Lev Semyonovich Pontryagin was a Soviet mathematician. He was born in Moscow and lost his eyesight due to a primus stove explosion when he was 14. Despite his blindness he was able to become one of the greatest mathematicians of the 20th century with the help of his mother Tatyana Andreevna who read mathematical books and papers to him, he made major discoveries in a number of fields of mathematics, including algebraic topology and differential topology. Pontryagin worked on duality theory for homology while still a student, he went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality. With René Thom, he is regarded as one of the co-founders of cobordism theory, co-discoverers of the central idea of this theory, that framed cobordism and stable homotopy are equivalent; this led to the introduction around 1940 of a theory of certain characteristic classes, now called Pontryagin classes, designed to vanish on a manifold, a boundary. In 1942 he introduced.
Moreover, in operator theory there are specific instances of Krein spaces called Pontryagin spaces. In his career he worked in optimal control theory, his maximum principle is fundamental to the modern theory of optimization. He introduced there the idea of a bang-bang principle, to describe situations where either the maximum'steer' should be applied to a system, or none. Pontryagin authored several influential monographs as well as popular textbooks in mathematics. Pontryagin's students include Dmitri Anosov, Vladimir Boltyansky, Revaz Gamkrelidze, Evgeni Mishchenko, Mikhail Postnikov, Vladimir Rokhlin, Mikhail Zelikin. Pontryagin was accused of anti-Semitism on several occasions. For example, he attacked Nathan Jacobson for being a "mediocre scientist" representing the "Zionism movement", while both men were vice-presidents of the International Mathematical Union, he rejected charges in anti-Semitism in an article published in Science in 1979, claiming that he struggled with Zionism, which he considered a form of racism.
When a prominent Soviet Jewish mathematician, Grigory Margulis, was selected by the IMU to receive the Fields Medal at the upcoming 1978 ICM, a member of the Executive Committee of the IMU at the time, vigorously objected. Although the IMU stood by its decision to award Margulis the Fields Medal, Margulis was denied a Soviet exit visa by the Soviet authorities and was unable to attend the 1978 ICM in person. Pontryagin participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair. Pontrjagin, L. Topological Groups, Princeton Mathematical Series, 2, Princeton: Princeton University Press, MR 0000265 1962 - Ordinary Differential Equations 1962 - The Mathematical Theory of Optimal Processes 1963 - Foundations of Combinatorial Topology Andronov–Pontryagin criterion Kuratowski's theorem called the Pontryagin–Kuratowski theorem Pontryagin class Pontryagin duality Pontryagin's minimum principle Lev Pontryagin at the Mathematics Genealogy Project O'Connor, John J..
Autobiography of Pontryagin
The Soviet Union the Union of Soviet Socialist Republics, was a socialist state in Eurasia that existed from 1922 to 1991. Nominally a union of multiple national Soviet republics, its government and economy were centralized; the country was a one-party state, governed by the Communist Party with Moscow as its capital in its largest republic, the Russian Soviet Federative Socialist Republic. Other major urban centres were Leningrad, Minsk, Alma-Ata, Novosibirsk, it spanned over 10,000 kilometres east to west across 11 time zones, over 7,200 kilometres north to south. It had five climate zones: tundra, steppes and mountains; the Soviet Union had its roots in the 1917 October Revolution, when the Bolsheviks, led by Vladimir Lenin, overthrew the Russian Provisional Government which had replaced Tsar Nicholas II during World War I. In 1922, the Soviet Union was formed by a treaty which legalized the unification of the Russian, Transcaucasian and Byelorussian republics that had occurred from 1918. Following Lenin's death in 1924 and a brief power struggle, Joseph Stalin came to power in the mid-1920s.
