Frankfurt is a metropolis and the largest city of the German federal state of Hesse, its 746,878 inhabitants make it the fifth-largest city of Germany after Berlin, Hamburg and Cologne. On the River Main, it forms a continuous conurbation with the neighbouring city of Offenbach am Main, its urban area has a population of 2.3 million. The city is at the centre of the larger Rhine-Main Metropolitan Region, which has a population of 5.5 million and is Germany's second-largest metropolitan region after the Rhine-Ruhr Region. Since the enlargement of the European Union in 2013, the geographic centre of the EU is about 40 km to the east of Frankfurt's central business district. Like France and Franconia, the city is named after the Franks. Frankfurt is the largest city in the Rhine Franconian dialect area. Frankfurt was a city state, the Free City of Frankfurt, for nearly five centuries, was one of the most important cities of the Holy Roman Empire, as a site of imperial coronations, it has been part of the federal state of Hesse since 1945.
A quarter of the population are foreign nationals, including many expatriates. Frankfurt is an alpha world city and a global hub for commerce, education and transportation, it is the site of many European corporate headquarters. Frankfurt Airport is among the world's busiest. Frankfurt is the major financial centre of the European continent, with the headquarters of the European Central Bank, Deutsche Bundesbank, Frankfurt Stock Exchange, Deutsche Bank, DZ Bank, KfW, several cloud and fintech startups and other institutes. Automotive and research, consulting and creative industries complement the economic base. Frankfurt's DE-CIX is the world's largest internet exchange point. Messe Frankfurt is one of the world's largest trade fairs. Major fairs include the Frankfurt Motor Show, the world's largest motor show, the Music Fair, the Frankfurt Book Fair, the world's largest book fair. Frankfurt is home to influential educational institutions, including the Goethe University, the UAS, the FUMPA, graduate schools like the Frankfurt School of Finance & Management.
Its renowned cultural venues include the concert hall Alte Oper, Europe's largest English theatre and many museums. Frankfurt's skyline is shaped by some of Europe's tallest skyscrapers; the city is characterised by various green areas and parks, including the central Wallanlagen, the City Forest and two major botanical gardens, the Palmengarten and the University's Botanical Garden. Important is the Frankfurt Zoo. In electronic music, Frankfurt has been a pioneering city since the 1980s, with renowned DJs including Sven Väth, Marc Trauner, Scot Project, Kai Tracid, the clubs Dorian Gray, U60311, Omen and Cocoon. In sports, the city is known as the home of the top tier football club Eintracht Frankfurt, the Löwen Frankfurt ice hockey team, the basketball club Frankfurt Skyliners, the Frankfurt Marathon and the venue of Ironman Germany. Frankfurt is the largest financial centre in continental Europe, it is home to the European Central Bank, Deutsche Bundesbank, Frankfurt Stock Exchange and several large commercial banks.
The Frankfurt Stock Exchange is one of the world's largest stock exchanges by market capitalization and accounts for more than 90 percent of the turnover in the German market. In 2010, 63 national and 152 international banks had their registered offices in Frankfurt, including Germany's major banks, notably Deutsche Bank, DZ Bank, KfW and Commerzbank, as well as 41 representative offices of international banks. Frankfurt is considered a global city. Among global cities it was ranked 10th by the Global Power City Index 2011 and 11th by the Global City Competitiveness Index 2012. Among financial centres it was ranked 8th by the International Financial Centers Development Index 2013 and 9th by the Global Financial Centres Index 2013, its central location within Germany and Europe makes Frankfurt a major air and road transport hub. Frankfurt Airport is one of the world's busiest international airports by passenger traffic and the main hub for Germany's flag carrier Lufthansa. Frankfurt Central Station is one of the largest rail stations in Europe and the busiest junction operated by Deutsche Bahn, the German national railway company, with 342 trains a day to domestic and European destinations.
