Coherence condition

In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.
An illustrative example: a monoidal category.
Part of the data of a monoidal category is a chosen morphism
formula_1, called the "associator":
for each triple of objects formula_3 in the category. Using compositions of these formula_1, one can construct a morphism
Actually, there are many ways to construct such a morphism as a composition of various formula_1. One coherence condition that is typically imposed is that these compositions are all equal.
Typically one proves a coherence condition using a coherence theorem, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects formula_7, the following diagram commutes.
Any pair of morphisms from formula_8 to formula_9 constructed as compositions of various formula_1 are equal.
Further examples.
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.
Identity.
Let be a morphism of a category containing two objects "A" and "B". Associated with these objects are the identity morphisms and . By composing these with "f", we construct two morphisms:
Both are morphisms between the same objects as "f". We have, accordingly, the following coherence statement:
Associativity of composition.
Let , and be morphisms of a category containing objects "A", "B", "C" and "D". By repeated composition, we can construct a morphism from "A" to "D" in two ways:
We have now the following coherence statement:
In these two particular examples, the coherence statements are "theorems" for the case of an abstract category, since they follow directly from the axioms; in fact, they "are" axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.


Dual q-Krawtchouk polynomials

In mathematics, the dual "q"-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions by


Frans Oort

Frans Oort (born 17 July 1935) is a Dutch mathematician, specializing in algebraic geometry.
Biography.
Oort studied from 1952 to 1958 at Leiden University, where he graduated with a thesis on elliptic curves. He received his doctorate in 1961 in Leiden from and Jaap Murre with thesis "Reducible and Multiple Algebraic Curves", but had previously studied under Jean-Pierre Serre in Paris and Aldo Andreotti in Pisa. Oort was from 1961 at the University of Amsterdam, where he became a professor in 1967. In 1977, until his retirement in 2000, he was a professor at Utrecht University. He was a visiting scholar at several academic institutions, including Harvard University (1966/67) and Aarhus University (1972/73). In 2008 he was the Eilenberg Professor at Columbia University.
His research deals with, among other topics, abelian varieties and their modules. In 1994 he formulated what is now known as the André–Oort conjecture (generalizing a conjecture made in 1989 by Yves André). In 2000 Oort proved a conjecture made by Grothendieck in 1970. Oort was a co-author of an article on Poncelet's closure theorem and the author of "An aspect of harmony in music of Johann Sebastian Bach".
In 1962 Oort made a short contribution "Multiple algebraic curves" at the International Congress of Mathematicians in Stockholm, but was not an invited speaker. In 2011 he was elected a member of Academia Europaea. In July 2013 he gave a talk at the International Congress of Chinese Mathematicians in Tapei.
His doctoral students include Michiel Hazewinkel, Aise Johan de Jong, Hendrik Lenstra and Joseph Steenbrink.
He was married to (later divorced) author (1936-2020), who wrote "Afscheid van Joke Smit" (1987 biography of Joke Smit), "Doezamand: Roman" (1994 novel based upon her life as a student), and "Eerlijk gezegd: Interviews met vrouwen" (1974 collection of interviews with women); the couple have three grown sons, among whom is the Ultimate Frisbee player Jeroen Oort.


Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics. (see also Outline_of_category_theory.)
Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.
The notations and the conventions used throughout the article are:
C.
 \times C \to X</math> is the dual of the end of "F" and is denoted by
For example, if "R" is a ring, "M" a right "R"-module and "N" a left "R"-module, then the tensor product of "M" and "N" is
E.
 \times C \to X</math> is the limit
where formula_5 is the category (called the subdivision category of "C") whose objects are symbols formula_6 for all objects "c" and all morphisms "u" in "C" and whose morphisms are formula_7 and formula_8 if formula_9 and where formula_10 is induced by "F" so that formula_11 would go to formula_12 and formula_13 would go to formula_14. For example, for functors formula_15,
L.
 \to \mathbf{Set}</math> is
T.
 \to B</math> and formula_36 is the coend:
Y.
, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)</math>


Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.
Main classes of groups.
The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.
Permutation groups.
The first class of groups to undergo a systematic study was permutation groups. Given any set "X" and a collection "G" of bijections of "X" into itself (known as "permutations") that is closed under compositions and inverses, "G" is a group acting on "X". If "X" consists of "n" elements and "G" consists of "all" permutations, "G" is the symmetric group S"n"; in general, any permutation group "G" is a subgroup of the symmetric group of "X". An early construction due to Cayley exhibited any group as a permutation group, acting on itself () by means of the left regular representation.
In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for , the alternating group A"n" is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree in radicals.
Matrix groups.
The next important class of groups is given by "matrix groups", or linear groups. Here "G" is a set consisting of invertible matrices of given order "n" over a field "K" that is closed under the products and inverses. Such a group acts on the "n"-dimensional vector space "K""n" by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group "G".
Transformation groups.
Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space "X" preserving its inherent structure. In the case of permutation groups, "X" is a set; for matrix groups, "X" is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of "all" transformations that preserve a certain structure.
The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.
Abstract groups.
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by "generators and relations",
A significant source of abstract groups is given by the construction of a "factor group", or quotient group, "G"/"H", of a group "G" by a normal subgroup "H". Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group "G" is a permutation group on a set "X", the factor group "G"/"H" is no longer acting on "X"; but the idea of an abstract group permits one not to worry about this discrepancy.
The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.
Groups with additional structure.
An important elaboration of the concept of a group occurs if "G" is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations "m" (multiplication) and "i" (inversion),
are compatible with this structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then "G" is a topological group, a Lie group, or an algebraic group.
The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group "Γ" can be realized as a lattice in a topological group "G", the geometry and analysis pertaining to "G" yield important results about "Γ". A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single "p"-adic analytic group "G" has a family of quotients which are finite "p"-groups of various orders, and properties of "G" translate into the properties of its finite quotients.
Branches of group theory.
Finite group theory.
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields.
Finite groups often occur when considering symmetry of mathematical or
physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups,
which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
Representation of groups.
Saying that a group "G" "acts" on a set "X" means that every element of "G" defines a bijective map on the set "X" in a way compatible with the group structure. When "X" has more structure, it is useful to restrict this notion further: a representation of "G" on a vector space "V" is a group homomorphism:
where GL("V") consists of the invertible linear transformations of "V". In other words, to every group element "g" is assigned an automorphism "ρ"("g") such that for any "h" in "G".
This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group "G": often, the group operation in "G" is abstractly given, but via "ρ", it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if "G" is finite, it is known that "V" above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole "V" (via Schur's lemma).
Given a group "G", representation theory then asks what representations of "G" exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value "1", acting on the "L"2-space of periodic functions.
Lie theory.
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term "groupes de Lie" first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3.
Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
Combinatorial and geometric group theory.
Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications . A more compact way of defining a group is by "generators and relations", also called the "presentation" of a group. Given any set "F" of generators formula_4, the free group generated by "F" surjects onto the group "G". The kernel of this map is called the subgroup of relations, generated by some subset "D". The presentation is usually denoted by formula_5 For example, the group presentation formula_6 describes a group which is isomorphic to formula_7 A string consisting of generator symbols and their inverses is called a "word".
Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free.
There are several natural questions arising from giving a group by its presentation. The "word problem" asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation formula_8 is isomorphic to the additive group Z of integers, although this may not be immediately apparent.
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group "G" acting in a reasonable manner on a metric space "X", for example a compact manifold, then "G" is quasi-isometric (i.e. looks similar from a distance) to the space "X".
Connection of groups and symmetry.
Given a structured object "X" of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.
Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.
The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.
Applications of group theory.
Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.
Galois theory.
Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, "S"5, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.
Algebraic topology.
Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.
Algebraic geometry.
Algebraic geometry likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.
Algebraic number theory.
Algebraic number theory makes uses of groups for some important applications. For example, Euler's product formula,
captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem.
Harmonic analysis.
Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.
Combinatorics.
In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.
Music.
The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. Transformational theory models musical transformations as elements of a mathematical group.
Physics.
In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group.
Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution.
Chemistry and materials science.
In chemistry and materials science, point groups are used to classify regular polyhedra, and the symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals.
Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule.
In chemistry, there are five important symmetry operations. They are identity operation (E), rotation operation or proper rotation (C"n"), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (S"n"). The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (C"n") consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/"n", where "n" is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, , since applying it twice produces the identity operation. In molecules with more than one rotation axis, the Cn axis having the largest value of n is the highest order rotation axis or principal axis. For example Borane (BH3), the highest order of rotation axis is C3, so Principal axis of rotation of axis is C3.
In the reflection operation (σ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σ"h" (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (σ"v") or dihedral (σ"d").
Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, methane and other tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (S"n") requires rotation of  360°/"n", followed by reflection through a plane perpendicular to the axis of rotation.
Cryptography.
Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.
History.
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry.
Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory.
The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.


