Costas array

In mathematics, a Costas array can be regarded geometrically as a set of "n" points, each at the center of a square in an "n"×"n" square tiling such that each row or column contains only one point, and all of the "n"("n" − 1)/2 displacement vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto-ambiguity function, making the arrays useful in applications such as sonar and radar. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional Golomb ruler construction, and, as well as being of mathematical interest, have similar applications in experimental design and phased array radar engineering.
Costas arrays are named after John P. Costas, who first wrote about them in a 1965 technical report. Independently, Edgar Gilbert also wrote about them in the same year, publishing what is now known as the logarithmic Welch method of constructing Costas arrays.
Numerical representation.
A Costas array may be represented numerically as an "n"×"n" array of numbers, where each entry is either 1, for a point, or 0, for the absence of a point. When interpreted as binary matrices, these arrays of numbers have the property that, since each row and column has the constraint that it only has one point on it, they are therefore also permutation matrices. Thus, the Costas arrays for any given "n" are a subset of the permutation matrices of order "n".
Arrays are usually described as a series of indices specifying the column for any row. Since it is given that any column has only one point, it is possible to represent an array one-dimensionally. For instance, the following is a valid Costas array of order "N" = 4:
There are dots at coordinates: (1,2), (2,1), (3,3), (4,4)
Since the "x"-coordinate increases linearly, we can write this in shorthand as the set of all "y"-coordinates. The position in the set would then be the "x"-coordinate. Observe: {2,1,3,4} would describe the aforementioned array. This makes it easy to communicate the arrays for a given order of "N".
Known arrays.
Costas array counts are known for orders 1 through 29 :
Enumeration of known Costas arrays to order 200, order 500 and to order 1030 are available. Although these lists and databases of these Costas arrays are likely near complete, other Costas arrays with orders above 29 that are not in these lists may exist.
Constructions.
Welch.
A Welch–Costas array, or just Welch array, is a Costas array generated using the following method, first discovered by Edgar Gilbert in 1965 and rediscovered in 1982 by Lloyd R. Welch.
The Welch–Costas array is constructed by taking a primitive root "g" of a prime number "p" and defining the array "A" by formula_1 if formula_2, otherwise 0. The result is a Costas array of size "p" − 1.
Example:
3 is a primitive element modulo 5.
Therefore, [3 4 2 1] is a Costas permutation. More specifically, this is an exponential Welch array. The transposition of the array is a logarithmic Welch array.
The number of Welch–Costas arrays which exist for a given size depends on the totient function.
Lempel–Golomb.
The Lempel–Golomb construction takes α and β to be primitive elements of the finite field GF("q") and similarly defines formula_1 if formula_4, otherwise 0. The result is a Costas array of size "q" − 2. If "α" + "β" = 1 then the first row and column may be deleted to form another Costas array of size "q" − 3: such a pair of primitive elements exists for every prime power "q>2".
Extensions by Taylor, Lempel, and Golomb.
Generation of new Costas arrays by adding or subtracting a row/column or two with a 1 or a pair of 1's in a corner were published in a paper focused on generation methods and in Golomb and Taylor's landmark 1984 paper.
More sophisticated methods of generating new Costas arrays by deleting rows and columns of existing Costas arrays that were generated by the Welch, Lempel or Golomb generators were published in 1992. There is no upper limit on the order for which these generators will produce Costas arrays.
Other methods.
Two methods that found Costas arrays up to order 52 using more complicated methods of adding or deleting rows and columns were published in 2004 and 2007.
Variants.
Costas arrays on a hexagonal lattice are known as "honeycomb arrays". It has been shown that there are only finitely many such arrays, which must have an odd number of elements, arranged in the shape of a hexagon. Currently, 12 such arrays (up to symmetry) are known, which has been conjectured to be the total number.


Heinrich Franz Friedrich Tietze

Heinrich Franz Friedrich Tietze (August 31, 1880 – February 17, 1964) was an Austrian mathematician, famous for the Tietze extension theorem on functions from topological spaces to the real numbers. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem. Tietze's graph is also named after him; it describes the boundaries of a subdivision of the Möbius strip into six mutually-adjacent regions, found by Tietze as part of an extension of the four color theorem to non-orientable surfaces.
Education and career.
Tietze was the son of Emil Tietze and the grandson of Franz Ritter von Hauer, both of whom were Austrian geologists.
He was born in Schleinz, Austria-Hungary, and studied mathematics at the Technische Hochschule in Vienna beginning in 1898. After additional studies in Munich, he returned to Vienna, completing his doctorate in 1904 and his habilitation in 1908.
From 1910 until 1918 Tietze taught mathematics in Brno, and was promoted to ordinary professor in 1913. He served in the Austrian army during World War I, and then returned to Brno, but in 1919 he took a position at the University of Erlangen, and then in 1925 moved again to the University of Munich, where he remained for the rest of his career. One of his doctoral students was Georg Aumann. Tietze retired in 1950, and died in Munich, West Germany.
Awards and honors.
Tietze was a fellow of the Bavarian Academy of Sciences and a fellow of the Austrian Academy of Sciences.


Leinster group

In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups.
The Leinster groups are named after Tom Leinster, a mathematician at the University of Edinburgh, who wrote about them in a paper written in 1996 but not published until 2001. He called them "perfect groups", and later "immaculate groups",
but they were renamed as the Leinster groups by , because "perfect group" already had a different meaning (a group that equals its commutator subgroup).
Leinster groups give a group-theoretic way of analyzing the perfect numbers and of approaching the still-unsolved problem of the existence of odd perfect numbers.
For a cyclic group, the orders of the subgroups are just the divisors of the order of the group,
so a cyclic group is a Leinster group if and only if its order is a perfect number. More strongly, as Leinster proved, an abelian group is a Leinster group if and only if it is a cyclic group whose order is a perfect number.
Examples.
The cyclic groups whose order is a perfect number are Leinster groups.
It is possible for a non-abelian Leinster group to have odd order; an example, of order 355433039577, was constructed by François Brunault.
Other examples of non-abelian Leinster groups include certain groups of the form formula_1, where formula_2 is an alternating group and formula_3 is a cyclic group . For instance, the groups formula_4, formula_5 , formula_6 and formula_7 are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form formula_8, such as formula_9.
The possible orders of Leinster groups form the integer sequence


Translation functor

In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by and . Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
Definition.
By the Harish-Chandra isomorphism, the characters of the center "Z" of the universal enveloping algebra of a complex reductive Lie algebra can be identified with the points of "L"⊗C/"W", where "L" is the weight lattice and "W" is the Weyl group. If λ is a point of "L"⊗C/"W" then write χλ for the corresponding character of "Z".
A representation of the Lie algebra is said to have central character χλ if every vector "v" is a generalized eigenvector of the center "Z" with eigenvalue χλ; in other words if "z"∈"Z" and "v"∈"V" then ("z" − χλ("z"))"n"("v")=0 for some "n".
The translation functor ψ takes representations "V" with central character χλ to representations with central character χμ. It is constructed in two steps:


