Efficient Hopfield pattern recognition on a scale-free neural network

Neural networks are supposed to recognise blurred images (or patterns) of N pixels (bits) each. Application of the network to an initial blurred version of one of P pre-assigned patterns should converge to the correct pattern. In the “standard" Hopfield model, the N “neurons” are connected to each other via N² bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with N². The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabási-Albert scale-free network, in which each newly added neuron is connected to only m other neurons, and at the end the number of neurons with q neighbours decays as 1/q ³. Although the quality of retrieval decreases for small m, we find good associative memory for 1 ≪ m ≪ N. Hence, these networks gain a factor N/m ≫ 1 in the computer memory and time. Received 12 January 2003 Published online 11 April 2003 RID="a" ID="a"e-mail: stauffer@thp.uni-koeln.de     

On the quasi-chemical theory of solutions

The classical canonical partition function for a one-dimensional, two-component system is examined for the case of nearest-neighbor interactions. The quasi-chemical model of solid solutions is found to be a natural consequence of the definition of the partition function when one replaces the normal configurational energy with a configurational free energy. Consequently, the quasi-chemical theory may be viewed as a consequence of the definition of the partition function and not as merely a physically pleasing model.     

Monte Carlo technique for very large ising models

Rebbi's multispin coding technique is improved and applied to the kinetic Ising model with size 600*600*600. We give the central part of our computer program (for a CDC Cyber 76), which will be helpful also in a simulation of smaller systems, and describe the other tricks necessary to go to large lattices. The magnetizationM atT=1.4*T _c is found to decay asymptotically as exp(-t/2.90) ift is measured in Monte Carlo steps per spin, and M(t = 0) = 1 initially.     

Spontaneous magnetization of the Ising model on a layered square lattice

We have studied the Ising model on a layered square lattice with four different coupling constants and two different magnetic moments. The partition function at zero magnetic field is derived exactly. We propose a formula for the spontaneous magnetization which agrees with the exact low-temperature series expansion up to the 16th order and reduces to the exact result of Au-Yang and McCoy in a special case.     

Ising模型的并行计算

以Ising模型为例,介绍有关格点系统的Monte Carlo数值模拟并行算法的设计和编程,并给出在本组建造的PC集群式高性能并行计算系统上的测量结果.本文的结果对格点量子色动力学的大规模数值模拟研究有一定的参考价值     

二维伊辛模型相变临界现象的元胞自动机模拟

用元胞自动机模型模拟二维伊辛模型的相变临界现象,得出了相变图和时空演化图;当不存在外场时,数值模拟得到的临界温度与理论值精确吻合。     

Cyano‐Bridged MnIII?MIII Single‐Chain Magnets with MIII=CoIII,FeIII, MnIII,and CrIII

A series of isostructural cyano‐bridged Mn^III(h.s.)–M^III(l.s.) alternating chains, [Mn^III(5‐TMAMsalen)M^III(CN)₆] ? 4H₂O (5‐TMAMsalen^2?=N,N′‐ethylenebis(5‐trimethylammoniomethylsalicylideneiminate), Mn^III(h.s.)=high‐spin Mn^III, M^III(l.s.)=low‐spin Co^III, Mn? Co ; Fe^III, Mn? Fe ; Mn^III, Mn? Mn ; Cr^III, Mn? Cr ) was synthesized by assembling [Mn^III(5‐TMAMsalen)]³⁺ and [M^III(CN)₆]^3?. The chains present in the four compounds, which crystallize in the monoclinic space group C2/c, are composed of an [‐Mn^III‐NC‐M^III‐CN‐] repeating motif, for which the ‐NC‐M^III‐CN‐ motif is provided by the [M^III(CN)₆]^3? moiety adopting a trans bridging mode between [Mn^III(5‐TMAMsalen)]³⁺ cations. The Mn^III and M^III ions occupy special crystallographic positions: a C₂ axis and an inversion center, respectively, forming a highly symmetrical chain with only one kind of cyano bridge. The Jahn–Teller axis of the Mn^III(h.s.) ion is perpendicular to the N₂O₂ plane formed by the 5‐TMAMsalen tetradentate ligand. These Jahn–Teller axes are all perfectly aligned along the unique chain direction without a bending angle, although the chains are corrugated with an Mn‐N_axis‐C angle of about 144°. In the crystal structures, the chains are well separated with the nearest inter‐chain M???M distance being relatively large at 9 Å due to steric hindrance of the bulky trimethylammoniomethyl groups of the 5‐TMAMsalen ligand. The magnetic properties of these compounds have been thoroughly studied. Mn? Fe and Mn? Mn display intra‐chain ferromagnetic interactions, whereas Mn? Cr is characterized by an antiferromagnetic exchange that induces a ferrimagnetic spin arrangement along the chain. Detailed analyses of both static and dynamic magnetic properties have demonstrated without ambiguity the single‐chain magnet (SCM) behavior of these three systems, whereas Mn? Co is merely paramagnetic with S_Mn=2 and D/k_B=?5.3 K (D being a zero‐field splitting parameter). At low temperatures, the Mn? M compounds with M=Fe, Mn, and Cr display remarkably large M versus H hysteresis loops for applied magnetic fields along the easy magnetic direction that corresponds to the chain direction. The temperature dependence of the associated relaxation time for this series of compounds systematically exhibits a crossover between two Arrhenius laws corresponding to infinite‐chain and finite‐chain regimes for the SCM behavior. These isostructural hetero‐spin SCMs offer a unique series of alternating [‐Mn‐NC‐M‐CN‐] chains, enabling physicists to test theoretical SCM models between the Ising and Heisenberg limits.     

Self-quenched dynamics

We introduce a model for the slow relaxation of an energy landscape caused by its local interaction with a random walker whose motion is dictated by the landscape itself. By choosing relevant measures of time and potential this self-quenched dynamics can be mapped on to the “True” Self-Avoiding Walk model. This correspondence reveals that the average distance of the walker at time t from its starting point is , where for one dimension and 1/2 for all higher dimensions. Furthermore, the evolution of the landscape is similar to that in growth models with extremal dynamics. Received 8 August 2000     

Quantum Computation in a Ising Spin Chain Taking into Account Second Neighbor Couplings

We consider the realization of a quantum computer in a chain of nuclear spins coupled by an Ising interaction. Quantum algorithms can be performed with the help of appropriate radio-frequency pulses. In addition to the standard nearest-neighbor Ising coupling, we also allow for a second neighbor coupling. It is shown, how to apply the 2π k method in this more general setting, where the additional coupling eventually allows to save a few pulses. We illustrate our results with two numerical simulations: the Shor prime factorization of the number 4 and the teleportation of a qubit along a chain of 3 qubits. In both cases, the optimal Rabi frequency (to suppress non-resonant effects) depends primarily on the strength of the second neighbor interaction.     

Linked-Cluster Expansion of the Ising Model

The linked-cluster expansion technique for the high-temperature expansion of spin modes is reviewed. A new algorithm for the computation of three-point and higher Green's functions is presented. Series are computed for all components of two-point Green's functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice.