The origin of degeneracies and crossings in the 1d Hubbard model

The paper is devoted to the connection between integrability of a finite quantum system and degeneracies of its energy levels. In particular, we analyse in detail the energy spectra of finite Hubbard chains. Heilmann and Lieb demonstrated that in these systems there are crossings of levels of the same parameter-independent symmetry. We show that this apparent violation of the Wigner-von Neumann noncrossing rule follows directly from the existence of nontrivial conservation laws and is a characteristic signature of quantum integrability. The energy spectra of Hubbard chains display many instances of permanent (at all values of the coupling) twofold degeneracies that cannot be explained by parameter-independent symmetries. We relate these degeneracies to the different transformation properties of the conserved currents under spatial reflections and the particle-hole transformation and estimate the fraction of doubly degenerate states. We also discuss multiply degenerate eigenstates of the Hubbard Hamiltonian. The wavefunctions of many of these states do not depend on the coupling, which suggests the existence of an additional parameter-independent symmetry.

Spherical piston problem in water

In this paper, we study the propagation of a shock wave in water, produced by the expansion of a spherical piston with a finite initial radius. The piston path in the x, t plane is a hyperbola. We have considered the following two cases: (i) the piston accelerates from a zero initial velocity and attains a finite velocity asymptotically as t tends to infinity, and (ii) the piston decelerates, starting from a finite initial velocity. Since an analytic approach to this problem is extremely difficult, we have employed the artificial viscosity method of von Neumann & Richtmyer after examining its applicability in water. For the accelerating piston case, we have studied the effect of different initial radii of the piston, different initial curvatures of the piston path in the x, t plane and the different asymptotic speeds of the piston. The decelerating case exhibits the interesting phenomenon of the formation of a cavity in water when the deceleration of the piston is sufficiently high. We have also studied the motion of the cavity boundary up to 550 cycles.

Entropy of quantum states: Ambiguities

The von Neumann entropy of a generic quantum state is not unique unless the state can be uniquely decomposed as a sum of extremal or pure states. As pointed out to us by Sorkin, this happens if the GNS representation (of the algebra of observables in some quantum state) is reducible, and some representations in the decomposition occur with non-trivial degeneracy. This non-unique entropy can occur at zero temperature. We will argue elsewhere in detail that the degeneracies in the GNS representation can be interpreted as an emergent broken gauge symmetry, and play an important role in the analysis of emergent entropy due to non-Abelian anomalies. Finally, we establish the analogue of an H-theorem for this entropy by showing that its evolution is Markovian, determined by a stochastic matrix.

Quantum entropy for the fuzzy sphere and its monopoles

Using generalized bosons, we construct the fuzzy sphere S-F(2) and monopoles on S-F(2) in a reducible representation of SU(2). The corresponding quantum states are naturally obtained using the GNS-construction. We show that there is an emergent nonabelian unitary gauge symmetry which is in the commutant of the algebra of observables. The quantum states are necessarily mixed and have non-vanishing von Neumann entropy, which increases monotonically under a bistochastic Markov map. The maximum value of the entropy has a simple relation to the degeneracy of the irreps that constitute the reducible representation that underlies the fuzzy sphere.

STOPPING THE CCR FLOW AND ITS ISOMETRIC COCYCLES

It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e. left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation.

Non-Abelian gauge redundancy and entropic ambiguities

The von Neumann entropy of a generic quantum state is not unique unless the state can be uniquely decomposed as a sum of extremal or pure states. Therefore one reaches the remarkable possibility that there may be many entropies for a given state. We show that this happens if the GNS representation (of the algebra of observables in some quantum state) is reducible, and some representations in the decomposition occur with non-trivial degeneracy. This ambiguity in entropy, which can occur at zero temperature, can often be traced to a gauge symmetry emergent from the non-trivial topological character of the configuration space of the underlying system. We also establish the analogue of an H-theorem for this entropy by showing that its evolution is Markovian, determined by a stochastic matrix. After demonstrating this entropy ambiguity for the simple example of the algebra of 2 x 2 matrices, we argue that the degeneracies in the GNS representation can be interpreted as an emergent broken gauge symmetry, and play an important role in the analysis of emergent entropy due to non-Abelian anomalies. We work out the simplest situation with such non-Abelian symmetry, that of an ethylene molecule.

