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.

Linda Dalrymple Henderson picks up on Duchamp's idea of inventing a " playful physics " and traces a quirky Victorian physics out of the notes and The Large Glass itself ; numerous mathematical and philosophical systems have been read out of ( or perhaps into ) its structures.

Large number co-incidences continue to fascinate many mathematical physicists.

Large eddy simulation ( LES ) is a mathematical model for turbulence used in computational fluid dynamics.

Although modern scholars have expressed surprise that `` the simple magic square of three '', a mere `` mathematical puzzle '', was able to exert a considerable influence on the minds and imaginations of the cultured Chinese for so many centuries, they could have found most of the answers right within the square itself.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

In the 1980s, Per Martin-Löf developed intuitionistic type theory ( also called Constructive type theory ), which associated functional programs with constructive proofs of arbitrarily complex mathematical propositions expressed as dependent types.

* In mathematical logic and computer science, lambda is used to introduce an anonymous function expressed with the concepts of lambda calculus.

In the computer science subfields of computer-aided design and computer graphics, the term B-spline frequently refers to a spline curve parametrized by spline functions that are expressed as linear combinations of B-splines ( in the mathematical sense above ).

This axis may be expressed in either mathematical terms ( event occurs once a year, once in ten years, once in 100 years etc.

" If we mistakenly confuse precision with accuracy, then we might be misled into thinking that an explanation expressed in precise mathematical or graphical terms is somehow more rigorous or useful than one that takes into account particulars of history, institutions or business strategy.

Theories in various fields of study are expressed in natural language, but are always constructed in such a way that their general form is identical to a theory as it is expressed in the formal language of mathematical logic.

For instance, the transform of a musical chord made up of pure notes ( without overtones ) expressed as amplitude as a function of time, is a mathematical representation of the amplitudes and phases of the individual notes that make it up.

This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical induction is expressed as an infinite set of axioms ( an axiom schema ).

Although to date, no single theory has successfully expressed these relationships in viable mathematical or physical terms.

His most important mathematical result, published in 1955, is the so-called " Lah identity " where he shows how the rising powers can be expressed in terms of falling powers.

Some of these are really coded mathematical problems, expressed using the geometry and pieces of the chessboard.

* Denotational semantics, in which the meaning of a system is expressed in the mathematical theory of domains.

By this Polykleitos meant that a statue should be composed of clearly definable parts, all related to one another through a system of ideal mathematical proportions and balance, no doubt expressed in terms of the ratios established by Pythagoras for the perfect intervals of the musical scale: 1: 2 ( octave ), 2: 3 ( harmonic fifth ), and 3: 4 ( harmonic fourth ).

Liu Hui expressed all of his mathematical results in the form of decimal fractions ( using metrological units ), yet the later Yang Hui ( c. 1238-1298 AD ) expressed his mathematical results in full decimal expressions.

During the modern evolutionary synthesis, these ideas were purged, and only evolutionary causes that could be expressed in the mathematical framework of population genetics were retained.

At least one mathematical model has been developed where both codon-usage and tRNA-expression co-evolve in feedback fashion ( i. e., codons already present in high frequencies drive up the expression of their corresponding tRNAs, and tRNAs normally expressed at high levels drive up the frequency of their corresponding codons ), however this model does not seem to yet have experimental confirmation.

Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols.

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole " charge " g. The Dirac string starts from, and terminates on, a magnetic monopole.

This corrected function, sometimes known as a " special field emission elliptic function ", was expressed as a function of a mathematical variable y known as the " Nordheim parameter ".