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The nth cubic number n^3 is a sum of n consecutive odd numbers, for example 1^3 = 1 (1) 2^3 = 3+5 (2) 3^3 = 7+9+11 (3) 4^3 = 13+15+17+19, (4) etc. This identity follows from ...

The geometry of the Lie group consisting of real matrices of the form [1 x y; 0 1 z; 0 0 1], i.e., the Heisenberg group.

Let N be a nilpotent, connected, simply connected Lie group, and let D be a discrete subgroup of N with compact right quotient space. Then N/D is called a nilmanifold.

An algebra, also called a nilalgebra, consisting only of nilpotent Elements.

A group G is nilpotent if the upper central sequence 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... of the group terminates with Z_n=G for some n. Nilpotent groups have the property that ...

There are two equivalent definitions for a nilpotent matrix. 1. A square matrix whose eigenvalues are all 0. 2. A square matrix A such that A^n is the zero matrix 0 for some ...

The set of nilpotent elements in a commutative ring is an ideal, and it is called the nilradical. Another equivalent description is that it is the intersection of the prime ...

A pile of counters in a game of nim. The nim-heap illustrated above corresponds to the game of Marienbad.

A conic section on which the midpoints of the sides of any complete quadrangle lie. The three diagonal points P, Q, and R also lie on this conic.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.