source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Proofs%20of%20quadratic%20reciprocity
|
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published.
Proof synopsis
Of the elementary combinatorial proofs, there are two which apply types of double counting. One by Gotthold Eisenstein counts lattice points. Another applies Zolotarev's lemma to , expressed by the Chinese remainder theorem as and calculates the signature of a permutation. The shortest known proof also uses a simplified version of double counting, namely double counting modulo a fixed prime.
Eisenstein's proof
Eisenstein's proof of quadratic reciprocity is a simplification of Gauss's third proof. It is more geometrically intuitive and requires less technical manipulation.
The point of departure is "Eisenstein's lemma", which states that for distinct odd primes p, q,
where denotes the floor function (the largest integer less than or equal to x), and where the sum is taken over the even integers u = 2, 4, 6, ..., p−1. For example,
This result is very similar to Gauss's lemma, and can be proved in a similar fashion (proof given below).
Using this representation of (q/p), the main argument is quite elegant. The sum counts the number of lattice points with even x-coordinate in the interior of the triangle ABC in the following diagram:
Because each column has an even number of points (namely q−1 points), the number of such lattice points in the region BCYX is the same modulo 2 as the number of such points in the region CZY:
Then by flipping the diagram in both axes, we see that the number of points with even x-coordinate inside CZY is the same as the number of points inside AXY having odd x-coordinates. This can be justified mathematically by noting that .
The conclusion is that
where μ is the total number of lattice points in the interior of AXY.
Switching p and q, the same argument shows that
where ν is the number of lattice points in the interior of WYA. Since there are no lattice points on the line AY itself (because p and q are relatively prime), and since the total number of points in the rectangle WYXA is
we obtain
Proof of Eisenstein's lemma
For an even integer u in the range 1 ≤ u ≤ p−1, denote by r(u) the least positive residue of qu modulo p. (For example, for p = 11, q = 7, we allow u = 2, 4, 6, 8, 10, and the corresponding values of r(u) are 3, 6, 9, 1, 4.) The numbers (−1)r(u)r(u), again treated as least positive residues modulo p, are all even (in our running example, they are 8, 6, 2, 10, 4.) Furthermore, they are all distinct, because if (−1)r(u)r(u) ≡ (−1)r(t)r(t) (mod p), then we may divide out by q to obtain u ≡ ±t (mod p). This forces u ≡ t (mod p), because both u and t are even, whereas p is odd. Since there exactly (p−1)/2 of them and they are distinct, they must be simply a rearrangement of the even integers 2, 4, ..., p−1. Multiplying them together, we obtain
Dividing ou
|
https://en.wikipedia.org/wiki/Logical%20equality
|
Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value true if both functional arguments have the same logical value, and false if they are different.
It is customary practice in various applications, if not always technically precise, to indicate the operation of logical equality on the logical operands x and y by any of the following forms:
Some logicians, however, draw a firm distinction between a functional form, like those in the left column, which they interpret as an application of a function to a pair of arguments — and thus a mere indication that the value of the compound expression depends on the values of the component expressions — and an equational form, like those in the right column, which they interpret as an assertion that the arguments have equal values, in other words, that the functional value of the compound expression is true.
Definition
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The truth table of p EQ q (also written as p = q, p ↔ q, Epq, p ≡ q, or p == q) is as follows:
{| class="wikitable" style="text-align:center"
|+ Logical equality
! p
! q
! p = q
|-
| 0 || 0 || 1
|-
| 0 || 1 || 0
|-
| 1 || 0 || 0
|-
| 1 || 1 || 1
|}
Alternative descriptions
The form (x = y) is equivalent to the form (x ∧ y) ∨ (¬x ∧ ¬y).
For the operands x and y, the truth table of the logical equality operator is as follows:
{| class="wikitable"
! colspan="2" rowspan="2" | !! colspan="2" | y
|-
! T !! F
|-
! rowspan="2" | x !! T
| style="padding: 1em;" | T
| style="padding: 1em;" | F
|-
! F
| style="padding: 1em;" | F
| style="padding: 1em;" | T
|}
Inequality
In mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of algebraic structure that is known as a field. For boolean algebra, this means that the logical operation signified by "+" is not the same as the inclusive disjunction signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the exclusive disjunction signified by "XOR" or "⊕". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality:
This explains why "EQ" is often called "XNOR" in the combinational logic of circuit engineers, since it is the negation of the XOR operation; "NXOR" is a less commonly used alternative. Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".
See also
Boolean function
|
https://en.wikipedia.org/wiki/Intermediate%20Jacobian
|
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus for n odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if M is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations.
A complex structure on a real vector space is given by an automorphism I with square . The complex structures on are defined using the Hodge decomposition
On the Weil complex structure is multiplication by , while the Griffiths complex structure is multiplication by if and if . Both these complex structures map into itself and so defined complex structures on it.
For the intermediate Jacobian is the Picard variety, and for it is the Albanese variety. In these two extreme cases the constructions of Weil and Griffiths are equivalent.
used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational.
See also
Deligne cohomology
References
Hodge theory
|
https://en.wikipedia.org/wiki/Degree%20of%20a%20continuous%20mapping
|
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.
The degree of a map was first defined by Brouwer, who showed that the degree is homotopy invariant (invariant among homotopies), and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number.
Definitions of the degree
From Sn to Sn
The simplest and most important case is the degree of a continuous map from the -sphere to itself (in the case , this is called the winding number):
Let be a continuous map. Then induces a homomorphism , where is the th homology group. Considering the fact that , we see that must be of the form for some fixed .
This is then called the degree of .
Between manifolds
Algebraic topology
Let X and Y be closed connected oriented m-dimensional manifolds. Orientability of a manifold implies that its top homology group is isomorphic to Z. Choosing an orientation means choosing a generator of the top homology group.
A continuous map f : X →Y induces a homomorphism f∗ from Hm(X) to Hm(Y). Let [X], resp. [Y] be the chosen generator of Hm(X), resp. Hm(Y) (or the fundamental class of X, Y). Then the degree of f is defined to be f*([X]). In other words,
If y in Y and f −1(y) is a finite set, the degree of f can be computed by considering the m-th local homology groups of X at each point in f −1(y).
Differential topology
In the language of differential topology, the degree of a smooth map can be defined as follows: If f is a smooth map whose domain is a compact manifold and p is a regular value of f, consider the finite set
By p being a regular value, in a neighborhood of each xi the map f is a local diffeomorphism. Diffeomorphisms can be either orientation preserving or orientation reversing. Let r be the number of points xi at which f is orientation preserving and s be the number at which f is orientation reversing. When the codomain of f is connected, the number r − s is independent of the choice of p (though n is not!) and one defines the degree of f to be r − s. This definition coincides with the algebraic topological definition above.
The same definition works for compact manifolds with boundary but then f should send the boundary of X to the boundary of Y.
One can also define degree modulo 2 (deg2(f)) the same way as before but taking the fundamental class in Z2 homology. In this case deg2(f) is an element of Z2 (the field with two elements), the manifolds need not be orientable and if n is the number of preimages of p as before then deg2(f) is n
|
https://en.wikipedia.org/wiki/Castelnuovo%E2%80%93de%20Franchis%20theorem
|
In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let
ω1 and ω2
be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pullback of an algebraic curve: there is a non-singular algebraic curve C, a morphism
φ: X → C,
and differentials of the first kind ω1 and ω2 on C such that
φ*(1) = ω1 and φ*(2) = ω2.
This result is due to Guido Castelnuovo and Michele de Franchis (1875–1946).
The converse, that two such pullbacks would have wedge 0, is immediate.
See also
de Franchis theorem
References
.
Algebraic surfaces
Theorems in geometry
|
https://en.wikipedia.org/wiki/De%20Franchis%20theorem
|
In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generally,
the set of non-constant morphisms from X to Y is finite;
fixing X, for all but a finite number of such Y, there is no non-constant morphism from X to Y.
These results are named for (1875–1946). It is sometimes referenced as the De Franchis-Severi theorem. It was used in an important way by Gerd Faltings to prove the Mordell conjecture.
See also
Castelnuovo–de Franchis theorem
References
M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat Palermo 36 (1913), 368
Algebraic curves
Riemann surfaces
Theorems in algebraic geometry
Theorems in algebraic topology
|
https://en.wikipedia.org/wiki/Enriques%20surface
|
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0.
Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by as an answer to a question discussed by about whether a surface with q = pg = 0 is necessarily rational, though some of the Reye congruences introduced earlier by are also examples of Enriques surfaces.
Enriques surfaces can also be defined over other fields.
Over fields of characteristic other than 2, showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular Enriques surfaces, described by . These two extra families are related to the two non-discrete algebraic group schemes of order 2 in characteristic 2.
Invariants of complex Enriques surfaces
The plurigenera Pn are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H2(X, Z) is isomorphic to the sum of the unique even unimodular lattice II1,9 of dimension 10 and signature -8 and a group of order 2.
Hodge diamond:
Marked Enriques surfaces form a connected 10-dimensional family, which showed is rational.
Characteristic 2
In characteristic 2 there are some new families of Enriques surfaces,
sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces. (The term "singular" does not mean that the surface has singularities, but means that the surface is "special" in some way.)
In characteristic 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10. (In characteristics other than 2 this is equivalent to the usual definition.) There are now 3 families of Enriques surfaces:
Classical: dim(H1(O)) = 0. This implies 2K = 0 but K is nonzero, and Picτ is Z/2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme μ2.
Singular: dim(H1(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K = 0, and Picτ is μ2. The surface is a quotient of a K3 surface by the group scheme Z/2Z.
Supersingular: dim(H1(O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K = 0, and Picτ is α2. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α2.
All Enriques surfaces are elliptic or quasi elliptic.
Examples
A Reye congruence is the family of lines contained in at least 2 quadrics of a given 3-dimensional linear system of quadr
|
https://en.wikipedia.org/wiki/Tate%20module
|
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.
Definition
Given an abelian group A and a prime number p, the p-adic Tate module of A is
where A[pn] is the pn torsion of A (i.e. the kernel of the multiplication-by-pn map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A[pn+1] → A[pn]. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Zp-module via
Examples
The Tate module
When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood). It is a free rank one module over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme Gm,K over K.
The Tate module of an abelian variety
Given an abelian variety G over a field K, the Ks-valued points of G are an abelian group. The p-adic Tate module Tp(G) of G is a Galois representation (of the absolute Galois group, GK, of K).
Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp(G) is a free module over Zp of rank 2d, where d is the dimension of G. In the other case, it is still free, but the rank may take any value from 0 to d (see for example Hasse–Witt matrix).
In the case where p is not equal to the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology .
A special case of the Tate conjecture can be phrased in terms of Tate modules. Suppose K is finitely generated over its prime field (e.g. a finite field, an algebraic number field, a global function field), of characteristic different from p, and A and B are two abelian varieties over K. The Tate conjecture then predicts that
where HomK(A, B) is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of GK-linear maps from Tp(A) to Tp(B). The case where K is a finite field was proved by Tate himself in the 1960s. Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".
In the case of a Jacobian over a curve C over a finite field k of characteristic prime to p, the Tate module can be identified with the Galois group of the composite extension
where is an extension of k containing all p-power roots of unity and A(p) is the maximal
|
https://en.wikipedia.org/wiki/Tate%20twist
|
In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules.
For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V⊗Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). Denoting by Qp(−1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as
References
Number theory
Algebraic geometry
|
https://en.wikipedia.org/wiki/Timothy%20Kanold
|
Dr. Timothy D. Kanold is a mathematics educator and author of textbooks. He was the president of the National Council of Supervisors of Mathematics (NCSM) from 2008 to 2009.
Dr. Kanold holds a bachelor's degree in Education and a master's degree in Mathematics from the University of Illinois, and a doctorate in Educational Leadership and Counseling Psychology from Loyola University Chicago. In 2007, he retired from his position as Superintendent at Adlai E. Stevenson High School in Lincolnshire, Illinois, where for 17 years, he served as Director of Mathematics and Science.
With Ron Larson, Dr. Kanold is co-author of 27 mathematics textbooks grades 6-12, written for Houghton Mifflin/McDougal Littell Publishing Company from 1988 to the present. Additionally, since 2001 he has authored and co-authored 18 books on K-12 mathematics and school leadership, published with Solution Tree Press. He continues to write and present for the National Council of Teachers of Mathematics on the Principles and Standards for School Mathematics, as well as for AASA and NASSP. He is the lead author for NCTM's update of the Teaching Performance Standards Document, and has presented more than 600 talks and seminars nationally and internationally over the past decade, with the primary focus on the creation of equitable learning experiences for all children in mathematics.
Dr. Kanold is the 1986 recipient of the Presidential Award for Excellence in Mathematics Teaching, the 1991 recipient of the Outstanding Young Alumni Award from Illinois State University, the 1994 recipient of the Outstanding School Administrator Award from the Illinois State Board of Education, and the 2001 recipient of the Outstanding Alumni Award from Addison Trail High School. He also is the developer and presenter for New Dimensions in Leadership: Leading in a Learning Organization, a training program for future school administrators. Considered to be a “Teacher of Leaders,” he currently provides training in mathematics program improvement, professional learning community development, and school leadership on behalf of Solution Tree. He also presents mathematics curriculum, instruction, and assessment workshops for NCTM and NCSM.
Dr. Kanold's daughter, Jessica McIntyre, taught mathematics at Aptakisic Junior High School, in Buffalo Grove, IL then served as the Principal from 2011–2016.
References
Esposito, Jennifer Chase (October 2006). "Perfect Equation". The Magazine of School District Management. District Administration.
Living people
Year of birth missing (living people)
University of Illinois alumni
Loyola University Chicago alumni
20th-century American mathematicians
21st-century American mathematicians
Schoolteachers from Illinois
American male non-fiction writers
|
https://en.wikipedia.org/wiki/Noether%27s%20theorem%20on%20rationality%20for%20surfaces
|
In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational.
See also
Hirzebruch surface
List of complex and algebraic surfaces
References
Castelnuovo’s Theorem
Notes
Algebraic surfaces
Theorems in algebraic geometry
|
https://en.wikipedia.org/wiki/Nehari
|
Zeev Nehari, mathematician
Nehari manifold in mathematics
Nihari, South Asian stew
|
https://en.wikipedia.org/wiki/Walter%20Plecker
|
Walter Ashby Plecker (April 2, 1861 – August 2, 1947) was an American physician and public health advocate who was the first registrar of Virginia's Bureau of Vital Statistics, serving from 1912 to 1946. He was a leader of the Anglo-Saxon Clubs of America, a white supremacist organization founded in Richmond, Virginia, in 1922. A eugenicist and proponent of scientific racism, Plecker drafted and lobbied for the passage of the Racial Integrity Act of 1924 by the Virginia legislature; it institutionalized the one-drop rule.
Plecker was killed after being struck by a car in 1947.
Early life and education
Plecker was born in Augusta County, the son of a returned Confederate veteran. Sent to Staunton as a boy, he graduated from Hoover Military Academy in 1880 and obtained a medical degree from the University of Maryland in 1885. He was a devout Presbyterian, and throughout his life he supported the denomination's fundamentalist Southern branch, funding missionaries who believed, as he later would, that God had destroyed Sodom and Gomorrah as punishment for racial intermixing.
Career
Plecker settled in Hampton, Virginia, in 1892, and before his mother's death in 1915, he worked with women of all races and became known for his active interest in obstetrics and public health issues. Plecker educated midwives, invented a home incubator, and prescribed home remedies for infants. Plecker became the public health officer for Elizabeth City County in 1902.
In 1912, Plecker became the first registrar of Virginia's newly created Bureau of Vital Statistics, a position he held until 1946. An avowed white supremacist and an advocate of eugenics, he became a leader of the Anglo-Saxon Clubs of America in 1922. He wanted to prevent miscegenation, or marriage between races, and he also thought that a decreasing number of mulattoes, as classified in the census, meant that more of them were passing as white.
With the help of John Powell and Earnest Sevier Cox, Plecker drafted and the state legislature passed the "Racial Integrity Act of 1924". It recognized only two races, "white" and "colored" (black). It essentially incorporated the one-drop rule, classifying any individual with any amount of African ancestry as "colored". This went beyond existing laws, which had classified persons who had one-sixteenth (equivalent to one great-great-grandparent) or less black ancestry as white. In 1967, the United States Supreme Court invalidated the law in Loving v. Virginia.
In particular, Plecker resented African Americans who passed as Native Americans, and he came to firmly believe that the state's Native Americans had been "mongrelized" with its African American population. In fact, since shortly after the Civil War, Native Americans from all over the country had been brought to the Hampton area to be educated alongside blacks, at times inter-marrying, although Hampton's Indian schools had closed down as racial discrimination against Native Americans and the eugenics mov
|
https://en.wikipedia.org/wiki/Axiality%20and%20rhombicity
|
In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry.
Let A denote a second-rank tensor in R3, which can be represented by a 3-by-3 matrix. We assume that A is symmetric. This implies that A has three real eigenvalues, which we denote by , and . We assume that they are ordered such that
The axiality of A is defined by
The rhombicity is the difference between the smallest and the second-smallest eigenvalue:
Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend on the context. For example, when using them as parameters in the irreducible spherical tensor expansion, it is most convenient to divide the above definition of axiality by and that of rhombicity by .
Applications
The description of physical interactions in terms of axiality and rhombicity is frequently encountered in spin dynamics and, in particular, in spin relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form
(hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators:
where are Wigner functions, are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are:
Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood.
References
D.M. Brink and G.R. Satchler, Angular momentum, 3rd edition, 1993, Oxford: Clarendon Press.
D.A. Varshalovich, A.N. Moskalev, V.K. Khersonski, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, 1988, Singapore: World Scientific Publications.
I. Kuprov, N. Wagner-Rundell, P.J. Hore, J. Magn. Reson., 2007 (184) 196-206. Article
Tensors
|
https://en.wikipedia.org/wiki/Fano%20variety
|
In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties.
Examples
The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of Pn over a field k is O(n+1), which is very ample (over the complex numbers, its curvature is n+1 times the Fubini–Study symplectic form).
Let D be a smooth codimension-1 subvariety in Pn. The adjunction formula implies that KD = (KX + D)|D = (−(n+1)H + deg(D)H)|D, where H is the class of a hyperplane. The hypersurface D is therefore Fano if and only if deg(D) < n+1.
More generally, a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if and only if the sum of their degrees is at most n.
Weighted projective space P(a0,...,an) is a singular (klt) Fano variety. This is the projective scheme associated to a graded polynomial ring whose generators have degrees a0,...,an. If this is well formed, in the sense that no n of the numbers a have a common factor greater than 1, then any complete intersection of hypersurfaces such that the sum of their degrees is less than a0+...+an is a Fano variety.
Every projective variety in characteristic zero that is homogeneous under a linear algebraic group is Fano.
Some properties
The existence of some ample line bundle on X is equivalent to X being a projective variety, so a Fano variety is always projective. For a Fano variety X over the complex numbers, the Kodaira vanishing theorem implies that the sheaf cohomology groups of the structure sheaf vanish for . In particular, the Todd genus automatically equals 1. The cases of this vanishing statement also tell us that the first Chern class induces an isomorphism .
By Yau's solution of the Calabi conjecture, a smooth complex variety admits Kähler metrics of positive
Ricci curvature if and only if it is Fano. Myers' theorem therefore tells us that the universal cover of a Fano manifold is compact, and so can only be a finite covering. However, we have just seen that the Todd genus of a Fano manifold must equal 1. Since this would also apply to the manifold's universal cover, and since the Todd genus is multiplicative under finite covers, it follows that any Fano manifold is simply connected.
A much easier fact is that every Fano variety has Kodaira dimension −∞.
Campana and Kollár–Miyaoka–Mori showed that a smooth Fano variety over an algebraically closed fiel
|
https://en.wikipedia.org/wiki/The%20Number%20Devil
|
The Number Devil: A Mathematical Adventure () is a book for children and young adults that explores mathematics. It was originally written in 1997 in German by Hans Magnus Enzensberger and illustrated by Rotraut Susanne Berner. The book follows a young boy named Robert, who is taught mathematics by a sly "number devil" called Teplotaxl over the course of twelve dreams.
The book was met with mostly positive reviews from critics, approving its description of math while praising its simplicity. Its colorful use of fictional mathematical terms and its creative descriptions of concepts have made it a suggested book for both children and adults troubled with math. The Number Devil was a bestseller in Europe, and has been translated into English by Michael Henry Heim.
Plot
Robert is a young boy who suffers from mathematical anxiety due to his boredom in school. His mother is Mrs. Wilson. He also experiences recurring dreams—including falling down an endless slide or being eaten by a giant fish—but is interrupted from this sleep habit one night by a small devil creature who introduces himself as the Number Devil. Although there are many Number Devils (from Number Heaven), Robert only knows him as the Number Devil before learning of his actual name, Teplotaxl, later in the story.
Over the course of twelve dreams, the Number Devil teaches Robert mathematical principles. On the first night, the Number Devil appears to Robert in an oversized world and introduces the number one. The next night, the Number Devil emerges in a forest of trees shaped like "ones" and explains the necessity of the number zero, negative numbers, and introduces hopping, a fictional term to describe exponentiation. On the third night, the Number Devil brings Robert to a cave and reveals how prima-donna numbers (prime numbers) can only be divided by themselves and one without a remainder. Later, on the fourth night, the Number Devil teaches Robert about rutabagas, another fictional term to depict square roots, at a beach.
For a time after the fourth night, Robert cannot find the Number Devil in his dreams; later, however, on the fifth night, Robert finds himself at a desert where the Number Devil teaches him about triangular numbers through the use of coconuts. On the sixth night, the Number Devil teaches Robert about the natural occurrence of Fibonacci numbers, which the Number Devil shortens to Bonacci numbers, by counting brown and white rabbits as they reproduce multiple times. By this dream, Robert's mother has noticed a visible change in Robert's mathematical interest, and Robert begins going to sleep earlier to encounter the Number Devil. The seventh night brings Robert to a bare, white room, where the Number Devil presents Pascal's triangle and the patterns that the triangular array displays. On the eighth night, Robert is brought to his classroom at school. The Number Devil arranges Robert's classmates in multiple ways, teaches him about permutations, and what the Number D
|
https://en.wikipedia.org/wiki/Zariski%20surface
|
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski surfaces are unirational. They were named by Piotr Blass in 1977 after Oscar Zariski who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational. (In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
Zariski surfaces are birational to surfaces in affine 3-space A3 defined by irreducible polynomials of the form
The following problem was posed by Oscar Zariski in 1971: Let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface? For p = 2 and for p = 3 the answer to the above problem is negative as shown in 1977 by Piotr Blass in his University of Michigan Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978. announced further examples giving a negative answer to Zariski's question in every characteristic p>0 .
His method however is non constructive at the moment and we do not have explicit equations for p>3.
See also
List of algebraic surfaces
References
Algebraic surfaces
University of Michigan
|
https://en.wikipedia.org/wiki/Kerala%20school%20of%20astronomy%20and%20mathematics
|
The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).
Background
Islamic scholars nearly developed a general formula for finding integrals of polynomials by 1000 AD —and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by the year 1600 able to use formula similar to ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed.
Contributions
Infinite series and calculus
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following infinite geometric series:
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result:
for large n.
They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for , , and . The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:
|
https://en.wikipedia.org/wiki/Gonality%20of%20an%20algebraic%20curve
|
In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C is defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field extensions
K(C)/K(f)
of the function field over its subfields generated by single functions f.
If K is algebraically closed, then the gonality is 1 precisely for curves of genus 0. The gonality is 2 for curves of genus 1 (elliptic curves) and for hyperelliptic curves (this includes all curves of genus 2). For genus g ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus g is the floor function of
(g + 3)/2.
Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of genus three and given by an equation
y3 = Q(x)
where Q is of degree 4.
The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of the algebraic curve C can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high degree. In many cases the gonality is two more than the Clifford index. The Green–Lazarsfeld conjecture is an exact formula in terms of the graded Betti numbers for a degree d embedding in r dimensions, for d large with respect to the genus. Writing b(C), with respect to a given such embedding of C and the minimal free resolution for its homogeneous coordinate ring, for the minimum index i for which βi, i + 1 is zero, then the conjectured formula for the gonality is
r + 1 − b(C).
According to the 1900 ICM talk of Federico Amodeo, the notion (but not the terminology) originated in Section V of Riemann's Theory of Abelian Functions. Amodeo used the term "gonalità" as early as 1893.
References
Algebraic curves
Homological algebra
|
https://en.wikipedia.org/wiki/K%C3%B6the%20conjecture
|
In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}.
This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive.
Equivalent formulations
The conjecture has several different formulations:
(Köthe conjecture) In any ring, the sum of two nil left ideals is nil.
In any ring, the sum of two one-sided nil ideals is nil.
In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring.
For any ring R and for any nil ideal J of R, the matrix ideal Mn(J) is a nil ideal of Mn(R) for every n.
For any ring R and for any nil ideal J of R, the matrix ideal M2(J) is a nil ideal of M2(R).
For any ring R, the upper nilradical of Mn(R) is the set of matrices with entries from the upper nilradical of R for every positive integer n.
For any ring R and for any nil ideal J of R, the polynomials with indeterminate x and coefficients from J lie in the Jacobson radical of the polynomial ring R[x].
For any ring R, the Jacobson radical of R[x] consists of the polynomials with coefficients from the upper nilradical of R.
Related problems
A conjecture by Amitsur read: "If J is a nil ideal in R, then J[x] is a nil ideal of the polynomial ring R[x]." This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz. While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false.
Kegel proved that a ring which is the direct sum of two nilpotent subrings is itself nilpotent. The question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when Kelarev produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings. This demonstrates that Kegel's question with "locally nilpotent" replacing "nilpotent" is answered in the negative.
The sum of a nilpotent subring and a nil subring is always nil.
References
External links
PlanetMath page
Survey paper (PDF)
Ring theory
Conjectures
Unsolved problems in mathematics
|
https://en.wikipedia.org/wiki/Nil%20ideal
|
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.
The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.
Commutative rings
In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if a and b are nilpotent elements of R with an = 0 and bm = 0, and r is any element of R, then (a·r)n = an·r n = 0, and by the binomial theorem, (a+b)m+n = 0. Therefore, the set of all nilpotent elements forms an ideal known as the nil radical of a ring. Because the nil radical contains every nilpotent element, an ideal of a commutative ring is nil if and only if it is a subset of the nil radical, and so the nil radical is maximal among non-nil ideals. Furthermore, for any nilpotent element a of a commutative ring R, the ideal aR is nil. For a non commutative ring however, it is not in general true that the set of nilpotent elements forms an ideal, or that a ·R is a nil (one-sided) ideal, even if a is nilpotent.
Noncommutative rings
The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the understanding of nil rings—rings whose every element is nilpotent—one may obtain a much better understanding of more general rings.
In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the Köthe conjecture. The Köthe conjecture was first posed in 1930 and yet remains unresolved as of 2023.
Relation to nilpotent ideals
The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent:
There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
The product of n nilpotent elements may be nonzero for arbitrarily high n.
Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.
In a right artinian ring, any nil ideal is nilpotent. This is proved by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical i
|
https://en.wikipedia.org/wiki/Locally%20nilpotent
|
In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a nilpotent ideal in Ap.
In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subalgebra or subgroup is nilpotent. The subgroup generated by the normal locally nilpotent subgroups is called the Hirsch–Plotkin radical and is the generalization of the Fitting subgroup to groups without the ascending chain condition on normal subgroups.
A locally nilpotent ring is one in which every finitely generated subring is nilpotent: locally nilpotent rings form a radical class, giving rise to the Levitzki radical.
Commutative algebra
|
https://en.wikipedia.org/wiki/Ruziewicz%20problem
|
In mathematics, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets.
This was answered affirmatively and independently for n ≥ 4 by Grigory Margulis and Dennis Sullivan around 1980, and for n = 2 and 3 by Vladimir Drinfeld (published 1984). It fails for the circle.
The problem is named after Stanisław Ruziewicz.
References
.
.
.
.
Survey of the area by Hee Oh
Measure theory
|
https://en.wikipedia.org/wiki/Ian%20G.%20Macdonald
|
Ian Grant Macdonald (11 October 1928 – 8 August 2023) was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
Early life and education
Born in London, he was educated at Winchester College and Trinity College, Cambridge, graduating in 1952.
Career
He then spent five years as a civil servant. He was offered a position at Manchester University in 1957 by Max Newman, on the basis of work he had done while outside academia. In 1960 he moved to the University of Exeter, and in 1963 became a Fellow of Magdalen College, Oxford. Macdonald became Fielden Professor at Manchester in 1972, and professor at Queen Mary College, University of London, in 1976.
He worked on symmetric products of algebraic curves, Jordan algebras and the representation theory of groups over local fields. In 1972 he proved the Macdonald identities, after a pattern known to Freeman Dyson. His 1979 book Symmetric Functions and Hall Polynomials has become a classic. Symmetric functions are an old theory, part of the theory of equations, to which both K-theory and representation theory lead. His was the first text to integrate much classical theory, such as Hall polynomials, Schur functions, the Littlewood–Richardson rule, with the abstract algebra approach. It was both an expository work and, in part, a research monograph, and had a major impact in the field. The Macdonald polynomials are now named after him. The Macdonald conjectures from 1982 also proved most influential.
Macdonald was elected a Fellow of the Royal Society in 1979. He was an invited speaker in 1970 at the International Congress of Mathematicians (ICM) in Nice and a plenary speaker in 1998 at the ICM in Berlin. In 1991 he received the Pólya Prize of the London Mathematical Society. He was awarded the 2009 Steele Prize for Mathematical Exposition. In 2012 he became a fellow of the American Mathematical Society.
Personal life and demise
Ian G. Macdonald died on 8 August 2023, at the age of 94.
Selected publications
Macdonald, I. G. Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, 2003. x+175 pp.
Macdonald, I. G. Symmetric Functions and Hall Polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.
Macdonald, I. G. Symmetric Functions and Orthogonal Polynomials. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 12. American Mathematical Society, Providence, Rhode Island, 1998. xvi+53 pp.
Atiyah, M. F.; Macdonald, I. G. Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. ix+128 pp. ; 1994 pbk edition
References
External links
Biographical notice
1928 b
|
https://en.wikipedia.org/wiki/Ore%20condition
|
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
General idea
The goal is to construct the right ring of fractions R[S−1] with respect to a multiplicative subset S. In other words, we want to work with elements of the form as−1 and have a ring structure on the set R[S−1]. The problem is that there is no obvious interpretation of the product (as−1)(bt−1); indeed, we need a method to "move" s−1 past b. This means that we need to be able to rewrite s−1b as a product b1s1−1. Suppose then multiplying on the left by s and on the right by s1, we get . Hence we see the necessity, for a given a and s, of the existence of a1 and s1 with and such that .
Application
Since it is well known that each integral domain is a subring of a field of fractions (via an embedding) in such a way that every element is of the form rs−1 with s nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a division ring (a noncommutative field) with the same property. It turns out that the answer is sometimes "no", that is, there are domains which do not have an analogous "right division ring of fractions".
For every right Ore domain R, there is a unique (up to natural R-isomorphism) division ring D containing R as a subring such that every element of D is of the form rs−1 for r in R and s nonzero in R. Such a division ring D is called a ring of right fractions of R, and R is called a right order in D. The notion of a ring of left fractions and left order are defined analogously, with elements of D being of the form s−1r.
It is important to remember that the definition of R being a right order in D includes the condition that D must consist entirely of elements of the form rs−1. Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean R is a left order in D, since it is possible D has an element which is not of the form s−1r. Thus it is possible for R to be a right-not-left Ore domain. Intuitively, the condition that all elements of D be of the form rs−1 says that R is a "big" R-submodule of D. In fact the condition ensures RR is an essential submodule of DR. Lastly, there is even an example of a domain in a division ring which satisfies neither Ore condition (see examples below).
Another natural question is: "When is a subring of a division ring right Ore?" One characterization is that a subring R of a division ring D is a right Ore domain if
|
https://en.wikipedia.org/wiki/Quasiregular%20representation
|
This article addresses the notion of quasiregularity in the context of representation theory and topological algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
In mathematics, quasiregular representation is a concept of representation theory, for a locally compact group G and a homogeneous space G/H where H is a closed subgroup.
In line with the concepts of regular representation and induced representation, G acts on functions on G/H. If however Haar measures give rise only to a quasi-invariant measure on G/H, certain 'correction factors' have to be made to the action on functions, for
L2(G/H)
to afford a unitary representation of G on square-integrable functions. With appropriate scaling factors, therefore, introduced into the action of G, this is the quasiregular representation or modified induced representation.
Unitary representation theory
Topological groups
|
https://en.wikipedia.org/wiki/Vector%20fields%20on%20spheres
|
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in -dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least such fields (see definition below). Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. Hence is the exact number of pointwise linearly independent vector fields that exist on an ()-dimensional sphere.
Technical details
In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the ()-sphere is exactly .
The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.
Radon–Hurwitz numbers
The Radon–Hurwitz numbers occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms. For written as the product of an odd number and a power of two , write
.
Then
.
The first few values of are (from ):
2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...
For odd , the value of the function is one.
These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics.
References
Differential topology
Theorems in topology
|
https://en.wikipedia.org/wiki/Barlow%20surface
|
In mathematics, a Barlow surface is one of the complex surfaces introduced by . They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:
See also
Hodge theory
References
Algebraic surfaces
Complex surfaces
|
https://en.wikipedia.org/wiki/Godeaux%20surface
|
In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931.
Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called numerical Godeaux surfaces.
Construction
The cyclic group of order 5 acts freely on the Fermat surface of points (w : x : y : z)
in P3 satisfying w5 + x5 + y5 + z5 = 0 by mapping (w : x : y : z) to (w:ρx:ρ2y:ρ3z) where ρ is a fifth root of 1. The quotient by this action is the original Godeaux surface.
Invariants
The fundamental group (of the original Godeaux surface) is cyclic of order 5.
It has invariants like rational surfaces do, though it is not rational. The square of the first Chern class (and moreover the canonical class is ample).
See also
Hodge theory
References
Algebraic surfaces
Complex surfaces
|
https://en.wikipedia.org/wiki/Stationary%20ergodic%20process
|
In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.
Stationarity is the property of a random process which guarantees that its statistical properties, such as the mean value, its moments and variance, will not change over time. A stationary process is one whose probability distribution is the same at all times. For more information see stationary process.
An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average. In practice this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process with no change in the measured result.
A simple example of a violation of ergodicity is a measured process which is the superposition of two underlying processes,
each with its own statistical properties. Although the measured process may be stationary in the long term, it is not appropriate to consider the sampled distribution to be the reflection of a single (ergodic) process: The ensemble average is meaningless. Also see ergodic theory and ergodic process.
See also
Measure-preserving dynamical system
References
Peebles,P. Z., 2001, Probability, Random Variables and Random Signal Principles, McGraw-Hill Inc, Boston,
Ergodic theory
pl:Proces ergodyczny
|
https://en.wikipedia.org/wiki/Horrocks%E2%80%93Mumford%20bundle
|
In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P4 introduced by . It is the only such bundle known, although a generalized construction involving Paley graphs produces other rank 2 sheaves (Sasukara et al. 1993). The zero sets of sections of the Horrocks–Mumford bundle are abelian surfaces of degree 10, called Horrocks–Mumford surfaces.
By computing Chern classes one sees that the second exterior power of the Horrocks–Mumford bundle F is the line bundle O(5) on P4. Therefore, the zero set V of a general section of this bundle is a quintic threefold called a Horrocks–Mumford quintic. Such a V has exactly 100 nodes; there exists a small resolution V′ which is a Calabi–Yau threefold fibered by Horrocks–Mumford surfaces.
See also
List of algebraic surfaces
References
Algebraic varieties
Vector bundles
Projective geometry of elliptic curves - contains chapter on constructions of the bundle
|
https://en.wikipedia.org/wiki/Vector%20area
|
In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an oriented area in three dimensions.
Every bounded surface in three dimensions can be associated with a unique area vector called its vector area. It is equal to the surface integral of the surface normal, and distinct from the usual (scalar) surface area.
Vector area can be seen as the three dimensional generalization of signed area in two dimensions.
Definition
For a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area:
For an orientable surface composed of a set of flat facet areas, the vector area of the surface is given by
where is the unit normal vector to the area .
For bounded, oriented curved surfaces that are sufficiently well-behaved, we can still define vector area. First, we split the surface into infinitesimal elements, each of which is effectively flat. For each infinitesimal element of area, we have an area vector, also infinitesimal.
where is the local unit vector perpendicular to . Integrating gives the vector area for the surface.
Properties
The vector area of a surface can be interpreted as the (signed) projected area or "shadow" of the surface in the plane in which it is greatest; its direction is given by that plane's normal.
For a curved or faceted (i.e. non-planar) surface, the vector area is smaller in magnitude than the actual surface area. As an extreme example, a closed surface can possess arbitrarily large area, but its vector area is necessarily zero. Surfaces that share a boundary may have very different areas, but they must have the same vector area—the vector area is entirely determined by the boundary. These are consequences of Stokes' theorem.
The vector area of a parallelogram is given by the cross product of the two vectors that span it; it is twice the (vector) area of the triangle formed by the same vectors. In general, the vector area of any surface whose boundary consists of a sequence of straight line segments (analogous to a polygon in two dimensions) can be calculated using a series of cross products corresponding to a triangularization of the surface. This is the generalization of the Shoelace formula to three dimensions.
Using Stokes' theorem applied to an appropriately chosen vector field, a boundary integral for the vector area can be derived:
where is the boundary of , i.e. one or more oriented closed space curves. This is analogous to the two dimensional area calculation using Green's theorem.
Applications
Area vectors are used when calculating surface integrals, such as when determining the flux of a vector field through a surface. The flux is given by the integral of the dot product of the field and the (infinitesimal) area vector. When the field is constant over the surface the integral simplifies to the dot product of the field and the vector area of the surface.
Pr
|
https://en.wikipedia.org/wiki/Contiguity
|
Contiguity or contiguous may refer to:
Contiguous data storage, in computer science
Contiguity (probability theory)
Contiguity (psychology)
Contiguous distribution of species, in biogeography
Geographic contiguity of territorial land
Contiguous zone in territorial waters
See also
|
https://en.wikipedia.org/wiki/Petr%20Vop%C4%9Bnka
|
Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of articles and monographs. Vopěnka’s name is associated with many mathematical achievements, including Vopěnka's principle.
Since the mid-eighties he concerned himself with philosophical questions of mathematics (particularly vis-à-vis Husserlian phenomenology).
Vopěnka served as the Minister of Education of the Czech Republic (then part of Czechoslovakia) from 1990 to 1992 within the government of Prime Minister Petr Pithart.
Biography
Petr Vopěnka grew up in small town of Dolní Kralovice. After finishing gymnasium in Ledeč nad Sázavou in 1953 he went to study mathematics at the Mathematics and Physics Faculty of Charles University in Prague, graduating in 1958. In 1962 he was made Candidate of Sciences (CSc) and in 1967 Doctor of Science (DrSc). His advisors were Eduard Čech and Ladislav Rieger.
Starting in 1958 Vopěnka taught at the Mathematics and Physics Faculty, since 1964 as lecturer, since 1965 as senior lecturer. In 1968 he was made professor but was prevented to take this title until 1990 due to political reasons. Between 1966 and 1969 Vopěnka served as Vice Dean of the faculty.
In 1967 Vopěnka became head of the newly established Department of Mathematical Logic. The department was abolished in 1970 and Vopěnka, though allowed to stay at the university, fell into disfavour with the regime, which limited his contacts with foreign mathematicians. During the 1970s and 1980s he concentrated on philosophy and history of mathematics and on phenomenology of infinity.
After the Velvet Revolution, in January 1990, Vopěnka became Deputy Rector of the Charles University. During the period June 1990 – July 1992 he served as Minister of Education of the Czech Republic (then part of Czechoslovakia). In this position he, without much of success and facing protests from the teachers, attempted to institute school reforms.
In 1992 the Department of Mathematical Logic was reopened and Vopěnka became its head. In 2000 he retired from the Charles University and the department was closed. Until 2009 Vopěnka worked as a professor at the Jan Evangelista Purkyně University in Ústí nad Labem, in the Department of Mathematics of the Faculty of Science.
Petr Vopěnka also participated in translation and publishing of early mathematical texts (such as works of Euclid and Al-Khwarezmi) into the Czech language, and then he worked at the Department of Philosophy and Department of Interdisciplinary Activities, University of West Bohemia in Pilsen.
Bibliography
See also
Semiset
Vopěnka's principle
Notes
References
Further reading
External links
Short biography in English
Documentary about Vopěnka (in Czech with English subtitles, freely downloadable)
1935 births
2015 deaths
Education ministers of the Czech Republic
Czech mathem
|
https://en.wikipedia.org/wiki/Klee%27s%20measure%20problem
|
In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers, which is a subset of Rd.
The problem is named after Victor Klee, who gave an algorithm for computing the length of a union of intervals (the case d = 1) which was later shown to be optimally efficient in the sense of computational complexity theory. The computational complexity of computing the area of a union of 2-dimensional rectangular ranges is now also known, but the case d ≥ 3 remains an open problem.
History and algorithms
In 1977, Victor Klee considered the following problem: given a collection of n intervals in the real line, compute the length of their union. He then presented an algorithm to solve this problem with computational complexity (or "running time") — see Big O notation for the meaning of this statement. This algorithm, based on sorting the intervals, was later shown by Michael Fredman and Bruce Weide (1978) to be optimal.
Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. He also obtained a complexity algorithm, now known as Bentley's algorithm, based on reducing the problem to n 1-dimensional problems: this is done by sweeping a vertical line across the area. Using this method, the area of the union can be computed without explicitly constructing the union itself. Bentley's algorithm is now also known to be optimal (in the 2-dimensional case), and is used in computer graphics, among other areas.
These two problems are the 1- and 2-dimensional cases of a more general question: given a collection of n d-dimensional rectangular ranges, compute the measure of their union. This general problem is Klee's measure problem.
When generalized to the d-dimensional case, Bentley's algorithm has a running time of . This turns out not to be optimal, because it only decomposes the d-dimensional problem into n (d-1)-dimensional problems, and does not further decompose those subproblems. In 1981, Jan van Leeuwen and Derek Wood improved the running time of this algorithm to for d ≥ 3 by using dynamic quadtrees.
In 1988, Mark Overmars and Chee Yap proposed an algorithm for d ≥ 3. Their algorithm uses a particular data structure similar to a kd-tree to decompose the problem into 2-dimensional components and aggregate those components efficiently; the 2-dimensional problems themselves are solved efficiently using a trellis structure. Although asymptotically faster than Bentley's algorithm, its data structures use significantly more space, so it is only used in problems where either n or d is large. In 1998, Bogdan Chlebus proposed a simpler algorithm with the same asymptotic running time for the common special cases where d is 3 or 4.
In 2013, Timothy M
|
https://en.wikipedia.org/wiki/Parallelizable%20manifold
|
In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields
on the manifold, such that at every point of the tangent vectors
provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on
A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .
Examples
An example with is the circle: we can take V1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in G).
A classical problem was to determine which of the spheres Sn are parallelizable. The zero-dimensional case S0 is trivially parallelizable. The case S1 is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that S2 is not parallelizable. However S3 is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is S7; this was proved in 1958, by Friedrich Hirzebruch, Michel Kervaire, and by Raoul Bott and John Milnor, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology.
The product of parallelizable manifolds is parallelizable.
Every orientable closed three-dimensional manifold is parallelizable.
Remarks
Any parallelizable manifold is orientable.
The term framed manifold (occasionally rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle.
A related notion is the concept of a π-manifold. A smooth manifold is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.
See also
Chart (topology)
Differentiable manifold
Frame bundle
Kervaire invariant
Orthonormal frame bundle
Principal bundle
Connection (mathematics)
G-structure
Notes
References
Differential topology
Fiber bundles
Manifolds
|
https://en.wikipedia.org/wiki/Stunted%20projective%20space
|
In mathematics, a stunted projective space is a construction on a projective space of importance in homotopy theory, introduced by . Part of a conventional projective space is collapsed down to a point.
More concretely, in a real projective space, complex projective space or quaternionic projective space
KPn,
where K stands for the real numbers, complex numbers or quaternions, one can find (in many ways) copies of
KPm,
where m < n. The corresponding stunted projective space is then
KPn,m = KPn/KPm,
where the notation implies that the KPm has been identified to a point. This makes a topological space that is no longer a manifold. The importance of this construction was realised when it was shown that real stunted projective spaces arose as Spanier–Whitehead duals of spaces of Ioan James, so-called quasi-projective spaces, constructed from Stiefel manifolds. Their properties were therefore linked to the construction of frame fields on spheres.
In this way the vector fields on spheres question was reduced to a question on stunted projective spaces: for RPn,m, is there a degree one mapping on the 'next cell up' (of the first dimension not collapsed in the 'stunting') that extends to the whole space? Frank Adams showed that this could not happen, completing the proof.
In later developments spaces KP∞,m and stunted lens spaces have also been used.
References
Homotopy theory
Differential topology
|
https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead%20duality
|
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds. The theory is also referred to as S-duality, but this can now cause possible confusion with the S-duality of string theory. It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955.
The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory.
Statement
Let X be a compact neighborhood retract in . Then and are dual objects in the category of pointed spectra with the smash product as a monoidal structure. Here is the union of and a point, and are reduced and unreduced suspensions respectively.
Taking homology and cohomology with respect to an Eilenberg–MacLane spectrum recovers Alexander duality formally.
References
Homotopy theory
Duality theories
|
https://en.wikipedia.org/wiki/Alexander%20duality
|
In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality.
General statement for spheres
Let be a compact, locally contractible subspace of the sphere of dimension n. Let be the complement of in . Then if stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism
for all . Note that we can drop local contractibility as part of the hypothesis if we use Čech cohomology, which is designed to deal with local pathologies.
Applications
This is useful for computing the cohomology of knot and link complements in . Recall that a knot is an embedding and a link is a disjoint union of knots, such as the Borromean rings. Then, if we write the link/knot as , we have
,
giving a method for computing the cohomology groups. Then, it is possible to differentiate between different links using the Massey products.
For example, for the Borromean rings , the homology groups are
Alexander duality for constructible sheaves
For smooth manifolds, Alexander duality is a formal consequence of Verdier duality for sheaves of abelian groups. More precisely, if we let denote a smooth manifold and we let be a closed subspace (such as a subspace representing a cycle, or a submanifold) represented by the inclusion , and if is a field, then if is a sheaf of -vector spaces we have the following isomorphism
,
where the cohomology group on the left is compactly supported cohomology. We can unpack this statement further to get a better understanding of what it means. First, if is the constant sheaf and is a smooth submanifold, then we get
,
where the cohomology group on the right is local cohomology with support in . Through further reductions, it is possible to identify the homology of with the cohomology of . This is useful in algebraic geometry for computing the cohomology groups of projective varieties, and is exploited for constructing a basis of the Hodge structure of hypersurfaces of degree using the Jacobian ring.
Alexander's 1915 result
Referring to Alexander's original work, it is assumed that X is a simplicial complex.
Alexander had little of the modern apparatus, and his result was only for the Betti numbers, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open if the other is closed, but this does not affect its homology. Each of the solid tori is from the homotopy point of view a circle. If we just write down the Betti numbers
1, 1, 0, 0
of the circle (up to , since we are in the 3-sphere), then reverse as
0, 0, 1,
|
https://en.wikipedia.org/wiki/Reduced%20homology
|
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres).
If P is a single-point space, then with the usual definitions the integral homology group
H0(P)
is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have
Hi(P) = {0}.
More generally if X is a simplicial complex or finite CW complex, then the group H0(X) is the free abelian group with the connected components of X as generators. The reduced homology should replace this group, of rank r say, by one of rank r − 1. Otherwise the homology groups should remain unchanged. An ad hoc way to do this is to think of a 0-th homology class not as a formal sum of connected components, but as such a formal sum where the coefficients add up to zero.
In the usual definition of homology of a space X, we consider the chain complex
and define the homology groups by .
To define reduced homology, we start with the augmented chain complex
where . Now we define the reduced homology groups by
for positive n and .
One can show that ; evidently for all positive n.
Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the tensor product, or reduced cohomology groups from the cochain complex made by using a Hom functor, can be applied.
References
Hatcher, A., (2002) Algebraic Topology Cambridge University Press, . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
Homology theory
de:Singuläre_Homologie#Reduzierte_Homologie
|
https://en.wikipedia.org/wiki/Statgraphics
|
Statgraphics is a statistics package that performs and explains basic and advanced statistical functions.
History
The software was created in 1980 by Dr. Neil W. Polhemus while on the faculty at the Princeton University School of Engineering and Applied Science for use as a teaching tool for his statistics students. It was made available to the public in 1982, becoming and early example of data science software designed for use on the PC.
Software
The flagship version of Statgraphics is Statgraphics Centurion, a Windows desktop application with capabilities for regression analysis, ANOVA, multivariate statistics, Design of Experiments, statistical process control, life data analysis, machine learning, and data visualization. The data analysis procedures include descriptive statistics, hypothesis testing, regression analysis, analysis of variance, survival analysis, time series analysis and forecasting, sample size determination, multivariate methods, machine learning and Monte Carlo techniques. The SPC menu includes many procedures for quality assessment, capability analysis, control charts, measurement systems analysis, and acceptance sampling. The program also features a DOE Wizard that creates and analyzes statistically designed experiments.
Applications
Statgraphics is frequently used for Six Sigma process improvement. The program has also been used in various health and nutrition-related studies. The software is heavily used in manufacturing chemicals, pharmaceuticals, medical devices, automobiles, food and consumer goods. It is also widely used in mining, environmental studies, and basic R&D.
Distribution
Statgraphics is distributed by Statgraphics Technologies, Inc., a privately held company based in The Plains, Virginia.
See also
List of statistical packages
Comparison of statistical packages
List of information graphics software
References
Statistical software
Science software for Windows
|
https://en.wikipedia.org/wiki/Nasrin%20Soltankhah
|
Nasrin Soltankhah () is an Iranian politician who was a Vice President under Mahmoud Ahmadinejad from 2009 to 2013.
Education
Soltankhan received a Bachelor of Science in Mathematics (1976), a Master of Science in Mathematics (1978), and a PhD in Mathematics (1994) from Sharif University of Technology.
Career
Cabinet position
Soltankhan was appointed to the Iranian Cabinet on September 25, 2005 by President Mahmoud Ahmadinejad. She was also president of Iran's National Elites Foundation.
Center for Women and Family Affairs
Soltankhan's portfolio includes both the position as head of the Center for Women and Family Affairs (formerly called the Center for Women's Participation, or CWP) and also the position of advisor to the President on issues pertaining to women.
Soltankhan has mentioned three main points for women-related policies which the center will be focusing on. These are, “upholding human dignity of women regardless of their gender,“ “capitalizing on women’s potentials in managerial and decision-making arenas,“ and “emphasizing on women’s key role in families.“ Soltankhah has also stated that the center is engaged in directing women’s capabilities into different social and cultural fields as well as generating jobs for them.
Political affiliation
Soltankhan is a member of the political organization called the Alliance of Builders of Islamic Iran.
City Council of Tehran
Apart from her work in the executive branch of the Iranian government, Nasrin Soltankhan was also on the City Council of Tehran having won a seat in 2003. The term of service for her council seat ended in 2007.
See also
Persian women
References
External links
http://www.iran-daily.com/1384/2391/html/panorama.htm
http://www.iran-daily.com/1384/2419/html/panorama.htm
https://web.archive.org/web/20160304031333/http://www.iranian.ws/iran_news/publish/article_9956.shtml
https://web.archive.org/web/20160304031333/http://www.iranian.ws/iran_news/publish/article_9956.shtml
http://www.mehrnews.com/en/NewsDetail.aspx?NewsID=39497
Living people
Alliance of Builders of Islamic Iran politicians
1963 births
Politicians from Tehran
Coalition of the Pleasant Scent of Servitude politicians
Tehran Councillors 2003–2007
Presidential advisers of Iran
Women vice presidents of Iran
Vice presidents of Iran
21st-century Iranian women politicians
21st-century Iranian politicians
|
https://en.wikipedia.org/wiki/OGF
|
OGF can refer to:
Open Gaming Foundation for role-playing games
Open Grid Forum for grid computing
Ordinary generating function in mathematics
Opioid growth factor, an alternative name for met-enkephalin
|
https://en.wikipedia.org/wiki/Vector%20calculus%20identities
|
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables , also called a scalar field, the gradient is the vector field:
where are orthogonal unit vectors in arbitrary directions.
As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.
For a vector field , also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:
For a tensor field of any order k, the gradient is a tensor field of order k + 1.
For a tensor field of order k > 0, the tensor field of order k + 1 is defined by the recursive relation
where is an arbitrary constant vector.
Divergence
In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function:
As the name implies the divergence is a measure of how much vectors are diverging.
The divergence of a tensor field of non-zero order k is written as , a contraction to a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,
where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
For a tensor field of order k > 1, the tensor field of order k − 1 is defined by the recursive relation
where is an arbitrary constant vector.
Curl
In Cartesian coordinates, for the curl is the vector field:
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.
As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.
In Einstein notation, the vector field has curl given by:
where = ±1 or 0 is the Levi-Civita parity symbol.
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used:
Specifically, for the outer product of two vectors,
Laplacian
In Cartesian coordinates, the Laplacian of a function is
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
When the Laplacian is equal to 0, the function is called a harmonic function. That is,
For a tensor field, , the Laplacian is generally written as:
and is a tensor field of the same order.
For a tensor field of order k > 0, the tensor field of order k is defined by the recursive relation
where is an arbitrary constant vector.
Special notatio
|
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres%20theorem
|
In mathematics, the Erdős–Szekeres theorem asserts that, given r, s, any sequence of distinct real numbers with length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same 1935 paper that mentions the Happy Ending problem.
It is a finitary result that makes precise one of the corollaries of Ramsey's theorem. While Ramsey's theorem makes it easy to prove that every infinite sequence of distinct real numbers contains a monotonically increasing infinite subsequence or a monotonically decreasing infinite subsequence, the result proved by Paul Erdős and George Szekeres goes further.
Example
For r = 3 and s = 2, the formula tells us that any permutation of three numbers has an increasing subsequence of length three or a decreasing subsequence of length two. Among the six permutations of the numbers 1,2,3:
1,2,3 has an increasing subsequence consisting of all three numbers
1,3,2 has a decreasing subsequence 3,2
2,1,3 has a decreasing subsequence 2,1
2,3,1 has two decreasing subsequences, 2,1 and 3,1
3,1,2 has two decreasing subsequences, 3,1 and 3,2
3,2,1 has three decreasing length-2 subsequences, 3,2, 3,1, and 2,1.
Alternative interpretations
Geometric interpretation
One can interpret the positions of the numbers in a sequence as x-coordinates of points in the Euclidean plane, and the numbers themselves as y-coordinates; conversely, for any point set in the plane, the y-coordinates of the points, ordered by their x-coordinates, forms a sequence of numbers (unless two of the points have equal x-coordinates). With this translation between sequences and point sets, the Erdős–Szekeres theorem can be interpreted as stating that in any set of at least rs − r − s + 2 points we can find a polygonal path of either r − 1 positive-slope edges or s − 1 negative-slope edges. In particular (taking r = s), in any set of at least n points we can find a polygonal path of at least ⌊⌋ edges with same-sign slopes. For instance, taking r = s = 5, any set of at least 17 points has a four-edge path in which all slopes have the same sign.
An example of rs − r − s + 1 points without such a path, showing that this bound is tight, can be formed by applying a small rotation to an (r − 1) by (s − 1) grid.
Permutation pattern interpretation
The Erdős–Szekeres theorem may also be interpreted in the language of permutation patterns as stating that every permutation of length at least rs + 1 must contain either the pattern 1, 2, 3, ..., r + 1 or the pattern s + 1, s, ..., 2, 1.
Proofs
The Erdős–Szekeres theorem can be proved in several different ways; surveys six different proofs of the Erdős–Szekeres theorem, including the following two.
Other proofs surveyed by Steele include the original proof by Erdős and Szekeres as well as those of , , and .
Pigeonhole principle
Given a sequence of length (r − 1)(s − 1) + 1, label each number ni in the se
|
https://en.wikipedia.org/wiki/Alfred%20Horn
|
Alfred Horn (February 17, 1918 – April 16, 2001) was an American mathematician notable for his work in lattice theory and universal algebra. His 1951 paper "On sentences which are true of direct unions of algebras" described Horn clauses and Horn sentences, which later would form the foundation of logic programming.
Biography
Horn was born on Lower East Side, Manhattan. His parents were both deaf, and his father died when Horn was three years old. At this point, the children moved in with their grandparents on the mother's side. They would later move to Brooklyn where Horn spent most of his childhood, raised by his extended family.
Horn attended the City College of New York, and later, New York University where he earned a Master's degree in mathematics. He went on to earn his Doctor of Philosophy at University of California, Berkeley in 1946. A year later, he started work at the University of California, Los Angeles, where he stayed until his retirement in 1988.
He died in 2001 in Pacific Palisades, Los Angeles after eight years of battling prostate cancer.
References
Alfred Horn, Palisadian Since 1954 and Noted UCLA Math Professor – obituary from UCLA
Publications of Alfred Horn – a list compiled by Dimiter Skordev
Alfred Horn – the information about him in the Mathematics Genealogy Project
1918 births
2001 deaths
20th-century American mathematicians
American logicians
Lattice theorists
City College of New York alumni
New York University alumni
University of California, Berkeley alumni
University of California, Los Angeles faculty
Philosophers from New York (state)
Philosophers from California
People from the Lower East Side
People from Brooklyn
|
https://en.wikipedia.org/wiki/Geometric%20genus
|
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one.
In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on . This definition, as the dimension of
then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle.
The geometric genus is the first invariant of a sequence of invariants called the plurigenera.
Case of curves
In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree .
The notion of genus features prominently in the statement of the Riemann–Roch theorem (see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus
where s is the number of singularities.
If is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree , then its normal line bundle is the Serre twisting sheaf , so by the adjunction formula, the canonical line bundle of is given by
Genus of singular varieties
The definition of geometric genus is carried over classically to singular curves , by decreeing that
is the geometric genus of the normalization . That is, since the mapping
is birational, the definition is extended by birational invariance.
See also
Genus (mathematics)
Arithmetic genus
Invariants of surfaces
Notes
References
Algebraic varieties
|
https://en.wikipedia.org/wiki/Tverberg%27s%20theorem
|
In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of
points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
The special case r = 2 was proved earlier by Radon, and it is known as Radon's theorem.
Examples
The case d = 1 states that any 2r-1 points on the real line can be partitioned into r subsets with intersecting convex hulls. Indeed, if the points are x1 < x2 < ... < x2r < x2r-1, then the partition into Ai = {xi, x2r-i} for i in 1,...,r satisfies this condition (and it is unique).
For r = 2, Tverberg's theorem states that any d + 2 points may be partitioned into two subsets with intersecting convex hulls. This is known as Radon's theorem. In this case, for points in general position, the partition is unique.
The case r = 3 and d = 2 states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustration shows an example in which the seven points are the vertices of a regular heptagon. As the example shows, there may be many different Tverberg partitions of the same set of points; these seven points may be partitioned in seven different ways that differ by rotations of each other.
Topological Tverberg Theorem
An equivalent formulation of Tverberg's theorem is:Let d, r be positive integers, and let N := (d+1)(r-1). If ƒ is any affine function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such that and .They are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an affine function from ΔN to Rd. Let be the vertices of ΔN, and let be their images under ƒ. By the original formulation, the can be partitioned into r disjoint subsets, e.g. ((xi)i in Aj)j in [r] with overlapping convex hull. Because f is affine, the convex hull of (xi)i in Aj is the image of the face spanned by the vertices (vi)i in Aj for all j in [r]. These faces are pairwise-disjoint, and their images under f intersect - as claimed by the new formulation.
The topological Tverberg theorem generalizes this formluation. It allows f to be any continuous function - not necessarily affine. But, currently it is proved only for the case where r is a prime power:Let d be a positive integer, and let r be a power of a prime number. Let N := (d+1)(r-1). If ƒ is any continuous function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is:
|
https://en.wikipedia.org/wiki/Bockstein%20homomorphism
|
In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence
of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one,
To be more precise, C should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with C (some flat module condition should enter). The construction of β is by the usual argument (snake lemma).
A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have
The Bockstein homomorphism associated to the coefficient sequence
is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:
,
;
in other words, it is a superderivation acting on the cohomology mod p of a space.
See also
Bockstein spectral sequence
References
.
Algebraic topology
Homological algebra
|
https://en.wikipedia.org/wiki/Cartan%20formula
|
In mathematics, Cartan formula can mean:
one in differential geometry: , where , and are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula. This formula is named after Élie Cartan.
one in algebraic topology, which is one of the five axioms of Steenrod algebra. It reads:
.
See Steenrod algebra for the detail. The name derives from Henri Cartan, son of Élie.
Footnotes
See also
List of things named after Élie Cartan
|
https://en.wikipedia.org/wiki/Vlastimil%20Pt%C3%A1k
|
Vlastimil Pták (; November 8, 1925 in Prague – May 5 1999) was a Czech mathematician, who worked in functional analysis, theoretical numerical analysis, and linear algebra. Notable early work include generalizations of the open mapping theorem .
During 1945–49 Vlastimil Pták studied mathematics and physics at the Charles University in Prague. Later, he worked at the university and since 1952 in Mathematical Institute of Czechoslovak Academy of Sciences. In 1965 he was named professor at the Charles University. He has published more than 160 mathematical research papers. He had three Ph.D. students, Nicholas Young, Michal Zajac and Miroslav Engliš.
Selected publications
Completeness and the open mapping theorem. Bull. Soc. Math. France 86 1958 41–74. Text online
On complete topological linear spaces. Czechoslovak Math. J. 3(78), (1953). 301–364.
On matrices with non-positive off-diagonal elements and positive principal minors. (with Miroslav Fiedler) Czechoslovak Math. J. 12 (87) 1962 382–400.
References
External links
Short obituary
Short biography (in Czech)
Overview of Pták's work
Seventy years of Professor Vlastimil Pták: Biography and interview (PDF or Postscript file, requires subscription)
Czechoslovak mathematicians
1999 deaths
1925 births
20th-century Czech mathematicians
Charles University alumni
Academic staff of Charles University
|
https://en.wikipedia.org/wiki/Complete%20intersection
|
In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials:
in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V.
Geometrically, each Fi defines a hypersurface; the intersection of these hypersurfaces should be V. The intersection of hypersurfaces will always have dimension at least m, assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to m, with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then V is automatically a hypersurface and there is nothing to prove.
Examples
Easy examples of complete intersections are given by hypersurfaces which are defined by the vanishing locus of a single polynomial. For example,
gives an example of a quintic threefold. It can be difficult to find explicit examples of complete intersections of higher dimensional varieties using two or more explicit examples (bestiary), but, there is an explicit example of a 3-fold of type given by
Non-examples
Twisted cubic
One method for constructing local complete intersections is to take a projective complete intersection variety and embed it into a higher dimensional projective space. A classic example of this is the twisted cubic in : it is a smooth local complete intersection meaning in any chart it can be expressed as the vanishing locus of two polynomials, but globally it is expressed by the vanishing locus of more than two polynomials. We can construct it using the very ample line bundle over giving the embedding
by
Note that . If we let the embedding gives the following relations:
Hence the twisted cubic is the projective scheme
Union of varieties differing in dimension
Another convenient way to construct a non complete intersection, which can never be a local complete intersection, is by taking the union of two different varieties where their dimensions do not agree. For example, the union of a line and a plane intersecting at a point is a classic example of this phenomenon. It is given by the scheme
Multidegree
A complete intersection has a multidegree, written as the tuple (properly though a multiset) of the degrees of defining hypersurfaces. For example, taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position is an elliptic curve. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira.
General position
For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a transversality condition (like their tangent spaces being in general positio
|
https://en.wikipedia.org/wiki/Complete%20intersection%20ring
|
In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as the local rings that can be defined using the "minimum possible" number of relations.
For Noetherian local rings, there is the following chain of inclusions:
Definition
A local complete intersection ring is a Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence. Taking the completion is a minor technical complication caused by the fact that not all local rings are quotients of regular ones. For rings that are quotients of regular local rings, which covers most local rings that occur in algebraic geometry, it is not necessary to take completions in the definition.
There is an alternative intrinsic definition that does not depend on embedding the ring in a regular local ring.
If R is a Noetherian local ring with maximal ideal m, then the dimension of m/m2 is called the embedding dimension emb dim (R) of R. Define a graded algebra H(R) as the homology of the Koszul complex with respect to a minimal system of generators of m/m2; up to isomorphism this only depends on R and not on the choice of the generators of m. The dimension of H1(R) is denoted by ε1 and is called the first deviation of R; it vanishes if and only if R is regular.
A Noetherian local ring is called a complete intersection ring if its
embedding dimension is the sum of the dimension and the first deviation:
emb dim(R) = dim(R) + ε1(R).
There is also a recursive characterization of local complete intersection rings that can be used as a definition, as follows. Suppose that R is a complete Noetherian local ring. If R has dimension greater than 0 and x is an element in the maximal ideal that is not a zero divisor then R is a complete intersection ring if and only if R/(x) is. (If the maximal ideal consists entirely of zero divisors then R is not a complete intersection ring.) If R has dimension 0, then showed that it is a complete intersection ring if and only if the Fitting ideal of its maximal ideal is non-zero.
Examples
Regular local rings
Regular local rings are complete intersection rings, but the converse is not true: the ring is a 0-dimensional complete intersection ring that is not regular.
Not a complete intersection
An example of a locally complete intersection ring which is not a complete intersection ring is given by which has length 3 since it is isomorphic as a vector space to .
Counterexample
Complete intersection local rings are Gorenstein rings, but the converse is not true: the ring is a 0-dimensional Gorenstein ring that is not a complete intersection ring. As a -vector space this ring is isomorphic to
, where , and
showing it is Gorenstein since the top-degree component is dimension and it satisfies the Poincare property. It is not a local complete intersection ring because the idea
|
https://en.wikipedia.org/wiki/Altitude%20%28disambiguation%29
|
Altitude is the height of an object over a datum.
It may also refer to:
Science and mathematics
Altitude (astronomy), one of the angular coordinates of the horizontal coordinate system
Altitude (triangle), in geometry, a line passing through one vertex of a triangle and perpendicular to the opposite side
Music
Altitude (ALT album), the collaborative album released by Andy White, Tim Finn and Liam O'Moanlai under the name ALT
Altitude (Autumn album), the album by Dutch rockband Autumn
Altitude (Yellow Second album), the album by pop punk band Yellow Second
Altitude (Joe Morris album), the album by jazz guitarist Joe Morris
Other uses
Altitude (building), a proposed skyscraper in Sri Lanka
Altitude (film), a 2010 Canadian horror film directed by Kaare Andrews
Altitude (computer game), a 2D aerial combat game released in 2009
Altitude (G.I. Joe), a fictional character in the G.I. Joe universe
Altitude Sports and Entertainment, a regional sports network in Colorado
See also
Altitude Film Entertainment, a British film production and distribution company
Major Altitude, a fictional character in the G.I. Joe universe
|
https://en.wikipedia.org/wiki/Open-access%20poll
|
An open-access poll is a type of opinion poll in which a nonprobability sample of participants self-select into participation. The term includes call-in, mail-in, and some online polls.
The most common examples of open-access polls ask people to phone a number, click a voting option on a website, or return a coupon cut from a newspaper. By contrast, professional polling companies use a variety of techniques to attempt to ensure that the polls they conduct are representative, reliable and scientific. The most glaring difference between an open-access poll and a scientific poll is that scientific polls typically randomly select their samples and sometimes use statistical weights to make them representative of the target population.
Advantages and disadvantages
Since participants in an open-access poll are volunteers rather than a random sample, such polls represent the most interested individuals, just as in voting. In the case of political polls, such participants might be more likely voters.
Because no sampling frame is used to draw the sample of participants, open-access polls may not have participants that represent the larger population. Indeed, they may be composed simply of individuals who happen to hear about the poll. As a consequence, the results of the poll cannot be generalized, but are only representative of the participants of the poll.
One example of an error produced by an open access-poll was one taken by The Literary Digest to predict the 1936 United States presidential election. Similar polls by the magazine had correctly predicted the outcome of the four earlier presidential elections. The magazine's 1936 poll suggested that Alfred Landon would defeat Franklin D. Roosevelt by an overwhelming margin. In fact, the opposite happened. Later studies suggested that the main reason for the error was that Roosevelt's opponents were more vocal and thus more willing to respond to the magazine, compared to the silent majority who supported Roosevelt. By contrast, scientific opinion polls taken by George Gallup correctly showed a clear lead for Roosevelt, albeit still noticeably lower than what he achieved.
A way to minimize that bias is to weigh the results in order to make them more representative of the overall population. This does not make the results of the poll completely representative of the population but it does help increase the chances of the results representing the overall population.
Online poll
An online poll is a survey in which participants communicate responses via the Internet, typically by completing a questionnaire in a web page. Online polls may allow anyone to participate, or they may be restricted to a sample drawn from a larger panel. The use of online panels has become increasingly popular and is now the single biggest research method in Australia.
Proponents of scientific online polling state that in practice their results are no less reliable than traditional polls, and that the problems faced by trad
|
https://en.wikipedia.org/wiki/Grassmann%20number
|
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.
Informal discussion
Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were defined as derivatives, and instead, simply contemplate a situation where one has objects that anti-commute, and have no other pre-defined or pre-supposed properties. Such objects form an algebra, and specifically the Grassmann algebra or exterior algebra.
The Grassmann numbers are elements of that algebra. The appellation of "number" is justified by the fact that they behave not unlike "ordinary" numbers: they can be added, multiplied and divided: they behave almost like a field. More can be done: one can consider polynomials of Grassmann numbers, leading to the idea of holomorphic functions. One can take derivatives of such functions, and then consider the anti-derivatives as well. Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics. The analogy does not stop there: one has an entire branch of supermathematics, where the analog of Euclidean space is superspace, the analog of a manifold is a supermanifold, the analog of a Lie algebra is a Lie superalgebra and so on. The Grassmann numbers are the underlying construct that make this all possible.
Of course, one could pursue a similar program for any other field, or even ring, and this is indeed widely and commonly done in mathematics. However, supermathematics takes on a special significance in physics, because the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle. Thus, the study of Grassmann numbers, and of supermathematics, in general, is strongly driven by their utility in physics.
Specifically, in quantum field theory, or more narrowly, second quantization, one works with ladder operators that create multi-particle quantum states. The ladder operators for fermions create field quanta that must necessarily have anti-symmetric wave functions, as this is forced by the Pauli exclusion principle. In this situation, a Grassmann number
|
https://en.wikipedia.org/wiki/Adjunction%20formula
|
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.
Adjunction for smooth varieties
Formula for a smooth subvariety
Let X be a smooth algebraic variety or smooth complex manifold and Y be a smooth subvariety of X. Denote the inclusion map by i and the ideal sheaf of Y in X by . The conormal exact sequence for i is
where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism
where denotes the dual of a line bundle.
The particular case of a smooth divisor
Suppose that D is a smooth divisor on X. Its normal bundle extends to a line bundle on X, and the ideal sheaf of D corresponds to its dual . The conormal bundle is , which, combined with the formula above, gives
In terms of canonical classes, this says that
Both of these two formulas are called the adjunction formula.
Examples
Degree d hypersurfaces
Given a smooth degree hypersurface we can compute its canonical and anti-canonical bundles using the adjunction formula. This reads aswhich is isomorphic to .
Complete intersections
For a smooth complete intersection of degrees , the conormal bundle is isomorphic to , so the determinant bundle is and its dual is , showingThis generalizes in the same fashion for all complete intersections.
Curves in a quadric surface
embeds into as a quadric surface given by the vanishing locus of a quadratic polynomial coming from a non-singular symmetric matrix. We can then restrict our attention to curves on . We can compute the cotangent bundle of using the direct sum of the cotangent bundles on each , so it is . Then, the canonical sheaf is given by , which can be found using the decomposition of wedges of direct sums of vector bundles. Then, using the adjunction formula, a curve defined by the vanishing locus of a section , can be computed as
Poincaré residue
The restriction map is called the Poincaré residue. Suppose that X is a complex manifold. Then on sections, the Poincaré residue can be expressed as follows. Fix an open set U on which D is given by the vanishing of a function f. Any section over U of can be written as s/f, where s is a holomorphic function on U. Let η be a section over U of ωX. The Poincaré residue is the map
that is, it is formed by applying the vector field ∂/∂f to the volume form η, then multiplying by the holomorphic function s. If U admits local coordinates z1, ..., zn such that for some i, ∂f/∂zi ≠ 0, then this can also be expressed as
Another way of viewing Poincaré residue first reinterprets the adjunction formula as an isomorphism
On an open set U as before, a section of is the product of a holomorphic function s with the form . The Poincaré residue is the map that takes the wedge pro
|
https://en.wikipedia.org/wiki/Rotation%20system
|
In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex.
A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface.
Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation-preserving topological equivalence). Conversely, any embedding of a connected multigraph G on an oriented closed surface defines a unique rotation system having G as its underlying multigraph. This fundamental equivalence between rotation systems and 2-cell-embeddings was first settled in a dual form by Lothar Heffter in the 1890s and extensively used by Ringel during the 1950s. Independently, Edmonds gave the primal form of the theorem and the details of his study have been popularized by Youngs. The generalization to multigraphs was presented by Gross and Alpert.
Rotation systems are related to, but not the same as, the rotation maps used by Reingold et al. (2002) to define the zig-zag product of graphs. A rotation system specifies a circular ordering of the edges around each vertex, while a rotation map specifies a (non-circular) permutation of the edges at each vertex. In addition, rotation systems can be defined for any graph, while as Reingold et al. define them rotation maps are restricted to regular graphs.
Formal definition
Formally, a rotation system is defined as a pair (σ, θ) where σ and θ are permutations acting on the same ground set B, θ is a fixed-point-free involution, and the group <σ, θ> generated by σ and θ acts transitively on B.
To derive a rotation system from a 2-cell embedding of a connected multigraph G on an oriented surface, let B consist of the darts (or flags, or half-edges) of G; that is, for each edge of G we form two elements of B, one for each endpoint of the edge. Even when an edge has the same vertex as both of its endpoints, we create two darts for that edge. We let θ(b) be the other dart formed from the same edge as b; this is clearly an involution with no fixed points. We let σ(b) be the dart in the clockwise position from b in the cyclic order of edges incident to the same vertex, where "clockwise" is defined by the orientation of the surface.
If a multigraph is embedded on an orientable but not oriented surface, it generally corresponds to two rotation systems, one for each of the two orientations of the surface. These two rotation systems have the same involution θ, but the permutation σ for one rotation system is the inverse of the corresponding permutation for the other rotation system.
Recovering the embedding from the rotation system
To recover a multigraph from a rotation system, we form a vertex for each orbit of σ, and an edge for
|
https://en.wikipedia.org/wiki/Peetre%20theorem
|
In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor, defined in a very general way, can in fact be shown to be a polynomial because of some extraneous condition or symmetry imposed upon it.
This article treats two forms of the Peetre theorem. The first is the original version which, although quite useful in its own right, is actually too general for most applications.
The original Peetre theorem
Let M be a smooth manifold and let E and F be two vector bundles on M. Let
be the spaces of smooth sections of E and F. An operator
is a morphism of sheaves which is linear on sections such that the support of D is non-increasing: supp Ds ⊆ supp s for every smooth section s of E. The original Peetre theorem asserts that, for every point p in M, there is a neighborhood U of p and an integer k (depending on U) such that D is a differential operator of order k over U. This means that D factors through a linear mapping iD from the k-jet of sections of E into the space of smooth sections of F:
where
is the k-jet operator and
is a linear mapping of vector bundles.
Proof
The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rn and E and F are trivial bundles. At this point, it relies primarily on two lemmas:
Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x and a positive integer k such that for any y∈V\{x} and for any section s of E whose k-jet vanishes at y (jks(y)=0), we have |Ds(y)|<C.
Lemma 2. The first lemma is sufficient to prove the theorem.
We begin with the proof of Lemma 1.
Suppose the lemma is false. Then there is a sequence xk tending to x, and a sequence of very disjoint balls Bk around the xk (meaning that the geodesic distance between any two such balls is non-zero), and sections sk of E over each Bk such that jksk(xk)=0 but |Dsk(xk)|≥C>0.
Let ρ(x) denote a standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B1/2(0), which vanishes to infinite order on the boundary of the unit ball.
Consider every other section s2k. At x2k, these satisfy
j2ks2k(x2k)=0.
Suppose that 2k is given. Then, since these functions are smooth and each satisfy j2k(s2k)(x2k)=0, it is possible to specify a smaller ball B′δ(x2k) such that the higher order derivatives obey the following estimate:
where
Now
is a standard bump function supported in B′δ(x2k), and the derivative of the product s2kρ2k is bounded in such a way that
As a result, because the following series and all of the partial sums of its derivatives converge uniformly
q(y) is a smooth fun
|
https://en.wikipedia.org/wiki/Planar%20lamina
|
In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integration.
Planar laminas can be used to determine moments of inertia, or center of mass of flat figures, as well as an aid in corresponding calculations for 3D bodies.
Definition
Basically, a planar lamina is defined as a figure (a closed set) of a finite area in a plane, with some mass .
This is useful in calculating moments of inertia or center of mass for a constant density, because the mass of a lamina is proportional to its area. In a case of a variable density, given by some (non-negative) surface density function the mass of the planar lamina is a planar integral of over the figure:
Properties
The center of mass of the lamina is at the point
where is the moment of the entire lamina about the y-axis and is the moment of the entire lamina about the x-axis:
with summation and integration taken over a planar domain .
Example
Find the center of mass of a lamina with edges given by the lines and where the density is given as .
For this the mass must be found as well as the moments and .
Mass is which can be equivalently expressed as an iterated integral:
The inner integral is:
Plugging this into the outer integral results in:
Similarly are calculated both moments:
with the inner integral:
which makes:
and
Finally, the center of mass is
References
Measure theory
|
https://en.wikipedia.org/wiki/Order%20dimension
|
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order.
first studied order dimension; for a more detailed treatment of this subject than provided here, see .
Formal definition
The dimension of a finite poset P is the least integer t for which there exists a family
of linear extensions of P so that, for every x and y in P, x precedes y in P if and only if it precedes y in all of the linear extensions. That is,
An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. if and only if for all i (, ).
Realizers
A family of linear orders on X is called a realizer of a poset P = (X, <P) if
,
which is to say that for any x and y in X,
x <P y precisely when x <1 y, x <2 y, ..., and x <t y.
Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P."
It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair (x,y) of P, y <i x for some order
<i in R.
Example
Let n be a positive integer, and let P be the partial order on the elements ai and bi (for 1 ≤ i ≤ n) in which ai ≤ bj whenever i ≠ j, but no other pairs are comparable. In particular, ai and bi are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.
Then, for each i, any realizer must contain a linear order that begins with all the aj except ai (in some order), then includes bi, then ai, and ends with all the remaining bj. This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have ai preceding bi, which would contradict the incomparability of ai and bi in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by Sn.
Order dimension two
The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different partial order . That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x, y of distinct elements is comparable in exactly one of these two partial orders. If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the per
|
https://en.wikipedia.org/wiki/Jan%20Mauersberger
|
Jan Mauersberger (born 17 June 1985) is a retired German footballer who played as a defender.
Mauersberger retired at the end of the 2018/19 season.
Career statistics
References
External links
1985 births
Living people
German men's footballers
Germany men's youth international footballers
FC Bayern Munich II players
SpVgg Greuther Fürth players
VfL Osnabrück players
Karlsruher SC players
Footballers from Munich
Men's association football defenders
2. Bundesliga players
3. Liga players
TSV 1860 Munich players
|
https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod%20axioms
|
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.
One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.
If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism.
Formal definition
The Eilenberg–Steenrod axioms apply to a sequence of functors from the category of pairs of topological spaces to the category of abelian groups, together with a natural transformation called the boundary map (here is a shorthand for . The axioms are:
Homotopy: Homotopic maps induce the same map in homology. That is, if is homotopic to , then their induced homomorphisms are the same.
Excision: If is a pair and U is a subset of A such that the closure of U is contained in the interior of A, then the inclusion map induces an isomorphism in homology.
Dimension: Let P be the one-point space; then for all .
Additivity: If , the disjoint union of a family of topological spaces , then
Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions and :
If P is the one point space, then is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
Consequences
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.
The homology of some relatively simple spaces, such as n-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (n − 1)-sphere is not a retract of the n-disk. This is used in a proof of the Brouwer fixed point theorem.
Dimension axiom
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory, which are extraordinary cohomology theories, and come with homology theories dual to them.
See also
Zig-zag lemma
Notes
References
Homology theory
Mathematical axioms
|
https://en.wikipedia.org/wiki/%E2%89%A4
|
≤ may refer to:
Inequality (mathematics), relation between values; a ≤ b means "a is less than or equal to b"
Subgroup, a subset of a given group in group theory; H ≤ G is read as "H is a subgroup of G"
|
https://en.wikipedia.org/wiki/Phil%20Moorby
|
Phil Moorby () was a British engineer and computer scientist. Moorby was born and brought up in Birmingham, England, and studied Mathematics at Southampton University, England. Moorby received his master's degree in computer science from Manchester University, England, in 1974. He moved to the United States in 1983.
While working in Gateway Design Automation, in 1984 he invented the Verilog hardware description language, and developed the first and industry standard simulator Verilog-XL. In 1990 Gateway was purchased by Cadence Design Systems.
In 1997, Moorby joined startup company SynaPix, where he worked on match moving and video tracking algorithms for automatically extracting 3D models from video frames, using techniques such as optical flow, motion field and point clouds.
Moorby joined Co-Design Automation in 1999, and in 2002 he joined Synopsys to work on SystemVerilog verification language.
On October 10, 2005, Moorby became the recipient of the 2005 Phil Kaufman Award for his contributions to the EDA industry, specifically for development and popularization of Verilog, one of the world's most popular tools of electronic design automation.
In April 2016, Moorby was made a Fellow of the Computer History Museum, "for his invention and promotion of the Verilog hardware description language."
Philip Raymond Moorby passed away on September 15, 2022 at the age of 69 in Rockport, MA.
References
American computer scientists
Electronic design automation people
People associated with the Department of Computer Science, University of Manchester
Living people
Year of birth missing (living people)
|
https://en.wikipedia.org/wiki/Weierstrass%20point
|
In mathematics, a Weierstrass point on a nonsingular algebraic curve defined over the complex numbers is a point such that there are more functions on , with their poles restricted to only, than would be predicted by the Riemann–Roch theorem.
The concept is named after Karl Weierstrass.
Consider the vector spaces
where is the space of meromorphic functions on whose order at is at least and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on ; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if is the genus of , the dimension from the -th term is known to be
for
Our knowledge of the sequence is therefore
What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: has dimension as most 1 because if and have the same order of pole at , then will have a pole of lower order if the constant is chosen to cancel the leading term). There are question marks here, so the cases or need no further discussion and do not give rise to Weierstrass points.
Assume therefore . There will be steps up, and steps where there is no increment. A non-Weierstrass point of occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like
Any other case is a Weierstrass point. A Weierstrass gap for is a value of such that no function on has exactly a -fold pole at only. The gap sequence is
for a non-Weierstrass point. For a Weierstrass point it contains at least one higher number. (The Weierstrass gap theorem or Lückensatz is the statement that there must be gaps.)
For hyperelliptic curves, for example, we may have a function with a double pole at only. Its powers have poles of order and so on. Therefore, such a has the gap sequence
In general if the gap sequence is
the weight of the Weierstrass point is
This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is
For example, a hyperelliptic Weierstrass point, as above, has weight Therefore, there are (at most) of them.
The ramification points of the ramified covering of degree two from a hyperelliptic curve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve of genus .
Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a numerical semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by R.-O. Buchweitz in 1980 and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16 (see ). A definition of Weierstrass point for a nonsingular curve ove
|
https://en.wikipedia.org/wiki/Tate%20conjecture
|
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.
Statement of the conjecture
Let V be a smooth projective variety over a field k which is finitely generated over its prime field. Let ks be a separable closure of k, and let G be the absolute Galois group Gal(ks/k) of k. Fix a prime number ℓ which is invertible in k. Consider the ℓ-adic cohomology groups (coefficients in the ℓ-adic integers Zℓ, scalars then extended to the ℓ-adic numbers Qℓ) of the base extension of V to ks; these groups are representations of G. For any i ≥ 0, a codimension-i subvariety of V (understood to be defined over k) determines an element of the cohomology group
which is fixed by G. Here Qℓ(i ) denotes the ith Tate twist, which means that this representation of the Galois group G is tensored with the ith power of the cyclotomic character.
The Tate conjecture states that the subspace WG of W fixed by the Galois group G is spanned, as a Qℓ-vector space, by the classes of codimension-i subvarieties of V. An algebraic cycle means a finite linear combination of subvarieties; so an equivalent statement is that every element of WG is the class of an algebraic cycle on V with Qℓ coefficients.
Known cases
The Tate conjecture for divisors (algebraic cycles of codimension 1) is a major open problem. For example, let f : X → C be a morphism from a smooth projective surface onto a smooth projective curve over a finite field. Suppose that the generic fiber F of f, which is a curve over the function field k(C), is smooth over k(C). Then the Tate conjecture for divisors on X is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian variety of F. By contrast, the Hodge conjecture for divisors on any smooth complex projective variety is known (the Lefschetz (1,1)-theorem).
Probably the most important known case is that the Tate conjecture is true for divisors on abelian varieties. This is a theorem of Tate for abelian varieties over finite fields, and of Faltings for abelian varieties over number fields, part of Faltings's solution of the Mordell conjecture. Zarhin extended these results to any finitely generated base field. The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves C1 × ... × Cn.
The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties. Namely, for any abelian varieties A and B over a finitely generated field k, the natural map
is an isomorphism. In particular, an abelian variety A is determined up to isogeny by the Galois representation on its Tate module H1(Aks, Zℓ).
The Tate co
|
https://en.wikipedia.org/wiki/Hedge%20%28linguistics%29
|
In the linguistic sub-fields of applied linguistics and pragmatics, a hedge is a word or phrase used in a sentence to express ambiguity, probability, caution, or indecisiveness about the remainder of the sentence, rather than full accuracy, certainty, confidence, or decisiveness. Hedges can also allow speakers and writers to introduce (or occasionally even eliminate) ambiguity in meaning and typicality as a category member. Hedging in category membership is used in reference to the prototype theory, to signify the extent to which items are typical or atypical members of different categories. Hedges might be used in writing, to downplay a harsh critique or a generalization, or in speaking, to lessen the impact of an utterance due to politeness constraints between a speaker and addressee.
Typically, hedges are adjectives or adverbs, but can also consist of clauses such as one use of tag questions. In some cases, a hedge could be regarded as a form of euphemism. Linguists consider hedges to be tools of epistemic modality; allowing speakers and writers to signal a level of caution in making an assertion. Hedges are also used to distinguish items into multiple categories, where items can be in a certain category to an extent.
Types of hedges
Hedges may take the form of many different parts of speech, for example:
There might just be a few insignificant problems we need to address. (adjective)
The party was somewhat spoiled by the return of the parents. (adverb)
I'm not an expert but you might want to try restarting your computer. (clause)
That's false, isn't it? (tag question clause)
Using hedges
Hedges are often used in everyday speech, and they can serve many different purposes. Below are a few ways to use hedges with examples to clarify these different functions.
Category membership
A very common use of hedges can be found in signaling typicality of category membership. Different hedges can signal prototypical membership in a category, meaning that member has most of the characteristics that are exemplary of the category. For example;
A robin is a bird par excellence.
This signifies that a robin has all of the typical characteristics of a bird, i.e. feathers, small, lives in a nest, etc.
Loosely speaking, a bat is a bird.
This sentence displays that a bat could technically be called a bird, but the hedge loosely speaking signifies that a bat has fringe membership in the category "bird".
Epistemic hedges
In some cases, "I don't know" functions as a prepositioned hedge—a forward-looking stance marker displaying that the speaker is not fully committed to what follows in their turn of talk.
Hedges may intentionally or unintentionally be employed in both spoken and written language since they are crucially important in communication. Hedges help speakers and writers indicate more precisely how the cooperative principle (expectations of quantity, quality, manner, and relevance) is observed in assessments. For example,
All I know is smoki
|
https://en.wikipedia.org/wiki/Azumaya%20algebra
|
In mathematics, an Azumaya algebra is a generalization of central simple algebras to -algebras where need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964–65. There are now several points of access to the basic definitions.
Over a ring
An Azumaya algebra
over a commutative ring is an -algebra obeying any of the following equivalent conditions:
There exists an -algebra such that the tensor product of -algebras is Morita equivalent to .
The -algebra is Morita equivalent to , where is the opposite algebra of .
The center of is , and is separable.
is finitely generated, faithful, and projective as an -module, and the tensor product is isomorphic to via the map sending to the endomorphism of .
Examples over a field
Over a field , Azumaya algebras are completely classified by the Artin–Wedderburn theorem since they are the same as central simple algebras. These are algebras isomorphic to the matrix ring for some division algebra over whose center is just . For example, quaternion algebras provide examples of central simple algebras.
Examples over local rings
Given a local commutative ring , an -algebra is Azumaya if and only if is free of positive finite rank as an -module, and the algebra is a central simple algebra over , hence all examples come from central simple algebras over .
Cyclic algebras
There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field , hence all elements in the Brauer group (defined below). Given a finite cyclic Galois field extension of degree , for every and any generator there is a twisted polynomial ring , also denoted , generated by an element such that
and the following commutation property holds:
As a vector space over , has basis with multiplication given by
Note that give a geometrically integral variety , there is also an associated cyclic algebra for the quotient field extension .
Brauer group of a ring
Over fields, there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined as the similarity classes of Azumaya algebras over a ring , where rings are similar if there is an isomorphism
of rings for some natural numbers . Then, this equivalence is in fact an equivalence relation, and if , , then , showing
is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted . Another definition is given by the torsion subgroup of the etale cohomology group
which is called the cohomological Brauer group. These two definitions agree when is a field.
Brauer group using Galois
|
https://en.wikipedia.org/wiki/Left-right%20planarity%20test
|
In graph theory, a branch of mathematics, the left-right planarity test
or de Fraysseix–Rosenstiehl planarity criterion is a characterization of planar graphs based on the properties of the depth-first search trees, published by and used by them with Patrice Ossona de Mendez to develop a linear time planarity testing algorithm. In a 2003 experimental comparison of six planarity testing algorithms, this was one of the fastest algorithms tested.
T-alike and T-opposite edges
For any depth-first search of a graph G, the edges
encountered when discovering a vertex for the first time define a depth-first search tree T of G. This is a Trémaux tree, meaning that the remaining edges (the cotree) each connect a pair of vertices that are related to each other as an ancestor and descendant in T. Three types of patterns can be used to define two relations between pairs of cotree edges, named the T-alike and T-opposite relations.
In the following figures, simple circle nodes represent vertices, double circle nodes represent subtrees, twisted segments represent tree paths, and curved arcs represent cotree edges. The root of each tree is shown at the bottom of the figure. In the first figure, the edges labeled and are T-alike, meaning that, at the endpoints nearest the root of the tree, they will both be on the same side of the tree in every planar drawing. In the next two figures, the edges with the same labels are T-opposite, meaning that they will be on different sides of the tree in every planar drawing.
The characterization
Let G be a graph and let T be a Trémaux tree of G. The graph G is planar if and only if there exists a partition of the cotree edges of G into two classes so that any two edges belong to a same class if they are T-alike and any two edges belong to different classes if they are T-opposite.
This characterization immediately leads to an (inefficient) planarity test: determine for all pairs of edges whether they are T-alike or T-opposite, form an auxiliary graph that has a vertex for each
connected component of T-alike edges and an edge for each pair of T-opposite edges, and check whether this auxiliary graph is bipartite. Making this algorithm efficient involves finding a subset of the T-alike and T-opposite pairs that is sufficient to carry out this method without determining the relation between all edge pairs in the input graph.
References
Further reading
Topological graph theory
Planar graphs
|
https://en.wikipedia.org/wiki/Scheinerman%27s%20conjecture
|
In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane . It was proven by .
For instance, the graph G shown below to the left may be represented as the intersection graph of the set of segments shown below to the right. Here, vertices of G are represented by straight line segments and edges of G are represented by intersection points.
Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directions
and no two segments belong to the same line, then the graph can be colored using k colors, one color for each direction. Therefore, if every planar graph can be represented in this way with only four directions,
then the four color theorem follows.
and proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also . proved that every triangle-free planar graph can be represented as an intersection graph of line segments having only three directions; this result implies Grötzsch's theorem that triangle-free planar graphs can be colored with three colors. proved that if a planar graph G can be 4-colored in such a way that no separating cycle uses all four colors, then G has a representation as an intersection graph of segments.
proved that planar graphs are in 1-STRING, the class of intersection graphs of simple curves in the plane that intersect each other in at most one crossing point per pair. This class is intermediate between the intersection graphs of segments appearing in Scheinerman's conjecture and the intersection graphs of unrestricted simple curves from the result of Ehrlich et al. It can also be viewed as a generalization of the circle packing theorem, which shows the same result when curves are allowed to intersect in a tangent. The proof of the conjecture by was based on an improvement of this result.
References
.
.
.
.
.
.
.
.
.
.
.
Planar graphs
Conjectures that have been proved
|
https://en.wikipedia.org/wiki/Whitney%27s%20planarity%20criterion
|
In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid).
In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G'=(V',E') and a bijective correspondence between the edges E' and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E-T form a spanning tree of G'.
Algebraic duals
An equivalent form of Whitney's criterion is that a graph G is planar if and only if it has a dual graph whose graphic matroid is dual to the graphic matroid of G.
A graph whose graphic matroid is dual to the graphic matroid of G is known as an algebraic dual of G. Thus, Whitney's planarity criterion can be expressed succinctly as: a graph is planar if and only if it has an algebraic dual.
Topological duals
If a graph is embedded into a topological surface such as the plane, in such a way that every face of the embedding is a topological disk, then the dual graph of the embedding is defined as the graph (or in some cases multigraph) H that has a vertex for every face of the embedding, and an edge for every adjacency between a pair of faces.
According to Whitney's criterion, the following conditions are equivalent:
The surface on which the embedding exists is topologically equivalent to the plane, sphere, or punctured plane
H is an algebraic dual of G
Every simple cycle in G corresponds to a minimal cut in H, and vice versa
Every simple cycle in H corresponds to a minimal cut in G, and vice versa
Every spanning tree in G corresponds to the complement of a spanning tree in H, and vice versa.
It is possible to define dual graphs of graphs embedded on nonplanar surfaces such as the torus, but these duals do not generally have the correspondence between cuts, cycles, and spanning trees required by Whitney's criterion.
References
Matroid theory
Planar graphs
|
https://en.wikipedia.org/wiki/Home%20runs%20allowed
|
In baseball statistics, home runs allowed (HRA) signifies the total number of home runs a pitcher allowed.
The Major League Baseball record for the most home runs allowed by any pitcher belongs to Jamie Moyer (522 in his career). He gave up home runs while pitching for eight different teams across both leagues. Warren Spahn gave up the most National League home runs (434) and the American League record is 422, held by Frank Tanana. The Minnesota Twins' Bert Blyleven set Major League Baseball's season record in 1986, allowing a total of 50 home runs to opposing batters.
References
External links
Home runs allowed records
Pitching statistics
|
https://en.wikipedia.org/wiki/Operad
|
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.
History
Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972.
Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:
"The name operad and the formal definition appear first in the early 1970's in J. Peter May's "The Geometry of Iterated Loop Spaces", but a year or more earlier, Boardman and Vogt described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane. In fact, there is an abundance of prehistory. Weibel [Wei] points out that the concept first arose a century ago in A.N. Whitehead's "A Treatise on Universal Algebra", published in 1898."
The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer).
Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology in the work of Maxim Kontsevich and Thomas Willwacher.
Intuition
Suppose is a set and for we define
,
the set of all functions from the cartesian product of copies of to .
We can compose these functions: given , , the function
is defined as follows: given arguments from , we divide them into blocks, the first one having arguments, the second one arguments, etc., and then apply to the first block, to the second block, etc. We then apply to the list of values obtained from in such a way.
We can also permute arguments, i.e. we have a right action of the symmetric group on , defined by
for , and .
The definition of a symmetric operad given below captures the essential properties of these two operations and .
Definition
Non-symmetric operad
A non-symmetric operad (sometimes called an operad without permutations, or a non- or plain operad) consists of the following:
a sequence of sets, whose elements are called -ary operations,
an element in called the identity,
for all positive integers , , a composition f
|
https://en.wikipedia.org/wiki/Beta%20prime%20distribution
|
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution. If has a beta distribution, then the odds has a beta prime distribution.
Definitions
Beta prime distribution is defined for with two parameters α and β, having the probability density function:
where B is the Beta function.
The cumulative distribution function is
where I is the regularized incomplete beta function.
The expected value, variance, and other details of the distribution are given in the sidebox; for , the excess kurtosis is
While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.
The mode of a variate X distributed as is .
Its mean is if (if the mean is infinite, in other words it has no well defined mean) and its variance is if .
For , the k-th moment is given by
For with this simplifies to
The cdf can also be written as
where is the Gauss's hypergeometric function 2F1 .
Alternative parameterization
The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ( p. 36).
Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and
β = 2 + ν. Under this parameterization
E[Y] = μ and Var[Y] = μ(1 + μ)/ν.
Generalization
Two more parameters can be added to form the generalized beta prime distribution :
shape (real)
scale (real)
having the probability density function:
with mean
and mode
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.
This generalization can be obtained via the following invertible transformation. If and for , then .
Compound gamma distribution
The compound gamma distribution is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by compounding two gamma distributions:
where is the gamma pdf with shape and inverse scale .
The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.
Another way to express the compounding is if and , then . (This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.)
Properties
If then .
If , and , then .
If then .
If and two iid variables, then with and , as the beta prime distribution is infinitely divisible.
More generally, let iid variables following the same beta prime distribution, i.e. , then the sum with and .
Related distributions
If
|
https://en.wikipedia.org/wiki/Join%20%28topology%29
|
In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in . The join of a space with itself is denoted by . The join is defined in slightly different ways in different contexts
Geometric sets
If and are subsets of the Euclidean space , then:,that is, the set of all line-segments between a point in and a point in .
Some authors restrict the definition to subsets that are joinable: any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if is in and is in , then and are joinable in . The figure above shows an example for m=n=1, where and are line-segments.
Examples
The join of two simplices is a simplex: the join of an n-dimensional and an m-dimensional simplex is an (m+n+1)-dimensional simplex. Some special cases are:
The join of two disjoint points is an interval (m=n=0).
The join of a point and an interval is a triangle (m=0, n=1).
The join of two line segments is homeomorphic to a solid tetrahedron or disphenoid, illustrated in the figure above right (m=n=1).
The join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex.
The join of a point and a polygon (or any polytope) is a pyramid, like the join of a point and square is a square pyramid. The join of a point and a cube is a cubic pyramid.
The join of a point and a circle is a cone, and the join of a point and a sphere is a hypercone.
Topological spaces
If and are any topological spaces, then:
where the cylinder is attached to the original spaces and along the natural projections of the faces of the cylinder:
Usually it is implicitly assumed that and are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder to the spaces and , these faces are simply collapsed in a way suggested by the attachment projections : we form the quotient space
where the equivalence relation is generated by
At the endpoints, this collapses to and to .
If and are bounded subsets of the Euclidean space , and and , where are disjoint subspaces of such that the dimension of their affine hull is (e.g. two non-intersecting non-parallel lines in ), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join":
Abstract simplicial complexes
If and are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:
The vertex set is a disjoint union of and .
The simplices of are all disjoint unions of a simplex of with a simplex of : (in the special case in which and are disjoint, the join is simply ).
Examples
|
https://en.wikipedia.org/wiki/Pl%C3%BCcker%20formula
|
In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its dual, is connected to the other invariants by similar formulae. These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values.
Plücker invariants and basic equations
A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane. Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C* called the dual curve. In the correspondence between the projective plane and its dual, points on C correspond to lines tangent C*, so the dual of C* can be identified with C.
The first two invariants covered by the Plücker formulas are the degree d of the curve C and the degree d*, classically called the class of C. Geometrically, d is the number of times a given line intersects C with multiplicities properly counted. (This includes complex points and points at infinity since the curves are taken to be subsets of the complex projective plane.) Similarly, d* is the number of tangents to C that are lines through a given point on the plane; so for example a conic section has degree and class both 2. If C has no singularities, the first Plücker equation states that
but this must be corrected for singular curves.
Of the double points of C, let δ be the number that are ordinary, i.e. that have distinct tangents (these are also called nodes) or are isolated points, and let κ be the number that are cusps, i.e. having a single tangent (spinodes). If C has higher order singularities then these are counted as multiple double points according to an analysis of the nature of the singularity. For example an ordinary triple point is counted as 3 double points. Again, complex points and points at infinity are included in these counts. The corrected form is of the first Plücker equation is
Similarly, let δ* be the number of ordinary double points, and κ* the number of cusps of C*. Then the second Plücker equation states
The geometric interpretation of an ordinary double point of C* is a line that is tangent to the curve at two points (double tangent) and the geometric interpretation of a cusp of C* is a point of inflection (stationary tangent).
Consider for instance, the case of a smooth cubic:
The above formula shows that it has
inflections. If the cubic degenerates and gets a double point, then 6 points converge to the singular point and only 3 inflection remain along the singular curve. If the cubic degenerates and gets a cusp then only one inflection remains.
Note that the first two Plücker equations have dual versions:
|
https://en.wikipedia.org/wiki/Dual%20curve
|
In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of , given in line coordinates, is known as the tangential equation of . Duality is an involution: the dual of the dual of is the original curve .
The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.
Equations
Let be the equation of a curve in homogeneous coordinates on the projective plane. Let be the equation of a line, with being designated its line coordinates in a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form which is the tangential equation of the curve.
At a point on the curve, the tangent is given by
So is a tangent to the curve if
Eliminating , , , and from these equations, along with , gives the equation in , and of the dual curve.
For example, let be the conic . The dual is found by eliminating , , , and from the equations
The first three equations are easily solved for , , , and substituting in the last equation produces
Clearing from the denominators, the equation of the dual is
Consider a parametrically defined curve in projective coordinates . Its projective tangent line is a linear plane spanned by the point of tangency and the tangent vector, with linear equation coefficients given by the cross product:which in affine coordinates is:
The dual of an inflection point will give a cusp and two points sharing the same tangent line will give a self-intersection point on the dual.
From the projective description, one may compute the dual of the dual:which is projectively equivalent to the original curve .
Properties of dual curve
Properties of the original curve correspond to dual properties on the dual curve. In the Introduction image, the red curve has three singularities – a node in the center, and two cusps at the lower right and lower left. The black curve has no singularities but has four distinguished points: the two top-most points correspond to the node (double point), as they both have the same tangent line, hence map to the same point in the dual curve, while the two inflection points correspond to the cusps, since the tangent lines first go one way then the other (slope increasing, then decreasing).
By contrast, on a smooth, convex curve the angle of the tangent line changes monotonically, and the resulting dual curve is also smooth and convex.
Further, both curves above have a reflectional symmetry: projective duality preserves symmetries a projective space, so dual curves have the same symmetry group. In this case both symmetries are realized as a left-right reflection; this is an
|
https://en.wikipedia.org/wiki/Vector%20bundles%20on%20algebraic%20curves
|
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points).
Some foundational results on classification were known in the 1950s. The result of , that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of on the Riemann–Hilbert problem.
gave the classification of vector bundles on elliptic curves.
The Riemann–Roch theorem for vector bundles was proved by , before the 'vector bundle' concept had really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. He was seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank. This idea would prove fruitful, in terms of moduli spaces of vector bundles. following on the work in the 1960s on geometric invariant theory.
See also
Hitchin system
References
Also in Collected Works vol. I
Algebraic curves
Vector bundles
|
https://en.wikipedia.org/wiki/Opposite%20ring
|
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in R. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see ).
Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.
Relation to automorphisms and antiautomorphisms
In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusion with some unary operation.
A ring having isomorphic opposite ring is called a self-opposite ring, which name indicates that is essentially the same as .
All commutative rings are self-opposite.
Let us define the antiisomorphism
, where for .
It is indeed an antiisomorphism, since .
The antiisomorphism can be defined generally for semigroups, monoids, groups, rings, rngs, algebras. In case of rings (and rngs) we obtain the general equivalence.
A ring is self-opposite if and only if it has at least one antiautomorphism.
Proof:
: Let be self-opposite. If is an isomorphism, then , being a composition of antiisomorphism and isomorphism, is an antiisomorphism from to itself, hence antiautomorphism.
: If is an antiautomorphism, then is an isomorphism as a composition of two antiisomorphisms. So is self-opposite.
and
If is self-opposite and the group of automorphisms is finite, then the number of antiautomorphisms equals the number of automorphisms.
Proof: By the assumption and the above equivalence there exist antiautomorphisms. If we pick one of them and denote it by , then the map , where runs over , is clearly injective but also surjective, since each antiautomorphism for some automorphism .
It can be proven in a similar way, that under the same assumptions the number of isomorphisms from to equals the number of antiautomorphisms of .
If some antiautomorphism is also an automorphism, then for each
Since is bijective, for all and , so the ring is commutative and all antiautomorphisms are automorphisms. By contraposition, if a ring is noncommutative (and self-opposite), then no antiautomorphism is an automorphism.
Denote by the group of all automorphisms together with all antiautomorphisms. The above remarks imply, that if a ring (or rng) is noncommutative and self-opposite. If it is commutative or non-self-opposite, then .
Examples
The smallest noncommutative ring with unity
The smallest such ring has eight elements and it is the only noncommutative ring among 11 rings with unity of order 8, up to isomorphism. It has the additive group . Obviously is antiisomorphic to , as is always the case, but it is also i
|
https://en.wikipedia.org/wiki/Quasi-Lie%20algebra
|
In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom
replaced by
(anti-symmetry).
In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers.
In a quasi-Lie algebra,
Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish.
See also
Whitehead product
References
Lie algebras
|
https://en.wikipedia.org/wiki/Whitehead%20product
|
In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in .
The relevant MSC code is: 55Q15, Whitehead products and generalizations.
Definition
Given elements , the Whitehead bracket
is defined as follows:
The product can be obtained by attaching a -cell to the wedge sum
;
the attaching map is a map
Represent and by maps
and
then compose their wedge with the attaching map, as
The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
Grading
Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so has degree ; equivalently, (setting L to be the graded quasi-Lie algebra). Thus acts on each graded component.
Properties
The Whitehead product satisfies the following properties:
Bilinearity.
Graded Symmetry.
Graded Jacobi identity.
Sometimes the homotopy groups of a space, together with the Whitehead product operation are called a graded quasi-Lie algebra; this is proven in via the Massey triple product.
Relation to the action of
If , then the Whitehead bracket is related to the usual action of on by
where denotes the conjugation of by .
For , this reduces to
which is the usual commutator in . This can also be seen by observing that the -cell of the torus is attached along the commutator in the -skeleton .
Whitehead products on H-spaces
For a path connected H-space, all the Whitehead products on vanish. By the previous subsection, this is a generalization of both the facts that the fundamental groups of H-spaces are abelian,
and that H-spaces are simple.
Suspension
All Whitehead products of classes , lie in the kernel of the suspension homomorphism
Examples
, where is the Hopf map.
This can be shown by observing that the Hopf invariant defines an isomorphism and explicitly calculating the cohomology ring of the cofibre of a map representing . Using the Pontryagin–Thom construction there is a direct geometric argument, using the fact that the preimage of a regular point is a copy of the Hopf link.
Applications to ∞-groupoids
Recall that an ∞-groupoid is an -category generalization of groupoids which is conjectured to encode the data of the homotopy type of in an algebraic formalism. The objects are the points in the space , morphisms are homotopy classes of paths between points, and higher morphisms are higher homotopies of those points.
The existence of the Whitehead product is the main reason why defining a notion of ∞-groupoids is such a demanding task. It was shown that any strict ∞-groupoid has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as .
See also
Generalised Whitehead product
Massey product
Toda bracket
References
Homotopy theory
Lie algebras
|
https://en.wikipedia.org/wiki/B%C3%A9zout%20domain
|
In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French mathematician Étienne Bézout.
Examples
All PIDs are Bézout domains.
Examples of Bézout domains that are not PIDs include the ring of entire functions (functions holomorphic on the whole complex plane) and the ring of all algebraic integers. In case of entire functions, the only irreducible elements are functions associated to a polynomial function of degree 1, so an element has a factorization only if it has finitely many zeroes. In the case of the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer. This shows in both cases that the ring is not a UFD, and so certainly not a PID.
Valuation rings are Bézout domains. Any non-Noetherian valuation ring is an example of a non-noetherian Bézout domain.
The following general construction produces a Bézout domain S that is not a UFD from any Bézout domain R that is not a field, for instance from a PID; the case is the basic example to have in mind. Let F be the field of fractions of R, and put , the subring of polynomials in F[X] with constant term in R. This ring is not Noetherian, since an element like X with zero constant term can be divided indefinitely by noninvertible elements of R, which are still noninvertible in S, and the ideal generated by all these quotients of is not finitely generated (and so X has no factorization in S). One shows as follows that S is a Bézout domain.
It suffices to prove that for every pair a, b in S there exist s, t in S such that divides both a and b.
If a and b have a common divisor d, it suffices to prove this for a/d and b/d, since the same s, t will do.
We may assume the polynomials a and b nonzero; if both have a zero constant term, then let n be the minimal exponent such that at least one of them has a nonzero coefficient of Xn; one can find f in F such that fXn is a common divisor of a and b and divide by it.
We may therefore assume at least one of a, b has a nonzero constant term. If a and b viewed as elements of F[X] are not relatively prime, there is a greatest common divisor of a and b in this UFD that has constant term 1, and therefore lies in S; we can divide by this factor.
We may therefore also assume that a and b are relatively prime in F[X], so that 1
|
https://en.wikipedia.org/wiki/Holomorphic%20functional%20calculus
|
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators.
This article will discuss the case where T is a bounded linear operator on some Banach space. In particular, T can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction.
Motivation
Need for a general functional calculus
In this section T will be assumed to be a n × n matrix with complex entries.
If a given function f is of certain special type, there are natural ways of defining f(T). For instance, if
is a complex polynomial, one can simply substitute T for z and define
where T0 = I, the identity matrix. This is the polynomial functional calculus. It is a homomorphism from the ring of polynomials to the ring of n × n matrices.
Extending slightly from the polynomials, if f : C → C is holomorphic everywhere, i.e. an entire function, with MacLaurin series
mimicking the polynomial case suggests we define
Since the MacLaurin series converges everywhere, the above series will converge, in a chosen operator norm. An example of this is the exponential of a matrix. Replacing z by T in the MacLaurin series of f(z) = ez gives
The requirement that the MacLaurin series of f converges everywhere can be relaxed somewhat. From above it is evident that all that is really needed is the radius of convergence of the MacLaurin series be greater than ǁTǁ, the operator norm of T. This enlarges somewhat the family of f for which f(T) can be defined using the above approach. However it is not quite satisfactory. For instance, it is a fact from matrix theory that every non-singular T has a logarithm S in the sense that eS = T. It is desirable to have a functional calculus that allows one to define, for a non-singular T, ln(T) such that it coincides with S. This can not be done via power series, for example the logarithmic series
converges only on the open unit disk. Substituting T for z in the series fails to give a well-defined expression for ln(T + I) for invertible T + I with ǁTǁ ≥ 1. Thus a more general functional calculus is needed.
Functional calculus and the spectrum
It is expected that a necessary condition for f(T) to make sense is f be defined on the spectrum of T. For example, the spectral theorem for normal matrices states every normal matrix is unitarily diagonalizable. This leads to a definition of f(T) when T is normal. One encounters difficulties if f(λ) is not defined for some eigenvalue λ of T.
Other indications a
|
https://en.wikipedia.org/wiki/Coframe
|
In mathematics, a coframe or coframe field on a smooth manifold is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of , one has a natural map from , given by . If is dimensional a coframe is given by a section of such that . The inverse image under of the complement of the zero section of forms a principal bundle over , which is called the coframe bundle.
References
See also
Frame fields in general relativity
Moving frame
Differential geometry
|
https://en.wikipedia.org/wiki/CR%20manifold
|
In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle such that
(L is formally integrable)
.
The subbundle L is called a CR structure on the manifold M.
The abbreviation CR stands for "Cauchy–Riemann" or "Complex-Real".
Introduction and motivation
The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.
Suppose for instance that M is the hypersurface of given by the equation
where z and w are the usual complex coordinates on . The holomorphic tangent bundle of consists of all linear combinations of the vectors
The distribution L on M consists of all combinations of these vectors which are tangent to M. The tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of
In particular, L consists of the holomorphic vector fields which annihilate F. Note that L gives a CR structure on M, for [L,L] = 0 (since L is one-dimensional) and since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.
More generally, suppose that M is a real hypersurface in with defining equation F(z1, ..., zn) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on :
which annihilate the defining function. In this case, for the same reason as before. Moreover, [L,L] ⊂ L since the commutator of holomorphic vector fields annihilating F is again a holomorphic vector field annihilating F.
Embedded and abstract CR manifolds
There is a sharp contrast between the theories of embedded CR manifolds (hypersurface and edges of wedges in complex space) and abstract CR manifolds (those given by the complex distribution L). Many of the formal geometrical features are similar. These include:
A notion of convexity (supplied by the Levi form)
A differential operator, analogous to the Dolbeault operator, and an associated cohomology (the tangential Cauchy–Riemann complex).
Embedded CR manifolds possess some additional structure, though: a Neumann and Dirichlet problem for the Cauchy–Riemann equations.
This article first treats the geometry of embedded CR manifolds, shows how to define these structures intrinsically, and then generalizes these to the abstract setting.
Embedded CR manifolds
Preliminaries
Embedded CR manifolds are, first and foremost, submanifolds of Define a pair of subbundles of the complexified tangent bundle by:
consists of the complex vectors annihilating the antiholomorphic funct
|
https://en.wikipedia.org/wiki/Barry%20Simon
|
Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics. He has authored more than 400 publications on mathematics and physics.
His work has focused on broad areas of mathematical physics and analysis covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics (including N-body systems and resonances), nonrelativistic quantum mechanics in electric and magnetic fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non-selfadjoint spectral theory.
Early life
Barry Simon's mother was a school teacher, his father was an accountant. Simon attended James Madison High School in Brooklyn.
Career
During his high school years, Simon started attending college courses for highly gifted pupils at Columbia University. In 1962, Simon won a MAA mathematics competition. The New York Times reported that in order to receive full credits for a faultless test result he had to make a submission with MAA. In this submission he proved that one of the problems posed in the test was ambiguous.
In 1962, Simon entered Harvard with a stipend. He became a Putnam Fellow in 1965 at 19 years old. He received his AB in 1966 from Harvard College and his PhD in Physics at Princeton University in 1970, supervised by Arthur Strong Wightman. His dissertation dealt with Quantum mechanics for Hamiltonians defined as quadratic forms.
Following his doctoral studies, Simon took a professorship at Princeton for several years, often working with colleague Elliott H. Lieb on the Thomas-Fermi Theory and Hartree-Fock Theory of atoms in addition to phase transitions and mentoring many of the same students as Lieb. He eventually was persuaded to take a post at Caltech, from which he retired in the summer of 2016.
