Marburg is a university town in the German federal state of Hesse, capital of the Marburg-Biedenkopf district. The town area spreads along the valley of the river Lahn and has a population of 72,000. Having been awarded town privileges in 1222, Marburg served as capital of the landgraviate of Hessen-Marburg during periods of the fifteenth to seventeenth centuries; the University of Marburg dominates the public life in the town to this day. Like many settlements, Marburg developed at the crossroads of two important early medieval highways: the trade route linking Cologne and Prague and the trade route from the North Sea to the Alps and on to Italy, the former crossing the river Lahn here; the settlement was protected and customs were raised by a small castle built during the ninth or tenth century by the Giso. Marburg has been a town since 1140. From the Gisos, it fell around that time to the Landgraves of Thuringia, residing on the Wartburg above Eisenach. In 1228, the widowed princess-landgravine of Thuringia, Elizabeth of Hungary, chose Marburg as her dowager seat, as she did not get along well with her brother-in-law, the new landgrave.
The countess dedicated her life to the sick and would become after her early death in 1231, aged 24, one of the most prominent female saints of the era. She was canonized in 1235. In 1264, St Elizabeth's daughter Sophie of Brabant, succeeded in winning the Landgraviate of Hessen, hitherto connected to Thuringia, for her son Henry. Marburg was one of the capitals of Hessen from that time until about 1540. Following the first division of the landgraviate, it was the capital of Hessen-Marburg from 1485 to 1500 and again between 1567 and 1605. Hessen was one of the more powerful second-tier principalities in Germany, its "old enemy" was the Archbishopric of Mainz, one of the prince-electors, who competed with Hessen in many wars and conflicts for coveted territory, stretching over several centuries. After 1605, Marburg became just another provincial town, known for the University of Marburg, it became a virtual backwater for two centuries after the Thirty Years' War, when it was fought over by Hessen-Darmstadt and Hesse-Kassel.
The Hessian territory around Marburg lost more than two-thirds of its population, more than in any wars combined. Marburg is the seat of the oldest Protestant-founded university in the world, the University of Marburg, founded in 1527, it is one of the smaller "university towns" in Germany: Greifswald, Jena, Tübingen, as well as the city of Gießen, located 30 km south of Marburg. In 1529, Philipp I of Hesse arranged the Marburg Colloquy, to propitiate Martin Luther and Huldrych Zwingli. Owing to its neglect during the entire eighteenth century, Marburg – like Rye or Chartres – survived as a intact Gothic town because there was no money spent on any new architecture or expansion; when Romanticism became the dominant cultural and artistic paradigm in Germany, Marburg became interesting once again, many of the leaders of the movement lived, taught, or studied in Marburg. They formed a circle of friends, of great importance in literature, philology and law; the group included Friedrich Karl von Savigny, the most important jurist of his day and father of the Roman Law adaptation in Germany.
Most famous internationally, were the Brothers Grimm, who collected many of their fairy tales here. The original building inspiring his drawing. Across the Lahn hills, in the area called Schwalm, the costumes of little girls included a red hood. In the Austro-Prussian War of 1866, the Prince-elector of Hessen had backed Austria. Prussia won and took the opportunity to invade and annex the Electorate of Hessen north of the Main River. However, the pro-Austrian Hesse-Darmstadt remained independent. For Marburg, this turn of events was positive, because Prussia decided to make Marburg its main administrative centre in this part of the new province Hessen-Nassau and to turn the University of Marburg into the regional academic centre. Thus, Marburg's rise as an administrative and university city began; as the Prussian university system was one of the best in the world at the time, Marburg attracted many respected scholars. However, there was hardly any industry to speak of, so students and civil servants – who had enough but not much money and paid little in taxes – dominated the town, which tended to be conservative.
