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Lattice path of length 5 in ℤ 2 with S = { (2,0), (1,1), (0,-1) }.. In combinatorics, a lattice path L in the d-dimensional integer lattice of length k with steps in the set S, is a sequence of vectors ,, …, such that each consecutive difference lies in S. [1]
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding ...
The (large) Schröder numbers count both types of paths, and the little Schröder numbers count only the paths that only touch the diagonal but have no movements along it. [ 3 ] Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the x {\displaystyle x} -axis.
The Narayana numbers also count the number of lattice paths from (,) to (,), with steps only northeast and southeast, not straying below the x-axis, with peaks. The following figures represent the Narayana numbers N ( 4 , k ) {\displaystyle \operatorname {N} (4,k)} , illustrating the above mentioned symmetries.
Laue equation. In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice. They are named after physicist Max von Laue (1879–1960).
An n-path from an n-tuple (,, …,) of vertices of G to an n-tuple (,, …,) of vertices of G will mean an n-tuple (,, …,) of paths in G, with each leading from to . This n -path will be called non-intersecting just in case the paths P i and P j have no two vertices in common (including endpoints) whenever i ≠ j {\displaystyle i\neq j} .
The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice. In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space.
The Born–Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918 [ 1 ] Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.
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