1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Order (group theory)
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In group theory, a branch of mathematics, the term order is used in two unrelated senses, The order of a group is its cardinality, i. e. the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e, if no such m exists, a is said to have infinite order. The ordering relation of a partially or totally ordered group and this article is about the first sense of order. The order of a group G is denoted by ord or | G |, the symmetric group S3 has the following multiplication table. This group has six elements, so ord =6, by definition, the order of the identity, e, is 1. Each of s, t, and w squares to e, completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. The order of a group and that of an element tend to speak about the structure of the group, roughly speaking, the more complicated the factorization of the order the more complicated the group. If the order of group G is 1, then the group is called a trivial group, given an element a, ord =1 if and only if a is the identity. If every element in G is the same as its inverse, then ord =2 and consequently G is abelian since a b = −1 = b −1 a −1 = b a by Elementary group theory. The converse of this statement is not true, for example, the cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3,2 +2 +2 =6 ≡0. The relationship between the two concepts of order is the following, if we write ⟨ a ⟩ = for the subgroup generated by a, for any integer k, we have ak = e if and only if ord divides k. In general, the order of any subgroup of G divides the order of G, more precisely, if H is a subgroup of G, then ord / ord =, where is called the index of H in G, an integer. As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord =6, the following partial converse is true for finite groups, if d divides the order of a group G and d is a prime number, then there exists an element of order d in G. The statement does not hold for composite orders, e. g. the Klein four-group does not have an element of order four) and this can be shown by inductive proof. The consequences of the include, the order of a group G is a power of a prime p if. If a has order, then all powers of a have infinite order as well. If a has order, we have the following formula for the order of the powers of a
3.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators
4.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
5.
Root system
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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in graph theory. As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right and these vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, moreover, the root to which it is sent equals α + nβ, where n is an integer. These six vectors satisfy the definition, and therefore they form a root system. Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by, in this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2, then they call root systems satisfying condition 2 reduced, in this article, all root systems are assumed to be reduced and crystallographic. In view of property 3, the integrality condition is equivalent to stating that β, Note that the operator ⟨ ⋅, ⋅ ⟩, Φ × Φ → Z defined by property 4 is not an inner product. It is not necessarily symmetric and is only in the first argument. The rank of a root system Φ is the dimension of V, two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible. Two root systems and are called if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots. The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ, as it acts faithfully on the finite set Φ, the Weyl group is always finite. The root lattice of a root system Φ is the Z-submodule of V generated by Φ, there is only one root system of rank 1, consisting of two nonzero vectors. This root system is called A1, in rank 2 there are four possibilities, corresponding to σ α = β + n α, where n =0,1,2,3. Whenever Φ is a system in V, and U is a subspace of V spanned by Ψ = Φ ∩ U. Thus, the exhaustive list of four systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank
6.
Dynkin diagram
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In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled. The multiple edges are, within certain constraints, directed, the main interest in Dynkin diagrams are as a means to classify semisimple Lie algebras over algebraically closed fields. This gives rise to Weyl groups, i. e. to many finite reflection groups, Dynkin diagrams may also arise in other contexts. The term Dynkin diagram can be ambiguous, in this article, Dynkin diagram means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named. The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields, one classifies such Lie algebras via their root system, which can be represented by a Dynkin diagram. One then classifies Dynkin diagrams according to the constraints they must satisfy, the central classification is that a simple Lie algebra has a root system, to which is associated an Dynkin diagram, all three of these may be referred to as Bn, for instance. The unoriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, thus Bn may refer to the unoriented diagram, the Weyl group, or the abstract Coxeter group. Note that while the Weyl group is isomorphic to the Coxeter group. Beware also that while Dynkin diagram notation is standardized, Coxeter diagram and group notation is varied and sometimes agrees with Dynkin diagram notation, lastly, sometimes associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include, The root lattice generated by the root system and this is naturally defined, but not one-to-one – for example, A2 and G2 both generate the hexagonal lattice. An associated polytope – for example Gosset 421 polytope may be referred to as the E8 polytope, as its vertices are derived from the E8 root system, an associated quadratic form or manifold – for example, the E8 manifold has intersection form given by the E8 lattice. These latter notations are used for objects associated with exceptional diagrams – objects associated to the regular diagrams instead have traditional names. However, n does not equal the dimension of the module of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, B4 corresponds to s o 2 ⋅4 +1 = s o 9, which acts on 9-dimensional space. The simply laced Dynkin diagrams, those with no multiple edges classify many further mathematical objects, for example, the symbol A2 may refer to, The Dynkin diagram with 2 connected nodes, which may also be interpreted as a Coxeter diagram. The root system with 2 simple roots at a 2 π /3 angle, the Lie algebra s l 2 +1 = s l 3 of rank 2. The Weyl group of symmetries of the roots, isomorphic to the symmetric group S3, the abstract Coxeter group, presented by generators and relations, ⟨ r 1, r 2 ∣2 =2 =3 =1 ⟩. Dynkin diagrams must satisfy certain constraints, these are essentially those satisfied by finite Coxeter–Dynkin diagrams, Dynkin diagrams are closely related to Coxeter diagrams of finite Coxeter groups, and the terminology is often conflated
7.
Root of unity
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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in branches of mathematics, and are especially important in number theory, the theory of group characters. In field theory and ring theory the notion of root of unity also applies to any ring with an identity element. Any algebraically closed field has exactly n nth roots of unity if n is not divisible by the characteristic of the field, an nth root of unity, where n is a positive integer, is a number z satisfying the equation z n =1. Without further specification, the roots of unity are complex numbers, however the defining equation of roots of unity is meaningful over any field F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either numbers, if the characteristic of F is 0, or, otherwise. Conversely, every element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details, an nth root of unity is primitive if it is not a kth root of unity for some smaller k, z k ≠1. Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n. In fact, if z1 =1 then z is a primitive first root of unity, otherwise if z2 =1 then z is a second root of unity. And, as z is a root of unity, one finds a first a such that za =1. If z is an nth root of unity and a ≡ b then za = zb, Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. Any integer power of an nth root of unity is also an nth root of unity, n = z k n = k =1 k =1. In particular, the reciprocal of an nth root of unity is its complex conjugate, let z be a primitive nth root of unity. Zn−1, zn = z0 =1 are all distinct, assume the contrary, that za = zb where 1 ≤ a < b ≤ n. But 0 < b − a < n, which contradicts z being primitive. Since an nth-degree polynomial equation can only have n distinct roots, from the preceding, it follows that if z is a primitive nth root of unity, z a = z b ⟺ a ≡ b. If z is not primitive there is only one implication, a ≡ b ⟹ z a = z b