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A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on ...

A proof of a formula on limits based on the epsilon-delta definition. An example is the following proof that every linear function f(x)=ax+b (a,b in R,a!=0) is continuous at ...

Erdős offered a $3000 prize for a proof of the proposition that "If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic ...

The Euler triangle formula states that the distance d between the incenter and circumcenter of a triangle is given by d^2=R(R-2r), where R is the circumradius and r is the ...

The extended mean-value theorem (Anton 1984, pp. 543-544), also known as the Cauchy mean-value theorem (Anton 1984, pp. 543) and Cauchy's mean-value formula (Apostol 1967, p. ...

Faà di Bruno's formula gives an explicit equation for the nth derivative of the composition f(g(t)). If f(t) and g(t) are functions for which all necessary derivatives are ...

Gauss's continued fraction is given by the continued fraction ...

Gauss's forward formula is f_p=f_0+pdelta_(1/2)+G_2delta_0^2+G_3delta_(1/2)^3+G_4delta_0^4+G_5delta_(1/2)^5+..., (1) for p in [0,1], where delta is the central difference and ...

Consider a clause (disjunction of literals) obtained from those of a first-order logic formula Phi in Skolemized form forall x_1... forall x_nS. Then an atomic statement ...

Let sumu_k be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for u_k. If f is decreasing and continuous for ...