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Isolated resonances in a dynamical system can cause considerable distortion of preserved tori in their neighborhood, but they do not introduce any chaos into a system. ...

The root separation (or zero separation) of a polynomial P(x) with roots r_1, r_2, ... is defined by Delta(P)=min_(i!=j)|r_i-r_j|. There are lower bounds on how close two ...

An n-dimensional open disk of radius r is the collection of points of distance less than r from a fixed point in Euclidean n-space. Krantz (1999, p. 3) uses the symbol D(x,r) ...

The term cube is used in topology to denote the Cartesian product of any (finite or infinite) number of copies of the closed interval [0,1] equipped with the product topology ...

If f is a continuous real-valued function on [a,b] and if any epsilon>0 is given, then there exists a polynomial p on [a,b] such that |f(x)-P(x)|<epsilon for all x in [a,b]. ...

Topology

Dini's theorem is a result in real analysis relating pointwise convergence of sequences of functions to uniform convergence on a closed interval. For an increasing sequence ...

If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the ...

Calculus I

Inverse function integration is an indefinite integration technique. While simple, it is an interesting application of integration by parts. If f and f^(-1) are inverses of ...