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The 57-cell, also called the pentacontaheptachoron, is a regular self-dual locally projective polytope with 57 hemidodecahedral facets described by Coxeter (1982) and also ...

The (n,k)-arrangement graph A_(n,k) is defined as the graph on the vertex set consisting of the permutations of {1,2,...,n} containing at most k elements where vertices are ...

The roots of a semisimple Lie algebra g are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely ...

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthogonal if <v_i,v_j>=0 when i!=j. That is, the vectors are mutually perpendicular. Note ...

Marion's theorem (Mathematics Teacher 1993, Maushard 1994, Morgan 1994) states that the area of the central hexagonal region determined by trisection of each side of a ...

Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms Delta:H->H tensor _RH (1) (comultiplication) and ...

The standard Lorentzian inner product on R^4 is given by -dx_0^2+dx_1^2+dx_2^2+dx_3^2, (1) i.e., for vectors v=(v_0,v_1,v_2,v_3) and w=(w_0,w_1,w_2,w_3), ...

Abstract Algebra

An operator A^~ is said to be antiunitary if it satisfies: <A^~f_1|A^~f_2> = <f_1|f_2>^_ (1) A^~[f_1(x)+f_2(x)] = A^~f_1(x)+A^~f_2(x) (2) A^~cf(x) = c^_A^~f(x), (3) where ...

The study of the implications of chaos for a system in the semiclassical (i.e., between classical and quantum mechanical) regime. In quantum chaos, trajectories do not ...