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A nilpotent Lie group is a Lie group G which is connected and whose Lie algebra is a nilpotent Lie algebra g. That is, its Lie algebra lower central series ...

A quantity that is nonzero everywhere is said to be nonvanishing. For instance, the values of x^2+1 are nonvanishing for real x, while those of x^2 are not (since x^2 ...

A sequence of n 0s and 1s is called an odd sequence if each of the n sums sum_(i=1)^(n-k)a_ia_(i+k) for k=0, 1, ..., n-1 is odd.

Let alpha and beta be any ordinal numbers, then ordinal exponentiation is defined so that if beta=0 then alpha^beta=1. If beta is not a limit ordinal, then choose gamma such ...

Two representations of a group chi_i and chi_j are said to be orthogonal if sum_(R)chi_i(R)chi_j(R)=0 for i!=j, where the sum is over all elements R of the representation.

Two vectors u and v whose dot product is u·v=0 (i.e., the vectors are perpendicular) are said to be orthogonal. In three-space, three vectors can be mutually perpendicular.

If f(x) is a monotonically increasing integrable function on [a,b] with f(b)<=0, then if g is a real function integrable on [a,b], ...

A fixed point of a linear transformation for which the rescaled variables satisfy (delta-alpha)^2+4betagamma=0.

The eigenvalues lambda satisfying P(lambda)=0, where P(lambda) is the characteristic polynomial, lie in the unions of the disks |z|<=1 |z+b_1|<=sum_(j=1)^n|b_j|.

The Poisson integral with n=0, J_0(z)=1/piint_0^picos(zcostheta)dtheta, where J_0(z) is a Bessel function of the first kind.