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The scalar form of Laplace's equation is the partial differential equation del ^2psi=0, (1) where del ^2 is the Laplacian. Note that the operator del ^2 is commonly written ...

In simple terms, let x, y, and z be members of an algebra. Then the algebra is said to be associative if x·(y·z)=(x·y)·z, (1) where · denotes multiplication. More formally, ...

If F(x) is a probability distribution with zero mean and rho=int_(-infty)^infty|x|^3dF(x)<infty, (1) where the above integral is a stieltjes integral, then for all x and n, ...

Let lambda be (possibly complex) eigenvalues of a set of random n×n real matrices with entries independent and taken from a standard normal distribution. Then as n->infty, ...

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was ...

An alternating group is a group of even permutations on a set of length n, denoted A_n or Alt(n) (Scott 1987, p. 267). Alternating groups are therefore permutation groups. ...

The identity element of an additive group G, usually denoted 0. In the additive group of vectors, the additive identity is the zero vector 0, in the additive group of ...

Let A be a matrix and x and b vectors. Then the system Ax=b, x>=0 has no solution iff the system A^(T)y>=0, b^(T)y<0 has a solution, where y is a vector (Fang and Puthenpura ...

An algebraic ring which appears in treatments of duality in algebraic geometry. Let A be a local Artinian ring with m subset A its maximal ideal. Then A is a Gorenstein ring ...

An inner product space is a vector space together with an inner product on it. If the inner product defines a complete metric, then the inner product space is called a ...