N-ary group

In mathematics, and in particular universal algebra, the concept of an n"-ary group (also called n"-group or multiary group) is a generalization of the concept of a group to a set "G" with an "n"-ary operation instead of a binary operation. By an "n"-ary operation is meant any map "f: Gn → G" from the "n"-th Cartesian power of "G" to "G". The axioms for an "n"-ary group are defined in such a way that they reduce to those of a group in the case . The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte; the first systematic account of (what were then called) polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the "Transactions of the American Mathematical Society".
Axioms.
Associativity.
The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity , i.e. the equality of the three possible bracketings of the string "abcde" in which any three consecutive symbols are bracketed. (Here it is understood that the equations hold for arbitrary choices of elements "a,b,c,d,e" in "G".) In general, "n"-ary associativity is the equality of the "n" possible bracketings of a string consisting of "n"+("n"-1) = "2n-1" distinct symbols with any "n" consecutive symbols bracketed. A set "G" which is closed under an associative "n"-ary operation is called an n"-ary semigroup. A set "G" which is closed under any (not necessarily associative) "n"-ary operation is called an n"-ary groupoid.
Inverses / unique solutions.
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means has a unique solution for "x", and likewise has a unique solution. In the ternary case we generalize this to , and each having unique solutions, and the "n"-ary case follows a similar pattern of existence of unique solutions and we get an "n"-ary quasigroup.
Definition of "n-ary" group.
An "n"-ary group is an "n"-ary semigroup which is also an "n"-ary quasigroup.
Identity / neutral elements.
In the 2-ary case, i.e. for an ordinary group, the existence of an identity element is a consequence of the associativity and inverse axioms, however in n-ary groups for n ≥ 3 there can be zero, one, or many identity elements.
An "n"-ary groupoid ("G", "ƒ") with "ƒ" = ("x"1 ◦ "x"2 ◦ . . . ◦ "x""n"), where ("G", ◦) is a group is called reducible or derived from the group ("G", ◦). In 1928 Dörnte published the first main results: An "n"-ary groupoid which is reducible is an "n"-ary group, however for all "n" > 2 there exist "n"-ary groups which are not reducible. In some "n"-ary groups there exists an element "e" (called an "n"-ary identity or neutral element) such that any string of "n"-elements consisting of all "e"'s, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity "e", "eeae" = "a" for every "a".
An "n"-ary group containing a neutral element is reducible. Thus, an "n"-ary group that is not reducible does not contain such elements. There exist "n"-ary groups with more than one neutral element. If the set of all neutral elements of an "n"-ary group is non-empty it forms an "n"-ary subgroup.
Some authors include an identity in the definition of an "n"-ary group but as mentioned above such "n"-ary operations are just repeated binary operations. Groups with intrinsically "n"-ary operations do not have an identity element.
Weaker axioms.
The axioms of associativity and unique solutions in the definition of an "n"-ary group are stronger than they need to be. Under the assumption of "n"-ary associativity it suffices to postulate the existence of the solution of equations with the unknown at the start or end of the string, or at one place other than the ends; e.g., in the 6-ary case, "xabcde"="f" and "abcdex"="f", or an expression like "abxcde"="f". Then it can be proved that the equation has a unique solution for "x" in any place in the string.
The associativity axiom can also be given in a weaker form.
Example.
The following is an example of a three element ternary group, one of four such groups
(n,m)-group.
The concept of an n-ary group can be further generalized to that of an (n,m)-group, also known as a vector valued group, which is a set G with a map "f: Gn → Gm" where n>m, subject to similar axioms as for an n-ary group except that the result of the map is a word consisting of m letters instead of a single letter. So an (n,1)-group is an n-ary group. (n,m)-groups were introduced by G Ĉupona in 1983.


Lifting property

In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.
Formal definition.
A morphism "i" in a category has the "left lifting property" with respect to a morphism "p", and "p" also has the "right lifting property" with respect to "i", sometimes denoted formula_1 or formula_2, iff the following implication holds for each morphism "f" and "g" in the category:
This is sometimes also known as the morphism "i" being "orthogonal to" the morphism "p"; however, this can also refer to
the stronger property that whenever "f" and "g" are as above, the diagonal morphism "h" exists and is also required to be unique.
For a class "C" of morphisms in a category, its "left orthogonal" formula_9 or formula_10 with respect to the lifting property, respectively its "right orthogonal" formula_11 or formula_12, is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class "C". In notation,
Taking the orthogonal of a class "C" is a simple way to define a class of morphisms excluding non-isomorphisms from "C", in a way which is useful in a diagram chasing computation.
Thus, in the category Set of sets, the right orthogonal formula_14 of the simplest non-surjection formula_15 is the class of surjections. The left and right orthogonals of formula_16 the simplest non-injection, are both precisely the class of injections,
It is clear that formula_18 and formula_19. The class formula_11 is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) and composition of morphisms, and contains all isomorphisms of C. Meanwhile, formula_9 is closed under retracts, pushouts, (small) coproducts and transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.
Examples.
A number of notions can be defined by passing to the left or right orthogonal several times starting from a list of explicit examples, i.e. as formula_22, where formula_23 is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class "C" is a kind of negation
of the property of being in "C", and that right-lifting is also a kind of negation. Hence the classes obtained from "C" by taking orthogonals an odd number of times, such as formula_24 etc., represent various kinds of negation of "C", so formula_24 each consists of morphisms which are far from having property formula_23.
Examples of lifting properties in algebraic topology.
A map formula_27 has the "path lifting property" iff formula_28 where formula_29 is the inclusion of one end point of the closed interval into the interval formula_30.
A map formula_27 has the homotopy lifting property iff formula_32 where formula_33 is the map formula_34.
Examples of lifting properties coming from model categories.
Fibrations and cofibrations.
Elementary examples in various categories.
In Set,
In the category "R"-Mod of modules over a commutative ring "R",
In the category Grp of groups,
For a finite group "G",
In the category Top of topological spaces, let formula_75, resp. formula_76 denote the discrete, resp. antidiscrete space with two points 0 and 1. Let formula_77 denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let formula_78 etc. denote the obvious embeddings.
In the category of metric spaces with uniformly continuous maps.


Unibranch local ring

In algebraic geometry, a local ring "A" is said to be unibranch if the reduced ring "A"red (obtained by quotienting "A" by its nilradical) is an integral domain, and the integral closure "B" of "A"red is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of "B" is a purely inseparable extension of the residue field of "A"red. A complex variety "X" is called topologically unibranch at a point "x" if for all complements "Y" of closed algebraic subsets of "X" there is a fundamental system of neighborhoods (in the classical topology) of "x" whose intersection with "Y" is connected.
In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:
Theorem Let "X" and "Y" be two integral locally noetherian schemes and formula_1 a proper dominant morphism. Denote their function fields by "K(X)" and "K(Y)", respectively. Suppose that the algebraic closure of "K(Y)" in "K(X)" has separable degree "n" and that formula_2 is unibranch. Then the fiber formula_3 has at most "n" connected components. In particular, if "f" is birational, then the fibers of unibranch points are connected.
In EGA, the theorem is obtained as a corollary of Zariski's main theorem.


Springer correspondence

In mathematics, the Springer representations are certain representations of the Weyl group "W" associated to unipotent conjugacy classes of a semisimple algebraic group "G". There is another parameter involved, a representation of a certain finite group "A"("u") canonically determined by the unipotent conjugacy class. To each pair ("u", φ) consisting of a unipotent element "u" of "G" and an irreducible representation "φ" of "A"("u"), one can associate either an irreducible representation of the Weyl group, or 0. The association
depends only on the conjugacy class of "u" and generates a correspondence between the irreducible representations of the Weyl group and the pairs ("u", φ) modulo conjugation, called the Springer correspondence. It is known that every irreducible representation of "W" occurs exactly once in the correspondence, although φ may be a non-trivial representation. The Springer correspondence has been described explicitly in all cases by Lusztig, Spaltenstein and Shoji. The correspondence, along with its generalizations due to Lusztig, plays a key role in Lusztig's classification of the irreducible representations of finite groups of Lie type.
Construction.
Several approaches to Springer correspondence have been developed. T. A. Springer's original construction (1976) proceeded by defining an action of "W" on the top-dimensional l-adic cohomology groups of the algebraic variety "B""u" of the Borel subgroups of "G" containing a given unipotent element "u" of a semisimple algebraic group "G" over a finite field. This construction was generalized by Lusztig (1981), who also eliminated some technical assumptions. Springer later gave a different construction (1978), using the ordinary cohomology with rational coefficients and complex algebraic groups.
Kazhdan and Lusztig found a topological construction of Springer representations using the Steinberg variety and, allegedly, discovered Kazhdan–Lusztig polynomials in the process. Generalized Springer correspondence has been studied by Lusztig-Spaltenstein (1985) and by Lusztig in his work on character sheaves. Borho and MacPherson (1983) gave yet another construction of the Springer correspondence.
Example.
For the special linear group "SL""n", the unipotent conjugacy classes are parametrized by partitions of "n": if "u" is a unipotent element, the corresponding partition is given by the sizes of the Jordan blocks of "u". All groups "A"("u") are trivial.
The Weyl group "W" is the symmetric group "S""n" on "n" letters. Its irreducible representations over a field of characteristic zero are also parametrized by the partitions of "n".
The Springer correspondence in this case is a bijection, and in the standard parametrizations, it is given by transposition of the partitions (so that the trivial representation of the Weyl group corresponds to the regular unipotent class, and the sign representation corresponds to the identity element of "G").
Applications.
Springer correspondence turned out to be closely related to the classification of primitive ideals in the universal enveloping algebra of a complex semisimple Lie algebra, both as a general principle and as a technical tool. Many important results are due to Anthony Joseph. A geometric approach was developed by Borho, Brylinski and MacPherson.


