0_Wizard
I always though I would enjoy studying archeology.
Archaeology, or archeology, is the study of human activity through the recovery and analysis of material culture.
chosen_Archaeology_0
Archaeology
chosen_topic_0_Archaeology
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.
Itô calculus
Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where "H" is a locally square-integrable process adapted to the filtration generated by "X" , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process.
Fourier transform
The Fourier transform (FT) decomposes a function of time (a "signal") into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the "frequency domain representation" of the original signal. The term "Fourier transform" refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.
Locally integrable function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions. Let be an open set in the Euclidean space and be a Lebesgue measurable function.
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. In the general theory of differential systems, there is "Frobenius integrability", which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical systems, there is the notion of "Liouville integrability". This causes the trajectories to be fixed to smaller submanifolds allowing the solution to be expressed with a sequence of integrals (the origin of the name integrable). More generally, in differentiable dynamical systems integrability relates to the existence of foliations by invariant submanifolds within the phase space.
Henstock–Kurzweil integral
In mathematics, the Henstock–Kurzweil integral or gauge integral (also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral) is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like  This function has a singularity at 0, and is not Lebesgue integrable.
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called "indefinite integral"), say "F", of some function "f" may be obtained as the integral of "f" with a variable bound of integration. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function "f" over some interval can be computed by using any one, say "F", of its infinitely many antiderivatives.