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For a real number x, the mantissa is defined as the positive fractional part x-|_x_|=frac(x), where |_x_| denotes the floor function. For example, for x=3.14159, the mantissa ...

A set of maximum degree to which all other degrees of recursively enumerable sets can be many-one reduced. If set A is many-one complete, then it is one-one complete, and ...

A map u:R^n->R^n from a domain G is called a map of class C^r if each component of u(x)=(u_1(x_1,...,x_n),...,u_m(x_1,...,x_n)) is of class C^r (0<=r<=infty or r=omega) in G, ...

The winding number W(theta) of a map f(theta) with initial value theta is defined by W(theta)=lim_(n->infty)(f^n(theta)-theta)/n, which represents the average increase in the ...

Let Y^X be the set of continuous mappings f:X->Y. Then the topological space for Y^X supplied with a compact-open topology is called a mapping space.

A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.

The Diophantine equation x^2+y^2+z^2=3xyz. The Markov numbers m are the union of the solutions (x,y,z) to this equation and are related to Lagrange numbers.

A type I move (conjugation) takes AB->BA for A, B in B_n where B_n is a braid group. A type II move (stabilization) takes A->Ab_n or A->Ab_n^(-1) for A in B_n, and b_n, Ab_n, ...

A random process whose future probabilities are determined by its most recent values. A stochastic process x(t) is called Markov if for every n and t_1<t_2...<t_n, we have ...

A sequence X_1, X_2, ... of random variates is called Markov (or Markoff) if, for any n, F(X_n|X_(n-1),X_(n-2),...,X_1)=F(X_n|X_(n-1)), i.e., if the conditional distribution ...