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An interesting example is the modular group = (): it acts on the tree given by the 1-skeleton of the associated tessellation of the hyperbolic plane and it has a finite index free subgroup (on two generators) of index 6 (for example the set of matrices in which reduce to the identity modulo 2 is such a group).
The group GL(2, Z) is the linear maps preserving the standard lattice Z 2, and SL(2, Z) is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the torus (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group of the torus, meaning that ...
This is based on the fact that a reciprocal lattice vector (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Fundamental parallelogram defined by a pair of vectors in the complex plane.
For example, the set of Miller indices (110) describes the set of parallel planes (family of lattice planes) parallel to the z-axis and cutting the x- and the y-axis once, such that every unit cell is bisected by precisely one of those planes in the x- and y-direction. An example of two surfaces with different surface free energies.
In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices). [1]
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
In situations where a balanced network is not appropriate, a single ended circuit operating with a ground plane is required. In such cases, the conversion of a lattice into a bridged T circuit is carried out, as described in the article Lattice network. The resulting unbalanced network has the same electrical characteristics as the balanced ...