Jacobi eigenvalue algorithm

In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers.
Description.
Let formula_1 be a symmetric matrix, and formula_2 be a Givens rotation matrix. Then:
is symmetric and similar to formula_1.
Furthermore, formula_5 has entries:
where formula_7 and formula_8.
Since formula_9 is orthogonal, formula_1 and formula_5 have the same Frobenius norm formula_12 (the square-root sum of squares of all components), however we can choose formula_13 such that formula_14, in which case formula_5 has a larger sum of squares on the diagonal:
Set this equal to 0, and rearrange:
if formula_18
In order to optimize this effect, "S""ij" should be the off-diagonal element with the largest absolute value, called the "pivot".
The Jacobi eigenvalue method repeatedly performs rotations until the matrix becomes almost diagonal. Then the elements in the diagonal are approximations of the (real) eigenvalues of "S".
Convergence.
If formula_20 is a pivot element, then by definition formula_21 for formula_22 . Let formula_23 denote the sum of squares of all off-diagonal entries of formula_1. Since formula_1 has exactly formula_26 off-diagonal elements, we have formula_27 or formula_28 . Now formula_29. This implies
 formula_30 or formula_31,
i.e. the sequence of Jacobi rotations converges at least linearly by a factor formula_32 to a diagonal matrix.
A number of formula_33 Jacobi rotations is called a sweep; let formula_34 denote the result. The previous estimate yields
i.e. the sequence of sweeps converges at least linearly with a factor ≈ formula_36 .
However the following result of Schönhage yields locally quadratic convergence. To this end let "S" have "m" distinct eigenvalues formula_37 with multiplicities formula_38 and let "d" > 0 be the smallest distance of two different eigenvalues. Let us call a number of
Jacobi rotations a Schönhage-sweep. If formula_40 denotes the result then
Thus convergence becomes quadratic as soon as
formula_42
Cost.
Each Jacobi rotation can be done in O("n") steps when the pivot element "p" is known. However the search for "p" requires inspection of all "N" ≈  "n"2 off-diagonal elements. We can reduce this to O("n") complexity too if we introduce an additional index array formula_43 with the property that formula_44 is the index of the largest element in row "i", ("i" = 1, ..., "n" − 1) of the current "S". Then the indices of the pivot ("k", "l") must be one of the pairs formula_45. Also the updating of the index array can be done in O("n") average-case complexity: First, the maximum entry in the updated rows "k" and "l" can be found in O("n") steps. In the other rows "i", only the entries in columns "k" and "l" change. Looping over these rows, if formula_46 is neither "k" nor "l", it suffices to compare the old maximum at formula_46 to the new entries and update formula_46 if necessary. If formula_46 should be equal to "k" or "l" and the corresponding entry decreased during the update, the maximum over row "i" has to be found from scratch in O("n") complexity. However, this will happen on average only once per rotation. Thus, each rotation has O("n") and one sweep O("n"3) average-case complexity, which is equivalent to one matrix multiplication. Additionally the formula_44 must be initialized before the process starts, which can be done in "n"2 steps.
Typically the Jacobi method converges within numerical precision after a small number of sweeps. Note that multiple eigenvalues reduce the number of iterations since formula_51.
Algorithm.
The following algorithm is a description of the Jacobi method in math-like notation.
It calculates a vector "e" which contains the eigenvalues and a matrix "E" which contains the corresponding eigenvectors, i.e. formula_52 is an eigenvalue and the column formula_53 an orthonormal eigenvector for formula_52, "i" = 1, ..., "n".
 procedure jacobi("S" ∈ R"n"×"n"; out "e" ∈ R"n"; out "E" ∈ R"n"×"n")
 var
 "i", "k", "l", "m", "state" ∈ N
 "s", "c", "t", "p", "y", "d", "r" ∈ R
 "ind" ∈ N"n"
 "changed" ∈ L"n"
 function maxind("k" ∈ N) ∈ N ! "index of largest off-diagonal element in row k"
 "m" := "k"+1
 for "i" := "k"+2 to "n" do
 if │"S""ki"│ > │"S""km"│ then "m" := "i" endif
 endfor
 return "m"
 endfunc
 procedure update("k" ∈ N; "t" ∈ R) ! "update ek and its status"
 "y" := "e""k"; "e""k" := "y"+"t"
 if "changed""k" and ("y"="e""k") then "changed""k" := false; "state" := "state"−1
 elsif (not "changed""k") and ("y"≠"e""k") then "changed""k" := true; "state" := "state"+1
 endif
 endproc
 procedure rotate("k","l","i","j" ∈ N) ! "perform rotation of Sij, Skl"
 │"S""kl"│ │"c" −"s"││"S""kl"│
 │"S""ij"│ │"s" "c"││"S""ij"│
 endproc
 ! "init e, E, and arrays ind, changed"
 "E" := "I"; "state" := "n"
 for "k" := 1 to "n" do "ind""k" := maxind("k"); "e""k" := "S""kk"; "changed""k" := true endfor
 while "state"≠0 do ! "next rotation"
 "m" := 1 ! "find index (k,l) of pivot p"
 for "k" := 2 to "n"−1 do
 if │"S""k" "ind""k"│ > │"S""m" "ind""m"│ then "m" := "k" endif
 endfor
 "k" := "m"; "l" := "ind""m"; "p" := "S""kl"
 ! "calculate c = cos φ, s = sin φ"
 "y" := ("e""l"−"e""k")/2; "d" := │"y"│+√("p"2+"y"2)
 "r" := √("p"2+"d"2); "c" := "d"/"r"; "s" := "p"/"r"; "t" := "p"2/"d"
 if "y"<0 then "s" := −"s"; "t" := −"t" endif
 "S""kl" := 0.0; update("k",−"t"); update("l","t")
 ! "rotate rows and columns k and l
 for "i" := 1 to "k"−1 do rotate("i","k","i","l") endfor
 for "i" := "k"+1 to "l"−1 do rotate("k","i","i","l") endfor
 for "i" := "l"+1 to "n" do rotate("k","i","l","i") endfor
 ! "rotate eigenvectors"
 for "i" := 1 to "n" do
 │"E""ik"│ │"c" −"s"││"E""ik"│
 │"E""il"│ │"s" "c"││"E""il"│
 endfor
 ! "rows k, l have changed, update rows indk, indl"
 "ind""k" := maxind("k"); "ind""l" := maxind("l")
 loop
 endproc
Notes.
1. The logical array "changed" holds the status of each eigenvalue. If the numerical value of formula_55 or formula_56 changes during an iteration, the corresponding component of "changed" is set to "true", otherwise to "false". The integer "state" counts the number of components of "changed" which have the value "true". Iteration stops as soon as "state" = 0. This means that none of the approximations formula_57 has recently changed its value and thus it is not very likely that this will happen if iteration continues. Here it is assumed that floating point operations are optimally rounded to the nearest floating point number.
2. The upper triangle of the matrix "S" is destroyed while the lower triangle and the diagonal are unchanged. Thus it is possible to restore "S" if necessary according to
 for "k" := 1 to "n"−1 do ! "restore matrix S"
 for "l" := "k"+1 to "n" do
 "S""kl" := "S""lk"
 endfor
 endfor
3. The eigenvalues are not necessarily in descending order. This can be achieved by a simple sorting algorithm.
 for "k" := 1 to "n"−1 do
 "m" := "k"
 for "l" := "k"+1 to "n" do
 if "e""l" > "e""m" then
 "m" := "l"
 endif
 endfor
 if "k" ≠ "m" then
 swap "e""m","e""k"
 swap "E""m","E""k"
 endif
 endfor
4. The algorithm is written using matrix notation (1 based arrays instead of 0 based).
5. When implementing the algorithm, the part specified using matrix notation must be performed simultaneously.
6. This implementation does not correctly account for the case in which one dimension is an independent subspace. For example, if given a diagonal matrix, the above implementation will never terminate, as none of the eigenvalues will change. Hence, in real implementations, extra logic must be added to account for this case.
Example.
Let
formula_58
Then "jacobi" produces the following eigenvalues and eigenvectors after 3 sweeps (19 iterations) :
formula_59
formula_60
formula_61
formula_62
formula_63
formula_64
formula_65
formula_66
Applications for real symmetric matrices.
When the eigenvalues (and eigenvectors) of a symmetric matrix are known, the following
values are easily calculated.
Generalizations.
The Jacobi Method has been generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices.
Since singular values of a real matrix are the square roots of the eigenvalues of the symmetric matrix formula_101 it can also be used for the calculation of these values. For this case, the method is modified in such a way that "S" must not be explicitly calculated which reduces the danger of round-off errors. Note that formula_102 with formula_103 .
The Jacobi Method is also well suited for parallelism.


Paranormal subgroup

In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it "within" that subgroup.
In symbols, formula_1 is paranormal in formula_2 if given any formula_3 in formula_2, the subgroup formula_5 generated by formula_1 and formula_7 is also equal to formula_8. Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups.
Here are some facts relating paranormality to other subgroup properties:


Cyclic module

In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is analogous to cyclic group, that is, a group that is generated by one element.
Definition.
A left "R"-module "M" is called cyclic if "M" can be generated by a single element i.e. for some "x" in "M". Similarly, a right "R"-module "N" is cyclic if for some .


