American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Francis for the Mathematical Association of America; the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals; the American Mathematical Monthly is the most read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997-2010 are available online; the MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the American Mathematical Monthly. 2017-: Susan Colley 2012-2016: Scott T. Chapman 2007-2011: Daniel J. Velleman 2002-2006: Bruce Palka 1997-2001: Roger A.
Horn 1992-1996: John H. Ewing 1987-1991: Herbert S. Wilf 1982-1986: Paul Richard Halmos 1978-1981: Ralph Philip Boas, Jr. 1977-1978: Alex Rosenberg and Ralph Philip Boas Jr. 1974-1976: Alex Rosenberg 1969-1973: Harley Flanders 1967-1968: Robert Abraham Rosenbaum 1962-1966: Frederick Arthur Ficken 1957-1961: Ralph Duncan James 1952-1956: Carl Barnett Allendoerfer 1947-1951: Carroll Vincent Newsom 1942-1946: Lester Randolph Ford 1937-1941: Elton James Moulton 1932-1936: Walter Buckingham Carver 1927-1931: William Henry Bussey 1923-1926: Walter Burton Ford 1922: Albert Arnold Bennett 1919-1921: Raymond Clare Archibald 1918: Robert Daniel Carmichael 1916-1917: Herbert Ellsworth Slaught 1914-1915: Board of editors: C. H. Ashton, R. P. Baker, W. C. Brenke, W. H. Bussey, W. DeW. Cairns, Florian Cajori, R. D. Carmichael, D. R. Curtiss, I. M. DeLong, B. F. Finkel, E. R. Hedrick, L. C. Karpinski, G. A. Miller, W. H. Roever, H. E. Slaught 1913: Herbert Ellsworth Slaught 1909-1912: Benjamin Franklin Finkel, Herbert Ellsworth Slaught, George Abram Miller 1907-1908: Benjamin Franklin Finkel, Herbert Ellsworth Slaught 1905-1906: Benjamin Franklin Finkel, Leonard Eugene Dickson, Oliver Edmunds Glenn 1904: Benjamin Franklin Finkel, Leonard Eugene Dickson, Saul Epsteen 1903: Benjamin Franklin Finkel, Leonard Eugene Dickson 1894-1902: Benjamin Franklin Finkel, John Marvin Colaw Mathematics Magazine Notices of the American Mathematical Society, another "most read mathematics journal in the world" American Mathematical Monthly homepage Archive of tables of contents with article summaries Mathematical Association of America American Mathematical Monthly on JSTOR The American mathematical monthly, hathitrust
Øystein Ore
Øystein Ore was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, the history of mathematics. Ore graduated from the University of Oslo with a Cand. Scient. Degree in mathematics. In 1924, the University of Oslo awarded him the Ph. D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem. Ore studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra, he was a fellow at the Mittag-Leffler Institute in Sweden, spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo. Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928 to full professor in 1929. In 1931, he became a Sterling Professor, a position he held until he retired in 1968. Ore gave an American Mathematical Society Colloquium lecture in 1941 and was a plenary speaker at the International Congress of Mathematicians in 1936 in Oslo.
He was elected to the American Academy of Arts and Sciences and the Oslo Academy of Science. He was a founder of the Econometric Society. Ore visited Norway nearly every summer. During World War II, he was active in the "American Relief for Norway" and "Free Norway" movements. In gratitude for the services rendered to his native country during the war, he was decorated in 1947 with the Order of St. Olav. In 1930, Ore married Gudrun Lundevall, they had two children. Ore had a passion for painting and sculpture, collected ancient maps, spoke several languages. Ore is known for his work in ring theory, Galois connections, most of all, graph theory, his early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring, he examined polynomial rings over skew fields, attempted to extend his work on factorisation to non-commutative rings. The Ore condition, which allows a ring of fractions to be defined, the Ore extension, a non-commutative analogue of rings of polynomials, are part of this work.