Admissible Hierachic Sets

In this paper we present a solution concept for abstract systems called the admissible hierarchic set. The solution we propose is a refinement of the hierarchic solution, a generalization of the von Neumann and Morgenstern solution. For finite abstract systems we show that the admissible hierarchic sets and the von Neumann and Morgenstern stable sets are the only outcomes of a coalition formation procedure (Wilson, 1972 and Roth, 1984). For coalitional games we prove that the core is either a vN&M stable set or an admissible hierarchic set.

The Supercore for Normal Form Games

We study the supercore of a system derived from a normal form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore of that system coincides with the set of Nash equilibrium strategy profiles of the last game in the sequence. This result is illustrated with the characterization of the supercore for the n-person prisoners’ dilemma. With regard to the mixed extension of a normal form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria. For games with an infinite number of Nash equilibria this need not be no longer the case. Yet, it is not difficult to find a binary relation which guarantees the coincidence of these two sets.

Numerical solution of parabolic equations by the box scheme

The box scheme proposed by H. B. Keller is a numerical method for solving parabolic partial differential equations. We give a convergence proof of this scheme for the heat equation, for a linear parabolic system, and for a class of nonlinear parabolic equations. Von Neumann stability is shown to hold for the box scheme combined with the method of fractional steps to solve the two-dimensional heat equation. Computations were performed on Burgers' equation with three different initial conditions, and Richardson extrapolation is shown to be effective.

液体多脉冲及2D NMR实验的计算机模拟

现代多脉冲及2D NMR技术是过去十年中发展起来的崭新的NMR实验方法。计算机模拟做为NMR实验的强有力分析手段已日益受到重视。国内这方面工作开展得尚很少；国外发表的工作主要采用的是数字模拟，存在分析结果不够直观、物理意义不够清晰等缺陷。本论文工作采用乘积算符方法研制出对分析多脉冲及2D NMR实验普适的模拟程序PROPER；在乘积算符基础上，针对磁等性自旋体系，提出了实用的对称化乘积算符及多量子积算符方法。一、多脉冲及2D NMR实验的计算机模拟 1. 采用乘积算符方法在本所PDP－11／23微机上研制了多脉冲及2D NMR实验的模拟程序PROPER。该程序对不超过4核（I ＝ 1／2）的同核及异核弱耦合自旋体系非选择性脉冲序列的分析是普遍适用的。受计算机内存的限制，PROPER程序所能处理的脉冲序列脉冲间隔数目一般不超过10。2. 应用PROPER模拟程序对INEP和DEPT脉冲序列进行了分析比较；特别对BIRD脉冲序列的各种相位变型进行了模拟分析，给出了分析结果，分析过程中考虑了影响BIRD作用效果的同核耦合因素。应用结果表明，PROPER程序计算正确、迅速、给出的模拟结果较通常的数字模拟方法简单、直观、物理意义清楚，便于分析。由于采用算符模拟，结果的输出打印比较费时。目前，PROPER程序正在改进和完善之中。二、多脉冲及2D NMR实验的密度算符描述 1. 针对磁等性自旋（I ＝ 1／2）体系，首次提出了对称化乘积算符描述方法。在通常的乘积算符基础上，引入了对称化乘积算符，并对其数理基础进行了详细论证。推导了算符循环对易关系决定的Liourill-Von Neumann方程的解，给出了算符间普遍存在的循环对易关系及其相应的演化公式。据此，以InS(I = 1/2, S = 1/2; n = 2,3)自旋体系为例，对DEPT脉冲序列进行了分析；结果表明，该方法较通常的乘积算符方法对磁等性自旋体系的分析要简单、实用，且物理意义更加明确。由于该方法涉及较多的算符对易关系，因此不易计算机编程。2. 在对称化乘积算符基础上引入了多量子积算符的概念。以In(I = 1/2; n = 2,3)体系为例，给出了两者的互换关系。推导出了具有标量耦合作用的两组合粒子体系普适的多量子积算符环对易关系及相应的演化解析式。多量子积算符方法可望将1／2－自旋磁等性组合粒子表象与自旋大于1／2的单粒子表象统一起来，并为计算机模拟提供新的数学方法。该方法尚有待于进一步研究。