Stalin committed the state's ideology to Marxism–Leninism and constructed a command economy which led to a period of rapid industrialization and collectivization. During his rule, political paranoia fermented and the Great Purge removed Stalin's opponents within and outside of the party via arbitrary arrests and persecutions of many people, resulting in at least 600,000 deaths. In 1933, a major famine struck the country. Before the start of World War II in 1939, the Soviets signed the Molotov–Ribbentrop Pact, agreeing to non-aggression with Nazi Germany, after which the USSR invaded Poland on 17 September 1939. In June 1941, Germany broke the pact and invaded the Soviet Union, opening the largest and bloodiest theatre of war in history. Soviet war casualties accounted for the highest proportion of the conflict in the effort of acquiring the upper hand over Axis forces at intense battles such as Stalingrad and Kursk; the territories overtaken by the Red Army became satellite states of the Soviet Union.
The post-war division of Europe into capitalist and communist halves would lead to increased tensions with the United States-led Western Bloc, known as the Cold War. Stalin died in 1953 and was succeeded by Nikita Khrushchev, who in 1956 denounced Stalin and began the de-Stalinization; the Cuban Missile Crisis occurred during Khrushchev's rule, among the many factors that led to his downfall in 1964. In the early 1970s, there was a brief détente of relations with the United States, but tensions resumed with the Soviet–Afghan War in 1979. In 1985, the last Soviet premier, Mikhail Gorbachev, sought to reform and liberalize the economy through his policies of glasnost and perestroika, which caused political instability. In 1989, Soviet satellite states in Eastern Europe overthrew their respective communist governments; as part of an attempt to prevent the country's dissolution due to rising nationalist and separatist movements, a referendum was held in March 1991, boycotted by some republics, that resulted in a majority of participating citizens voting in favor of preserving the union as a renewed federation.
Gorbachev's power was diminished after Russian President Boris Yeltsin's high-profile role in facing down a coup d'état attempted by Communist Party hardliners. In late 1991, Gorbachev resigned and the Supreme Soviet of the Soviet Union met and formally dissolved the Soviet Union; the remaining 12 constituent republics emerged as independent post-Soviet states, with the Russian Federation—formerly the Russian SFSR—assuming the Soviet Union's rights and obligations and being recognized as the successor state. The Soviet Union was a powerhouse of many significant technological achievements and innovations of the 20th century, including the world's first human-made satellite, the first humans in space and the first probe to land on another planet, Venus; the country had the largest standing military in the world. The Soviet Union was recognized as one of the five nuclear weapons states and possessed the largest stockpile of weapons of mass destruction, it was a founding permanent member of the United Nations Security Council as well as a member of the Organization for Security and Co-operation in Europe, the World Federation of Trade Unions and the leading member of the Council for Mutual Economic Assistance and the Warsaw Pact.
The word "Soviet" is derived from a Russian word сове́т meaning council, advice, harmony and all deriving from the proto-Slavic verbal stem of vět-iti, related to Slavic věst, English "wise", the root in "ad-vis-or", or the Dutch weten. The word sovietnik means "councillor". A number of organizations in Russian history were called "council". For example, in the Russian Empire the State Council, which functioned from 1810 to 1917, was referred to as a Council of Ministers after the revolt of 1905. During the Georgian Affair, Vladimir Lenin envisioned an expression of Great Russian ethnic chauvinism by Joseph Stalin and his supporters, calling for these nation-states to join Russia as semi-independent parts of a greater union, which he named as the Union of Soviet Republics of Europe and Asia. Stalin resisted the proposal, but accepted it, although with Lenin's agreement changed the name of the newly proposed sta
U. S. R. Murty
Uppaluri Siva Ramachandra Murty, or U. S. R. Murty, is a Professor Emeritus of the Department of Combinatorics and Optimization, University of Waterloo. U. S. R. Murty received his Ph. D. in 1967 from the Indian Statistical Institute, with a thesis on extremal graph theory. Murty is well known for his work in matroid theory and graph theory, for being a co-author with J. A. Bondy of a textbook on graph theory. Murty has served as a managing editor and co-editor-in-chief of the Journal of Combinatorial Theory, Series B. John Adrian Bondy and U. S. R. Murty, Graph Theory with Applications. North-Holland. Book's page at the University of Paris VI. John Adrian Bondy and U. A. R. Murty, "Graph Theory and Related Topics." Academic Press Inc. ISBN 978-0121143503. U. S. R. Murty How Many Magic Configurations are There? The American Mathematical Monthly. U. S. R. Murty Equicardinal matroids. Journal of Combinatorial Theory, Series B U. S. R. Murty Matroids with Sylvester property. Aequationes Mathematicae. Murty, U. S. R.
