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A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ...

It is always possible to "fairly" divide a cake among n people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes ...

Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces. The group is most commonly N, R, R-+, ...

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces. For example, the orthogonal group O(n) has an irreducible ...

The lines containing the three points of the intersection of the three pairs of opposite sides of a (not necessarily regular) hexagon. There are 6! (i.e., 6 factorial) ...

Propositional calculus is the formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies." Many systems of propositional ...

Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an ...

If f is continuous on a closed interval [a,b], then there is at least one number x^* in [a,b] such that int_a^bf(x)dx=f(x^*)(b-a). The average value of the function (f^_) on ...

Two open manifolds M and M^' are cobordant if there exists a manifold with boundary W^(n+1) such that an acceptable restrictive relationship holds.

An immediate consequence of a result already proved. Corollaries usually state more complicated theorems in a language simpler to use and apply.