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
Jakob Nielsen (mathematician)
For other people with similar names see Jakob Nielsen. Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces, he was born in the village Mjels on the island of Als in modern-day Denmark. His mother died when he was 3, in 1900 he went to live with his aunt and was enrolled in the Realgymnasium. In 1907 he was expelled for membership to an illicit student club, he matriculated at the University of Kiel in 1908. Nielsen completed his doctoral dissertation in 1913. Soon thereafter, he was drafted into the German Imperial Navy, he was assigned to coastal defense. In 1915 he was sent to Constantinople as a military adviser to the Turkish Government. After the war, in the spring of 1919, Nielsen married Carola von Pieverling, a German medical doctor. In 1920 Nielsen took a position at the Technical University of Breslau; the next year he published a paper in Mathematisk Tidsskrift in which he proved that any subgroup of a finitely generated free group is free. In 1926 Otto Schreier would generalize this result by removing the condition that the free group be finitely generated.
In 1921 Nielsen moved to the Royal Veterinary and Agricultural University in Copenhagen, where he would stay until 1925, when he moved to the Technical University in Copenhagen. He proved the Dehn–Nielsen theorem on mapping class groups. Nielsen was a Plenary Speaker of the ICM in 1936 in Oslo. During World War II some efforts were made to bring Nielsen to the United States as it was feared that he would be assaulted by the Nazis. Nielsen would, in fact, stay in Denmark during the war without being harassed by the Nazis. In 1951 Nielsen became professor of mathematics at the University of Copenhagen, taking the position vacated by the death of Harald Bohr, he resigned this position in 1955 because of his international undertakings, in particular with UNESCO, where he served on the executive board from 1952 to 1958. Fenchel, Werner. Nielsen, Hansen, Vagn Lundsgaard, ed. Jakob Nielsen: collected mathematical papers. Vol. 1, Contemporary Mathematicians, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3140-6, MR 0865335 Nielsen, Hansen, Vagn Lundsgaard, ed. Jakob Nielsen: collected mathematical papers.
Vol. 2, Contemporary Mathematicians, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3151-2, MR 0865336 O'Connor, John J.. Fenchel–Nielsen coordinates Nielsen transformation Nielsen theory Nielsen–Thurston classification Nielsen realization problem O'Connor, John J.. Jakob Nielsen at the Mathematics Genealogy Project
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Americans are nationals and citizens of the United States of America. Although nationals and citizens make up the majority of Americans, some dual citizens and permanent residents may claim American nationality; the United States is home to people of many different ethnic origins. As a result, American culture and law does not equate nationality with race or ethnicity, but with citizenship and permanent allegiance. English-speakers, speakers of many other languages use the term "American" to mean people of the United States; the word "American" can refer to people from the Americas in general. The majority of Americans or their ancestors immigrated to America or are descended from people who were brought as slaves within the past five centuries, with the exception of the Native American population and people from Hawaii, Puerto Rico and the Philippine Islands, who became American through expansion of the country in the 19th century, additionally America expanded into American Samoa, the U. S. Virgin Islands and Northern Mariana Islands in the 20th century.
Despite its multi-ethnic composition, the culture of the United States held in common by most Americans can be referred to as mainstream American culture, a Western culture derived from the traditions of Northern and Western European colonists and immigrants. It includes influences of African-American culture. Westward expansion integrated the Creoles and Cajuns of Louisiana and the Hispanos of the Southwest and brought close contact with the culture of Mexico. Large-scale immigration in the late 19th and early 20th centuries from Southern and Eastern Europe introduced a variety of elements. Immigration from Asia and Latin America has had impact. A cultural melting pot, or pluralistic salad bowl, describes the way in which generations of Americans have celebrated and exchanged distinctive cultural characteristics. In addition to the United States and people of American descent can be found internationally; as many as seven million Americans are estimated to be living abroad, make up the American diaspora.
The United States of America is a diverse country and ethnically. Six races are recognized by the U. S. Census Bureau for statistical purposes: White, American Indian and Alaska Native, Black or African American, Native Hawaiian and Other Pacific Islander, people of two or more races. "Some other race" is an option in the census and other surveys. The United States Census Bureau classifies Americans as "Hispanic or Latino" and "Not Hispanic or Latino", which identifies Hispanic and Latino Americans as a racially diverse ethnicity that comprises the largest minority group in the nation. People of European descent, or White Americans, constitute the majority of the 308 million people living in the United States, with 72.4% of the population in the 2010 United States Census. They are considered people who trace their ancestry to the original peoples of Europe, the Middle East, North Africa. Of those reporting to be White American, 7,487,133 reported to be Multiracial. Additionally, there are Latinos.
