The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

We review part of the classical theory of curves and surfaces in 3-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

/pdf/differential-geometry-of-curves-and-surfaces-in-lorentz-5f7aizouyb.pdf

Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space

We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E3 and 3-dimensional Minkowski space E
 1
 3
 .

Translation surfaces with constant mean curvature in 3-dimensional spaces

Associated curves of a Frenet curve and their applications

In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space $${\mathbb{E}}^{3}$$
 and Minkowski space $${\mathbb{E}}^{3}_{1}$$
 , respectively. Some examples are also given.

/pdf/mannheim-partner-curves-in-3-space-1qlwyv5hd5.pdf

Mannheim partner curves in 3-space ∗

In this paper, we study curves in the lightlike cone. We first obtain the conformally invariant arc length in the (n+1)-dimensional lightlike cone and then characterize some curves in the 2-dimensional lightlike cone and 3-dimensional lightlike cone.

/pdf/curves-in-the-lightlike-cone-3ijbqst0ui.pdf

Curves in the Lightlike Cone

In this paper we study an interesting operator, the Tchebychev operator, which has properties similar to the shape operator in Euclidean differential geometry and equiaftine differential geometry, respectively.

The Centroaffine Tchebychev Operator

We introduce the concept of a relative Tchebychev hypersurface which extends that of affine spheres in equiaffine geometry and also that of centroaffine Tchebychev hypersurfaces and give partial local and global classifications for this new class. Our tools concern a new operator and interesting properties of the traceless part of the cubic form.

Huili Liu

Papers

Mannheim partner curves in 3-space ∗

Translation surfaces with constant mean curvature in 3-dimensional spaces

Curves in the Lightlike Cone

The Centroaffine Tchebychev Operator

Cubic Form Methods and Relative Tchebychev Hypersurfaces