"On some extremal graphs", Acta Mathematica Academiae Scientiarum Hungaricae, 19: 69–74, doi:10.1007/BF01894681, MR 0224509 de Carvalho, Marcelo H.. "On a conjecture of Lovász concerning bricks. II. Bricks of finite characteristic", Journal of Combinatorial Theory, Series B, 85: 137–180, doi:10.1006/jctb.2001.2092, MR 1900684
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph; this drawing should not be confused with the graph itself: different layouts can correspond to the same graph. In the abstract, all that matters is. In the concrete, the arrangement of these vertices and edges within a drawing affects its understandability, fabrication cost, aesthetics; the problem gets worse if the graph changes over time by adding and deleting edges and the goal is to preserve the user's mental map. Graphs are drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane.
Node–link diagrams can be traced back to the 13th century work of Ramon Llull, who drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts. In the case of directed graphs, arrowheads form a used graphical convention to show their orientation. Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary. Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions. Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures; the crossing number of a drawing is the number of pairs of edges.
If the graph is planar it is convenient to draw it without any edge intersections. However, nonplanar graphs arise in applications, so graph drawing algorithms must allow for edge crossings; the area of a drawing is the size of its smallest bounding box, relative to the closest distance between any two vertices. Drawings with smaller area are preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly; the aspect ratio of the bounding box may be important. Symmetry display is the problem of finding symmetry groups within a given graph, finding a drawing that displays as much of the symmetry as possible; some layout methods automatically lead to symmetric drawings. It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its number of bends, many methods aim to provide drawings with few total bends or few bends per edge.
For spline curves the complexity of an edge may be measured by the number of control points on the edge. Several used quality measures concern lengths of edges: it is desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than varied. Angular resolution is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high degree it will have small angular resolution, but the angular resolution can be bounded below by a function of the degree; the slope number of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges. Cubic graphs have slope number at most four, but graphs of degree five may have unbounded slope number. There are many different graph layout strategies: In force-based layout systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of springs or molecular mechanics.
These systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform gradient descent based minimization of an energy function, or they may translate the forces directly into velocities or accelerations for the moving vertices. Spectral layout methods use as coordinates the eigenvectors of a matrix such as the Laplacian derived from the adjacency matrix of t
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Kazimierz Kuratowski was born in Warsaw, Vistula Land, on 2 February 1896, into an assimilated Jewish family, he was a son of Marek Kuratow, a barrister, Róża Karzewska. He completed a Warsaw secondary school, named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian, he completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics, he obtained his Ph. D. in 1921, in newly independent Poland. In autumn 1921 Kuratowski was awarded the Ph. D. degree for his groundbreaking work. His thesis statement consisted of two parts.
One was devoted to an axiomatic construction of topology via the closure axioms. This first part has been cited in hundreds of scientific articles; the second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Stefan Mazurkiewicz. Kuratowski's thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin. Two years in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University, he was appointed a full professor of mathematics at Lwów Polytechnic in Lwów, in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific Society While Kuratowski associated with many of the scholars of the Lwów School of Mathematics, such as Stefan Banach and Stanislaw Ulam, the circle of mathematicians based around the Scottish Café he kept close connections with Warsaw.
Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun, hence did not contribute any problems to it. He did however, collaborate with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University. A year Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. During World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968.
After World War II, Kuratowski was involved in the rebuilding of scientific life in Poland. He helped to establish the State Mathematical Institute, incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, was a long-time chairman of the Polish and International Mathematics Societies, he was president of the Scientific Council of the State Institute of Mathematics. From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski. Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów's Scottish Café, he was a member of the Warsaw Scientific Society. What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin, he was one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences.
