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Let X and Y be Banach spaces and let f:X->Y be a function between them. f is said to be Gâteaux differentiable if there exists an operator T_x:X->Y such that, for all v in X, ...

A graph or directed graph together with a function which assigns a positive real number to each edge (Harary 1994, p. 52).

A phase portrait is a plot of multiple phase curves corresponding to different initial conditions in the same phase plane (Tabor 1989, p. 14). Phase portraits for simple ...

Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the differential ...

A vector field is a map f:R^n|->R^n that assigns each x a vector f(x). Several vector fields are illustrated above. A vector field is uniquely specified by giving its ...

Percolation, the fundamental notion at the heart of percolation theory, is a difficult idea to define precisely though it is quite easy to describe qualitatively. From the ...

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston's geometrization conjecture in ...

The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a C^1 ...

Let f(x,y)=u(x,y)+iv(x,y), (1) where z=x+iy, (2) so dz=dx+idy. (3) The total derivative of f with respect to z is then (df)/(dz) = ...

Newton's method for finding roots of a complex polynomial f entails iterating the function z-[f(z)/f^'(z)], which can be viewed as applying the Euler backward method with ...