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https://en.wikipedia.org/wiki/Dirichlet%27s%20test
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In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
Statement
The test states that if is a sequence of real numbers and a sequence of complex numbers satisfying
is monotonic
for every positive integer N
where M is some constant, then the series
converges.
Proof
Let and .
From summation by parts, we have that . Since is bounded by M and , the first of these terms approaches zero, as .
We have, for each k, .
Since is monotone, it is either decreasing or increasing:
<li>
If is decreasing,
which is a telescoping sum that equals and therefore approaches as . Thus, converges.
<li>
If is increasing,
which is again a telescoping sum that equals and therefore approaches as . Thus, again, converges.
So, the series converges, by the absolute convergence test. Hence converges.
Applications
A particular case of Dirichlet's test is the more commonly used alternating series test for the case
Another corollary is that converges whenever is a decreasing sequence that tends to zero. To see that
is bounded, we can use the summation formula
Improper integrals
An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals, and g is a non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.
Notes
References
Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13–15) .
External links
PlanetMath.org
Convergence tests
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https://en.wikipedia.org/wiki/Glossary%20of%20arithmetic%20and%20diophantine%20geometry
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This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields. Of those, only the complex numbers are algebraically closed; over any other K the existence of points of V with coordinates in K is something to be proved and studied as an extra topic, even knowing the geometry of V.
Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers. Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in number theory.
A
B
C
D
E
F
G
H
I
K
L
M
N
O
Q
R
S
T
U
V
W
See also
Arithmetic topology
Arithmetic dynamics
References
Further reading
Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore,
Diophantine geometry
Algebraic geometry
Geometry
Wikipedia glossaries using description lists
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https://en.wikipedia.org/wiki/Height%20function
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A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers.
For instance, the classical or naive height over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale.
Significance
Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's theorem in transcendental number theory which was proved by .
In other cases, height functions can distinguish some objects based on their complexity. For instance, the subspace theorem proved by demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation.
Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by and respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
History
An early form of height function was proposed by Giambattista Benedetti (c. 1563), who argued that the consonance of a musical interval could be measured by the product of its numerator and denominator (in reduced form); see .
Heights in Diophantine geometry were initially developed by André Weil and Douglas Northcott beginning in the 1920s. Innovations in 1960s were the Néron–Tate height and the realization that heights were linked to projective representations in much the same way that ample line bundles are in other parts of algebraic geometry. In the 1970s, Suren Arakelov developed Arakelov heights in Arakelov theory. In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.
Height functions in Diophantine geometry
Naive height
Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates. It is typically a logarithmic scale and therefore can be viewed as being proportional to the "algebraic complexity" or number of bits needed to store a point. It is typically defined to be the logarithm of the maximum absolute value of the vector of coprime integers obtained by multiplying through by a
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https://en.wikipedia.org/wiki/Bombieri%E2%80%93Lang%20conjecture
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In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
Statement
The weak Bombieri–Lang conjecture for surfaces states that if is a smooth surface of general type defined over a number field , then the points of do not form a dense set in the Zariski topology on .
The general form of the Bombieri–Lang conjecture states that if is a positive-dimensional algebraic variety of general type defined over a number field , then the points of do not form a dense set in the Zariski topology.
The refined form of the Bombieri–Lang conjecture states that if is an algebraic variety of general type defined over a number field , then there is a dense open subset of such that for all number field extensions over , the set of points in is finite.
History
The Bombieri–Lang conjecture was independently posed by Enrico Bombieri and Serge Lang. In a 1980 lecture at the University of Chicago, Enrico Bombieri posed a problem about the degeneracy of rational points for surfaces of general type. Independently in a series of papers starting in 1971, Serge Lang conjectured a more general relation between the distribution of rational points and algebraic hyperbolicity, formulated in the "refined form" of the Bombieri–Lang conjecture.
Generalizations and implications
The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points.
If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem, as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational.
In 1997, Lucia Caporaso, Barry Mazur, Joe Harris, and Patricia Pacelli showed that the Bombieri–Lang conjecture implies a uniform boundedness conjecture for rational points: there is a constant depending only on and such that the number of rational points of any genus curve over any degree number field is at most .
References
Diophantine geometry
Unsolved problems in geometry
Conjectures
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https://en.wikipedia.org/wiki/Positive%20definiteness
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In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
Positive-definite bilinear form
Positive-definite function
Positive-definite function on a group
Positive-definite functional
Positive-definite kernel
Positive-definite matrix
Positive-definite quadratic form
References
.
.
Quadratic forms
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https://en.wikipedia.org/wiki/Jacobi%20sum
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In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by
where the summation runs over all residues (for which neither a nor is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. Jacobi sums J can be factored generically into products of powers of Gauss sums g. For example, when the character χψ is nontrivial,
analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums g have absolute value p, it follows that also has absolute value p when the characters χψ, χ, ψ are nontrivial. Jacobi sums J lie in smaller cyclotomic fields than do the nontrivial Gauss sums g. The summands of for example involve no pth root of unity, but rather involve just values which lie in the cyclotomic field of th roots of unity. Like Gauss sums, Jacobi sums have known prime ideal factorisations in their cyclotomic fields; see Stickelberger's theorem.
When χ is the Legendre symbol,
In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the formula for the number of points on a conic section that is a projective line over the field of p elements. A paper of André Weil from 1949 very much revived the subject. Indeed, through the Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more.
As well as pointing out the possibility of writing down local zeta-functions for diagonal hypersurfaces by means of general Jacobi sums, Weil (1952) demonstrated the properties of Jacobi sums as Hecke characters. This was to become important once the complex multiplication of abelian varieties became established. The Hecke characters in question were exactly those one needs to express the Hasse–Weil L-functions of the Fermat curves, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.
References
Cyclotomic fields
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https://en.wikipedia.org/wiki/Thin%20set%20%28Serre%29
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In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within K a polynomial that does not always factorise. One is also allowed to take finite unions.
Formulation
More precisely, let V be an algebraic variety over K (assumptions here are: V is an irreducible set, a quasi-projective variety, and K has characteristic zero). A type I thin set is a subset of V(K) that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than d, the dimension of V. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic variety V′, that maps essentially onto V as a ramified covering with degree e > 1. Saying this more technically, a thin set of type II is any subset of
φ(V′(K))
where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer's point of view. At the level of function fields we therefore have
[K(V): K(V′)] = e > 1.
While a typical point v of V is φ(u) with u in V′, from v lying in K(V) we can conclude typically only that the coordinates of u come from solving a degree e equation over K. The whole object of the theory of thin sets is then to understand that the solubility in question is a rare event. This reformulates in more geometric terms the classical Hilbert irreducibility theorem.
A thin set, in general, is a subset of a finite union of thin sets of types I and II .
The terminology thin may be justified by the fact that if A is a thin subset of the line over Q then the number of points of A of height at most H is ≪ H: the number of integral points of height at most H is , and this result is best possible.
A result of S. D. Cohen, based on the large sieve method, extends this result, counting points by height function and showing, in a strong sense, that a thin set contains a low proportion of them (this is discussed at length in Serre's Lectures on the Mordell-Weil theorem). Let A be a thin set in affine n-space over Q and let N(H) denote the number of integral points of naive height at most H. Then
Hilbertian fields
A Hilbertian variety V over K is one for which V(K) is not thin: this is a birational invariant of V. A Hilbertian field K is one for which there exists a Hilbertian variety of positive dimension over K: the term was introduced by Lang in 1962. If K is Hilbertian then the projective line over K is Hilbertian, so this may be taken as the definition.
The rational number field Q is Hilbertian, because Hilbert's irreducibility theorem has as a corollary that the projective line over Q is Hilbertian: indeed, any algebraic number field is Hilbertian, aga
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https://en.wikipedia.org/wiki/Euclid%27s%20theorem
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Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.
Euclid's proof
Euclid offered a proof published in his work Elements (Book IX, Proposition 20), which is paraphrased here.
Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then q is either prime or not:
If q is prime, then there is at least one more prime that is not in the list, namely, q itself.
If q is not prime, then some prime factor p divides q. If this factor p were in our list, then it would divide P (since P is the product of every number in the list); but p also divides P + 1 = q, as just stated. If p divides P and also q, then p must also divide the difference of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, p cannot be in the list. This means that at least one more prime number exists beyond those in the list.
This proves that for every finite list of prime numbers there is a prime number not in the list. In the original work, as Euclid had no way of writing an arbitrary list of primes, he used a method that he frequently applied, that is, the method of generalizable example. Namely, he picks just three primes and using the general method outlined above, proves that he can always find an additional prime. Euclid presumably assumes that his readers are convinced that a similar proof will work, no matter how many primes are originally picked.
Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method. The philosopher Torkel Franzén, in a book on logic, states, "Euclid's proof that there are infinitely many primes is not an indirect proof [...] The argument is sometimes formulated as an indirect proof by replacing it with the assumption 'Suppose are all the primes'. However, since this assumption isn't even used in the proof, the reformulation is pointless."
Variations
Several variations on Euclid's proof exist, including the following:
The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence is either prime or divisible by a prime larger than n. In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.
Euler's proof
Another proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a uni
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https://en.wikipedia.org/wiki/Belyi%27s%20theorem
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In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.
This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data.
Quotients of the upper half-plane
It follows that the Riemann surface in question can be taken to be the quotient
H/Γ
(where H is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.
Belyi functions
A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Möbius transformation may be taken to be . Belyi functions may be described combinatorially by dessins d'enfants.
Belyi functions and dessins d'enfants – but not Belyi's theorem – date at least to the work of Felix Klein; he used them in his article to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).
Applications
Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
References
Further reading
Algebraic curves
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Spring%20%28mathematics%29
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In geometry, a spring is a surface in the shape of a coiled tube, generated by sweeping a circle about the path of a helix.
Definition
A spring wrapped around the z-axis can be defined parametrically by:
where
is the distance from the center of the tube to the center of the helix,
is the radius of the tube,
is the speed of the movement along the z axis (in a right-handed Cartesian coordinate system, positive values create right-handed springs, whereas negative values create left-handed springs),
is the number of rounds in a spring.
The implicit function in Cartesian coordinates for a spring wrapped around the z-axis, with = 1 is
The interior volume of the spiral is given by
Other definitions
Note that the previous definition uses a vertical circular cross section. This is not entirely accurate as the tube becomes increasingly distorted as the Torsion increases (ratio of the speed and the incline of the tube).
An alternative would be to have a circular cross section in the plane perpendicular to the helix curve. This would be closer to the shape of a physical spring. The mathematics would be much more complicated.
The torus can be viewed as a special case of the spring obtained when the helix degenerates to a circle.
References
See also
Spiral
Helix
Surfaces
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https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch%20theorem%20for%20surfaces
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In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by , after preliminary versions of it were found by and . The sheaf-theoretic version is due to Hirzebruch.
Statement
One form of the Riemann–Roch theorem states that if D is a divisor on a non-singular projective surface then
where χ is the holomorphic Euler characteristic, the dot . is the intersection number, and K is the canonical divisor. The constant χ(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + pa, where pa is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve states that χ(D) = χ(0) + deg(D).
Noether's formula
Noether's formula states that
where χ=χ(0) is the holomorphic Euler characteristic, c12 = (K.K) is a Chern number and the self-intersection number of the canonical class K, and e = c2 is the topological Euler characteristic. It can be used to replace the
term χ(0) in the Riemann–Roch theorem with topological terms; this gives the Hirzebruch–Riemann–Roch theorem for surfaces.
Relation to the Hirzebruch–Riemann–Roch theorem
For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces combined with the Noether formula. To see this, recall that for each divisor D on a surface there is an invertible sheaf L = O(D) such that the linear system of D is more or less the space of sections of L.
For surfaces the Todd class is , and the Chern character of the sheaf L is just , so the Hirzebruch–Riemann–Roch theorem states that
Fortunately this can be written in a clearer form as follows. First putting D = 0 shows that
(Noether's formula)
For invertible sheaves (line bundles) the second Chern class vanishes. The products of second cohomology classes can be identified with intersection numbers in the Picard group, and we get a more classical version of Riemann Roch for surfaces:
If we want, we can use Serre duality to express h2(O(D)) as h0(O(K − D)), but unlike the case of curves there is in general no easy way to write the h1(O(D)) term in a form not involving sheaf cohomology (although in practice it often vanishes).
Early versions
The earliest forms of the Riemann–Roch theorem for surfaces were often stated as an inequality rather than an equality, because there was no direct geometric description of first cohomology groups.
A typical example is given by , which states that
where
r is the dimension of the complete linear system |D| of a divisor D (so r = h0(O(D)) −1)
n is the virtual degree of D, given by the self-intersection number (D.D)
π is the virtual genus of D, equal to 1 + (D.D + K.D)/2
pa is the arithmetic genus χ(OF) − 1 of the surface
i is the index of speciality of D, equal to dim H0(O(K − D)) (which by Serre duality is the same as dim H2(O(D))).
The difference between the two sides of this inequality was called the supera
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https://en.wikipedia.org/wiki/Madelung
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Madelung is a German surname. It is also the name of multiple terms in mathematics and science based on people named Madelung.
People
Erwin Madelung (1881–1972), German physicist
Georg Hans Madelung (1889–1972), German aeronautical engineer
Otto Wilhelm Madelung (1846–1926), German surgeon
Wilferd Madelung (1930–2023), German-American author and scholar of Islamic history
Mathematics and science
Madelung constant, chemical energy of an ion in a crystal, named after Erwin Madelung
Madelung equations, Erwin Madelung's equivalent alternative formulation of the Schrödinger equation
Medicine
Madelung's deformity, characterized by malformed wrists and wrist bones, and short stature, named after Otto Wilhelm Madelung
Madelung's syndrome, also known as "benign symmetric lipomatosis", named after Otto Wilhelm Madelung
German-language surnames
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https://en.wikipedia.org/wiki/Schur%20polynomial
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In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.
Definition (Jacobi's bialternant formula)
Schur polynomials are indexed by integer partitions. Given a partition ,
where , and each is a non-negative integer, the functions
are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes sign under any transposition of the variables.
Since they are alternating, they are all divisible by the Vandermonde determinant
The Schur polynomials are defined as the ratio
This is known as the bialternant formula of Jacobi. It is a special case of the Weyl character formula.
This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.
Properties
The degree Schur polynomials in variables are a linear basis for the space of homogeneous degree symmetric polynomials in variables.
For a partition , the Schur polynomial is a sum of monomials,
where the summation is over all semistandard Young tableaux of shape . The exponents give the weight of , in other words each counts the occurrences of the number in . This can be shown to be equivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page).
Schur polynomials can be expressed as linear combinations of monomial symmetric functions with non-negative integer coefficients called Kostka numbers,
The Kostka numbers are given by the number of semi-standard Young tableaux of shape λ and weight μ.
Jacobi−Trudi identities
The first Jacobi−Trudi formula expresses the Schur polynomial as a determinant
in terms of the complete homogeneous symmetric polynomials,
where .
The second Jacobi-Trudi formula expresses the Schur polynomial as
a determinant in terms of the elementary symmetric polynomials,
where
and is the conjugate partition to .
In both identities, functions with negative subscripts are defined to be zero.
The Giambelli identity
Another determinantal identity is Giambelli's formula, which expresses the Schur function for an arbitrary partition in terms of those for the hook partitions contained within the Young
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https://en.wikipedia.org/wiki/Semimodule
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In mathematics, a semimodule over a semiring R is like a module over a ring except that it is only a commutative monoid rather than an abelian group.
Definition
Formally, a left R-semimodule consists of an additively-written commutative monoid M and a map from to M satisfying the following axioms:
.
A right R-semimodule can be defined similarly. For modules over a ring, the last axiom follows from the others. This is not the case with semimodules.
Examples
If R is a ring, then any R-module is an R-semimodule. Conversely, it follows from the second, fourth, and last axioms that (-1)m is an additive inverse of m for all , so any semimodule over a ring is in fact a module.
Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself. Every commutative monoid is uniquely an -semimodule in the same way that an abelian group is a -module.
References
Algebraic structures
Module theory
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https://en.wikipedia.org/wiki/Intersection%20theory
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In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.
There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.
Topological intersection form
For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in . Stated precisely, there is a bilinear form
given by
with
This is a symmetric form for even (so doubly even), in which case the signature of is defined to be the signature of the form, and an alternating form for odd (so is singly even). These can be referred to uniformly as ε-symmetric forms, where respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an -quadratic form, though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with coefficients instead.
These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism.
By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative -dimensional submanifolds , for the Poincaré duals of and . Then is the oriented intersection number of and , which is well-defined because since dimensions of and sum to the total dimension of they generically intersect at isolated points. This explains the terminology intersection form.
Intersection theory in algebraic geometry
William Fulton in Intersection Theory (1984) writes
... if and are subvarieties of a non-singular variety , the intersection product should be an equivalence class of algebraic cycles closely related to the geometry of how , and are situated in . Two extreme cases have been most familiar. If the intersection is proper, i.e. , then is a linear combination of the irreducible components of , with coefficients the intersection multiplicities. At the other extreme, if is a non-singular subvariety, the self-intersection formula says that is represented by the top Chern class of the normal bundle of in .
To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the I
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https://en.wikipedia.org/wiki/St%20Paul%27s%20School%20for%20Girls%2C%20Birmingham
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St Paul's School For Girls is a voluntary aided, comprehensive, girls' school in Edgbaston, Birmingham, UK,
Admissions
It is a Roman Catholic school, and became a specialist school in maths and computing in September 2005. It is ethnically diverse, with a mixture of Black and White English/Irish pupils.
It is situated just north of the A456 and B4125, just south of Edgbaston Reservoir. The school is named after St Paul's Convent, and the headmistresses were nuns until 1998.
History
It was founded on 7 October 1908, from an earlier establishment based on Whittall Street. St Paul's Convent had been founded at Selly Park in 1864. In the 1940s, it became a Grammar School with selective entry based on the 11-plus. In 1975, it became a Comprehensive School.
Part of the school was destroyed by fire in November 1973. Mother Teresa visited on 12 September 1974. In 2010 a new Maths and English building was constructed named 'The Sister Suite' in reference to the nuns who founded the school, certain rooms inside the building were named after notable Sisters in the school's history.
Academic performance
It has relatively high GCSE pass rates for similar schools in the Birmingham LEA and in England. In 2009 it got the second best A-level results for comprehensive schools in Birmingham, with results similar to a grammar school.
Notable former pupils
Clare Short former Labour MP
Sarah Smart Actress
Julie Walters Actress
See also
Highclare School, an independent school in Sutton Coldfield founded by St Paul's Convent.
References
External links
St Paul's School for Girls
Sisters of St Paul Selly Park
Ofsted
Girls' schools in the West Midlands (county)
Secondary schools in Birmingham, West Midlands
Educational institutions established in 1908
Catholic secondary schools in the Archdiocese of Birmingham
1908 establishments in England
Edgbaston
Voluntary aided schools in England
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https://en.wikipedia.org/wiki/Hecke%20character
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In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.).
Definition using ideles
A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map.
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in C ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character χ is the largest ideal m such that χ is a Hecke character mod m. Here we say that χ is a Hecke character mod m if χ (considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 + mOv.