In more elementary number theory, Ore's harmonic numbers are the numbers whose divisors have an integer harmonic mean. As a teacher, Ore is notable for teaching mathematics to two doctoral students who would make contributions to science and mathematics, Grace Hopper, who would become a United States rear admiral and computer scientist, a pioneer in the development of the first computers, Marshall Hall, Jr. an American mathematician who did important research in group theory and combinatorics. In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether, he turned his attention to lattice theory becoming, together with Garrett Birkhoff, one of the two founders of American expertise in the subject. Ore's early work on lattice theory led him to the study of equivalence and closure relations, Galois connections, to graph theory, which occupied him to the end of his life, he wrote one on the theory of graphs and another on their applications.
Within graph theory, Ore's theorem is one of several results proving that sufficiently dense graphs contain Hamiltonian cycles. Ore had a lively interest in the history of mathematics, was an unusually able author of books for laypeople, such as his biographies of Cardano and Niels Henrik Abel. Les Corps Algébriques et la Théorie des Idéaux L'Algèbre Abstraite Number Theory and its History Cardano, the Gambling Scholar Niels Henrik Abel, Mathematician Extraordinary Theory of Graphs Graphs and Their Uses The Four-Color Problem Invitation to Number Theory O'Connor, John J.. The source for much of this entry. Øystein Ore at the Mathematics Genealogy Project
Norway
Norway the Kingdom of Norway, is a Nordic country in Northern Europe whose territory comprises the western and northernmost portion of the Scandinavian Peninsula. The Antarctic Peter I Island and the sub-Antarctic Bouvet Island are dependent territories and thus not considered part of the kingdom. Norway lays claim to a section of Antarctica known as Queen Maud Land. Norway has a total area of 385,207 square kilometres and a population of 5,312,300; the country shares a long eastern border with Sweden. Norway is bordered by Finland and Russia to the north-east, the Skagerrak strait to the south, with Denmark on the other side. Norway has an extensive coastline, facing the Barents Sea. Harald V of the House of Glücksburg is the current King of Norway. Erna Solberg has been prime minister since 2013. A unitary sovereign state with a constitutional monarchy, Norway divides state power between the parliament, the cabinet and the supreme court, as determined by the 1814 constitution; the kingdom was established in 872 as a merger of a large number of petty kingdoms and has existed continuously for 1,147 years.
From 1537 to 1814, Norway was a part of the Kingdom of Denmark-Norway, from 1814 to 1905, it was in a personal union with the Kingdom of Sweden. Norway was neutral during the First World War. Norway remained neutral until April 1940 when the country was invaded and occupied by Germany until the end of Second World War. Norway has both administrative and political subdivisions on two levels: counties and municipalities; the Sámi people have a certain amount of self-determination and influence over traditional territories through the Sámi Parliament and the Finnmark Act. Norway maintains close ties with both the United States. Norway is a founding member of the United Nations, NATO, the European Free Trade Association, the Council of Europe, the Antarctic Treaty, the Nordic Council. Norway maintains the Nordic welfare model with universal health care and a comprehensive social security system, its values are rooted in egalitarian ideals; the Norwegian state has large ownership positions in key industrial sectors, having extensive reserves of petroleum, natural gas, lumber and fresh water.
The petroleum industry accounts for around a quarter of the country's gross domestic product. On a per-capita basis, Norway is the world's largest producer of oil and natural gas outside of the Middle East; the country has the fourth-highest per capita income in the world on the World IMF lists. On the CIA's GDP per capita list which includes autonomous territories and regions, Norway ranks as number eleven, it has the world's largest sovereign wealth fund, with a value of US$1 trillion. Norway has had the highest Human Development Index ranking in the world since 2009, a position held between 2001 and 2006, it had the highest inequality-adjusted ranking until 2018 when Iceland moved to the top of the list. Norway ranked first on the World Happiness Report for 2017 and ranks first on the OECD Better Life Index, the Index of Public Integrity, the Democracy Index. Norway has one of the lowest crime rates in the world. Norway has two official names: Norge in Noreg in Nynorsk; the English name Norway comes from the Old English word Norþweg mentioned in 880, meaning "northern way" or "way leading to the north", how the Anglo-Saxons referred to the coastline of Atlantic Norway similar to scientific consensus about the origin of the Norwegian language name.
