Ore condition

In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The "right Ore condition" for a multiplicative subset "S" of a ring "R" is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
General idea.
The goal is to construct the right ring of fractions "R"["S"−1] with respect to a multiplicative subset "S". In other words, we want to work with elements of the form "as"−1 and have a ring structure on the set "R"["S"−1]. The problem is that there is no obvious interpretation of the product ("as"−1)("bt"−1); indeed, we need a method to "move" "s"−1 past "b". This means that we need to be able to rewrite "s"−1"b" as a product "b"1"s"1−1. Suppose then multiplying on the left by "s" and on the right by "s"1, we get . Hence we see the necessity, for a given "a" and "s", of the existence of "a"1 and "s"1 with and such that .
Application.
Since it is well known that each integral domain is a subring of a field of fractions (via an embedding) in such a way that every element is of the form "rs"−1 with "s" nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a division ring (a noncommutative field) with the same property. It turns out that the answer is sometimes "no", that is, there are domains which do not have an analogous "right division ring of fractions".
For every right Ore domain "R", there is a unique (up to natural "R"-isomorphism) division ring "D" containing "R" as a subring such that every element of "D" is of the form "rs"−1 for "r" in "R" and "s" nonzero in "R". Such a division ring "D" is called a ring of right fractions of "R", and "R" is called a right order in "D". The notion of a ring of left fractions and left order are defined analogously, with elements of "D" being of the form "s"−1"r".
It is important to remember that the definition of "R" being a right order in "D" includes the condition that "D" must consist entirely of elements of the form "rs"−1. Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean "R" is a left order in "D", since it is possible "D" has an element which is not of the form "s"−1"r". Thus it is possible for "R" to be a right-not-left Ore domain. Intuitively, the condition that all elements of "D" be of the form "rs"−1 says that "R" is a "big" "R"-submodule of "D". In fact the condition ensures "R""R" is an essential submodule of "D""R". Lastly, there is even an example of a domain in a division ring which satisfies "neither" Ore condition (see examples below).
Another natural question is: "When is a subring of a division ring right Ore?" One characterization is that a subring "R" of a division ring "D" is a right Ore domain if and only if "D" is a flat left "R"-module .
A different, stronger version of the Ore conditions is usually given for the case where "R" is not a domain, namely that there should be a common multiple
with "u", "v" not zero divisors. In this case, Ore's theorem guarantees the existence of an over-ring called the (right or left) classical ring of quotients.
Examples.
Commutative domains are automatically Ore domains, since for nonzero "a" and "b", "ab" is nonzero in . Right Noetherian domains, such as right principal ideal domains, are also known to be right Ore domains. Even more generally, Alfred Goldie proved that a domain "R" is right Ore if and only if "R""R" has finite uniform dimension. It is also true that right Bézout domains are right Ore.
A subdomain of a division ring which is not right or left Ore: If "F" is any field, and formula_1 is the free monoid on two symbols "x" and "y", then the monoid ring formula_2 does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring of a division ring, by .
Multiplicative sets.
The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in and . A subset "S" of a ring "R" is called a right denominator set if it satisfies the following three conditions for every "a", "b" in "R", and "s", "t" in "S":
If "S" is a right denominator set, then one can construct the ring of right fractions "RS"−1 similarly to the commutative case. If "S" is taken to be the set of regular elements (those elements "a" in "R" such that if "b" in "R" is nonzero, then "ab" and "ba" are nonzero), then the right Ore condition is simply the requirement that "S" be a right denominator set.
Many properties of commutative localization hold in this more general setting. If "S" is a right denominator set for a ring "R", then the left "R"-module "RS"−1 is flat. Furthermore, if "M" is a right "R"-module, then the "S"-torsion, is an "R"-submodule isomorphic to , and the module is naturally isomorphic to a module "MS"−1 consisting of "fractions" as in the commutative case.


Nondegenerate form


Griess algebra

In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group "M" as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct "M". The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space.
Griess's construction was later simplified by Jacques Tits and John H. Conway.
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products.


Bertram Huppert

Bertram Huppert (born 22 October 1927 in Worms, Germany) is a German mathematician specializing in group theory and the representation theory of finite groups. His "Endliche Gruppen" (finite groups) is an influential textbook in group theory, and he has over 50 doctoral descendants.
Life.
Education.
Bertram Huppert went to school in Bonn from 1934 until 1945. In 1950, he wrote his diploma thesis in mathematics at the University of Mainz. The thesis discussed ""nicht fortsetzbare Potenzreihen" (discontinuous power series), and was written under the direction of Helmut Wielandt.
When Wielandt moved to the University of Tübingen in April 1951, Huppert followed him later in the year, and gained his doctorate (as Wielandt's first doctoral student) with the work "Produkte von paarweise vertauschbaren zyklischen Gruppen"" (products of pairwise permutable cyclic groups), in which he showed, among other things, that such groups were supersoluble. This was the first of more than forty further scientific works, not including his books and monographs. The focus of the dissertation was very close to Wielandt's interests at the time, whose 1951 work shows that the product of pairwise permutable nilpotent groups is solvable.
Academic career.
Huppert spent the years 1963/64 as a visiting professor at the University of Illinois at Urbana-Champaign and at the California Institute of Technology (Caltech) in Pasadena. In January 1965, he became a professor of pure mathematics at the University of Mainz, where he later became a professor emeritus in 1994. He put a lot of effort into building up the Mainz group theory and abstract algebra research groups.
Following an assignment, he wrote a monumental standard text in the theory of finite groups, "Endliche Gruppen I". The group around Wolfgang Gaschütz in Kiel provided important contributions in discussions to that volume. Volumes II and III appeared 14 years later in English with co-author Norman Blackburn.
In 1984, Huppert founded, together with Gerhard Michler, the first Deutsche Forschungsgemeinschaft priority programme in Mathematics at the German universities of Aachen, Bielefeld, Essen and Mainz.
From 1964 to 1985, Huppert was a member of the editorial board of the Journal of Algebra. Together with Wolfgang Gaschütz and Karl W. Gruenberg, he organised Oberwolfach workshops on group theory over many years, and with Michler the Oberwolfach workshop on representation theory.
He was a founding member of the Institute of Experimental Mathematics of the University of Essen and is a member of the "Akademie gemeinnütziger Wissenschaften zu Erfurt".


