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The French mathematician Gaston Julia (1893–1978) published his famous Mémoire sur l’itération des fonctions rationelles in 1918 [26]. He was 25. The paper investigates the behavior of the iterates z _ n +1 = f ( z _n), where f is a rational function and, in particular, the set of points whose nth iterates remain bounded as n tends to infinity.

Johannes Kepler (1571–1630) came from a modest family but, thanks to a scholarship, he could attend the Lutheran Seminary at the University of Tübingen. At Tübingen, Kepler was taught astronomy by one of the leading astronomers of the day, Michael Maestlin (1550–1631). Maestlin lent Kepler his own annotated copy of Copernicus’ book. Kepler quickly grasped the essential ideas of the Copernican system and built up a cosmological theory based on the five regular polyhedra. He collaborated with the Danish astronomer Tycho Brahe (1546–1601) who was one of the most prominent observational astronomer of the time, and succeeded him, when he died in 1601, as Imperial Mathematician (they were both living in the Holy Roman Empire). Among his many achievements, Kepler was probably the first to explicitly use the concept of observational error. He is chiefly known for the three laws bearing his name which Isaac Newton (1643–1727) was able to derive from his gravitational law. The first two laws were published in Astronomia Nova (A New Astronomy) in 1609, and the third law in Harmonices Mundi (The Harmony of the World) in 1619.

Lindenmayer systems (usually referred to as L-systems) are rewriting techniques developed by the biologist Aristid Lindenmayer (1925–1989) in 1968. They are actually symbolic dynamical systems whose evolution can be represented graphically. Thanks to this feature, they have found several applications in computer graphics such as the generation of fractals and realistic models of plants.

The discrete logistic map is the simplest nonlinear map. Its properties are, however, far from being trivial. In 1962, Myrberg [39] already mentioned the existence of considerable difficulties: En nous limitant dans notre travail au cas le plus simple non linéaire, c’est-à-dire aux polynômes réels du second degré, nous observons que même dans ce cas spécial on rencontre des difficultés considerables, dont l’explication exigera des recherches ultérieures. Limiting ourselves to the simplest nonlinear case, that is, to quadratic real polynomials, it is observed that even in this special case considerable difficulties are encountered, whose explanation will require more work in the future .

In his historical paper, published in 1963, Lorenz [31] derived, from a model of fluid convection, a three-parameter family of three ordinary differential equations that appeared, when integrated numerically, to have extremely complicated solutions. In particular, he discovered that all nonperiodic solutions of his deterministic model were bounded but showed irregular fluctuations. Thirty years later [32] he described how he had come across a phenomenon that later came to be called “chaos” — — seemingly random and unpredictable behavior that nevertheless proceeds according to precise and often easily expressed rules .

In a paper published in July 1929 Philip Morse [38] used the potential $$ V(r) = De^{ - 2a(r - r_0 )} - 2De^{ - a(r - r_0 )} $$ to model the vibrational energy of a diatomic molecule, where r is the bond length, r _0 the equilibrium bond length, a a parameter controlling the potential width, and D the dissociation energy of the molecule measured from the potential minimum.

A positive integer p is a prime (number) if, and only if, it has only two distinct divisors: 1 and p itself. The only divisors of 227 are 1 and 227 making 227 a prime whereas 237, which has four distinct divisors 1, 3, 79, and 237, is not a prime.

Public-key encryption is a cryptographic system that allows users to communicate securely without having prior access to a shared secret key. It uses two keys: a public key known to everyone and a private or secret key known only to the recipient of the message. These two keys are related mathematically.

In 430 BC, Hippias (460 BC–400 BC) of Elis (in the Peloponnese, Greece), a contemporary of Socrates, discovered the quadratrix, a curve he used for trisecting an angle. As a matter of fact, the quadratrix may be used for dividing an angle into any number of equal parts. In 350 BC Dinostratus (390BC–320BC) used the quadratrix to square the circle.^1

After having finished his chemistry studies, Erwin Schrödinger (1887–1961) devoted himself to Italian painting for many years and then took up botany and published a series of papers on plant phylogeny. During the years 1906 to 1910, as a student at the University of Vienna, he was greatly influenced by Fritz Hasenöhrl’s lectures on theoretical physics. He then acquired a mastery of eigenvalue problems in the physics of continuous media, thus laying the foundation of his future important work. Moving very often, he occupied many academic positions starting as assistant to Max Wien (1866–1938). His most fruitful period took place when he replaced Max von Laue (1879–1960) at the University of Zürich, where he enjoyed contacts, in particular, with Hermann Weyl (1885–1955) who was to provide the deep mathematical knowledge that would prove so helpful to Schrödinger. Having never been very satisfied by the quantum condition on orbits in Niels Bohr’s (1885–1962) atomic model, he believed that atomic spectra should be determined by some kind of eigenvalue problem. In 1926, he discovered the wave equation that bears his name. In 1933, “for the discovery of new productive forms of atomic theory,” he shared with Paul Adrien Dirac (1902–1984) the Nobel Prize in Physics.