In mathematics, especially in geometry and group theory, a lattice in is a discrete subgroup of which spans the real vector space . Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell.
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "frame work" of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.
Read more about Lattice (group): Symmetry Considerations and Examples, Dividing Space According To A Lattice, Lattice Points in Convex Sets, Computing With Lattices, Lattices in Two Dimensions: Detailed Discussion, Lattices in Three Dimensions, Lattices in Complex Space, In Lie Groups, Lattices in General Vector-spaces
Other articles related to "lattices, lattice, group":
... Whilst we normally consider latticesin this concept can be generalized to any finite dimensional vector space over any field ... Then the R latticein V generated by B is given by Different bases B will in general generate different lattices ... However, if the transition matrix T between the bases is in - the general linear groupof R (in simple terms this means that all the entries of T are in R and all the entries of are in R - which is ...