Dynamical systems theoryDynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.
Dynamical systemIn mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.
Leonhard EulerLeonhard Euler (ˈɔɪlər , ˈleːɔnhaʁt ˈɔʏlɐ; 15 April 1707 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function.
Euler–Lagrange equationIn the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it.
Newton's laws of motionNewton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force. When a body is acted upon by a force, the time rate of change of its momentum equals the force. If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.
Roman provinceThe Roman provinces (Latin: provincia, pl. provinciae) were the administrative regions of Ancient Rome outside Roman Italy that were controlled by the Romans under the Roman Republic and later the Roman Empire. Each province was ruled by a Roman appointed as governor. For centuries it was the largest administrative unit of the foreign possessions of ancient Rome. With the administrative reform initiated by Diocletian, it became a third level administrative subdivision of the Roman Empire, or rather a subdivision of the imperial dioceses (in turn subdivisions of the imperial prefectures).
Roman armyThe Roman army (Latin: exercitus Romanus) was the armed forces deployed by the Romans throughout the duration of Ancient Rome, from the Roman Kingdom (753 BC–509 BC) to the Roman Republic (509 BC–27 BC) and the Roman Empire (27 BC–476 AD), and its medieval continuation, the Eastern Roman Empire. It is thus a term that may span approximately 2,205 years (753 BC–1453 AD), during which the Roman armed forces underwent numerous permutations in size, composition, organisation, equipment and tactics, while conserving a core of lasting traditions.
Roman legionThe Roman legion (legiō, ˈɫɛɡioː), the largest military unit of the Roman army, comprised 5,200 infantry and 300 equites (cavalry) in the period of the Roman Republic (509 BC–27 BC) and 5,600 infantry and 200 auxilia in the period of the Roman Empire (27 BC – AD 476). The size of a typical legion varied throughout the history of ancient Rome, with complements ranging from 4,200 legionaries and 300 equites (drawn from the wealthier classes – in early Rome all troops provided their own equipment) in the Republic, to 5,500 in the Imperial period.
Roman mythologyRoman mythology is the body of myths of ancient Rome as represented in the literature and visual arts of the Romans. One of a wide variety of genres of Roman folklore, Roman mythology may also refer to the modern study of these representations, and to the subject matter as represented in the literature and art of other cultures in any period. Roman mythology draws from the mythology of the Italic peoples and ultimately from Proto-Indo-European mythology.
Roman Britain'Roman Britain' was the territory that became the Roman Province of Britannia after the Roman conquest of Britain, consisting of a large part of the island of Great Britain. The occupation lasted from AD 43 to AD 410. Julius Caesar invaded Britain in 55 and 54 BC as part of his Gallic Wars. According to Caesar, the Britons had been overrun or culturally assimilated by Belgic tribes during the British Iron Age and had been aiding Caesar's enemies. The Belgae were the only Celtic tribe to cross the sea into Britain, for to all other Celtic tribes this land was unknown.
Roman citizenshipCitizenship in ancient Rome (Latin: civitas) was a privileged political and legal status afforded to free individuals with respect to laws, property, and governance. Citizenship in Ancient Rome was complex and based upon many different laws, traditions, and cultural practices. There existed several different types of citizenship, determined by one's gender, class, and political affiliations, and the exact duties or expectations of a citizen varied throughout the history of the Roman Empire.