Weak factorization system


Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in formula_1 or a plane in Euclidean space formula_2 or a hyperplane in higher dimensions. It is primarily used for calculating distances (see point-plane distance and point-line distance).
It is written in vector notation as
The dot formula_4 indicates the scalar product or dot product.
The vector formula_5 represents the unit normal vector of "E" or "g", that points from the origin of the coordinate system to the plane (or line, in 2D). The distance formula_6 is the distance from the origin to the plane (or line).
This equation is satisfied by all points "P", lying precisely in the plane "E" (or in 2D, on the line "g"), described by the location vector formula_7 that points from the origin of the coordinate system to "P".
Derivation/Calculation from the normal form.
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
a plane is given by a normal vector formula_9 as well as an arbitrary position vector formula_10 of a point formula_11. The direction of formula_9 is chosen to satisfy the following inequality
By dividing the normal vector formula_9 by its magnitude formula_15, we obtain the unit (or normalized) normal vector
and the above equation can be rewritten as
Substituting
we obtain the Hesse normal form
In this diagram, "d" is the distance from the origin. Because formula_20 holds for every point in the plane, it is also true at point "Q" (the point where the vector from the origin meets the plane E), with formula_21, per the definition of the Scalar product
The magnitude formula_23 of formula_24 is the shortest distance from the origin to the plane.


Glossary of module theory

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.
See also: Glossary of ring theory, Glossary of representation theory.


Hyperdeterminant

In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an "n" × "n" square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes.
There are at least three definitions of hyperdeterminant. The first was discovered by Arthur Cayley in 1843 presented to the Cambridge Philosophical Society. It is in two parts and Cayley's first hyperdeterminant is covered in the second part. It is usually denoted by det0. The second Cayley hyperdeterminant originated in 1845 and is often denoted "Det". This definition is a discriminant for a singular point on a scalar valued multilinear map.
Cayley's first hyperdeterminant is defined only for hypercubes having an even number of dimensions (although variations exist in odd dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions). The third hyperdeterminant, most recently defined by Glynn, occurs only for fields of prime characteristic "p". It is denoted by det"p" and acts on all hypercubes over such a field.
Only the first and third hyperdeterminants are "multiplicative," except for the second hyperdeterminant in the case of "boundary" formats. The first and third hyperdeterminants also have closed formulae as polynomials and therefore their degrees are known, whereas the second one does not appear to have a closed formula or degree in all cases that are known.
The notation for determinants can be extended to hyperdeterminants without change or ambiguity. Hence the hyperdeterminant of a hypermatrix "A" may be written using the vertical bar notation as |"A"| or as "det"("A").
A standard modern textbook on Cayley's second hyperdeterminant Det (as well as many other results) is "Discriminants, Resultants and Multidimensional Determinants" by Gel'fand, Kapranov and Zelevinsky. Their notation and terminology is followed in the next section.
Cayley's second hyperdeterminant Det.
In the special case of a 2 × 2 × 2 hypermatrix the hyperdeterminant is known as Cayley's hyperdeterminant after the British mathematician Arthur Cayley who discovered it. The quartic expression for the Cayley's hyperdeterminant of hypermatrix "A" with components "a""ijk", ∊ } is given by
This expression acts as a discriminant in the sense that it is zero "if and only if" there is a non-zero solution in six unknowns "x""i", "y""i", "z""i", (with superscript "i" = 0 or 1) of the following system of equations
The hyperdeterminant can be written in a more compact form using the Einstein convention for summing over indices and the Levi-Civita symbol which is an alternating tensor density with components ε"ij" specified by ε00 = ε11 = 0, ε01 = −ε10 = 1:
Using the same conventions we can define a multilinear form
Then the hyperdeterminant is zero if and only if there is a non-trivial point where all partial derivatives of "f" vanish.
As a tensor expression.
The above determinant can be written in terms of a generalisation of the Levi-Civita symbol:
where "f" is a generalisation of the Levi-Civita symbol which allows two indices to be the same:
where the "f" satisfy:
As a discriminant.
For symmetric 2 × 2 × 2 × ⋯ hypermatrices, the hyperdeterminant is the discriminant of a polynomial. For example,
Then Det("A") is the discriminant of formula_9
Other general hyperdeterminants related to Cayley's Det.
Definitions.
In the general case a hyperdeterminant is defined as a discriminant for a multilinear map "f" from finite-dimensional vector spaces "V""i" to their underlying field "K" which may be formula_10 or formula_11.
"f" can be identified with a tensor in the tensor product of each dual space "V"*"i"
By definition a hyperdeterminant "Det"("f") is a polynomial in components of the tensor "f" which is zero if and only if the map "f" has a non-trivial point where all partial derivatives with respect to the components of its vector arguments vanish (a non-trivial point means that none of the vector arguments are zero.)
The vector spaces "V""i" need not have the same dimensions and the hyperdeterminant is said to be of format ("k"1, …, "k""r") "k""i" > 0, if the dimension of each space "V""i" is "k""i" + 1. It can be shown that the hyperdeterminant exists for a given format and is unique up to a scalar factor, if and only if the largest number in the format is less than or equal to the sum of the other numbers in the format.
This definition does not provide a means to construct the hyperdeteriminant and in general this is a difficult task. For hyperdeterminants with formats where "r" ≥ 4 the number of terms is usually too large to write out the hyperdeterminant in full. For larger "r" even the degree of the polynomial increases rapidly and does not have a convenient general formula.
Examples.
The case of formats with "r" = 1 deals with vectors of length "k"1 + 1. In this case the sum of the other format numbers is zero and "k"1 is always greater than zero so no hyperdeterminants exist.
The case of "r" = 2 deals with ("k"1 + 1) × ("k"2 + 1) matrices. Each format number must be greater than or equal to the other, therefore only square matrices "S" have hyperdeterminants and they can be identified with the determinant det("S"). Applying the definition of the hyperdeterminant as a discriminant to this case requires that det("S") is zero when there are vectors "X" and "Y" such that the matrix equations "SX" = 0 and "YS" = 0 have solutions for non-zero "X" and "Y".
For "r" > 2 there are hyperdeterminants with different formats satisfying the format inequality. e.g. Cayley's hyperdeterminant has format (1, 1, 1) and a hyperdeterminant of format (1, 1, 2) also exists. However a 2 × 2 × 4 hyperdeterminant would have format (1, 1, 3) but 3 > 1 + 1 so it does not exist.
Degree.
Since the hyperdeterminant is homogeneous in its variables it has a well-defined degree that is a function of the format and is written "N"("k"1, …, "k""r"). In special cases we can write down an expression for the degree. For example, a hyperdeterminant is said to be of boundary format when the largest format number is the sum of the others and in this case we have
For hyperdeterminants of dimensions 2"r", a convenient generating formula for the degrees "N""r" is
In particular for "r" = 2,3,4,5,6 the degree is respectively 2, 4, 24, 128, 880 and then grows very rapidly.
Three other special formulae for computing the degree of hyperdeterminants are given in
for 2 × "m" × "m" use "N"(1, "m" − 1, "m" − 1) = 2"m"("m" − 1)
for 3 × "m" × "m" use "N"(2, "m" − 1, "m" − 1) = 3"m"("m" − 1)2
for 4 × "m" × "m" use "N"(3, "m" − 1, "m" − 1) = (2/3)"m"("m" − 1)("m" − 2)(5"m" − 3)
A general result that follows from the hyperdeterminants product rule and invariance properties listed below is that the least common multiple of the dimensions of the vector spaces on which the linear map acts divides the degree of the hyperdeterminant, i.e.
Properties of hyperdeterminants.
Hyperdeterminants generalise many of the properties of determinants. The property of being a discriminant is one of them and it is used in the definition above.
Multiplicative properties.
One of the most familiar properties of determinants is the multiplication rule which is sometimes known as the Binet-Cauchy formula. For square "n" × "n" matrices "A" and "B" the rule says that
This is one of the harder rules to generalize from determinants to hyperdeterminants because generalizations of products of hypermatrices can give hypermatrices of different sizes. The full domain of cases in which the product rule can be generalized is still a subject of research. However, there are some basic instances that can be stated.
Given a multilinear form "f"(x1, …, x"r") we can apply a linear transformation on the last argument using an "n" × "n" matrix "B", y"r" = "B" x"r". This generates a new multilinear form of the same format,
In terms of hypermatrices this defines a product which can be written "g" = "f"."B"
It is then possible to use the definition of the hyperdeterminant to show that
where "n" is the degree of the hyperdeterminant. This generalises the product rule for matrices.
Further generalizations of the product rule have been demonstrated for appropriate products of hypermatrices of boundary format.
Invariance properties.
A determinant is not usually considered in terms of its properties as an algebraic invariant but when determinants are generalized to hyperdeterminants the invariance is more notable. Using the multiplication rule above on the hyperdeterminant of a hypermatrix "H" times a matrix "S" with determinant equal to one gives
In other words, the hyperdeterminant is an algebraic invariant under the action of the special linear group "SL"("n") on the hypermatrix. The transformation can be equally well applied to any of the vector spaces on which the multilinear map acts to give another distinct invariance. This leads to the general result,
E.g. the determinant of an "n" × "n" matrix is an "SL"("n")2 invariant and Cayley's hyperdeterminant for a 2 × 2 × 2 hypermatrix is an "SL"(2)3 invariant.
A more familiar property of a determinant is that if you add a multiple of a row (or column) to a different row (or column) of a square matrix then its determinant is unchanged. This is a special case of its invariance in the case where the special linear transformation matrix is an identity matrix plus a matrix with only one non-zero off-diagonal element. This property generalizes immediately to hyperdeterminants implying invariance when you add a multiple of one slice of a hypermatrix to another parallel slice.
A hyperdeterminant is not the only polynomial algebraic invariant for the group acting on the hypermatrix. For example, other algebraic invariants can be formed by adding and multiplying hyperdeterminants. In general the invariants form a ring algebra and it follows from Hilbert's basis theorem that the ring is finitely generated. In other words, for a given hypermatrix format, all the polynomial algebraic invariants with integer coefficients can be formed using addition, subtraction and multiplication starting from a finite number of them. In the case of a 2 × 2 × 2 hypermatrix, all such invariants can be generated in this way from Cayley's second hyperdeterminant alone, but this is not a typical result for other formats. For example, the second hyperdeterminant for a hypermatrix of format 2 × 2 × 2 × 2 is an algebraic invariant of degree 24 yet all the invariants can be generated from a set of four simpler invariants of degree 6 and less.
History and applications.
The second hyperdeterminant was invented and named by Arthur Cayley in 1845, who was able to write down the expression for the 2 × 2 × 2 format, but Cayley went on to use the term for any algebraic invariant and later abandoned the concept in favour of a general theory of polynomial forms which he called "quantics". For the next 140 years there were few developments in the subject and hyperdeterminants were largely forgotten until they were rediscovered by Gel'fand, Kapranov and Zelevinsky in the 1980s as an offshoot of their work on generalized hypergeometric functions. This led to them writing their textbook in which the hyperdeterminant is reintroduced as a discriminant. Indeed, Cayley's first hyperdeterminant is more fundamental than his second, since it is a straightforward generalization the ordinary determinant, and has found recent applications in the Alon-Tarsi conjecture.
Since then the hyperdeterminant has found applications over a wide range of disciplines including algebraic geometry, number theory, quantum computing and string theory.
In "algebraic geometry" the second hyperdeterminant is studied as a special case of an X-discriminant. A principal result is that there is a correspondence between the vertices of the Newton polytope for hyperdeterminants and the "triangulation" of a cube into simplices.
In "quantum computing" the invariants on hypermatrices of format 2"N" are used to study the entanglement of "N" qubits.
In "string theory" the hyperdeterminant first surfaced in connection with string dualities and black hole entropy.
Further reading.
For other historical developments not contained in the book from Gel'fand,Kapranov and Zelevinsky, see:


