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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.[1] Topology
Topology
developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation.[2] Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place")
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The Spectator
The Spectator
The Spectator
is a weekly British magazine on politics, culture, and current affairs.[1] It was first published on 6 July 1828.[2] It is currently owned by David and Frederick Barclay
David and Frederick Barclay
who also own The Daily Telegraph newspaper, via Press Holdings. Its principal subject areas are politics and culture. Its editorial outlook is generally supportive of the Conservative Party, although regular contributors include some outside that fold, such as Frank Field, Rod Liddle and Martin Bright. The magazine also contains arts pages on books, music, opera, and film and TV reviews. In late 2008, Spectator Australia was launched
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Lawvere–Tierney Topology
In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves
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Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology
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Trivial Knot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity element with respect to the knot sum operation.Contents1 Unknotting problem 2 Examples 3 Invariants 4 See also 5 References 6 External linksUnknotting problem[edit] Main article: Unknotting problem Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram
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Greek Language
Greek (Modern Greek: ελληνικά [eliniˈka], elliniká, "Greek", ελληνική γλώσσα [eliniˈci ˈɣlosa] ( listen), ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece
Greece
and other parts of the Eastern Mediterranean
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Jacques Hadamard
Jacques Salomon Hadamard ForMemRS[2] (French: [adamaʁ]; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.[3][4][5]Contents1 Biography 2 On creativity 3 Publications 4 See also 5 References 6 Further reading 7 External linksBiography[edit] The son of a teacher, Amédée Hadamard, of Jewish
Jewish
descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne
Lycée Charlemagne
and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard
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Cesare Arzelà
Cesare Arzelà
Cesare Arzelà
(6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna
University of Bologna
and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continuous functions, generalizing the one given earlier by Giulio Ascoli
Giulio Ascoli
in the Arzelà-Ascoli theorem.Contents1 Life 2 Works 3 See also 4 Further reading 5 External linksLife[edit] He was a pupil of the Scuola Normale Superiore
Scuola Normale Superiore
of Pisa
Pisa
where he graduated in 1869
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Vito Volterra
Vito Volterra
Vito Volterra
(3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations,[2][3] being one of the founders of functional analysis.[4]Contents1 Biography 2 Selected writings by Volterra 3 See also 4 Notes 5 Biographical references 6 General references 7 External linksBiography[edit] Born in Ancona, then part of the Papal States, into a very poor Jewish family, Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics in 1883. He immediately started work developing his theory of functionals which led to his interest and later contributions in integral and integro-differential equations
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Augustin-Louis Cauchy
Baron
Baron
Augustin-Louis Cauchy
Augustin-Louis Cauchy
FRS FRSE
FRSE
(/koʊˈʃiː/;[1] French: [oɡystɛ̃ lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician and physicist who made pioneering contributions to mathematical analysis
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Ludwig Schläfli
Ludwig Schläfli
Ludwig Schläfli
(15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces. The concept of multidimensionality has come to play a pivotal role in physics, and is a common element in science fiction.Contents1 Life and career1.1 Youth and education 1.2 Teaching 1.3 Later life2 Higher dimensions 3 Polytopes 4 See also 5 References 6 External linksLife and career[edit] Youth and education[edit] Ludwig spent most of his life in Switzerland. He was born in Grasswil (now part of Seeberg), his mother's hometown. The family then moved to the nearby Burgdorf, where his father worked as a tradesman
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Nature (journal)
Nature is a British multidisciplinary scientific journal, first published on 4 November 1869.[1] It was ranked the world's most cited scientific journal by the Science Edition of the 2010 Journal Citation Reports and is ascribed an impact factor of 40.137 , making it one of the world's top academic journals.[2][3] It is one of the few remaining academic journals that publishes original research across a wide range of scientific fields.[3][4] Research
Research
scientists are the primary audience for the journal, but summaries and accompanying articles are intended to make many of the most important papers understandable to scientists in other fields and the educated public. Towards the front of each issue are editorials, news and feature articles on issues of general interest to scientists, including current affairs, science funding, business, scientific ethics and research breakthroughs. There are also sections on books and arts
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Giulio Ascoli
Giulio Ascoli
Giulio Ascoli
(20 January 1843, Trieste
Trieste
– 12 July 1896, Milan) was an Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868. In 1872 he became Professor of Algebra and Calculus of the Politecnico di Milano University. From 1879 he was professor of mathematics at the Reale Istituto Tecnico Superiore, where, in 1901, was affixed a plaque that remembers him. He was also corresponding member of Istituto Lombardo. He made contributions to the theory of functions of a real variable and to Fourier series
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Topography
Topography
Topography
is the study of the shape and features of the surface of the Earth
Earth
and other observable astronomical objects including planets, moons, and asteroids. The topography of an area could refer to the surface shapes and features themselves, or a description (especially their depiction in maps). This field of geoscience and planetary science is concerned with local detail in general, including not only relief but also natural and artificial features, and even local history and culture. This meaning is less common in the United States, where topographic maps with elevation contours have made "topography" synonymous with relief. The older sense of topography as the study of place still has currency in Europe. Topography
Topography
in a narrow sense involves the recording of relief or terrain, the three-dimensional quality of the surface, and the identification of specific landforms
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Trefoil Knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop
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Enrico Betti
Enrico Betti
Enrico Betti
Glaoui (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations, giving early expositions of Galois theory. He also discovered Betti's theorem, a result in the theory of elasticity.Contents1 Biography 2 Works 3 See also 4 Notes 5 Further reading 6 External linksBiography[edit] Betti was born in Pistoia, Tuscany. He graduated from the University of Pisa
Pisa
in 1846 under Giuseppe Doveri (it) (1792–1857).[1] In Pisa, he was also a student of Ottaviano Fabrizio Mossotti
Ottaviano Fabrizio Mossotti
and Carlo Matteucci. After a time teaching, he held an appointment there from 1857. In 1858 he toured Europe with Francesco Brioschi
Francesco Brioschi
and Felice Casorati, meeting Bernhard Riemann
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