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In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space.
For a 3-dimensional lattice, the steps are analogous, but in step 2 instead of drawing perpendicular lines, perpendicular planes are drawn at the midpoint of the lines between the lattice points. As in the case of all primitive cells, all area or space within the lattice can be filled by Wigner–Seitz cells and there will be no gaps.
The history of aperiodic crystals can be traced back to the early 20th century, when the science of X-ray crystallography was in its infancy. At that time, it was generally accepted that the ground state of matter was always an ideal crystal with three-dimensional space group symmetry, or lattice periodicity.
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators.It was proposed by Pascual Jordan and Eugene Wigner [1] for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created.
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E 8 lattice and the Leech lattice are two famous examples.
A three-dimensional lattice filled with two molecules A and B, here shown as black and white spheres. Lattices such as this are used - for example - in the Flory–Huggins solution theory In mathematical physics , a lattice model is a mathematical model of a physical system that is defined on a lattice , as opposed to a continuum , such as the ...
For example, a crystal, viewed as a lattice with a single kind of atom located at every lattice point (the simplest basis form), may also be viewed as a lattice with a basis of two atoms. In this case, a primitive unit cell is a unit cell having only one lattice point in the first way of describing the crystal in order to ensure the smallest ...
The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality. [3]Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon [4] proved that the two point spin correlation of the ferromagnetics XY model in dimension D, coupling J > 0 and inverse temperature β is dominated by (i.e ...