Frankfurter Kreuz, the Autobahn interchange close to the airport, is the most used interchange in the EU, used by 320,000 cars daily. In 2011 human-resource-consulting firm Mercer ranked Frankfurt as seventh in its annual'Quality of Living' survey of cities around the world. According to The Economist cost-of-living survey, Frankfurt is Germany's most expensive city and the world's 10th most expensive. Frankfurt has many high-rise buildings in the city centre, forming the Frankfurt skyline, it is one of the few cities in the European Union to have such a skyline and because of it Germans sometimes refer to Frankfurt as Mainhattan, a portmanteau of the local Main River and Manhattan. The other well known and obvious nickname is Bankfurt. Before World War II the city was globally noted for its unique old town with timber-framed buildings, the largest timber-framed old town in Europe; the Römer area was rebuilt and is popular with visitors and for eve
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t
Walter Bendix Schönflies Benjamin was a German Jewish philosopher, cultural critic and essayist. An eclectic thinker, combining elements of German idealism, Western Marxism, Jewish mysticism, Benjamin made enduring and influential contributions to aesthetic theory, literary criticism, historical materialism, he was associated with the Frankfurt School, maintained formative friendships with thinkers such as playwright Bertolt Brecht and Kabbalah scholar Gershom Scholem. He was related by law to German political theorist and philosopher Hannah Arendt through her first marriage to Benjamin's cousin, Günther Anders. Among Benjamin's best known works are the essays "The Task of the Translator", "The Work of Art in the Age of Mechanical Reproduction", "Theses on the Philosophy of History", his major work as a literary critic included essays on Baudelaire, Kafka, Leskov, Proust and translation theory. He made major translations into German of the Tableaux Parisiens section of Baudelaire's Les Fleurs du mal and parts of Proust's À la recherche du temps perdu.
In 1940, at the age of 48, Benjamin committed suicide at Portbou on the French–Spanish border while attempting to escape from invading Nazi forces. Though popular acclaim eluded him during his life, the decades following his death won his work posthumous renown. Benjamin and his younger siblings and Dora, were born to a wealthy business family of assimilated Ashkenazi Jews in the Berlin of the German Empire; the patriarch of Walter Benjamin's family, Emil Benjamin, was a banker in Paris who had relocated from France to Germany, where he worked as an antiques trader in Berlin. He owned a number of investments in Berlin, including ice skating rinks. Benjamin's uncle William Stern was a prominent German child psychologist who developed the concept of the intelligence quotient, Benjamin's cousin Günther Anders was a German philosopher and anti-nuclear activist who studied under Edmund Husserl and Martin Heidegger. Through his mother, his great-uncle was the classical archaeologist Gustav Hirschfeld.
In 1902, ten-year-old Walter was enrolled to the Kaiser Friedrich School in Charlottenburg. Walter Benjamin was a boy of fragile health and so in 1905 the family sent him to Hermann-Lietz-Schule Haubinda, a boarding school in the Thuringian countryside, for two years. In 1912, at the age of twenty, he enrolled at the University of Freiburg, but, at summer semester's end, returned to Berlin matriculated into the University of Berlin, to continue studying philosophy. Here Benjamin had his first exposure to the ideas of Zionism, which had not been part of his liberal upbringing; this exposure gave him occasion to formulate his own ideas about the meaning of Judaism. Benjamin distanced himself from political and nationalist Zionism, instead developing in his own thinking what he called a kind of "cultural Zionism"—an attitude which recognized and promoted Judaism and Jewish values. In Benjamin's formulation his Jewishness meant a commitment to the furtherance of European culture. Benjamin expressed "My life experience led me to this insight: the Jews represent an elite in the ranks of the spiritually active...
For Judaism is to me in no sense an end in itself, but the most distinguished bearer and representative of the spiritual." This was a position that Benjamin held lifelong. Elected president of the Freie Studentenschaft, Benjamin wrote essays arguing for educational and general cultural change; when not re-elected as student association president, he returned to Freiburg University to study, with particular attention to the lectures of Heinrich Rickert. In 1914, at the outbreak of the First World War, Benjamin began faithfully translating the works of the 19th-century French poet Charles Baudelaire; the next year, 1915, he moved to Munich, continued his schooling at the University of Munich, where he met Rainer Maria Rilke and Gershom Scholem. In that year, Benjamin wrote about the 18th-century Romantic German poet Friedrich Hölderlin. In 1917 he transferred to the University of Bern, they had Stefan Rafael. In 1919 Benjamin earned his Ph. D. cum laude with the dissertation Begriff der Kunstkritik in der Deutschen Romantik.
Unable to support himself and family, he returned to Berlin and resided with his parents. In 1921 he published the essay Kritik der Gewalt. At this time Benjamin first became acquainted with Leo Strauss, Benjamin would remain an admirer of Strauss and of his work throughout his life. In 1923, when the Institut für Sozialforschung was founded to become home to the Frankfurt School, Benjamin published Charles Baudelaire, Tableaux Parisiens. At that time he became acquainted with Theodor Adorno and befriended Georg Lukács, whose The Theory of the Novel much influenced him. Meanwhile, the inflation in the Weimar Republic consequent to the First World War made it difficult for the father Emil Benjamin to continue supporting his son's family. At the end of 1923 his best friend Gershom Scholem immigrated to Palestine, a country under the British Mandate of P
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Walther Hermann Nernst, was a German chemist known for his work in thermodynamics, physical chemistry and solid state physics. His formulation of the Nernst heat theorem helped pave the way for the third law of thermodynamics, for which he won the 1920 Nobel Prize in Chemistry, he is known for developing the Nernst equation in 1887. Nernst was born in Briesen in West Prussia to Ottilie Nerger, his father was a country judge. Nernst had one younger brother, his third sister died of cholera. Nernst went to elementary school at Graudenz, he studied physics and mathematics at the universities of Zürich, Graz and Würzburg, where he received his doctorate 1887. In 1889, he finished his habilitation at University of Leipzig, it was said that Nernst was mechanically minded in that he was always thinking of ways to apply new discoveries to industry. His hobbies included fishing, his friend Albert Einstein was amused by "his childlike vanity and self-complacency" "His own study and laboratory always presented aspects of extreme chaos which his coworkers termed appropriately'the state of maximum entropy'".
Nernst married Emma Lohmeyer in 1892 with whom he had three daughters. Both of Nernst's sons died fighting in World War I, he was a friend and colleague of Svante Arrhenius, suggested setting fire to unused coal seams to increase the global temperature. He was a vocal critic of Adolf Hitler and Nazism, two of his three daughters married Jewish men. After Hitler came to power they one to England and the other to Brazil. Nazism ended Nernst's career as a scientist. Nernst had a severe heart attack in 1939, he is buried near Max Planck, Otto Hahn and Max von Laue in Göttingen, Germany. Nernst started university at Zurich in 1883 after an interlude in Berlin, he returned to Zurich, he wrote his thesis at Graz where Boltzmann was professor, though he worked under the direction of Ettinghausen. They discovered the Nernst effect: that a magnetic field applied perpendicular to a metallic conductor in a temperature gradient gives rise to an electrical potential difference. Next, he moved to Würzburg under Kohlrausch where he defended his thesis.
Ostwald recruited him to the first department of physical chemistry at Leipzig. Nernst moved there as an assistant, working on the thermodynamics of electrical currents in solutions. Promoted to lecturer, he taught at Heidelberg and moved to Göttingen. Three years he was offered a professorship in Munich, to keep him in Prussia the government created a chair for him at Göttingen. There, he wrote a celebrated textbook Theoretical Chemistry, translated into English and Russian, he derived the Nernst equation for the electrical potential generated by unequal concentrations of an ion separated by a membrane, permeable to the ion. His equation is used in cell physiology and neurobiology; the carbon electric filament lamp in use was dim and expensive because it required a vacuum in its bulb. Nernst invented a solid-body radiator with a filament of rare-earth oxides, known as the Nernst glower, it is still important in the field of infrared spectroscopy. Continuous ohmic heating of the filament results in conduction.
The glower operates best in wavelengths from 2 to 14 micrometers. It gives a bright light but only after a warm-up period. Nernst sold the patent for one million marks, wisely not opting for royalties because soon the tungsten filament lamp filled with inert gas was introduced. With his riches, Nernst in 1898 bought the first of the eighteen automobiles he owned during his lifetime and a country estate of more than a five hundred hectares for hunting, he increased the power of his early automobiles by carrying a cylinder of nitrous oxide that he could inject into the carburetor. After eighteen productive years at Göttingen, investigating osmotic pressure and electrochemistry and presenting a theory of how nerves conduct, he moved to Berlin, was awarded the title Geheimrat In 1905, he proposed his "New Heat Theorem" known as the Third law of thermodynamics, he showed that as the temperature approached absolute zero, the entropy approaches zero — while the free energy remains above zero. This is the work for which he is best remembered, as it enabled chemists to determine free energies of chemical reactions from heat measurements.
Theodore Richards claimed that Nernst had stolen his idea, but Nernst is universally credited with the discovery. Nernst became friendly with Kaiser Wilhelm, whom he persuaded to found the Kaiser Wilhelm Gesellschaft for the Advancement of the Sciences with an initial capital of eleven million marks. Nernst's laboratory discovered that at low temperatures specific heats fell markedly and would disappear at absolute zero; this fall was predicted for liquids and solids in a 1909 paper of Einstein's on the quantum mechanics of specific heats at cryogenic temperatures. Nernst was so impressed that he traveled all the way to Zurich to visit Einstein, unknown in Zurich in 1909, so people said: "Einstein must be a clever fellow if the great Nernst comes all the way from Berlin to Zurich to talk to him." Nernst and Planck lobbied to establish a special professorship in Berlin and Nernst donated to its endowment. In 1913 they traveled to Switzerland to persuade Einstein to accept it. In 1911, Nernst and Max Planck organized the first Solvay Conference in Brussels.