Franz von Papen, vice-chancellor of Germany in 1934, delivered an anti-Nazi speech at the University of Marburg on 17 June. From 1942 to 1945, the whole city of Marburg was turned into a hospital with schools and government buildings turned into wards to augment the existing hospitals. By the spring of 1945, there were over 20,000 patients – wounded German soldiers; as a result of its being designated a hospital city, there was not much damage from bombings except along the railroad tracks. In 1945, the Elisabethkirche in Marburg became the final resting place of Field Marshal and President Paul von Hindenburg, he is an honorary citizen of the town. As a larger mid-sized city, like six other such cities in Hessen, has a special status as compared to the other municipalities in the district; this means that the city takes on tasks more performed by the district so that in many ways it is comparable to an urban
Erich Hecke was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students. Hecke was born in Buk, Posen and died in Copenhagen, Denmark, his early work included establishing the functional equation for the Dedekind zeta function, with a proof based on theta functions. The method extended to the L-functions associated to a class of characters now known as Hecke characters or idele class characters, he devoted most of his research to the theory of modular forms, creating the general theory of cusp forms, as it is now understood in the classical setting. He was a Plenary Speaker of the ICM in 1936 in Oslo. List of things named after Erich Hecke Hecke algebra Tate's thesis Erich Hecke at the Mathematics Genealogy Project O'Connor, John J..
Wilhelm Johann Eugen Blaschke was an Austrian differential and integral geometer. His students included Luis Santaló, Gheorghe Gheorghiev and Emanuel Sperner. In 1916 Blaschke published one of the first books devoted to convex sets: Sphere. Drawing on dozens of sources, Blaschke made a thorough review of the subject with citations within the text to attribute credit in a classical area of mathematics. In 1933 Blaschke signed the Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State. Wilhelm Blaschke joined the NSDAP in 1937. Kreis und Kugel, Veit 1916. Berlin, de Gruyter 1956 Vorlesungen über Differentialgeometrie, 3 vols. Springer, Grundlehren der mathematischen Wissenschaften 1921-1929 with G. Bol: Geometrie der Gewebe. Berlin: Springer 1938 Ebene Kinematik. Leipzig: B. G. Teubner 1938, 2nd expanded edn. with Hans Robert Müller, Oldenbourg, München 1956 Nicht-Euklidische Geometrie und Mechanik I, II, III. Leipzig: B. G. Teubner Zur Bewegungsgeometrie auf der Kugel. In: Sitzungsberichte der Heidelberger Akademie der Wissenschaften Einführung in die Differentialgeometrie.
Springer 1950, 2nd expanded edn. with H. Reichardt 1960 with Kurt Leichtweiß: Elementare Differentialgeometrie. Berlin: Springer Reden und Reisen eines Geometers. Berlin: VEB Deutscher Verlag der Wissenschaften Mathematik und Leben, Steiner 1951 Griechische und anschauliche Geometrie, Oldenbourg 1953 Projektive Geometrie, 3rd edn, Birkhäuser 1954 Analytische Geometrie, 2nd edn. Birkhäuser 1954 Einführung in die Geometrie der Waben, Birkhäuser 1955 Vorlesungen über Integralgeometrie, VEB, Berlin 1955 Kinematik und Quaternionen. Berlin: VEB Deutscher Verlag der Wissenschaften Gesammelte Werke, Essen 1985 A number of mathematical theorems and concepts is associated with the name of Blaschke. Blaschke selection theorem Blaschke–Lebesgue theorem Blaschke product Blaschke condition Blaschke–Santaló inequality Blaschke conjecture: "The only Wiedersehen manifolds in any dimension are the standard Euclidean spheres." Blaschke, W.. Reden und Reisen eines Geometers. East Berlin. O'Connor, John J.. Wilhelm Blaschke at the Mathematics Genealogy Project
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself. Knots can be described in various ways. Given a method of description, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this problem exists. In practice, knots are distinguished by using a knot invariant, a "quantity", the same when computed from different descriptions of a knot.
Important invariants include knot polynomials, knot groups, hyperbolic invariants. The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century. To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space. Archaeologists have discovered. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC; the endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures representing strength in unity.
The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss, who defined the linking integral. In the 1860s, Lord Kelvin's theory that atoms were knots in the aether led to Peter Guthrie Tait's creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, what came to be known as the Tait conjectures; this record motivated the early knot theorists, but knot theory became part of the emerging subject of topology. These topologists in the early part of the 20th century—Max Dehn, J. W. Alexander, others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial.
