Vertical versus horizontal Sobolev spaces

Vertical perimeter versus horizontal perimeter

Given k ∈ N, the k’th discrete Heisenberg group, denoted H Z , is the group generated by the elements a1, b1, . . . , ak, bk, c, subject to the commutator relations [a1, b1] = . . . = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ {1, . . . , k} we have [ai, aj ] = [bi, bj ] = [ai, bj ] = [ai, c] = [bi, c] = ...

Horizontal versus vertical plate motions

We review both present and past motions at major plate boundaries, which have the horizontal component in average 10 to 100 times faster (10–100 mm/yr) than the vertical component (0.01–1 mm/yr) in all geodynamic settings. The steady faster horizontal velocity of the lithosphere with respect to the upward or downward velocities at plate 5 boundaries supports dominating tangential forces acting ...

Sobolev Spaces

Complementarity among species in horizontal versus vertical rooting space

Aims: Many experiments have shown a positive effect of species richness on productivity in grassland plant communities. However, it is poorly understood how environmental conditions affect this relationship. We aimed to test whether deep soil and limiting nutrient conditions increase the complementarity effect (CE) of species richness due to enhanced potential for resource partitioning. Methods...

Vertical versus Horizontal Incentives in Education: Evidence from Randomized Trials

This paper describes randomized field experiments in eighty-four urban public schools in two cities designed to understand the impact of aligned incentives on student achievement. In Washington DC, incentives were “horizontal” – provided to one agent (students) for various inputs in the education production function (i.e. attendance, behavior, interim assessments, homework, and uniforms). In Ho...