LIBPATH (environment variable)


Alternating sign matrix

In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.
Example.
An example of an alternating sign matrix (that is not also a permutation matrix) is
Alternating sign matrix theorem.
The "alternating sign matrix theorem" states that the number of formula_2 alternating sign matrices is
The first few terms in this sequence for "n" = 0, 1, 2, 3, … are
This theorem was first proved by Doron Zeilberger in 1992. In 1995, Greg Kuperberg gave a short proof based on the Yang–Baxter equation for the six-vertex model with domain-wall boundary conditions, that uses a determinant calculation due to Anatoli Izergin. A third proof was given by Ilse Fischer using what is called the "operator method".
Razumov–Stroganov problem.
In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.
This conjecture was proved in 2010 by Cantini and Sportiello.


Fake projective space

In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it.
There are exactly 50 fake projective planes. found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.


Luigi Cremona

Antonio Luigi Gaudenzio Giuseppe Cremona (7 December 1830 – 10 June 1903) was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his "Introduzione ad una teoria geometrica delle curve piane". He notably enriched our knowledge of algebraic curves and algebraic surfaces.
Biography.
Luigi Cremona was born in Pavia (Lombardy), then part of the Austrian-controlled Kingdom of Lombardy-Venetia. His youngest brother was the painter Tranquillo Cremona.
In 1848, when Milan and Venice rose against Austria, Cremona, then only seventeen, joined the ranks of the Italian volunteers. He remained with them, fighting on behalf of his country's freedom, until, in 1849, the capitulation of Venice put an end to the campaign.
He then returned to Pavia, where he pursued his studies at the university under Francesco Brioschi, and determined to seek a career as teacher of mathematics. He graduated in 1853 as "dottore negli studi di ingegnere civile e architetto".
Cremona is noted for the important role he played in bringing about the great geometrical advances in Italy. While, at the beginning of the nineteenth century, Italy had very little mathematical standing, the end of the century found Italy in the lead along geometric lines, largely as a result of the work of Cremona. He was very influential in bringing about reforms in the secondary schools of Italy and became a leader in questions of mathematical pedagogy as well as in those relating to the advancement of knowledge. The mathematical advances which Italy made since the middle of the nineteenth century were largely guided by Cremona, Brioschi, and Beltrami.
His first appointment was as elementary mathematical master at the gymnasium and lyceum of Cremona, and he afterwards obtained a similar post at Milan. In 1860 he was appointed to the professorship of higher geometry at the University of Bologna, and in 1866 to that of higher geometry and graphical statics at the higher technical college of Milan. In this same year he competed for the Steiner Prize of the Berlin Academy, with a treatise entitled "Memoria sulle superfici del terzo ordine", and shared the award with J. C. F. Sturm. Two years later the same prize was conferred on him without competition.
As early as 1856 Cremona had begun to contribute to the "Annali di scienze matematiche e fisiche", and to the "Annali di matematica", of which he became afterwards joint editor. Papers by him appeared in the mathematical journals of Italy, France, Germany and England, and he published several important works, many of which have been translated into other languages. His manual "Graphical Statics" and his "Elements of Projective Geometry" (translated by Thomas Hudson Beare and C. Leudesdorf respectively) were published in English by the Clarendon Press.
In 1873 he was called to Rome to organize the Royal College of Engineering, and was also appointed professor of higher mathematics at the university. Cremona's reputation had now become European, and in 1879 he was elected a corresponding member of the Royal Society. In the same year he became a member of the senate of the Kingdom of Italy. In 1898 he was briefly minister for education. The Royal Swedish Academy of Sciences elected Cremona as member in 1901. The following year, he was awarded the German Pour le Mérite for Sciences and Arts.
He died in Rome in 1903.


Discrete least squares meshless method

The discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. While most of the existing meshless methods need background cells for numerical integration, DLSM did not require a numerical integration procedure due to the use of the discrete least squares method to discretize the governing differential equation. A Moving least squares (MLS) approximation method is used to construct the shape function, making the approach a fully least squares-based approach.
Arzani and Afshar developed the DLSM method in 2006 for the solution of Poisson's equation. Firoozjaee and Afshar proposed the collocated discrete least squares meshless (CDLSM) method to solve elliptic partial differential equations, and studied the effect of the collocation points on the convergence and accuracy of the method. The method can be considered as an extension the earlier method of DLSM by the introduction of a set of collocation points for the calculation of the least squares functional.
CDLSM was later used by Naisipour et al. to solve elasticity problems regarding the irregular distribution of nodal points. Afshar and Lashckarbolok used the CDLSM method for the adaptive simulation of hyperbolic problems. A simple a posteriori error indicator based on the value of the least squares functional and a node moving strategy was used and tested on 1-D hyperbolic problems. Shobeyri and Afshar simulated free surface problems using the DLSM method.
The method was then extended for adaptive simulation of two-dimensional shocked hyperbolic problems by Afshar and Firoozjaee. Also, adaptive node-moving refinement and multi-stage node enrichment adaptive refinement are formulated in the DLSM for the solution of elasticity problems.
Amani, Afshar and Naisipour. proposed mixed discrete least squares meshless (MDLSM) formulation for solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet type, which is easily incorporated via a penalty method. Because this is a least squares based algorithm of the MDLSM method, the proposed method does not need to be satisfied by the Ladyzhenskaya–Babuška–Brezzi (LBB) condition.


LOCALAPPDATA (environment variable)


Category of uniform spaces


Complete category

In mathematics, a complete category is a category in which all small limits exist. That is, a category "C" is complete if every diagram "F" : "J" → "C" (where "J" is small) has a limit in "C". Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete.
The existence of "all" limits (even when "J" is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.
A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category "J"). Dually, a category is finitely cocomplete if all finite colimits exist.
Theorems.
It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products. Since equalizers may be constructed from pullbacks and binary products (consider the pullback of ("f", "g") along the diagonal Δ), a category is complete if and only if it has pullbacks and products.
Dually, a category is cocomplete if and only if it has coequalizers and all (small) coproducts, or, equivalently, pushouts and coproducts.
Finite completeness can be characterized in several ways. For a category "C", the following are all equivalent:
The dual statements are also equivalent.
A small category "C" is complete if and only if it is cocomplete. A small complete category is necessarily thin.
A posetal category vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness. Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.


