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The process of finding a reduced set of basis vectors for a given lattice having certain special properties. Lattice reduction algorithms are used in a number of modern ...

A Lorentz transformation is a four-dimensional transformation x^('mu)=Lambda^mu_nux^nu, (1) satisfied by all four-vectors x^nu, where Lambda^mu_nu is a so-called Lorentz ...

Let B_t={B_t(omega)/omega in Omega}, t>=0, be one-dimensional Brownian motion. Integration with respect to B_t was defined by Itô (1951). A basic result of the theory is that ...

The total number of contravariant and covariant indices of a tensor. The rank R of a tensor is independent of the number of dimensions N of the underlying space. An intuitive ...

Transitivity is a result of the symmetry in the group. A group G is called transitive if its group action (understood to be a subgroup of a permutation group on a set Omega) ...

There exist polyhedra which can be plaited (braided). Examples include a plaited cube and plaited icosahedron illustrated above (Pargeter 1959, Wells 1991). In the above ...

Consider n intersecting circles. The maximal number of regions into which these divide the plane are N(n)=n^2-n+2, giving values for n=1, 2, ... of 2, 4, 8, 14, 22, 32, 44, ...

The number of ways N(m,n) of finding a subrectangle with an m×n rectangle can be computed by counting the number of ways in which the upper right-hand corner can be selected ...

The number of staircase walks on a grid with m horizontal lines and n vertical lines is given by (m+n; m)=((m+n)!)/(m!n!) (Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch ...

Triangle picking is the process of picking triangles at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a random ...