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Take K a number field and m a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m (I_K^m) that contains ...

On the class of topological spaces, a homeomorphism class is an equivalence class under the relation of being homeomorphic. For example, the open interval (-pi/2,pi/2) and ...

Asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence relation.

Given two topological spaces M and N, place an equivalence relationship on the continuous maps f:M->N using homotopies, and write f_1∼f_2 if f_1 is homotopic to f_2. Roughly ...

The tangent space at a point p in an abstract manifold M can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called ...

Given any set B, the associated pair groupoid is the set B×B with the maps alpha(a,b)=a and beta(a,b)=b, and multiplication (a,b)·(b,c)=(a,c). The inverse is ...

If A=>B and B=>A (i.e., A=>B ^ B=>A, where => denotes implies), then A and B are said to be equivalent, a relationship which is written symbolically in this work as A=B. The ...

Two elements alpha, beta of a field K, which is an extension field of a field F, are called conjugate (over F) if they are both algebraic over F and have the same minimal ...

Given an expression involving known constants, integration in finite terms, computation of limits, etc., the constant problem is the determination of if the expression is ...

A map is called bijective if it is both injective and surjective. A bijective map is also called a bijection. A function f admits an inverse f^(-1) (i.e., "f is invertible") ...