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Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in mathematical analysis, analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups, and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simone Weil
Simone Adolphine Weil ( ; ; 3 February 1909 – 24 August 1943) was a French philosopher, mystic and political activist. Despite her short life, her ideas concerning religion, spirituality, and politics have remained widely influential in contemporary philosophy. She was born in Paris to an Alsatian Jewish family. Her elder brother, André, would later become a renowned mathematician. After her graduation from formal education, Weil became a teacher. She taught intermittently throughout the 1930s, taking several breaks because of poor health and in order to devote herself to political activism. She assisted in the trade union movement, taking the side of the anarchists known as the Durruti Column in the Spanish Civil War. During a twelve-month period she worked as a labourer, mostly in car factories, so that she could better understand the working class. Weil became increasingly religious and inclined towards mysticism as her life progressed. She died of heart failure in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Strasbourg
The University of Strasbourg (, Unistra) is a public research university located in Strasbourg, France, with over 52,000 students and 3,300 researchers. Founded in the 16th century by Johannes Sturm, it was a center of intellectual life during the Age of Enlightenment. The old university was split into three separate entities in the 1970s before merging back together in 2009. Today, the University of Strasbourg comprises 35 academic faculties, schools, and institutes, as well as 71 research laboratories spread across six campuses, including the historic site in the Neustadt. Throughout its existence, Unistra alumni, faculty, or researchers have included 18 Nobel laureates, two Fields Medalists and a wide range of notable individuals in their respective fields. Among them are Goethe, statesman Robert Schuman, historian Marc Bloch and several chemists such as Louis Pasteur. History The university emerged from the Jean Sturm Gymnasium, a gymnasium of Lutheran and humanist ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First World War
World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting took place mainly in European theatre of World War I, Europe and the Middle Eastern theatre of World War I, Middle East, as well as in parts of African theatre of World War I, Africa and the Asian and Pacific theatre of World War I, Asia-Pacific, and in Europe was characterised by trench warfare; the widespread use of Artillery of World War I, artillery, machine guns, and Chemical weapons in World War I, chemical weapons (gas); and the introductions of Tanks in World War I, tanks and Aviation in World War I, aircraft. World War I was one of the List of wars by death toll, deadliest conflicts in history, resulting in an estimated World War I casualties, 10 million military dead and more than 20 million wounded, plus some 10 million civilian de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Éléments De Mathématique
''Éléments de mathématique'' (English: ''Elements of Mathematics'') is a series of mathematics books written by the pseudonymous French collective Nicolas Bourbaki. Begun in 1939, the series has been published in several volumes, and remains in progress. The series is noted as a large-scale, self-contained, formal treatment of mathematics. The members of the Bourbaki group originally intended the work as a textbook on analysis, with the working title ''Traité d'analyse'' (''Treatise on Analysis''). While planning the structure of the work they became more ambitious, expanding its scope to cover several branches of modern mathematics. Once the plan of the work was expanded to treat other fields in depth, the title ''Éléments de mathématique'' was adopted. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. The unusual singular "mathématique" (mathematic) of the title is deliberate, meant to convey the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners (who could be independent learners outside of formal education). Schoolbooks are textbooks and other books used in schools. Today, many textbooks are published in both print and digital formats. History The history of textbooks dates back to ancient civilizations. For example, Ancient Greeks wrote educational texts. The modern textbook has its roots in the mass production made possible by the printing press. Johannes Gutenberg himself may have printed editions of ''Ars Minor'', a schoolbook on Latin grammar by Aelius Donatus. Early textbooks were used by tutors and teachers (e.g. alphabet books), as well as by individuals who taught themselves. The Greek philosopher Socrates lamented the loss of knowledge because the media of transmis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