Definition using ideals
The original definition of a Hecke character, going back to Hecke, was in terms of
a character on fractional ideals. For a number field K, let
m = mfm∞ be a
K-modulus, with mf, the "finite part", being an integral ideal of K and m∞, the "infinite part", being a (formal) product of real places of K. Let Im
denote the group of fractional ideals of K relatively prime to mf and
let Pm denote the subgroup of principal fractional ideals (a)
where a is near 1 at each place of m in accordance with the multiplicities of
its factors: for each finite place v in mf, ordv(a − 1) is at least as large as the exponent for v in mf, and a is positive under each real embedding in m∞. A Hecke character with modulus m
is a group homomorphism from Im into the nonzero complex numbers
such that on ideals (a) in Pm its value is equal to the
value at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of K where each local component of the homomorphism has the same real part (in the exponent). (Here we embed a into the product of Archimedean completions of K using embeddings corresponding to the various Archimedean places on K.) Thus a Hecke character may be defined on the ray class group modulo m, which is the quotient Im/Pm.
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked w
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https://en.wikipedia.org/wiki/Tate%27s%20thesis
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In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, of a number field to a zeta integral and study its properties. Using harmonic analysis, more precisely the Poisson summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the Hecke L-function. He also located the poles of the twisted zeta function. His work can be viewed as an elegant and powerful reformulation of a work of Erich Hecke on the proof of the functional equation of the Hecke L-function. Erich Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers.
Iwasawa–Tate theory
Kenkichi Iwasawa independently discovered essentially the same method (without an analog of the local theory in Tate's thesis) during the Second World War and announced it in his 1950 International Congress of Mathematicians paper and his letter to Jean Dieudonné written in 1952. Hence this theory is often called Iwasawa–Tate theory. Iwasawa in his letter to Dieudonné derived on several pages not only the meromorphic continuation and functional equation of the L-function, he also proved finiteness of the class number and Dirichlet's theorem on units as immediate byproducts of the main computation. The theory in positive characteristic was developed one decade earlier by Ernst Witt, Wilfried Schmid, and Oswald Teichmüller.
Iwasawa-Tate theory uses several structures which come from class field theory, however it does not use any deep result of class field theory.
Generalisations
Iwasawa–Tate theory was extended to the general linear group GL(n) over an algebraic number field and automorphic representations of its adelic group by Roger Godement and Hervé Jacquet in 1972 which formed the foundations of the Langlands correspondence. Tate's thesis can be viewed as the GL(1) case of the work by Godement–Jacquet.
See also
Basic Number Theory
References
Algebraic number theory
Zeta and L-functions
1950 in science
1950 documents
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https://en.wikipedia.org/wiki/Hecke%20L-function
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In mathematics, a Hecke L-function may refer to:
an L-function of a modular form
an L-function of a Hecke character
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https://en.wikipedia.org/wiki/Deviation%20analysis
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Deviation analysis may mean;
in statistics; measurement of the absolute difference between any one number in a set and the mean of the set.
in social psychology; monitoring of the behavior of people or objects within systems to measure compliance with expected or desired norms in order to trigger alerts, identity users or spot anomalies.
Analysis
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https://en.wikipedia.org/wiki/P%20series
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P series or P-series may refer to:
the p-series in mathematics, related to convergence of certain series
P-series fuels, blends of fuels
Huawei P series, mobile phone series by Huawei
IBM pSeries, computer series by IBM
Ruger P series – pistols
ThinkPad P series, mobile workstation line by Lenovo
Sony Cybershot P-series digital cameras, see Cyber-shot
Sony Vaio P series – notebook computers
Sony Ericsson P series, a series of cell phones
Vespa P-series motor scooters
See also
O series (disambiguation)
Q series (disambiguation)
T series (disambiguation)
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https://en.wikipedia.org/wiki/Small%20set%20%28category%20theory%29
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In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is the category of all sets one cares to consider. This is used when one does not wish to bother with set-theoretic concerns of what is and what is not considered a set, which concerns would arise if one tried to speak of the category of "all sets".
A small set is not to be confused with a small category, which is a category in which the collection of arrows (and therefore also the collection of objects) is a set.
In other choices of foundations, such as Grothendieck universes, there exist both sets that belong to the universe, called “small sets” and sets that do not, such as the universe itself, “large sets”. We gain an intermediate notion of moderate set: a subset of the universe, which may be small or large. Every small set is moderate, but not conversely.
Since in many cases the choice of foundations is irrelevant, it makes sense to always say “small set” for emphasis even if one has in mind a foundation where all sets are small.
Similarly, a small family is a family indexed by a small set; the axiom of replacement (if it applies in the foundation in question) then says that the image of the family is also small.
See also
Category of sets
References
S. Mac Lane, Ieke Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, , , the chapter on "Categorical preliminaries"
Categories in category theory
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https://en.wikipedia.org/wiki/Large%20set%20%28combinatorics%29
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In combinatorial mathematics, a large set of positive integers
is one such that the infinite sum of the reciprocals
diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges.
Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions.
Examples
Every finite subset of the positive integers is small.
The set of all positive integers is a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form an + b with a ≥ 1, b ≥ 1 and n = 0, 1, 2, 3, ...) is a large set.
The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer values of any polynomial of degree 2 or larger forms a small set.
The set {1, 2, 4, 8, ...} of powers of 2 is a small set, and so is any geometric progression (i.e., a set of numbers of the form of the form abn with a ≥ 1, b ≥ 2 and n = 0, 1, 2, 3, ...).
The set of prime numbers is large. The set of twin primes is small (see Brun's constant).
The set of prime powers which are not prime (i.e., all numbers of the form pn with n ≥ 2 and p prime) is small although the primes are large. This property is frequently used in analytic number theory. More generally, the set of perfect powers is small; even the set of powerful numbers is small.
The set of numbers whose expansions in a given base exclude a given digit is small. For example, the set
of integers whose decimal expansion does not include the digit 7 is small. Such series are called Kempner series.
Any set whose upper asymptotic density is nonzero, is large.
Properties
Every subset of a small set is small.
The union of finitely many small sets is small, because the sum of two convergent series is a convergent series. (In set theoretic terminology, the small sets form an ideal.)
The complement of every small set is large.
The Müntz–Szász theorem states that a set is large if and only if the set of polynomials spanned by is dense in the uniform norm topology of continuous functions on a closed interval in the positive real numbers. This is a generalization of the Stone–Weierstrass theorem.
Open problems involving large sets
Paul Erdős conjectured that all large sets contain arbitrarily long arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his other conjectures, and joked that this prize offer violated the minimum wage law. The question is still open.
It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.
See also
List of sums of reciprocals
Notes
References
A. D. Wadhwa (1975). An interesting subseries of the harmonic series. American Mathematical Monthly 82 (9) 931–933.
Combinatorics
Intege
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https://en.wikipedia.org/wiki/Geometry%20Wars%3A%20Retro%20Evolved
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Geometry Wars is a video game made by Bizarre Creations. Initially a minigame in Project Gotham Racing 2, an updated version, titled Retro Evolved, was eventually released for the Xbox 360. That version, at one point, held the record for the most downloaded Xbox Live Arcade Game.
Retro Evolved was later included in the 2007 game Geometry Wars: Galaxies, the first game in the series to be released for non-Microsoft platforms.
Gameplay
The object of Geometry Wars is to survive as long as possible and score as many points as possible by destroying an ever-increasing swarm of enemies. The game takes place on a rectangular playfield and the player controls a claw-shaped "ship" that can move in any direction using the left thumbstick, and can fire in any direction independently using the right thumbstick. The player also has a limited number of bombs that can be detonated and destroy all enemies on the playfield. As the game progresses, the player can earn extra lives and additional bombs at set score increments, and the primary weapon changes at regular intervals (10,000 points). Also, enemies spawn in progressively larger quantities and at greater frequency as the game progresses. If an enemy touches the player's ship, the ship explodes and a life is lost, plus the multiplier worked up by how many enemies are killed in one life is also lost. The game is over when the player runs out of lives.
The Evolved version of the game takes place on a playfield that is slightly larger than the display area of the TV screen, and the camera follows the player's movements. A background grid pattern adds to the graphical effects by warping in reaction to player shots and the behavior of certain enemies. This version introduces new enemies and a score multiplier that increases as the player destroys enemies without losing a life.
Development
The game initially started out as a way for the team at Bizarre Creations to test out the Xbox controller while making Project Gotham Racing. The team included the game as an extra in the sequel not expecting very much. When the creators realized how popular the game was they decided to work on a stand-alone game for the 360's Live Arcade. For the standalone version, which eventually grew to become Retro Evolved, creator Stephen Cakebread initially wanted to make the game have a more linear structure where the players would progress through levels. However, Cakebread soon became aware of the game Mutant Storm and realized that a level-based structure would make Retro Evolved almost identical and thus decided to drop it.
The soundtrack was composed by Chris Chudley from Audioantics who created the music for all of the series up until Dimensions.
Reception
Geometry Wars: Retro Evolved received "generally favorable reviews" on both platforms according to the review aggregation website Metacritic.
GameSpots Carrie Gouskos praised the Xbox 360 version's gameplay, graphics, controls, sound effects, and its low launch price
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https://en.wikipedia.org/wiki/Ostrowski%20Prize
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The Ostrowski Prize is a mathematics award given every odd year for outstanding mathematical achievement judged by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands. Alexander Ostrowski, a longtime professor at the University of Basel, left his estate to the foundation in order to establish a prize for outstanding achievements in pure mathematics and the foundations of numerical mathematics. It currently carries a monetary award of 100,000 Swiss francs.