三维双楔面定常激波相互作用

本文通过理论分析,将三维对称双楔面上定常激波相互作用简化到二维截面进行分析,利用二维非定常激波楔面反射理论求解该三维激波相互作用结构。同时,通过采用二阶精度的NND格式求解三维Euler方程,对该理论分析结果做出数值模拟验证。理论分析和数值模拟结果显示,对于三维双楔面超声速定常流动在取定的二维截面上激波结构兼有二维非定常及定常激波反射的性质,即形成了类似于二维非定常激波楔面反射的规则反射、单马赫反射、过渡马赫反射及双马赫反射等结构,同时其规则—马赫反射转变却遵循适用于二维定常激波反射的von Neumann准则。理论分析得出的各种反射结构的存在范围与数值模拟结果吻合良好。同时,探讨了两楔面间夹角以及楔面前缘后掠角对该激波结构的影响。

压缩真空场与原子非线性作用过程中的纠缠与消纠缠

用VonNeumann熵研究了附加克尔介质的压缩真空场与二能级原子依赖强度耦合相互作用量子体系的量子纠缠特性 .讨论了初始压缩真空场的压缩度以及克尔非线性作用的强度对该量子体系纠缠特性的影响 .结果表明 ,克尔介质的非线性作用的强弱可以改变体系量子纠缠的周期性 ;在初始压缩度较大 (r =5 )时 ,克尔介质的非线性作用可导致原子与场持续地处于最大纠缠态 ,无消纠缠态或持续地处于消纠缠态 .

Intelligence Without Reason

Computers and Thought are the two categories that together define Artificial Intelligence as a discipline. It is generally accepted that work in Artificial Intelligence over the last thirty years has had a strong influence on aspects of computer architectures. In this paper we also make the converse claim; that the state of computer architecture has been a strong influence on our models of thought. The Von Neumann model of computation has lead Artificial Intelligence in particular directions. Intelligence in biological systems is completely different. Recent work in behavior-based Artificial Intelligenge has produced new models of intelligence that are much closer in spirit to biological systems. The non-Von Neumann computational models they use share many characteristics with biological computation.

Approximately Uniform Random Sampling in Sensor Networks

Recent work in sensor databases has focused extensively on distributed query problems, notably distributed computation of aggregates. Existing methods for computing aggregates broadcast queries to all sensors and use in-network aggregation of responses to minimize messaging costs. In this work, we focus on uniform random sampling across nodes, which can serve both as an alternative building block for aggregation and as an integral component of many other useful randomized algorithms. Prior to our work, the best existing proposals for uniform random sampling of sensors involve contacting all nodes in the network. We propose a practical method which is only approximately uniform, but contacts a number of sensors proportional to the diameter of the network instead of its size. The approximation achieved is tunably close to exact uniform sampling, and only relies on well-known existing primitives, namely geographic routing, distributed computation of Voronoi regions and von Neumann's rejection method. Ultimately, our sampling algorithm has the same worst-case asymptotic cost as routing a point-to-point message, and thus it is asymptotically optimal among request/reply-based sampling methods. We provide experimental results demonstrating the effectiveness of our algorithm on both synthetic and real sensor topologies.

[Book review] Theory of Games and Economic Behaviour

Tony Mann provides a review of the book: Theory of Games and Economic Behavior, John von Neumann and Oskar Morgenstern, Princeton University Press, 1944.