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The dot product can be defined for two vectors X and Y by X·Y=|X||Y|costheta, (1) where theta is the angle between the vectors and |X| is the norm. It follows immediately ...

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy [f,f]=0 (1) [f+g,h]=[f,h]+[g,h], (2) and ...

The Smale horseshoe map consists of a sequence of operations on the unit square. First, stretch in the y direction by more than a factor of two, then compress in the x ...

A variable is a symbol on whose value a function, polynomial, etc., depends. For example, the variables in the function f(x,y) are x and y. A function having a single ...

The circle through the cusp of the arbelos and the tangent points of the first Pappus circle, which is congruent to the two Archimedes' circles. If AB=r and AC=1, then the ...

A graph with graph edges of two possible "colors," usually identified as red and blue. For a bichromatic graph with R red graph edges and B blue graph edges, R+B>=2.

A finite or infinite square matrix with rational entries. (If the matrix is infinite, all but a finite number of entries in each row must be 0.) The sum or product of two ...

A paradox arising in the use of comparative adjectives. Suppose you have exactly two brothers, both of whom are older than you are. Then the following apparently false ...

Also called a chain map. Given two chain complexes C_* and D_*, a chain homomorphism is given by homomorphisms alpha_i:C_i->D_i such that alpha degreespartial_C=partial_D ...

Two angles alpha and beta are said to be complementary if alpha+beta=pi/2. In other words, alpha and beta are complementary angles if they produce a right angle when combined.