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The brachistocrone (from the Greek, brakhisto meaning “shortest” and chronos meaning “time”), is the planar curve on which a body subjected only to the force of gravity slides without friction between two points in the least possible time. Finding the curve was a problem first posed by Galileo (1564–1642). In June 1696 the Swiss mathematician Johann Bernoulli (1667–1748), father of Daniel (1700–1782), another famous Bernoulli, who was the first to find the correct solution in 1696, challenged his contemporaries in Acta Eruditorum to find the solution. Correct solutions came from his brother Jakob (1654–1705), Gottfried Wilhelm von Leibniz (1646–1716), Guillaume de l’Hôpital (1661–1704), and Isaac Newton (1643–1727). The solution of this famous problem led to the development of the calculus of variations.

It is common practice to represent numbers using positional number systems. In such a system, each number has a unique representation by an ordered sequence of symbols, the value of the number being determined by the position of the symbols and the base b of the system. If b is a positive integer we need b different symbols which are digits if 2 ≤ b ≤ 10 and extra symbols if b ≻ 10. A number N _b is then represented by the sequence of digits d _n d _n−1... d 1 d _0 such that N _ b = d _ n × b _ n + d _ n −1× b ^ n −1+...+ b _1× b + d _0 The concept of base can be extended to negative and even complex bases.

If f and g are two functions defined on the semi-infinite interval [0, ∞[, the convolution of these two functions is defined by $$ \int_0^t {f(t - u} )g(u) du. $$ .

A double pendulum is a simple dynamical system that exhibits a complex dynamical behavior. It consists of two mass points at the end of rigid massless rods, one suspended from a fixed pivot and the second one suspended from the bob of the first. The system is free to oscillate in a vertical plane.

The Duffing equation is $$ x^{11} - ax + bx^3 = 0, $$ where a and b ≻ 0 are two parameters. Depending upon the sign of a we have either a single-well ( a ≺ 0) or a double-well ( a ≻ 0) anharmonic potential V defined by $$ V(x) = - \frac{a} {2}x^2 + \frac{b} {4}x^4 . $$ We can ask Mathematica to plot the potential when a is either negative or positive.

In 1858 the Scottish antiquarian Alexander Henry Rhind (1833–1863), traveling in Egypt, bought in Luxor an ancient scroll that has been the source of much information about Egyptian mathematics. This important document, known as the Rhind Mathematical Papyrus , contains, in particular, tables to help find a representation of rational numbers less than one as sums of different unit fractions, that is, with numerators equal to 1, as, for example, $$ \frac{2} {{29}} = \frac{1} {{24}} + \frac{1} {{58}} + \frac{1} {{174}} + \frac{1} {{232}}. $$ .

Electostatics is the study of time-independent distributions of electric charges.

The stars appear to move in circles about a line through the poles of the earth as if they were attached to a sphere rotating about the earth.

Fractals are exotic sets that first appeared in the mathematical literature at the end of the 19th century. They were devised by Georg Ferdinand Ludwig Philipp Cantor (1845–1918), Giuseppe Peano (1858–1932), David Hilbert (1862–1943), Henri Léon Lebesgue (1875–1941), Arnaud Denjoy (1884–1974), George Pólya (1887–1985), Wacław Sierpiński (1882–1969), and many others. There is no precise definition but most authors agree to call fractals sets possessing certain characteristic properties such as self-similarity illustrated in the examples presented below. The idea of self-similarity originated implicitly in a paper of Niels Fabian Helge von Koch (1870–1924) (see the von Koch curve below), and was formulated explicitly by Ernesto Cesàro (1859–1906). The word fractal was coined by Benoît Mandelbrot (born 1924) who wrote a few books [34,35] and many articles on fractal geometry, drawing attention to its relevance in such diverse fields as fluid mechanics, geomorphology, economics, and linguistics.

Let { f _i | 1 ≤ i ≤ n } be a finite set of n mappings from a complete metric space (E, d) onto itself; { p _i | 1 ≤ i ≤ n } a discrete probability distribution, that is, a set of nonnegative real numbers such that $$ \sum\nolimits_{i = 1}^n {pi} = 1 $$ ; and F : E : E the mapping defined by x : F ( x ) = f _i( x ) with probability p _i. The dynamical system (( E,d ), F ) is called an iterated function system or IFS .