Here using the standard coordinate system of Euclidean 3-space [E.sup.3], a surface r(u, v) in [E.sup.3] will be written as r(u, v) = (x(u, v), y(u, v), z(u, v)).The surface which can be written as (1.2) is usually called translation surface in Euclidean 3-space [E.sup.3] (, ).
An affine translation surface in Euclidean 3-space [E.sup.3] is defined as a parameter surface r(u, v) in [E.sup.3] which can be written as
The constant c = 0 means that f" = g" = 0 and the affine translation surface r(u, v) is a plane.
Let r(x, y)=(x, y, z(x, y)) be a minimal affine translation surface. Then either z(x, y) is linear or can be written as
By comparing the two texts, the sociolinguistic virtues of Poniatowska's translation surface
, as do the cultural inflections of a Mexican Spanish translation.
Yoon, "Minimal translation surfaces
in [H.sup.2] x R" Taiwanese Journal of Mathematics, vol.
Affine translation surfaces
in Huili LIU and Yanhua YU Euclidean 3-space On the equivalence of several Shintaro KUROKI and Li YU definitions of compact infra- solvmanifolds Duc Tai PHO Alexander polynomials of certain Above three, communicated by dual of smooth quartics Kenji FUKAYA, M.J.A.
In particular Lopez  proved that the only minimal translation surfaces in hyperbolic space are totally geodesic planes.
Zafindratafa, A generalization of the translation surfaces of Scherk, Diff.
Jiu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J.
Lopez, Minimal translation surfaces in hyperbolic space, To appear in Contributions to Algebra and Geometry.
Some of the topics include ergodic natural measures, odometers and Toeplitz flows, an approach to non-singular entropy, the exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries, elements of the theory of unimodular Pisot substitutions with an application to beta-shifts, anisotropic Sobolev spaces and dynamical transfer operators, Hausdorff dimensions for Martin metrics, twistwise flow equivalence (and beyond), dynamical zeta functions and symplectic Floer homology, periodic geodesics on genetic translation surfaces
, actions in the infinite-dimensional nilpotent group, orbit counting, spaces of elliptic differentials, and interactions between dynamics, arithmetics and combinatorics (the good, the bad and the ugly).