These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

J. Desaulx suggested in 1877 that the phenomenon was caused by the thermal motion of water molecules, and in 1905 Albert Einstein produced the first mathematical analysis of the motion.

Grothendieck's early mathematical work was in functional analysis.

The use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations form the scientific basis for heavier-than-air flight and a number of other technologies.

B *- algebras were mathematical structures studied in functional analysis.

A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

In addition to mathematical analysis of cryptographic algorithms, cryptanalysis also includes the study of side-channel attacks that do not target weaknesses in the cryptographic algorithms themselves, but instead exploit weaknesses in their implementation.

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

A system can be mechanical, electrical, fluid, chemical, financial and even biological, and the mathematical modeling, analysis and controller design uses control theory in one or many of the time, frequency and complex-s domains, depending on the nature of the design problem.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

In mathematical analysis, a metric space M is called complete ( or a Cauchy space ) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it.

The mathematical study of Diophantine problems Diophantus initiated is now called " Diophantine analysis ".

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function.

Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure ( e. g. inner product, norm, topology, etc.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

He worked on a great variety of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.

Galileo, however, felt that the descriptive content of the technical disciplines warranted philosophical interest, particularly because mathematical analysis of astronomical observations — notably the radical analysis offered by astronomer Nicolaus Copernicus concerning the relative motions of the Sun, Earth, Moon, and planets — indicated that philosophers ' statements about the nature of the universe could be shown to be in error.

He advised the doctoral thesis of several important German mathematicians, as Gotthold Eisenstein, Leopold Kronecker, Rudolf Lipschitz and Carl Wilhelm Borchardt, while being influential in the mathematical formation of many other scientists, including Elwin Bruno Christoffel, Wilhelm Weber, Eduard Heine, Ludwig von Seidel and Julius Weingarten.

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity.

In mathematical analysis, semi-continuity ( or semicontinuity ) is a property of extended real-valued functions that is weaker than continuity.

A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.

A continuity equation is the mathematical way to express this kind of statement.

Other equations in physics, such as Gauss's law of the electric field and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually called by the term " continuity equation ", because j in those cases does not represent the flow of a real physical quantity.

The mathematical study of continuity was revived in the fourteenth century by the Oxford Calculators and French collaborators such as Nicole Oresme.

The situation of Buridan's ass was given a mathematical basis in a 1984 paper by American computer scientist Leslie Lamport, in which Lamport presents an argument that, given certain assumptions about continuity in a simple mathematical model of the Buridan's ass problem, there will always be some starting conditions under which the ass will starve to death, no matter what strategy it takes.

In mathematical analysis, a modulus of continuity is a function

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity.

These paradoxes are sometimes seen as relating to Zeno's paradoxes which all deal with the physical manifestations of mathematical properties of continuity, infinitesimals, and infinities often associated with space and time.

In particular, this field contains all the numbers named in the mathematical constants article, and all algebraic numbers ( and therefore all rational numbers ).

The Dyson series, the formal solution of an explicitly time-dependent Schrödinger equation by iteration, and the corresponding Dyson time-ordering operator an entity of basic importance in the mathematical formulation of quantum mechanics, are also named after Dyson.

* Josephus problem — a mathematical problem named after Josephus.

Albert Einstein's mathematical description of how the photoelectric effect was caused by absorption of quanta of light ( now called photons ), was in one of his 1905 papers, named " On a Heuristic Viewpoint Concerning the Production and Transformation of Light ".

A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star ( Kleene closure ), Kleene's recursion theorem and the Kleene fixpoint theorem.

Some of the notable mathematical concepts named after Banach include Banach spaces, Banach algebras, the Banach – Tarski paradox, the Hahn – Banach theorem, the Banach – Steinhaus theorem, the Banach-Mazur game, the Banach – Alaoglu theorem and the Banach fixed-point theorem.

The De Bruijn – Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function H ( λ, z ), where λ is a real parameter and z is a complex variable.

The Fourier transform, named after Joseph Fourier, is a mathematical transform with many applications in physics and engineering.

MATHLAB (" mathematical laboratory ") should not be confused with MATLAB (" matrix laboratory ") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly.

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states.

A simple mathematical representation of Brownian motion, the Wiener equation, named after Wiener, assumes the current velocity of a fluid particle fluctuates.

Wiener took a great interest in the mathematical theory of Brownian motion ( named after Robert Brown ) proving many results now widely known such as the non-differentiability of the paths.

In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:

Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville:

" It has been known that fleas do not use muscle power but energy stored in a protein named resilin but the researchers used high-speed video technology and mathematical models to discover where the spring action actually happens.

In mathematical optimization, the method of Lagrange multipliers ( named after Joseph Louis Lagrange ) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener, assumes the current velocity of a fluid particle fluctuates randomly:

In the mathematical areas of order and lattice theory, the Knaster – Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:

In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of L < sup > p </ sup > spaces.

More than 180 named mathematical facts have so benefited from formal codification.

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form.

Also in 1996 he was named research vice president in charge of computing and mathematical sciences and, additionally, in 1997, chief technical officer for Lucent ’ s Communications Software Group.

In computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge.