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Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...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] |
Well-defined Expression
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well defined (and thus not a function). The term ''well-defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if f(x)=\frac, then even though f(0) is undefined, this does not mean that the function is ''not'' well defined; rather, 0 is not in the domain of f. Example Let A_0,A_1 be sets, let A = A_0 \cup A_1 and "define" f: A \rightarrow \ as f(a)=0 if a \in A_0 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...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] |
Providence, Rhode Island
Providence () is the List of capitals in the United States, capital and List of municipalities in Rhode Island, most populous city of the U.S. state of Rhode Island. The county seat of Providence County, Rhode Island, Providence County, it is one of the oldest cities in New England, founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port, as it is situated at the mouth of the Providence River at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and List of colleges and universities in Rhode Island#Institutions, eight instit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mineola, New York
Mineola is a Administrative divisions of New York#Village, village and the county seat of Nassau County, New York, Nassau County, on Long Island, Long Island, New York, United States. The population was 20,800 at the time of the 2020 United States census, 2020 census. The village's name is derived from an Algonquin language, Algonquin Chief, Miniolagamika, which means "pleasant village". The Incorporated Village of Mineola is located primarily in the North Hempstead, New York, Town of North Hempstead, with the exception being a small portion of its southern edge within the Hempstead, New York, Town of Hempstead. especially see page 5 Old Country Road runs along the village's southern border. The area serviced by the Mineola Post Office extends farther south into the adjacent village of Garden City, New York, Garden City, where the Old Nassau County Courthouse (New York), Old Nassau County Courthouse is located. Offices of many Nassau County agencies are in both Mineola and Garde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Buchsbaum
David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory. He proved the Auslander–Buchsbaum formula and the Auslander–Buchsbaum theorem. Career Buchsbaum earned his Ph.D. under Samuel Eilenberg in 1954 from Columbia University with thesis ''Exact Categories and Duality''. Among his doctoral students are Peter J. Freyd and Hema Srinivasan. In 1995, he was elected to the American Academy of Arts and Sciences. In 2012, he became a fellow of the 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, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Auslander
Maurice Auslander (August 3, 1926 – November 18, 1994) was an American mathematician who worked on commutative algebra, homological algebra and the representation theory of Artin algebras (e.g. finite-dimensional associative algebras over a field). He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and, in collaboration with Idun Reiten, introduced Auslander–Reiten theory and Auslander algebras. Born in Brooklyn, New York, Auslander received his bachelor's degree and his Ph.D. (1954) from Columbia University. He was a visiting scholar at the Institute for Advanced Study in 1956-57. He was a professor at Brandeis University from 1957 until his death in Trondheim, Norway aged 68. He was elected a Fellow of the American Academy of Arts and Sciences in 1971. Upon his death he was survived by his mother, his widow, a daughter, and a son. His widow Bernice L. Auslander (November 21, 1931 - June 18, 2022) wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''. An element ''x'' of ''G'' is called positive if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''+, and is called the positive cone of ''G''. By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ -''a'' + ''b''. So we can reduce the partial order to a monadic property: if and only if For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * 0 ∈ ''H'' * if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H'' * if ''a'' ∈ ''H'' then -''x'' + ''a'' + ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-ring
In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups. Definition A set ''N'' together with two binary operations + (called ''addition'') and ⋅ (called ''multiplication'') is called a (right) ''near-ring'' if: * ''N'' is a group (not necessarily abelian) under addition; * multiplication is associative (so ''N'' is a semigroup under multiplication); and * multiplication ''on the right'' distributes over addition: for any ''x'', ''y'', ''z'' in ''N'', it holds that (''x'' + ''y'')⋅''z'' = (''x''⋅''z'') + (''y''⋅''z'').G. Pilz, (1982), "Near-Rings: What They Are and What They Are Good For" in ''Contemp. Math.'', 9, pp. 97–119. Amer. Math. Soc., Providence, R.I., 1981. Similarly, it is possible to define a '' left near-ring'' by replacing the right distributive law by the corresponding left distributive law. Both right and left near-rings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
