Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and from 1945 to 1948 at the École Normale Supérieure in Paris, he was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994, his wife, Professor Josiane Heulot-Serre, was a chemist. Their daughter is the former French diplomat and writer Claudine Monteil; the French mathematician Denis Serre is his nephew. He practices skiing, table tennis, rock climbing. From a young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques.
Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. At an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures; the problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.
Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite étale map – are important; this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne. From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" on Galois cohomology. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free; this question led to a great deal of activity in commutative algebra, was answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976.
This result is now known as the Quillen–Suslin theorem. Serre, at twenty-seven in 1954, is the youngest to be awarded the Fields Medal, he went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre, he has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society. Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour and Grand Cross of the Legion of Merit. List of things named after Jean-Pierre Serre Nicolas Bourbaki Groupes Algébriques et Corps de Classes, translated into English as Algebraic Groups and Class Fields, Springer-Verlag Corps Locaux, Hermann, as Local Fields, Springer-Verlag Cohomologie Galoisienne Collège de France course 1962–63, as Galois Cohomology, Springer-Verlag Algèbre Locale, Multiplicités Collège de France course 1957–58, as Local Algebra, Springer-Verlag "Lie algebras and Lie groups" Harvard Lectures, Springer-Verlag.
Algèbres de Lie Semi-simples Complexes, as Complex Semisimple Lie Algebras, Springer-Verlag Abelian ℓ-Adic Representations and Elliptic Curves, CRC Press, reissue. Addison-Wesley. 1989. Cours d'arithmétique, PUF, as A Course in Arithmetic, Springer-Verlag Représentations linéaires des groupes finis, Hermann, as Linear Represent
Élie Joseph Cartan, ForMemRS was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems, differential geometry. He made significant contributions to general relativity and indirectly to quantum mechanics, he is regarded as one of the greatest mathematicians of the twentieth century. Cartan's recognition as a first–rate mathematician came to him only in his old age; this was due to his extreme modesty and to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since his influence has been increasing, with the exception of Poincaré and Hilbert no one else has done so much to give the mathematics of 20th century its shape and viewpoints. Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan and Anne Cottaz.
Joseph Cartan was the village blacksmith. Élie had an elder sister Jeanne-Marie. Élie Cartan was the best student in the school. One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, this was combined with an excellent memory". Antonin Dubost the representative of Isère, visited the school and was impressed by Cartan's unusual abilities, he recommended Cartan to participate in a contest for a scholarship in a lycée. Cartan prepared for the contest under the supervision of M. Dupuis and passed at the age of ten years, he spent five years at the College of Vienne and two years at the Lycée of Grenoble. In 1887 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years. Cartan enrolled in the École Normale Supérieure in 1888, he attended there lectures by Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard, Edouard Goursat, Henri Poincaré whose lectures were what Cartan thought most of.
After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant. For next two years Cartan returned to ENS and, following the advice of his classmate Arthur Tresse who studied under Sophus Lie in the years 1888–1889, worked on the subject of classification of simple Lie groups, started by Wilhelm Killing. In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, met Cartan for the first time. Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the University of Montpellier. In 1903, while in Lyons, Cartan married Marie-Louise Bianconi. In 1904, Cartan's first son, Henri Cartan, who became an influential mathematician, was born. In 1909 Cartan moved his family to Paris and worked as a lecturer in the Faculty of Sciences in the Sorbonne. In 1912 Cartan became Professor there, based on the reference he received from Poincaré.