Non-Hispanic Whites are the majority in 46 states. There are four minority-majority states: California, New Mexico, Hawaii. In addition, the District of Columbia has a non-white majority; the state with the highest percentage of non-Hispanic White Americans is Maine. The largest continental ancestral group of Americans are that of Europeans who have origins in any of the original peoples of Europe; this includes people via African, North American, Central American or South American and Oceanian nations that have a large European descended population. The Spanish were some of the first Europeans to establish a continuous presence in what is now the United States in 1565. Martín de Argüelles born 1566, San Agustín, La Florida a part of New Spain, was the first person of European descent born in what is now the United States. Twenty-one years Virginia Dare born 1587 Roanoke Island in present-day North Carolina, was the first child born in the original Thirteen Colonies to English parents. In the 2017 American Community Survey, German Americans, Irish Americans, English Americans and Italian Americans were the four largest self-reported European ancestry groups in the United States forming 35.1% of the total population.
However, the English Americans and British Americans demography is considered a serious under-count as they tend to self-report and identify as "Americans" due to the length of time they have inhabited America. This is over-represented in the Upland South, a region, settled by the British. Overall, as the largest group, European Americans have the lowest poverty rate and the second highest educational attainment levels, median household income, median personal income of any racial demographic in the nation. According to the American Jewish Archives and the Arab American National Museum, some of the first Middle Easterners and North Africans arrived in the Americas between the late 15th and mid-16th centuries. Many were fleeing ethnic or ethnoreligious persecution during the Spanish Inquisition, a few were taken to the Americas as slaves. In 2014, The United States Census Bureau began finalizing the ethnic classification of MENA populations. According to the Arab American Institute, Arab
St. John's College (Annapolis/Santa Fe)
St. John's College is a private liberal arts college with dual campuses in Annapolis and Santa Fe, New Mexico, which are ranked separately by U. S. News & World Report within the top 100 National Liberal Arts Colleges, it is known for its distinctive curriculum centered on reading and discussing the Great Books of Western Civilization. St. John's has no religious affiliation. St. John's claims to be of the oldest institutions of higher learning in the United States,as the successor institution of King William's School, a preparatory school founded in 1696. In 1937, St. John's adopted a Great Books curriculum based on discussion of works from the Western canon of philosophical, historical, mathematical and literary works; the school grants only one bachelor's degree, in "Liberal Arts." Two master's degrees are available through the college's Graduate Institute—one in "Liberal Arts,", a modified version of the undergraduate curriculum, one in "Eastern Classics," which applies most of the features of the undergraduate curriculum to a list of classic works from India and Japan.
The Master of Arts in Eastern Classics is only available at the Santa Fe campus. The average admittance rate for Fall 2018 undergraduate students was 60 percent: Santa Fe campus and the Annapolis campus. St. John's College traces its origins to King William's School, founded in 1696. In 1784, Maryland chartered St. John's College, which absorbed King William's School when it opened 1785; the college took up residence in a building known as Bladen's Folly, built to be the Maryland governor's mansion, but was not completed. There was some association with the Freemasons early in the college's history, leading to speculation that it was named after Saint John the Evangelist; the college's original charter, reflecting the Masonic value of religious tolerance as well as the religious diversity of the founders stated that "youth of all religious denominations shall be and liberally admitted". The college always maintained a small size enrolling fewer than 500 men at a time. In its early years, the college was at least nominally public—the college's founders had envisaged it as the Western Shore branch of a proposed “University of Maryland” but a lack of enthusiasm from the Maryland General Assembly and its Eastern Shore counterpart, Washington College, made this a paper institution.
After years of inconsistent funding and litigation, the college accepted a smaller annual grant in lieu of being funded through the state's annual appropriations process. During the Civil War, the college closed and its campus was used as a military hospital. In 1907 it became the undergraduate college of a loosely organized "University of Maryland" that included the professional schools located in Baltimore. By 1920, when Maryland State College became the University of Maryland at College Park, St. John's was a free-standing private institution; the college curriculum has taken various forms throughout its history. It began with a general program of study in the liberal arts, but St. John's was a military school for much of the late 19th century and early 20th century, it ended compulsory military training with Major Enoch Garey's accession as president in 1923. Garey and the Navy instituted a Naval Reserve unit in September 1924, creating the first-ever collegiate Department of Naval Science in the United States.
But despite St. John's pioneering the entire NROTC movement, student interest waned, the voluntary ROTC disappeared in 1926 with Garey's departure, the Naval Reserve unit followed by 1929. In 1936, the college lost its accreditation; the Board of Visitors and Governors, faced with dire financial straits caused by the Great Depression, invited educational innovators Stringfellow Barr and Scott Buchanan to make a fresh start. They introduced a new program of study. Buchanan became dean of the college. In his guide Cool Colleges, Donald Asher writes that the New Program was implemented to save the college from closing: "Several benefactors convinced the college to reject a watered-down curriculum in favor of becoming a distinctive academic community, thus this great institution was reborn as a survival measure."In 1938, Walter Lippman wrote a column praising liberal arts education as a bulwark against fascism, said "in the future, men will point to St. John's College and say that there was the seed-bed of the American renaissance."In 1940, national attention was attracted to St. John's by a story in Life entitled "The Classics: At St. John's They Come into Their Own Once More".
