Direction-preserving function

In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacent points. It can be considered a discrete analogue of a continuous function. The concept was first defined by Iimura. Some variants of it were later defined by Yang, Chen and Deng, Herings, van-der-Laan, Talman and Yang, and others.

In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacent points. It can be considered a discrete analogue of a continuous function. The concept was first defined by Iimura. Some variants of it were later defined by Yang, Chen and Deng, Herings, van-der-Laan, Talman and Yang, and others. Specific definitions are presented below. All examples below are for n=2 dimensions and for X = { (2,6), (2,7), (3, 6), (3, 7) }. Some authors use a variant requiring that, for any pair of cell-connected points x,y in X, for all i\in [n] : (f_i(x) - x_i)\cdot (f_i(y)-y_i) \geq 0 . A function f(x) is HDP by the second variant, iff the function g(x):=f(x)-x is HDP by the first variant. f is called hypercubic gross direction preserving (HGDP), or locally gross direction preserving (LGDP), if for any pair of cell-connected points x,y in X, f(x)\cdot f(y) \geq 0 . Every HDP function is HGDP, but the converse is not true. The function f b is HGDP, since the scalar product of every two vectors in the table is non-negative. But it is not HDP, since the second component switches sign between (2,6) and (3,6): f^b_2(2,6)\cdot f^b_2(3,6) = -1 . Some authors use a variant requiring that, for any pair of cell-connected points x,y in X, (f(x)-x)\cdot (f(y)-y) \geq 0 . A function f(x) is HGDP by the second variant, iff the function g(x):=f(x)-x is HGDP by the first variant. Every HGDP function is SGDP, but HGDP is much stronger: it is equivalent to SGDP w.r.t. all possible integral triangulations of ch(X), whereas SGDP relates to a single triangulation. As an example, the function f c on the right is SGDP by the triangulation that partitions the cell into the two triangles {(2,6),(2,7),(3,7)} and {(2,6),(3,6),(3,7)}, since in each triangle, the scalar product of every two vectors is non-negative. But it is not HGDP, since f^c(3,6)\cdot f^c(2,7) = -1 .

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