Kyoji Saito

Kyōji Saitō (齋藤 恭司, Saitō Kyōji; born 25 December 1944) is a Japanese mathematician, specializing in algebraic geometry and complex analytic geometry.
Educationa and career.
Saito received in 1971 his promotion Ph.D. from the University of Göttingen under Egbert Brieskorn, with thesis "Quasihomogene isolierte Singularitäten von Hyperflächen" (Quasihomogeneous isolated singularities of hypersurfaces). Saito is a professor at the Research Institute for Mathematical Sciences (RIMS) of Kyoto University.
Saito's research deals with the interplay among Lie algebras, reflection groups (Coxeter groups), braid groups, and singularities of hypersurfaces.
From the 1980s, he did research on underlying symmetries of period integrals in complex hypersurfaces. Saito introduced higher-dimensional generalizations of elliptic integrals. These generalizations are integrals of "primitive forms", first considered in the study of the unfolding of isolated singularities of complex hypersurfaces, associated with infinite-dimensional Lie algebras. He also studied the corresponding new automorphic forms. The theory has a geometric connection to "flat structures" (now called "Saito Frobenius manifolds"), mirror symmetry, Frobenius manifolds, and Gromov–Witten theory in algebraic geometry and various topics in mathematical physics related to string theory.
Saito supervised the thesis of 7 Ph.D. students at Kyoto University, including Hiroaki Terao and Masahiko Yoshinaga.
He was an Invited Speaker with talk "The limit element in the configuration algebra for a discrete group: a précis" at the International Congress of Mathematicians 1990 in Kyoto. In 2011 he was awarded the Geometry Prize of the Mathematical Society of Japan.


Orthogonal transformation

In linear algebra, an orthogonal transformation is a linear transformation "T" : "V" → "V" on a real inner product space "V", that preserves the inner product. That is, for each pair of elements of "V", we have
Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases.
Orthogonal transformations are injective: if formula_2 then formula_3, hence formula_4, so the kernel of formula_5 is trivial.
Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions.
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of "V". The columns of the matrix form another orthonormal basis of "V".
If an orthogonal transformation is invertible (which is always the case when "V" is finite-dimensional) then its inverse is another orthogonal transformation. Its matrix representation is the transpose of the matrix representation of the original transformation.
Examples.
Consider the inner-product space formula_6 with the standard euclidean inner product and standard basis. Then, the matrix transformation
is orthogonal. To see this, consider
Then,
The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on formula_10:


Pseudo algebraically closed field

In mathematics, a field formula_1 is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.
Formulation.
A field "K" is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:


Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and :
or in matrix form:
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix formula_2 is denoted by formula_3, then the Hermitian property can be written concisely as
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are formula_4, although note that in quantum mechanics, formula_5 typically means the complex conjugate only, and not the conjugate transpose.
Alternative characterizations.
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
Equality with the adjoint.
A square matrix formula_2 is Hermitian if and only if it is equal to its adjoint, that is, it satisfies
formula_7
for any pair of vectors formula_8, where formula_9 denotes the inner product operation.
This is also the way that the more general concept of self-adjoint operator is defined.
Reality of quadratic forms.
A square matrix formula_2 is Hermitian if and only if
formula_11
Spectral properties.
A square matrix formula_2 is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.
Applications.
Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.
Examples.
In this section, the conjugate transpose of matrix formula_13 is denoted as formula_14, the transpose of matrix formula_13 is denoted as formula_16 and conjugate of matrix formula_13 is denoted as formula_18.
See the following example:
The diagonal elements must be real, as they must be their own complex conjugate.
Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients, which results in skew-Hermitian matrices.
Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix formula_13 equals the multiplication of a matrix and its conjugate transpose, that is, formula_21, then formula_13 is a Hermitian positive semi-definite matrix. Furthermore, if formula_23 is row full-rank, then formula_13 is positive definite.
Decomposition into Hermitian and skew-Hermitian.
Additional facts related to Hermitian matrices include:
Rayleigh quotient.
In mathematics, for a given complex Hermitian matrix "M" and nonzero vector x, the Rayleigh quotient formula_59, is defined as:
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose formula_61 to the usual transpose formula_62. Note that formula_63 for any non-zero real scalar formula_64. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value formula_65 (the smallest eigenvalue of M) when formula_66 is formula_67 (the corresponding eigenvector). Similarly, formula_68 and formula_69.
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, formula_70 is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.


Nagata ring

In commutative algebra, an N−1 ring is an integral domain "A" whose integral closure in its quotient field is a finitely generated "A" module. It is called a Japanese ring (or an N−2 ring) if for every finite extension "L" of its quotient field "K", the integral closure of "A" in "L" is a finitely generated "A"-module (or equivalently a finite "A"-algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, (or a pseudo–geometric ring) if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N−2 rings.) A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring , but this concept is not used much.
Examples.
Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important example is a Noetherian integrally closed domain (e.g. a Dedekind domain) having a perfect field of fractions. On the other hand, a PID or even a DVR is not necessarily Japanese.
Any quasi-excellent ring is a Nagata ring, so in particular almost all Noetherian rings that occur in algebraic geometry are Nagata rings.
The first example of a Noetherian domain that is not a Nagata ring was given by .
Here is an example of a discrete valuation ring that is not a Japanese ring. Choose a prime "p" and an infinite degree field extension "K" of a characteristic "p" field "k", such that "K""p"⊆"k". Let the discrete valuation ring "R" be the ring of formal power series over "K" whose coefficients generate a finite extension of "k". If "y" is any formal power series not in "R" then the ring "R"["y"] is not an N−1 ring (its integral closure is not a finitely generated module) so "R" is not a Japanese ring.
If "R" is the subring of the polynomial ring "k"["x"1,"x"2...] in infinitely many generators generated by the squares and cubes of all generators, and "S" is obtained from "R" by adjoining inverses to all elements not in any of the ideals generated by some "x""n", then "S" is a 1-dimensional Noetherian domain that is not an N−1 ring, in other words its integral closure in its quotient field is not a finitely generated "S"-module. Also "S" has a cusp singularity at every closed point, so the set of singular points is not closed.


John R. Stallings

John Robert Stallings Jr. (July 22, 1935 – November 24, 2008) was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.
Biographical data.
John Stallings was born on July 22, 1935, in Morrilton, Arkansas.
Stallings received his B.Sc. from University of Arkansas in 1956 (where he was one of the first two graduates in the university's Honors program) and he received a Ph.D. in Mathematics from Princeton University in 1959 under the direction of Ralph Fox.
After completing his PhD, Stallings held a number of postdoctoral and faculty positions, including being an NSF postdoctoral fellow at the University of Oxford as well as an instructorship and a faculty appointment at Princeton. Stallings joined the University of California at Berkeley as a faculty member in 1967 where he remained until his retirement in 1994. Even after his retirement, Stallings continued supervising UC Berkeley graduate students until 2005. Stallings was an Alfred P. Sloan Research fellow from 1962 to 1965 and a Miller Institute fellow from 1972 to 1973.
Over the course of his career, Stallings had 22 doctoral students including Marc Culler, Stephen M. Gersten, and J. Hyam Rubinstein and 100 doctoral descendants. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds.
Stallings delivered an invited address as the International Congress of Mathematicians in Nice in 1970 and a James K. Whittemore Lecture at Yale University in 1969.
Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970.
The conference "Geometric and Topological Aspects of Group Theory", held at the Mathematical Sciences Research Institute in Berkeley in May 2000, was dedicated to the 65th birthday of Stallings.
In 2002 a special issue of the journal "Geometriae Dedicata" was dedicated to Stallings on the occasion of his 65th birthday. Stallings died from prostate cancer on November 24, 2008.
Mathematical contributions.
Most of Stallings' mathematical contributions are in the areas of geometric group theory and low-dimensional topology (particularly the topology of 3-manifolds) and on the interplay between these two areas.
An early significant result of Stallings is his 1960 proof of the Poincaré conjecture in dimensions greater than six. (Stallings' proof was obtained independently from and shortly after the different proof of Stephen Smale who established the same result in dimensions bigger than four).
Using "engulfing" methods similar to those in his proof of the Poincaré conjecture for "n" > 6, Stallings proved that ordinary Euclidean "n"-dimensional space has a unique piecewise linear, hence also smooth, structure, if "n" is not equal to 4. This took on added significance when, as a consequence of work of Michael Freedman and Simon Donaldson in 1982, it was shown that 4-space has exotic smooth structures, in fact uncountably many such.
In a 1963 paper Stallings constructed an example of a finitely presented group with infinitely generated 3-dimensional integral homology group and, moreover, not of the type formula_1, that is, not admitting a classifying space with a finite 3-skeleton. This example came to be called the "Stallings group" and is a key example in the study of homological finiteness properties of groups. Robert Bieri later showed that the Stallings group is exactly the kernel of the homomorphism from the direct product of three copies of the free group formula_2 to the additive group formula_3 of integers that sends to formula_4 the six elements coming from the choice of free bases for the three copies of formula_2. Bieri also showed that the Stallings group fits into a sequence of examples of groups of type formula_6 but not of type formula_7. The Stallings group is a key object in the version of discrete Morse theory for cubical complexes developed by Mladen Bestvina and Noel Brady and in the study of subgroups of direct products of limit groups.
Stallings' most famous theorem in group theory is an algebraic characterization of groups with more than one end (that is, with more than one "connected component at infinity"), which is now known as Stallings' theorem about ends of groups. Stallings proved that a finitely generated group "G" has more than one end if and only if this group admits a nontrivial splitting as an amalgamated free product or as an HNN extension over a finite group (that is, in terms of Bass–Serre theory, if and only if the group admits a nontrivial action on a tree with finite edge stabilizers). More precisely, the theorem states that a finitely generated group "G" has more than one end if and only if either "G" admits a splitting as an amalgamated free product formula_8, where the group "C" is finite and formula_9, formula_10, or "G" admits a splitting as an HNN extension formula_11 where formula_12 are finite subgroups of "H".
Stallings proved this result in a series of works, first dealing with the torsion-free case (that is, a group with no nontrivial elements of finite order) and then with the general case. Stalling's theorem yielded a positive solution to the long-standing open problem about characterizing finitely generated groups of cohomological dimension one as exactly the free groups. Stallings' theorem about ends of groups is considered one of the first results in geometric group theory proper since it connects a geometric property of a group (having more than one end) with its algebraic structure (admitting a splitting over a finite subgroup). Stallings' theorem spawned many subsequent alternative proofs by other mathematicians (e.g.) as well as many applications (e.g.). The theorem also motivated several generalizations and relative versions of Stallings' result to other contexts, such as the study of the notion of relative ends of a group with respect to a subgroup, including a connection to CAT(0) cubical complexes. A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of C. T. C. Wall.
Another influential paper of Stallings is his 1983 article "Topology on finite graphs". Traditionally, the algebraic structure of subgroups of free groups has been studied in combinatorial group theory using combinatorial methods, such as the Schreier rewriting method and Nielsen transformations. Stallings' paper put forward a topological approach based on the methods of covering space theory that also used a simple graph-theoretic framework. The paper introduced the notion of what is now commonly referred to as "Stallings subgroup graph" for describing subgroups of free groups, and also introduced a foldings technique (used for approximating and algorithmically obtaining the subgroup graphs) and the notion of what is now known as a "Stallings folding". Most classical results regarding subgroups of free groups acquired simple and straightforward proofs in this set-up and Stallings' method has become the standard tool in the theory for studying the subgroup structure of free groups, including both the algebraic and algorithmic questions (see ). In particular, Stallings subgroup graphs and Stallings foldings have been the used as a key tools in many attempts to approach the Hanna Neumann conjecture.
Stallings subgroup graphs can also be viewed as finite-state automata and they have also found applications in semigroup theory and in computer science.
Stallings' foldings method has been generalized and applied to other contexts, particularly in Bass–Serre theory for approximating group actions on trees and studying the subgroup structure of the fundamental groups of graphs of groups. The first paper in this direction was written by Stallings himself, with several subsequent generalizations of Stallings' folding methods in the Bass–Serre theory context by other mathematicians.
Stallings' 1991 paper "Non-positively curved triangles of groups"" introduced and studied the notion of a triangle of groups. This notion was the starting point for the theory of complexes of groups (a higher-dimensional analog of Bass–Serre theory), developed by André Haefliger and others. Stallings' work pointed out the importance of imposing some sort of "non-positive curvature" conditions on the complexes of groups in order for the theory to work well; such restrictions are not necessary in the one-dimensional case of Bass–Serre theory.
Among Stallings' contributions to 3-manifold topology, the most well-known is the "Stallings fibration theorem". The theorem states that if "M" is a compact irreducible 3-manifold whose fundamental group contains a normal subgroup, such that this subgroup is finitely generated and such that the quotient group by this subgroup is infinite cyclic, then "M" fibers over a circle. This is an important structural result in the theory of Haken manifolds that engendered many alternative proofs, generalizations and applications (e.g. ), including a higher-dimensional analog.
A 1965 paper of Stallings "How not to prove the Poincaré conjecture" gave a group-theoretic reformulation of the famous Poincaré conjecture. The paper began with a humorous admission: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now, no one has known about it." Despite its ironic title, Stallings' paper informed much of the subsequent research on exploring the algebraic aspects of the Poincaré conjecture (see, for example,).


Takashi Ono (mathematician)

 is a retired Japanese-born American mathematician, specializing in number theory and algebraic groups.
Early life and education.
Takashi Ono was born in Nishinomiya, Japan. He received his Ph.D. in 1958 at Nagoya University.
Career.
Ono immigrated to the United States after receiving an invitation from J. Robert Oppenheimer to work at the Institute for Advanced Study with a fellowship for the two academic years 1959–1961 and then went to the University of British Columbia to work as a mathematics professor from 1961 to 1964.
From 1964 to 1969 Ono was a tenured professor at the University of Pennsylvania. From 1969 to his retirement in 2011, he was a professor at Johns Hopkins University.
In 1966 he was an invited speaker at the International Congress of Mathematicians in Moscow. In 2012 he was elected a Fellow of the American Mathematical Society.
Personal life.
Ono's youngest son, Ken Ono, is also a mathematician and professor at the University of Virginia as well as a former triathlete. His middle son, Santa J. Ono, is the President and Vice-Chancellor of the University of British Columbia (previously, the 28th President of the University of Cincinnati) and a biomedical researcher. His eldest son, Momoro Ono, is a music professor at Creighton University.


Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
The prehistory of Dedekind domains.
In the 19th century it became a common technique to gain insight into integer solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer formula_1. In the attempt to determine which integers are represented by the quadratic form formula_2, it is natural to factor the quadratic form into formula_3, the factorization taking place in the ring of integers of the quadratic field formula_4. Similarly, for a positive integer formula_5 the polynomial formula_6 (which is relevant for solving the Fermat equation formula_7) can be factored over the ring formula_8, where formula_9 is a primitive "n"-th root of unity.
For a few small values of formula_1 and formula_5 these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat (formula_12) and Euler (formula_13). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field formula_14 is a PID was well known to the quadratic form theorists. Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of formula_15 for which the ring of integers is a PID and conjectured that there were no further values. (Gauss' conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.) However, this was understood (only) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all formula_16; that is, that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring formula_8 is a UFD. Ernst Kummer had shown three years before that this was not the case already for formula_18 (the full, finite list of values for which formula_8 is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents formula_5 using what we now recognize as the fact that the ring formula_8 is a Dedekind domain. In fact Kummer worked not with ideals but with "ideal numbers", and the modern definition of an ideal was given by Dedekind.
By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID, but as seen above the ring formula_22 of algebraic integers in a number field formula_23 need not be a PID. In fact, although Gauss also conjectured that there are infinitely many primes formula_24 such that the ring of integers of formula_25 is a PID, we do not even know whether there are infinitely many number fields formula_23 (of arbitrary degree) such that formula_22 is a PID! On the other hand, the ring of integers in a number field is always a Dedekind domain.
Another illustration of the delicate/robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property: a Noetherian domain formula_28 is Dedekind iff for every maximal ideal formula_29 of formula_28 the localization formula_31 is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.
Alternative definitions.
For an integral domain formula_28 that is not a field, all of the following conditions are equivalent:
Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste. In practice, it is often easiest to verify (DD4).
A Krull domain is a higher-dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension 1. This notion can be used to study the various characterizations of a Dedekind domain. In fact, this is the definition of a Dedekind domain used in Bourbaki's "Commutative algebra".
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
Some examples of Dedekind domains.
All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.
The ring formula_46 of algebraic integers in a number field "K" is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal "I" of "R", "R"/"I" is a finite set, and recall that a finite integral domain is a field; so by (DD4) "R" is a Dedekind domain. As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples.
The other class of Dedekind rings that is arguably of equal importance comes from geometry: let "C" be a nonsingular geometrically integral affine algebraic curve over a field "k". Then the coordinate ring "k"["C"] of regular functions on "C" is a Dedekind domain. This is largely clear simply from translating geometric terms into algebra: the coordinate ring of any affine variety is, by definition, a finitely generated "k"-algebra, hence Noetherian; moreover "curve" means "dimension one" and "nonsingular" implies (and, in dimension one, is equivalent to) "normal", which by definition means "integrally closed".
Both of these constructions can be viewed as special cases of the following basic result:
Theorem: Let "R" be a Dedekind domain with fraction field "K". Let "L" be a finite degree field extension of "K" and denote by "S" the integral closure of "R" in "L". Then "S" is itself a Dedekind domain.
Applying this theorem when "R" is itself a PID gives us a way of building Dedekind domains out of PIDs. Taking "R" = Z, this construction says precisely that rings of integers of number fields are Dedekind domains. Taking "R" = "k"["t"], one obtains the above case of nonsingular affine curves as branched coverings of the affine line.
Zariski and Samuel were sufficiently taken with this construction to ask whether every Dedekind domain arises from it; that is, by starting with a PID and taking the integral closure in a finite degree field extension. A surprisingly simple negative answer was given by L. Claborn.
If the situation is as above but the extension "L" of "K" is algebraic of infinite degree, then it is still possible for the integral closure "S" of "R" in "L" to be a Dedekind domain, but it is not guaranteed. For example, take again "R" = Z, "K" = Q and now take "L" to be the field formula_47 of all algebraic numbers. The integral closure is nothing else than the ring formula_48 of all algebraic integers. Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that formula_48 is not even Noetherian! In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Prüfer domain; it turns out that the ring of algebraic integers is slightly more special than this: it is a Bézout domain.
Fractional ideals and the class group.
Let "R" be an integral domain with fraction field "K". A fractional ideal is a nonzero "R"-submodule "I" of "K" for which there exists a nonzero "x" in "K" such that formula_50
Given two fractional ideals "I" and "J", one defines their product "IJ" as the set of all finite sums formula_51: the product "IJ" is again a fractional ideal. The set Frac("R") of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal "R".
For any fractional ideal "I", one may define the fractional ideal
One then tautologically has formula_53. In fact one has equality if and only if "I", as an element of the monoid of Frac("R"), is invertible. In other words, if "I" has any inverse, then the inverse must be formula_54.
A principal fractional ideal is one of the form formula_55 for some nonzero "x" in "K". Note that each principal fractional ideal is invertible, the inverse of formula_55 being simply formula_57. We denote the subgroup of principal fractional ideals by Prin("R").
A domain "R" is a PID if and only if every fractional ideal is principal. In this case, we have Frac("R") = Prin("R") = formula_58, since two principal fractional ideals formula_55 and formula_60 are equal iff formula_61 is a unit in "R".
For a general domain "R", it is meaningful to take the quotient of the monoid Frac("R") of all fractional ideals by the submonoid Prin("R") of principal fractional ideals. However this quotient itself is generally only a monoid. In fact it is easy to see that the class of a fractional ideal I in Frac("R")/Prin("R") is invertible if and only if I itself is invertible.
Now we can appreciate (DD3): in a Dedekind domain (and only in a Dedekind domain) every fractional ideal is invertible. Thus these are precisely the class of domains for which Frac("R")/Prin("R") forms a group, the ideal class group Cl("R") of "R". This group is trivial if and only if "R" is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.
We note that for an arbitrary domain one may define the Picard group Pic("R") as the group of invertible fractional ideals Inv("R") modulo the subgroup of principal fractional ideals. For a Dedekind domain this is of course the same as the ideal class group. However, on a more general class of domains, including Noetherian domains and Krull domains, the ideal class group is constructed in a different way, and there is a canonical homomorphism
which is however generally neither injective nor surjective. This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety.
A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group "G" whatsoever, there exists a Dedekind domain "R" whose ideal class group is isomorphic to "G". Later, C.R. Leedham-Green showed that such an "R" may constructed as the integral closure of a PID in a quadratic field extension (Leedham-Green 1972). In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an "elliptic" construction should be possible for a general abelian group (Rosen 1976). Rosen's conjecture was proven in 2008 by P.L. Clark (Clark 2009).
In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.
Finitely generated modules over a Dedekind domain.
In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.
Let us briefly recall the structure theory in the case of a finitely generated module formula_29 over a PID formula_28. We define the torsion submodule formula_64 to be the set of elements formula_1 of formula_29 such that formula_67 for some nonzero formula_68 in formula_28. Then:
(M1) formula_64 can be decomposed into a direct sum of cyclic torsion modules, each of the form formula_71 for some nonzero ideal formula_37 of formula_28. By the Chinese Remainder Theorem, each formula_71 can further be decomposed into a direct sum of submodules of the form formula_75, where formula_76 is a power of a prime ideal. This decomposition need not be unique, but any two decompositions
differ only in the order of the factors.
(M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule formula_78 of formula_29 such that formula_80.
(M3PID) formula_78 isomorphic to formula_82 for a uniquely determined non-negative integer formula_5. In particular, formula_78 is a finitely generated free module.
Now let formula_29 be a finitely generated module over an arbitrary Dedekind domain formula_28. Then (M1) and (M2) hold verbatim. However, it follows from (M3PID) that a finitely generated torsionfree module formula_78 over a PID is free. In particular, it asserts that all fractional ideals are principal, a statement that is false whenever formula_28 is not a PID. In other words, the nontriviality of the class group Cl(R) causes (M3PID) to fail. Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain. Over an arbitrary Dedekind domain one has
(M3DD) formula_78 is isomorphic to a direct sum of rank one projective modules: formula_90. Moreover, for any rank one projective modules formula_91, one has
if and only if
and
Rank one projective modules can be identified with fractional ideals, and the last condition can be rephrased as
Thus a finitely generated torsionfree module of rank formula_96 can be expressed as formula_97, where formula_37 is a rank one projective module. The Steinitz class for "P" over "R" is the class formula_99 of formula_37 in Cl(R): it is uniquely determined. A consequence of this is:
Theorem: Let "R" be a Dedekind domain. Then formula_101, where K0("R") is the Grothendieck group of the commutative monoid of finitely generated projective "R" modules.
These results were established by Ernst Steinitz in 1912.
An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.
Locally Dedekind rings.
There exist integral domains formula_28 that are locally but not globally Dedekind: the localization of formula_28 at each maximal ideal is a Dedekind ring (equivalently, a DVR) but formula_28 itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called "proper almost Dedekind rings."


Barbara L. Osofsky

Barbara L. Osofsky (born 1937) is a retired professor of mathematics at Rutgers University. Her research concerns abstract algebra. Osofsky's contributions to mathematics include her characterization of semisimple rings in terms of properties of cyclic modules. Osofsky also established a logical equivalence between the continuum hypothesis and statements about the global dimension of associative rings.
Career.
Osofsky received her Ph.D. from Rutgers University in 1964. She then worked at Rutgers until 2004, when she retired. She served as acting chair of the Rutgers mathematics department in 1978.
Awards and honors.
In 1973, Osofsky addressed a national meeting of the AMS. She was the first woman in 50 years to do so. She became the first female editor of an AMS journal in 1974 when she became the editor of Proceedings of the American Mathematical Society.
From 2000 to 2002, Osofsky served as First Vice-President of the Mathematical Association of America. In 2005, she was awarded the MAA meritorious service award.
In 2012, Osofsky became a fellow of the American Mathematical Society.


Pidduck polynomials

In mathematics, the Pidduck polynomials "s""n"("x") are polynomials introduced by given by the generating function


Kan extension

Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.
An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors.
In "Categories for the Working Mathematician" Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that
A comprehensive tutorial on Kan extensions can be found in “All Concepts are Kan Extensions” by M.C.Lehner.
Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on constrained optimization.
Definition.
A Kan extension proceeds from the data of three categories
and two functors
and comes in two varieties: the "left" Kan extension and the "right" Kan extension of formula_3 along formula_4.
The right Kan extension amounts to finding the dashed arrow and the natural transformation formula_5 in the following diagram:
Formally, the right Kan extension of formula_3 along formula_4 consists of a functor formula_10 and a natural transformation formula_11 that is couniversal with respect to the specification, in the sense that for any functor formula_12 and natural transformation formula_13, a unique natural transformation formula_14 is defined and fits into a commutative diagram:
where formula_15 is the natural transformation with formula_16 for any object formula_17 of formula_18
The functor "R" is often written formula_19.
As with the other universal constructs in category theory, the "left" version of the Kan extension is dual to the "right" one and is obtained by replacing all categories by their opposites.
The effect of this on the description above is merely to reverse the direction of the natural transformations.
This gives rise to the alternate description: the left Kan extension of formula_3 along formula_4 consists of a functor formula_29 and a natural transformation formula_30 that are universal with respect to this specification, in the sense that for any other functor formula_12 and natural transformation formula_32, a unique natural transformation formula_33 exists and fits into a commutative diagram:
where formula_34 is the natural transformation with formula_35 for any object formula_17 of formula_37.
The functor "L" is often written formula_38.
The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique up to unique isomorphism. In this case, that means that (for left Kan extensions) if formula_39 are two left Kan extensions of formula_3 along formula_4, and formula_42 are the corresponding transformations, then there exists a unique "isomorphism" of functors formula_33 such that the second diagram above commutes. Likewise for right Kan extensions.
Properties.
Kan extensions as (co)limits.
Suppose formula_44 and formula_45 are two functors. If A is small and C is cocomplete, then there exists a left Kan extension formula_38 of formula_3 along formula_4, defined at each object "b" of B by
where the colimit is taken over the comma category formula_50, where formula_51 is the constant functor. Dually, if A is small and C is complete, then right Kan extensions along formula_4 exist, and can be computed as the limit
over the comma category formula_54.
Kan extensions as (co)ends.
Suppose formula_55 and formula_56 are two functors such that for all objects "m" and "m′" of M and all objects "c" of C, the copowers formula_57 exist in A. Then the functor "T" has a left Kan extension "L" along "K", which is such that, for every object "c" of C,
when the above coend exists for every object "c" of C.
Dually, right Kan extensions can be computed by the end formula
Limits as Kan extensions.
The limit of a functor formula_60 can be expressed as a Kan extension by
where formula_62 is the unique functor from formula_63 to 𝟙 (the category with one object and one arrow, a terminal object in formula_64). The colimit of formula_4 can be expressed similarly by
Adjoints as Kan extensions.
A functor formula_67 possesses a left adjoint if and only if the right Kan extension of formula_68 along formula_4 exists and is preserved by formula_4. In this case, a left adjoint is given by formula_71 and this Kan extension is even preserved by any functor formula_72 whatsoever, i.e. is an "absolute" Kan extension.
Dually, a right adjoint exists if and only if the left Kan extension of the identity along formula_4 exists and is preserved by formula_4.
Applications.
The codensity monad of a functor formula_75 is a right Kan extension of "G" along itself.


G-ring

In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck.
A ring that is a both G-ring and a J-2 ring is called a quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring.
Examples.
Here is an example of a discrete valuation ring "A" of characteristic "p">0 which is not a G-ring. If "k" is any field of characteristic "p" with ["k":"k""p"] = ∞ and "R"="k""x" and "A" is the subring of power series Σ"a"i"x""i" such that ["k""p"("a"0,"a"1...):"k""p" ] is finite then the formal fiber of "A" over the generic point is not geometrically regular so "A" is not a G-ring. Here "k""p" denotes the image of "k" under the Frobenius morphism "a"→"a""p".


Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.
In general the map from the Pin group to the orthogonal group is "not" onto or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.
The non-trivial element of the kernel is denoted formula_1 which should not be confused with the orthogonal transform of reflection through the origin, generally denoted formula_2
Definite form.
The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form "Q" and for its negative −"Q" are not isomorphic, but the orthogonal groups are.
In terms of the standard forms, O("n", 0) = O(0, "n"), but Pin("n", 0) and Pin(0, "n") are in general not isomorphic. Using the "+" sign convention for Clifford algebras (where formula_3), one writes
and these both map onto O("n") = O("n", 0) = O(0, "n").
By contrast, we have the natural isomorphism Spin("n", 0) ≅ Spin(0, "n") and they are both the (unique) non-trivial double cover of the special orthogonal group SO("n"), which is the (unique) universal cover for "n" ≥ 3.
Indefinite form.
There are as many as eight different double covers of O("p", "q"), for "p", "q" ≠ 0, which correspond to the extensions of the center (which is either C2 × C2 or C4) by C2. Only two of them are pin groups—those that admit the Clifford algebra as a representation. They are called Pin("p", "q") and Pin("q", "p") respectively.
As topological group.
Every connected topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group. For a disconnected topological group, there is a unique universal cover of the identity component of the group, and one can take the same cover as topological spaces on the other components (which are principal homogeneous spaces for the identity component) but the group structure on other components is not uniquely determined in general.
The Pin and Spin groups are "particular" topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.
In 2001, Andrzej Trautman found the set of all 32 inequivalent double covers of O("p") x O("q"), the maximal compact subgroup of O("p", "q") and an explicit construction of 8 double covers of the same group O("p", "q").
Construction.
The two pin groups correspond to the two central extensions
The group structure on Spin("V") (the connected component of determinant 1) is already determined; the group structure on the other component is determined up to the center, and thus has a ±1 ambiguity.
The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin("V") → SO("V")), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in O("V"), "r"2 = 1, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of Spin±("V") → SO("V"), so formula_6, and either choice determines a pin group (since all reflections are conjugate by an element of SO("V"), which is connected, all reflections must square to the same value).
Concretely, in Pin+, formula_7 has order 2, and the preimage of a subgroup {1, "r"} is C2 × C2: if one repeats the same reflection twice, one gets the identity.
In Pin−, formula_7 has order 4, and the preimage of a subgroup {1, "r"} is C4: if one repeats the same reflection twice, one gets "a rotation by 2π"—the non-trivial element of Spin("V") → SO("V") can be interpreted as "rotation by 2π" (every axis yields the same element).
Low dimensions.
In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups:
In 2 dimensions, the distinction between Pin+ and Pin− mirrors the distinction between the dihedral group of a 2"n"-gon and the dicyclic group of the cyclic group C2"n".
In Pin+, the preimage of the dihedral group of an "n"-gon, considered as a subgroup Dih"n" < O(2), is the dihedral group of a 2"n"-gon, Dih2"n" < Pin+(2), while in Pin−, the preimage of the dihedral group is the dicyclic group Dic"n" < Pin−(2).
The resulting commutative square of subgroups for Spin(2), Pin+(2), SO(2), O(2) – namely C2"n", Dih2"n", C"n", Dih"n" – is also obtained using the projective orthogonal group (going down from O by a 2-fold quotient, instead of up by a 2-fold cover) in the square SO(2), O(2), PSO(2), PO(2), though in this case it is also realized geometrically, as "the projectivization of a 2"n"-gon in the circle is an "n"-gon in the projective line".
In 3 dimensions the situation is as follows. The Clifford algebra generated by 3 anticommuting square roots of +1 is the algebra of 2×2 complex matrices, and Pin+(3) is isomorphic to SO(3) × C4. The Clifford algebra generated by 3 anticommuting square roots of -1 is the algebra formula_10, and Pin−(3) is isomorphic to SU(2) × C2. These groups are nonisomorphic because the center of Pin+(3) is C4 while the center of Pin−(3) is C2 × C2.
Center.
The center of Pin(n,0) = Pin+ is C2 when n is even, C2 × C2 when n = 1 mod 4, and C4 when "n" = 3 mod 4. The center of Pin(0,n) = Pin− is C2 when n is even, C4 when n = 1 mod 4, and C2 × C2 when "n" = 3 mod 4. For "p", "q" ≠ 0 the center of Pin(p,q) is an extension of either C2 × C2 or C4 by C2.
Name.
The name was introduced in , where they state "This joke is due to J-P. Serre".
It is a back-formation from Spin: "Pin is to O("n") as Spin is to SO("n")", hence dropping the "S" from "Spin" yields "Pin".


Ray Kunze

Ray Alden Kunze (March 7, 1928 – May 21, 2014) was an American mathematician who chaired the mathematics departments at the University of California, Irvine and the University of Georgia. His mathematical research concerned the representation theory of groups and noncommutative harmonic analysis.
Kunze was born in Des Moines, Iowa and grew up near Milwaukee, Wisconsin. He began his undergraduate studies at Denison University but transferred to the University of Chicago after two years, and earned bachelor's and master's degrees in mathematics. After working as a military mathematical analyst, he returned to the University of Chicago, and earned his Ph.D. in 1957 with a dissertation on Fourier transformations supervised by Irving Segal. As well as his positions at UCI and Georgia, he also worked at the Institute for Advanced Study, Massachusetts Institute of Technology, Brandeis University, and Washington University in St. Louis. He has over 50 academic descendants, many of them through his students Paul Sally at Brandeis and Edward N. Wilson at Washington University.
With his advisor Irving Segal, Kunze was the author of the textbook "Integrals and Operators" (McGraw-Hill, 1968; 2nd ed., Grundlehren der Mathematischen Wissenschaften 228, Springer, 1978). With Kenneth M. Hoffman he was the author of "Linear Algebra" (Prentice-Hall, 1961; 2nd ed., Pearson, 1971).
In 1994, a special session on representation theory and harmonic analysis was held in honor of Kunze as part of the 889th meeting of the American Mathematical Society, and the papers from the session were published as a festschrift.
In 2012, Kunze was recognized as one of the inaugural fellows of the American Mathematical Society.


Thomas W. Hungerford

Thomas William Hungerford (March 21, 1936 – November 28, 2014) was an American mathematician who worked in algebra and mathematics education. He is the author or coauthor of several widely used and widely cited textbooks covering high-school to graduate-level mathematics. From 1963 until 1980 he taught at the University of Washington and then at Cleveland State University until 2003. From 2003–2014 he was at Saint Louis University. Hungerford had a special interest in promoting the use of technology to teach mathematics.
Hungerford did his undergraduate work at the College of the Holy Cross and defended his Ph.D. thesis at the University of Chicago in 1963 (advised by Saunders Mac Lane). Throughout his career he wrote more than a dozen widely used mathematics textbooks, ranging from high school to graduate level.


Illumination angle

In computer graphics and geography, the illumination angle of a surface with a light source (such as the Earth's surface and the Sun) is the angle between the inward surface normal and the direction of light. It can also be equivalently described as the angle between the tangent plane of the surface and another plane at right angles to the light rays. This means that the illumination angle of a certain point on Earth's surface is 0° if the Sun is precisely overhead and that it is 90° at sunset or sunrise.


Atriphtaloid

An atriphtaloid, also called an atriphtothlassic curve, is type of sextic plane curve. It is given by the equation
where "a" and "b" are positive numbers.


N-transform

In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem has renamed it the natural transform and has proposed a detail theory and applications.
Formal definition.
The natural transform of a function "f"("t"), defined for all real numbers "t" ≥ "0", is the function "R"("u", "s"), defined by:
Khan showed that the above integral converges to Laplace transform when "u" = 1, and into Sumudu transform for "s" = 1.


Admissible representation

In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.
Real or complex reductive Lie groups.
Let "G" be a connected reductive (real or complex) Lie group. Let "K" be a maximal compact subgroup. A continuous representation (π, "V") of "G" on a complex Hilbert space "V" is called admissible if π restricted to "K" is unitary and each irreducible unitary representation of "K" occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of "G".
An admissible representation π induces a formula_1-module which is easier to deal with as it is an algebraic object. Two admissible representations are said to be infinitesimally equivalent if their associated formula_1-modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of formula_1-modules. This reduces the study of the equivalence classes of irreducible unitary representations of "G" to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by Robert Langlands and is called the Langlands classification.
Totally disconnected groups.
Let "G" be a locally compact totally disconnected group (such as a reductive algebraic group over a nonarchimedean local field or over the finite adeles of a global field). A representation (π, "V") of "G" on a complex vector space "V" is called smooth if the subgroup of "G" fixing any vector of "V" is open. If, in addition, the space of vectors fixed by any compact open subgroup is finite dimensional then π is called admissible. Admissible representations of "p"-adic groups admit more algebraic description through the action of the Hecke algebra of locally constant functions on "G".
Deep studies of admissible representations of "p"-adic reductive groups were undertaken by Casselman and by Bernstein and Zelevinsky in the 1970s. Progress was made more recently by Howe and Moy and Bushnell and Kutzko, who developed a "theory of types" and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.


E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras formula_1, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see ). This classifies Lie algebras into four infinite series labeled A"n", B"n", C"n", D"n", and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.
The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional.
In particle physics, E6 plays a role in some grand unified theories.
Real and complex forms.
There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) of E6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.
As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:
The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. The compact real form of E6 is the isometry group of a 32-dimensional Riemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E7 and E8 are known as the Rosenfeld projective planes, and are part of the Freudenthal magic square.
E6 as an algebraic group.
By means of a Chevalley basis for the Lie algebra, one can define E6 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) adjoint form of E6. Over an algebraically closed field, this and its triple cover are the only forms; however, over other fields, there are often many other forms, or “twists” of E6, which are classified in the general framework of Galois cohomology (over a perfect field "k") by the set "H"1("k", Aut(E6)) which, because the Dynkin diagram of E6 (see below) has automorphism group Z/2Z, maps to "H"1("k", Z/2Z) = Hom (Gal("k"), Z/2Z) with kernel "H"1("k", E6,ad).
Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E6 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E6 have fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); the further non-compact real Lie group forms of E6 are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E6 as well as the noncompact forms EI=E6(6) and EIV=E6(-26) are said to be "inner" or of type 1E6 meaning that their class lies in "H"1("k", E6,ad) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be "outer" or of type 2E6.
Over finite fields, the Lang–Steinberg theorem implies that "H"1("k", E6) = 0, meaning that E6 has exactly one twisted form, known as 2E6: see below.
Automorphisms of an Albert Algebra.
Similar to how the algebraic group G2 is the automorphism group of the octonions and the algebraic group F4 is the automorphism group of an Albert algebra, an exceptional Jordan algebra, the algebraic group E6 is the group of linear automorphisms of an Albert algebra that preserve a certain cubic form, called the "determinant".
Algebra.
Dynkin diagram.
The Dynkin diagram for E6 is given by , which may also be drawn as .
Roots of E6.
Although they span a six-dimensional space, it is much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space. Then one can take the roots to be
plus all 27 combinations of formula_2 where formula_3 is one of formula_4
plus all 27 combinations of formula_5 where formula_6 is one of formula_7
Simple roots
One possible selection for the simple roots of E6 is:
E6 roots derived from the roots of E8.
E6 is the subset of E8 where a consistent set of three coordinates are equal (e.g. first or last). This facilitates explicit definitions of E7 and E6 as:
The following 72 E6 roots are derived in this manner from the split real even E8 roots. Notice the last 3 dimensions being the same as required:
An alternative description.
An alternative (6-dimensional) description of the root system, which is useful in considering E6 × SU(3) as a subgroup of E8, is the following:
All formula_9 permutations of
and all of the following roots with an odd number of plus signs
Thus the 78 generators consist of the following subalgebras:
One choice of simple roots for E6 is given by the rows of the following matrix, indexed in the order :
Weyl group.
The Weyl group of E6 is of order 51840: it is the automorphism group of the unique simple group of order 25920 (which can be described as any of: PSU4(2), PSΩ6−(2), PSp4(3) or PSΩ5(3)).
Important subalgebras and representations.
The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) × SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2).
In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional "vector" representations.
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are :
The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E6 (equivalently, those whose weights belong to the root lattice of E6), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E6.
The symmetry of the Dynkin diagram of E6 explains why many dimensions occur twice, the corresponding representations being related by the non-trivial outer automorphism; however, there are sometimes even more representations than this, such as four of dimension 351, two of which are fundamental and two of which are not.
The fundamental representations have dimensions 27, 351, 2925, 351, 27 and 78 (corresponding to the six nodes in the Dynkin diagram in the order chosen for the Cartan matrix above, i.e., the nodes are read in the five-node chain first, with the last node being connected to the middle one).
E6 polytope.
The E6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions; its symmetry group contains the Coxeter group for E6 as an index 2 subgroup.
Chevalley and Steinberg groups of type E6 and 2E6.
The groups of type "E"6 over arbitrary fields (in particular finite fields) were introduced by .
The points over a finite field with "q" elements of the (split) algebraic group E6 (see above), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group. This is closely connected to the group written E6("q"), however there is ambiguity in this notation, which can stand for several things:
From the finite group perspective, the relation between these three groups, which is quite analogous to that between SL("n,q"), PGL("n,q") and PSL("n,q"), can be summarized as follows: E6("q") is simple for any "q", E6,sc("q") is its Schur cover, and E6,ad("q") lies in its automorphism group; furthermore, when "q"−1 is not divisible by 3, all three coincide, and otherwise (when "q" is congruent to 1 mod 3), the Schur multiplier of E6("q") is 3 and E6("q") is of index 3 in E6,ad("q"), which explains why E6,sc("q") and E6,ad("q") are often written as 3·E6("q") and E6("q")·3. From the algebraic group perspective, it is less common for E6("q") to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraic group over F"q" unlike E6,sc("q") and E6,ad("q").
Beyond this “split” (or “untwisted”) form of E6, there is also one other form of E6 over the finite field F"q", known as 2E6, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E6. Concretely, 2E6("q"), which is known as a Steinberg group, can be seen as the subgroup of E6("q"2) fixed by the composition of the non-trivial diagram automorphism and the non-trivial field automorphism of F"q"2. Twisting does not change the fact that the algebraic fundamental group of 2E6,ad is Z/3Z, but it does change those "q" for which the covering of 2E6,ad by 2E6,sc is non-trivial on the F"q"-points. Precisely: 2E6,sc("q") is a covering of 2E6("q"), and 2E6,ad("q") lies in its automorphism group; when "q"+1 is not divisible by 3, all three coincide, and otherwise (when "q" is congruent to 2 mod 3), the degree of 2E6,sc("q") over 2E6("q") is 3 and 2E6("q") is of index 3 in 2E6,ad("q"), which explains why 2E6,sc("q") and 2E6,ad("q") are often written as 3·2E6("q") and 2E6("q")·3.
Two notational issues should be raised concerning the groups 2E6("q"). One is that this is sometimes written 2E6("q"2), a notation which has the advantage of transposing more easily to the Suzuki and Ree groups, but the disadvantage of deviating from the notation for the F"q"-points of an algebraic group. Another is that whereas 2E6,sc("q") and 2E6,ad("q") are the F"q"-points of an algebraic group, the group in question also depends on "q" (e.g., the points over F"q"2 of the same group are the untwisted E6,sc("q"2) and E6,ad("q"2)).
The groups E6("q") and 2E6("q") are simple for any "q", and constitute two of the infinite families in the classification of finite simple groups. Their order is given by the following formula :
. The order of E6,sc("q") or E6,ad("q") (both are equal) can be obtained by removing the dividing factor gcd(3,"q"−1) from the first formula , and the order of 2E6,sc("q") or 2E6,ad("q") (both are equal) can be obtained by removing the dividing factor gcd(3,"q"+1) from the second .
The Schur multiplier of E6("q") is always gcd(3,"q"−1) (i.e., E6,sc("q") is its Schur cover). The Schur multiplier of 2E6("q") is gcd(3,"q"+1) (i.e., 2E6,sc("q") is its Schur cover) outside of the exceptional case "q"=2 where it is 22·3 (i.e., there is an additional 22-fold cover). The outer automorphism group of E6("q") is the product of the diagonal automorphism group Z/gcd(3,"q"−1)Z (given by the action of E6,ad("q")), the group Z/2Z of diagram automorphisms, and the group of field automorphisms (i.e., cyclic of order "f" if "q"="pf" where "p" is prime). The outer automorphism group of 2E6("q") is the product of the diagonal automorphism group Z/gcd(3,"q"+1)Z (given by the action of 2E6,ad("q")) and the group of field automorphisms (i.e., cyclic of order "f" if "q"="p""f" where "p" is prime).
Importance in physics.
 supergravity in five dimensions, which is a dimensional reduction from dimensional supergravity, admits an bosonic global symmetry and an bosonic local symmetry. The fermions are in representations of , the gauge fields are in a representation of , and the scalars are in a representation of both (Gravitons are singlets with respect to both). Physical states are in representations of the coset .
In grand unification theories, appears as a possible gauge group which, after its breaking, gives rise to gauge group of the standard model. One way of achieving this is through breaking to . The adjoint representation breaks, as explained above, into an adjoint , spinor and as well as a singlet of the subalgebra. Including the charge we have
Where the subscript denotes the charge.
Likewise, the fundamental representation and its conjugate break into a scalar , a vector and a spinor, either or :
Thus, one can get the Standard Model's elementary fermions and Higgs boson.


Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
A common way of expressing this property is to say that the field has no sources or sinks.
Properties.
The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
where formula_2 is the outward normal to each surface element.
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:
automatically results in the identity (as can be shown, for example, using Cartesian coordinates):
The converse also holds: for any solenoidal v there exists a vector potential A such that formula_5 (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)
Etymology.
"Solenoidal" has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume.


Dual representation

In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:
The dual representation is also known as the contragredient representation.
If is a Lie algebra and is a representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:
The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation.
In both cases, the dual representation is a representation in the usual sense.
Properties.
Irreducibility and second dual.
If a (finite-dimensional) representation is irreducible, then the dual representation is also irreducible—but not necessarily isomorphic to the original representation. On the other hand, the dual of the dual of any representation is isomorphic to the original representation.
Unitary representations.
Consider a "unitary" representation formula_1 of a group formula_2, and let us work in an orthonormal basis. Thus, formula_1 maps formula_2 into the group of unitary matrices. Then the abstract transpose in the definition of the dual representation may be identified with the ordinary matrix transpose. Since the adjoint of a matrix is the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, formula_5 is the complex conjugate of the adjoint of the inverse of formula_6. But since formula_6 is assumed to be unitary, the adjoint of the inverse of formula_6 is just formula_6.
The upshot of this discussion is that when working with unitary representations in an orthonormal basis, formula_5 is just the complex conjugate of formula_6.
The SU(2) and SU(3) cases.
In the representation theory of SU(2), the dual of each irreducible representation does turn out to be isomorphic to the representation. But for the representations of SU(3), the dual of the irreducible representation with label formula_12 is the irreducible representation with label formula_13. In particular, the standard three-dimensional representation of SU(3) (with highest weight formula_14) is not isomorphic to its dual. In the theory of quarks in the physics literature, the standard representation and its dual are called "formula_15" and "formula_16."
General semisimple Lie algebras.
More generally, in the representation theory of semisimple Lie algebras (or the closely related representation theory of compact Lie groups), the weights of the dual representation are the "negatives" of the weights of the original representation. (See the figure.) Now, for a given Lie algebra, if it should happen that operator formula_17 is an element of the Weyl group, then the weights of every representation are automatically invariant under the map formula_18. For such Lie algebras, "every" irreducible representation will be isomorphic to its dual. (This is the situation for SU(2), where the Weyl group is formula_19.) Lie algebras with this property include the odd orthogonal Lie algebras formula_20 (type formula_21) and the symplectic Lie algebras formula_22 (type formula_23).
If, for a given Lie algebra, formula_17 is "not" in the Weyl group, then the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a unique Weyl group element formula_25 mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then if we have an irreducible representation with highest weight formula_26, the "lowest" weight of the dual representation will be formula_27. It then follows that the "highest" weight of the dual representation will be formula_28. Since we are assuming formula_17 is not in the Weyl group, formula_25 cannot be formula_17, which means that the map formula_32 is not the identity. Of course, it may still happen that for certain special choices of formula_26, we might have formula_34. The adjoint representation, for example, is always isomorphic to its dual.
In the case of SU(3) (or its complexified Lie algebra, formula_35), we may choose a base consisting of two roots formula_36 at an angle of 120 degrees, so that the third positive root is formula_37. In this case, the element formula_25 is the reflection about the line perpendicular to formula_39. Then the map formula_32 is the reflection about the line "through" formula_39. The self-dual representations are then the ones that lie along the line through formula_39. These are the representations with labels of the form formula_43, which are the representations whose weight diagrams are "regular" hexagons.
Motivation.
In representation theory, both vectors in and linear functionals in are considered as "column vectors" so that the representation can act (by matrix multiplication) from the "left". Given a basis for and the dual basis for , the action of a linear functional on , can be expressed by matrix multiplication,
where the superscript is matrix transpose. Consistency requires
With the definition given,
For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if is a representation of a Lie group, then given by
is a representation of its Lie algebra. If is dual to , then its corresponding Lie algebra representation is given by
Example.
Consider the group formula_49 of complex numbers of absolute value 1. The irreducible representations are all one dimensional, as a consequence of Schur's lemma. The irreducible representations are parameterized by integers formula_50 and given explicitly as
The dual representation to formula_52 is then the inverse of the transpose of this one-by-one matrix, that is,
That is to say, the dual of the representation formula_52 is formula_55.
Generalization.
A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.


Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set "X" which maps the elements of some subset "S" of "X" to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of "X". If "S" has "k" elements, the cycle is called a "k"-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.
For example, given "X" = {1, 2, 3, 4}, the permutation (1, 3, 2, 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 (so "S" = "X") is a 4-cycle, and the permutation (1, 3, 2) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 (so "S" = {1, 2, 3} and 4 is a fixed element) is a 3-cycle. On the other hand, the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs {1, 3} and {2, 4}.
The set "S" is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits.
The cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or "fixed point") and the third is composed of two 2-cycles, and denoted (1, 3) (2, 4).
Definition.
A permutation is called a cyclic permutation if and only if it has a single nontrivial cycle (a cycle of length > 1).
For example, the permutation, written in two-line notation (in two ways) and also cycle notations,
is a six-cycle; its cycle diagram is shown at right.
Some authors restrict the definition to only those permutations which consist of one nontrivial cycle (that is, no fixed points allowed).
For example, the permutation
is a cyclic permutation under this more restrictive definition, while the preceding example is not.
More formally, a permutation formula_3 of a set "X", viewed as a bijective function formula_4, is called a cycle if the action on "X" of the subgroup generated by formula_3 has at most one orbit with more than a single element. This notion is most commonly used when "X" is a finite set; then of course the largest orbit, "S", is also finite. Let formula_6 be any element of "S", and put formula_7 for any formula_8. If "S" is finite, there is a minimal number formula_9 for which formula_10. Then formula_11, and formula_3 is the permutation defined by
and formula_14 for any element of formula_15. The elements not fixed by formula_3 can be pictured as
A cycle can be written using the compact cycle notation formula_18 (there are no commas between elements in this notation, to avoid confusion with a "k"-tuple). The "length" of a cycle is the number of elements of its largest orbit. A cycle of length "k" is also called a "k"-cycle.
The orbit of a 1-cycle is called a "fixed point" of the permutation, but as a permutation every 1-cycle is the identity permutation. When cycle notation is used, the 1-cycles are often suppressed when no confusion will result.
Basic properties.
One of the basic results on symmetric groups is that any permutation can be expressed as the product of disjoint cycles (more precisely: cycles with disjoint orbits); such cycles commute with each other, and the expression of the permutation is unique up to the order of the cycles. The multiset of lengths of the cycles in this expression (the cycle type) is therefore uniquely determined by the permutation, and both the signature and the conjugacy class of the permutation in the symmetric group are determined by it.
The number of "k"-cycles in the symmetric group "S""n" is given, for formula_19, by the following equivalent formulas
A "k"-cycle has signature (−1)"k" − 1.
The inverse of a cycle formula_18 is given by reversing the order of the entries: formula_22. In particular, since formula_23, every two-cycle is its own inverse. Since disjoint cycles commute, the inverse of a product of disjoint cycles is the result of reversing each of the cycles separately.
Transpositions.
A cycle with only two elements is called a transposition. For example, the permutation formula_24 that swaps 2 and 4.
Properties.
Any permutation can be expressed as the composition (product) of transpositions—formally, they are generators for the group. In fact, when the set being permuted is for some integer , then any permutation can be expressed as a product of formula_25 and so on. This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions. Concretely, one can express the transposition formula_26 where formula_27 by moving to one step at a time, then moving back to where was, which interchanges these two and makes no other changes:
The decomposition of a permutation into a product of transpositions is obtained for example by writing the permutation as a product of disjoint cycles, and then splitting iteratively each of the cycles of length 3 and longer into a product of a transposition and a cycle of length one less:
This means the initial request is to move formula_30 to formula_31 formula_32 to formula_33 formula_34 to formula_35 and finally formula_36 to formula_37 Instead one may roll the elements keeping formula_30 where it is by executing the right factor first (as usual in operator notation, and following the convention in the article on Permutations). This has moved formula_36 to the position of formula_31 so after the first permutation, the elements formula_30 and formula_36 are not yet at their final positions. The transposition formula_43 executed thereafter, then addresses formula_36 by the index of formula_32 to swap what initially were formula_30 and formula_47
In fact, the symmetric group is a Coxeter group, meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form.
One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions. This permits the parity of a permutation to be a well-defined concept.


Malnormal subgroup

In mathematics, in the field of group theory, a subgroup formula_1 of a group formula_2 is termed malnormal if for any formula_3 in formula_2 but not in formula_1, formula_1 and formula_7 intersect in the identity element.
Some facts about malnormality:
When "G" is finite, a malnormal subgroup "H" distinct from 1 and "G" is called a "Frobenius complement". The set "N" of elements of "G" which are, either equal to 1, or non-conjugate to any
element of "H", is a normal subgroup of "G", called the "Frobenius kernel", and "G" is the semi-direct product of "H" and "N" (Frobenius' theorem).


K-frame

In linear algebra, a branch of mathematics, a k"-frame is an ordered set of "k" linearly independent vectors in a vector space; thus "k" ≤ "n", where "n" is the dimension of the space, and if "k" = "n" an n"-frame is precisely an ordered basis.
If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame, respectively.


Zak transform

In mathematics, the Zak transform is a certain operation which takes as input a function of one variable and produces as output a function of two variables. The output function is called the Zak transform of the input function. The transform is defined as an infinite series in which each term is a product of a dilation of a translation by an integer of the function and an exponential function. In applications of Zak transform to signal processing the input function represents a signal and the transform will be a mixed time–frequency representation of the signal. The signal may be real valued or complex-valued, defined on a continuous set (for example, the real numbers) or a discrete set (for example, the integers or a finite subset of integers). The Zak transform is a generalization of the discrete Fourier transform.
The Zak transform had been discovered by several people in different fields and was called by different names. It was called the "Gel'fand mapping" because I.M. Gel'fand introduced it in his work on eigenfunction expansions. The transform was rediscovered independently by Joshua Zak in 1967 who called it the "k-q representation". There seems to be a general consent among experts in the field to call it the Zak transform, since Zak was the first to systematically study that transform in a more general setting and recognize its usefulness.
Continuous-time Zak transform: Definition.
In defining the continuous-time Zak transform, the input function is a function of a real variable. So, let "f"("t") be a function of a real variable "t". The continuous-time Zak transform of "f"("t") is a function of two real variables one of which is "t". The other variable may be denoted by "w". The continuous-time Zak transform has been defined variously.
Definition 1.
Let "a" be a positive constant. The Zak transform of "f"("t"), denoted by "Z""a"["f"], is a function of "t" and "w" defined by
Definition 2.
The special case of Definition 1 obtained by taking "a" = 1 is sometimes taken as the definition of the Zak transform. In this special case, the Zak transform of "f"("t") is denoted by "Z"["f"].
Definition 3.
The notation "Z"["f"] is used to denote another form of the Zak transform. In this form, the Zak transform of "f"("t") is defined as follows:
Definition 4.
Let "T" be a positive constant. The Zak transform of "f"("t"), denoted by "Z""T"["f"], is a function of "t" and "w" defined by
Here "t" and "w" are assumed to satisfy the conditions 0 ≤ "t" ≤ "T" and 0 ≤ "w" ≤ 1/"T".
Example.
The Zak transform of the function
is given by
where formula_7 denotes the smallest integer not less than formula_8 (the ceil function).
Properties of the Zak transform.
In the following it will be assumed that the Zak transform is as given in Definition 2.
1. Linearity
Let "a" and "b" be any real or complex numbers. Then
2. Periodicity
3. Quasi-periodicity
4. Conjugation
5. Symmetry
6. Convolution
Let formula_15 denote convolution with respect to the variable "t".
Inversion formula.
Given the Zak transform of a function, the function can be reconstructed using the following formula:
Discrete Zak transform: Definition.
In defining the discrete Zak transform, the input function is a function of an integer variable. So, let "f"("n") be a function of an integer variable "n" ("n" taking all positive, zero and negative integers as values). The discrete Zak transform of "f"("n") is a function of two real variables one of which is the integer variable "n". The other variable is a real variable which may be denoted by "w". The discrete Zak transform has also been defined variously. However, only one of the definitions is given below.
Definition.
The discrete Zak transform of the function "f"("n") where "n" is an integer variable, denoted by "Z"["f"], is defined by
Inversion formula.
Given the discrete transform of a function "f"("n"), the function can be reconstructed using the following formula:
Applications.
The Zak transform has been successfully used in physics in quantum field theory, in electrical engineering in time-frequency representation of signals, and in digital data transmission. The Zak transform has also applications in mathematics. For example, it has been used in the Gabor representation problem.


GotoBLAS

In scientific computing, GotoBLAS and GotoBLAS2 are open source implementations of the BLAS (Basic Linear Algebra Subprograms) API with many hand-crafted optimizations for specific processor types. GotoBLAS was developed by Kazushige Goto at the Texas Advanced Computing Center. , it was used in seven of the world's ten fastest supercomputers.
GotoBLAS remains available, but development ceased with a final version touting optimal performance on Intel's Nehalem architecture (contemporary in 2008).
OpenBLAS is an actively maintained fork of GotoBLAS, developed at the Lab of Parallel Software and Computational Science, ISCAS.
GotoBLAS was written by Goto during his sabbatical leave from the Japan Patent Office in 2002. It was initially optimized for the Pentium 4 processor and managed to immediately boost the performance of a supercomputer based on that CPU from 1.5 TFLOPS to 2 TFLOPS. , the library was available at no cost for noncommercial use. A later open source version was released under the terms of the BSD license.
GotoBLAS's matrix-matrix multiplication routine, called GEMM in BLAS terms, is highly tuned for the x86 and AMD64 processor architectures by means of handcrafted assembly code. It follows a similar decomposition into smaller "kernel" routines that other BLAS implementations use, but where earlier implementations streamed data from the L1 processor cache, GotoBLAS uses the L2 cache.
The kernel used for GEMM is a routine called GEBP, for "General block-times-panel multiply", which was experimentally found to be "inherently superior" over several other kernels that were considered in the design.
Several other BLAS routines are, as is customary in BLAS libraries, implemented in terms of GEMM.