William Messing

William Messing is an American mathematician who works in the field of arithmetic algebraic geometry.
Messing received his doctorate in 1971 at Princeton University under the supervisions of Alexander Grothendieck (and Nicholas Katz) with his thesis entitled "The Crystals Associated to Barsotti–Tate Groups: With Applications to Abelian Schemes." In 1972, he was a C.L.E. Moore instructor at Massachusetts Institute of Technology. He is currently a professor at the University of Minnesota (Minneapolis).
In his thesis, Messing elaborated on Grothendieck's 1970 lecture at the International Congress of Mathematicians in Nice on p-divisible groups (Barsotti–Tate groups) that are important in algebraic geometry in prime characteristic, which were introduced in the 1950s by Dieudonné in his study of Lie algebras over fields of finite characteristic. Messing worked together with Pierre Berthelot, Barry Mazur and Aise Johan de Jong.
References.
The original article was a Google-aided translation of the corresponding article in German Wikipedia.


Mott polynomials

In mathematics the Mott polynomials "s""n"("x") are polynomials introduced by who applied them to a problem in the theory of electrons.
They are given by the exponential generating function
Because the factor in the exponential has the power series
in terms of Catalan numbers formula_3, the coefficient in front of formula_4 of the polynomial can be written as
according to the general formula for generalized Appell polynomials,
where the sum is over all compositions formula_6 of formula_7 into formula_8 positive odd integers. The empty product appearing for formula_9 equals 1. Special values, where all contributing Catalan numbers equal 1, are
By differentiation the recurrence for the first derivative becomes
The first few of them are
The polynomials "s""n"("x") form the associated Sheffer sequence for –2"t"/(1–t2) .
 give an explicit expression for them in terms of the generalized hypergeometric function 3F0:


Francis Dominic Murnaghan (mathematician)

Francis Dominic Murnaghan (August 4, 1893 – March 24, 1976) was an Irish mathematician, former head of the mathematics department at Johns Hopkins University. His name is attached to developments in group theory and mathematics applied to continuum mechanics (Murnaghan and Birch–Murnaghan equations of state).
Biography.
Frank Murnaghan was born in Omagh, Co. Tyrone, Ireland, seventh of the nine children of George Murnaghan, Nationalist MP representing Mid Tyronne constituency. He graduated from Irish Christian Brothers secondary school in 1910, and University College Dublin with first-class honors BSc in Mathematical Sciences in 1913. Following an MSc in 1914, he was awarded a National University of Ireland (NUI) Travelling Studentship, which funded him to pursue his doctorate at Johns Hopkins University. In 1916, after just two years working under department chair Frank Morley's new PhD student Harry Bateman, he was awarded the Ph.D.
He then lectured at Rice University, and returned to Johns Hopkins in the rank of Associate Professor at the young age of 25. In 1928 he was promoted to Professor and became only the fourth head of the Department of Mathematics (after J.J. Sylvester, Simon Newcomb and Frank Morley). After his retirement in 1949, he worked at the Instituto Tecnológico de Aeronáutica near São Paulo, Brasil, but returned to Baltimore in 1959. He continued working as a consultant for the Marine Engineering Laboratory; his last publication appeared in 1972.
Murnaghan was a member of US National Academy of Sciences, American Philosophical Society, Royal Irish Academy, and Brazilian Academy of Sciences. He wrote 15 books, some in English and some in Portuguese, and over 90 papers.
He was the father of Francis Dominic Murnaghan, Jr., former U.S. federal judge and uncle of Northern Irish barrister and politician Sheelagh Murnaghan.


Reducible variety


Edmonds matrix

In graph theory, the Edmonds matrix formula_1 of a balanced bipartite graph formula_2 with sets of vertices formula_3 and formula_4 is defined by
where the "x"ij are indeterminates. One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det("A"ij) in the "x"ij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det("A"), and is also equal to the permanent of formula_1. In addition, rank of formula_1 is equal to the maximum matching size of formula_8.
The Edmonds matrix is named after Jack Edmonds. The Tutte matrix is a generalisation to non-bipartite graphs.


Mohammad Reza Darafsheh

Mohammad Reza Darafsheh (born 31 August 1950) is an Iranian Mathematician and Professor of Mathematics at the University of Tehran, and a member of the executive committee of the Iranian Mathematical Society.
He received a B.Sc. in Mathematics from the University of Tehran with first rank in 1974. And his M.Sc. and Ph.D. degrees from University of Birmingham in 1975 and 1978 respectively under the supervision of Donald Livingston. His dissertation was titled "On some Subgroups of C_1." He has been a faculty member at the Shahid Chamran University of Ahvaz from 1978 to 1989, and from 1989 to present at the University of Tehran, where he chaired the mathematics department from 1995 to 1999
he has advised 33 Ph.D. students.
His research area includes the representations and characters of finite groups, symmetry class of tensors, Thomson's conjecture, characterization of simple groups by the set of elements order. He has solved the Feit's conjecture on the p-Steinberg characters of finite groups. He has contributed to finding the character tables of linear groups. He has authored the books "An introduction to group theory", "Linear groups", and a 3 volume book titled "Algebra" which was included in "Eighty Treasures" in 2008 as part of the 80th anniversary of University of Tehran's establishment
He was awarded the Abdus-Salam prize in Mathematics by International Centre for Theoretical Physics in 1990, and the Khwarizmi International Award in 1997


Affine Lie algebra

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra formula_1, one considers the loop algebra, formula_2, formed by the formula_1-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra formula_4 is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a quantum anomaly (in this case, the anomaly of the WZW model) and mathematicians a central extension. More generally,
if σ is an automorphism of the simple Lie algebra formula_1 associated to an automorphism of its Dynkin diagram, the twisted loop algebra formula_6 consists of formula_1-valued functions "f" on the real line which satisfy
the twisted periodicity condition . Their central extensions are precisely the twisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group.
Affine Lie algebras from simple Lie algebras.
Definition.
If formula_1 is a finite-dimensional simple Lie algebra, the corresponding
affine Lie algebra formula_4 is constructed as a central extension of the infinite-dimensional Lie algebra formula_10, with one-dimensional center formula_11
As a vector space,
where formula_13 is the complex vector space of Laurent polynomials in the indeterminate "t". The Lie bracket is defined by the formula
for all formula_15 and formula_16, where formula_17 is the Lie bracket in the Lie algebra formula_1 and formula_19 is the Cartan-Killing form on formula_20
The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by
The corresponding affine Kac-Moody algebra is defined as a semidirect product by adding an extra generator "d" that satisfies ["d", "A"] = "δ"("A").
Constructing the Dynkin diagrams.
The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras.
Classifying the central extensions.
The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra formula_22. In this case one also needs to add "n" further central elements for the "n" abelian generators.
The second integral cohomology of the loop group of the corresponding simple compact Lie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first Chern class of the fibration. Therefore, the central extensions of an affine Lie group are classified by a single parameter "k" which is called the "level" in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when "k" is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component.
Representation theory.
The representation theory for affine Lie algebras is usually developed using Verma modules. Just as in the case of semi-simple Lie algebras, these can be gotten as the highest weight modules. There are no finite-dimensional representations; this follows from the fact that the null vectors of a finite-dimensional Verma module are necessarily zero; whereas those for the affine Lie algebras are not. Roughly speaking, this follows because the Killing form is Lorentzian in the formula_23 directions, thus formula_24 are sometimes called "lightcone coordinates" on the string. The "radially ordered" current operator products can be understood to be time-like normal ordered by taking formula_25 with formula_26 the time-like direction along the string world sheet and formula_27 the spatial direction.
Weyl group and characters.
The Weyl group of an affine Lie algebra can be written as a semi-direct product of the Weyl group of the zero-mode algebra (the Lie algebra used to define the loop algebra) and the coroot lattice.
The Weyl character formula of the algebraic characters of the affine Lie algebras generalizes to the Weyl-Kac character formula. A number of interesting constructions follow from these. One may construct generalizations of the Jacobi theta function. These theta functions transform under the modular group. The usual denominator identities of semi-simple Lie algebras generalize as well; because the characters can be written as "deformations" or q-analogs of the highest weights, this led to many new combinatoric identities, include many previously unknown identities for the Dedekind eta function. These generalizations can be viewed as a practical example of the Langlands program.
Applications.
Due to the Sugawara construction, the universal enveloping algebra of any affine Lie algebra has the Virasoro algebra as a subalgebra. This allows affine Lie algebras to serve as symmetry algebras of conformal field theories such as WZW models or coset models. As a consequence, affine Lie algebras also appear in the worldsheet description of string theory.