This would be the main approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants; the discovery of the Jones polynomial by Vaughan Jones in 1984, subsequent contributions from Edward Witten, Maxim Kontsevich, others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since utilizing sophisticated tools such as quantum groups and Floer homology. In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine. Tangles, strings with both ends fixed in place, have been used in studying the action of topoisomerase on DNA.
Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation. A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, fusing its two free ends together to form a closed loop. We can say a knot K is a "simple closed curve" or " Jordan curve" — that is: a "nearly" injective and continuous function K: → R 3, with the only "non-injectivity" being K = K. Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot; the idea of knot equivalence is to give a precise definitio
Royal Library of the Netherlands
The Royal Library of the Netherlands is based in The Hague and was founded in 1798. The mission of the Royal Library of the Netherlands, as presented on the library's web site, is to provide "access to the knowledge and culture of the past and the present by providing high-quality services for research and cultural experience"; the initiative to found a national library was proposed by representative Albert Jan Verbeek on August 17 1798. The collection would be based on the confiscated book collection of William V; the library was founded as the Nationale Bibliotheek on November 8 of the same year, after a committee of representatives had advised the creation of a national library on the same day. The National Library was only open to members of the Representative Body. King Louis Bonaparte gave the national library its name of the Royal Library in 1806. Napoleon Bonaparte transferred the Royal Library to The Hague as property, while allowing the Imperial Library in Paris to expropriate publications from the Royal Library.
In 1815 King William I of the Netherlands confirmed the name of'Royal Library' by royal resolution. It has been known as the National Library of the Netherlands since 1982, when it opened new quarters; the institution became independent of the state in 1996, although it is financed by the Department of Education and Science. In 2004, the National Library of the Netherlands contained 3,300,000 items, equivalent to 67 kilometers of bookshelves. Most items in the collection are books. There are pieces of "grey literature", where the author, publisher, or date may not be apparent but the document has cultural or intellectual significance; the collection contains the entire literature of the Netherlands, from medieval manuscripts to modern scientific publications. For a publication to be accepted, it must be from a registered Dutch publisher; the collection is accessible for members. Any person aged 16 years or older can become a member. One day passes are available. Requests for material take 30 minutes.
The KB hosts several open access websites, including the "Memory of the Netherlands". List of libraries in the Netherlands European Library Nederlandse Centrale Catalogus Books in the Netherlands Media related to Koninklijke Bibliotheek at Wikimedia Commons Official website
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act. Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects; this is done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is new, became a identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are substantial connections with complexity theory, mathematical logic, the study of Lie Groups and their discrete subgroups, dynamical systems, probability theory, K-theory, other areas of mathematics.
In the introduction to his book Topics in Geometric Group Theory, Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, reminds me of several things that Georges de Rham practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend". Geometric group theory grew out of combinatorial group theory that studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups. Combinatorial group theory as an area is subsumed by geometric group theory. Moreover, the term "geometric group theory" came to include studying discrete groups using probabilistic, measure-theoretic, arithmetic and other approaches that lie outside of the traditional combinatorial group theory arsenal.
In the first half of the 20th century, pioneering work of Max Dehn, Jakob Nielsen, Kurt Reidemeister and Otto Schreier, J. H. C. Whitehead, Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups. Other precursors of geometric group theory include Bass -- Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp, it studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie Groups Mostow rigidity theorem, the study of Kleinian groups, the progress achieved in low-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by William Thurston's Geometrization program.
The emergence of geometric group theory as a distinct area of mathematics is traced to the late 1980s and early 1990s. It was spurred by the 1987 monograph of Mikhail Gromov "Hyperbolic groups" that introduced the notion of a hyperbolic group, which captures the idea of a finitely generated group having large-scale negative curvature, by his subsequent monograph Asymptotic Invariants of Infinite Groups, that outlined Gromov's program of understanding discrete groups up to quasi-isometry; the work of Gromov had a transformative effect on the study of discrete groups and the phrase "geometric group theory" started appearing soon afterwards.. Notable themes and developments in geometric group theory in 1990s and 2000s include: Gromov's program to study quasi-isometric properties of groups. A influential broad theme in the area is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry.
This program involves: The study of properties. Examples of such properties of finitely generated groups include: the growth rate of a finitely generated group. Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space; this dire