Recipients
1989: Louis de Branges (France / United States)
1991: Jean Bourgain (Belgium)
1993: Miklós Laczkovich (Hungary) and Marina Ratner (Russia / United States)
1995: Andrew J. Wiles (UK)
1997: Yuri V. Nesterenko (Russia) and Gilles I. Pisier (France)
1999: Alexander A. Beilinson (Russia / United States) and Helmut H. Hofer (Switzerland / United States)
2001: Henryk Iwaniec (Poland / United States) and Peter Sarnak (South Africa / United States) and Richard L. Taylor (UK / United States)
2003: Paul Seymour (UK)
2005: Ben Green (UK) and Terence Tao (Australia / United States)
2007: Oded Schramm (Israel / United States)
2009: Sorin Popa (Romania / United States)
2011: Ib Madsen (Denmark), David Preiss (UK) and Kannan Soundararajan (India / United States)
2013: Yitang Zhang (United States)
2015: Peter Scholze (Germany)
2017: Akshay Venkatesh (India / Australia)
2019: Assaf Naor (Israel / USA)
2021: (UK)
See also
List of mathematics awards
References
Mathematics awards
Awards established in 1989
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https://en.wikipedia.org/wiki/Robert%20M.%20Solovay
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Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Biography
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann–Roch theorem. Solovay has spent his career at the University of California at Berkeley, where his Ph.D. students include W. Hugh Woodin and Matthew Foreman.
Work
Solovay's theorems include:
Solovay's theorem showing that, if one assumes the existence of an inaccessible cardinal, then the statement "every set of real numbers is Lebesgue measurable" is consistent with Zermelo–Fraenkel set theory without the axiom of choice;
Isolating the notion of 0#;
Proving that the existence of a real-valued measurable cardinal is equiconsistent with the existence of a measurable cardinal;
Proving that if is a strong limit singular cardinal, greater than a strongly compact cardinal then holds;
Proving that if is an uncountable regular cardinal, and is a stationary set, then can be decomposed into the union of disjoint stationary sets;
With Stanley Tennenbaum, developing the method of iterated forcing and showing the consistency of Suslin's hypothesis;
With Donald A. Martin, showed the consistency of Martin's axiom with arbitrarily large cardinality of the continuum;
Outside of set theory, developing (with Volker Strassen) the Solovay–Strassen primality test, used to identify large natural numbers that are prime with high probability. This method has had implications for cryptography;
Regarding the P versus NP problem, he proved with T. P. Baker and J. Gill that relativizing arguments cannot prove .
Proving that GL (the normal modal logic which has the instances of the schema as additional axioms) completely axiomatizes the logic of the provability predicate of Peano arithmetic;
With Alexei Kitaev, proving that a finite set of quantum gates can efficiently approximate an arbitrary unitary operator on one qubit in what is now known as Solovay–Kitaev theorem.
Selected publications
See also
Provability logic
References
External links
American logicians
Members of the United States National Academy of Sciences
20th-century American mathematicians
Set theorists
1938 births
Living people
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https://en.wikipedia.org/wiki/Gaspar%20Schott
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Gaspar Schott (German: Kaspar (or Caspar) Schott; Latin: Gaspar Schottus; 5 February 1608 – 22 May 1666) was a German Jesuit and scientist, specializing in the fields of physics, mathematics and natural philosophy, and known for his industry.
Biography
He was born at Bad Königshofen im Grabfeld. It is probable, but not certain, that his early education was at the Jesuit College at Würzburg. In any case, at the age of 19 he joined the Society of Jesus, entering the novitiate at Trier on 30 October 1627. After two years of novitiate training, he matriculated at the University of Würzburg on 6 November 1629 to begin a three-year study of Philosophy, following the normal academic path prescribed for Jesuit seminarians. Owing to the Swedish invasion of Würzburg in October 1631, the Jesuit community fled the city. Schott went, first to the Jesuit seminary of Tournai in Belgium, and subsequently, in 1633, to Caltagirone in Sicily, where he continued his study of Theology. After two years at Caltagirone, he was transferred to Palermo for his final year study of Theology after which, in 1637, he was ordained a priest. For the next fifteen years he held a range of teaching and pastoral positions in various Jesuit colleges in Sicily. In 1652, following correspondence with his old mathematics teacher at Würzburg, Fr. Athanasius Kircher, now an internationally acclaimed scholar at the Collegio Romano, Schott was transferred to the Collegio to work as Kircher's assistant. He was to spend the next two and a half years assisting Kircher, but also assembling material of his own for which he would later seek a publisher.
In 1655 Schott returned to Germany, first to Mainz, and later the same year to Würzburg where he was to remain until his death. His return to Germany appears to have been partly motivated by the desire of his Jesuit superiors to mollify the Archbishop-Elector of Mainz, Johann von Schönborn, with whom relations had been strained.
Works
Schott was the author of numerous works from the fields of mathematics, physics, and magic. However, those works were mostly compilations of reports, articles or books he read and his own repeated experiments; he did little, if any, original research.
Schott is most widely known for his works on hydraulic and mechanical instruments. His treatise on "chronometric marvels" is the first work describing a universal joint and providing the classification of gear teeth.
Among his most famous works is the book Magia universalis naturæ et artis (4 vols., Würtzburg, 1657–1659), filled with many mathematical problems and physical experiments, mostly from the areas of optics and acoustics. His Mechanica hydraulica-pneumatica (Würtzburg, 1657) contains the first description of von Guericke's air pump. He also published Pantometrum Kircherianum (Würtzburg, 1660); Physica curiosa (Würtzburg, 1662), a supplement to the Magia universalis; Anatomia physico-hydrostatica fontium et fluminum (Würtzburg, 1663), Technica Curiosa (16
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https://en.wikipedia.org/wiki/Lower%20convex%20envelope
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In mathematics, the lower convex envelope of a function defined on an interval is defined at each point of the interval as the supremum of all convex functions that lie under that function, i.e.
See also
Convex hull
Lower envelope
Convex analysis
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https://en.wikipedia.org/wiki/List%20of%20Braunschweig%20University%20of%20Technology%20people
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Among the people who have taught or studied at the Braunschweig University of Technology or its precursor, the Collegium Carolinum, are the following:
Natural sciences and mathematics
Ewald Banse — Geography
Ernst Otto Beckmann — Chemistry
August Wilhelm Heinrich Blasius — Zoology and Botany
Johann Heinrich Blasius — Zoology
Rudolf Blasius — Bacteriology
Caesar Rudolf Boettger — Zoology
Victor von Bruns — Medicine
Lorenz Florenz Friedrich von Crell — Chemistry and Metallurgy
Julius Wilhelm Richard Dedekind — Mathematics
Carl Georg Oscar Drude — Botany
Manfred Eigen — Biophysical chemistry — Nobel Prize in Chemistry 1967
Theodor Engelbrecht — Physiology
Herbert Freundlich — Chemistry
Robert Fricke — Mathematics
Kurt Otto Friedrichs — Mathematics
Karl Theophil Fries — Chemistry
Gustav Gassner — Botany
Carl Friedrich Gauß — Mathematics
Karl Heinrich Gräffe — Mathematics
Heiko Harborth — Mathematics
Wolfgang Hahn — Mathematics
Robert Hartig — Forestry
Theodor Hartig — Forestry
Johann Christian Ludwig Hellwig — Entomology
Adolph Henke — Pharmacology
Wilhelm Henneberg — Chemistry
Nikolaus Hofreiter — Mathematics
Johann Karl Wilhelm Illiger — Zoology
Henning Kagermann — Physics
Klaus von Klitzing — Physics — Nobel Prize in Physics 1985
Friedrich Ludwig Knapp — Chemistry
August Wilhelm Knoch — Physics
William F. Martin — Botany
Rainer Moormann — Physical chemistry
Justus Mühlenpfordt — Nuclear physics
Adolph Nehrkorn — Ornithology
Agnes Pockels — Chemistry
Friedrich Carl Alwin Pockels — Physics
Mark Ronan — Mathematics
Ferdinand Schneider — Chemistry
Gerhard Schrader — Chemistry
Kornelia Smalla — Chemistry
Sami Solanki — Astronomy
Hans Sommer — Mathematics
Ferdinand Tiemann — Chemistry
Heinrich Emil Timerding — Mathematics
Julius Tröger — Chemistry
Gerd Wedler — Chemistry
Christian Rudolph Wilhelm Wiedemann — Anatomy and Entomology
Arend Friedrich Wiegmann — Botany
Georg Wittig — Chemistry — Nobel Prize in Chemistry 1979
Eberhard August Wilhelm von Zimmermann — Zoogeography
Humanities and theology
Johann Joachim Eschenburg — Literary history
Ernst Ludwig Theodor Henke — Theology and Philosophy
Christoph Luetge — Philosophy
Willy Moog — Philosophy
Neal R. Norrick — English Linguistics
Werner Pöls — History
Nina Ruge — German language and literature
Gustav Anton von Seckendorff — Philosophy and Aesthetics
Gerhard Vollmer — Philosophy
Justus Friedrich Wilhelm Zachariae — Poetry
Social sciences
Theodor Geiger — Sociology
Jakob Mauvillon — Military science
Ulrich Menzel — Political science
Alfred Vierkandt — Ethnology and Sociology
Architecture, engineering sciences, and environmental sciences
Oliver Blume — Mechanical engineering
Adolf Busemann — Aerospace engineering
Heinrich Büssing — Engineering
Meinhard von Gerkan — Architecture
Walter Henn — Architecture
Friedrich Wilhelm Kraemer — Architecture
Boris Laschka — Aeronautical engineering
Erwin Otto Marx — Electrical engineering
Dieter Oesterlen — Architecture
August Orth — Architecture
Carl Theod
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https://en.wikipedia.org/wiki/Regular%20singular%20point
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In mathematics, in the theory of ordinary differential equations in the complex plane , the points of are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.
Formal definitions
More precisely, consider an ordinary linear differential equation of -th order
with meromorphic functions.