David M. Goldschmidt

David M. Goldschmidt (born 21 May 1942, New York City) is an American mathematician specializing in group theory.
Goldschmidt received in 1969 from the University of Chicago a Ph.D. under John Griggs Thompson with thesis "On the 2-exponent of a finite group". From 1969 to 1971 he was a Gibbs Instructor at Yale University. From 1971 to 1989 he was on the faculty of the mathematics department at the University of California, Berkeley. In 1989–1991 he was Deputy Director and in 1991 he became Director of the Institute for Defense Analyses's Center for Communication Research in Princeton, New Jersey.
Goldschmidt published his amalgam method for finite groups in 1980. In the 1980s this method was important for new developments in the local structure theory of finite groups. In addition to his work on the amalgam method, Goldschmidt has done research on the representation theory of finite groups and has written a book on algebraic curves without the sophisticated approach of modern algebraic geometry.
In 2012 he was elected a Fellow of the American Mathematical Society. His doctoral students include Jeffrey Shallit.


Linearised polynomial

In mathematics, a linearised polynomial (or "q"- polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of "q" and the coefficients come from some extension field of the finite field of order "q".
We write a typical example as
This special class of polynomials is important from both a theoretical and an applications viewpoint. The highly structured nature of their roots makes these roots easy to determine.
Symbolic multiplication.
In general, the product of two linearised polynomials will not be a linearized polynomial, but since the composition of two linearised polynomials results in a linearised polynomial, composition may be used as a replacement for multiplication and, for this reason, composition is often called symbolic multiplication in this setting. Notationally, if "L"1("x") and "L"2("x") are linearised polynomials we define
when this point of view is being taken.
Associated polynomials.
The polynomials L("x") and
are "q - associates" (note: the exponents ""q""i" " of "L"("x") have been replaced by "i" in "l"("x")). More specifically, "l(x}" is called the "conventional q-associate" of "L(x)", and "L(x)" is the "linearised q-associate" of "l(x)".
q-polynomials over F"q".
Linearised polynomials with coefficients in F"q" have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorization. Two important examples of this type of linearised polynomial are the Frobenius automorphism formula_7 and the trace function formula_8.
In this special case it can be shown that, as an operation, symbolic multiplication is commutative, associative and distributes over ordinary addition. Also, in this special case, we can define the operation of symbolic division. If "L"("x") and "L"1("x") are linearised polynomials over F"q", we say that "L"1("x") "symbolically divides" "L"("x") if there exists a linearised polynomial "L"2("x") over F"q" for which:
If "L"1("x") and "L"2("x") are linearised polynomials over F"q" with conventional q-associates "l"1("x") and "l"2("x") respectively, then "L"1("x") symbolically divides "L"2("x") if and only if "l"1("x") divides "l"2("x"). Furthermore,
"L"1("x") divides "L"2("x") in the ordinary sense in this case.
A linearised polynomial "L"("x") over F"q" of degree > 1 is "symbolically irreducible" over F"q" if the only symbolic decompositions
with "L""i" over F"q" are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always reducible in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor "x". A linearised polynomial "L"("x") over F"q" is symbolically irreducible if and only if its conventional "q"-associate "l"("x") is irreducible over F"q".
Every "q"-polynomial "L"("x") over F"q" of degree > 1 has a "symbolic factorization" into symbolically irreducible polynomials over F"q" and this factorization is essentially unique (up to rearranging factors and multiplying by nonzero elements of F"q".)
For example, consider the 2-polynomial "L"("x") = "x"16 + "x"8 + "x"2 + "x" over F2 and its conventional 2-associate "l"("x") = "x"4 + "x"3 + "x" + 1. The factorization into irreducibles of "l"("x") = ("x"2 + "x" + 1)("x" + 1)2 in F2["x"], gives the symbolic factorization
Affine polynomials.
Let "L" be a linearised polynomial over formula_2. A polynomial of the form formula_13 is an "affine polynomial" over formula_2.
Theorem: If "A" is a nonzero affine polynomial over formula_2 with all of its roots lying in the field formula_3 an extension field of formula_2, then each root of "A" has the same multiplicity, which is either 1, or a positive power of "q".


Fuzzy category


Septic function


Dihedral group of order 6

In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abelian group.
This page illustrates many group concepts using this group as example.
Symmetry groups.
The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders.
The dihedral group D3 is isomorphic to two other symmetry groups in three dimensions:
Permutations of a set of three objects.
Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The symmetric group S3 is then the group of all possible rearrangements of these blocks.
If we denote by "a" the action "swap the first two blocks", and by "b" the action "swap the last two blocks", we can write all possible permutations in terms of these two actions.
In multiplicative form, we traditionally write "xy" for the combined action "first do "y", then do "x""; so that "ab" is the action , i.e., "take the last block and move it to the front".
If we write "e" for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
The notation in brackets is the cycle notation.
Note that the action "aa" has the effect , leaving the blocks as they were; so we can write .
Similarly,
so each of the above actions has an inverse.
By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that
The group is non-abelian since, for example, . Since it is built up from the basic actions "a" and "b", we say that the set "generates" it.
The group has presentation
where "a" and "b" are swaps and is a cyclic permutation. Note that the second presentation means that the group is a Coxeter group. (In fact, all dihedral and symmetry groups are Coxeter groups.)
Summary of group operations.
With the generators "a" and "b", we define the additional shorthands , and , so that "a, b, c, d, e", and "f" are all the elements of this group. We can then summarize the group operations in the form of a Cayley table:
Note that non-equal non-identity elements only commute if they are each other's inverse. Therefore, the group is centerless, i.e., the center of the group consists only of the identity element.
Conjugacy classes.
We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group:
For example, (RG) and (RB) are both of the form ("x" "y"); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB). Therefore, if we apply (GB), then (RB), and then the inverse of (GB), which is also (GB), the resulting permutation is (RG).
Note that conjugate group elements always have the same order, but in general two group elements that have the same order need not be conjugate.
Subgroups.
From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.
The first-mentioned is the alternating group A3.
The left cosets and the right cosets of A3 coincide (as they do for any subgroup of index 2) and consist of A3 and the set of three swaps }.
The left cosets of are:
The right cosets of are:
Thus A3 is normal, and the other three non-trivial subgroups are not. The quotient group is isomorphic with "C"2.
formula_5, a semidirect product, where "H" is a subgroup of two elements: () and one of the three swaps. This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem.
In terms of permutations the two group elements of are the set of even permutations and the set of odd permutations.
If the original group is that generated by a 120°-rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image".
Note that for the symmetry group of a "square", an uneven permutation of vertices does "not" correspond to taking a mirror image, but to operations not allowed for "rectangles", i.e. 90° rotation and applying a diagonal axis of reflection.
Semidirect products.
formula_6 is formula_7 if both "φ"(0) and "φ"(1) are the identity.
The semidirect product is isomorphic to the dihedral group of order 6 if "φ"(0) is the identity and "φ"(1) is the non-trivial automorphism of C3, which inverses the elements.
Thus we get:
for all "n"1, "n"2 in C3 and "h"2 in C2.
More concisely,
for all "n"1, "n"2 in C3 and "h"1, "h"2 in C2.
In a Cayley table:
Note that for the second digit we essentially have a 2×2 table, with 3×3 equal values for each of these 4 cells. For the first digit the left half of the table is the same as the right half, but the top half is different from the bottom half.
For the "direct" product the table is the same except that the first digits of the bottom half of the table are the same as in the top half.
Group action.
Consider "D"3 in the geometrical way, as a symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes.
This section illustrates group action concepts for this case.
The action of "G" on "X" is called
Orbits and stabilizers.
The orbit of a point "x" in "X" is the set of elements of "X" to which "x" can be moved by the elements of "G". The orbit of "x" is denoted by "Gx":
The orbits are and The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same.
The set of all orbits of "X" under the action of "G" is written as .
If "Y" is a subset of "X", we write "GY" for the set We call the subset "Y" "invariant under G" if (which is equivalent to . In that case, "G" also operates on "Y". The subset "Y" is called "fixed under G" if for all "g" in "G" and all "y" in "Y". The union of e.g. two orbits is invariant under "G", but not fixed.
For every "x" in "X", we define the stabilizer subgroup of "x" (also called the isotropy group or little group) as the set of all elements in "G" that fix "x":
If "x" is a reflection point , its stabilizer is the group of order two containing the identity and the reflection in "x". In other cases the stabilizer is the trivial group.
For a fixed "x" in "X", consider the map from "G" to "X" given by . The image of this map is the orbit of "x" and the coimage is the set of all left cosets of "Gx". The standard quotient theorem of set theory then gives a natural bijection between and "Gx". Specifically, the bijection is given by . This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial.
If two elements "x" and "y" belong to the same orbit, then their stabilizer subgroups, "G""x" and "G""y", are isomorphic. More precisely: if "y" = "g" · "x", then "G""y" = "gG""x" "g"−1. In the example this applies e.g. for 5 and 25, both reflection points. Reflection about 25 corresponds to a rotation of 10, reflection about 5, and rotation of −10.
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
where "X""g" is the set of points fixed by "g". I.e., the number of orbits is equal to the average number of points fixed per group element.
For the identity all 30 points are fixed, for the two rotations none, and for the three reflections two each: and Thus, the average is six, the number of orbits.
Representation theory.
Up to isomorphism, this group has three irreducible complex unitary representations, which we will call formula_12 (the trivial representation), formula_13 and formula_14, where the subscript indicates the dimension. By its definition as a permutation group over the set with three elements, the group has a representation on formula_15 by permuting the entries of the vector, the fundamental representation. This representation is not irreducible, as it decomposes as a direct sum of formula_12 and formula_14. formula_12 appears as the subspace of vectors of the form formula_19 and formula_14 is the representation on its orthogonal complement, which are vectors of the form formula_21.
The nontrivial one-dimensional representation formula_13 arises through the groups formula_23 grading: The action is multiplication by the sign of the permutation of the group element. Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action. Counting the square dimensions of the representations (formula_24, the order of the group), we see these must be all of the irreducible representations.
A 2-dimensional irreducible linear representation yields a 1-dimensional projective representation (i.e., an action on the projective line, an embedding in the Möbius group ), as elliptic transforms. This can be represented by matrices with entries 0 and ±1 (here written as fractional linear transformations), known as the anharmonic group:
and thus descends to a representation over any field, which is always faithful/injective (since no two terms differ only by only a sign). Over the field with two elements, the projective line has only 3 points, and this is thus the exceptional isomorphism formula_28 In characteristic 3, this embedding stabilizes the point formula_29 since formula_30 (in characteristic greater than 3 these points are distinct and permuted, and are the orbit of the harmonic cross-ratio). Over the field with three elements, the projective line has 4 elements, and since is isomorphic to the symmetric group on 4 elements, S4, the resulting embedding formula_31 equals the stabilizer of the point formula_32.