High quality research monographs of the representatives of Warsaw's and Lwów’s School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, including the Royal Society of Edinburgh, Germany, Hungary and the Union of Soviet Socialist Republics. In 1981, IMPAN, the Polish Mathematical Society, Kuratowski's daughter Zofia Kuratowska established a prize in his name for achievements in mathematics to people under the age of 30 years; the prize is considered the most prestigious of awards for young Polish mathematicians. Kuratowski’s research focused on abstract topological and metric structures, he implemented the closure axioms. This was fundamental for the development of topological space theory and irreducible continuum theory between two points; the most valuable results, which were obtained by Kazimierz K
Three utilities problem
The classical mathematical puzzle known as the three utilities problem. Without using a third dimension or sending any of the connections through another company or cottage, is there a way to make all nine connections without any of the lines crossing each other? The problem is an abstract mathematical puzzle which imposes constraints that would not exist in a practical engineering situation, it is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar; this graph is referred to as the utility graph in reference to the problem. A review of the history of the three utilities problem is given by Kullman, he states that most published references to the problem characterize it as "very ancient". In the earliest publication found by Kullman, Henry Dudeney names it "water and electricity". However, Dudeney states that the problem is "as old as the hills...much older than electric lighting, or gas".
Dudeney published the same puzzle in The Strand Magazine in 1913. Another early version of the problem involves connecting three houses to three wells, it is stated to a different puzzle that involves three houses and three fountains, with all three fountains and one house touching a rectangular wall. Mathematically, the problem can be formulated in terms of graph drawings of the complete bipartite graph K3,3; this graph makes an early appearance in Henneberg. It has six vertices, split into two subsets of three vertices, nine edges, one for each of the nine ways of pairing a vertex from one subset with a vertex from the other subset; the three utilities problem is the question of. As it is presented, the solution to the utility puzzle is "no": there is no way to make all nine connections without any of the lines crossing each other. In other words, the graph K3,3 is not planar. Kazimierz Kuratowski stated in 1930 that K3,3 is nonplanar, from which it follows that the problem has no solution. Kullman, states that "Interestingly enough, Kuratowski did not publish a detailed proof that non-planar".
One proof of the impossibility of finding a planar embedding of K3,3 uses a case analysis involving the Jordan curve theorem. In this solution, one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding. Alternatively, it is possible to show that any bridgeless bipartite planar graph with V vertices and E edges has E ≤ 2V − 4 by combining the Euler formula V − E + F = 2 with the observation that the number of faces is at most half the number of edges. In the utility graph, E = 9 and 2V − 4 = 8, violating this inequality, so the utility graph cannot be planar. Two important characterizations of planar graphs, Kuratowski's theorem that the planar graphs are the graphs that contain neither K3,3 nor the complete graph K5 as a subdivision, Wagner's theorem that the planar graphs are the graphs that contain neither K3,3 nor K5 as a minor, make use of and generalize the non-planarity of K3,3.
K3,3 is a toroidal graph. In terms of the three cottage problem this means the problem can be solved by punching two holes through the plane and connecting them with a tube; this changes the topological properties of the surface and using the tube allows the three cottages to be connected without crossing lines. An equivalent statement is that the graph genus of the utility graph is one, therefore it cannot be embedded in a surface of genus less than one. A surface of genus one is equivalent to a torus. A toroidal embedding of K3,3 may be obtained by replacing the crossing by a tube, as described above, in which the two holes where the tube connects to the plane are placed along one of the crossing edges on either side of the crossing. Another way of changing the rules of the puzzle is to allow utility lines to pass through the cottages or utilities. Pál Turán's "brick factory problem" asks more for a formula for the minimum number of crossings in a drawing of the complete bipartite graph Ka,b in terms of the numbers of vertices a and b on the two sides of the bipartition.
The utility graph K3,3 may be drawn with only one crossing, but not with zero crossings, so its crossing number is one. The utility graph K3,3 is a circulant graph, it is the smallest triangle-free cubic graph. Like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two maximal independent sets are the two sides of the bipartition, they are equal. K3,3 is one of only seven 3-regular 3-connected well-covered graphs, it is a Laman graph, meaning that it forms a minimally rigid system when it is embedded in the plane. It is the smallest example of a nonplanar
In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment