Our result thus covers arrival processes which can be ruled out in the work of Chang [5]. The result is then applied to obtain the exponential decay rate of the queue length probability in an intree network as was obtained by Chang [5], who considered the arrival process satisfying the sample path LDP.