Glossary of representation theory

This is a glossary of representation theory in mathematics.
The term "module" is often used synonymously for a representation; for the module-theoretic terminology, see also glossary of module theory.
See also Glossary of Lie groups and Lie algebras, list of representation theory topics and .
Notations: We write formula_1. Thus, for example, a one-representation (i.e., a character) of a group "G" is of the form formula_2.


Michel Lazard

Michel Paul Lazard (5 December 1924 – 15 September 1987) was a French mathematician who worked in the theory of Lie groups in the context of p-adic analysis.
Born in Paris, Lazard studied at the University of Paris–Sorbonne, where he obtained his Ph.D. in 1954 under the direction of Albert Châtelet, with thesis titled "Sur les groupes nilpotents et les anneaux de Lie". Subsequently he was a Professor at the University of Poitiers and the University of Paris 7.
His work took on a life of its own in the hands of Daniel Quillen in the late 20th century. Quillen's discovery, that a ring Lazard used to classify formal group laws was isomorphic to an important ring in topology, led to the subject of chromatic homotopy theory. Lazard's self-contained treatise on one-dimensional formal groups also gave rise to the field of p-divisible groups. His major contributions were:


Real radical

In algebra, the real radical of an ideal "I" in a polynomial ring with real coefficients is the largest ideal containing "I" with the same vanishing locus.
It plays a similar role in real algebraic geometry that the radical of an ideal plays in algebraic geometry over an algebraically closed field.
More specifically, the Nullstellensatz says that when "I" is an ideal in a polynomial ring with coefficients coming from an algebraically closed field, the radical of "I" is the set of polynomials vanishing on the vanishing locus of "I". In real algebraic geometry, the Nullstellensatz fails as the real numbers are not algebraically closed. However, one can recover a similar theorem, the "real Nullstellensatz", by using the real radical in place of the (ordinary) radical.
Definition.
The real radical of an ideal "I" in a polynomial ring formula_1 over the real numbers, denoted by formula_2, is defined as
The Positivstellensatz then implies that formula_2 is the set of all polynomials that vanish on the real variety defined by the vanishing of formula_5.


Thomas J. Laffey

Thomas J. Laffey (born December, 1943) is an Irish mathematician known for his contributions to group theory and matrix theory. His entire career has been spent at University College Dublin (UCD), where he served two terms as head of the school of mathematics. While he formally retired in 2009, he remains active in research and publishing. The journal Linear Algebra and Its Applications had a special issue (April 2009) to mark his 65th birthday. He received the Hans Schneider Prize in 2013. In May 2019 at UCD, the International Conference on Linear Algebra and Matrix Theory held a celebration to honor Professor Laffey on his 75th birthday.
Education and career.
Tom Laffey was born in Cross, County Mayo. His parents were farmers, and the family had no tradition of education. His own early schooling was entirely through the Irish language, and in mathematics and physics he was more or less self-taught. The technical books he had to study were in English, which at first he found challenging. Nobody at his school had attempted honours Leaving Cert maths before. However, he got one of the highest marks in the country in the 1961 Leaving Certificate mathematics examination, thereby earning a state scholarship to university.
He attended University College Galway, earning bachelor's (1964) and master's (1965) degrees in mathematical science and also winning a National University of Ireland Traveling Studentship Prize. In 1968 he was awarded the D.Phil. by the University of Sussex for a thesis on "Structure Theorems for Linear Groups" done under advisor Walter Ledermann.
He immediately joined the staff at University College Dublin, from which he officially retired in 2009, but he has continued to publish regularly. His research has focussed on group theory, and later linear algebra too, and he has supervised five Ph.D. students. He has also played a significant role in the establishment of the Irish Mathematical Olympiad, and had frequently served the BT Young Scientists Exhibition as a judge and reviewer.
Early in his career, he developed a strong interest in matrix theory, due to the influence of Olga Taussky-Todd, with whom he often corresponded. This was cemented by a 1972–3 sabbatical spent at Northern Illinois University.
He received the Hans Schneider Prize in 2013 in recognition of his constructive solution to the NIEP (Non-negative inverse eigenvalue problem) for non-zero spectra.


Function field (scheme theory)

The sheaf of rational functions "KX" of a scheme "X" is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open set "U" the ring of all rational functions on that open set; in other words, "KX"("U") is the set of fractions of regular functions on "U". Despite its name, "KX" does not always give a field for a general scheme "X".
Simple cases.
In the simplest cases, the definition of "KX" is straightforward. If "X" is an (irreducible) affine algebraic variety, and if "U" is an open subset of "X", then "KX"("U") will be the field of fractions of the ring of regular functions on "U". Because "X" is affine, the ring of regular functions on "U" will be a localization of the global sections of "X", and consequently "KX" will be the constant sheaf whose value is the fraction field of the global sections of "X".
If "X" is integral but not affine, then any non-empty affine open set will be dense in "X". This means there is not enough room for a regular function to do anything interesting outside of "U", and consequently the behavior of the rational functions on "U" should determine the behavior of the rational functions on "X". In fact, the fraction fields of the rings of regular functions on any open set will be the same, so we define, for any "U", "KX"("U") to be the common fraction field of any ring of regular functions on any open affine subset of "X". Alternatively, one can define the function field in this case to be the local ring of the generic point.
General case.
The trouble starts when "X" is no longer integral. Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
The correct solution is to proceed as follows:
Further issues.
Once "KX" is defined, it is possible to study properties of "X" which depend only on "KX". This is the subject of birational geometry.
If "X" is an algebraic variety over a field "k", then over each open set "U" we have a field extension "KX"("U") of "k". The dimension of "U" will be equal to the transcendence degree of this field extension. All finite transcendence degree field extensions of "k" correspond to the rational function field of some variety.
In the particular case of an algebraic curve "C", that is, dimension 1, it follows that any two non-constant functions "F" and "G" on "C" satisfy a polynomial equation "P"("F","G") = 0.