Group of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase "group of Lie type" does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type.
Classical groups.
An initial approach to this question was the definition and detailed study of the so-called "classical groups" over finite and other fields by . These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series A"n", B"n", C"n", D"n",2A"n", 2D"n" of Chevalley and Steinberg groups.
Chevalley groups.
Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of algebraic groups, and the work of on Lie algebras, by means of which the "Chevalley group" concept was isolated. Chevalley constructed a Chevalley basis (a sort of integral form but over finite fields) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras A"n", B"n", C"n", D"n" this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras E6, E7, E8, F4, and G2. The ones of type G2 (sometimes called "Dickson groups") had already been constructed by , and the ones of type E6 by .
Steinberg groups.
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. found a modification of Chevalley's construction that gave these groups and two new families 3D4, 2E6, the second of which was discovered at about the same time from a different point of view by . This construction generalizes the usual construction of the unitary group from the general linear group.
The unitary group arises as follows: the general linear group over the complex numbers has a "diagram automorphism" given by reversing the Dynkin diagram A"n" (which corresponds to taking the transpose inverse), and a "field automorphism" given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
These gave:
The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the D4 diagram also give rise to triality.
Suzuki–Ree groups.
 found a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups. knew that the algebraic group B2 had an "extra" automorphism in characteristic 2 whose square was the Frobenius automorphism. He found that if a finite field of characteristic 2 also has an automorphism whose square was the Frobenius map, then an analogue of Steinberg's construction gave the Suzuki groups. The fields with such an automorphism are those of order 22"n"+1, and the corresponding groups are the Suzuki groups
(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the Frobenius group of order 20.) Ree was able to find two new similar families
and
of simple groups by using the fact that F4 and G2 have extra automorphisms in characteristic 2 and 3. (Roughly speaking, in characteristic "p" one is allowed to ignore the arrow on bonds of multiplicity "p" in the Dynkin diagram when taking diagram automorphisms.) The smallest group 2F4(2) of type 2F4 is not simple, but it has a simple subgroup of index 2, called the "Tits group" (named after the mathematician Jacques Tits). The smallest group 2G2(3) of type 2G2 is not simple, but it has a simple normal subgroup of index 3, isomorphic to A1(8). In the classification of finite simple groups, the Ree groups
are the ones whose structure is hardest to pin down explicitly. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form Z/2Z × PSL(2, "q") for "q" = 3"n", and by investigating groups with an involution centralizer of the similar form Z/2Z × PSL(2, 5) Janko found the sporadic group "J"1.
The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3. They have order 22(2"n"+1)(22(2"n"+1) + 1)(2(2"n"+1) − 1).
Relations with finite simple groups.
Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, "p") being constructed by Évariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, "q") is simple for "q" ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL("n", "q") of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field "k", leading to construction of what are now called "Chevalley groups". Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups ("Tits simplicity theorem"). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their "geometry" in the sense of Tits.
The belief has now become a theorem – the classification of finite simple groups. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group, and the 26 sporadic simple groups.
Small groups of Lie type.
In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universal central extension of a simple group, so is perfect and has trivial Schur multiplier. However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected".
Cases where the group is not perfect include
Some cases where the group is perfect but has a Schur multiplier that is larger than expected include:
There is a bewildering number of "accidental" isomorphisms between various small groups of Lie type (and alternating groups). For example, the groups SL(2, 4), PSL(2, 5), and the alternating group on 5 points are all isomorphic.
For a complete list of these exceptions see the list of finite simple groups. Many of these special properties are related to certain sporadic simple groups.
Alternating groups sometimes behave as if they were groups of Lie type over the field with one element. Some of the small alternating groups also have exceptional properties. The alternating groups usually have an outer automorphism group of order 2, but the alternating group on 6 points has an outer automorphism group of order 4. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a .
Notation issues.
There is no standard notation for the finite groups of Lie type, and the literature contains dozens of incompatible and confusing systems of notation for them.


Glossary of field theory

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field.
A field is a commutative ring ("F",+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field "F" form an abelian group under multiplication; this group is typically denoted by "F"×;
The ring of polynomials in the variable "x" with coefficients in "F" is denoted by "F"["x"].
Field extensions.
Let "E" / "F" be a field extension.


Fitting ideal

In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced by .
Definition.
If "M" is a finitely generated module over a commutative ring "R" generated by elements "m"1...,"m""n"
with relations
then the "i"th Fitting ideal Fitt"i"("M") of "M" is generated by the minors (determinants of submatrices) of order "n" − "i" of the matrix "a""jk".
The Fitting ideals do not depend on the choice of generators and relations of "M".
Some authors defined the Fitting ideal "I"("M") to be the first nonzero Fitting ideal Fitt"i"("M").
Properties.
The Fitting ideals are increasing
If "M" can be generated by "n" elements then Fitt"n"("M") = "R", and if "R" is local the converse holds. We have Fitt0("M") ⊆ Ann("M") (the annihilator of "M"), and Ann("M")Fitt"i"("M") ⊆ Fitt"i"−1("M"), so in particular if "M" can be generated by "n" elements then Ann("M")"n" ⊆ Fitt0("M").
Examples.
If "M" is free of rank "n" then the Fitting ideals Fitt"i"("M") are zero for "i"<"n" and "R" for "i" ≥ "n".
If "M" is a finite abelian group of order |"M"| (considered as a module over the integers) then the Fitting ideal Fitt0("M") is the ideal (|"M"|).
The Alexander polynomial of a knot is a generator of the Fitting ideal of the first homology of the infinite abelian cover of the knot complement.
Fitting image.
The zeroth Fitting ideal can be used also to give a definition of scheme-theoretic image of morphisms, which behaves well in families. Given a morphism of schemes formula_2, the Fitting image of "f" is defined to be the closed subscheme associated to the sheaf of ideals formula_3, where formula_4 is seen as a formula_5-module via the canonical morphism formula_6.


Monster Lie algebra

In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.
Structure.
The monster Lie algebra "m" is a "Z2"-graded Lie algebra. The piece of degree ("m", "n") has dimension "c""mn" if ("m", "n") ≠ (0, 0) and dimension 2 if ("m", "n") = (0, 0).
The integers "cn" are the coefficients of "q""n" of the "j"-invariant as elliptic modular function
The Cartan subalgebra is the 2-dimensional subspace of degree (0, 0), so the monster Lie algebra has rank 2.
The monster Lie algebra has just one real simple root, given by the vector (1, −1), and the Weyl group has order 2, and acts by mapping ("m", "n") to ("n", "m"). The imaginary simple roots are the vectors (1, "n") for "n" = 1, 2, 3, ..., and they have multiplicities "c""n".
The denominator formula for the monster Lie algebra is the product formula for the "j"-invariant:
The denominator formula (sometimes called the Koike-Norton-Zagier infinite product identity) was discovered in the 1980s. Several mathematicians, including Masao Koike, Simon P. Norton, and Don Zagier, independently made the discovery.
Construction.
There are two ways to construct the monster Lie algebra. As it is a generalized Kac–Moody algebra whose simple roots are known, it can be defined by explicit generators and relations; however, this presentation does not give an action of the monster group on it.
It can also be constructed from the monster vertex algebra by using the Goddard–Thorn theorem of string theory. This construction is much harder, but also proves that the monster group acts naturally on it.


Orthogonal matrix

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
where is the transpose of and is the identity matrix.
This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse:
where is the inverse of .
An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint (conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation.
The set of orthogonal matrices forms a group, , known as the orthogonal group. The subgroup consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.
Overview.
An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product, so, for vectors and in an -dimensional real Euclidean space
where is an orthogonal matrix. To see the inner product connection, consider a vector in an -dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of is . If a linear transformation, in matrix form , preserves vector lengths, then
Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
Orthogonal matrices are important for a number of reasons, both theoretical and practical. The orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by , which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.
Examples.
Below are a few examples of small orthogonal matrices and possible interpretations.
Elementary constructions.
Lower dimensions.
The simplest orthogonal matrices are the matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin.
The matrices have the form
which orthogonality demands satisfy the three equations
In consideration of the first equation, without loss of generality let , ; then either , or , . We can interpret the first case as a rotation by (where is the identity), and the second as a reflection across a line at an angle of .
The special case of the reflection matrix with generates a reflection about the line at 45° given by and therefore exchanges and ; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):
The identity is also a permutation matrix.
A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.
Higher dimensions.
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
represent an "inversion" through the origin and a "rotoinversion", respectively, about the -axis.
Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a rotation matrix in terms of an axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a plane of rotation.
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
Primitives.
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any permutation matrix can be constructed as a product of no more than transpositions.
A Householder reflection is constructed from a non-null vector as
Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of . This is a reflection in the hyperplane perpendicular to (negating any vector component parallel to ). If is a unit vector, then suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size can be constructed as a product of at most such reflections.
A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size can be constructed as a product of at most such rotations. In the case of matrices, three such rotations suffice; and by fixing the sequence we can thus describe all rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles.
A Jacobi rotation has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a symmetric submatrix.
Properties.
Matrix properties.
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of . It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy , with a diagonal matrix.
The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:
The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.
With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.
Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.
Group properties.
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension , called the orthogonal group and denoted by .
The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of of index 2, the special orthogonal group of rotations. The quotient group is isomorphic to , with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, is a semidirect product of by . In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with matrices. If is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.
Now consider orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an orthogonal matrix; thus is a subgroup of (and of all higher groups).
Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of in ; in this way is a bundle over the unit sphere with fiber .
Similarly, is a subgroup of ; and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: . A single rotation can produce a zero in the first row of the last column, and series of rotations will zero all but the last row of the last column of an rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, therefore has
degrees of freedom, and so does .
Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order symmetric group . By the same kind of argument, is a subgroup of . The even permutations produce the subgroup of permutation matrices of determinant +1, the order alternating group.
Canonical form.
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if is special orthogonal then one can always find an orthogonal matrix , a (rotational) change of basis, that brings into block diagonal form:
where the matrices are rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, . Thus, negating one column if necessary, and noting that a reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form
The matrices give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If is odd, there is at least one real eigenvalue, +1 or −1; for a rotation, the eigenvector associated with +1 is the rotation axis.
Lie algebra.
Suppose the entries of are differentiable functions of , and that gives . Differentiating the orthogonality condition
yields
Evaluation at () then implies
In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).
For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra tangent to . Given , with being a unit vector, the correct skew-symmetric matrix form of is
The exponential of this is the orthogonal matrix for rotation around axis by angle ; setting , ,
Numerical linear algebra.
Benefits.
Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.
Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of indices.
Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order to a much more efficient order . When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following , we do "not" store a rotation angle, which is both expensive and badly behaved.)
Decompositions.
A number of important matrix decompositions involve orthogonal matrices, including especially:
Examples.
Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write , where is , .
A decomposition reduces to upper triangular . For example, if is then has the form
The linear least squares problem is to find the that minimizes , which is equivalent to projecting to the subspace spanned by the columns of . Assuming the columns of (and hence ) are independent, the projection solution is found from . Now is square () and invertible, and also equal to . But the lower rows of zeros in are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing to , but also for allowing solution without magnifying numerical problems.
In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful. With factored as , a satisfactory solution uses the Moore-Penrose pseudoinverse, , where merely replaces each non-zero diagonal entry with its reciprocal. Set to .
The case of a square invertible matrix also holds interest. Suppose, for example, that is a rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so has gradually lost its true orthogonality. A Gram–Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique "closest" orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton's method" approach due to (1990), repeatedly averaging the matrix with its inverse transpose. has published an accelerated method with a convenient convergence test.
For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
and which acceleration trims to two steps (with = 0.353553, 0.565685).
Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.
Randomization.
Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformly distributed orthogonal matrices, but the decomposition of independent normally distributed random entries does, as long as the diagonal of contains only positive entries . replaced this with a more efficient idea that later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an orthogonal matrix, take an one and a uniformly distributed unit vector of dimension . Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).
Nearest orthogonal matrix.
The problem of finding the orthogonal matrix nearest a given matrix is related to the Orthogonal Procrustes problem. There are several different ways to get the unique solution, the simplest of which is taking the singular value decomposition of and replacing the singular values with ones. Another method expresses the explicitly but requires the use of a matrix square root:
This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
where .
These iterations are stable provided the condition number of is less than three.
Using a first-order approximation of the inverse and the same initialization results in the modified iteration:
Spin and pin.
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, , is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO("n"), the spin group, . Likewise, has covering groups, the pin groups, Pin("n"). For , Spin("n") is simply connected and thus the universal covering group for . By far the most famous example of a spin group is , which is nothing but , or the group of unit quaternions.
The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.
Rectangular matrices.
If is not a square matrix, then the conditions and are not equivalent. The condition says that the columns of "Q" are orthonormal. This can only happen if is an matrix with (due to linear dependence). Similarly, says that the rows of are orthonormal, which requires .
There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".


Heinrich Martin Weber

Heinrich Martin Weber (5 March 1842, Heidelberg, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He is best known for his text "Lehrbuch der Algebra" published in 1895 and much of it is his original research in algebra and number theory. His work "Theorie der algebraischen Functionen einer Veränderlichen" (with Dedekind) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann-Roch theorem. Weber's research papers were numerous, most of them appearing in "Crelle's Journal" or "Mathematische Annalen". He was the editor of Riemann's collected works.
Weber was born in Heidelberg, Baden, and entered the University of Heidelberg in 1860. In 1866 he became a privatdozent, and in 1869 he was appointed as extraordinary professor at that school. Weber also taught in Zurich at the Federal Polytechnic Institute, today the ETH Zurich, at the University of Königsberg, and at the Technische Hochschule in Charlottenburg. His final post was at the Kaiser-Wilhelm-Universität Straßburg, Alsace-Lorraine, where he died.
In 1893 in Chicago, his paper "Zur Theorie der ganzzahligen algebraischen Gleichungen" was read (but not by him) at the International Mathematical Congress held in connection with the World's Columbian Exposition. In 1895 and in 1904 he was president of the Deutsche Mathematiker-Vereinigung. His doctoral students include Heinrich Brandt, E. V. Huntington, Louis Karpinski, and Friedrich Levi.


Weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the variance of observations is incorporated into the regression.
WLS is also a specialization of generalized least squares.
Introduction.
A special case of generalized least squares called weighted least squares occurs when all the off-diagonal entries of "Ω" (the correlation matrix of the residuals) are null; the variances of the observations (along the covariance matrix diagonal) may still be unequal (heteroscedasticity).
The fit of a model to a data point is measured by its residual, formula_1, defined as the difference between a measured value of the dependent variable, formula_2 and the value predicted by the model, formula_3:
If the errors are uncorrelated and have equal variance, then the minimum of the function
is found when formula_6 (defining formula_7).
The Gauss–Markov theorem shows that, when this is so, formula_8 is a best linear unbiased estimator (BLUE). If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted. Aitken showed that when a weighted sum of squared residuals is minimized, formula_8 is the BLUE if each weight is equal to the reciprocal of the variance of the measurement
If the errors are correlated, the resulting estimator is the BLUE if the weight matrix is equal to the inverse of the variance-covariance matrix of the observations.
When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as formula_11. The normal equations can then be written in the same form as ordinary least squares:
where we define the following scaled matrix and vector:
This is a type of whitening transformation; the last expression involves an entrywise division.
For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.
Note that for empirical tests, the appropriate W is not known for sure and must be estimated. For this feasible generalized least squares (FGLS) techniques may be used; in this case it is specialized for a diagonal covariance matrix, thus yielding a feasible weighted least squares solution.
If the uncertainty of the observations is not known from external sources, then the weights could be estimated from the given observations. This can be useful, for example, to identify outliers. After the outliers have been removed from the data set, the weights should be reset to one.
Motivation.
In some cases the observations may be weighted—for example, they may not be equally reliable. In this case, one can minimize the weighted sum of squares:
where "w""i" > 0 is the weight of the "i"th observation, and "W" is the diagonal matrix of such weights.
The weights should, ideally, be equal to the reciprocal of the variance of the measurement. (This implies that the observations are uncorrelated. If the observations are correlated, the expression formula_16 applies. In this case the weight matrix should ideally be equal to the inverse of the variance-covariance matrix of the observations).
The normal equations are then:
This method is used in iteratively reweighted least squares.
Parameter errors and correlation.
The estimated parameter values are linear combinations of the observed values
Therefore, an expression for the estimated variance-covariance matrix of the parameter estimates can be obtained by error propagation from the errors in the observations. Let the variance-covariance matrix for the observations be denoted by "M" and that of the estimated parameters by "Mβ". Then
When "W" = "M"−1, this simplifies to
When unit weights are used ("W" = "I", the identity matrix), it is implied that the experimental errors are uncorrelated and all equal: "M" = "σ"2"I", where "σ"2 is the "a priori" variance of an observation. In any case, "σ"2 is approximated by the reduced chi-squared formula_21:
where "S" is the minimum value of the (weighted) objective function:
The denominator, formula_24, is the number of degrees of freedom; see effective degrees of freedom for generalizations for the case of correlated observations.
In all cases, the variance of the parameter estimate formula_25 is given by formula_26 and the covariance between the parameter estimates formula_25 and formula_28 is given by formula_29. The standard deviation is the square root of variance, formula_30, and the correlation coefficient is given by formula_31. These error estimates reflect only random errors in the measurements. The true uncertainty in the parameters is larger due to the presence of systematic errors, which, by definition, cannot be quantified.
Note that even though the observations may be uncorrelated, the parameters are typically correlated.
Parameter confidence limits.
It is often "assumed", for want of any concrete evidence but often appealing to the central limit theorem—see Normal distribution#Occurrence and applications—that the error on each observation belongs to a normal distribution with a mean of zero and standard deviation formula_32. Under that assumption the following probabilities can be derived for a single scalar parameter estimate in terms of its estimated standard error formula_33 (given here):
The assumption is not unreasonable when "m" » "n". If the experimental errors are normally distributed the parameters will belong to a Student's t-distribution with "m" − "n" degrees of freedom. When "m" ≫ "n" Student's t-distribution approximates a normal distribution. Note, however, that these confidence limits cannot take systematic error into account. Also, parameter errors should be quoted to one significant figure only, as they are subject to sampling error.
When the number of observations is relatively small, Chebychev's inequality can be used for an upper bound on probabilities, regardless of any assumptions about the distribution of experimental errors: the maximum probabilities that a parameter will be more than 1, 2, or 3 standard deviations away from its expectation value are 100%, 25% and 11% respectively.
Residual values and correlation.
The residuals are related to the observations by
where "H" is the idempotent matrix known as the hat matrix:
and "I" is the identity matrix. The variance-covariance matrix of the residuals, "M" r is given by
Thus the residuals are correlated, even if the observations are not.
When formula_40,
The sum of weighted residual values is equal to zero whenever the model function contains a constant term. Left-multiply the expression for the residuals by "X" "W":
Say, for example, that the first term of the model is a constant, so that formula_43 for all "i". In that case it follows that
Thus, in the motivational example, above, the fact that the sum of residual values is equal to zero is not accidental, but is a consequence of the presence of the constant term, α, in the model.
If experimental error follows a normal distribution, then, because of the linear relationship between residuals and observations, so should residuals, but since the observations are only a sample of the population of all possible observations, the residuals should belong to a Student's t-distribution. Studentized residuals are useful in making a statistical test for an outlier when a particular residual appears to be excessively large.


Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form
Univariate polynomials.
If a polynomial has only one indeterminate (univariate polynomial), then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in "x" of degree "n" then takes the general form displayed above, where
are constants, the coefficients of the polynomial.
Here the term "c""n""x""n" is called the "leading term", and its coefficient "c""n" the "leading coefficient"; if the leading coefficient , the univariate polynomial is called monic.
Properties.
Multiplicatively closed.
The set of all monic polynomials (over a given (unitary) ring "A" and for a given variable "x") is closed under multiplication, since the product of the leading terms of two monic polynomials is the leading term of their product. Thus, the monic polynomials form a multiplicative semigroup of the polynomial ring "A"["x"]. Actually, since the constant polynomial 1 is monic, this semigroup is even a monoid.
Partially ordered.
The restriction of the divisibility relation to the set of all monic polynomials (over the given ring) is a partial order, and thus makes this set to a poset. The reason is that if "p"("x") divides "q"("x") and "q"("x") divides "p"("x") for two monic polynomials "p" and "q", then "p" and "q" must be equal. The corresponding property is not true for polynomials in general, if the ring contains invertible elements other than 1.
Polynomial equation solutions.
In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ring "A". If "A" is a field, then every non-zero polynomial "p" has exactly one associated monic polynomial "q": "p" divided by its leading coefficient. In this manner, then, any non-trivial polynomial equation "p"("x") = 0 may be replaced by an equivalent monic equation "q"("x") = 0. For example, the general real second degree equation
may be replaced by
by substituting  "p" = "b"/"a"  and  "q" = "c"/"a". Thus, the equation
is equivalent to the monic equation
The general quadratic solution formula is then the slightly more simplified form of:
Integrality.
On the other hand, if the coefficient ring is not a field, there are more essential differences. For example, a monic polynomial equation with integer coefficients cannot have rational solutions which are not integers. Thus, the equation
possibly might have some rational root, which is not an integer, (and incidentally one of its roots is −1/2); while the equations
and
can only have integer solutions or irrational solutions.
The roots of monic polynomials with integer coefficients are called algebraic integers.
The solutions to monic polynomial equations over an integral domain are important in the theory of integral extensions and integrally closed domains, and hence for algebraic number theory. In general, assume that "A" is an integral domain, and also a subring of the integral domain "B". Consider the subset "C" of "B", consisting of those "B" elements, which satisfy monic polynomial equations over "A":
The set "C" contains "A", since any "a" ∈ "A" satisfies the equation "x" − "a" = 0. Moreover, it is possible to prove that "C" is closed under addition and multiplication. Thus, "C" is a subring of "B". The ring "C" is called the [[integral closure] of "A" in "B"; or just the integral closure of "A", if "B" is the [[fraction field]] of "A"; and the elements of "C" are said to be "[[integral element|integral]]" over "A". If here formula_12 (the ring of [[integer]]s) and formula_13 (the field of [[complex number]]s), then "C" is the ring of "[[algebraic integers]]".
Irreduciblity.
If is a [[prime number]], the number of monic [[irreducible polynomial]]s of degree over a [[finite field]] formula_14 with elements is equal to the [[Necklace (combinatorics)|necklace counting function]] .
If one removes the constraint of being monic, this number becomes .
The total number of roots of these monic irreducible polynomials is . This is the number of elements of the field (with elements) that do not belong to any smaller field.
For , such polynomials are commonly used to generate [[pseudorandom binary sequence]]s.
Multivariate polynomials.
Ordinarily, the term "monic" is not employed for polynomials of several variables. However, a polynomial in several variables may be regarded as a polynomial in only "the last" variable, but with coefficients being polynomials in the others. This may be done in several ways, depending on which one of the variables is chosen as "the last one". E.g., the real polynomial
is monic, considered as an element in R["y"]["x"], i.e., as a univariate polynomial in the variable "x", with coefficients which themselves are univariate polynomials in "y":
but "p"("x","y") is not monic as an element in R["x"]["y"], since then the highest degree coefficient (i.e., the "y"2 coefficient) is  2"x" − 1.
There is an alternative convention, which may be useful e.g. in [[Gröbner basis]] contexts: a polynomial is called monic, if its leading coefficient (as a multivariate polynomial) is 1. In other words, assume that "p = p"("x"1"...,xn") is a non-zero polynomial in "n" variables, and that there is a given [[monomial order]] on the set of all ("monic") monomials in these variables, i.e., a total order of the free commutative [[monoid]] generated by "x"1"...,xn", with the unit as lowest element, and respecting multiplication. In that case, this order defines a highest non-vanishing term in "p", and "p" may be called monic, if that term has coefficient one.
"Monic multivariate polynomials" according to either definition share some properties with the "ordinary" (univariate) monic polynomials. Notably, the product of monic polynomials again is monic.
References.
[[Category:Polynomials]]


Martin Isaacs

I. Martin "Marty" Isaacs is a group theorist and representation theorist and professor emeritus of mathematics at the University of Wisconsin–Madison. He currently lives in Berkeley, California and is an occasional participant on MathOverflow.
Academic biography.
Isaacs completed his Ph.D. from Harvard University in 1964 under Richard Brauer. From at least 1979 until 2011, he was a professor at the University of Wisconsin–Madison. In May 2011, he retired and became a professor emeritus. The Mathematics Genealogy Project lists him as having had 28 doctoral students.
Research.
Isaacs is most famous for formulating the Isaacs–Navarro conjecture along with Gabriel Navarro, a widely cited generalization of the McKay conjecture.
Books.
Isaacs is famous as the author of "Character Theory of Finite Groups" (first published in 1976), one of the most well-known graduate student-level introductory books in character theory and representation theory of finite groups.
Isaacs is also the author of the book "Algebra: A Graduate Course" (first published in 1994; republished in 2009), which received highly positive reviews. Additionally, he is the author of "Finite Group Theory" (published in 2008).
Honors.
Isaacs is a professor emeritus at the University of Wisconsin–Madison. He retired in 2011.
In 2009, a conference was held at the Universitat de Valencia in Spain to honor his contributions.
Isaacs is also a Fellow of the American Mathematical Society.
Isaacs was a Pólya lecturer for the Mathematical Association of America. He received the Benjamin Smith Reynolds award for teaching engineering students at the University of Wisconsin and a UW Madison campus teaching award. He was also the recipient of a Sloan Foundation research award.


Infinite-dimensional vector function

Infinite-dimensional vector function refers to a function whose values lie in an infinite-dimensional vector space, such as a Hilbert space or a Banach space.
Such functions are applied in most sciences including physics.
Example.
Set formula_1 for every positive integer "k" and every real number "t". Then the function f, defined for real numbers t by the formula
takes values that lie in the infinite-dimensional vector space "X" (or formula_3) of real-valued sequences. For example,
As a number of different topologies can be defined on the space "X", we cannot talk about the derivative of "f" without first
defining the topology of "X" or the concept of a limit in "X".
Moreover, for any set "A", there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of "A" (e.g., the space of functions formula_5 with finitely-many nonzero elements, where "K" is the desired field of scalars). Furthermore, the argument "t" could lie in any set instead of the set of real numbers.
Integral and derivative.
If, e.g., formula_6, where "X" is a Banach space or another topological vector space, the derivative of "f" can be defined in the standard way: formula_7.
The measurability of "f" can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.
The most important integrals of "f" are called Bochner integral (when "X" is a Banach space) and Pettis integral (when "X" is a topological vector space). Both these integrals commute with linear functionals. Also formula_8 spaces have been defined for such functions.
Most theorems on integration and differentiation of scalar functions can be generalized to vector-valued functions, often using essentially the same proofs. Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, e.g., "X" is a Hilbert space); see Radon–Nikodym theorem
Derivative.
Functions with values in a Hilbert space.
If "f" is a function of real numbers with values in a Hilbert space "X", then the derivative of "f" at a point "t" can be defined as in the finite-dimensional case:
Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis mutandis. Differentiation can also be defined to functions of several variables (e.g., formula_10 or even formula_11, where "Y" is an infinite-dimensional vector space).
N.B. If "X" is a Hilbert space, then one can easily show that any derivative (and any other limit) can be computed componentwise: if
(i.e., formula_13, where formula_14 is an orthonormal basis of the space "X"), and formula_15 exists, then
However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.
Other infinite-dimensional vector spaces.
Most of the above hold for other topological vector spaces "X" too. However, not as many classical results hold in the Banach space setting, e.g., an absolutely continuous function with values in a suitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.


Algebraically closed field

In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples.
As an example, the field of real numbers is not algebraically closed, because the polynomial equation "x"2 + 1 = 0  has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field "F" is algebraically closed, because if "a"1, "a"2, ..., "an" are the elements of "F", then the polynomial ("x" − "a"1)("x" − "a"2) ⋯ ("x" − "a""n") + 1
has no zero in "F". By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.
Equivalent properties.
Given a field "F", the assertion ""F" is algebraically closed" is equivalent to other assertions:
The only irreducible polynomials are those of degree one.
The field "F" is algebraically closed if and only if the only irreducible polynomials in the polynomial ring "F"["x"] are those of degree one.
The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If "F" is algebraically closed and "p"("x") is an irreducible polynomial of "F"["x"], then it has some root "a" and therefore "p"("x") is a multiple of "x" − "a". Since "p"("x") is irreducible, this means that "p"("x") = "k"("x" − "a"), for some "k" ∈ "F" \ {0}. On the other hand, if "F" is not algebraically closed, then there is some non-constant polynomial "p"("x") in "F"["x"] without roots in "F". Let "q"("x") be some irreducible factor of "p"("x"). Since "p"("x") has no roots in "F", "q"("x") also has no roots in "F". Therefore, "q"("x") has degree greater than one, since every first degree polynomial has one root in "F".
Every polynomial is a product of first degree polynomials.
The field "F" is algebraically closed if and only if every polynomial "p"("x") of degree "n" ≥ 1, with coefficients in "F", splits into linear factors. In other words, there are elements "k", "x"1, "x"2, ..., "xn" of the field "F" such that "p"("x") = "k"("x" − "x"1)("x" − "x"2) ⋯ ("x" − "xn").
If "F" has this property, then clearly every non-constant polynomial in "F"["x"] has some root in "F"; in other words, "F" is algebraically closed. On the other hand, that the property stated here holds for "F" if "F" is algebraically closed follows from the previous property together with the fact that, for any field "K", any polynomial in "K"["x"] can be written as a product of irreducible polynomials.
Polynomials of prime degree have roots.
If every polynomial over "F" of prime degree has a root in "F", then every non-constant polynomial has a root in "F". It follows that a field is algebraically closed if and only if every polynomial over "F" of prime degree has a root in "F".
The field has no proper algebraic extension.
The field "F" is algebraically closed if and only if it has no proper algebraic extension.
If "F" has no proper algebraic extension, let "p"("x") be some irreducible polynomial in "F"["x"]. Then the quotient of "F"["x"] modulo the ideal generated by "p"("x") is an algebraic extension of "F" whose degree is equal to the degree of "p"("x"). Since it is not a proper extension, its degree is 1 and therefore the degree of "p"("x") is 1.
On the other hand, if "F" has some proper algebraic extension "K", then the minimal polynomial of an element in "K" \ "F" is irreducible and its degree is greater than 1.
The field has no proper finite extension.
The field "F" is algebraically closed if and only if it has no proper finite extension because if, within the previous proof, the term "algebraic extension" is replaced by the term "finite extension", then the proof is still valid. (Note that finite extensions are necessarily algebraic.)
Every endomorphism of "Fn" has some eigenvector.
The field "F" is algebraically closed if and only if, for each natural number "n", every linear map from "Fn" into itself has some eigenvector.
An endomorphism of "Fn" has an eigenvector if and only if its characteristic polynomial has some root. Therefore, when "F" is algebraically closed, every endomorphism of "Fn" has some eigenvector. On the other hand, if every endomorphism of "Fn" has an eigenvector, let "p"("x") be an element of "F"["x"]. Dividing by its leading coefficient, we get another polynomial "q"("x") which has roots if and only if "p"("x") has roots. But if "q"("x") = "xn" + "a""n" − 1"x""n" − 1+ ⋯ + "a"0, then "q"("x") is the characteristic polynomial of the "n×n" companion matrix
Decomposition of rational expressions.
The field "F" is algebraically closed if and only if every rational function in one variable "x", with coefficients in "F", can be written as the sum of a polynomial function with rational functions of the form "a"/("x" − "b")n, where "n" is a natural number, and "a" and "b" are elements of "F".
If "F" is algebraically closed then, since the irreducible polynomials in "F"["x"] are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition.
On the other hand, suppose that the property stated above holds for the field "F". Let "p"("x") be an irreducible element in "F"["x"]. Then the rational function 1/"p" can be written as the sum of a polynomial function "q" with rational functions of the form "a"/("x" − "b")n. Therefore, the rational expression
can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since "p"("x") is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.
Relatively prime polynomials and roots.
For any field "F", if two polynomials "p"("x"),"q"("x") ∈ "F"["x"] are relatively prime then they do not have a common root, for if "a" ∈ "F" was a common root, then "p"("x") and  "q"("x") would both be multiples of "x" − "a" and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields.
If the field "F" is algebraically closed, let "p"("x") and "q"("x") be two polynomials which are not relatively prime and let "r"("x") be their greatest common divisor. Then, since "r"("x") is not constant, it will have some root "a", which will be then a common root of "p"("x") and "q"("x").
If "F" is not algebraically closed, let "p"("x") be a polynomial whose degree is at least 1 without roots. Then "p"("x") and "p"("x") are not relatively prime, but they have no common roots (since none of them has roots).
Other properties.
If "F" is an algebraically closed field and "n" is a natural number, then "F" contains all "n"th roots of unity, because these are (by definition) the "n" (not necessarily distinct) zeroes of the polynomial "xn" − 1. A field extension that is contained in an extension generated by the roots of unity is a "cyclotomic extension", and the extension of a field generated by all roots of unity is sometimes called its "cyclotomic closure". Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form "xn" − "a" splits into linear factors is not enough to assure that the field is algebraically closed.
If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number "N" such that the proposition is valid for every algebraically closed field with characteristic "p" when "p" > "N".
Every field "F" has some extension which is algebraically closed. Such an extension is called an algebraically closed extension. Among all such extensions there is one and only one (up to isomorphism, but not unique isomorphism) which is an algebraic extension of "F"; it is called the algebraic closure of "F".
The theory of algebraically closed fields has quantifier elimination.


Register (keyword)

In the C programming language, codice_1 is a reserved word (or keyword), type modifier, storage class, and hint. The codice_1 keyword was deprecated in C++, until it became reserved and unused in C++17. It "suggests" that the compiler store a declared variable in a CPU register (or some other faster location) instead of in RAM. If possible, depending of type of CPU and complexity of the program code, it will optimize access to that variable, and will hence improve the execution time of a program. In C (but not C++ where the keyword is essentially ignored) the location of a variable declared with codice_1 cannot be accessed, but the codice_4 operator can be applied. Aside from this limitation, codice_1 is essentially meaningless in modern compilers due to optimization which will place variables in a register if appropriate regardless of whether the hint is given. For programming of embedded systems codice_1 may still be significant; for example the Microchip MPLAB XC32 compiler allows the programmer to specify a particular register with the keyword; however, this is discouraged in favor of the compiler's optimizations. When used, codice_1 is typically for loop counters, or possibly for other very frequently used variables in the code.
Examples.
/* store integer variable "i" in RAM, register, or other location as compiler sees fit */
int i;
/* suggests storing integer variable "i" in a CPU register or other fast location */
register int i;


Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring formula_1 of a ring formula_2, such that
The last two conditions can be stated in less formal terms: Additively, formula_1 is a free abelian group generated by a basis for "formula_2" over formula_4.
More generally for "formula_14" an integral domain contained in a field "formula_15", we define formula_1 to be an "formula_14"-order in a "formula_15"-algebra "formula_2" if it is a subring of "formula_2" which is a full "formula_14"-lattice.
When "formula_2" is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings.
Examples.
Some examples of orders are:
A fundamental property of "formula_14"-orders is that every element of an "formula_14"-order is integral over "formula_14".
If the integral closure "formula_33" of "formula_14" in "formula_2" is an "formula_14"-order then this result shows that "formula_33" must be the maximal "formula_14"-order in "formula_2". However this hypothesis is not always satisfied: indeed "formula_33" need not even be a ring, and even if "formula_33" is a ring (for example, when "formula_2" is commutative) then "formula_33" need not be an "formula_14"-lattice.
Algebraic number theory.
The leading example is the case where "formula_2" is a number field "formula_15" and formula_1 is its ring of integers. In algebraic number theory there are examples for any "formula_15" other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension "formula_69" of Gaussian rationals over formula_4, the integral closure of "formula_9" is the ring of Gaussian integers "formula_72" and so this is the unique "maximal" "formula_9"-order: all other orders in "formula_2" are contained in it. For example, we can take the subring of the complex numbers in the form formula_75, with formula_38 and formula_77 integers.
The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.


Charles Royal Johnson

Charles Royal Johnson (born January 28, 1948) is an American mathematician specializing in linear algebra. He is a Class of 1961 professor of mathematics at College of William and Mary. The books "Matrix Analysis" and "Topics in Matrix Analysis", co-written by him with Roger Horn, are standard texts in advanced linear algebra.
Career.
Charles R. Johnson received a B.A. with distinction in Mathematics and Economics from Northwestern University in 1969. In 1972, he received a Ph.D. in Mathematics and Economics from the California Institute of Technology, where he was advised by Olga Taussky Todd; his dissertation was entitled "Matrices whose Hermitian Part is Positive Definite". Johnson held various professorships over ten years at the University of Maryland, College Park starting in 1974. He was a professor at Clemson University from 1984–1987. In 1987, he became a professor of mathematics at the College of William and Mary, where he remains today.


Chow group of a stack

In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack formula_1, the Chow group of "X" is the same as the "G"-equivariant Chow group of "Y".
A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers).
Definitions.
 develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence.
If a stack "X" can be written as the quotient stack formula_1 for some quasi-projective variety "Y" with a linearized action of a linear algebraic group "G", then the Chow group of "X" is defined as the "G"-equivariant Chow group of "Y". This approach is introduced and developed by Dan Edidin and William A. Graham, as well as Burt Totaro. later extended the theory to a stack admitting a stratification by quotient stacks.
For higher Chow groups (precursor of motivic homologies) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II.
Examples.
The calculations depend on definitions. Thus, here, we proceed somehow axiomatically. Specifically, we assume: given an algebraic stack "X" locally of finite type over a base field "k",
These properties are valid if "X" is Deligne–Mumford and are expected to hold for any other reasonable theory.
We take "X" to be the classifying stack formula_5, the stack of principal "G"-bundles for a smooth linear algebraic group "G". By definition, it is the quotient stack formula_6, where * is viewed as the stack associated to * = Spec "k". We approximate it as follows. Given an integer "p", choose a representation formula_7 such that there is a "G"-invariant open subset "U" of "V" on which "G" acts freely and the complement formula_8 has codimension formula_9. Let formula_10 be the quotient of formula_11 by the action formula_12. Note the action is free and so formula_10 is a vector bundle over formula_14. By Property 1 applied to this vector bundle,
Then, since formula_16, by Property 2,
since formula_18.
As a concrete example, let formula_19 and let it act on formula_20 by scaling. Then formula_21 acts freely on formula_22. By the above calculation, for each pair of integers "n", "p" such that formula_23,
In particular, for every integer "p" ≥ 0, formula_25. In general, formula_26 for the hyperplane class "h", formula_27 "k"-times self-intersection and formula_28 for negative "k" and so
where the right-hand side is independent of models used in the calculation (since different "h"s correspond under the projections between projective spaces.) For formula_30, the class formula_31, any "n", may be thought of as the fundamental class of formula_32.
Similarly, we have
where formula_34 is the first Chern class of "h" (and "c" and "h" are identified when Chow groups and Chow rings of projective spaces are identified). Since formula_35, we have that formula_36 is the free formula_37-module generated by formula_38.
Virtual fundamental class.
The notion originates in the Kuranishi theory in symplectic geometry.
In § 2. of , given a DM stack "X" and "C""X" the intrinsic normal cone to "X", K. Behrend defines the virtual fundamental class of "X" as
where "s"0 is the zero-section of the cone determined by the perfect obstruction theory and "s"0! is the refined Gysin homomorphism defined just as in Fulton's "Intersection theory". The same paper shows that the degree of this class, morally the integration over it, is equal to the weighted Euler characteristic of the Behrend function of "X".
More recent (circa 2017) approaches do this type of construction in the context of derived algebraic geometry.


Q-Laguerre polynomials

In mathematics, the "q"-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials "P"("x";"q") are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by . give a detailed list of their properties.
Definition.
The "q"-Laguerre polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
Orthogonality.
DEFINED BY THE UNIMONO NATURE OF THE POLYNOMIALS CONVERGANCE AT BOUNDARIES IN INTEGRAL FORM.


Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class C if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.
Properties.
Let "M" be a compact manifold of Fujiki class C, and
formula_1 its complex subvariety. Then "X"
is also in Fujiki class C (, Lemma 4.6). Moreover, the Douady space of "X" (that is, the moduli of deformations of a subvariety formula_1, "M" fixed) is compact and in Fujiki class C.
Conjectures.
J.-P. Demailly and M. Pǎun have
shown that a manifold is in Fujiki class C if and only
if it supports a Kähler current.
They also conjectured that a manifold "M" is in Fujiki class C if it admits a nef current which is "big", that is, satisfies
For a cohomology class formula_4 which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle "L" with first Chern class
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
is generically finite onto its image, which is algebraic, and therefore Kähler.
Fujiki and Ueno asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun


Max Kelly

Gregory Maxwell "Max" Kelly (5 June 1930 – 26 January 2007) was an Australian mathematician who worked on category theory.
Biography.
Kelly was born in Bondi, New South Wales, Australia, on 5 June 1930. He obtained his PhD at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, "Single-space axioms for homology theory". He taught in the Pure Mathematics department at the University of Sydney from 1957 to 1966, rising from lecturer to reader. During 1963–1965 he was a visiting fellow at Tulane University and the University of Illinois, where with Samuel Eilenberg he formalized and developed the notion of an enriched category based on intuitions then in the air about making the homsets of a category just as abstract as the objects themselves.
He subsequently developed the notion in considerably more detail in his 1982 monograph "Basic Concepts of Enriched Category Theory". Let formula_1 be a monoidal category, and denote by formula_1-Cat the category of formula_1-enriched categories. Among other things, Kelly showed that formula_1-Cat has all weighted limits and colimits even when formula_1 does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extensions, density of the Yoneda embedding, and essentially algebraic theories.
In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1992. In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate in the department as professorial fellow and professor emeritus until his death at age 76 on 26 January 2007.
Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of collaborations. His PhD students include Ross Street.


Block Lanczos algorithm

In computer science, the block Lanczos algorithm is an algorithm for finding the nullspace of a matrix over a finite field, using only multiplication of the matrix by long, thin matrices. Such matrices are considered as vectors of tuples of finite-field entries, and so tend to be called 'vectors' in descriptions of the algorithm.
The block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization algorithms such as the quadratic sieve and number field sieve, and its development has been entirely driven by this application.
Parallelization issues.
The algorithm is essentially not parallel: it is of course possible to distribute the matrix–'vector' multiplication, but the whole vector must be available for the combination step at the end of each iteration, so all the machines involved in the calculation must be on the same fast network. In particular, it is not possible to widen the vectors and distribute slices of vectors to different independent machines.
The block Wiedemann algorithm is more useful in contexts where several systems each large enough to hold the entire matrix are available, since in that algorithm the systems can run independently until a final stage at the end.
History.
The block Lanczos algorithm was developed by Peter Montgomery and published in 1995; it is based on, and bears a strong resemblance to, the Lanczos algorithm for finding eigenvalues of large sparse real matrices.


Games graph

In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non-adjacent pair of vertices have exactly 20 shared neighbors. It is named after Richard A. Games, who suggested its construction in an unpublished communication and wrote about related constructions.
Construction.
The construction of this graph involves the unique (up to symmetry) 56-point cap set (a subset of points with no three in line) in formula_1, the five-dimensional projective geometry over a three-element field. The six-dimensional projective geometry, formula_2, can be partitioned into a six-dimensional affine space formula_3 and a copy of formula_1 (the points at infinity with respect to the affine space). The Games graph has as its vertices the 729 points of the affine space formula_3. Each line in the affine space goes through three of these points, and through a fourth point at infinity. The graph contains a triangle for every line of three affine points that passes through a point of the cap set.
Properties.
Several of the graph's properties follow immediately from this construction. It has formula_6 vertices, because the number of points in an affine space is the size of the base field to the power of the dimension. For each affine point, there are 56 lines through cap set points, 56 triangles containing the corresponding vertex, and formula_7 neighbors of the vertex. And there can be no triangles other than the ones coming from the construction, because any other triangle would have to come from three different lines meeting in a common plane of formula_2, and the three cap set points of the three lines would all lie on the intersection of this plane with formula_1, which is a line. But this would violate the defining property of a cap set that it has no three points on a line, so no such extra triangle can exist. The remaining property of strongly regular graphs, that all non-adjacent pairs of points have the same number of shared neighbors, depends on the specific properties of the 5-dimensional cap set.
Related graphs.
With the formula_10 Rook's graph and the Brouwer–Haemers graph, the Games graph is one of only three possible strongly regular graphs whose parameters have the form formula_11.
The same properties that produce a strongly regular graph from a cap set can also be used with an 11-point cap set in formula_12, producing a smaller strongly regular graph with parameters (243,22,1,2).
This graph is the Berlekamp–Van Lint–Seidel graph.


Conjugacy class sum

In abstract algebra, a conjugacy class sum, or simply class sum, is a function defined for each conjugacy class of a finite group "G" as the sum of the elements in that conjugacy class. The class sums of a group form a basis for the center of the associated group algebra.
Definition.
Let "G" be a finite group, and let "C1...,Ck" be the distinct conjugacy classes of "G". For 1≤ "i"≤ "k", define
The functions formula_2 are the class sums of "G".
In the Group Algebra.
Let CG" be the complex group algebra over "G". Then the center of CG", denoted "Z"(C"G"), is defined by
This is equal to the set of all class functions (functions which are constant on conjugacy classes). To see this, note that "f" is central if and only if "f"("yx")="f"("xy") for all "x,y" ∈ "G". Replacing "y" by "yx−1", this condition becomes
The class sums are a basis for the set of all class functions, and thus they are a basis for the center of the algebra.
In particular, this shows that the dimension of "Z"(C"G") is equal to the number of class sums of "G".


(B, N) pair

In mathematics, a ("B", "N") pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems.
Definition.
A ("B", "N") pair is a pair of subgroups "B" and "N" of a group "G" such that the following axioms hold:
The idea of this definition is that "B" is an analogue of the upper triangular matrices of
the general linear group "GL""n"("K"), "H" is an analogue of the diagonal matrices, and "N" is an analogue of the normalizer of "H".
The subgroup "B" is sometimes called the Borel subgroup, "H" is sometimes called the Cartan subgroup, and "W" is called the Weyl group. The pair ("W","S") is a "Coxeter system".
The number of generators is called the rank.
Properties of groups with a BN pair.
The map taking "w" to "BwB" is an isomorphism from the set of elements of "W" to the set of double cosets of "B"; this is the Bruhat decomposition "G" = "BWB".
If "T" is a subset of "S" then let "W"("T") be the subgroup of "W" generated by "T": we define and "G"("T") = "BW"("T")"B" to be the "standard parabolic subgroup" for "T".
The subgroups of "G" containing conjugates of "B" are the "parabolic subgroups"; conjugates of "B" are called Borel subgroups (or minimal parabolic subgroups). These are precisely the standard parabolic subgroups.
Applications.
BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if "G" has a "BN"-pair such that "B" is a solvable group, the intersection of all conjugates of "B" is trivial, and the set of generators of "W" cannot be decomposed into two non-empty commuting sets, then "G" is simple whenever it is a perfect group. In practice all of these conditions except for "G" being perfect are easy to check. Checking that "G" is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showing that a group is perfect is usually far easier than showing it is simple.