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Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played an important role in the modernization of knot theory and of bringing it into the mainstream. Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, ''On the Lusternick–Schnirelmann Category'', was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morrisville, Bucks County, Pennsylvania
Morrisville (, ) is a borough in Bucks County, Pennsylvania, United States. It is located just below the falls of the Delaware River opposite Trenton, New Jersey. The population was 9,809 in the 2020 census. Morrisville is located northeast of Philadelphia and southeast of Allentown. History The earliest known settlement in what is now Morrisville, was a trading post of the Dutch West India Company operating from 1624 to 1627 on an island in the Delaware River. In its early days, the area was known as Crewcorne and was a part of Falls Township. Later, one of the first ferries to cross the Delaware was established at the site. By the late 18th century, a settlement was forming at the ferry crossing then known as Colvin's Ferry. The settlement incorporated into a borough in 1804, taking the name of Morrisville, after Founding Father Robert Morris. In that same year, the first bridge began being built across the Delaware connected Morrisville to Trenton. It would welcome traf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fox–Artin Arc
In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. found the first example of a wild arc. found another example, called the Fox-Artin arc, whose complement is not simply connected. Fox-Artin arcs Two very similar wild arcs appear in the article. Example 1.1 (page 981) is most generally referred to as the Fox-Artin wild arc. The crossings have the regular sequence over/over/under/over/under/under when following the curve from left to right. The left end-point 0 of the closed unit interval ,1/math> is mapped by the arc to the left limit point of the curve, and 1 is mapped to the right limit point. The range of the arc lies in the Euclidean space \mathbb^3 or the 3-sphere S^3. Fox-Artin arc variant Example 1.1* has the crossing sequence over/under/over/under/over/under. According to , pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Spaces
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if and also itself are vector spaces over , the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact-open Topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945. If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain. Definition Let and be two topological spaces, and let denote the set of all continuous maps between and . Given a compact subset of and an open subset of , let denote the set of all functions such that In other words, V(K, U) = C(K, U) \times_ C(X, Y). Then the collection of all such is a subbase for the compact- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Differential Calculus
In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953. Definition If G is a free group with identity element e and generating set of a group, generators g_i, then the Fox derivative with respect to g_i is a function from G into the integral group ring \Z G which is denoted \frac, and obeys the following axioms: * \frac(g_j) = \delta_, where \delta_ is the Kronecker delta * \frac(e) = 0 * \frac(uv) = \frac(u) + u\frac(v) for any elements u and v of G. The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge, Massachusetts
Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. It is a suburb in the Greater Boston metropolitan area, located directly across the Charles River from Boston. The city's population as of the 2020 United States census, 2020 U.S. census was 118,403, making it the most populous city in the county, the List of municipalities in Massachusetts, fourth-largest in Massachusetts behind Boston, Worcester, Massachusetts, Worcester, and Springfield, Massachusetts, Springfield, and List of cities in New England by population, ninth-most populous in New England. The city was named in honor of the University of Cambridge in Cambridge, England, which was an important center of the Puritans, Puritan theology that was embraced by the town's founders. Harvard University, an Ivy League university founded in Cambridge in 1636, is the oldest institution of higher learning in the United States. The Massachusetts Institute of Technology (MIT), Lesley University, and Hult Inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before 2022 as the Nevanlinna Prize), the Carl Friedrich Gauss Prize, Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being List of International Congresses of Mathematicians Plenary and Invited Speakers, invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History German mathematicians Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review. ''CMS Notes'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guillermo Torres-Diaz
Guillermo () is the Spanish form of the male given name William. The name is also commonly shortened to 'Guille' or, in Latin America, to nickname 'Memo'. People * Guillermo Amador (born 1974), American musician * Guillermo Amor (born 1967), Spanish football manager and former player *Guillermo Arévalo (born 1952), a Shipibo shaman and ''curandero'' (healer) of the Peruvian Amazon; among the Shipibo he is known as Kestenbetsa * Guillermo Barros Schelotto (born 1973), Argentine former football player * Guillermo Bermejo (born 1975), Peruvian politician * Guillermo C. Blest (1800–1884), Anglo-Irish physician settled in Chile *Guillermo Cañas, Argentine tennis player * Guillermo Chong, Chilean geologist *Guillermo Coria, another Argentine tennis player *Guillermo Dávila, Venezuelan actor and singer *Guillermo Díaz (actor) (born 1975), American actor of Cuban descent * Guillermo Diaz (basketball), Puerto Rican basketball player for the Los Angeles Clippers *Guillermo del Toro, Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lusternik–Schnirelmann Category
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X is the homotopy invariant defined to be the smallest integer number k such that there is an open covering \_ of X with the property that each inclusion map U_i\hookrightarrow X is nullhomotopic. For example, if X is a sphere, this takes the value two. Sometimes a different normalization of the invariant is adopted, which is one less than the definition above. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below). In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of X a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philadelphia
Philadelphia ( ), colloquially referred to as Philly, is the List of municipalities in Pennsylvania, most populous city in the U.S. state of Pennsylvania and the List of United States cities by population, sixth-most populous city in the United States, with a population of 1,603,797 in the 2020 United States census, 2020 census. The city is the urban core of the Philadelphia metropolitan area (sometimes called the Delaware Valley), the nation's Metropolitan statistical area, seventh-largest metropolitan area and ninth-largest combined statistical area with 6.245 million residents and 7.379 million residents, respectively. Philadelphia was founded in 1682 by William Penn, an English Americans, English Quakers, Quaker and advocate of Freedom of religion, religious freedom, and served as the capital of the Colonial history of the United States, colonial era Province of Pennsylvania. It then played a historic and vital role during the American Revolution and American Revolutionary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