Hahn polynomials

In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.
Hahn polynomials are defined in terms of generalized hypergeometric functions by
 give a detailed list of their properties.
If formula_2, these polynomials are identical to the Discrete Chebyshev polynomials except for a scale factor.
Closely related polynomials include the dual Hahn polynomials "R""n"("x";γ,δ,"N"), the continuous Hahn polynomials "p""n"("x","a","b", , ), and the continuous dual Hahn polynomials "S""n"("x";"a","b","c"). These polynomials all have "q"-analogs with an extra parameter "q", such as the q-Hahn polynomials "Q""n"("x";α,β, "N";"q"), and so on.
Orthogonality.
where "δx,y" is the Kronecker delta function and the weight functions are
and


Brieskorn manifold

In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by , is the intersection of a small sphere around the origin with the singular hypersurface
studied by .
Brieskorn manifolds give examples of exotic spheres.


Walter Edwin Arnoldi

Walter Edwin Arnoldi (New York, December 14, 1917 – October 5, 1995) was an American engineer mainly known for the Arnoldi iteration, an eigenvalue algorithm used in numerical linear algebra. Arnoldi graduated in mechanical engineering at the Stevens Institute of Technology in 1937 and attended a Master of Science course at Harvard. He worked at United Aircraft Corp. from 1939 to 1977. His main research interests included modelling vibrations, acoustics and aerodynamics of aircraft propellers. His 1951 paper "The principle of minimized iterations in the solution of the eigenvalue problem" is one of the most cited papers in numerical linear algebra.


Morton L. Curtis

Morton Landers Curtis (November 11, 1921 – February 4, 1989) was an American mathematician, an expert on group theory and the W. L. Moody, Jr. Professor of Mathematics at Rice University.
Born in Texas, Curtis earned a bachelor's degree in 1948 from Texas A&I University, and received his Ph.D. in 1951 from the University of Michigan under the supervision of Raymond Louis Wilder. Subsequently, he taught mathematics at Florida State University before moving to Rice. At Rice, he was the Ph.D. advisor of well-known mathematician John Morgan.
Curtis is, with James J. Andrews, the namesake of the Andrews–Curtis conjecture concerning Nielsen transformations of balanced group presentations. Andrews and Curtis formulated the conjecture in a 1965 paper; it remains open. Together with Gustav A. Hedlund and Roger Lyndon, he proved the Curtis–Hedlund–Lyndon theorem characterizing cellular automata as being defined by continuous equivariant functions on a shift space.
Curtis was the author of two books, "Matrix Groups" (Springer-Verlag, 1979), and "Abstract Linear Algebra" (Springer-Verlag, 1990).


Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general.
Definition.
Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism "F" is defined by
for all "r" in "R". It respects the multiplication of "R":
and is clearly 1 also. What is interesting, however, is that it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients
if . Therefore, the coefficients of all the terms except and are divisible by , the characteristic, and hence they vanish. Thus
This shows that "F" is a ring homomorphism.
If is a homomorphism of rings of characteristic , then
If and are the Frobenius endomorphisms of and , then this can be rewritten as:
This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic rings to itself.
If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: means , which by definition means that is nilpotent of order at most . In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective.
The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in . Then the image of does not contain . If it did, then there would be a rational function whose -th power would equal . But the degree of this -th power is , which is a multiple of . In particular, it can't be 1, which is the degree of . This is a contradiction; so is not in the image of .
A field is called "perfect" if either it is of characteristic zero or it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect.
Fixed points of the Frobenius endomorphism.
Consider the finite field . By Fermat's little theorem, every element of satisfies . Equivalently, it is a root of the polynomial . The elements of therefore determine roots of this equation, and because this equation has degree it has no more than roots over any extension. In particular, if is an algebraic extension of (such as the algebraic closure or another finite field), then is the fixed field of the Frobenius automorphism of .
Let be a ring of characteristic . If is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if is not a domain, then may have more than roots; for example, this happens if .
A similar property is enjoyed on the finite field formula_7 by the "n"th iterate of the Frobenius automorphism: Every element of formula_7 is a root of formula_9, so if is an algebraic extension of formula_7 and is the Frobenius automorphism of , then the fixed field of is formula_7. If "R" is a domain which is an formula_7-algebra, then the fixed points of the "n"th iterate of Frobenius are the elements of the image of formula_7.
Iterating the Frobenius map gives a sequence of elements in :
This sequence of iterates is used in defining the Frobenius closure and the tight closure of an ideal.
As a generator of Galois groups.
The Galois group of an extension of finite fields is generated by an iterate of the Frobenius automorphism. First, consider the case where the ground field is the prime field . Let be the finite field of elements, where . The Frobenius automorphism of fixes the prime field , so it is an element of the Galois group . In fact, since formula_15 is cyclic with elements,
we know that the Galois group is cyclic and is a generator. The order of is because acts on an element by sending it to , and this is the identity on elements of . Every automorphism of is a power of , and the generators are the powers with coprime to .
Now consider the finite field as an extension of , where as above. If , then the Frobenius automorphism of does not fix the ground field , but its th iterate does. The Galois group is cyclic of order and is generated by . It is the subgroup of generated by . The generators of are the powers where is coprime to .
The Frobenius automorphism is not a generator of the absolute Galois group
because this Galois group is isomorphic to the profinite integers
which are not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of , it is a generator of every finite quotient of the absolute Galois group. Consequently, it is a topological generator in the usual Krull topology on the absolute Galois group.
Frobenius for schemes.
There are several different ways to define the Frobenius morphism for a scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations.
The absolute Frobenius morphism.
Suppose that is a scheme of characteristic . Choose an open affine subset of . The ring is an -algebra, so it admits a Frobenius endomorphism. If is an open affine subset of , then by the naturality of Frobenius, the Frobenius morphism on , when restricted to , is the Frobenius morphism on . Consequently, the Frobenius morphism glues to give an endomorphism of . This endomorphism is called the absolute Frobenius morphism of , denoted . By definition, it is a homeomorphism of with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of -schemes to itself.
If is an -scheme and the Frobenius morphism of is the identity, then the absolute Frobenius morphism is a morphism of -schemes. In general, however, it is not. For example, consider the ring formula_18. Let and both equal with the structure map being the identity. The Frobenius morphism on sends to . It is not a morphism of formula_19-algebras. If it were, then multiplying by an element in formula_19 would commute with applying the Frobenius endomorphism. But this is not true because:
The former is the action of in the formula_19-algebra structure that begins with, and the latter is the action of formula_19 induced by Frobenius. Consequently, the Frobenius morphism on is not a morphism of formula_19-schemes.
The absolute Frobenius morphism is a purely inseparable morphism of degree . Its differential is zero. It preserves products, meaning that for any two schemes and , .
Restriction and extension of scalars by Frobenius.
Suppose that is the structure morphism for an -scheme . The base scheme has a Frobenius morphism "F""S". Composing with "F""S" results in an -scheme "X""F" called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an -morphism induces an -morphism .
For example, consider a ring "A" of characteristic and a finitely presented algebra over "A":
The action of "A" on "R" is given by:
where α is a multi-index. Let . Then is the affine scheme , but its structure morphism , and hence the action of "A" on "R", is different:
Because restriction of scalars by Frobenius is simply composition, many properties of are inherited by "X""F" under appropriate hypotheses on the Frobenius morphism. For example, if and "S""F" are both finite type, then so is "X""F".
The extension of scalars by Frobenius is defined to be:
The projection onto the factor makes an -scheme. If is not clear from the context, then is denoted by . Like restriction of scalars, extension of scalars is a functor: An -morphism determines an -morphism .
As before, consider a ring "A" and a finitely presented algebra "R" over "A", and again let . Then:
A global section of is of the form:
where "α" is a multi-index and every "a""iα" and "b""i" is an element of "A". The action of an element "c" of "A" on this section is:
Consequently, is isomorphic to:
where, if:
then:
A similar description holds for arbitrary "A"-algebras "R".
Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does . Furthermore, being a base change means that extension of scalars preserves properties such as being of finite type, finite presentation, separated, affine, and so on.
Extension of scalars is well-behaved with respect to base change: Given a morphism , there is a natural isomorphism:
Relative Frobenius.
Let be an -scheme with structure morphism . The relative Frobenius morphism of is the morphism:
defined by the universal property of the pullback (see the diagram above):
Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of -schemes.
Consider, for example, the "A"-algebra:
We have:
The relative Frobenius morphism is the homomorphism defined by:
Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of and , we have:
Relative Frobenius is a universal homeomorphism. If is an open immersion, then it is the identity. If is a closed immersion determined by an ideal sheaf "I" of , then is determined by the ideal sheaf and relative Frobenius is the augmentation map .
"X" is unramified over if and only if "F""X"/"S" is unramified and if and only if "F""X"/"S" is a monomorphism. "X" is étale over if and only if "F""X"/"S" is étale and if and only if "F""X"/"S" is an isomorphism.
Arithmetic Frobenius.
The arithmetic Frobenius morphism of an -scheme is a morphism:
defined by:
That is, it is the base change of "F""S" by 1"X".
Again, if:
then the arithmetic Frobenius is the homomorphism:
If we rewrite as:
then this homomorphism is:
Geometric Frobenius.
Assume that the absolute Frobenius morphism of is invertible with inverse formula_49. Let formula_50 denote the -scheme formula_51. Then there is an extension of scalars of by formula_49:
If:
then extending scalars by formula_49 gives:
If:
then we write:
and then there is an isomorphism:
The geometric Frobenius morphism of an -scheme is a morphism:
defined by:
It is the base change of formula_49 by .
Continuing our example of "A" and "R" above, geometric Frobenius is defined to be:
After rewriting "R" in terms of formula_64, geometric Frobenius is:
Arithmetic and geometric Frobenius as Galois actions.
Suppose that the Frobenius morphism of is an isomorphism. Then it generates a subgroup of the automorphism group of . If is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, and may be identified with . The arithmetic and geometric Frobenius morphisms are then endomorphisms of , and so they lead to an action of the Galois group of "k" on "X".
Consider the set of "K"-points . This set comes with a Galois action: Each such point "x" corresponds to a homomorphism from the structure sheaf to "K", which factors via "k(x)", the residue field at "x", and the action of Frobenius on "x" is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism
is the same as the composite morphism:
by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of "X".
Frobenius for local fields.
Given an unramified finite extension of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.
Suppose is an unramified extension of local fields, with ring of integers "OK" of such that the residue field, the integers of modulo their unique maximal ideal , is a finite field of order , where is a power of a prime. If is a prime of lying over , that is unramified means by definition that the integers of modulo , the residue field of , will be a finite field of order extending the residue field of where is the degree of . We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that
Frobenius for global fields.
In algebraic number theory, Frobenius elements are defined for extensions of global fields that are finite Galois extensions for prime ideals of that are unramified in . Since the extension is unramified the decomposition group of is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of as in the local case, by
where is the order of the residue field .
Lifts of the Frobenius are in correspondence with p-derivations.
Examples.
The polynomial
has discriminant
and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root of it to the field of -adic numbers gives an unramified extension of . We may find the image of under the Frobenius map by locating the root nearest to , which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the -adic integers . Modulo this polynomial is
This is algebraic over and is the correct global Frobenius image in terms of the embedding of into ; moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if -adic results will suffice.
If is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime in the base field . For an example, consider the extension of obtained by adjoining a root satisfying
to . This extension is cyclic of order five, with roots
for integer . It has roots which are Chebyshev polynomials of :
give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form (which split). It is immediately apparent how the Frobenius map gives a result equal mod to the -th power of the root .


