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Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.

In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.

In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co

Liberty Bell Trail

The Liberty Bell Trail is a suburban rail trail under construction in southeastern Pennsylvania. When complete it will cover 25 miles in suburban southeastern Pennsylvania, traveling from East Norriton Township in Montgomery County to Quakertown in Bucks County, it was proposed in 1996, follows the former path of the now defunct Liberty Bell Trolley Route, operated by the Lehigh Valley Transit Company from around 1900 to 1951. The tram route was named by the company for the Liberty Bell because a branch of it followed Bethlehem Pike, the road along which the bell was transported in October 1777 when it was being moved from Philadelphia to Northamptontown for safekeeping shortly before the British occupation of Philadelphia during the American Revolutionary War; the trail has been named for both the bell. The route passes through 15 municipalities: East Norriton Township Whitpain Township Upper Gwynedd Township Lansdale Borough Hatfield Township Hatfield Borough Franconia Township Souderton Borough Telford Borough West Rockhill Township Hilltown Township Sellersville Borough Perkasie Borough Richland Township Quakertown Borough Liberty Bell Trail home Maps of the route

Miss Ecuador 2001

The Miss Ecuador 2001 contest was held on March 22, 2001. There were 13 candidates for the national title, the crown passed from Carolina Alfonso from Pichincha to Jéssica Bermúdez from Guayas, but the new Miss Ecuador 2001 was crowned by Ximena Aulestia, the 1st Runner-up at Miss Ecuador 1969 and competed at Miss World 1969. Jéssica competed at Miss Universe 2001. Last competed in: 1999 Chimborazo Azuay Esmeraldas Imbabura Loja

William Long (surgeon)

William Long FRS, FSA was an English surgeon. Born in Salisbury, Wiltshire, he was the youngest of ten children of Walter Long of Preshaw and Philippa Blackall, he was eminent in his profession, for thirty-three years, from 1784 to 1807, was surgeon at St Bartholomew's Hospital in London. He was appointed Master of the Royal College of Surgeons in 1800 and was among those who gave a donation to help fund their new surgical library, he was on the College's list of first Governors, first Examiners of Surgeons and the first Court of Assistants. He wrote several papers, including one entitled "The Effects of Cancer", he lived in London's Chancery Lane, at Lincoln's Inn Fields, developed close friendships with the painter George Romney, sculptor John Flaxman, writers William Hayley, Isaac Reed and William Blake, like Long, were members of the Unincreasable Club, at nearby Queens Head, London. Long sat for Romney as his first subject for a portrait, done for his friend Hayley. Subsequently Long acquired many of Romney's paintings, which were sold by Christie's on behalf of the family, in 1890.

William Long purchased Marwell Hall near Winchester, Hampshire about 1798, between 1812-1816 made considerable alterations, resulting in what is now the house as it stands today. He was a man of compassion and generosity, when resident at his country seat away from London, he always gave his advice and medicine gratuitously to the poor of the surrounding neighbourhood, he and his wife Alice had no children, in his will Long made generous bequests to his nephews and nieces. After his death on 24 March 1818 his collections of preserved medical specimens and surgical instruments were donated by his executors to the Royal College of Surgeons Museum in London. Alice continued to live at Marwell Hall, during the Owslebury riots of 1830 a mob of rioters, accompanied by John Boyes, a local farmer, arrived at the house; the mob demanded money from Alice and John Boyce demanded a reduction in the rents of her farm tenants, so they could pay their agricultural labourers higher wages.. William Long is buried in Salisbury Cathedral and his widow erected a monument to'perpetuate the memory of a much esteemed husband'.

Part of the epitaph, written in Latin, says: Alice died 18 September 1840, leaving numerous charitable bequests. Inheriting the Earth: The Long Family's 500 Year Reign in Wiltshire.

Onorio Longhi

Onorio Longhi was an Italian architect, the father of Martino Longhi the Younger and the son of Martino Longhi the Elder. Born in Viggiù, Longhi began as assistant for his father, inherited the latter's commission at his death in 1591, he is described by contemporary sources as a ruthless figure, a companion of Caravaggio, together with whom he was tried for homicide in Rome in 1606, subsequently exiled. Returning to Lombardy he executed several unfinished plans for the Duomo of Milan and other churches, until a Papal amnesty allowed him to come back to Rome in 1611. Here he designed the first plan for the Milanese national church in Rome, San Carlo al Corso, completed by his son and by Pietro da Cortona. Other Longhi's works include the church of Santa Maria Liberatrice in the Roman Forum and the Santoro Chapel in St. John Lateran, he died in Rome in 1619. Longhi, Onorio. Discorso di Honorio Lunghi. Del Teuere, della sua inondazione, & de' suoi rimedij. In Milano: appresso Girolamo Bordoni. Baglione, Giovanni.

Giovanni Battista Passari. Le Vite de’ Pittori, Architetti, ed Intagliatori dal Pontificato di Gregorio XII del 1572. Fino a’ tempi de Papa Urbano VIII. nel 1642.. 1731 edition. P. 149

Karl Vossler

Karl Vossler was a German linguist and scholar, a leading Romanist. Vossler was known for his interest in Italian thought, as a follower of Benedetto Croce, he declared his support of the German military by signing the Manifesto of the Ninety-Three in 1914. However, he opposed the Nazi government, supported many Jewish intellectuals at that time. In 1897 he received his doctorate from the University of Heidelberg, in 1909 was named a professor of Romance studies at the University of Würzburg. From 1911 onward, he taught classes at the University of Munich. "Mediaeval culture. "The spirit of language in civilization". "Jean Racine". Karl-Vossler-Preis Newspaper clippings about Karl Vossler in the 20th Century Press Archives of the ZBW Critical edition of Karl Vossler's translation of the Divine Comedy in German at