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A fiber of a map f:X->Y is the preimage of an element y in Y. That is, f^(-1)(y)={x in X:f(x)=y}. For instance, let X and Y be the complex numbers C. When f(z)=z^2, every ...

An infinite sequence of homomorphisms of modules or additive Abelian groups ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->... (1) such that, for all indices i in Z, ...

The partial order width of a set P is equal to the minimum number of chains needed to cover P. Equivalently, if a set P of ab+1 elements is partially ordered, then P contains ...

Let a^p+b^p=c^p be a solution to Fermat's last theorem. Then the corresponding Frey curve is y^2=x(x-a^p)(x+b^p). (1) Ribet (1990a) showed that such curves cannot be modular, ...

Let gamma be a path in C, w=f(z), and theta and phi be the tangents to the curves gamma and f(gamma) at z_0 and w_0. If there is an N such that f^((N))(z_0) != 0 (1) ...

In general, there is no unique matrix solution A to the matrix equation y=Ax. Even in the case of y parallel to x, there are still multiple matrices that perform this ...

A Gaussian integer is a complex number a+bi where a and b are integers. The Gaussian integers are members of the imaginary quadratic field Q(sqrt(-1)) and form a ring often ...

Let A be a matrix and x and b vectors. Then the system Ax=b, x>=0 has no solution iff the system A^(T)y>=0, b^(T)y<0 has a solution, where y is a vector (Fang and Puthenpura ...

The correspondence which relates the Hanoi graph to the isomorphic graph of the odd binomial coefficients in Pascal's triangle, where the adjacencies are determined by ...

A collection of moves together with a corresponding set of weights which are followed probabilistically in the playing of a game. The minimax theorem of game theory states ...