Monoidal functor

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two "coherence maps"—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors
Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.
Definition.
Let formula_1 and formula_2 be monoidal categories. A lax monoidal functor from formula_3 to formula_4 (which may also just be called a monoidal functor) consists of a functor formula_5 together with a natural transformation
between formula_7 functors and a morphism
called the coherence maps or structure morphisms, which are such that for every three objects formula_9, formula_10 and formula_11 of formula_3 the diagrams
commute in the category formula_4. Above, the various natural transformations denoted using formula_14 are parts of the monoidal structure on formula_3 and formula_4.
Monoidal functors and adjunctions.
Suppose that a functor formula_5 is left adjoint to a monoidal formula_42. Then formula_43 has a comonoidal structure formula_44 induced by formula_45, defined by
and
If the induced structure on formula_43 is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint of a monoidal adjunction is always a strong monoidal functor.
Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.


Ore extension

In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials.
Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
Definition.
Suppose that "R" is a (not necessarily commutative) ring, formula_1 is a ring homomorphism, and formula_2 is a "σ"-derivation of "R", which means that formula_3 is a homomorphism of abelian groups satisfying
Then the Ore extension formula_5, also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials formula_6 a new multiplication, subject to the identity
If "δ" = 0 (i.e., is the zero map) then the Ore extension is denoted "R"["x"; "σ"]. If "σ" = 1 (i.e., the identity map) then the Ore extension is denoted "R"["x","δ"] and is called a differential polynomial ring.
Examples.
The Weyl algebras are Ore extensions, with "R" any commutative polynomial ring, "σ" the identity ring endomorphism, and "δ" the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.
Elements.
An element "f" of an Ore ring "R" is called


Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial "splits" or decomposes into linear factors.
Definition.
A splitting field of a polynomial "p"("X") over a field "K" is a field extension "L" of "K" over which "p" factors into linear factors
where formula_2 and for each formula_3 we have formula_4 with "ai" not necessarily distinct and such that the roots "ai" generate "L" over "K". The extension "L" is then an extension of minimal degree over "K" in which "p" splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of "p" (if we assume it is separable).
Properties.
An extension "L" which is a splitting field for a set of polynomials "p"("X") over "K" is called a normal extension of "K".
Given an algebraically closed field "A" containing "K", there is a unique splitting field "L" of "p" between "K" and "A", generated by the roots of "p". If "K" is a subfield of the complex numbers, the existence is immediate. On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid circular reasoning.
Given a separable extension "K"′ of "K", a Galois closure "L" of "K"′ is a type of splitting field, and also a Galois extension of "K" containing "K"′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials "p" over "K" that are minimal polynomials over "K" of elements "a" of "K"′.
Constructing splitting fields.
Motivation.
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as over , the real numbers, have no roots. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
The construction.
Let "F" be a field and "p"("X") be a polynomial in the polynomial ring "F"["X"] of degree "n". The general process for constructing "K", the splitting field of "p"("X") over "F", is to construct a chain of fields formula_5 such that "Ki" is an extension of "K""i"−1 containing a new root of "p"("X"). Since "p"("X") has at most "n" roots the construction will require at most "n" extensions. The steps for constructing "Ki" are given as follows:
The irreducible factor "fi"("X") used in the quotient construction may be chosen arbitrarily. Although different choices of factors may lead to different subfield sequences, the resulting splitting fields will be isomorphic.
Since "f"("X") is irreducible, ("f"("X")) is a maximal ideal of "K""i"["X"] and "K""i"["X"]/("f"("X")) is, in fact, a field. Moreover, if we let formula_7 be the natural projection of the ring onto its quotient then
so π("X") is a root of "f"("X") and of "p"("X").
The degree of a single extension formula_9 is equal to the degree of the irreducible factor "f"("X"). The degree of the extension ["K" : "F"] is given by formula_10 and is at most "n"!.
The field "K""i"["X"]/("f"("X")).
As mentioned above, the quotient ring "K""i"+1 = "K""i"["X"]/("f"("X")) is a field when "f"("X") is irreducible. Its elements are of the form
where the "cj" are in "Ki" and α = π("X"). (If one considers "K""i"+1 as a vector space over "Ki" then the powers α"j" for form a basis.)
The elements of "K""i"+1 can be considered as polynomials in α of degree less than "n". Addition in "K""i"+1 is given by the rules for polynomial addition and multiplication is given by polynomial multiplication modulo "f"("X"). That is, for "g"(α) and "h"(α) in "K""i"+1 the product "g"(α)"h"(α) = "r"(α) where "r"("X") is the remainder of "g"("X")"h"("X") divided by "f"("X") in "Ki"["X"].
The remainder "r"("X") can be computed through long division of polynomials, however there is also a straightforward reduction rule that can be used to compute "r"(α) = "g"(α)"h"(α) directly. First let
The polynomial is over a field so one can take "f"("X") to be monic without loss of generality. Now α is a root of "f"("X"), so
If the product "g"(α)"h"(α) has a term α"m" with it can be reduced as follows:
As an example of the reduction rule, take "Ki" = Q["X"], the ring of polynomials with rational coefficients, and take "f"("X") = "X"7 − 2. Let formula_15 and "h"(α) = α3 +1 be two elements of Q["X"]/("X"7 − 2). The reduction rule given by "f"("X") is α7 = 2 so
Examples.
The complex numbers.
Consider the polynomial ring R["x"], and the irreducible polynomial The quotient ring is given by the congruence As a result, the elements (or equivalence classes) of are of the form where "a" and "b" belong to R. To see this, note that since it follows that , , , etc.; and so, for example
The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo , i.e. using the fact that , , , , etc. Thus:
If we identify with ("a","b") then we see that addition and multiplication are given by
We claim that, as a field, the quotient is isomorphic to the complex numbers, C. A general complex number is of the form , where "a" and "b" are real numbers and Addition and multiplication are given by
If we identify with ("a","b") then we see that addition and multiplication are given by
The previous calculations show that addition and multiplication behave the same way in and C. In fact, we see that the map between and C given by is a homomorphism with respect to addition "and" multiplication. It is also obvious that the map is both injective and surjective; meaning that is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed:
In 1847, Cauchy used this approach to "define" the complex numbers.
Cubic example.
Let be the rational number field and . Each root of equals times a cube root of unity. Therefore, if we denote the cube roots of unity by
any field containing two distinct roots of will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ω2 or formula_28. It follows that a splitting field of will contain ω2, as well as the real cube root of 2; conversely, any extension of containing these elements contains all the roots of . Thus
Note that applying the construction process outlined in the previous section to this example, one begins with formula_30 and constructs the field formula_31. This field is not the splitting field, but contains one (any) root. However, the polynomial formula_32 is not irreducible over formula_33 and in fact:
Note that formula_35 is not an indeterminate, and is in fact an element of formula_33. Now, continuing the process, we obtain formula_37 which is indeed the splitting field and is spanned by the formula_38-basis formula_39. Notice that if we compare this with formula_40 from above we can identify formula_41 and formula_42.


