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A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1g_2)=f(g_1)f(g_2) for all g_1,g_2 in G, where the product on the ...

A (2n)×(2n) complex matrix A in C^(2n×2n) is said to be Hamiltonian if J_nA=(J_nA)^(H), (1) where J_n in R^(2n×2n) is the matrix of the form J_n=[0 I_n; I_n 0], (2) I_n is ...

Two topological spaces X and Y are homotopy equivalent if there exist continuous maps f:X->Y and g:Y->X, such that the composition f degreesg is homotopic to the identity ...

A matrix whose entries are all integers. Special cases which arise frequently are those having only (-1,1) as entries (e.g., Hadamard matrix), (0,1)-matrices having only ...

"The" Jacobi identity is a relationship [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0,, (1) between three elements A, B, and C, where [A,B] is the commutator. The elements of a Lie algebra ...

A nonassociative algebra named after physicist Pascual Jordan which satisfies xy=yx (1) and (xx)(xy)=x((xx)y)). (2) The latter is equivalent to the so-called Jordan identity ...

A representation of a Lie algebra g is a linear transformation psi:g->M(V), where M(V) is the set of all linear transformations of a vector space V. In particular, if V=R^n, ...

Given a map f from a space X to a space Y and another map g from a space Z to a space Y, a lift is a map h from X to Z such that gh=f. In other words, a lift of f is a map h ...

A monoid is a set that is closed under an associative binary operation and has an identity element I in S such that for all a in S, Ia=aI=a. Note that unlike a group, its ...

cos(20 degrees)cos(40 degrees)cos(80 degrees)=1/8. An identity communicated to Feynman as a child by a boy named Morrie Jacobs (Gleick 1992, p. 47). Feynman remembered this ...