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The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of a topological space X is ...

Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms Delta:H->H tensor _RH (1) (comultiplication) and ...

The study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations. Analytic geometry ...

Two elements x and y of a set S are said to be commutative under a binary operation * if they satisfy x*y=y*x. (1) Real numbers are commutative under addition x+y=y+x (2) and ...

An early name for calculus of variations. The term is also sometimes used in place of predicate calculus.

Given three circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy (ace)/(bdf)=1 (Honsberger 1995).

A quantified system of real algebraic equations and inequalities in variables {x_1,...,x_n} is an expression QS=Q_1(y_1)Q_2(y_2)...Q_m(y_m)S(x_1,...,x_n;y_1,...,y_m), where Q ...

The triquetra is a geometric figure consisting of three mutually intersecting vesica piscis lens shapes, as illustrated above. The central region common to all three lenses ...

A polygon whose interior consists of all points in the plane which are closer to a particular lattice point than to any other. The generalization to n dimensions is called a ...

The arf invariant is a link invariant that always has the value 0 or 1. A knot has Arf invariant 0 if the knot is "pass equivalent" to the unknot and 1 if it is pass ...