Wigner distribution function

The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis.
The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: "Wigner quasi-probability distribution", also called the "Wigner function" or the "Wigner–Ville distribution").
Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of the uncertainty principle. The downside is the introduction of large cross terms between every pair of signal components and between positive and negative frequencies, which makes the original formulation of the function a poor fit for most analysis applications. Subsequent modifications have been proposed which preserve the sharpness of the Wigner distribution function but largely suppress cross terms.
Mathematical definition.
There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. Given the time series formula_1, its non-stationary autocorrelation function is given by
where formula_3 denotes the average over all possible realizations of the process and formula_4 is the mean, which may or may not be a function of time. The Wigner function formula_5 is then given by first expressing the autocorrelation function in terms of the average time formula_6 and time lag formula_7, and then Fourier transforming the lag.
So for a single (mean-zero) time series, the Wigner function is simply given by
The motivation for the Wigner function is that it reduces to the spectral density function at all times formula_10 for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function. Therefore, the Wigner function tells us (roughly) how the spectral density changes in time.
Time-frequency analysis example.
Here are some examples illustrating how the WDF is used in time-frequency analysis.
Constant input signal.
When the input signal is constant, its time-frequency distribution is a horizontal line along the time axis. For example, if "x"("t") = 1, then
Sinusoidal input signal.
When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency. For example, if , then
Chirp input signal.
When the input signal is a linear chirp function, the instantaneous frequency is a linear function. This means that the time frequency distribution should be a straight line. For example, if
then its instantaneous frequency is
and its WDF
Delta input signal.
When the input signal is a delta function, since it is only non-zero at t=0 and contains infinite frequency components, its time-frequency distribution should be a vertical line across the origin. This means that the time frequency distribution of the delta function should also be a delta function. By WDF
The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. For those signals, WDF can exactly generate the time frequency distribution of the input signal.
Boxcar function.
the rectangular function   ⇒
Cross term property.
The Wigner distribution function is not a linear transform. A cross term ("time beats") occurs when there is more than one component in the input signal, analogous in time to frequency beats. In the ancestral physics Wigner quasi-probability distribution, this term has important and useful physics consequences, required for faithful expectation values. By contrast, the short-time Fourier transform does not have this feature. Negative features of the WDF are reflective of the Gabor limit of the classical signal and physically unrelated to any possible underlay of quantum structure.
The following are some examples that exhibit the cross-term feature of the Wigner distribution function.
In order to reduce the cross-term difficulty, several approaches have been proposed in the literature, some of them leading to new transforms as the modified Wigner distribution function, the Gabor–Wigner transform, the Choi-Williams distribution function and Cohen's class distribution.
Properties of the Wigner distribution function.
The Wigner distribution function has several evident properties listed in the following table.


Generalized minimal residual method

In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of a nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.
The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986.
GMRES is a generalization and improvement of the MINRES method developed by Chris Paige and Michael Saunders in 1975. GMRES also is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is also applicable to non-linear systems.
The method.
Denote the Euclidean norm of any vector v by formula_1. Denote the (square) system of linear equations to be solved by
The matrix "A" is assumed to be invertible of size "m"-by-"m". Furthermore, it is assumed that b is normalized, i.e., that formula_3.
The "n"-th Krylov subspace for this problem is
where formula_5 is the initial error given an initial guess formula_6. Clearly formula_7 if formula_8.
GMRES approximates the exact solution of formula_9 by the vector formula_10 that minimizes the Euclidean norm of the residual formula_11.
The vectors formula_12 might be close to linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors formula_13 which form a basis for formula_14. In particular, formula_15.
Therefore, the vector formula_10 can be written as formula_17 with formula_18, where formula_19 is the "m"-by-"n" matrix formed by formula_20.
The Arnoldi process also produces an (formula_21)-by-formula_22 upper Hessenberg matrix formula_23 with
Because columns of formula_25 are orthonormal, we have
where
is the first vector in the standard basis of formula_28, and
formula_30 being the first trial vector (usually zero). Hence, formula_31 can be found by minimizing the Euclidean norm of the residual
This is a linear least squares problem of size "n".
This yields the GMRES method. On the formula_22-th iteration:
At every iteration, a matrix-vector product formula_38 must be computed. This costs about formula_39 floating-point operations for general dense matrices of size formula_40, but the cost can decrease to formula_41 for sparse matrices. In addition to the matrix-vector product, formula_42 floating-point operations must be computed at the "n" -th iteration.
Convergence.
The "n"th iterate minimizes the residual in the Krylov subspace formula_14. Since every subspace is contained in the next subspace, the residual does not increase. After "m" iterations, where "m" is the size of the matrix "A", the Krylov space "K""m" is the whole of R"m" and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to "m"), the vector "x""n" is already a good approximation to the
exact solution.
This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence "a"1, …, "a""m"−1, "a""m" = 0, one can find a matrix "A" such that the ||"r""n"|| = "a""n" for all "n", where "r""n" is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for "m" − 1 iterations, and only drops to zero at the last iteration.
In practice, though, GMRES often performs well. This can be proven in specific situations. If the symmetric part of "A", that is formula_44, is positive definite, then
where formula_46 and formula_47 denote the smallest and largest eigenvalue of the matrix formula_48, respectively.
If "A" is symmetric and positive definite, then we even have
where formula_50 denotes the condition number of "A" in the Euclidean norm.
In the general case, where "A" is not positive definite, we have
where "P""n" denotes the set of polynomials of degree at most "n" with "p"(0) = 1, "V" is the matrix appearing in the spectral decomposition of "A", and "σ"("A") is the spectrum of "A". Roughly speaking, this says that fast convergence occurs when the eigenvalues of "A" are clustered away from the origin and "A" is not too far from normality.
All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate "x""n" and the exact solution.
Extensions of the method.
Like other iterative methods, GMRES is usually combined with a preconditioning method in order to speed up convergence.
The cost of the iterations grow as O("n"2), where "n" is the iteration number. Therefore, the method is sometimes restarted after a number, say "k", of iterations, with "x""k" as initial guess. The resulting method is called GMRES("k") or Restarted GMRES. For non-positive definite matrices, this method may suffer from stagnation in convergence as the restarted subspace is often close to the earlier subspace.
The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR.
Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.
Comparison with other solvers.
The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.
Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed.
The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynomials of the iteration sequence suitably.
None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.
Solving the least squares problem.
One part of the GMRES method is to find the vector formula_52 which minimizes
Note that formula_23 is an ("n" + 1)-by-"n" matrix, hence it gives an over-constrained linear system of "n"+1 equations for "n" unknowns.
The minimum can be computed using a QR decomposition: find an ("n" + 1)-by-("n" + 1) orthogonal matrix Ω"n" and an ("n" + 1)-by-"n" upper triangular matrix formula_55 such that
The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as
where formula_58 is an "n"-by-"n" (thus square) triangular matrix.
The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:
where "h""n+1" = ("h"1,"n+1", …, "h""n+1,n+1")T. This implies that premultiplying the Hessenberg matrix with Ω"n", augmented with zeroes and a row with multiplicative identity, yields almost a triangular matrix:
This would be triangular if σ is zero. To remedy this, one needs the Givens rotation
where
With this Givens rotation, we form
Indeed,
is a triangular matrix.
Given the QR decomposition, the minimization problem is easily solved by noting that
Denoting the vector formula_66 by
with "g""n" ∈ R"n" and γ"n" ∈ R, this is
The vector "y" that minimizes this expression is given by
Again, the vectors formula_70 are easy to update.
Example code.
Regular GMRES (MATLAB / GNU Octave).
function [x, e] = gmres( A, b, x, max_iterations, threshold)
 n = length(A);
 m = max_iterations;
 % use x as the initial vector
 r = b - A * x;
 b_norm = norm(b);
 error = norm(r) / b_norm;
 % initialize the 1D vectors
 sn = zeros(m, 1);
 cs = zeros(m, 1);
 %e1 = zeros(n, 1);
 e1 = zeros(m+1, 1);
 e1(1) = 1;
 e = [error];
 r_norm = norm(r);
 Q(:,1) = r / r_norm;
 beta = r_norm * e1; %Note: this is not the beta scalar in section "The method" above but the beta scalar multiplied by e1
 for k = 1:m
 % run arnoldi
 [H(1:k+1, k) Q(:, k+1)] = arnoldi(A, Q, k);
 % eliminate the last element in H ith row and update the rotation matrix
 [H(1:k+1, k) cs(k) sn(k)] = apply_givens_rotation(H(1:k+1,k), cs, sn, k);
 % update the residual vector
 beta(k + 1) = -sn(k) * beta(k);
 beta(k) = cs(k) * beta(k);
 error = abs(beta(k + 1)) / b_norm;
 % save the error
 e = [e; error];
 if (error <= threshold)
 break;
 end
 end
 % if threshold is not reached, k = m at this point (and not m+1)
 % calculate the result
 y = H(1:k, 1:k) \ beta(1:k);
 x = x + Q(:, 1:k) * y;
end
% Arnoldi Function %
function [h, q] = arnoldi(A, Q, k)
 q = A*Q(:,k); % Krylov Vector
 for i = 1:k % Modified Gram-Schmidt, keeping the Hessenberg matrix
 h(i) = q' * Q(:, i);
 q = q - h(i) * Q(:, i);
 end
 h(k + 1) = norm(q);
 q = q / h(k + 1);
end
% Applying Givens Rotation to H col %
function [h, cs_k, sn_k] = apply_givens_rotation(h, cs, sn, k)
 % apply for ith column
 for i = 1:k-1
 temp = cs(i) * h(i) + sn(i) * h(i + 1);
 h(i+1) = -sn(i) * h(i) + cs(i) * h(i + 1);
 h(i) = temp;
 end
 % update the next sin cos values for rotation
 [cs_k sn_k] = givens_rotation(h(k), h(k + 1));
 % eliminate H(i + 1, i)
 h(k) = cs_k * h(k) + sn_k * h(k + 1);
 h(k + 1) = 0.0;
end
%%----Calculate the Given rotation matrix----%%
function [cs, sn] = givens_rotation(v1, v2)
% if (v1 == 0)
% cs = 0;
% sn = 1;
% else
 t = sqrt(v1^2 + v2^2);
% cs = abs(v1) / t;
% sn = cs * v2 / v1;
 cs = v1 / t; % see http://www.netlib.org/eispack/comqr.f
 sn = v2 / t;
% end
end


