Dupin hypersurfaces with four principal Curvatures,II

If M is an isoparametric hypersurface in a sphere S ⁿ with four distinct principal curvatures, then the principal curvatures κ₁, . . . , κ₄ can be ordered so that their multiplicities satisfy m ₁ = m ₂ and m ₃ = m ₄, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R ⁿ (or S ⁿ ) with four distinct principal curvatures with multiplicities m ₁ = m ₂ ≥ 1 and m ₃ = m ₄ = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.      

Pairwise Correlations in a Multicomponent Anisotropic Liquid

We investigate a multicomponent anisotropic liquid system. The first spatial moment of the direct correlation function is taken into account to obtain asymptotic expressions for the pairwise correlation functions. In this approximation, we obtain the pairwise correlation functions that describe the system behavior not only in the critical-state neighborhood but also in the noncritical domain. We show that the critical parameters for the anisotropic system differ from those for the isotropic system.     

Homogeneity and Canonical Connections of Isoparametric Manifolds

On every isoparametric submanifold M a connection with parallel second fundamental form is constructed geometrically such that M is an orbit of an s-representation if and only if the connection is a canonical one. If the rank of M is greater than one this connection is in case of homogeneity the canonical connection of the reductive decomposition given by the orbit of s-representation. This revised version was published online in June 2006 with corrections to the Cover Date.     

De Sitter空间中具有常均曲率的类空超曲面（Ⅱ）

设Ｍ＾ｎ是Ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１中具有常数平均曲率且第二基本形式长度的平方为常数的完备类空超曲面，若Ｓ≤２（ｎ－１）＾１／２，则Ｍ＾ｎ是等参超曲面。     

An algorithm for determining dynamically defined reaction paths (DDRP)

Summary A numerically stable and well-parallelizable curve variational algorithm is described for determining tangent curves of vector fields between two given stationary points. In particular, the method is suitable for finding reaction paths and saddle points on potential energy hypersurfaces (PHS). The stability of the procedure is illustrated by an artificial mathematical function, showing phases of following the reaction on the PHS.Dedicated to Professor Zoltán G. Szabó, the great teacher and scientist in reaction kinetics and in many other fields of physical chemistry, on his 84th birthday.     

Crystal Structures of Twisted Di(2-Pyridyl)ketone and Planar Azo-di(2-Pyridine) and Their Discussion Based on PM3 Enthalpy Hypersurfaces for the Isoelectronic Molecules

Single crystal structure determinations prove the two pyridine substituents in di(2-pyridyl)ketone (H₄C₄NC)₂C=O to be twisted out of the carbonyl skeleton plane by torsion angles (OCCN) of 41° and –163°, in contrast to their planar arrangement in azo-di(2-pyridine) H₄C₄NC)-N=N-(CNC₄H₄). In order to rationalize the surprising difference between the two isoelectronic molecules, approximate PM3 enthalpy of formation hypersurfaces have been calculated for each of the two ring torsions, which are assumed to be the dominant ones among the 3N – 6 = 60 degrees of freedom. For both the ketone and the azo derivative, global minima are calculated, the torsion angles of which deviate from the crystal structure results, and, therefore, support the assumption that both the experimentally determined twisting of di(2-pyridyl)ketone as well as the flattening of azo-di(2-pyridine) might be affected by the crystal packing.     

The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a Rippled surface are presented.     

Phenomenology of nonlocal cellular automata

Dynamical systems with nonlocal connections have potential applications to economic and biological systems. This paper studies the dynamics of nonlocal cellular automata. In particular, all two-state, three-input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings, although it is observed that sometimes a rule can have different dynamical behaviors with different wirings. The nonlocal cellular automata rule space is studied using a mean-field parametrization which is ideal for the situation of random wiring. Nonlocal cellular automata can be considered as computers carrying out computation at the level of each component. Their computational abilities are studied from the point of view of whether they contain many basic logical gates. In particular, I ask the question of whether a three-input cellular automaton rule contains the three fundamental logical gates: two-input rules AND and OR, and one-input rule NOT. A particularly interesting edge-of-chaos nonlocal cellular automaton, the rule 184, is studied in detail. It is a system of coupled selectors or multiplexers. It is also part of the Fredkin's gate—a proposed fundamental gate for conservative computations. This rule exhibits irregular fluctuations of density, large coherent structures, and long transient times.     

Willmore submanifolds in the unit sphere via isoparametric functions

In this paper, we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore, hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.     

THE GENERALIZED MAXIMUM ANGLE CONDITION FOR THE Q1 ISOPARAMETRIC ELEMENT

We consider the quadrilateral Q1 isoparametric element and establish an optimal error estimate in H^1 norm for the interpolation operator under a weaker mesh condition which admits anisotropic quadrilaterals and allows the quadrilateral to become a regular triangle in the sense of maximum angle condition [5, 11].