|
Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville.. He was christened "Leslie Saunders MacLane", but "Leslie" fell into disuse because his parents, Donald MacLane and Winifred Saunders, came to dislike it. He began inserting a space into his surname because his first wife found it difficult to type the name without a space. He was the oldest of three brothers; one of his brothers, Gerald MacLane, also became a mathematics professor at Rice University and Purdue University. Another sister died as a baby. His father and grandfather were both ministers; his grandfather had been a Presbyterian, but was kicked out of the church for believing in evolution, and his father was a Congregationalist. His mother, Winifred, studied at Mount Holyoke College and taught English, Latin, and mathematics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taftville, Connecticut
Taftville is a small village in eastern Connecticut. It is a neighborhood of Norwich but has its own post office ( ZIP Code 06380). It was established in 1866 as site for the large Taftville Mill, later Ponemah Mill. The village is listed on the National Register of Historic Places as Taftville and as alternative name Taftville/Ponemah Mill National Register Historic District. Currently redevelopment of the large mill is being conducted by The O'Neill Group in conjunction with OneKey LLC. The National Park Service will oversee the historic preservation of the structure, to ensure the historic elements are sustained. The Ponemah Mill is being converted into luxury apartments and commercial space. History of the Ponemah Mills The Ponemah Mills, a cotton textile factory, was built on the Shetucket River where a large dam could be built to provide power. The large mill building (Building No. 1) was purported to be the largest weave-shed under one roof at that time. The original work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Alvin Howard
William Alvin Howard (born 1926) is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He has also been active in the theory of proof-theoretic ordinals. He earned his Ph.D. at the University of Chicago in 1956 for his dissertation "k-fold recursion and well-ordering". He was a student of Saunders Mac Lane. The Howard ordinal (also known as the Bachmann–Howard ordinal) was named after him. He was elected to the 2018 class of fellows of the American Mathematical Society. References External linksEntry for ''William Alvin Howard''at the Mathematics Genealogy Project The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a ty .... * 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semimodular Lattice
In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: ;Semimodular law: ''a'' ∧ ''b'' <: ''a'' implies ''b'' <: ''a'' ∨ ''b''. The notation ''a'' <: ''b'' means that ''b'' covers ''a'', i.e. ''a'' < ''b'' and there is no element ''c'' such that ''a'' < ''c'' < ''b''. An atomistic (hence ) semimodular bounded lattice is called a |
Mac Lane Set Theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. The axioms of Zermelo set theory The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mac Lane Coherence Theorem
In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. More precisely (cf. #Counter-example), it states every formal diagram commutes, where "formal diagram" is an analog of well-formed formulae and terms in proof theory. Counter-example It is ''not'' reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell. Let \mathsf_0 \subset \mathsf be a skeleton of the category of sets and ''D'' a unique countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numb ... in it; note D \times D = D by uniqueness. Let p : D = D \times D \to D be the projection onto the first factor. For any functions f, g: D \to D, we have f \circ p = p \circ (f \times g). No ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Complex
In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by and and has since been generalized in many ways. The name "bar complex" comes from the fact that used a vertical bar , as a shortened form of the tensor product \otimes in their notation for the complex. Definition If ''A'' is an associative algebra over a field ''K'', the standard complex is :\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A \rightarrow 0\,, with the differential given by :d(a_0\otimes \cdots\otimes a_)=\sum_^n (-1)^i a_0\otimes\cdots\otimes a_ia_\otimes\cdots\otimes a_\,. If ''A'' is a unital ''K''-algebra, the standard complex is exact. Moreover, cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A/math> is a free ''A''-bimodule resolution of the ''A''-bimodule ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shuffle Algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product ''X'' ⧢ ''Y'' of two words ''X'', ''Y'': the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative. Shuffle product The shuffle product of words of lengths ''m'' and ''n'' is a sum over the ways of interleaving the two words, as shown in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Model
In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process. It can be used to prove the Eilenberg–Zilber theorem; this leads to the idea of the model category. Statement of the theorem Let \mathcal be an arbitrary category and \mathcal(R) be the category of chain complexes of R-modules over some ring R. Let F,V : \mathcal \to \mathcal(R) be covariant functors such that: * F_i = V_i = 0 for i 0 and all M \in \mathcal_k \cup \mathcal_. Then the following assertions hold: * Every natural transformation \varphi : H_0(F) \to H_0(V) induces a natural chain map f : F \to V. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John G
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory. Biography Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on ''A Functorial Form of the Differentiable Riemann–Roch theorem''. Solovay has spent his career at the University of California at Berkeley, where his Ph.D. students include W. Hugh Woodin and Matthew Foreman. Work Solovay's theorems include: * Solovay's theorem showing that, if one assumes the existence of an inaccessible cardinal, then the statement "every set of real numbers is Lebesgue measurable" is consistent with Zermelo–Fraenkel set theory without the axiom of choice; * Isolating the notion of 0#; * Proving that the existence of a real-valued measurable cardinal is equiconsistent with the existence of a measurable cardinal; * Proving that if \lambda is a strong limit singular cardinal, greater than a strongly compact cardinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anil Nerode
Anil Nerode (born 1932) is an American mathematician. He received his undergraduate education and a Ph.D. in mathematics from the University of Chicago, the latter under the directions of Saunders Mac Lane. He enrolled in the Hutchins College at the University of Chicago in 1947 at the age of 15, and received his Ph.D. in 1956. His Ph.D. thesis was on an algebraic abstract formulation of substitution in many-sorted free algebras and its relation to equational definitions of the partial recursive functions. While in graduate school, beginning in 1954, he worked at Professor Walter Bartky's Institute for Air Weapons Research, which did classified work for the US Air Force. He continued to work there following the completion of his Ph.D., from 1956 to 1957. In the summer of 1957 he attended the Cornell NSF Summer 1957 Institute in Logic. In 1958 to 1959 he went to the Institute for Advanced Study in Princeton, New Jersey, where he worked with Kurt Gödel. He also did post ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