Classic works unavailable in English translation were translated by faculty members, typed and bound. They were sold to the general public as well as to students, by 1941 the St. John's College bookshop was famous as the only source for English translations of works such as Copernicus's De revolutionibus orbium coelestium, St. Augustine's De Musica, Ptolemy's Almagest; the wartime years were difficult for the all-male St. John's. Enlistment and the draft all but emptied the college. From 1940 to 1946, St. John's was confronted with threats of its land being seized by the Navy for expansion of the neighboring U. S. Naval Academy, James Forrestal, S
Norwegian Institute of Technology
The Norwegian Institute of Technology was a science institute in Trondheim, Norway. It was established in 1910, existed as an independent technical university for 58 years, after which it was merged into the University of Trondheim as an independent college. In 1996 NTH ceased to exist as an organizational superstructure when the university was restructured and rebranded; the former NTH departments are now basic building blocks of the Norwegian University of Science and Technology. NTH was a polytechnic institute, educating master level engineers as well as architects. In 1992 NTH had 1591 employees. Ing.. The operating budget was equivalent to US$100M, the total premises amounted to around 260,000 m2. Since the merger, it forms a part of the university campus known as Gløshaugen, from the geographical area in which it is situated; the decision to establish a Norwegian national college of technology was made by the Norwegian parliament, the Storting, in 1900, after years of heated debate on where the institution should be located.
However Den Tekniske Høgskole was located in the geographically central city of Trondheim, based on an emerging policy of decentralisation as well as the city's existing and esteemed technical college, Trondhjems Tekniske Læreanstalt. Hovedbygningen, the building of Norges tekniske høgskole was designed by architect Bredo Greve, it was built of granite block construction in the National Romantic style of architecture. Five academical departments were present in the parliament's resolution of 31 May 1900: Architecture and city planning Civil engineering Mechanical engineering Electrical engineering Chemistry This section is in its early stages; this will at least entail: 1) early years, pre-WWII history, incl Samfundet. Design Town and Regional Planning Faculty of Applied Earth Science and Metallurgy, with 3 Departments: Metallurgy Geology and Mineral Resources Petroleum Technology and Applied Geophysics Faculty of Civil Engineering, with 8 Departments: Building and Construction Engineering Geotechnical Engineering Road and Railway Engineering Transportation Engineering Hydraulic and Sanitary Engineering Building Materials Structural Engineering Geodesy and Photogrammetry Faculty of Electrical Engineering and Computer Science, with 5 Departments: Electrical Power Engineering Telecommunications Engineering Cybernetics Physical Electronics Computer Systems and Telematics Faculty of Chemistry and Chemical Engineering, with 7 Departments: Inorganic Chemistry Organic Chemistry Physical Chemistry Chemical Engineering Industrial Chemistry Industrial Biochemistry Biotechnology Faculty of Mechanical Engineering, with 6 Departments: Thermal Energy and Hydropower Machine Design and Materials Technology Production and Quality Engineering Applied Mechanics, Thermo- and Fluid Dynamics Heating and Ventilation Refrigeration Engineering Faculty of Physics and Mathematics, with two Departments: Mathematics Sciences Physics Faculty of Marine Technology, with 4 Departments: Marine Systems Design Marine Structures Marine Hydrodynamics Marine Engineering Faculty of Economics and Industrial Management, with two Departments: Economics Organisation and Work Science Center for Management Education Technical University Library of Norway The national resource library of technology and architecture Locations: Technical Main Library as well as six Faculty Libraries on campus Jens G. Balchen, electronics engr. professor, "father of Norwegian cybernetics", IEEE fellow Alf Egil Bogen, electronics engr. co-inventor of Atmel AVR µcontroller, co-founder of Atmel Norway Ivar Brandvold, Chief Operating Officer of DNO ASA Helmer Dahl, electronics engr.
World War II radar and ASDIC pioneer and industry mentor, technology historian Johannes Falnes, wave energy researcher Asbjorn Folling - chemical engr. Discovery of Phenylketonuria, Jahreprisen 1960 Ivar Giaever, mechanical engr. physicist, 1973 Nobel laureate Bjarne Hurlen, mechanical engr. army officer, defence industry executive Ralph Høibakk, computer industry executive, adventurer Fred Kavli, innovator, business leader, philanthropist Paal Kibsgaard, petroleum engineer, chairman and CEO of Schlumberger Arne Korsmo - architect, Norwegian National Academy of Craft and Art Industry Olav Landsverk, electronics engr. military weapon systems computer pioneer, professor John M. Lervik, electronics engr. co-founder and CEO of cXense, co-founder and former CEO of Fast Search & Transfer Finn Lied, electronics engr. World War II resistance agent, defence research director, Minister of Industry Terje Michalsen, electronics engr. Venture capitalist Lars Monrad Krohn, electronics engr. Industrialist Ingvild Myhre, electronics engr. telecom industry executiv