The equation should be studied on the Riemann sphere to include the point at infinity as a possible singular point. A Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below.
Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers near any given in the complex plane where need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from , or on a Riemann surface of some punctured disc around . This presents no difficulty for an ordinary point (Lazarus Fuchs 1866). When is a regular singular point, which by definition means that
has a pole of order at most at , the Frobenius method also can be made to work and provide independent solutions near .
Otherwise the point is an irregular singularity. In that case the monodromy group relating solutions by analytic continuation has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions. The irregularity of an irregular singularity is measured by the Poincaré rank ().
The regularity condition is a kind of Newton polygon condition, in the sense that the allowed poles are in a region, when plotted against , bounded by a line at 45° to the axes.
An ordinary differential equation whose only singular points, including the point at infinity, are regular singular points is called a Fuchsian ordinary differential equation.
Examples for second order differential equations
In this case the equation above is reduced to:
One distinguishes the following cases:
Point is an ordinary point when functions and are analytic at .
Point is a regular singular point if has a pole up to order 1 at and has a pole of order up to 2 at .
Otherwise point is an irregular singular point.
We can check whether there is an irregular singular point at infinity by using the substitution and the relations:
We can thus transform the equat
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https://en.wikipedia.org/wiki/Nicolaas%20Govert%20de%20Bruijn
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Nicolaas Govert "Dick" de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic.
Biography
De Bruijn was born in The Hague where he attended elementary school between 1924 and 1930 and secondary school until 1934. He started studies in mathematics at Leiden University in 1936 but his studies were interrupted by the outbreak of World War II in 1939. He became a full-time Assistant in the Department of Mathematics of the Technological University of Delft in September 1939 while continuing his studies. He completed his undergraduate studies at the University of Leiden in 1941. He received his PhD in 1943 from the Vrije Universiteit Amsterdam with a thesis entitled "Over modulaire vormen van meer veranderlijken" advised by Jurjen Ferdinand Koksma.
From June 1944 he was a Scientific Associate working in Philips Research Laboratories in Eindhoven.
He married Elizabeth de Groot on 30 August 1944. The couple had four children: Jorina Aleida (born 19 January 1947), Frans Willem (born 13 April 1948), Elisabeth (born 24 November 1950), and Judith Elizabeth (born 31 March 1963).
De Bruijn started his academic career at the University of Amsterdam, where he was Professor of Mathematics from 1952 to 1960. In 1960 he moved to the Technical University Eindhoven where he was Professor of Mathematics until his retirement in 1984. Among his graduate students were Johannes Runnenburg (1960), Antonius Levelt (1961), S. Ackermans (1964), Jozef Beenakker (1966), W. van der Meiden (1967), Matheus Hautus (1970), Robert Nederpelt Lazarom (1973), Lambert van Benthem Jutting (1977), A. Janssen (1979), Diederik van Daalen (1980), and Harmannus Balsters (1986).
In 1957 he was appointed member of the Royal Netherlands Academy of Arts and Sciences. He was Knighted with the Order of the Netherlands Lion.
Work
De Bruijn covered many areas of mathematics. He is especially noted for:
the discovery of the De Bruijn sequence,
discovering an algebraic theory of the Penrose tiling and, more generally, discovering the "projection" and "multigrid" methods for constructing quasi-periodic tilings,
the De Bruijn–Newman constant,
the De Bruijn–Erdős theorem, in graph theory,
a different theorem of the same name: the De Bruijn–Erdős theorem, in incidence geometry,
the BEST theorem in graph theory, and
De Bruijn indices.
He wrote one of the standard books in advanced asymptotic analysis (De Bruijn, 1958).
In the late sixties, he designed the Automath language for representing mathematical proofs, so that they could be verified automatically (see automated theorem checking). Shortly before his death, he had been working on models for the human brain.
Publications
Books, a selection:
1943. Over modulaire vormen van meer veranderlijken
1958. Asymptotic Methods in Analysis, North-Holland, Amsterdam.
Articles, a selection:
de Bruijn, Nicolaas Govert. "A combinato
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https://en.wikipedia.org/wiki/Francesco%20Paolo%20Cantelli
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Francesco Paolo Cantelli (20 December 187521 July 1966) was an Italian mathematician. He made contributions to celestial mechanics, probability theory, and actuarial science.
Biography
Cantelli was born in Palermo. He received his doctorate in mathematics in 1899 from the University of Palermo with a thesis on celestial mechanics and continued his interest in astronomy by working until 1903 at Palermo Astronomical Observatory (osservatorio astronomico cittadino), which was under the direction of Annibale Riccò. Cantelli's early papers were on problems in astronomy and celestial mechanics.
From 1903 to 1923 Cantelli worked at the Istituto di Previdenza della Cassa Depositi e Prestiti (Pension Fund for the Government Deposits and Loans Bank). During these years he did research on the mathematics of finance theory and actuarial science, as well as the probability theory. Cantelli's later work was all on probability theory. Borel–Cantelli lemma, Cantelli's inequality and the Glivenko–Cantelli theorem are result of his work in this field. In 1916–1917 he made contributions to the theory of stochastic convergence. In 1923 he resigned his actuarial position when he was appointed professor of actuarial mathematics at the University of Catania. From there, he went to the University of Naples, where he worked as a professor and then in 1931 to the Sapienza University of Rome where he remained until his retirement in 1951. He died in Rome.
Cantelli made fundamental contributions to the foundations of probability theory and to the clarification of different types of probabilistic convergence. He also made seminal contributions to actuarial science. He was the founder of the Istituto Italiano degli Attuari for the applications of mathematics and probability to economics. Cantelli was the editor of the Giornale dell'Istituto Italiano degli Attuari (GIIA) from 1930 to 1958.
Works
Sull'adattamento delle curve ad una serie di misure o di osservazioni, Palermo, 1905
Genesi e costruzione delle tavole di mutualità, 1914
Sulla legge dei grandi numeri, 1916
La tendenza a un limite nel senso del calcolo delle probabilità, 1916
Sulla probabilità come limite della frequenza in "Rendiconti della Reale Accademia dei Lincei", 1917
Una teoria astratta del calcolo delle probabilità, GIIA, vol. 3, pp. 257–265, Roma, 1932
Considerazioni sulla legge uniforme dei grandi numeri e sulla generalizzazione di un fondamentale teorema del Sig. Paul Levy, 1933
Sulla determinazione empirica delle leggi di probabilità, 1933
Su una teoria astratta del calcolo delle probabilità e sulla sua applicazione al teorema detto "delle probabilità zero e uno", 1939
See also
Cantelli's inequality
Borel–Cantelli lemma
Glivenko–Cantelli theorem
References
External links
1875 births
1966 deaths
19th-century Italian mathematicians
20th-century Italian mathematicians
Mathematicians from Sicily
Italian statisticians
University of Palermo alumni
Academic staff of the University of Catan
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https://en.wikipedia.org/wiki/Lan
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Lan or LAN may also refer to:
Science and technology
Local asymptotic normality, a fundamental property of regular models in statistics
Longitude of the ascending node, one of the orbital elements used to specify the orbit of an object in space
Łan, unit of measurement in Poland
Local area network, a computer network that interconnects within a limited area such as one or more buildings
Lan blood group system, a human blood group
Places
Lancashire (Chapman code), England
Lancaster railway station (National Rail station code), England
Capital Region International Airport (IATA airport code), Lansing, Michigan, US
Lan County, Shanxi, China
Łan, Lublin Voivodeship, Poland
Lan (river), Belarus
Llan (placename), a placename element known in Breton as lan
Airlines
LAN Airlines, former name of LATAM Chile, an airline in Chile, with a stake in other airlines:
LAN Peru, an airline based in Peru
LAN Ecuador, an airline based in Quito, Ecuador
LAN Argentina, a defunct Argentine airline
LAN Dominicana, a defunct Dominican airline
LAN Colombia, an airline based in Bogotá, Colombia
Arts
Lan (film), a 2009 Chinese film
Lan Mandragoran, a fictional character from Robert Jordan's The Wheel of Time series
Lan Hikari, protagonist of the Mega Man: Battle Network series of video games and animated television shows
People
Lan (surname 蓝), a Chinese surname
Lan (surname 兰), a Chinese surname
Lan (given name), a given name (including a list of people with the name)
Lan (tribe), ethnic group in Han dynasty China
David Lan (born 1952), South African-born British playwright
Donald Lan (1930–2019), American politician
Phạm Chi Lan, Vietnamese economist
Other uses
Län, administrative division used in Sweden and until 2009 in Finland
Lan (tribe), a tribe of Eastern Huns
Lan, a Cantonese profanity
Lockwood, Andrews & Newnam, an American civil engineering company
Typhoon Lan, a tropical cyclone of the 2017 Pacific typhoon season
See also
Lymphadenopathy
LAN party, a social happening on a local area network e.g multiplayer games.
Lans (disambiguation)
Ian
Llan (disambiguation)
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https://en.wikipedia.org/wiki/Generator%20%28mathematics%29
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In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set.
List of generators
A list of examples of generating sets follow.
Generating set or spanning set of a vector space: a set that spans the vector space
Generating set of a group: A subset of a group that is not contained in any subgroup of the group other than the entire group
Generating set of a ring: A subset S of a ring A generates A if the only subring of A containing S is A
Generating set of an ideal in a ring
Generating set of a module
A generator, in category theory, is an object that can be used to distinguish morphisms
In topology, a collection of sets that generate the topology is called a subbase
Generating set of a topological algebra: S is a generating set of a topological algebra A if the smallest closed subalgebra of A containing S is A
Differential equations
In the study of differential equations, and commonly those occurring in physics, one has the idea of a set of infinitesimal displacements that can be extended to obtain a manifold, or at least, a local part of it, by means of integration. The general concept is of using the exponential map to take the vectors in the tangent space and extend them, as geodesics, to an open set surrounding the tangent point. In this case, it is not unusual to call the elements of the tangent space the generators of the manifold. When the manifold possesses some sort of symmetry, there is also the related notion of a charge or current, which is sometimes also called the generator, although, strictly speaking, charges are not elements of the tangent space.
