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Raymond Louis Wilder
Raymond Louis Wilder (3 November 1896 in Palmer, Massachusetts – 7 July 1982 in Santa Barbara, California) was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests. Life Wilder's father was a printer. Raymond was musically inclined. He played cornet in the family orchestra, which performed at dances and fairs, and accompanied silent films on the piano. He entered Brown University in 1914, intending to become an actuary. During World War I, he served in the U.S. Navy as an ensign. Brown awarded him his first degree in 1920, and a master's degree in actuarial mathematics in 1921. That year, he married Una Maude Greene; they had four children, thanks to whom they have ample descent. Wilder chose to do his Ph.D. at the University of Texas at Austin, the most fateful decision of his life. At Texas, Wilder discovered pure mathematics and topology, thanks to the remarkable influence of Robert Lee Moore, the founder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Steenrod
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled ''Universal homology groups''. Steenrod held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children. Work Thanks to Lefschetz and others, the cup product structure of cohomology was understood by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of Abelian group, abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for Topological space, topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a Cochain complexes, cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation,Gerard O'Regan (2016): ''Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications'' or the reciprocal L^ of the relation L. Other notations for the converse relation include L^, L^, \breve, L^, or L^. The notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complementary Domain
Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visual arts Biology and medicine *Complement system (immunology), a cascade of proteins in the blood that form part of innate immunity *Complementary DNA, DNA reverse transcribed from a mature mRNA template *Complementarity (molecular biology), a property whereby double stranded nucleic acids pair with each other *Complementation (genetics), a test to determine if independent recessive mutant phenotypes are caused by mutations in the same gene or in different genes Grammar and linguistics * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase adding to a clause's subject after a linking verb **Object complement, a word or phrase adding to the direct object of a verb phrase. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Closed Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Curve Theorem
In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be confused with the interior (topology), interior of a set) and an "exterior" region containing all of the nearby and far away exterior points. Every path (topology), continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. The Jordan curve theorem is named after the mathematician Camil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Moritz Schönflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoenflies was born in Landsberg an der Warthe (modern Gorzów, Poland). Arthur Schoenflies married Emma Levin (1868–1939) in 1896. He studied under Ernst Kummer and Karl Weierstrass, and was influenced by Felix Klein. The Schoenflies problem is to prove that an (n - 1)-sphere in Euclidean ''n''-space bounds a topological ball, however embedded. This question is much more subtle than it initially appears. He studied at the University of Berlin from 1870 to 1875. He obtained a doctorate in 1877, and in 1878 he was a teacher at a school in Berlin. In 1880, he went to Colmar to teach. Schoenflies was a frequent contributor to Klein's ''Encyclopedia of Mathematical Sciences'': In 1898 he wrote on set theory, in 1902 on kinematics, and on project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Michigan
The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Michigan is one of the earliest American research universities and is a founding member of the Association of American Universities. In the fall of 2023, the university employed 8,189 faculty members and enrolled 52,065 students in its programs. The university is Carnegie Classification of Institutions of Higher Education, classified among "R1: Doctoral Universities – Very high research activity". It consists of nineteen colleges and offers 250 degree programs at the undergraduate and graduate levels. The university is Higher education accreditation in the United States, accredited by the Higher Learning Commission. In 2021, it ranked third among American universities in List of countries by research and development spending, research expe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States National Academy Of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the National Academy of Medicine (NAM). As a national academy, new members of the organization are elected annually by current members, based on their distinguished and continuing achievements in original research. Election to the National Academy is one of the highest honors in the scientific field in the United States. Members of the National Academy of Sciences serve '' pro bono'' as "advisers to the nation" on science, engineering, and medicine. The group holds a congressional charter under Title 36 of the United States Code. Congress legislated and President Abraham Lincoln signed an Act of Congress (1863) establishing the National Academy of Sciences as an independent, trusted nongovernmental institution, created for the purpose of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ..., college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 11 Dupont in the Dupont Circle, Washington, D.C., Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the ''American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