Vincent Pilloni

Vincent Pilloni (born 1982) is a French mathematician, specializing in arithmetic geometry and the Langlands program.
Career.
Pilloni studied at the École Normale Supérieure and received his doctorate in 2009 from Université Sorbonne Paris Nord with thesis advisor Jacques Tilouine and thesis "Arithmétique des variétés de Siegel".
His research deals with, among other topics, the question of how the modularity theorem for elliptic curves over the rational numbers (which led to the proof of Fermat's Last Theorem) can be extended to abelian varieties. With Fabrizio Andreatta and Adrian Iovita, he worked on general modularity conjectures (following Fontaine-Mazur, Langlands, Clozel, and others).
Pilloni is a "Chargé de recherche" of CNRS at the École normale supérieure de Lyon (UMPA).
In 2018 he was an invited speaker, with Fabrizio Andreatta and Adrian Iovita, at the International Congress of Mathematicians in Rio de Janeiro. In 2018 Pilloni received the Prix Élie Cartan.


Myles Tierney

Myles Tierney (September 1937 – 5 October 2017) was an American mathematician and Professor at Rutgers University who founded the theory of elementary toposes with William Lawvere.
Tierney obtained his B.A. from Brown University in 1959 and his Ph.D. from Columbia University in 1965. His dissertation, "On the classifying spaces for K-Theory mod p", was written under the supervision of Samuel Eilenberg. Following positions at Rice University (1965–66) and ETH Zurich (1966–68), he became an associate professor at Rutgers in 1968.
Tierney was named a Fellow of the American Mathematical Society.


Efim Zelmanov

Efim Isaakovich Zelmanov (; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in 1994.
Zelmanov was born into a Jewish family in Khabarovsk, Soviet Union (now in Russia). He entered Novosibirsk State University in 1972, when he was 17 years old. He obtained a doctoral degree at Novosibirsk State University in 1980, and a higher degree at Leningrad State University in 1985. He had a position in Novosibirsk until 1987, when he left the Soviet Union.In 1990 he moved to the United States, becoming a professor at the University of Wisconsin–Madison. He was at the University of Chicago in 1994/5, then at Yale University. As of 2011, he is a professor at the University of California, San Diego and a Distinguished Professor at the Korea Institute for Advanced Study.
Zelmanov was elected a member of the U.S. National Academy of Sciences in 2001, becoming, at the age of 47, the youngest member of the mathematics section of the academy.
He is also an elected member of the American Academy of Arts and Sciences (1996) and a foreign member of the Korean Academy of Science and Engineering and of the Spanish Royal Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society.
Zelmanov gave invited talks at the International Congress of Mathematicians in Warsaw (1983), Kyoto (1990) and Zurich (1994). He delivered the 2004 Turán Memorial Lectures. He was awarded Honorary Doctor degrees from the University of Alberta, Canada (2011), Taras Shevchenko National University of Kyiv, Ukraine (2012), the Universidad Internacional Menéndez Pelayo in Santander, Spain (2015) and the University of Lincoln, UK (2016).
Zelmanov's early work was on Jordan algebras in the case of infinite dimensions. He was able to show that Glennie's identity in a certain sense generates all identities that hold. He then showed that the Engel identity for Lie algebras implies nilpotence, in the case of infinite dimensions.


Konhauser polynomials

In mathematics, the Konhauser polynomials, introduced by , are biorthogonal polynomials for the distribution function of the Laguerre polynomials.


Peter Scholze

Peter Scholze (born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. He has been called one of the leading mathematicians in the world. He won the Fields Medal in 2018, which is regarded as the highest professional honor in mathematics.
Early life and education.
Scholze was born in Dresden and grew up in Berlin. His father is a physicist, his mother a computer scientist, and his sister studied chemistry. He attended the in Berlin-Friedrichshain, a gymnasium devoted to mathematics and science. As a student, Scholze participated in the International Mathematical Olympiad, winning three gold medals and one silver medal.
He studied at the University of Bonn and completed his Bachelor's degree in three semesters and his Master's degree in two further semesters. He obtained his Ph.D. in 2012 under the supervision of Michael Rapoport.
Career.
From July 2011 until 2016, Scholze was a Research Fellow of the Clay Mathematics Institute in New Hampshire. In 2012, shortly after completing his PhD, he was made full professor at the University of Bonn, becoming at the age of 24 the youngest full professor in Germany.
In Fall 2014, Scholze was appointed the Chancellor's Professor at University of California, Berkeley, where he taught a course on "p"-adic geometry.
In 2018, Scholze was appointed as a director of the Max Planck Institute for Mathematics in Bonn.
Work.
Scholze's work has concentrated on purely local aspects of arithmetic geometry such as "p"-adic geometry and its applications. He presented in a more compact form some of the previous fundamental theories pioneered by Gerd Faltings, Jean-Marc Fontaine and later by Kiran Kedlaya. His PhD thesis on perfectoid spaces yields the solution to a special case of the weight-monodromy conjecture.
Scholze and Bhargav Bhatt have developed a theory of prismatic cohomology, which has been described as progress towards motivic cohomology by unifying singular cohomology, de Rham cohomology, ℓ-adic cohomology, and crystalline cohomology.
Awards.
In 2012, he was awarded the Prix and Cours Peccot. He was awarded the 2013 SASTRA Ramanujan Prize. In 2014, he received the Clay Research Award. In 2015, he was awarded the Frank Nelson Cole Prize in Algebra, and the Ostrowski Prize.
He received the Fermat Prize 2015 from the Institut de Mathématiques de Toulouse. In 2016, he was awarded the Leibniz Prize 2016 by the German Research Foundation. He declined the $100,000 "New Horizons in Mathematics Prize" of the 2016 Breakthrough Prizes. His turning down of the prize received little media attention.
He was awarded the Fields Medal in 2018, for "transforming arithmetic algebraic geometry over "p"-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories".
In 2019, Scholze received the Great Cross of Merit of the Order of Merit of the Federal Republic of Germany.
Personal life.
Scholze is married to a fellow mathematician and has a daughter.


Chiral Lie algebra

In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an formula_1-algebra via the Riemann–Hilbert correspondence.


Shinichi Mochizuki

 is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory, which has attracted a high level of attention of non-mathematicians due to claims it provides a resolution of the abc conjecture.
Biography.
Early life.
Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki. When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974-76). Mochizuki attended Phillips Exeter Academy and graduated in 1985. He entered Princeton University as an undergraduate at age 16 and graduated salutatorian in 1988. He then received a Ph.D. under the supervision of Gerd Faltings at age 23. After his PhD, Mochizuki spent two years at Harvard and then in 1994 moved back to Japan to join the Research Institute for Mathematical Sciences (RIMS) at Kyoto University in 1992, and was promoted to professor in 2002.
Career.
Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998. In 2000-2008 he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve.
On August 30, 2012 Mochizuki released four preprints, whose total size was about 500 pages, that develop inter-universal Teichmüller theory and apply it to attempt to prove several very famous problems in Diophantine geometry. These include the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. The preprints have not been published. In September 2018, Mochizuki posted a report on his work by Peter Scholze and Jakob Stix asserting that the third preprint contains an irreparable flaw; he also posted several documents containing his rebuttal of their criticism. The majority of number theorists have found Mochizuki's preprints very difficult to follow and have not accepted the conjectures as settled, although there are a few prominent exceptions, including Go Yamashita, Ivan Fesenko, and Yuichiro Hoshi, who vouch for the work and have written expositions of the theory.
On April 3, 2020, two Japanese mathematicians, Masaki Kashiwara and Akio Tamagawa, announced that Mochizuki's claimed proof of the abc conjecture would be published in "Publications of the Research Institute for Mathematical Sciences" (RIMS), a journal of which Mochizuki is chief editor. The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by "Nature" as "unlikely to move many researchers over to Mochizuki's camp."