Alternating permutation

In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set {1, 2, 3, ..., "n"} is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, the five alternating permutations of {1, 2, 3, 4} are:
This type of permutation was first studied by Désiré André in the 19th century.
Different authors use the term alternating permutation slightly differently: some require that the second entry in an alternating permutation should be larger than the first (as in the examples above), others require that the alternation should be reversed (so that the second entry is smaller than the first, then the third larger than the second, and so on), while others call both types by the name alternating permutation.
The determination of the number "A""n" of alternating permutations of the set {1, ..., "n"} is called André's problem. The numbers "A""n" are known as Euler numbers, zigzag numbers, or up/down numbers. When "n" is even the number "A""n" is known as a secant number, while if "n" is odd it is known as a tangent number. These latter names come from the study of the generating function for the sequence.
Definitions.
A permutation is said to be "alternating" if its entries alternately rise and descend. Thus, each entry other than the first and the last should be either larger or smaller than both of its neighbors. Some authors use the term alternating to refer only to the "up-down" permutations for which , calling the "down-up" permutations that satisfy by the name "reverse alternating". Other authors reverse this convention, or use the word "alternating" to refer to both up-down and down-up permutations.
There is a simple one-to-one correspondence between the down-up and up-down permutations: replacing each entry with reverses the relative order of the entries.
By convention, in any naming scheme the unique permutations of length 0 (the permutation of the empty set) and 1 (the permutation consisting of a single entry 1) are taken to be alternating.
André's theorem.
The determination of the number "A""n" of alternating permutations of the set {1, ..., "n"} is called "André's problem". The numbers "A""n" are variously known as "Euler numbers", "zigzag numbers", "up/down numbers", or by some combinations of these names. The name Euler numbers in particular is sometimes used for a closely related sequence. The first few values of "A""n" are 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, ... .
These numbers satisfy a simple recurrence, similar to that of the Catalan numbers: by splitting the set of alternating permutations (both down-up and up-down) of the set { 1, 2, 3, ..., "n", "n" + 1 } according to the position "k" of the largest entry , one can show that
for all . used this recurrence to give a differential equation satisfied by the exponential generating function
for the sequence . In fact, the recurrence gives:
where we substitute formula_4 and formula_5. This gives the integral equation
which after differentiation becomes formula_7.
This differential equation can be solved by separation of variables (using the initial condition formula_8), and simplified using a tangent half-angle formula, giving the final result
the sum of the secant and tangent functions. This result is known as "André's theorem".
It follows from André's theorem that the radius of convergence of the series is /2. This allows one to compute the asymptotic expansion
Related integer sequences.
The odd-indexed zigzag numbers (i.e., the tangent numbers) are closely related to Bernoulli numbers. The relation is given by the formula
for "n" > 0.
If "Z""n" denotes the number of permutations of {1, ..., "n"} that are either up-down or down-up (or both, for "n" < 2) then it follows from the pairing given above that "Z""n" = 2"A""n" for "n" ≥ 2. The first few values of "Z""n" are 1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, ... .
The Euler zigzag numbers are related to Entringer numbers, from which the zigzag numbers may be computed. The Entringer numbers can be defined recursively as follows:
The "n"th zigzag number is equal to the Entringer number "E"("n", "n").
The numbers "A"2"n" with even indices are called secant numbers or zig numbers: since the secant function is even and tangent is odd, it follows from André's theorem above that they are the numerators in the Maclaurin series of . The first few values are 1, 1, 5, 61, 1385, 50521, ... .
Secant numbers are related to the signed Euler numbers (Taylor coefficients of hyperbolic secant) by the formula "E"2"n" = (−1)"n""A"2"n". ("E""n" = 0 when "n" is odd.)
Correspondingly, the numbers "A"2"n"+1 with odd indices are called tangent numbers or zag numbers. The first few values are 1, 2, 16, 272, 7936, ... .
Explicit formula in terms of Stirling numbers of the second kind.
The relationships of Euler zigzag numbers with the Euler numbers, and the Bernoulli numbers can be used to prove the following
where
formula_17 denotes the rising factorial, and formula_18 denotes Stirling numbers of the second kind.


Peak algebra

In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group "S""n", studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2...,"n"} is an index "i" such that σ("i"–1)<σ("i")>σ("i"+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all "n" has a natural structure of a Hopf algebra.


Monoidal natural transformation

Suppose that formula_1 and formula_2 are two monoidal categories and
are two lax monoidal functors between those categories.
A monoidal natural transformation
between those functors is a natural transformation formula_6 between the underlying functors such that the diagrams
commute for every objects formula_7 and formula_8 of formula_9 (see Definition 11 in ).
A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors.


Generalized Petersen graph

In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins.
Definition and notation.
In Watkins' notation, "G"("n", "k") is a graph with vertex set
and edge set
where subscripts are to be read modulo "n" and "k" < "n"/2. Some authors use the notation "GPG"("n", "k"). Coxeter's notation for the same graph would be {"n"} + {"n"/"k"}, a combination of the Schläfli symbols for the regular "n"-gon and star polygon from which the graph is formed. The Petersen graph itself is "G"(5, 2) or {5} + {5/2}.
Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other edge.
Examples.
Among the generalized Petersen graphs are the "n"-prism "G"("n", 1), the Dürer graph "G"(6, 2), the Möbius-Kantor graph "G"(8, 3), the dodecahedron "G"(10, 2), the Desargues graph "G"(10, 3) and the Nauru graph "G"(12, 5).
Four generalized Petersen graphs – the 3-prism, the 5-prism, the Dürer graph, and "G"(7, 2) – are among the seven graphs that are cubic, 3-vertex-connected, and well-covered (meaning that all of their maximal independent sets have equal size).
Properties.
This family of graphs possesses a number of interesting properties. For example:
Isomorphisms.
"G"("n", "k") is isomorphic to "G"("n", "l") if and only if "kl" ≡ ±1 (mod "n").
Girth.
The girth of "G"("n", "k") is at least 3 and at most 8, in particular:
A table with exact girth values:
Chromatic number and chromatic index.
Being regular, according to Brooks' theorem their chromatic number can not be larger than their degree. Generalized Petersen graphs are cubic, so their chromatic number can be either 2 or 3. More exactly:
Where formula_5 denotes the logical AND, while formula_6 the logical OR. For example, the chromatic number of formula_7 is 3.
Petersen graph, being a snark, has a chromatic index of 4. All other generalized Petersen graph has chromatic index 3.
The generalized Petersen graph "G"(9, 2) is one of the few graphs known to have only one 3-edge-coloring.
The Petersen graph itself is the only generalized Petersen graph that is not 3-edge-colorable.


