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Transfinite induction, like regular induction, is used to show a property P(n) holds for all numbers n. The essential difference is that regular induction is restricted to ...

Transitivity is a result of the symmetry in the group. A group G is called transitive if its group action (understood to be a subgroup of a permutation group on a set Omega) ...

Let A={A_1,A_2,...,A_n} be a union-closed set, then the union-closed set conjecture states that an element exists which belongs to at least n/2 of the sets in A. Sarvate and ...

The unknot, also called the trivial knot (Rolfsen 1976, p. 51), is a closed loop that is not knotted. In the 1930s Reidemeister first proved that knots exist which are ...

The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function psi and a constant vector c such that M = del x(cpsi) (1) = psi(del ...

A schematic diagram used in logic theory to depict collections of sets and represent their relationships. The Venn diagrams on two and three sets are illustrated above. The ...

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as ...

The quasiperiodic function defined by d/(dz)lnsigma(z;g_2,g_3)=zeta(z;g_2,g_3), (1) where zeta(z;g_2,g_3) is the Weierstrass zeta function and lim_(z->0)(sigma(z))/z=1. (2) ...

A pair of closed form functions (F,G) is said to be a Wilf-Zeilberger pair if F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k). (1) The Wilf-Zeilberger formalism provides succinct proofs of ...

Calculus I