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Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

Place 2n balls in a bag and number them 1 to 2n, then pick half of them at random. The number of different possible sums for n=1, 2, 3, ... are then 2, 5, 10, 17, 26, ... ...

The number sign # sometimes used in mathematics to indicate the number of a quantity satisfying some condition, e.g., #{n:n>1}. The symbol is also used to denote a primorial.

The illusion illustrated above in which the bounding rectangle and inner square both appear distorted.

Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate.

A function which has infinitely many derivatives at a point. If a function is not polygenic, it is monogenic.

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...

A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in a neighborhood of every point.

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of ...

S_n(z) = zj_n(z)=sqrt((piz)/2)J_(n+1/2)(z) (1) C_n(z) = -zn_n(z)=-sqrt((piz)/2)N_(n+1/2)(z), (2) where j_n(z) and n_n(z) are spherical Bessel functions of the first and ...