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.

Other Flexowriter characters that were found a use in Atlas Autocode were: in floating-point numbers, e. g. for modern ; to mean " the second half of an Atlas memory word "; for the mathematical pi number.

In Thomson's mathematical model the " corpuscles " ( or modern electrons ) were arranged non-randomly, in rotating rings.

Sets are of great importance in mathematics ; in fact, in modern formal treatments, most mathematical objects ( numbers, relations, functions, etc.

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.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics.

The idea of the Erdős number was created by friends as a humorous tribute to the enormous output of Erdős, one of the most prolific modern writers of mathematical papers, and has become well known in scientific circles as a tongue-in-cheek measurement of mathematical prominence.

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.

The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic.

As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction.

Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales.

There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics.

He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.

While some of Euler's proofs are not acceptable by modern standards of mathematical rigour ( in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

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.

A practical method for investigating neutrino oscillations was first suggested by Bruno Pontecorvo in 1957 using an analogy with kaon oscillations ; over the subsequent 10 years he developed the mathematical formalism and the modern formulation of vacuum oscillations.

By revealing ( in modern terms ) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history ; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.

Many see an attempt to model a system as complex as a modern economy by a mathematical model as unrealistic and doomed to failure.

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.

A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:

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.