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Let x:p(x)->xp(x), then for any operator T, T^'=Tx-xT is called the Pincherle derivative of T. If T is a shift-invariant operator, then its Pincherle derivative is also a ...

D_P(x)=lim_(epsilon->0)(lnmu(B_epsilon(x)))/(lnepsilon), where B_epsilon(x) is an n-dimensional ball of radius epsilon centered at x and mu is the probability measure.

Rényi entropy is defined as: H_alpha(p_1,p_2,...,p_n)=1/(1-alpha)ln(sum_(i=1)^np_i^alpha), where alpha>0, alpha!=1. As alpha->1, H_alpha(p_1,p_2,...,p_n) converges to ...

A Cantor set C in R^3 is said to be scrawny if for each neighborhood U of an arbitrary point p in C, there is a neighborhood V of p such that every map f:S^1->V subset C ...

A set function is a function whose domain is a collection of sets. In many instances in real analysis, a set function is a function which associates an affinely extended real ...

A function f(x) is said to be square integrable if int_(-infty)^infty|f(x)|^2dx is finite.

The topological entropy of a map M is defined as h_T(M)=sup_({W_i})h(M,{W_i}), where {W_i} is a partition of a bounded region W containing a probability measure which is ...

Differential Geometry

The term Borel hierarchy is used to describe a collection of subsets of R defined inductively as follows: Level one consists of all open and closed subsets of R, and upon ...

A subset F subset R of the real numbers is said to be an F_sigma set provided F is the countable union of closed sets. The name F_sigma comes from French: The F stands for ...