De Lempicka was criticized and admired for her ' perverse Ingrism ', referring to her modern restatement of the master Jean Auguste Dominique Ingres, as displayed in her work Group of Four Nudes, 1925.

Finally, it is a basic tool for proving theorems in modern number theory, such as Lagrange's four-square theorem and the fundamental theorem of arithmetic ( unique factorization ).

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

This modern form of Stokes ' theorem is a vast generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850.

This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory.

Bayes ' theorem was named after the Reverend Thomas Bayes ( 1702 – 61 ), who studied how to compute a distribution for the probability parameter of a binomial distribution ( in modern terminology ).

Bertrand considers only special cases of problems from electrodynamics and heat conduction, but his article contains in distinct terms all basic ideas of modern proof of the pi – theorem and clear indication of the use of the pi – theorem for modelling physical phenomena.

The statement that ƒ can be reconstructed from ƒ ̂ is known as the Fourier integral theorem, and was first introduced in Fourier's Analytical Theory of Heat,, although what would be considered a proof by modern standards was not given until much later.

In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal theory.

From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold.

Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations.

Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used version of Noether's theorem.

Arguments for the mean speed theorem ( above ) require the modern mathematical concept of limit, so Bradwardine had to use arguments of his day.

The above equation is a modern statement of the theorem.

The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions.

Propositions 30 and 32 together are essentially equivalent to the fundamental theorem of arithmetic stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.

In modern mathematical language, the theorem may be stated as follows:

E is a modern, high performance theorem prover for full first-order logic with equality.

The Modigliani – Miller theorem ( of Franco Modigliani, Merton Miller ) forms the basis for modern thinking on capital structure.

The chord function can be related to the modern sine function, by taking one of the points to be ( 1, 0 ), and the other point to be ( cos, sin ), and then using the Pythagorean theorem to calculate the chord length:

Based on modern thermodynamics, Carnot's theorem is a result of the second law of thermodynamics.

Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.

Alexander Grothendieck (; ; born 28 March 1928 ) is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry.

His conjectural theory of motives has been a driving force behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.

Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

In the modern times, the works by J. J. Sylvester ( late 19th century ) and Percy MacMahon ( early 20th century ) laid the foundation for enumerative and algebraic combinatorics.

In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.

This led to modern abstract algebraic notions such as Euclidean domains.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

In modern language, what Diophantus does is to find rational parametrisations of many varieties ; in other words, he shows how to obtain infinitely many rational numbers satisfying a system of equations by giving a procedure that can be made into an algebraic expression

The name " transcendental " comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x. Euler was probably the first person to define transcendental numbers in the modern sense.

It is an important tool that underlies several modern developments in algebraic geometry and representation theory.

However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article .< ref > A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud ( 1995 ), which uses " h < sub > A </ sub >" to mean the covariant hom-functor.

A modern, geometrically flavored introduction to algebraic topology.

In modern commentary, the columns ( called files ) are labeled by the letters a to h from left to right from the white player's point of view, and the rows ( called ranks ) by the numbers 1 to 8, with 1 being closest to the white player, thus providing a standard notation called algebraic chess notation.

In algebraic notation, the usual modern way of recording chess games, the letter N stands for the knight ( K is reserved for the king ); in descriptive chess notation, Kt is sometimes used instead, mainly in older literature.

In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes.

Beyond Thom's contributions in algebraic topology, his influence on modern differential geometry, through the intensive study of generic properties, can hardly be exaggerated.

The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology.

And when he retold the legend of Gregorius he interpolated a modern version in which the medieval players speak contemporary thoughts in archaic language ; ;

According to Akbar S. Ahmed, like modern anthropologists, he engaged in extensive participant observation with a given group of people, learnt their language and studied their primary texts, and presented his findings with objectivity and neutrality using cross-cultural comparisons.

* Salvatore Iro, artist, author of Manifesto Art is ... Shit, a collection of works conceived to decrypt the language of modern and contemporary art, in 2012 creates the artwork Ceci est une pomme, marked by the subheading: " non aristotelian obviousness ".

Altogether, Starostin concluded that the Altaic grouping was substantiated, though " older than most other language families in Eurasia, such as Indo-European or Finno-Ugric, and this is the reason why the modern Altaic languages preserve few common elements ".

The use of a single official language, which modern scholarship has dubbed Official Aramaic or Imperial Aramaic, can be assumed to have greatly contributed to the astonishing success of the Achaemenids in holding their far-flung empire together for as long as they did.

Avicenna created new scientific vocabulary that had not existed before in the modern Persian language.

The programming language to be employed by users was akin to modern day assembly languages.

The modern computer programming language Ada is named in her honour.

Although there have been attempts to show that the Ainu language and the Japanese language are related, modern scholars have rejected that the relationship goes beyond contact, such as the mutual borrowing of words between Japanese and Ainu.

The death of Arcadius in May 408 caused milder counsel to prevail in the western court, but Alaric, who had actually entered Epirus, demanded in a somewhat threatening manner that if he were thus suddenly requested to desist from war, he should be paid handsomely for what modern language would call the " expenses of mobilization ".

One of these was the property, an essential universal true of the species, but not in the definition ( in modern terms, some examples would be grammatical language, a property of man, or a spectral pattern characteristic of an element, both of which are defined in other ways ).

As a national revival occurred towards the end of the period of Ottoman rule ( mostly during the 19th century ), a modern Bulgarian literary language gradually emerged which drew heavily on Church Slavonic / Old Bulgarian ( and to some extent on literary Russian, which had preserved many lexical items from Church Slavonic ) and later reduced the number of Turkish and other Balkanic loans.

A wax sculpture of Chuon Nath, the conservator of the modern Khmer language, at the Cambodian Cultural Village

It has made its presence firmly known in the philosophy of language and epistemology-a modern revival of rationalism-as well as constituting a substantial wing of modern linguistics.

While the language has continued to change in all of these places, modern Canadian English has inherited significant vocabulary and spelling from the shared political and social institutions of Commonwealth countries.

The COBOL 2002 standard includes support for object-oriented programming and other modern language features.

However, the best modern analysis of the sources of the creed ( by A. de Halleux, in Revue Theologique de Louvain 7, 1976 ) and a reading of the acts, or proceedings, of the council ( recently translated into English ) show that the bishops considered Cyril the great authority and that even the language of ' two natures ' derives from him.

Therefore, it is clear there is no real consensus of what the term crannog actually implies, although the modern adoption in the English language broadly refers to a partially or completely artificial islet which saw use from the prehistoric to the Post-Medieval period in Ireland and Scotland.

In the Welsh language who's origins, like Cornish is from the ancient British or Brythonic language line, ' Cist ' is also used for such ancient graves, but in modern use, can also mean a chest, a coffer, a box, or even the boot / trunk of a car.

In 18th and 19th century Germany, several thousand local languages of the continental west Germanic dialect continuum were reclassified as dialects of modern New High German although the vast majority of them were ( and still are ) mutually incomprehensible, despite the fact that they all existed long before New High German, which had at least in part been shaped as a compromise or mediative language between these local languages.

In one opinion, this pattern is clearly present among the modern Romance tongues, with Italian and Spanish having a high degree of mutual comprehensibility, which neither language shares with French, despite some claiming that both languages are genetically closer to French than to each other: In fact, French-Italian and French-Spanish relative mutual incomprehensibility is due to French having undergone more rapid and more pervasive phonological change than have Spanish and Italian, not to real or imagined distance in genetic relationship.