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.     

Improved error performance for generalised spatial modulation with enhanced spectral efficiency

The existing analytical average bit error rate (ABER) expression of conventional generalised spatial modulation (CGSM) does not agree well with the Monte Carlo simulation results in the low signal‐to‐noise ratio (SNR) region. Hence, the first contribution of this paper is to derive a new and easy way to evaluate analytical ABER expression that improves the validation of the simulation results at low SNRs. Secondly, a novel system termed CGSM with enhanced spectral efficiency (CGSM‐ESE) is presented. This system is realised by applying a rotation angle to one of the two active transmit antennas. As a result, the overall spectral efficiency is increased by 1 bit/s/Hz when compared with the equivalent CGSM system. In order to validate the simulation results of CGSM‐ESE, the third contribution is to derive an analytical ABER expression. Finally, to improve the ABER performance of CGSM‐ESE, three link adaptation algorithms are developed. By assuming full knowledge of the channel at the receiver, the proposed algorithms select a subset of channel gain vector (CGV) pairs based on the Euclidean distance between all CGV pairs, CGV splitting, CGV amplitudes, or a combination of these.     

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas–Weyl(α, β)-metrics. Suppose that a regular(α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.     

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.     

小型机载激光通信系统设计与验证

针对空空宽带高速通信的需求，设计了小型化机载激光通信系统。仿真分析了300 km、2.5 Gb/s无线激光链路性能，并通过运动仿真台模拟机动环境测试了系统的跟踪与通信性能，其中粗跟踪误差为533.2 μrad(1σ)，精跟踪误差为3.6 μrad(1σ)，测试数据传输240 s，通信误码率为2.82×10-9。仿真与实验验证了该系统用于远距离空空无线激光通信的可行性。     

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.     

The Szegö kernel on a class of non-compact CR manifolds of high codimension

We generalize Nagel’s formula for the Szegö kernel and use it to compute the Szegö kernel on a class of non-compact CR manifolds whose tangent space decomposes into one complex direction and several totally real directions. We also discuss the control metric on these manifolds and relate it to the size of the Szegö kernel.     

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.     

理论、实践经验和实践能力

掌握了理论知识并不等于有了实战经验,更不等于有实战能力。从理论到有实战能力的飞跃,只有亲临生产线,在不断解决问题过程中实现。     

Uplink Scheduling in CDMA Packet-Data Systems

Uplink scheduling in wireless systems is gaining importance due to arising uplink intensive data services (ftp, image uploads etc.), which could be hampered by the currently in-built asymmetry in favor of the downlink. In this work, we propose and study algorithms for efficient uplink packet-data scheduling in a CDMA cell. The algorithms attempt to maximize system throughput under transmit power limitations on the mobiles assuming instantaneous knowledge of user queues and channels. However no channel statistics or traffic characterization is necessary. Apart from increasing throughput, the algorithms also improve fairness of service among users, hence reducing chances of buffer overflows for poorly located users. The major observation arising from our analysis is that it is advantageous on the uplink to schedule “strong” users one-at-a-time, and “weak” users in larger groups. This contrasts with the downlink where one-at-a-time transmission for all users has shown to be the preferred mode in much previous work. Based on the optimal schedules, we propose less complex and more practical approximate methods, both of which offer significant performance improvement compared to one-at-a-time transmission, and the widely acclaimed Proportional Fair (PF) algorithm, in simulations. When queue content cannot be fed back, we propose a simple modification of PF, Uplink PF (UPF), that offers similar improvement. Hereafter, we refer to users with low recieved power at the base even when transmitting at peak transmit power as “weak” users, and the strongly recieved users at the base as “strong” users. Krishnan Kumaran is currently a member of the Complex Systems Modeling section in the Corporate Strategic Research of ExxonMobil Corp., Clinton, NJ. Formerly, he was a Member of Technical Staff in the Mathematics of Networks and Systems Research Department at Bell Labs in Murray Hill, NJ, where his research interests were in modeling, analysis and simulation of design, resource management and scheduling issues in telecommunication networks. Lijun Qian is an assistant professor in the Department of Electrical Engineering at Prairie View A&M University. He received his B.S. from Tsinghua University in Beijing, M.S. from Technion-Israel Institute of Technology, and Ph.D. from WINLAB, Rutgers University, all in electrical engineering. Before joining PVAMU, he was a researcher at Networks and Systems Research Department of Bell Labs in Murray Hill, NJ. His major research interests are in wireless communications and networking technologies, especially in radio resource management, protocol design, TCP/RLP optimization and MPLS traffic engineering.