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Combinatorial Topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group. While this kind of topology was still "combinatorial" in 1942, it had become "algebraic" by 1944. This corresponds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azriel Rosenfeld
Azriel Rosenfeld (February 19, 1931 – February 22, 2004) was an American Research Professor, a Distinguished University Professor, and Director of the Center for Automation Research at the University of Maryland, College Park, Maryland, where he also held affiliate professorships in the Departments of Computer Science, Electrical Engineering, and Psychology. He was a leading researcher in the field of computer image analysis. Over a period of nearly 40 years, he made many fundamental and pioneering contributions to nearly every area of that field. He wrote the first textbook in the field (1969); was founding editor of its first journal, '' Computer Graphics and Image Processing'' (1972); and was co-chairman of its first international conference (1987). He published over 30 books and over 600 book chapters and journal articles, and directed nearly 60 Ph.D. dissertations. Rosenfeld's research on digital image analysis (specifically on digital geometry and digital topology, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bar-Ilan University
Bar-Ilan University (BIU, , ''Universitat Bar-Ilan'') is a public research university in the Tel Aviv District city of Ramat Gan, Israel. Established in 1955, Bar Ilan is Israel's second-largest academic university institution. It has 20,000 students and 1,350 faculty members. Bar-Ilan's mission is to "blend Jewish tradition with modern technologies and scholarship and the university endeavors to ... teach the Jewish heritage to all its students while providing nacademic education." The university is among the best in the Middle East in the fields of computer science, engineering, engineering physics and applied physics. In 2024, the university was donated $260 million, one of the biggest donations to a university in Israeli history. History Bar-Ilan University has Jewish-American roots: It was conceived in Atlanta in a meeting of the American Mizrahi organization in 1950, and was founded by Professor Pinkhos Churgin, an American Orthodox rabbi and educator, who was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Graph Theory
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem. Other applications can be found in printing electronic circuits where the aim is to print (embed) a circuit (the graph) on a circuit board (the surface) without two connections crossing each other and resulting in a short circuit. Graphs as topological spaces To an undirected graph we may associate an abstract simplicial complex ''C'' with a single-element set per vertex and a two-element set per edge. The geometric realization , ''C'', of the complex consists of a copy of the unit interval ,1per edge, with the endpoints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory, Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a ''k''-connected graph ''G'', ''k'' points v_1,\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hauptvermutung
The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two Triangulation (topology), triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, but it is now known to be false. History The non-manifold version was disproved by John Milnor in 1961 using Analytic torsion, Reidemeister torsion. The manifold version is true in dimensions m\le 3. The cases m = 2 and 3 were mathematical proof, proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively. An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–69 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H^3(M;\mathbb/2\mathbb). In dimension m \ge 5, a homeomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Cell Topology
The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics. The elements of the ''n''-dimensional grid cell topology (''n'' ≥ 1) are all ''n''-dimensional grid cubes and their ''k''-dimensional faces ( for 0 ≤ ''k'' ≤ ''n''−1); between these a partial order ''A'' ≤ ''B'' is defined if ''A'' is a subset of ''B'' (and thus also dim(''A'') ≤ dim(''B'')). The grid cell topology is the Alexandrov topology (open sets are up-sets) with respect to this partial order. (See also poset topology.) Alexandrov and Hopf first introduced the grid cell topology, for the two-dimensional case, within an exercise in their text ''Topologie'' I (1935). A recursive method to obtain ''n''-dimensional grid cells and an intuitive definition for grid cell manifolds can be found in Chen, 2004. It is related to digital manifold In mathematics, a digital manifold is a special kind of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a Surface (topology), surface to its underlying topology. In the simplest application, the case of a triangle Euclidean geometry, on a plane, the Sum of angles of a triangle, sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a Compact space, compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the volume element, element of area of the surface, and is the line element along the bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