Jacob Lurie

Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow.
Life.
When he was a student in the Science, Mathematics, and Computer Science Magnet Program at Montgomery Blair High School, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994. In 1996 he took first place in the Westinghouse Science Talent Search and was featured in a front-page story in the "Washington Times".
Lurie earned his Bachelor's degree in mathematics from Harvard College in 2000 and was awarded in the same year the Morgan Prize for his undergraduate thesis on Lie algebras. He earned his Ph.D. from the Massachusetts Institute of Technology under supervision of Michael J. Hopkins, in 2004 with a thesis on derived algebraic geometry. In 2007, he became associate professor at MIT, and in 2009 he became professor at Harvard University. In 2019, he joined the Institute for Advanced Study as a permanent faculty member in mathematics.
Mathematical work.
Lurie's research interests started with logic and the theory of surreal numbers while he was still in high school. He is best known for his work, starting with his thesis, on infinity categories and derived algebraic geometry. Derived algebraic geometry is a way of infusing homotopical methods into algebraic geometry, with two purposes: deeper insight into algebraic geometry (e.g. into intersection theory) and the use of methods of algebraic geometry in stable homotopy theory. The latter area is the topic of Lurie's work on elliptic cohomology. Infinity categories (in the form of André Joyal's quasi-categories) are a convenient framework to do homotopy theory in abstract settings. They are the main topic of his book "Higher Topos Theory".
Another part of Lurie's work is his article on topological field theories, where he sketches a classification of extended field theories using the language of infinity categories (cobordism hypothesis). In joint work with Dennis Gaitsgory, he used his non-abelian Poincaré duality in an algebraic-geometric setting, to prove the Siegel mass formula for function fields.
Lurie was one of the inaugural winners of the Breakthrough Prize in Mathematics in 2014, "for his work on the foundations of higher category theory and derived algebraic geometry; for the classification of fully extended topological quantum field theories; and for providing a moduli-theoretic interpretation of elliptic cohomology." Lurie was also awarded a MacArthur "Genius Grant" Fellowship in 2014.


Maximal subgroup

In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroup "H" of a group "G" is a proper subgroup, such that no proper subgroup "K" contains "H" strictly. In other words, "H" is a maximal element of the partially ordered set of subgroups of "G" that are not equal to "G". Maximal subgroups are of interest because of their direct connection with primitive permutation representations of "G". They are also much studied for the purposes of finite group theory: see for example Frattini subgroup, the intersection of the maximal subgroups.
In semigroup theory, a maximal subgroup of a semigroup "S" is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of "S" which is not properly contained in another subgroup of "S". Notice that, here, there is no requirement that a maximal subgroup be proper, so if "S" is in fact a group then its unique maximal subgroup (as a semigroup) is "S" itself. Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory. There is a one-to-one correspondence between idempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is the identity element of a unique maximal subgroup.
Existence of maximal subgroup.
Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finite partially ordered set under inclusion. There are, however, infinite abelian groups that contain no maximal subgroups, for example the Prüfer group.
Maximal normal subgroup.
Similarly, a normal subgroup "N" of "G" is said to be a maximal normal subgroup (or maximal proper normal subgroup) of "G" if "N" < "G" and there is no normal subgroup "K" of "G" such that "N" < "K" < "G". We have the following theorem:
Hasse diagrams.
These Hasse diagrams show the lattices of subgroups of S4, Dih4 and Z23.<br>
The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram.