His status is legendary in mathematical physics and he is renowned for his ability to write scientific manuscripts "in five percent of the time ordinary mortals need to write such papers."
A former graduate student of Simon's, in a tale revealing of his brilliance, once stated:
Honors and awards
1974: Invited Speaker at the International Congress of Mathematicians in Vancouver
1981: Elected fellow of the American Physical Society
1990: Elected correspondent member of the Austrian Academy of Sciences
2005: Elected fellow of the American Academy of Arts and Sciences
2012: Elected fellow of the American Mathematical Society
2012: Awarded the Henri Poincaré Prize
2015: Awarded the Bolyai Prize of the Hungarian Academy of Sciences
2016: Awarded the Steele Prize for Lifetime achievements
2018: Dannie Heineman Prize for Ma
|
https://en.wikipedia.org/wiki/Nakai%20conjecture
|
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.
It states that if V is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then V is a smooth variety. The converse statement, that smooth algebraic varieties have rings of differential operators that are generated by their derivations, is a result of Alexander Grothendieck.
The Nakai conjecture is known to be true for algebraic curves and Stanley–Reisner rings. A proof of the conjecture would also establish the Zariski–Lipman conjecture, for a complex variety V with coordinate ring R. This conjecture states that if the derivations of R are a free module over R, then V is smooth.
References
Algebraic geometry
Singularity theory
Conjectures
Unsolved problems in geometry
|
https://en.wikipedia.org/wiki/Mordell%20curve
|
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer.
These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He showed that every Mordell curve contains only finitely many integer points (x, y). In other words, the differences of perfect squares and perfect cubes tend to infinity. The question of how fast was dealt with in principle by Baker's method. Hypothetically this issue is dealt with by Marshall Hall's conjecture.
Properties
If (x, y) is an integer point on a Mordell curve, then so is (x, -y).
There are certain values of n for which the corresponding Mordell curve has no integer solutions; these values are:
6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... .
−3, −5, −6, −9, −10, −12, −14, −16, −17, −21, −22, ... .
The specific case where n = −2 is also known as Fermat's Sandwich Theorem.
List of solutions
The following is a list of solutions to the Mordell curve y2 = x3 + n for |n| ≤ 25. Only solutions with y ≥ 0 are shown.
In 1998, J. Gebel, A. Pethö, H. G. Zimmer found all integers points for 0 < |n| ≤ 104.
In 2015, M. A. Bennett and A. Ghadermarzi computed integer points for 0 < |n| ≤ 107.
References
External links
J. Gebel, Data on Mordell's curves for –10000 ≤ n ≤ 10000
M. Bennett, Data on Mordell curves for –107 ≤ n ≤ 107
Algebraic curves
Diophantine equations
Elliptic curves
|
https://en.wikipedia.org/wiki/Zariski%20geometry
|
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.
Definition
A Zariski geometry consists of a set X and a topological structure on each of the sets
X, X2, X3, ...
satisfying certain axioms.
(N) Each of the Xn is a Noetherian topological space, of dimension at most n.
Some standard terminology for Noetherian spaces will now be assumed.
(A) In each Xn, the subsets defined by equality in an n-tuple are closed. The mappings
Xm → Xn
defined by projecting out certain coordinates and setting others as constants are all continuous.
(B) For a projection
p: Xm → Xn
and an irreducible closed subset Y of Xm, p(Y) lies between its closure Z and Z \ where is a proper closed subset of Z. (This is quantifier elimination, at an abstract level.)
(C) X is irreducible.
(D) There is a uniform bound on the number of elements of a fiber in a projection of any closed set in Xm, other than the cases where the fiber is X.
(E) A closed irreducible subset of Xm, of dimension r, when intersected with a diagonal subset in which s coordinates are set equal, has all components of dimension at least r − s + 1.
The further condition required is called very ample (cf. very ample line bundle). It is assumed there is an irreducible closed subset P of some Xm, and an irreducible closed subset Q of P× X2, with the following properties:
(I) Given pairs (x, y), (, ) in X2, for some t in P, the set of (t, u, v) in Q includes (t, x, y) but not (t, , )
(J) For t outside a proper closed subset of P, the set of (x, y) in X2, (t, x, y) in Q is an irreducible closed set of dimension 1.
(K) For all pairs (x, y), (, ) in X2, selected from outside a proper closed subset, there is some t in P such that the set of (t, u, v) in Q includes (t, x, y) and (t, , ).
Geometrically this says there are enough curves to separate points (I), and to connect points (K); and that such curves can be taken from a single parametric family.
Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.
References
Model theory
Algebraic curves
Vector bundles
|
https://en.wikipedia.org/wiki/Theta%20divisor
|
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1.
Classical theory
Classical results of Bernhard Riemann describe Θ in another way, in the case that A is the Jacobian variety J of an algebraic curve (compact Riemann surface) C. There is, for a choice of base point P on C, a standard mapping of C to J, by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is an algebraic group, C may be added to itself k times on J, giving rise to subvarieties Wk.
If g is the genus of C, Riemann proved that Θ is a translate on J of Wg − 1. He also described which points on Wg − 1 are non-singular: they correspond to the effective divisors D of degree g − 1 with no associated meromorphic functions other than constants. In more classical language, these D do not move in a linear system of divisors on C, in the sense that they do not dominate the polar divisor of a non constant function.
Riemann further proved the Riemann singularity theorem, identifying the multiplicity of a point p = class(D) on Wg − 1 as the number of linearly independent meromorphic functions with pole divisor dominated by D, or equivalently as h0(O(D)), the number of linearly independent global sections of the holomorphic line bundle associated to D as Cartier divisor on C.
Later work
The Riemann singularity theorem was extended by George Kempf in 1973, building on work of David Mumford and Andreotti - Mayer, to a description of the singularities of points p = class(D) on Wk for 1 ≤ k ≤ g − 1. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D (Riemann-Kempf singularity theorem).
More precisely, Kempf mapped J locally near p to a family of matrices coming from an exact sequence which computes h0(O(D)), in such a way that Wk corresponds to the locus of matrices of less than maximal rank. The multiplicity then agrees with that of the point on the corresponding rank locus. Explicitly, if
h0(O(D)) = r + 1,
the multiplicity of Wk at class(D) is the binomial coefficient
When k = g − 1, this is r + 1, Riemann's formula.
Notes
References
Theta functions
Algebraic curves
Bernhard Riemann
|
https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Katz%20p-curvature%20conjecture
|
In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form.
The general case remains unsolved, despite recent progress; it has been linked to geometric investigations involving algebraic foliations.
Formulation
In a simplest possible statement the conjecture can be stated in its essentials for a vector system written as
for a vector v of size n, and an n×n matrix A of algebraic functions with algebraic number coefficients. The question is to give a criterion for when there is a full set of algebraic function solutions, meaning a fundamental matrix (i.e. n vector solutions put into a block matrix). For example, a classical question was for the hypergeometric equation: when does it have a pair of algebraic solutions, in terms of its parameters? The answer is known classically as Schwarz's list. In monodromy terms, the question is of identifying the cases of finite monodromy group.
By reformulation and passing to a larger system, the essential case is for rational functions in A and rational number coefficients. Then a necessary condition is that for almost all prime numbers p, the system defined by reduction modulo p should also have a full set of algebraic solutions, over the finite field with p elements.
Grothendieck's conjecture is that these necessary conditions, for almost all p, should be sufficient. The connection with p-curvature is that the mod p condition stated is the same as saying the p-curvature, formed by a recurrence operation on A, is zero; so another way to say it is that p-curvature of 0 for almost all p implies enough algebraic solutions of the original equation.
Katz's formulation for the Galois group
Nicholas Katz has applied Tannakian category techniques to show that this conjecture is essentially the same as saying that the differential Galois group G (or strictly speaking the Lie algebra g of the algebraic group G, which in this case is the Zariski closure of the monodromy group) can be determined by mod p information, for a certain wide class of differential equations.
Progress
A wide class of cases has been proved by Benson Farb and Mark Kisin; these equations are on a locally symmetric variety X subject to some group-theoretic conditions. This work is based on the previous results of Katz for Picard–Fuchs equations (in the contemporary sense of the Gauss–Manin connection), as amplified in the Tannakian direction by André. It also applies a version of superrigidity particular to arithmetic groups. Other progress has been by arithmetic methods.
History
Nicholas Katz related some cases to deformation theory in 1972, in a paper where the conjecture was published. Since then, reformul
|
https://en.wikipedia.org/wiki/Principal%20ideal%20theorem
|
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called principalization, or sometimes capitulation.
Formal statement
For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then
is a principal ideal αOL, for OL the ring of integers of L and some element α in it.
History
The principal ideal theorem was conjectured by , and was the last remaining aspect of his program on class fields to be completed, in 1929.
reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial. This result was proved by Philipp Furtwängler (1929).
References
Ideals (ring theory)
Group theory
Homological algebra
Theorems in algebraic number theory
|
https://en.wikipedia.org/wiki/Jet%20group
|
In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
Overview
The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ(0)=0.
The following is a more precise definition of the jet group.
Let k ≥ 2. The differential of a function f: Rk → R can be interpreted as a section of the cotangent bundle of RK given by df: Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm(Rk) = Rk × W, where
Here R* is the dual vector space to R, and Si denotes the i-th symmetric power. A smooth function f: Rk → R has a prolongation jmf: Rk → Jm(Rk) defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si((R*)k) component of W.
Consider a point . There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is, . The differential data x′ may be transferred to lie over another point y ∈ Rn as jmfp(y) , the partials of fp over y.
Provide Jm(Rn) with a group structure by taking
With this group structure, Jm(Rn) is a Carnot group of class m + 1.
Because of the properties of jets under function composition, Gnk is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations.
Notes
References
Lie groups
|
https://en.wikipedia.org/wiki/Kaufmann%20vortex
|
The Kaufmann vortex, also known as the Scully model, is a mathematical model for a vortex taking account of viscosity. It uses an algebraic velocity profile. This vortex is not a solution of the Navier–Stokes equations.
Kaufmann and Scully's model for the velocity in the Θ direction is:
The model was suggested by W. Kaufmann in 1962, and later by Scully and Sullivan in 1972 at the Massachusetts Institute of Technology.
See also
Rankine vortex – a simpler, but more crude, approximation for a vortex.
Lamb–Oseen vortex – the exact solution for a free vortex decaying due to viscosity.
References
Equations of fluid dynamics
Vortices
|
https://en.wikipedia.org/wiki/Symmetric%20product%20of%20an%20algebraic%20curve
|
In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product
C × C × ... × C
or Cn by the group action of the symmetric group Sn on n letters permuting the factors. It exists as a smooth algebraic variety denoted by ΣnC. If C is a compact Riemann surface, ΣnC is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients.
For C the projective line (say the Riemann sphere ∪ {∞} ≈ S2), its nth symmetric product ΣnC can be identified with complex projective space of dimension n.
If G has genus g ≥ 1 then the ΣnC are closely related to the Jacobian variety J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors.
For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil's technique of constructing J as an abstract variety from 'birational data'. Other ways of constructing J, for example as a Picard variety, are preferred now but this does mean that for any rational function F on C
F(x1) + ... + F(xg)
makes sense as a rational function on J, for the xi staying away from the poles of F.
For n > g the mapping from ΣnC to J by addition fibers it over J; when n is large enough (around twice g) this becomes a projective space bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai.
Betti numbers and the Euler characteristic of the symmetric product
Let C be a smooth projective curve of genus g over the complex numbers C. The Betti numbers bi(ΣnC) of the symmetric products ΣnC for all n = 0, 1, 2, ... are given by the generating function
and their Euler characteristics e(ΣnC) are given by the generating function
Here we have set u = -1 and y = -p in the previous formula.
Notes
References
Algebraic curves
Symmetric functions
|
https://en.wikipedia.org/wiki/Linearly%20disjoint
|
In mathematics, algebras A, B over a field k inside some field extension of k are said to be linearly disjoint over k if the following equivalent conditions are met:
(i) The map induced by is injective.
(ii) Any k-basis of A remains linearly independent over B.
(iii) If are k-bases for A, B, then the products are linearly independent over k.
Note that, since every subalgebra of is a domain, (i) implies is a domain (in particular reduced). Conversely if A and B are fields and either A or B is an algebraic extension of k and is a domain then it is a field and A and B are linearly disjoint. However, there are examples where is a domain but A and B are not linearly disjoint: for example, A = B = k(t), the field of rational functions over k.
One also has: A, B are linearly disjoint over k if and only if subfields of generated by , resp. are linearly disjoint over k. (cf. Tensor product of fields)
Suppose A, B are linearly disjoint over k. If , are subalgebras, then and are linearly disjoint over k. Conversely, if any finitely generated subalgebras of algebras A, B are linearly disjoint, then A, B are linearly disjoint (since the condition involves only finite sets of elements.)
See also
Tensor product of fields
References
P.M. Cohn (2003). Basic algebra
Algebra
|
https://en.wikipedia.org/wiki/Prime%20decomposition%20of%203-manifolds
|
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds.
A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension. This condition is necessary since for any manifold M of dimension it is true that
(where means the connected sum of and ). If is a prime 3-manifold then either it is or the non-orientable bundle over
or it is irreducible, which means that any embedded 2-sphere bounds a ball. So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of over
The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable bundles over This sum is unique as long as we specify that each summand is either irreducible or a non-orientable bundle over
The proof is based on normal surface techniques originated by Hellmuth Kneser. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by John Milnor.
References
3-manifolds
Manifolds
Theorems in differential geometry
|
https://en.wikipedia.org/wiki/Child%20mortality
|
Child mortality is the mortality of children under the age of five. The child mortality rate (also under-five mortality rate) refers to the probability of dying between birth and exactly five years of age expressed per 1,000 live births.
It encompasses neonatal mortality and infant mortality (the probability of death in the first year of life).
Reduction of child mortality is reflected in several of the United Nations' Sustainable Development Goals. Target 3.2 is "by 2030, end preventable deaths of newborns and children under 5 years of age, with all countries aiming to reduce … under‑5 mortality to at least as low as 25 per 1,000 live births."
Child mortality rates have decreased in the last 40 years. Rapid progress has resulted in a significant decline in preventable child deaths since 1990, with the global under-5 mortality rate declining by over half between 1990 and 2016. While in 1990, 12.6 million children under age five died, in 2016 that number fell to 5.6 million children, and then in 2020, the global number fell again to 5 million. However, despite advances, there are still 15,000 under-five deaths per day from largely preventable causes. About 80 per cent of these occur in sub-Saharan Africa and South Asia, and just 6 countries account for half of all under-five deaths: China, India, Pakistan, Nigeria, Ethiopia and the Democratic Republic of the Congo. 45% of these children died during the first 28 days of life. Death rates were highest among children under age 1, followed by children ages 15 to 19, 1 to 4, and 5 to 14.
Types of Child Mortality
Child mortality refers to number of child deaths under the age of 5 per 1000 live births. More specific terms include:
Perinatal mortality rate: Number of child deaths within first week of birth ÷ total number of births.
Neonatal mortality rate: Number of child deaths within first 28 days of life ÷ total number of births.
Infancy mortality rate: Number of child deaths within first 12 months of life ÷ total number of births.
Under 5 mortality rates: Number of child deaths within 5th birthday ÷ total number of births.
Child Mortality refers to the premature deaths of any child under the age of 5 years old. However, within those 5 years, there are 5 smaller groups. Perinatal refers to a fetus, a living organism, but not yet born. Typically, peri neonate deaths are due to premature birth or birth defects. Neonatal refers to child death within one month, or 28 days, of birth. Neonate deaths are reflected in the type of care the hospital is providing, as well as birth defects and complications. Infant refers to the death of a child before their first birthday or within 12 months of life. Some of the main causes include premature birth, SIDS, low birth weight, malnutrition, and infectious diseases. And lastly, the under-5 mortality rate refers to children who die under the age of 5 years old or within the first 5 years of life.
Causes
The leading causes of death of children under five i
|
https://en.wikipedia.org/wiki/Normal%20surface
|
In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting.
Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above.
The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surface and spun normal surface.
The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost normal surfaces is due to Hyam Rubinstein. The notion of spun normal surface is due to Bill Thurston.
Regina is software which enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other things.
References
Hatcher, Notes on basic 3-manifold topology, available online
Gordon, ed. Kent, The theory of normal surfaces,
Hempel, 3-manifolds, American Mathematical Society,
Jaco, Lectures on three-manifold topology, American Mathematical Society,
R. H. Bing, The Geometric Topology of 3-Manifolds, (1983) American Mathematical Society Colloquium Publications Volume 40, Providence RI, .
Further reading
3-manifolds
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.