A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL ^d lattice scales like . I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.     

On a point of order

A method of using algebraic curves to obtain estimates of critical points accurate enough to identify them as simple algebraic numbers (if that is what they are) is discussed and illustrated with an application to the (q-state Potts model on the triangular lattice for cases of pure two-site interactions and pure three-site interactions. In the latter case the critical point is conjectured to be . In a similar conjecture for the critical percolation probability on thedirected square lattice,q _c ^1/2 (q _c +3)=2(q _c +p _c =1).     

Hamiltonian limit of the 3D Zamolodchikov model

A two-dimensional quantum Hamiltonian _N,M commuting with the layer-to-layer transfer matrix of the three-dimensional Zamolodchikov model is derived. This Hamiltonian is defined on a lattice ofN×M sites. The special casesN×2, 2×M, and 3×M are studied.This paper is dedicated to Cyril Domb.     

血液净化用吸附剂脱附过程的数学模型研究

对已吸附了低密度脂蛋白（ＬＤＬ）及极低密度脂蛋白（ＶＬＤＬ）达到饱和程度的吸附柱，用１ｍｏｌ／Ｌ的生理盐水进行脱附，采用修正Ｌａｎｇｍｕｉｒ吸附式建立了连续脱附过程的数学模型，计算结果与实验测定值吻合，表明该数学模型适用于描述类似于ＬＤＬ及ＶＬＤＬ这类蛋白质大分子的脱附过程。     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

A class of interaction-round-a-face models and its equivalence with an ice-type model

A new model (called the Temperley-Lieb interactions model) is introduced, in two-dimensional lattice statistics, on a square lattice . The Temperley-Lieb equivalence of this model to the six-vertex, self-dual Potts, critical hard-hexagons and critical nonintersecting string models is established. A graphical equivalence of this model to the six-vertex model generalizes this equivalence to noncritical cases of the above models. The order parameters of a specialization of this model are studied.     

Selection of the number of regression variables; A minimax choice of generalized FPE

Summary A generalized Final Prediction Error (FPE_α)_ criterion is considered. Based onn observations, the numberk of regression variables is selected from a given range 0≦k≦K, so as to minimize . It is shown that if α tends to infinity withn, the selection is consistent but the maximum of the mean squared error of estimates of parameters diverges to infinity with the same order of divergence as that of α. A meaningful minimax choice of α exists for a regret type mean squared error, while for simple mean squared error it is trivially 0. The minimax regret choice of α converges to a constant, approximately 3.5 forK≧8 ifn−K increases simultaneously withn, otherwise it diverges to infinity withn.     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

Sampling and local models for multivariate image analysis

Local models are a very important concept for microscopic and macroscopic imaging. Different methods of sub-sampling a multivariate image are described both in general and for three examples. The need for sub-sampling and its influence on multivariate image analysis and visualization are studied. Examples from MRI (256 × 256), satellite imaging (7 × 512 × 512) and biofuel studies (6 × 512 × 512) are used to illustrate some of the principles involved.     

Photoelectron Spectra of Nonmetal Compounds and Their Interpretation by MO Models

The results of (low energy) photoelectron spectroscopy render possible a better appreciation of the “Nature of the Chemical Bond”. The application of this new experimental method is demonstrated utilizing representative compounds of the nonmetal elements, and a close symbiosis delineated with molecular orbital models. In particular, general consequences are discussed concerning electron deficiency, σ- and π-interactions, electron pair delocalization, and substituent effects or geometric perturbations. Photoelectron spectroscopic ionization energies permit evaluation of parameters for specified molecular groups, allow correlation with numerous other experimental data, and are didactically valuable in the teaching of general chemistry.