Proj construction

In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory.
In this article, all rings will be assumed to be commutative and with identity.
Proj of a graded ring.
Proj as a set.
Let formula_1 be a graded ring, whereformula_2is the direct sum decomposition associated with the gradation. The irrelevant ideal of formula_1 is the ideal of elements of positive degreeformula_4We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set,formula_5For brevity we will sometimes write formula_6 for formula_7.
Proj as a topological space.
We may define a topology, called the Zariski topology, on formula_7 by defining the closed sets to be those of the form
where formula_10 is a homogeneous ideal of formula_1. As in the case of affine schemes it is quickly verified that the formula_12 form the closed sets of a topology on formula_6.
Indeed, if formula_14 are a family of ideals, then we have formula_15 and if the indexing set "I" is finite, then formula_16.
Equivalently, we may take the open sets as a starting point and define
A common shorthand is to denote "D"("Sf") by "D"("f"), where "Sf" is the ideal generated by "f". For any ideal "a", the sets "D"("a") and "V"("a") are complementary, and hence the same proof as before shows that the sets "D"("a") form a topology on formula_7. The advantage of this approach is that the sets "D"("f"), where "f" ranges over all homogeneous elements of the ring "S", form a base for this topology, which is an indispensable tool for the analysis of formula_7, just as the analogous fact for the spectrum of a ring is likewise indispensable.
Proj as a scheme.
We also construct a sheaf on formula_7, called the “structure sheaf” as in the affine case, which makes it into a scheme. As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set formula_21 of formula_7 (which is by definition a set of homogeneous prime ideals of "formula_1" not containing formula_24) we define the ring formula_25 to be the set of all functions
(where formula_27 denotes the subring of the ring of fractions formula_28 consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal formula_29 of formula_21:
It follows immediately from the definition that the formula_25 form a sheaf of rings formula_43 on formula_7, and it may be shown that the pair (formula_7, formula_43) is in fact a scheme (this is accomplished by showing that each of the open subsets formula_47 is in fact an affine scheme).
The sheaf associated to a graded module.
The essential property of "formula_1" for the above construction was the ability to form localizations formula_27 for each prime ideal formula_29 of formula_1. This property is also possessed by any graded module formula_52 over "formula_1", and therefore with the appropriate minor modifications the preceding section constructs for any such formula_52 a sheaf, denoted formula_55, of formula_43-modules on formula_7. This sheaf is quasicoherent by construction. If "formula_1" is generated by finitely many elements of degree formula_59 (e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on formula_7 arise from graded modules by this construction. The corresponding graded module is not unique.
The twisting sheaf of Serre.
A special case of the sheaf associated to a graded module is when we take "formula_52" to be "formula_1" itself with a different grading: namely, we let the degree formula_63 elements of formula_52 be the degree formula_65 elements of "formula_1", soformula_67and denote formula_68. We then obtain formula_55 as a quasicoherent sheaf on formula_7, denoted formula_71 or simply formula_72, called the twisting sheaf of Serre. It can be checked that formula_72 is in fact an invertible sheaf.
One reason for the utility of formula_72 is that it recovers the algebraic information of "formula_1" that was lost when, in the construction of formula_43, we passed to fractions of degree zero. In the case Spec "A" for a ring "A", the global sections of the structure sheaf form "A" itself, whereas the global sections of formula_77 here form only the degree-zero elements of "formula_1". If we define
then each formula_80 contains the degree-formula_81 information about formula_1, denoted formula_83, and taken together they contain all the grading information that was lost. Likewise, for any sheaf of graded formula_77-modules formula_85 we define
and expect this “twisted” sheaf to contain grading information about "formula_85". In particular, if formula_85 is the sheaf associated to a graded formula_1-module formula_52 we likewise expect it to contain lost grading information about "formula_52". This suggests, though erroneously, that "formula_1" can in fact be reconstructed from these sheaves; asformula_93however, this is true in the case that "formula_1" is a polynomial ring, below. This situation is to be contrasted with the fact that the spec functor is adjoint to the global sections functor in the category of locally ringed spaces.
Projective "n"-space.
If "formula_95" is a ring, we define projective "n"-space over formula_95 to be the scheme
The grading on the polynomial ring formula_98 is defined by letting each formula_99 have degree one and every element of "formula_95", degree zero. Comparing this to the definition of formula_72, above, we see that the sections of formula_72 are in fact linear homogeneous polynomials, generated by the formula_103 themselves. This suggests another interpretation of formula_72, namely as the sheaf of “coordinates” for formula_7, since the formula_103 are literally the coordinates for projective formula_81-space.
Examples of Proj.
Proj over the affine line.
If we let the base ring be formula_108, thenformula_109has a canonical projective morphism to the affine line formula_110 whose fibers are elliptic curves except at the points formula_111 where the curves degenerate into nodal curves. So there is a fibrationformula_112which is also a smooth morphism of schemes (which can be checked using the Jacobian criterion).
Projective hypersurfaces and varieties.
The projective hypersurface formula_113 is an example of a Fermat quintic threefold which is also a Calabi–Yau manifold. In addition to projective hypersurfaces, any projective variety cut out by a system of homogeneous polynomialsformula_114in formula_115-variables can be converted into a projective scheme using the proj construction for the graded algebraformula_116giving an embedding of projective varieties into projective schemes.
Weighted projective space.
Weighted projective spaces can be constructed using a polynomial ring whose variables have non-standard degrees. For example, the weighted projective space formula_117 corresponds to taking formula_118 of the ring formula_119 where formula_120 have weight formula_59 while formula_122 has weight 2.
Bigraded rings.
The proj construction extends to bigraded and multigraded rings. Geometrically, this corresponds to taking products of projective schemes. For example, given the graded ringsformula_123with the degree of each generator formula_59. Then, the tensor product of these algebras over formula_125 gives the bigraded algebraformula_126where the formula_127 have weight formula_128 and the formula_129 have weight formula_130. Then the proj construction givesformula_131which is a product of projective schemes. There is an embedding of such schemes into projective space by taking the total graded algebraformula_132where a degree formula_133 element is considered as a degree formula_134 element. This means the formula_135-th graded piece of formula_136 is the moduleformula_137In addition, the scheme formula_138 now comes with bigraded sheaves formula_139 which are the tensor product of the sheaves formula_140 whereformula_141and formula_142are the canonical projections coming from the injections of these algebras from the tensor product diagram of commutative algebras.
Global Proj.
A generalization of the Proj construction replaces the ring "S" with a sheaf of algebras and produces, as the end result, a scheme which might be thought of as a fibration of Proj's of rings. This construction is often used, for example, to construct projective space bundles over a base scheme.
Assumptions.
Formally, let "X" be any scheme and "S" be a sheaf of graded formula_43-algebras (the definition of which is similar to the definition of formula_43-modules on a locally ringed space): that is, a sheaf with a direct sum decomposition
where each formula_146 is an formula_43-module such that for every open subset "U" of "X", "S"("U") is an formula_25-algebra and the resulting direct sum decomposition
is a grading of this algebra as a ring. Here we assume that formula_150. We make the additional assumption that "S" is a quasi-coherent sheaf; this is a “consistency” assumption on the sections over different open sets that is necessary for the construction to proceed.
Construction.
In this setup we may construct a scheme formula_151 and a “projection” map "p" onto "X" such that for every open affine "U" of "X",
This definition suggests that we construct formula_151 by first defining schemes formula_154 for each open affine "U", by setting
and maps formula_156, and then showing that these data can be glued together “over” each intersection of two open affines "U" and "V" to form a scheme "Y" which we define to be formula_151. It is not hard to show that defining each formula_158 to be the map corresponding to the inclusion of formula_25 into "S"("U") as the elements of degree zero yields the necessary consistency of the formula_158, while the consistency of the formula_154 themselves follows from the quasi-coherence assumption on "S".
The twisting sheaf.
If "S" has the additional property that formula_162 is a coherent sheaf and locally generates "S" over formula_163 (that is, when we pass to the stalk of the sheaf "S" at a point "x" of "X", which is a graded algebra whose degree-zero elements form the ring formula_164 then the degree-one elements form a finitely-generated module over formula_164 and also generate the stalk as an algebra over it) then we may make a further construction. Over each open affine "U", Proj "S"("U") bears an invertible sheaf O(1), and the assumption we have just made ensures that these sheaves may be glued just like the formula_154 above; the resulting sheaf on formula_151 is also denoted "O"(1) and serves much the same purpose for formula_151 as the twisting sheaf on the Proj of a ring does.
Proj of a quasi-coherent sheaf.
Let formula_169 be a quasi-coherent sheaf on a scheme formula_6. The sheaf of symmetric algebras formula_171 is naturally a quasi-coherent sheaf of graded formula_43-modules, generated by elements of degree 1. The resulting scheme is denoted by formula_173. If formula_169 is of finite type, then its canonical morphism formula_175 is a projective morphism.
For any formula_176, the fiber of the above morphism over formula_177 is the projective space formula_178 associated to the dual of the vector space formula_179 over formula_180.
If formula_181 is a quasi-coherent sheaf of graded formula_43-modules, generated by formula_183 and such that formula_183 is of finite type, then formula_185 is a closed subscheme of formula_186 and is then projective over formula_6. In fact, every closed subscheme of a projective formula_173 is of this form.
Projective space bundles.
As a special case, when formula_169 is locally free of rank formula_190, we get a projective bundle formula_173 over formula_6 of relative dimension formula_81. Indeed, if we take an open cover of "X" by open affines formula_194 such that when restricted to each of these, formula_169 is free over "A", then
and hence formula_173 is a projective space bundle. Many families of varieties can be constructed as subschemes of these projective bundles, such as the Weierstrass family of elliptic curves. For more details, see the main article.
Example of Global Proj.
Global proj can be used to construct Lefschetz pencils. For example, let formula_198 and take homogeneous polynomials formula_199 of degree k. We can consider the ideal sheaf formula_200 of formula_201 and construct global proj of this quotient sheaf of algebras formula_202. This can be described explicitly as the projective morphism formula_203.


Quadratically closed field

In mathematics, a quadratically closed field is a field in which every element has a square root.
Quadratic closure.
A quadratic closure of a field "F" is a quadratically closed field containing "F" which embeds in any quadratically closed field containing "F". A quadratic closure for any given "F" may be constructed as a subfield of the algebraic closure "F"alg of "F", as the union of all iterated quadratic extensions of "F" in "F"alg.


Local criterion for flatness

In algebra, the local criterion for flatness gives conditions one can check to show flatness of a module.
Statement.
Given a commutative ring "A", an ideal "I" and an "A"-module "M", suppose either
or
Then the following are equivalent:
The assumption that “"A" is a Noetherian ring” is used to invoke the Artin–Rees lemma and can be weakened; see
Proof.
Following SGA 1, Exposé IV, we first prove a few lemmas, which are interesting themselves. (See also this blog post by Akhil Mathew for a proof of a special case.)
"Proof": The equivalence of the first two can be seen by studying the Tor spectral sequence. Here is a direct proof: if 1. is valid and formula_3 is an injection of formula_4-modules with cokernel "C", then, as "A"-modules,
Since formula_6 and the same for formula_7, this proves 2. Conversely, considering formula_8 where "F" is "B"-free, we get:
Here, the last map is injective by flatness and that gives us 1. To see the "Moreover" part, if 1. is valid, then formula_10 and so
By descending induction, this implies 3. The converse is trivial. formula_12
"Proof": The assumption implies that formula_13 and so, since tensor product commutes with base extension,
For the second part, let formula_15 denote the exact sequence formula_16 and formula_17. Consider the exact sequence of complexes:
Then formula_19 (it is so for large formula_20 and then use descending induction). 3. of Lemma 1 then implies that formula_21 is flat. formula_12
"Proof of the main statement".
formula_23: If formula_24 is nilpotent, then, by Lemma 1, formula_25 and formula_21 is flat over formula_27. Thus, assume that the first assumption is valid. Let formula_28 be an ideal and we shall show formula_29 is injective. For an integer formula_30, consider the exact sequence
Since formula_32 by Lemma 1 (note formula_33 kills formula_34), tensoring the above with formula_21, we get:
Tensoring formula_21 with formula_38, we also have:
We combine the two to get the exact sequence:
Now, if formula_41 is in the kernel of formula_29, then, a fortiori, formula_41 is in formula_44. By the Artin–Rees lemma, given formula_45, we can find formula_30 such that formula_47. Since formula_48, we conclude formula_49.
formula_50 follows from Lemma 2.
formula_51: Since formula_52, the condition 4. is still valid with formula_53 replaced by formula_54. Then Lemma 2 says that formula_55 is flat over formula_56.
formula_57 Tensoring formula_58 with "M", we see formula_59 is the kernel of formula_60. Thus, the implication is established by an argument similar to that of formula_23formula_12
Application: characterization of an étale morphism.
The local criterion can be used to prove the following:
"Proof": Assume that formula_63 is an isomorphism and we show "f" is étale. First, since formula_64 is faithfully flat (in particular is a pure subring), we have:
Hence, formula_66 is unramified (separability is trivial). Now, that formula_67 is flat follows from (1) the assumption that the induced map on completion is flat and (2) the fact that flatness descends under faithfully flat base change (it shouldn’t be hard to make sense of (2)).
Next, we show the converse: by the local criterion, for each "n", the natural map formula_68
is an isomorphism. By induction and the five lemma, this implies formula_69 is an isomorphism for each "n". Passing to limit, we get the asserted isomorphism. formula_12
Mumford’s Red Book gives an extrinsic proof of the above fact (Ch. III, § 5, Theorem 3).
Miracle flatness theorem.
B. Conrad calls the next theorem "the miracle flatness theorem".


Polynomial identity ring

In mathematics, in the subfield of ring theory, a ring "R" is a polynomial identity ring if there is, for some "N" > 0, an element "P" other than 0 of the free algebra, Z, over the ring of integers in "N" variables "X"1, "X"2, ..., "X""N" such that for all "N"-tuples "r"1, "r"2, ..., "r""N" taken from "R" it happens that
Strictly the "X""i" here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring "S" may be used, and gives the concept of PI-algebra.
If the degree of the polynomial "P" is defined in the usual way, the polynomial "P" is called monic if at least one of its terms of highest degree has coefficient equal to 1.
Every commutative ring is a PI-ring, satisfying the polynomial identity "XY" - "YX" = 0. Therefore, PI-rings are usually taken as "close generalizations of commutative rings". If the ring has characteristic "p" different from zero then it satisfies the polynomial identity "pX" = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
PI-rings as generalizations of commutative rings.
Among noncommutative rings, PI-rings satisfy the [[Köthe conjecture]]. [[Finitely generated algebra|Affine]] PI-algebras over a [[field (mathematics)|field]] satisfy the [[Kurosh conjecture]], the [[Nullstellensatz]] and the [[catenary ring|catenary property]] for [[prime ideals]].
If "R" is a PI-ring and "K" is a subring of its center such that "R" is [[integral extension|integral over]] "K" then the [[Going up and going down|going up and going down properties]] for prime ideals of "R" and "K" are satisfied. Also the "lying over" property (If "p" is a prime ideal of "K" then there is a prime ideal "P" of "R" such that formula_7 is minimal over formula_8) and the "incomparability" property (If "P" and "Q" are prime ideals of "R" and formula_9 then formula_10) are satisfied.
The set of identities a PI-ring satisfies.
If "F" := Z is the free algebra in "N" variables and "R" is a PI-ring satisfying the polynomial "P" in "N" variables, then "P" is in the [[Kernel (algebra)|kernel]] of any homomorphism
An ideal "I" of "F" is called T-ideal if formula_13 for every [[endomorphism]] "f" of "F".
Given a PI-ring, "R", the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|ideal]] but even more it is a T-ideal. Conversely, if "I" is a T-ideal of "F" then "F"/"I" is a PI-ring satisfying all identities in "I". It is assumed that "I" contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.
External links.
[[Category:Ring theory]]


Thin group (combinatorial group theory)

In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is not thin.
Given any generating set of the group, we can consider a graph whose vertices are elements of the group with two vertices adjacent if their ratio is in the generating set. The graph is connected and vertex transitive. paths in the graph correspond to words in the generators.
If the graph has a cycle of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity. A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity.
If the graph has no cycles, its girth is set to be infinity.
The girth depends on the choice of generating set. A "thin" group is a group where the girth has an upper bound for all finite generating sets.
Some facts about thin and fat groups and about girths:


Lamb vector

In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb. The Lamb vector is defined as
where formula_2 is the velocity field and formula_3 is the vorticity field of the flow. It appears in the Navier–Stokes equations through the material derivative term, specifically via convective acceleration term,
In irrotational flows, the Lamb vector is zero, so does in Beltrami flows. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to electric field, when the Navier–Stokes equation is compared with Maxwell's equations.
Properties of Lamb vector.
The divergence of the lamb vector can be derived from vector identities,
At the same time, the divergence can also be obtained from Navier–Stokes equation by taking its divergence. In particular, for incompressible flow, where formula_6, with body forces given by formula_7, the Lamb vector divergence reduces to
where
In regions where formula_10, there is tendency for formula_11 to accumulate there and vice versa.


Tower of objects

In category theory, a branch of abstract mathematics, a tower is defined as follows. Let formula_1 be the poset
of whole numbers in reverse order, regarded as a category. A (countable) tower of objects in a category formula_3 is a functor from formula_1 to formula_3.
In other words, a tower (of formula_3) is a family of objects formula_7 in formula_3 where there exists a map
and the composition
is the map formula_12
Example.
Let formula_13 for some formula_14-module formula_15. Let formula_16 be the identity map for formula_10. Then formula_18 forms a tower of modules.


Branched covering

In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology.
In topology, a map is a "branched covering" if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle.
In algebraic geometry.
In algebraic geometry, the term branched covering is used to describe morphisms formula_1 from an algebraic variety formula_2 to another one formula_3, the two dimensions being the same, and the typical fibre of formula_1 being of dimension 0.
In that case, there will be an open set formula_5 of formula_3 (for the Zariski topology) that is dense in formula_3, such that the restriction of formula_1 to formula_5 (from formula_10 to formula_5, that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability). Generically, then, such a morphism resembles a covering space in the topological sense. For example, if formula_2 and formula_3 are both Riemann surfaces, we require only that formula_1 is holomorphic and not constant, and then there is a finite set of points formula_15 of formula_3, outside of which we do find an honest covering
Ramification locus.
The set of exceptional points on formula_3 is called the ramification locus (i.e. this is the complement of the largest possible open set formula_5). In general monodromy occurs according to the fundamental group of formula_5 acting on the sheets of the covering (this topological picture can be made precise also in the case of a general base field).
Kummer extensions.
Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic examples.
Unramified covering.
An unramified covering then is the occurrence of an empty ramification locus.
Examples.
Elliptic curve.
Morphisms of curves provide many examples of ramified coverings. For example, let be the elliptic curve of equation
The projection of onto the -axis is a ramified cover with ramification locus given by
This is because for these three values of the fiber is the double point formula_23 while for any other value of , the fiber consists of two distinct points (over an algebraically closed field).
This projection induces an algebraic extension of degree two of the function fields:
Also, if we take the fraction fields of the underlying commutative rings, we get the morphism
Hence this projection is a degree 2 branched covering. This can be homogenized to construct a degree 2 branched covering of the corresponding projective elliptic curve to the projective line.
Plane algebraic curve.
The previous example may be generalized to any algebraic plane curve in the following way.
Let be a plane curve defined by the equation , where is a separable and irreducible polynomial in two indeterminates. If is the degree of in , then the fiber consists of distinct points, except for a finite number of values of . Thus, this projection is a branched covering of degree .
The exceptional values of are the roots of the coefficient of formula_25 in , and the roots of the discriminant of with respect to .
Over a root of the discriminant, there is at least a ramified point, which is either a critical point or a singular point. If is also a root of the coefficient of formula_25 in , then this ramified point is "at infinity".
Over a root of the coefficient of formula_25 in , the curve has an infinite branch, and the fiber at has less than points. However, if one extends the projection to the projective completions of and the -axis, and if is not a root of the discriminant, the projection becomes a covering over a neighbourhood of .
The fact that this projection is a branched covering of degree may also be seen by considering the function fields. In fact, this projection corresponds to the field extension of degree
Varying Ramifications.
We can also generalize branched coverings of the line with varying ramification degrees. Consider a polynomial of the form
as we choose different points formula_30, the fibers given by the vanishing locus of formula_31 vary. At any point where the multiplicity of one of the linear terms in the factorization of formula_31 increases by one, there is a ramification.
Scheme Theoretic Examples.
Elliptic Curves.
Morphisms of curves provide many examples of ramified coverings of schemes. For example, the morphism from an affine elliptic curve to a line
is a ramified cover with ramification locus given by
This is because at any point of formula_35 in formula_36 the fiber is the scheme
Also, if we take the fraction fields of the underlying commutative rings, we get the field homomorphism
which is an algebraic extension of degree two;
hence we got a degree 2 branched covering of an elliptic curve to the affine line. This can be homogenized to construct a morphism of a projective elliptic curve to formula_39.
Hyperelliptic curve.
A hyperelliptic curve provides a generalization of the above degree formula_40 cover of the affine line, by considering the affine scheme defined over formula_41 by a polynomial of the form
Higher Degree Coverings of the Affine Line.
We can generalize the previous example by taking the morphism
where formula_46 has no repeated roots. Then the ramification locus is given by
where the fibers are given by
Then, we get an induced morphism of fraction fields
There is an formula_50-module isomorphism of the target with
Hence the cover is of degree formula_52.
Superelliptic Curves.
Superelliptic curves are a generalization of hyperelliptic curves and a specialization of the previous family of examples since they are given by affine schemes formula_53 from polynomials of the form
Ramified Coverings of Projective Space.
Another useful class of examples come from ramified coverings of projective space. Given a homogeneous polynomial formula_57 we can construct a ramified covering of formula_58 with ramification locus
by considering the morphism of projective schemes
Again, this will be a covering of degree formula_1.
Applications.
Branched coverings formula_62 come with a symmetry group of transformations formula_63. Since the symmetry group has stabilizers at the points of the ramification locus, branched coverings can be used to construct examples of orbifolds, or Deligne–Mumford stacks.


Pure submodule

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences, each defined by a direct summand.
Definition.
Let "R" be a ring (associative, with 1), and let "M" and "P" be modules over "R". If "i": "P" → "M" is injective then "P" is a pure submodule of "M" if, for any "R"-module "X", the natural induced map on tensor products "i" ⊗ id"X" : "P" ⊗ "X" → "M" ⊗ "X" is injective.
Analogously, a short exact sequence
of "R"-modules is pure exact if the sequence stays exact when tensored with any "R"-module "X". This is equivalent to saying that "f"("A") is a pure submodule of "B".
Purity can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, "P" is pure in "M" if and only if the following condition holds: for any "m"-by-"n" matrix ("a""ij") with entries in "R", and any set "y"1, ..., "y""m" of elements of "P", if there exist elements "x"1, ..., "x""n" in "M" such that
then there also exist elements "x"1′, ..., "x""n"′ in "P" such that
Examples.
is a short exact sequence of "R"-modules, then:
Equivalent characterization.
A sequence is pure exact if and only if it is the filtered colimit (also known as direct limit) of split exact sequences


Continuous big q-Hermite polynomials

In mathematics, the continuous big "q"-Hermite 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.
formula_1


Strong monomorphism


Suzuki groups

In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22"n"+1), 2"B"2(22"n"+1), Suz(22"n"+1), or "G"(22"n"+1), form an infinite family of groups of Lie type found by , that are simple for "n" ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.
Constructions.
Suzuki.
 originally constructed the Suzuki groups as subgroups of SL4(F22"n"+1) generated by certain explicit matrices.
Ree.
Ree observed that the Suzuki groups were the fixed points of exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families of simple groups, called the Ree groups. In the lowest case the symplectic group B2(2)≈S6; its exceptional automorphism fixes the subgroup Sz(2) or 2B2(2), of order 20.
 gave a detailed exposition of Ree's observation.
Tits.
 constructed the Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective space over a field of characteristic 2.
Wilson.
 constructed the Suzuki groups as the subgroup of the symplectic group in 4 dimensions preserving a certain product on pairs of orthogonal vectors.
Properties.
Let q = 22n+1, r = 2n, n a non-negative integer.
The Suzuki groups Sz(q) or 2"B"2(q) are simple for "n"≥1. The group Sz(2) is solvable and is the Frobenius group of order 20.
The Suzuki groups Sz(q) have orders "q"2("q"2+1)("q"−1). These groups have orders divisible by 5, not by 3.
The Schur multiplier is trivial for "n">1, Klein 4-group for "n"=1, i. e. Sz(8).
The outer automorphism group is cyclic of order 2"n"+1, given by automorphisms of the field of order "q".
Suzuki group are Zassenhaus groups acting on sets of size (22"n"+1)2+1, and have 4-dimensional representations over the field with 22"n"+1 elements.
Suzuki groups are CN-groups: the centralizer of every non-trivial element is nilpotent.
Subgroups.
When n is a positive integer. Sz(q) has at least 4 types of maximal subgroups.
The diagonal subgroup is cyclic, of order q - 1.
Either q+2r+1 or q-2r+1 is divisible by 5, so that Sz(q) contains the Frobenius group C5:4.
Conjugacy classes.
 showed that the Suzuki group has "q"+3 conjugacy classes. Of these "q"+1 are strongly real, and the other two are classes of elements of order 4.
The normalizers of all these subgroups are Frobenius groups.
Characters.
 showed that the Suzuki group has "q"+3 irreducible representations over the complex numbers, 2 of which are complex and the rest of which are real. They are given as follows:


Representation up to homotopy

A Representation up to homotopy has several meanings. One of the earliest appeared in the `physical' context of constrained Hamiltonian systems. The essential idea is lifting a non-representation on a quotient to a representation up to strong homotopy on a resolution of the quotient.
As a concept in differential geometry, it generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. As such, it was introduced by Abad and Crainic.
As a motivation consider a regular Lie algebroid ("A","ρ",[..]) (regular meaning that the anchor "ρ" has constant rank) where we have two natural "A"-connections on "g"("A") = ker "ρ" and "ν"("A")= "TM"/im "ρ" respectively:
In the deformation theory of the Lie algebroid "A" there is a long exact sequence
This suggests that the correct cohomology for the deformations (here denoted as "H"def) comes from the direct sum of the two modules "g"("A") and "ν"("A") and should be called adjoint representation. Note however that in the more general case where "ρ" does not have constant rank we cannot easily define the representations "g"("A") and "ν"("A"). Instead we should consider the 2-term complex "A"→"TM" and a representation on it. This leads to the notion explained here.
Definition.
Let ("A","ρ",[..]) be a Lie algebroid over a smooth manifold "M" and let Ω("A") denote its Lie algebroid complex. Let further "E" be a ℤ-graded vector bundle over "M" and Ω("A","E") = Ω("A") ⊗ Γ("E") be its ℤ-graded "A"-cochains with values in "E". A representation up to homotopy of "A" on "E" is a differential operator "D" that maps
fulfills the Leibniz rule
and squares to zero, i.e. "D"2 = 0.
Homotopy operators.
A representation up to homotopy as introduced above is equivalent to the following data
The correspondence is characterized as
Homomorphisms.
A homomorphism between representations up to homotopy ("E","DE") and ("F","DF") of the same Lie algebroid "A" is a degree 0 map Φ:Ω("A","E") → Ω("A","F") that commutes with the differentials, i.e.
An isomorphism is now an invertible homomorphism.
We denote Rep∞ the category of equivalence classes of representations up to homotopy together with equivalence classes of homomorphisms.
In the sense of the above decomposition of "D" into a cochain map ∂, a connection ∇, and higher homotopies, we can also decompose the Φ as Φ0 + Φ1 + … with
and then the compatibility condition reads
Examples.
Examples are usual representations of Lie algebroids or more specifically Lie algebras, i.e. modules.
Another example is given by a "p"-form "ω""p" together with "E" = M × ℝ[0] ⊕ ℝ["p"] and the operator "D" = ∇ + "ω""p" where ∇ is the flat connection on the trivial bundle "M" × ℝ.
Given a representation up to homotopy as "D" = ∂ + ∇ + "ω"2 + … we can construct a new representation up to homotopy by conjugation, i.e.
Adjoint representation.
Given a Lie algebroid ("A","ρ",[..]) together with a connection ∇ on its vector bundle we can define two associated "A"-connections as follows
Moreover, we can introduce the mixed curvature as
This curvature measures the compatibility of the Lie bracket with the connection and is one of the two conditions of "A" together with "TM" forming a matched pair of Lie algebroids.
The first observation is that this term decorated with the anchor map "ρ", accordingly, expresses the curvature of both connections ∇bas. Secondly we can match up all three ingredients to a representation up to homotopy as:
Another observation is that the resulting representation up to homotopy is independent of the chosen connection ∇, basically because the difference between two "A"-connections is an ("A" − 1 -form with values in End("E").