Coherent ring

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.
Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings.
Every left Noetherian ring is left-coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left-coherent ring that is not left Noetherian.
A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.


Flower snark

In the mathematical field of graph theory, the flower snarks form an infinite family of snarks introduced by Rufus Isaacs in 1975.
As snarks, the flower snarks are connected, bridgeless cubic graphs with chromatic index equal to 4. The flower snarks are non-planar and non-hamiltonian. The flower snarks J5 and J7 have book thickness 3 and queue number 2.
Construction.
The flower snark J"n" can be constructed with the following process :
By construction, the Flower snark J"n" is a cubic graph with 4"n" vertices and 6"n" edges. For it to have the required properties, "n" should be odd.
Special cases.
The name flower snark is sometimes used for J5, a flower snark with 20 vertices and 30 edges. It is one of 6 snarks on 20 vertices . The flower snark J5 is hypohamiltonian.
J3 is a trivial variation of the Petersen graph formed by replacing one of its vertices by a triangle. This graph is also known as the Tietze's graph. In order to avoid trivial cases, snarks are generally restricted to have girth at least 5. With that restriction, J3 is not a snark.


Boris Moishezon

Boris Gershevich Moishezon () (October 26, 1937 – August 25, 1993) was a Soviet mathematician. He left the Soviet Union in 1972 for Tel Aviv, and in 1977 moved to Columbia University, where he was a professor of mathematics until his death sixteen years later. He was a Guggenheim Fellow in 1983.
A resident of Leonia, New Jersey, Moishezon died at Holy Name Hospital in Teaneck, New Jersey on August 25, 1993, due to a heart attack he suffered while jogging.


Near-semiring

In mathematics, a near-semiring (also "seminearring") is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.
Definition.
A near-semiring is a set "S" with two binary operations "+" and "·", and a constant 0 such that ("S", +, 0) is a monoid (not necessarily commutative), ("S", ·) is a semigroup, these structures are related by a single (right or left) distributive law, and accordingly 0 is a one-sided (right or left, respectively) absorbing element.
Formally, an algebraic structure ("S", +, ·, 0) is said to be a near-semiring if it satisfies the following axioms:
Near-semirings are a common abstraction of semirings and near-rings [Golan, 1999; Pilz, 1983]. The standard examples of near-semirings are typically of the form "M"(Г), the set of all mappings on a monoid (Г; +, 0), equipped with composition of mappings, pointwise addition of mappings, and the zero function. Subsets of "M"(Г) closed under the operations provide further examples of near-semirings. Another example is the ordinals under the usual operations of ordinal arithmetic (here Clause 3 should be replaced with its symmetric form "c" · ("a" + "b") = "c" · "a" + "c" · "b". Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a "class near-semiring". We get a near-semiring in the standard sense if we restrict to those ordinals strictly less than some multiplicatively indecomposable ordinal.


Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra formula_1 of a Lie group formula_2 to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when formula_2 is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
Definitions.
Let formula_2 be a Lie group and formula_1 be its Lie algebra (thought of as the tangent space to the identity element of formula_2). The exponential map is a map
which can be defined in several different ways. The typical modern definition is this:
It follows easily from the chain rule that formula_13. The map formula_14 may be constructed as the integral curve of either the right- or left-invariant vector field associated with formula_12. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
We have a more concrete definition in the case of a matrix Lie group. The exponential map coincides with the matrix exponential and is given by the ordinary series expansion:
where formula_17 is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra formula_1 of formula_2.
Comparison with Riemannian exponential map.
If "G" is compact, it has a Riemannian metric invariant under left "and" right translations, and the Lie-theoretic exponential map for "G" coincides with the exponential map of this Riemannian metric.
For a general "G", there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will "not" in general agree with the exponential map in the Lie group sense. That is to say, if "G" is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of "G" .
Other definitions.
Other equivalent definitions of the Lie-group exponential are as follows:
Properties.
Elementary properties of the exponential.
For all formula_8, the map formula_41 is the unique one-parameter subgroup of formula_2 whose tangent vector at the identity is formula_12. It follows that:
More generally:
It is important to emphasize that the preceding identity does not hold in general; the assumption that formula_12 and formula_48 commute is important.
The image of the exponential map always lies in the identity component of formula_2.
The exponential near the identity.
The exponential map formula_7 is a smooth map. Its differential at zero, formula_51, is the identity map (with the usual identifications).
It follows from the inverse function theorem that the exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in formula_1 to a neighborhood of 1 in formula_2.
It is then not difficult to show that if "G" is connected, every element "g" of "G" is a "product" of exponentials of elements of formula_1:formula_55.
Globally, the exponential map is not necessarily surjective. Furthermore, the exponential map may not be a local diffeomorphism at all points. For example, the exponential map from formula_56(3) to SO(3) is not a local diffeomorphism; see also cut locus on this failure. See derivative of the exponential map for more information.
Surjectivity of the exponential.
In these important special cases, the exponential map is known to always be surjective:
For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.
The image of the exponential map of the connected but non-compact group "SL"2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix formula_58. (Thus, the image excludes matrices with real, negative eigenvalues, other than formula_58.)
Exponential map and homomorphisms.
Let formula_60 be a Lie group homomorphism and let formula_61 be its derivative at the identity. Then the following diagram commutes:
In particular, when applied to the adjoint action of a Lie group formula_2, since formula_63, we have the useful identity:
Logarithmic coordinates.
Given a Lie group formula_2 with Lie algebra formula_66, each choice of a basis formula_67 of formula_66 determines a coordinate system near the identity element "e" for "G", as follows. By the inverse function theorem, the exponential map formula_69 is a diffeomorphism from some neighborhood formula_70 of the origin to a neighborhood formula_71 of formula_72. Its inverse:
is then a coordinate system on "U". It is called by various names such as logarithmic coordinates, exponential coordinates or normal coordinates. See the closed-subgroup theorem for an example of how they are used in applications.
Remark: The open cover formula_74 gives a structure of a real-analytic manifold to "G" such that the group operation formula_75 is real-analytic.


Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of "small" transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix "A". It is not the matrix of an actual rotation in space; but for small real values of a parameter ε the transformation
is a small rotation, up to quantities of order ε2.
History.
A comprehensive theory of infinitesimal transformations was first given by Sophus Lie. This was at the heart of his work, on what are now called Lie groups and their accompanying Lie algebras; and the identification of their role in geometry and especially the theory of differential equations. The properties of an abstract Lie algebra are exactly those definitive of infinitesimal transformations, just as the axioms of group theory embody symmetry. The term "Lie algebra" was introduced in 1934 by Hermann Weyl, for what had until then been known as the "algebra of infinitesimal transformations" of a Lie group.
Examples.
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the cross product, once a skew-symmetric matrix has been identified with a 3-vector. This amounts to choosing an axis vector for the rotations; the defining Jacobi identity is a well-known property of cross products.
The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions. Here it is stated that a function "F" of "n" variables "x"1, ..., "x""n" that is homogeneous of degree "r", satisfies
with
the Theta operator. That is, from the property
it is possible to differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a smooth function "F" to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysis considerations here). This setting is typical, in that there is a one-parameter group of scalings operating; and the information is coded in an infinitesimal transformation that is a first-order differential operator.
Operator version of Taylor's theorem.
The operator equation
where
is an operator version of Taylor's theorem — and is therefore only valid under "caveats" about "f" being an analytic function. Concentrating on the operator part, it shows that "D" is an infinitesimal transformation, generating translations of the real line via the exponential. In Lie's theory, this is generalised a long way. Any connected Lie group can be built up by means of its infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the Baker–Campbell–Hausdorff formula.


Ind-scheme

In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.


Nilpotent ideal

In mathematics, more specifically ring theory, an ideal "I" of a ring "R" is said to be a nilpotent ideal if there exists a natural number "k" such that "I""k" = 0. By "I""k", it is meant the additive subgroup generated by the set of all products of "k" elements in "I". Therefore, "I" is nilpotent if and only if there is a natural number "k" such that the product of any "k" elements of "I" is 0.
The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem. The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
Relation to nil ideals.
The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than one reason. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.
In a right Artinian ring, any nil ideal is nilpotent. This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis), the result follows. In fact, this can be generalized to right Noetherian rings; this result is known as Levitzky's theorem.


Moufang set

In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
Definition.
A Moufang set is a pair formula_1 where "X" is a set and formula_2 is a family of subgroups of the symmetric group formula_3 indexed by the elements of "X". The system satisfies the conditions
Examples.
Let "K" be a field and "X" the projective line P1("K") over "K". Let "U""x" be the stabiliser of each point "x" in the group PSL2("K"). The Moufang set determines "K" up to isomorphism or anti-isomorphism: an application of Hua's identity.
A quadratic Jordan division algebra gives rise to a Moufang set structure. If "U" is the quadratic map on the unital algebra "J", let τ denote the permutation of the additive group ("J",+) defined by
Then τ defines a Moufang set structure on "J". The Hua maps "h""a" of the Moufang structure are just the quadratic "U""a" . Note that the link is more natural in terms of J-structures.