Elements of the Lie algebra to a Lie group are sometimes referred to as "generators of the group," especially by physicists. The Lie algebra can be thought of as the infinitesimal vectors generating the group, at least locally, by means of the exponential map, but the Lie algebra does not form a generating set in the strict sense.
In stochastic analysis, an Itō diffusion or more general Itō process has an infinitesimal generator.
The generator of any continuous symmetry implied by Noether's theorem, the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, examples include:
angular momentum as the generator of rotations,
l
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https://en.wikipedia.org/wiki/Integral%20symbol
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The integral symbol:
is used to denote integrals and antiderivatives in mathematics, especially in calculus.
History
The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings; it first appeared publicly in the article "" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686. The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.
Typography in Unicode and LaTeX
Fundamental symbol
The integral symbol is in Unicode and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (named entity).
The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.
The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh").
Extensions of the symbol
Related symbols include:
Typography in other languages
In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.
Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol:
By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing, but is more compact horizontally, especially when longer expressions are used in the limits:
See also
Capital sigma notation
Capital pi notation
Notes
References
External links
Fileformat.info
History of calculus
Mathematical symbols
Gottfried Wilhelm Leibniz
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https://en.wikipedia.org/wiki/Hyperbolic%20group
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In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.
Definition
Let be a finitely generated group, and be its Cayley graph with respect to some finite set of generators. The set is endowed with its graph metric (in which edges are of length one and the distance between two vertices is the minimal number of edges in a path connecting them) which turns it into a length space. The group is then said to be hyperbolic if is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a such that any geodesic triangle in is -thin, as illustrated in the figure on the right (the space is then said to be -hyperbolic).
A priori this definition depends on the choice of a finite generating set . That this is not the case follows from the two following facts:
the Cayley graphs corresponding to two finite generating sets are always quasi-isometric one to the other;
any geodesic space which is quasi-isometric to a geodesic Gromov-hyperbolic space is itself Gromov-hyperbolic.
Thus we can legitimately speak of a finitely generated group being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a -hyperbolic space is itself -hyperbolic for some but the latter depends on both the original and on the quasi-isometry, thus it does not make sense to speak of being -hyperbolic.
Remarks
The Švarc–Milnor lemma states that if a group acts properly discontinuously and with compact quotient (such an action is often called geometric) on a proper length space , then it is finitely generated, and any Cayley graph for is quasi-isometric to . Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space.
If is a subgroup with finite index (i.e., the set is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of into any locally finite Cayley graph of . Thus is hyperbolic if and only if itself is. More generally, if two groups are commensurable, then one is hyperbolic if and only if the oth
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https://en.wikipedia.org/wiki/Inductive%20dimension
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In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.
The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to the Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.
Formal definition
We want the dimension of a point to be 0, and a point has empty boundary, so we start with
Then inductively, ind(X) is the smallest n such that, for every and every open set U containing x, there is an open set V containing x, such that the closure of V is a subset of U, and the boundary of V has small inductive dimension less than or equal to n − 1. (If X is a Euclidean n-dimensional space, V can be chosen to be an n-dimensional ball centered at x.)
For the large inductive dimension, we restrict the choice of V still further; Ind(X) is the smallest n such that, for every closed subset F of every open subset U of X, there is an open V in between (that is, F is a subset of V and the closure of V is a subset of U), such that the boundary of V has large inductive dimension less than or equal to n − 1.
Relationship between dimensions
Let be the Lebesgue covering dimension. For any topological space X, we have
if and only if
Urysohn's theorem states that when X is a normal space with a countable base, then
Such spaces are exactly the separable and metrizable X (see Urysohn's metrization theorem).
The Nöbeling–Pontryagin theorem then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the Euclidean spaces, with their usual topology. The Menger–Nöbeling theorem (1932) states that if is compact metric separable and of dimension , then it embeds as a subspace of Euclidean space of dimension . (Georg Nöbeling was a student of Karl Menger. He introduced Nöbeling space, the subspace of consisting of points with at least co-ordinates being irrational numbers, which has universal properties for embedding spaces of dimension .)
Assuming only X metrizable we have (Miroslav Katětov)
ind X ≤ Ind X = dim X;
or assuming X compact and Hausdorff (P. S. Aleksandrov)
dim X ≤ ind X ≤ Ind X.
Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.
A separa
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https://en.wikipedia.org/wiki/Free%20Lie%20algebra
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In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity.
Definition
The definition of the free Lie algebra generated by a set X is as follows:
Let X be a set and a morphism of sets (function) from X into a Lie algebra L. The Lie algebra L is called free on X if is the universal morphism; that is, if for any Lie algebra A with a morphism of sets , there is a unique Lie algebra morphism such that .
Given a set X, one can show that there exists a unique free Lie algebra generated by X.
In the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functor.
The free Lie algebra on a set X is naturally graded. The 1-graded component of the free Lie algebra is just the free vector space on that set.
One can alternatively define a free Lie algebra on a vector space V as left adjoint to the forgetful functor from Lie algebras over a field K to vector spaces over the field K – forgetting the Lie algebra structure, but remembering the vector space structure.
Universal enveloping algebra
The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. By the Poincaré–Birkhoff–Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
Ernst Witt showed that the number of basic commutators of degree k in the free Lie algebra on an m-element set is given by the necklace polynomial:
where is the Möbius function.
The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra. This essentially follows because universal enveloping algebras have the structure of a Hopf algebra, and the shuffle product describes the action of comultiplication in this algebra. See tensor algebra for a detailed exposition of the inter-relation between the shuffle product and comultiplication.
Hall sets
An explicit basis of the free Lie algebra can be given in terms of a Hall set, which is a particular kind of subset inside the free magma on X. Elements of the free magma are binary trees, with their leaves labelled by elements of X. Hall sets were introduced by based on work of Philip Hall on groups. Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory due to Philip Hall and Witt.
Lyndon basis
The Lyndon words are a special case of the Hall
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https://en.wikipedia.org/wiki/Rizza
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Rizza may refer to
Rizza (surname)
A frazione of Villafranca di Verona in the province of Verona, Italy
Isola Rizza, a commune in the province of Verona, Italy
Rizza manifold in differential geometry
Rizza Islam (born 1990), member of the Nation of Islam and social media influencer
RZA, an American rapper and record producer (pronounced "rizza")
Riza, a metal covering protecting an icon
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https://en.wikipedia.org/wiki/Morgan%20Quitno%20Press
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Morgan Quitno Press is a research and publishing company founded in 1989 and based in Lawrence, Kansas. The company compiled annual reference books of US state and city statistics. Its primary volumes included State Rankings, Health Care State Rankings, Education State Rankings, Crime State Rankings, City Crime Rankings, and State Trends. In 2007, Morgan Quitno Press was acquired by CQ Press, a division of Congressional Quarterly Inc. CQ Press later was acquired by Sage Publications which incorporated the Morgan Quitno statistics into its Sage Stats database.
Products
State Rankings – comprehensive livability of each State. Annual Most Livable State Award is given to No. 1 State in this ranking.
Health Care State Ranking – limited to and more detailed in health care. No. 1 State is awarded annual Healthiest State Award each year.
Crime State Rankings – limited to and more detailed in crime. Best State is awarded annual Safest State Award while worst State is tabbed as Most Dangerous State.
Education State Rankings – limited to and more detailed in K-12 education. Best State is awarded annual Smartest State Award.
City Crime Rankings – city and metro area version of Crime State Rankings mentioned above. Safest City, Safest Metro, Most Dangerous City, and Most Dangerous Metro are ranked based on the data on this rankings.
State Trends – shows trend of each State data. Most Improved State decided based on the data. Last edition was in 2007.
America's safest (most dangerous) cities
Evaluated cities and metro areas
Cities assessed – population of 75,000+
Metro areas assessed – all, as defined by US Census Bureau (no minimum population)
Assessed crimes
murder
rape
robbery
assault
burglary
automobile theft
Methodology
Calculate crime rate (number of cases per pop. 100,000) of each 6 crimes based on previous year's FBI data.
Compare these rates with U.S. average
Calculate Crime Index based on Morgan Quitno's equation.
Rank cities and metros by their Crime Index.
Criticism
The FBI recommends against use of its crime statistics for directly comparing cities as Morgan Quitno does in its "Most Dangerous Cities" rankings. This is due to the many factors that influence crime, such as population density and the degree of urbanization, modes of transportation of highway system, economic conditions, and citizens' attitudes toward crime.
Cities of Illinois are not included in this ranking, due to a disparity in the way Illinois State police and the FBI report rape cases. Other cities may be excluded because of lack of some data.
