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If p_1, ..., p_n are positive numbers which sum to 1 and f is a real continuous function that is convex, then f(sum_(i=1)^np_ix_i)<=sum_(i=1)^np_if(x_i). (1) If f is concave, ...

A univariate function f(x) is said to be odd provided that f(-x)=-f(x). Geometrically, such functions are symmetric about the origin. Examples of odd functions include x, ...

The signature of a non-degenerate quadratic form Q=y_1^2+y_2^2+...+y_p^2-y_(p+1)^2-y_(p+2)^2-...-y_r^2 of rank r is most often defined to be the ordered pair (p,q)=(p,r-p) of ...

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an ...

Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass ...

Let two points x and y be picked randomly from a unit n-dimensional hypercube. The expected distance between the points Delta(n), i.e., the mean line segment length, is then ...

A perspective collineation with center O and axis o not incident is called a geometric homology. A geometric homology is said to be harmonic if the points A and A^' on a line ...

Zeno's paradoxes are a set of four paradoxes dealing with counterintuitive aspects of continuous space and time. 1. Dichotomy paradox: Before an object can travel a given ...

Let {a_i}_(i=1)^n be a set of positive numbers. Then sum_(i=1)^n(a_1a_2...a_i)^(1/i)<=esum_(i=1)^na_i (which is given incorrectly in Gradshteyn and Ryzhik 2000). Here, the ...

The curve formed by the intersection of a cylinder and a sphere is known as Viviani's curve. The problem of finding the lateral surface area of a cylinder of radius r ...