Coset

In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint equal-size subsets called cosets. There are "left cosets" and "right cosets". Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by .
Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
Definition.
Let be a subgroup of the group whose operation is written multiplicatively (juxtaposition means apply the group operation). Given an element of , the left cosets of in are the sets obtained by multiplying each element of by a fixed element of (where is the left factor). In symbols these are,
The right cosets are defined similarly, except that the element is now a right factor, that is,
As varies through the group, it would appear that many cosets (right or left) would be generated. Nevertheless, it turns out that any two left cosets (respectively right cosets) are either disjoint or are identical as sets.
If the group operation is written additively, as is often the case when the group is abelian, the notation used changes to or , respectively.
First example.
Let be the dihedral group of order six. Its elements may be represented by }. In this group, and . This is enough information to fill in the entire multiplication table:
Let be the subgroup }. The (distinct) left cosets of are:
Since all the elements of have now appeared in one of these cosets, generating any more can not give new cosets, since a new coset would have to have an element in common with one of these and therefore be identical to one of these cosets. For instance, .
The right cosets of are:
In this example, except for , no left coset is also a right coset.
Let be the subgroup . The left cosets of are and . The right cosets of are and . In this case, every left coset of is also a right coset of .
Let be a subgroup of a group and suppose that , . The following statements are equivalent:
Properties.
Because is a subgroup, it contains the group's identity element, with the result that the element belongs to the coset . If belongs to then . Thus every element of belongs to exactly one left coset of the subgroup .
The identity is in precisely one left or right coset, namely itself. Thus is both a left and right coset of itself.
Elements and belong to the same left coset of , that is, if and only if belongs to . More can be said here. Define two elements of , say and , to be equivalent with respect to the subgroup if belongs to . This is then an equivalence relation on and the equivalence classes of this relation are the left cosets of . As with any set of equivalence classes, they form a partition of the underlying set. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.
Similar statements apply to right cosets.
If is an abelian group, then for every subgroup of and every element of . For general groups, given an element and a subgroup of a group , the right coset of with respect to is also the left coset of the conjugate subgroup with respect to , that is, .
Normal subgroups.
A subgroup of a group is a normal subgroup of if and only if for all elements of the corresponding left and right cosets are equal, that is, . This is the case for the subgroup in the first example above. Furthermore, the cosets of in form a group called the quotient group or factor group.
If is not normal in , then its left cosets are different from its right cosets. That is, there is an in such that no element satisfies . This means that the partition of into the left cosets of is a different partition than the partition of into right cosets of . This is illustrated by the subgroup in the first example above. ("Some" cosets may coincide. For example, if is in the center of , then .)
On the other hand, if the subgroup is normal the set of all cosets form a group called the quotient group with the operation ∗ defined by . Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".
Index of a subgroup.
Every left or right coset of has the same number of elements (or cardinality in the case of an infinite ) as itself. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of in "G", written as . Lagrange's theorem allows us to compute the index in the case where and are finite:
This equation also holds in the case where the groups are infinite, although the meaning may be less clear.
More examples.
Integers.
Let be the additive group of the integers, and the subgroup . Then the cosets of in are the three sets , , and , where }. These three sets partition the set , so there are no other right cosets of . Due to the commutivity of addition and . That is, every left coset of is also a right coset, so is a normal subgroup. (The same argument shows that every subgroup of an Abelian group is normal.)
This example may be generalized. Again let be the additive group of the integers, , and now let the subgroup , where is a positive integer. Then the cosets of in are the sets , , ..., , where }. There are no more than cosets, because . The coset is the congruence class of modulo . The subgroup is normal in , and so, can be used to form the quotient group the group of integers mod "m".
Vectors.
Another example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector space form an abelian group under vector addition. The subspaces of the vector space are subgroups of this group. For a vector space , a subspace , and a fixed vector in , the sets
are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin. For example, consider the plane . If is a line through the origin , then is a subgroup of the abelian group . If is in , then the coset is a line parallel to and passing through .
Matrices.
Let be the multiplicative group of matrices,
and the subgroup of ,
For a fixed element of consider the left coset
That is, the left cosets consist of all the matrices in having the same upper-left entry. This subgroup is normal in , but the subgroup
is not normal in .
As orbits of a group action.
A subgroup of a group can be used to define an action of on in two natural ways. A "right action", given by or a "left action", given by . The orbit of under the right action is the left coset , while the orbit under the left action is the right coset .
History.
The concept of a coset dates back to Galois's work of 1830–31. He introduced a notation but did not provide a name for the concept. The term "co-set" appears for the first time in 1910 in a paper by G. A. Miller in the "Quarterly Journal of Mathematics" (vol. 41, p. 382). Various other terms have been used including the german "Nebengruppen" (Weber) and "conjugate group" (Burnside).
Galois was concerned with deciding when a given polynomial equation was solvable by radicals. A tool that he developed was in noting that a subgroup of a group of permutations induced two decompositions of (what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over instead of . Camille Jordan in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term.
Calling the coset the "left coset" of with respect to , while most common today, has not been universally true in the past. For instance, would call a "right coset", emphasizing the subgroup being on the right.
An application from coding theory.
A binary linear code is an -dimensional subspace of an -dimensional vector space over the binary field . As is an additive abelian group, is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a "codeword" (element of ) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted "received word" could have started out as. This procedure is called "decoding" and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of (a received word could be any element of ) into a standard array. A standard array is a coset decomposition of put into tabular form in a certain way. Namely, the top row of the array consists of the elements of , written in any order, except that the zero vector should be written first. Then, an element of with a minimal number of ones that does not already appear in the top row is selected and the coset of containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of directly above it). This element is called a coset leader and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of containing it is the next row. The process ends when all the vectors of have been sorted into the cosets.
An example of a standard array for the 2-dimensional code in the 5-dimensional space (with 32 vectors) is as follows:
The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in. Since in binary arithmetic adding is the same operation as subtracting, this always results in an element of . In the event that the transmission errors occurred in precisely the non-zero positions of the coset leader the result will be the right codeword. In this example, if a single error occurs, the method will always correct it, since all possible coset leaders with a single one appear in the array.
Syndrome decoding can be used to improve the efficiency of this method. It is a method of computing the correct coset (row) that a received word will be in. For an -dimensional code in an -dimensional binary vector space, a parity check matrix is an matrix having the property that if and only if is in . The vector is called the "syndrome" of , and by linearity, every vector in the same coset will have the same syndrome. To decode, the search is now reduced to finding the coset leader that has the same syndrome as the received word.
Double cosets.
Given two subgroups, and (which need not be distinct) of a group , the double cosets of and in are the sets of the form . These are the left cosets of and right cosets of when and respectively.
Two double cosets and are either disjoint or identical. The set of all double cosets for fixed and form a partition of .
A double coset contains the complete right cosets of (in ) of the form , with an element of and the complete left cosets of (in ) of the form , with in .
Notation.
Let be a group with subgroups and . Several authors working with these sets have developed a specialized notation for their work, where


Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition.
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one.
In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on . This definition, as the dimension of
then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle.
The geometric genus is the first invariant of a sequence of invariants called the plurigenera.
Case of curves.
In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree .
The notion of genus features prominently in the statement of the Riemann–Roch theorem (see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree "d" has geometric genus
where "s" is the number of singularities when properly counted
If is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree , then its normal line bundle is the Serre twisting sheaf , so by the adjunction formula, the canonical line bundle of is given by
Genus of singular varieties.
The definition of geometric genus is carried over classically to singular curves , by decreeing that
is the geometric genus of the normalization . That is, since the mapping
is birational, the definition is extended by birational invariance.


Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
History.
The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, "Algebraic combinatorics", of the AMS Mathematics Subject Classification, introduced in 1991.
Scope.
Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.
Important topics.
Symmetric functions.
The ring of symmetric functions is a specific limit of the rings of symmetric polynomials in "n" indeterminates, as "n" goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number "n" of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric groups.
Association schemes.
An association scheme is a collection of binary relations satisfying certain compatibility conditions. Association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
Strongly regular graphs.
A strongly regular graph is defined as follows. Let "G" = ("V","E") be a regular graph with "v" vertices and degree "k". "G" is said to be strongly regular if there are also integers λ and μ such that:
A graph of this kind is sometimes said to be an srg("v", "k", λ, μ).
Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs, and their complements, the Turán graphs.
Young tableaux.
A Young tableau (pl.: "tableaux") is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
Matroids.
A matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.
Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory.
Finite geometries.
A finite geometry is any geometric system that has only a finite number of points.
The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries.


Martin Kassabov

Martin Dimitrov Kassabov (, born 1977) is a Bulgarian mathematician who works as a professor of mathematics at Cornell University, specializing in combinatorial group theory.
Kassabov earned a master's degree from Sofia University in 1998.
He completed his Ph.D. from Yale University in 2003 under the supervision of Efim Zelmanov. After postdoctoral research at the University of Alberta he joined Cornell as an H. C. Wang Assistant Professor in 2004.
He was the 2007–2008 winner of the AMS Centennial Fellowship.
He was an invited speaker at the International Congress of Mathematicians in 2014,
and in the same year won the Mathematics Prize of the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences.
In 2015 he was elected as a fellow of the American Mathematical Society.