In October 2007 the American Society of Criminology, the U.S. Conference of Mayors, and the Federal Bureau of Investigation asked the publisher to reconsider promotion of the book – specifically, "their inaccurate and inflammatory press release labeling cities as 'safest' and 'most dangerous'" – because the rankings are "baseless and damaging." In November 2007 the executive board of the American Society of Criminology (ASC) ap
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https://en.wikipedia.org/wiki/Side
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Side or Sides may refer to:
Geometry
Edge (geometry) of a polygon (two-dimensional shape)
Face (geometry) of a polyhedron (three-dimensional shape)
Places
Side, Turkey, a city in Turkey
Side (Ainis), a town of Ainis, ancient Thessaly, Greece
Side (Caria), a town of ancient Caria, Anatolia
Side (Laconia), a town of ancient Laconia, Greece
Side (Pontus), a town of ancient Pontus, Anatolia
Side (Ukraine), a village in Ukraine
Side, Iran, a village in Iran
Side, Gloucestershire, or Syde, a village in England
Music
Side (recording), the A-side or B-side of a record
The Side, a Scottish rock band
Sides (album), a 1979 album by Anthony Phillips
Sides, a 2020 album by Emily King
"Side" (song), a 2001 song by Travis
"Sides", a song by Flobots from the album The Circle in the Square, 2012
"Sides", a song by Allday from the album Speeding, 2017
Teams
Side (cue sports technique)
Side, a team, in particular:
Sports team
Other uses
Side (gay sex), a sex role
Side (mythology), in Greek mythology
Side, a Morris dance team
Sideboard (cards), known as a "side" in some collectible card games
Sides (surname), a surname
Side dish, a food item accompanying a main course
School of Isolated and Distance Education, a public school in Perth, Western Australia
Secretariat of Intelligence, an Argentinian intelligence agency
Social identity model of deindividuation effects, in social psychology
See also
Relative direction, left and right
Syde
Cide (disambiguation)
Sidle (disambiguation)
Site (disambiguation)
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https://en.wikipedia.org/wiki/Maxime%20B%C3%B4cher
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Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as Trigonometry and Analytic Geometry. Bôcher's theorem, Bôcher's equation, and the Bôcher Memorial Prize are named after him.
Life
Bôcher was born in Boston, Massachusetts. His parents were Caroline Little and Ferdinand Bôcher. Maxime's father was professor of modern languages at the Massachusetts Institute of Technology when Maxime was born, and became Professor of French at Harvard University in 1872.
Bôcher received an excellent education from his parents and from a number of public and private schools in Massachusetts. He graduated from the Cambridge Latin School in 1883. He received his first degree from Harvard in 1888. At Harvard, he studied a wide range of topics, including mathematics, Latin, chemistry, philosophy, zoology, geography, geology, meteorology, Roman art, and music.
Bôcher was awarded many prestigious prizes, which allowed him to travel to Europe to do research. The University of Göttingen was then the leading mathematics university, and he attended there lectures by Felix Klein, Arthur Moritz Schoenflies, Hermann Schwarz, Issai Schur and Woldemar Voigt. He was awarded a doctorate in 1891 for his dissertation Über die Reihenentwicklungen der Potentialtheorie (German for "On the Development of the Potential Function into Series"); he was encouraged to study this topic by Klein. He received a Göttingen university prize for this work.
Bocher was elected to the American Academy of Arts and Sciences in 1899, the United States National Academy of Sciences in 1909, and the American Philosophical Society in 1916.
In Göttingen he met Marie Niemann, and they were married in July 1891. They had three children, Helen, Esther, and Frederick. He returned with his wife to Harvard where he was appointed as an instructor. In 1894 he was promoted to assistant professor, due to his impressive record. He became a full professor of mathematics in 1904. He was president of the American Mathematical Society from 1908 to 1910.
Although he was only 46 years old, there were already signs that his weak health was failing. He died at his Cambridge home after suffering a prolonged illness.
Bôcher's theorem
Bôcher's theorem states that the finite zeros of the derivative of a non-constant rational function that are not multiple zeros of are the positions of equilibrium in the field of force due to particles of positive mass at the zeros of and particles of negative mass at the poles of , with masses numerically equal to the respective multiplicities, where each particle repels with a force equal to the mass times the inverse distance.
Bôcher's equation
Bôcher's equation is a second-order ordinary differential equation of the form:
The Bôcher Memorial Prize
The Bôcher Memorial Prize is awarded by the American Mathematical Society every five
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https://en.wikipedia.org/wiki/Kafr%20Abbush
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Kafr 'Abbush () is a Palestinian town in the Tulkarm Governorate in the northwestern West Bank. According to the Palestinian Central Bureau of Statistics, Kafr 'Abbush had a population of approximately 1,488 inhabitants in mid-year 2006 and 1,739 by 2017. 24.8% of the population of Kafr 'Abbush were refugees in 1997. The healthcare facilities for Kafr 'Abbush are based in Kafr 'Abbush, where the facilities are designated as MOH level 2.
History
Archeological findings from Kafr 'Abbush include potsherds from the Byzantine era and two menorahs carved in stone.
Ottoman era
Kafr 'Abbush was incorporated into the Ottoman Empire in 1517 with all of Palestine, and in 1596 it appeared under the name of Abbus in the tax registers as being in the Nahiya of Bani Sa'b, part of Nablus Sanjak. It had a population of 19 Muslim households. The villagers paid a fixed tax rate of 33.3% on various agricultural products, such as wheat, barley, summer crops, olive trees, goats and/or beehives, in addition to "occasional revenues" and a press for olive oil or grape syrup; a total of 4,974 akçe.
In 1838, Robinson noted Kefr 'Abush as a village in Beni Sa'ab district, west of Nablus.In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of Bani Sa'b.
In the 1860s, the Ottoman authorities granted the village an agricultural plot of land called Ghabat Kafr 'Abbush in the former confines of the Forest of Arsur (Ar. Al-Ghaba) in the coastal plain, west of the village.
In 1882, the PEF's Survey of Western Palestine (SWP) described Kafr Abbush as: "a stone village of moderate size, on steep round hill, with a few olives. It is supplied by cisterns. The ground is very rugged near it."
British Mandate era
In the 1922 census of Palestine conducted by the British Mandate authorities, Kafr Abbush had a population of 263 Muslims, increasing in the 1931 census to 360 Muslims, in 63 houses.
In the 1945 statistics the population of Kafr Abbush was 480 Muslims, with 4,923 dunams of land according to an official land and population survey. Of this, 952 dunams were plantations and irrigable land, 1,047 were used for cereals, while 11 dunams were built-up (urban) land.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Kafr Abbush came under Jordanian rule.
In 1961, the population of Kafr Abbush was 704.
Post 1967
Since the Six-Day War in 1967, Kafr Abbush has been under Israeli occupation.
Footnotes
Bibliography
External links
Welcome To Kafr Abbush
Survey of Western Palestine, Map 11: IAA, Wikimedia commons
Villages in the West Bank
Tulkarm Governorate
Municipalities of the State of Palestine
Ancient Samaritan settlements
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https://en.wikipedia.org/wiki/Ore%20extension
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In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials.
Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
Definition
Suppose that R is a (not necessarily commutative) ring, is a ring homomorphism, and is a σ-derivation of R, which means that is a homomorphism of abelian groups satisfying
.
Then the Ore extension , also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials a new multiplication, subject to the identity
.
If δ = 0 (i.e., is the zero map) then the Ore extension is denoted R[x; σ]. If σ = 1 (i.e., the identity map) then the Ore extension is denoted R[ x, δ ] and is called a differential polynomial ring.
Examples
The Weyl algebras are Ore extensions, with R any commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.
Properties
An Ore extension of a domain is a domain.
An Ore extension of a skew field is a non-commutative principal ideal domain.
If σ is an automorphism and R is a left Noetherian ring then the Ore extension R[ λ; σ, δ ] is also left Noetherian.
Elements
An element f of an Ore ring R is called
twosided (or invariant ), if R·f = f·R, and
central, if g·f = f·g for all g in R.
Further reading
Azeddine Ouarit (1992) Extensions de ore d'anneaux noetheriens á i.p, Comm. Algebra, 20 No 6,1819-1837. https://zbmath.org/?q=an:0754.16014
Azeddine Ouarit (1994) A remark on the Jacobson property of PI Ore extensions. (Une remarque sur la propriété de Jacobson des extensions de Ore a I.P.) (French) Zbl 0819.16024. Arch. Math. 63, No.2, 136-139 (1994). https://zbmath.org/?q=an:00687054
References
Ring theory
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https://en.wikipedia.org/wiki/Analytic%20manifold
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In mathematics, an analytic manifold, also known as a manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.
For , the space of analytic functions, , consists of infinitely differentiable functions , such that the Taylor series
converges to in a neighborhood of , for all . The requirement that the transition maps be analytic is significantly more restrictive than that they be infinitely differentiable; the analytic manifolds are a proper subset of the smooth, i.e. , manifolds. There are many similarities between the theory of analytic and smooth manifolds, but a critical difference is that analytic manifolds do not admit analytic partitions of unity, whereas smooth partitions of unity are an essential tool in the study of smooth manifolds. A fuller description of the definitions and general theory can be found at differentiable manifolds, for the real case, and at complex manifolds, for the complex case.
See also
Complex manifold
Analytic variety
References
Structures on manifolds
Manifolds
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https://en.wikipedia.org/wiki/William%20Smyth%20%28professor%29
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William Smyth (February 2, 1797 – April 3, 1868) was an American academic and writer on mathematics and other subjects.
Early life
William Smyth was born in Pittston, Maine on February 2, 1797. He graduated from Bowdoin College in 1822, then studied theology at Andover Theological Seminary.
Career
In 1825, he became a professor of mathematics at Bowdoin College, and in 1846 became an associate professor of natural philosophy. The Bowdoin College Department of Mathematics Smyth Prize is named in his honor.
Smyth was an ardent abolitionist of slavery and supporter of the temperance movement. While at Bowdoin, Smyth supported the effort to the First Parish Church, which is now on the National Register of Historic Places.
Personal life
Smyth married Harriet Porter, daughter of Mary (née Porter) and Nathaniel Coffin. They had nine children.
He died in Brunswick, Maine in April 1868. He is interred at Pine Grove Cemetery in Brunswick.
