对称等仿射球和极小对称Lagrange子流形的对应

利用对称空间的对偶性,本文建立局部强凸对称等仿射球之集与某复空间形式中的极小对称Lagrange子流形之集间的对应关系,在自然定义的等价意义下,这是一一对应关系.作为这种对应关系的直接应用,本文用完全不同的方法重新证明胡泽军等人最近建立的一个重要定理.该定理对具有平行Fubini-Pick形式的局部强凸等仿射球进行了完全分类.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

A characterization of Möbius transformations

Let be an integer and let be a domain of . Let be an injective mapping which takes hyperspheres whose interior is contained in to hyperspheres in . Then is the restriction of a Möbius transformation.

     

A generalization of the Carnahan–Starling approach with applications to four- and five-dimensional hard spheres

Development of good equations of state for hard spheres is an important task in the study of real fluids. In a way consistent with other theoretical results, we generalize the famous Carnahan–Starling approach for arbitrary dimensions and apply it to four- and five-dimensional hard spheres. We obtain simple and integer representations for virial coefficients of lower orders and accurate equations of state. Since theoretically and practically validated, these results improve understanding of hard sphere fluids.     

Graph immersions with parallel cubic form

We consider non-degenerate graph immersions into affine space ${\mathbb{A}}^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,\gamma )$, where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.     

Characterization of the generalized Calabi composition of affine hyperspheres

In this paper, continuing with Hu–Li–Vrancken and the recent work of Anti′c–DillenSchoels–Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex affine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.     

Gheorghe Ţiţeica and the origins of affine differential geometry

We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced by ?i?eica in 1907, and this discovery has been viewed by many as a consequence of Klein's Erlangen program. We thus present the starting point of affine differential geometry, as it was discovered by ?i?eica after his years in the Ph.D. program in Paris (1896–1899) under the guidance of Gaston Darboux.     

Monte Carlo Simulation of Hard Hyperspheres in Six,Seven and Eight Dimensions for Low to Moderate Densities

The pair correlation function of hard hyperspheres in six, seven and eight dimensions is obtained from Monte Carlo simulations. The value of the pair correlation function at contact is compared with the results from molecular dynamics calculations and a variety of theoretical approaches. Remarkably good agreement is found with the simple, closed-form equations of Y. Song, E. A. Mason and R. M. Stratt, J. Phys. Chem., 93:6916–6919 (1989). The Monte Carlo results for the equation of state are compared with the theoretical expressions of M. Baus and J. L. Colot, Phys. Rev. A, 36:3912 (1987), M. Luban and J. P. J. Michels, Phys. Rev A, 41:6796 (1990), and high order virial expansions. In addition, in seven dimensions, comparisons are made with the exact PY solution provided by M. Robles, M. L. de Haro and A. Santos, J. Chem. Phys., 120:9113 (2004). Very good agreement was observed between theory and computer simulation in all dimensions.     

An Extremal Class of Three-Dimensional Hyperbolic Affine Spheres

In analogy to an inequality of Chen [2], Scharlach and co-workers [7] have found a new, optimal inequality for (equi-) affine spheres. We classify those three-dimensional hyperbolic affine spheres for which the corresponding equality is assumed. This complements the classification of the elliptic case [3].