Spaltenstein variety

In algebraic geometry, a Spaltenstein variety is a variety given by the fixed point set of a nilpotent transformation on a flag variety. They were introduced by . In the special case of full flag varieties the Spaltenstein varieties are Springer varieties.


Gary Seitz

Gary Michael Seitz (born 1943) is an American mathematician, a Fellow of the American Mathematical Society and a College of Arts and Sciences Distinguished Professor Emeritus in Mathematics at the University of Oregon. He received his Ph.D. from the University of Oregon in 1968, where his adviser was Charles W. Curtis. Seitz specializes in the study of algebraic and finite groups. Seitz has been active in the effort to exploit the relationship between algebraic groups and the finite groups of Lie type, in order to study the structure and representations of groups in the latter class. Such information is important in its own right, but was also critical in the classification of the finite simple groups, a major achievement of 20th century mathematics. Seitz made contributions to the classification of finite simple groups, such as those containing standard subgroups of Lie type. Following the classification, he pioneered the study of the subgroup structure of simple algebraic groups, and as an application went a long way towards solving the maximal subgroup problem for finite groups. For this work he received the Creativity Award from the National Science Foundation in 1991.
Early life.
He was born in Santa Monica, California, graduated from University High School in 1961. In high school he was a bowling champion with an average of 194 and a pool shark.
Education and career.
Seitz received his B.A. and Masters at University of California, Berkeley. He went on to earn his Ph.D. at the University of Oregon in 1968. He joined the faculty of the University of Illinois at Chicago in 1968. In 1970, he moved to University of Oregon as assistant professor, where he became a full professor in 1977 and a CAS Distinguished Professor in 2000. In his tenure at University of Oregon he also served as the Department Head of Mathematics and Associate Dean for the sciences in the College of Arts and Sciences. He served as thesis advisor for ten PhD students of whom eight currently hold academic positions. He has been visiting professor at the Institute for Advanced Study, California Institute of Technology, Cambridge University, Institut des Hautes Etudes Scientifiques, Imperial College, Tel Aviv University, Utrecht University, Tokyo University, among other institutions. In 2004, about a dozen mathematicians gathered for an international conference at the University of Oregon to honor his work.


Aschbacher block

In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher.
Definition.
A group "L" is called short if it has the following properties :
An example of a short group is the semidirect product of a quasisimple group with an irreducible module over the 2-element field F2
A block of a group "G" is a short subnormal subgroup.


Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns. An "n"-by-"n" matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if formula_1 is a square matrix representing a rotation (rotation matrix) and formula_2 is a column vector describing the position of a point in space, the product formula_3 yields another column vector describing the position of that point after that rotation. If formula_2 is a row vector, the same transformation can be obtained using where formula_5 is the transpose of
Main diagonal.
The entries formula_6 ("i" = 1, …, "n") form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , .
The diagonal of a square matrix from the top right to the bottom left corner is called "antidiagonal" or "counterdiagonal".
Special kinds.
Diagonal or triangular matrix.
If all entries outside the main diagonal are zero, formula_7 is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero, formula_7 is called a lower (or upper) triangular matrix.
Identity matrix.
The identity matrix formula_9 of size formula_10 is the formula_11 matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
It is a square matrix of order and also a special kind of diagonal matrix. It is called identity matrix because multiplication with it leaves a matrix unchanged:
Invertible matrix and its inverse.
A square matrix formula_7 is called "invertible" or "non-singular" if there exists a matrix formula_14 such that
If formula_14 exists, it is unique and is called the "inverse matrix" of denoted
Symmetric or skew-symmetric matrix.
A square matrix formula_7 that is equal to its transpose, i.e., is a symmetric matrix. If instead then formula_7 is called a skew-symmetric matrix.
For a complex square matrix often the appropriate analogue of the transpose is the conjugate transpose defined as the transpose of the complex conjugate of A complex square matrix formula_7 satisfying formula_20 is called a Hermitian matrix. If instead then formula_7 is called a skew-Hermitian matrix.
By the spectral theorem, real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.
Definite matrix.
A symmetric "n"×"n"-matrix is called" positive-definite" (respectively negative-definite; indefinite), if for all nonzero vectors formula_22 the associated quadratic form given by
takes only positive values (respectively only negative values; both some negative and some positive values). If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive. The table at the right shows two possibilities for 2×2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to "A":
Orthogonal matrix.
An "orthogonal matrix" is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix "A" is orthogonal if its transpose is equal to its inverse:
which entails
where "I" is the identity matrix.
An orthogonal matrix "A" is necessarily invertible (with inverse ), unitary (), and normal (). The determinant of any orthogonal matrix is either +1 or −1. The special orthogonal group formula_25 consists of the orthogonal matrices with determinant +1.
The complex analogue of an orthogonal matrix is a unitary matrix.
Normal matrix.
A real or complex square matrix formula_7 is called "normal" if If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds.
Operations.
Trace.
The trace, tr("A") of a square matrix "A" is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors:
This is immediate from the definition of matrix multiplication:
Also, the trace of a matrix is equal to that of its transpose, i.e.,
Determinant.
The "determinant" formula_30 or formula_31 of a square matrix formula_7 is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in formula_33) or volume (in formula_34) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2×2 matrices is given by
The determinant of 3×3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions.
The determinant of a product of square matrices equals the product of their determinants:
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1×1 matrix, which is its unique entry, or even the determinant of a 0×0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
Eigenvalues and eigenvectors.
A number λ and a non-zero vector formula_2 satisfying
are called an "eigenvalue" and an "eigenvector" of respectively. The number λ is an eigenvalue of an "n"×"n"-matrix "A" if and only if is not invertible, which is equivalent to
The polynomial "p""A" in an indeterminate "X" given by evaluation of the determinant is called the characteristic polynomial of "A". It is a monic polynomial of degree "n". Therefore the polynomial equation has at most "n" different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of "A" are real. According to the Cayley–Hamilton theorem, , that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.


Sophus Lie

Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Biography.
His first mathematical work, "Repräsentation der Imaginären der Plangeometrie", was published in 1869 by the Academy of Sciences in Christiania and also by "Crelle's Journal". That same year he received a scholarship and traveled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends. When he left Berlin, Lie traveled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie left for Fontainebleau where after a while he was arrested under suspicion of being a German spy, an event that made him famous in Norway. He was released from prison after a month, thanks to the intervention of Darboux.
Lie obtained his PhD at the University of Christiania (present day Oslo) in 1871 with a thesis entitled "Over en Classe geometriske Transformationer" (On a Class of Geometric Transformations).
It would be described by Darboux as "one of the most handsome discoveries of modern Geometry". The next year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program.
In 1872 Lie spent eight years together with Peter Ludwig Mejdell Sylow, editing and publishing the mathematical works of their countryman, Niels Henrik Abel.
At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880) and Herman (b. 1884).
In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at Leipzig, by then). Engel would help Lie to write his most important treatise, "Theorie der Transformationsgruppen", published in Leipzig in three volumes from 1888 to 1893. Decades later, Engel would also be one of the two editors of Lie's collected works.
In 1886 Lie became professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. After that, he returned to his post, but over the years his anaemia progressed to the point where he decided to return to his homeland. Consequently, in 1898 he tendered his resignation in May, and left for home (for good) in September the same year. He died the following year, 1899.
He was made Honorary Member of the London Mathematical Society in 1878, Member of the French Academy of Sciences in 1892, Foreign Member of the Royal Society of London in 1895 and foreign associate of the National Academy of Sciences of the United States of America in 1895.
Sophus Lie died at the age of 56, due to pernicious anemia, a disease caused by impaired absorption of vitamin B12.
Legacy.
Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groups) could be better understood by "linearizing" them, and studying the corresponding generating vector fields (the so-called infinitesimal generators). The generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra.
Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.
However, the subject of Lie groups as it is studied today is vastly different from what the research by Sophus Lie was about and "among the 19th century masters, Lie's work is "in detail" certainly the least known today".
Sophus Lie was an eager proponent in the establishment of the Abel Prize. Inspired by the Nansen fund named after Fridtjof Nansen, and the lack of a prize for mathematics in the Nobel Prize. He gathered support for the establishment of an award for outstanding work in pure mathematics.
Lie advised many doctoral students who went on to become successful mathematicians. Élie Cartan became widely regarded as one of the greatest mathematicians of the twentieth century. Kazimierz Żorawski's work was proved to be of importance to a variety of fields. Hans Frederick Blichfeldt made great contributions to various fields of mathematics.