The Anglo-Saxons of Britain referred to the kingdom of Norway in 880 as Norðmanna land. There is some disagreement about whether the native name of Norway had the same etymology as the English form. According to the traditional dominant view, the first component was norðr, a cognate of English north, so the full name was Norðr vegr, "the way northwards", referring to the sailing route along the Norwegian coast, contrasting with suðrvegar "southern way" for, austrvegr "eastern way" for the Baltic. In the translation of Orosius for Alfred, the name is Norðweg, while in younger Old English sources the ð is gone. In the 10th century many Norsemen settled in Northern France, according to the sagas, in the area, called Normandy from norðmann, although not a Norwegian possession. In France normanni or northmanni referred to people of Sweden or Denmark; until around 1800 inhabitants of Western Norway where referred to as nordmenn while inhabitants of Eastern Norway where referred to as austmenn. According to another theory, the first component was a word nór, meaning "narrow" or "northern", referring to the inner-archipelago sailing route through the land.
The interpretation as "northern", as reflected in the English and Latin forms of the name, would have been due to folk etymology. This latter view originated with philologist Niels Halvorsen Trønnes in 1847; the form Nore is still used in placenames such as the village of Nore and lake Norefjorden in Buskerud county, still has the same meaning. Among other arguments in favour of the theor
Strongly connected component
In the mathematical theory of directed graphs, a graph is said to be connected or diconnected if every vertex is reachable from every other vertex. The connected components or diconnected components of an arbitrary directed graph form a partition into subgraphs that are themselves connected, it is possible to test the strong connectivity of a graph, or to find its connected components, in linear time. A directed graph is called connected if there is a path in each direction between each pair of vertices of the graph; that is, a path exists from the first vertex in the pair to the second, another path exists from the second vertex to the first. In a directed graph G that may not itself be connected, a pair of vertices u and v are said to be connected to each other if there is a path in each direction between them; the binary relation of being connected is an equivalence relation, the induced quotientgraphs of its equivalence classes are called connected components. Equivalently, a connected component of a directed graph G is a subgraph, connected, is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being connected.
The collection of connected components forms a partition of the set of vertices of G. If each connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no connected subgraphs with more than one vertex, because a directed cycle is connected and every nontrivial connected component contains at least one directed cycle. Several algorithms based on depth first search compute connected components in linear time. Kosaraju's algorithm uses two passes of depth first search; the first, in the original graph, is used to choose the order in which the outer loop of the second depth first search tests vertices for having been visited and recursively explores them if not. The second depth first search is on the transpose graph of the original graph, each recursive exploration finds a single new connected component, it is named after S. Rao Kosaraju, who described it in 1978. Tarjan's connected components algorithm, published by Robert Tarjan in 1972, performs a single pass of depth first search.
It maintains a stack of vertices that have been explored by the search but not yet assigned to a component, calculates "low numbers" of each vertex which it uses to determine when a set of vertices should be popped off the stack into a new component. The path-based strong component algorithm uses a depth first search, like Tarjan's algorithm, but with two stacks. One of the stacks is used to keep track of the vertices not yet assigned to components, while the other keeps track of the current path in the depth first search tree; the first linear time version of this algorithm was published by Edsger W. Dijkstra in 1976. Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two. Previous linear-time algorithms are based on depth-first search, considered hard to parallelize. Fleischer et al. in 2000 proposed a divide-and-conquer approach based on reachability queries, such algorithms are called reachability-based SCC algorithms.
The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a connected component has to be contained in one of the subsets; the vertex subset reached by both searches forms a connected components, the algorithm recurses on the other 3 subsets. The expected sequential running time of this algorithm is shown to be O, a factor of O more than the classic algorithms; the parallelism comes from: the reachability queries can be parallelized more easily. This algorithm performs well on real-world graphs, but does not have theoretical guarantee on the parallelism. Blelloch et al. in 2016 shows that if the reachability queries are applied in a random order, the cost bound of O still holds. Furthermore, the queries can be batched in a prefix-doubling manner and run in one round; the overall span of this algorithm is log2 n reachability queries, the optimal parallelism that can be achieved using the reachability-based approach.
Algorithms for finding connected components may be used to solve 2-satisfiability problems: as Aspvall, Plass & Tarjan showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable v such that v and its complement are both contained in the same connected component of the implication graph of the instance. Connected components are used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph. A directed graph is connected