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The Weyl group of SO(2n + 1) is the semidirect product {} of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1} n acts on the corresponding circle factor of T × {1} by inversion, and the symmetric group S n acts on both {±1} n and T × {1} by permuting factors.
In mechanics and geometry, the 3D rotation group, often denoted SO (3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [1] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation ...
The first four partial sums of the series 1 + 2 + 3 ... The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 ...
Spin group. In mathematics the spin group, denoted Spin (n), [1][2] is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO (n) = SO (n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2) The group multiplication law on the double cover is given by lifting the multiplication on .
The group SO(1, 1) may be identified with the group of unit split-complex numbers. In terms of being a group of Lie type – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are Chevalley groups , while the non-split orthogonal groups require a slightly more complicated construction, and are Steinberg groups .
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
Power of two. A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent. Powers of two with non-negative exponents are integers: 20 = 1, 21 = 2, and 2n is two multiplied by itself n times. [1][2] The first ten powers of 2 for non-negative ...
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.