Luc Illusie

Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.
Biography.
Luc Illusie entered the École Normale Supérieure in 1959. At first a student of the mathematician Henri Cartan, he participated in the Cartan–Schwartz seminar of 1963–1964. In 1964, following Cartan’s advice, he began to work with Alexandre Grothendieck, collaborating with him on two volumes of the latter’s Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex.
A researcher in the Centre national de la recherche scientifique from 1964 to 1976, Illusie then became a professor at the University of Paris-Sud, retiring as emeritus professor in 2005. Between 1984 and 1995, he was the director of the arithmetic and algebraic geometry group in the department of mathematics of that university. and Gérard Laumon are among his students.
Thesis.
In May 1971, Illusie defended a state doctorate ( Thèse d’État) entitled "Cotangent complex; application to the theory of deformations" at the University of Paris-Sud, in front of a jury composed of
Alexander Grothendieck, Michel Demazure and Jean-Pierre Serre and presided by Henri Cartan.
The thesis was published in French by Springer-Verlag as a two-volume book (in 1971 & 1972). The main results of the thesis are summarized in a paper in English (entitled "Cotangent complex and Deformations of torsors and group schemes") presented in Halifax, at Dalhousie University, on January 1971 as part of a colloquium on algebraic geometry. This paper, originally published by Springer-Verlag in 1972, also exists in a slightly extended version.
Illusie's construction of the cotangent complex generalizes that of Michel André and Daniel Quillen to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to
various first-order deformation problems: schemes, morphisms of schemes,
group schemes and torsors under group schemes. Results concerning commutative
group schemes in particular were the key tool in Grothendieck's proof of his
existence and structure theorem for infinitesimal deformations of Barsotti–Tate groups, an ingredient in Gerd Faltings' proof of the Mordell conjecture. In Chapter VIII of the second volume of the thesis, Illusie introduces
and studies derived de Rham complexes.
Awards.
Illusie has received the Langevin Prize of the French Academy of Sciences in 1977 and, in 2012, the Émile Picard Medal of the French Academy of Sciences for "his fundamental work on the cotangent complex, the Picard–Lefschetz formula, Hodge theory and logarithmic geometry".


Wagner congruence


Appell sequence

In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence formula_1 satisfying the identity
and in which formula_3 is a non-zero constant.
Among the most notable Appell sequences besides the trivial example formula_4 are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
Equivalent characterizations of Appell sequences.
The following conditions on polynomial sequences can easily be seen to be equivalent:
Recursion formula.
Suppose
where the last equality is taken to define the linear operator formula_16 on the space of polynomials in formula_17. Let
be the inverse operator, the coefficients formula_19 being those of the usual reciprocal of a formal power series, so that
In the conventions of the umbral calculus, one often treats this formal power series formula_21 as representing the Appell sequence formula_22. One can define
by using the usual power series expansion of the formula_24 and the usual definition of composition of formal power series. Then we have
In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.
Subgroup of the Sheffer polynomials.
The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose formula_27 and formula_28 are polynomial sequences, given by
Then the umbral composition formula_30 is the polynomial sequence whose formula_31th term is
(the subscript formula_31 appears in formula_22, since this is the formula_31th term of that sequence, but not in formula_36, since this refers to the sequence as a whole rather than one of its terms).
Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form
and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator formula_26.
Different convention.
Another convention followed by some authors (see "Chihara") defines this concept in a different way, conflicting with Appell's original definition, by using the identity
instead.


Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory formula_1 of a category formula_2 is said to be isomorphism closed or replete if every formula_2-isomorphism formula_4 with formula_5 belongs to formula_6 This implies that both formula_7 and formula_8 belong to formula_1 as well.
A subcategory that is isomorphism closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every formula_2-object that is isomorphic to an formula_1-object is also an formula_1-object.
This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties. Every topological property corresponds to a strictly full subcategory of formula_13


Pincherle polynomials

In mathematics, the Pincherle polynomials P"n"("x") are polynomials introduced by given by the generating function
Humbert polynomials are a generalization of Pincherle polynomials


Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group ("G", ∗) in which there exists at least one pair of elements "a" and "b" of "G", such that "a" ∗ "b" ≠ "b" ∗ "a". This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute).
Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them the other way round).
Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.


Codensity monad

In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors.
Definition.
The codensity monad of a functor formula_1 is defined to be the right Kan extension of "G" along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor
The codensity monad exists whenever "D" is a small category (has only a set, as opposed to a proper class, of morphisms) and "C" possesses all (small, i.e., set-indexed) limits. It also exists whenever "G" has a left adjoint.
By the general formula computing right Kan extensions in terms of ends, the codensity monad is given by the following formula:
where formula_5 denotes the set of morphisms in "C" between the indicated objects and the integral denotes the end. The codensity monad therefore amounts to considering maps from "c" to an object in the image of "G", and maps from the set of such morphisms to "G"("d"), compatible for all the possible "d". Thus, as is noted by Avery, codensity monads share some kinship with the concept of integration and double dualization.
Examples.
Codensity monads of right adjoints.
If the functor "G" admits a left adjoint "F", the codensity monad is given by the composite formula_6, together with the standard unit and multiplication maps.
Concrete examples for functors not admitting a left adjoint.
In several interesting cases, the functor "G" is an inclusion of a full subcategory not admitting a left adjoint. For example, the codensity monad of the inclusion of FinSet into Set is the ultrafilter monad associating to any set "M" the set of ultrafilters on "M". This was proven by Kennison and Gildenhuys, though without using the term "codensity". In this formulation, the statement is reviewed by Leinster.
A related example is discussed by Leinster: the codensity monad of the inclusion of finite-dimensional vector spaces (over a fixed field "k") into all vector spaces is the double dualization monad given by sending a vector space "V" to its double dual
Thus, in this example, the end formula mentioned above simplifies to considering (in the notation above) only one object "d", namely a one-dimensional vector space, as opposed to considering all objects in "D". Adámek show that, in a number of situations, the codensity monad of the inclusion
of finitely presented objects (also known as compact objects) is a double dualization monad with respect to a sufficiently nice cogenerating object. This recovers both the inclusion of finite sets in sets (where a cogenerator is the set of two elements), and also the inclusion of finite-dimensional vector spaces in vector spaces (where the cogenerator is the ground field).
Sipoş showed that the algebras over the codensity monad of the inclusion of finite sets (regarded as discrete topological spaces) into topological spaces are equivalent to Stone spaces.
Avery shows that the Giry monad arises as the codensity monad of natural forgetful functors between certain categories of convex vector spaces to measurable spaces.
Relation to Isbell duality.
Di Liberti shows that the codensity monad is closely related to Isbell duality: for a given small category "C", Isbell duality refers to the adjunction
between the category of presheaves on "C" (i.e., functors from the opposite category of "C" to sets) and the opposite category of copresheaves on "C". The monad
induced by this adjunction is shown to be the codensity monad of the Yoneda embedding
Conversely, the codensity monad of a full small dense subcategory "K" in a cocomplete category "C" is shown to be induced by Isbell duality.
References.
Footnotes


Geronimus polynomials

In mathematics, Geronimus polynomials may refer to one of the several different families of orthogonal polynomials studied by Yakov Lazarevich Geronimus.