He remained in Sorbonne until his retirement in 1940 and spent the last years of his life teaching mathematics at the École Normale Supérieure for girls. In 1921 he became a foreign member of the Polish Academy of Learning and in 1937 a foreign member of the Royal Netherlands Academy of Arts and Sciences. In 1938 he participated in the International Committee composed to organise the International Congresses for the Unity of Science, he died in 1951 in Paris after a long illness. In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are: Lie theory Representations of Lie groups Hypercomplex numbers, division algebras Systems of PDEs, Cartan–Kähler theorem Theory of equivalence Integrable systems, theory of prolongation and systems in involution Infinite-dimensional groups and pseudogroups Differential geometry and moving frames Generalised spaces with structure groups and connections, Cartan connection, Weyl tensor Geometry and topology of Lie groups Riemannian geometry Symmetric spaces Topology of compact groups and their homogeneous spaces Integral invariants and classical mechanics Relativity, spinorsCartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now conside
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the Journal Citation Reports, the journal has a 2013 impact factor of 0.840. Proceedings of the American Mathematical Society publishes articles from all areas of pure and applied mathematics, including topology, analysis, number theory, logic and statistics; this journal is indexed in the following databases: Mathematical Reviews Zentralblatt MATH Science Citation Index Science Citation Index Expanded ISI Alerting Services CompuMath Citation Index Current Contents / Physical, Chemical & Earth Sciences. Bulletin of the American Mathematical Society Memoirs of the American Mathematical Society Notices of the American Mathematical Society Journal of the American Mathematical Society Transactions of the American Mathematical Society Official website Proceedings of the American Mathematical Society on JSTOR
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra, algebraic geometry, functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, as a historian of mathematics in the fields of functional analysis and algebraic topology. His work on the classical groups, on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields, he was born and brought up in Lille, with a formative stay in England where he was introduced to algebra. In 1924 he was admitted to the École Normale Supérieure, he began working, conventionally enough, in complex analysis. In 1934 he was one of the group of normaliens convened by Weil, which would become'Bourbaki', he served in the French Army during World War II, taught in Clermont-Ferrand until the liberation of France. After holding professorships at the University of São Paulo, the University of Nancy and the University of Michigan, he joined the Department of Mathematics at Northwestern University in 1953, before returning to France as a founding member of the Institut des Hautes Études Scientifiques.
He moved to the University of Nice to found the Department of Mathematics in 1964, retired in 1970. He was elected as a member of the Académie des Sciences in 1968. Dieudonné drafted much of the Bourbaki series of texts, the many volumes of the EGA algebraic geometry series, nine volumes of his own Éléments d'Analyse; the first volume of the Traité is a French version of the book Foundations of Modern Analysis, which had become a graduate textbook on functional analysis. He wrote individual monographs on Infinitesimal Calculus, Linear Algebra and Elementary Geometry, invariant theory, commutative algebra, algebraic geometry, formal groups. With Laurent Schwartz he supervised the early research of Alexander Grothendieck. From 1959 to 1964 he was at the Institut des Hautes Études Scientifiques alongside Grothendieck, collaborating on the expository work needed to support the project of refounding algebraic geometry on the new basis of schemes. Sur les groupes classiques. Paris: Hermann. 1948. Dieudonné, Jean, La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 5, New York: Springer-Verlag, ISBN 978-0-387-05391-2, MR 0072144 9 volumes of Éléments d'analyse, éd.
Gauthier-VillarsFoundations of Modern Analysis. Academic Press. 1960. Algèbre linéaire et géométrie élémentaire. Hermann. 1964.. 1969. "The work of Nicolas Bourbaki". Amer. Math. Monthly. 77: 134–145. 1970. Doi:10.2307/2317325. Dieudonné, Jean A.. "Invariant theory and new", Advances in Mathematics, Boston, MA: Academic Press, 4: 1–80, doi:10.1016/0001-870890015-0, ISBN 978-0-12-215540-6, MR 0279102 Historical development of algebraic geometry. American Mathematical Monthly. 79. Oct 1972. Pp. 827–866. Doi:10.2307/2317664. Introduction to the theory of formal groups. Dekker. 1973. Cours de géométrie algébrique I. P. U. F. 1974.. Wadsworth Inc. 1985. Cours de géométrie algébrique II. P. U. F. 1974. Dieudonné, Jean Alexandre, A panorama of pure mathematics and Applied Mathematics, 97, London: Academic Press Inc. ISBN 978-0-12-215560-4, MR 0478177 Dieudonné, Choix d'œuvres mathématiques. Tome I, Paris: Hermann, ISBN 978-2-7056-5922-6, MR 0611149 Dieudonné, Choix d'œuvres mathématiques. Tome II, Paris: Hermann, ISBN 978-2-7056-5923-3, MR 0611150 History of functional analysis.
North-Holland. 1981. Pour l'honneur de l'esprit humain: les mathématiques aujourd'hui. Hachette. 1987. A History of Algebraic and Differential Topology 1900-1960. Birkhäuser Boston. 1988. Mathematics - the music of reason. Springer. 1992. Dugac, Jean Dieudonné: Mathématicien complet, Editions Jacques Gabay, ISBN 978-2-87647-156-6 O'Connor, John J.. "Jean Dieudonné", MacTutor History of Mathematics archive, University of St Andrews. Jean Dieudonné at the Mathematics Genealogy Project Dieudonné determinant Dieudonné plank Dieudonné's theorem A talk on the history of Algebraic Geometry given by Jean Dieudonné at the Department of Mathematics of the University of Wisconsin-Milwaukee in 1972 has been restored and is available here Dieudonné appears in the Horizon BBC documentary A Mathematical Mystery Tour
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the