Simplification and normalization of indexed differentials involving coordinate transformation

In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulation of such indexed functions is one of the oldest research topics in computer algebra, so far there exists no normal form reduction algorithm to judge whether two indexed polynomials involving indices of different coordinate systems are equal or not. It is a challenging task in computer algebra...     

A survey of average case complexity for linear multivariate problems

We survey recent results on the average case complexity for linear multivariate problems. Our emphasis is on problems defined on spaces of functions of d variables with large d. We present the sharp order of the average case complexity for a number of linear multivariate problems as well as necessary and sufficient conditions for the average case complexity not to be exponential in d. Dedicated to the 50th anniversary of the journal. The text was submitted by the authors in English.     

Cartan connection and its geodesics on a manifold of nondegenerate affinor fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.     

Estimation of spinodals from the density profile of the vapor–liquid interface

An extension of a recently proposed method for the calculation of the spinodals in pure fluid systems from the interfacial properties is elaborated, which requires the density profile as only input. The foundation of this approach is the so-called Fuchs-transformation which gives an estimate for the tangential pressure profile from the density profile. Using molecular dynamics simulation data for argon and carbon dioxide as well as lattice Boltzmann simulation data for the argon-like Shan–Chen fluid, the accuracy of the approach is analyzed. The Fuchs-transformation is qualitative, however it is possible to estimate the temperature–density projection of the spinodal. Depending on the underlying correlation function for the interfacial density profile reasonable results are obtained for the liquid and the vapor spinodal. The advantage of this method is that equilibrium data can be used to estimate the spinodal which is experimentally impossible to access because it is a highly non-equilibrium property. In the final consequence of this approach only the coexistence vapor and liquid densities are required to estimate the temperature–density projection of the spinodals.     

Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in is an infinite absolute convex combination of points of the form with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of . D. García and M. Maestre were supported by MEC and FEDER Project MTM2005-08210. B. C. Grecu wishes to acknowledge the financial support of a Marie Curie Intra European Fellowship (MEIF-CT-2005-006958).     

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from its Satake diagram, in a way that is suited for the use with computer algebra systems; an example implementation for Maple Version 10 can be found on . As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified.      

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

复合材料刚度性能表征的协同损伤力学模型

针对复合材料层合板的弥散型损伤,提出一个刚度性能表征的协同损伤力学模型. 该模型兼顾了微观物理损伤响应和宏观材料刚度性能表征. 从微观角度,建立细观RVE 模型求解裂纹表面张开位移和滑开位移,以此定义损伤张量,并在宏观上通过对材料应变和损伤表面位移进行均匀化处理,建立单向板或层合板的损伤刚度矩阵和损伤张量之间的联系. 以基体裂纹为例,详细分析并建立了横向裂纹和纵向裂纹的损伤本构. 计算了[±θ/90₄]_S 铺层层合板中基体横向裂纹对刚度性能的影响,结果表明该方法能够准确地预测复合材料层合板由损伤导致的刚度性能衰减.      

关于一族模的任意重张量积的实性

引进一族模的任意重张量积的概念.通过建立一个充分必要条件,模的任意重张量积的半实性被得到刻画。此外,本文给出了一族模的张量积具有序的一些充分必要条件。     

Pseudo anti‐commuting Ricci tensor and Ricci soliton real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti commuting Ricci tensor for real hypersurfaces in complex hyperbolic two‐plane Grassmannians and prove a complete classification theorem that such a hypersurface must be a tube over a totally real totally geodesic , , a horosphere whose center at the infinity is singular or an exceptional case.