In the following year, the impressionist painter Max Lieb
Max Horkheimer was a German philosopher and sociologist, famous for his work in critical theory as a member of the'Frankfurt School' of social research. Horkheimer addressed authoritarianism, economic disruption, environmental crisis, the poverty of mass culture using the philosophy of history as a framework; this became the foundation of critical theory. His most important works include Eclipse of Reason, Between Philosophy and Social Science and, in collaboration with Theodor Adorno, Dialectic of Enlightenment. Through the Frankfurt School, Horkheimer planned and made other significant works possible. On February 14, 1895, Horkheimer was born the only son of Babetta Horkheimer. Horkheimer was born into a wealthy Orthodox Jewish family, his father was a successful businessman who owned several textile factories in the Zuffenhausen district of Stuttgart, where Max was born. Moritz expected his son to own the family business. Max was taken out of school in 1910 to work in the family business, where he became a junior manager.
During this period he would begin two relationships. First, he met Friedrich Pollock, who would become a close academic colleague, who would remain Max's closest friend, he met Rose Riekher, his father's personal secretary. Eight years Max's senior, a gentile, of an economically lower class, Riekher was not considered a suitable match by Moritz Horkheimer. Despite this and Maidon would marry in 1926 and remain together until her death in 1969. In 1917, his manufacturing career ended and his chances of taking over his family business were interrupted when he was drafted into World War I. However, Horkheimer was denied service on medical grounds. In the spring of 1919, after failing an army physical, Horkheimer enrolled at Munich University. While living in Munich, he was mistaken for the revolutionary playwright Ernst Toller and arrested and imprisoned. After being released, Horkheimer moved to Frankfurt am Main, where he studied philosophy and psychology under the respectable Hans Cornelius. There, he met Theodor Adorno, several years his junior, with whom he would strike a lasting friendship and a collaborative relationship.
After an abortive attempt at writing a dissertation on gestalt psychology, with Cornelius's direction, completed his doctorate in philosophy with a 78-page dissertation titled The Antinomy of Teleological Judgment. In 1925, Horkheimer was habilitated with a dissertation entitled Kant's Critique of Judgement as Mediation between Practical and Theoretical Philosophy. Here, he met Friedrich Pollock; the following year, Max was appointed Privatdozent. Shortly after, in 1926, Horkheimer married Rose Riekher. In 1926 Horkheimer was an "unsalaried lecturer in Frankfurt." Shortly after, in 1930, he was promoted to professor of philosophy at Frankfurt University. In the same year, when the Institute for Social Research's directorship became vacant, after the departure of Carl Grünberg, Horkheimer was elected to the position "by means of an endowment from a wealthy businessman"; the Institute had had its beginnings in a Marxist study group started by Felix Weil, a one-time student of political science at Frankfurt who used his inheritance to fund the group as a way to support his leftist academic aims.
Pollock and Horkheimer were partners with Weil in the early activities of the Institute. Horkheimer worked to make the Institute a purely academic enterprise; as director, he changed Frankfurt from an orthodox Marxist school to a heterodox school for critical social research. The following year publication of the Institute's Zeitschrift für Sozialforschung began, with Horkheimer as its editor. Horkheimer intellectually reoriented the Institute, proposing a programme of collective research aimed at specific social groups that would highlight the problem of the relationship of history and reason; the Institute focused on integrating the views of Sigmund Freud. The Frankfurt School attempted this by systematically hitching together the different conceptual structures of historical materialism and psychoanalysis. During the time between Horkheimer's being named Professor of Social Philosophy and director of the Institute in 1930, the Nazis became the second largest party in the Reichstag. In the midst of the violence surrounding the Nazis' rise and his associates began to prepare for the possibility of moving the Institute out of Germany.
Horkheimer's venia legendi was revoked by the new Nazi government because of the Marxian nature of the Institute's ideas as well as its prominent Jewish association. When Hitler was named the Chancellor in 1933, the Institute was thus forced to close its location in Germany, he emigrated to Geneva, Switzerland and to New York City the following year, where Horkheimer met with the president of Columbia University to discuss hosting the Institute. To Horkheimer's surprise, the president agreed to host the Institute in exile as well as offer Horkheimer a building for the Institute. In July 1934 Horkheimer accepted an offer from Columbia to relocate the Institute to one of their buildings. In 1940, Horkheimer received American citizenship and moved to the Pacific Palisades district of Los Angeles, where his collaboration with Adorno would yield the Dialectic of Enlightenment. In 1942, Horkheimer assumed t