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Let q be a positive integer, then Gamma_0(q) is defined as the set of all matrices [a b; c d] in the modular group Gamma Gamma with c=0 (mod q). Gamma_0(q) is a subgroup of ...

Let X be a set. Then a sigma-algebra F is a nonempty collection of subsets of X such that the following hold: 1. X is in F. 2. If A is in F, then so is the complement of A. ...

A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose ...

If sets E and F are independent, then so are E and F^', where F^' is the complement of F (i.e., the set of all possible outcomes not contained in F). Let union denote "or" ...

A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q!=0. A rational number p/q is said to have numerator p and denominator ...

Let O be an incidence geometry, i.e., a set with a symmetric, reflexive binary relation I. Let e and f be elements of O. Let an incidence plane be an incidence geometry whose ...

In nonstandard analysis, the limitation to first-order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the ...

Mergelyan's theorem can be stated as follows (Krantz 1999). Let K subset= C be compact and suppose C^*\K has only finitely many connected components. If f in C(K) is ...

A member of a collection of sets is said to be maximal if it cannot be expanded to another member by addition of any element. Maximal sets are important in graph theory since ...

Let C=C^+ union C^- (where C^+ intersection C^-=emptyset) be the disjoint union of two finite components C^+ and C^-. Let alpha and beta be two involutions on C, each of ...