Bibliography
Smyth wrote several widely used textbooks:
Elements of Algebra (1833) digitized version
Elementary Algebra for Schools (1850) digitized version
Treatise on Algebra" (1852) digitized version
Trigonometry, Surveying, and Navigation(1855) digitized version
Elements of Analytical Geometry" (1855)
Elements of the Differential and Integral Calculus" (1856; 2d ed., 1859) digitized version
Lectures on Modern History, edited by Jared Sparks (1849) digitized version
References
Bowdoin College Catalogue 1840-1848. Bowdoin College Catalogue. George J. Mitchell Department of Special Collections & Archives. Bowdoin College Library.
External links
Smyth Prize
An article whose original source is the controversial Appleton's Cyclopedia of American Biography, originally published in 1887-1889, and republished in 1999.
1797 births
1868 deaths
19th-century American mathematicians
Bowdoin College alumni
19th-century American male writers
People from Kennebec County, Maine
People from Brunswick, Maine
Writers from Maine
Burials at Pine Grove Cemetery (Brunswick, Maine)
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https://en.wikipedia.org/wiki/Macaulay2
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Macaulay2 is a free computer algebra system created by Daniel Grayson (from the University of Illinois at Urbana–Champaign) and Michael Stillman (from Cornell University) for computation in commutative algebra and algebraic geometry.
Overview
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar. In addition, the system has been extended by a large number of packages. Nearly 200 packages are included in the distribution of Macaulay2 as of 2019, and notable package authors include Craig Huneke and Frank-Olaf Schreyer. The Journal of Software for Algebra and Geometry has published numerous packages and programs for Macaulay2.
Macaulay2 has an interactive command-line interface used from the terminal (see ). It can also use emacs or GNU TeXmacs as a user interface.
Macaulay2 uses its own interpreted high-level programming language both from the command line and in saved programs. This language is intended to be easy to use for mathematicians, and many parts of the system are indeed written in the Macaulay2 language. The algebraic algorithms that form the core functionality are written in C++ for speed. The interpreter itself is written in a custom type safety layer over C. Both the system and the programming language are published under the GNU General Public License version 2 or 3.
History
Stillman, along with Dave Bayer had authored the predecessor system, Macaulay, beginning in 1983. They named Macaulay after Francis Sowerby Macaulay, an English mathematician who made significant contributions to algebraic geometry. The Macaulay system showed that it was possible to solve actual problems in algebraic geometry using Gröbner basis techniques, but by the early 1990s, limitations in its architecture were becoming an obstruction. Using the experience with Macaulay, Grayson and Stillman began work on Macaulay2 in 1993. The Macaulay2 language and design has a number of improvements over that of Macaulay, allowing for infinite coefficient rings, new data types, and other useful features.
Macaulay continued to be updated and used for some time after the 1993 introduction of Macaulay2. The last released version was 3.1, from August 2000. The Macaulay webpage currently recommends switching to Macaulay2.
Macaulay2 has been updated regularly since its introduction. David Eisenbud has been listed as a collaborator on the project since 2007.
Sample session
The following session defines a polynomial ring , an ideal inside , and the quotient ring . The text i1 : is the 1st input prompt in a session, while o1 is the corresponding output.
i1 : S=QQ[a,b,c,d,e]
o1 = S
o1 : PolynomialRing
i2 : I=ideal(a^3-b^3, a+b+c+d+e)
3 3
o2 = ideal (a - b , a +
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https://en.wikipedia.org/wiki/Bruno%20Mathsson
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Bruno Mathsson (13 January 190717 August 1988) was a Swedish furniture designer and architect whose ideas aligned with functionalism, modernism, as well as old Swedish crafts tradition.
Biography
Mathsson was raised in the town of Värnamo in the Småland region of Sweden, the son of a master cabinet maker. After a short time of education in school, he started to work in his father's gallery. He soon found a great interest in furniture and especially chairs, their function and design. In the 1920s and 30s he developed a techniques for building bentwood chairs with hemp webbing. The first model, called the Grasshopper, was used at Värnamo Hospital in 1931.
Edgar Kaufmann Jr., director of the Industrial Design Department at the Museum of Modern Art (MOMA), collected Mathsson's chairs and included them in several exhibitions in the 1940s. Kaufmann considered Mathsson's importance in furniture design on par with that of Alvar Aalto. Kaufmann and his family also had a Mathsson chair at their house Fallingwater.
Mathsson was also an accomplished architect; he completed about 100 structures in the 1940s and 50s. He was the first architect in Sweden to build all-glass structures with heated floors. His furniture showroom in Värnamo (1950) was a significant example; it is well-preserved and open to the public today. For his glass houses, he developed double- and triple-pane insulated glass units called "Bruno-Pane".
He traveled extensively in the United States and was strongly influenced by the solar houses of George Fred Keck. Mathsson's architecture was also influenced by a visit to the Eames House by Charles and Ray Eames in March 1949 just as it was being completed.
Works
Furniture
Grasshopper (1931)
Mimat (1932)
Pernilla (1934)
The Eva Chair (1935)
Folding table (1935)
Paris Daybed (1937)
Swivel chair (1939-1940)
Pernilla Lounge
Jetson Chair
Super-Ellipse™ table series, with Piet Hein (1966)
Annika nesting tables (1968)
Architecture
Bruno Mathsson furniture showroom, Värnamo (1950)
house at Danderyd (1955)
Villa Prenker, Kungsör (1955)
Kosta Glassworks exhibition hall and residences, Kosta (1956)
weekend cottage at Frösakull (1960)
"one of the most daring examples of his glass houses."
Södrakull, outside Värnamo (1965)
References
External links
1907 births
1988 deaths
Swedish furniture designers
20th-century Swedish architects
Recipients of the Prince Eugen Medal
People from Värnamo Municipality
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https://en.wikipedia.org/wiki/Ikeda%20map
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In physics and mathematics, the Ikeda map is a discrete-time dynamical system given by the complex map
The original map was proposed first by Kensuke Ikeda as a model of light going around across a nonlinear optical resonator (ring cavity containing a nonlinear dielectric medium) in a more general form. It is reduced to the above simplified "normal" form by Ikeda, Daido and Akimoto stands for the electric field inside the resonator at the n-th step of rotation in the resonator, and and are parameters which indicate laser light applied from the outside, and linear phase across the resonator, respectively. In particular the parameter is called dissipation parameter characterizing the loss of resonator, and in the limit of the Ikeda map becomes a conservative map.
The original Ikeda map is often used in another modified form in order to take the saturation effect of nonlinear dielectric medium into account:
A 2D real example of the above form is:
where u is a parameter and
For , this system has a chaotic attractor.
Attractor
This shows how the attractor of the system changes as the parameter is varied from 0.0 to 1.0 in steps of 0.01. The Ikeda dynamical system is simulated for 500 steps, starting from 20000 randomly placed starting points. The last 20 points of each trajectory are plotted to depict the attractor. Note the bifurcation of attractor points as is increased.
Point trajectories
The plots below show trajectories of 200 random points for various values of . The inset plot on the left shows an estimate of the attractor while the inset on the right shows a zoomed in view of the main trajectory plot.
Octave/MATLAB code for point trajectories
The Octave/MATLAB code to generate these plots is given below:
% u = ikeda parameter
% option = what to plot
% 'trajectory' - plot trajectory of random starting points
% 'limit' - plot the last few iterations of random starting points
function ikeda(u, option)
P = 200; % how many starting points
N = 1000; % how many iterations
Nlimit = 20; % plot these many last points for 'limit' option
x = randn(1, P) * 10; % the random starting points
y = randn(1, P) * 10;
for n = 1:P,
X = compute_ikeda_trajectory(u, x(n), y(n), N);
switch option
case 'trajectory' % plot the trajectories of a bunch of points
plot_ikeda_trajectory(X); hold on;
case 'limit'
plot_limit(X, Nlimit); hold on;
otherwise
disp('Not implemented');
end
end
axis tight; axis equal
text(- 25, - 15, ['u = ' num2str(u)]);
text(- 25, - 18, ['N = ' num2str(N) ' iterations']);
end
% Plot the last n points of the curve - to see end point or limit cycle
function plot_limit(X, n)
plot(X(end - n:end, 1), X(end - n:end, 2), 'ko');
end
% Plot the whole trajectory
function plot_ikeda_trajectory(X)
plot(X(:, 1), X(:, 2), 'k');
% hold on; plot(X(1,1), X(1,2), 'bo', 'markerfacecolor', 'g'); hold off
end
% u is the ikeda parameter
% x,y is the starting poin
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https://en.wikipedia.org/wiki/Constance%20Kamii
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Constance Kamii was a Swiss-Japanese-American mathematics education scholar and psychologist. She was a professor in the Early Childhood Education Program
Department of Curriculum and Instruction at the University of Alabama in Birmingham, Alabama.
Overview
Constance Kamii was born in Geneva, Switzerland, and attended elementary schools there and in Japan. She finished high school in Los Angeles, attended Pomona College, and received her Ph.D. in education and psychology from the University of Michigan.
She was a professor of early childhood education at the University of Alabama in Birmingham. A major concern of hers since her work on the Perry Preschool Project in the 1960s was the conceptualization of goals and objectives for early childhood education on the basis of a scientific theory explaining children’s sociological and intellectual development. Convinced that the only theory in existence that explains this development from the first day of life to adolescence was that of Jean Piaget, she studied under him on and off for 15 years.
When she was not studying under Piaget in Geneva, she worked closely with teachers in the United States to develop practical ways of using his theory in classrooms. The outcome of this classroom research can be seen in Physical Knowledge in Preschool Education and Group Games in Early Education, which she wrote with Rheta DeVries. Since 1980, she had been extending this curriculum research to the primary grades and wrote Young Children Reinvent Arithmetic (about first grade), Young Children Continue to Reinvent Arithmetic, 2nd Grade, and Young Children Continue to Reinvent Arithmetic, 3rd Grade. In all these books, she emphasized the long-range, over-all aim of education envisioned by Piaget, which is children’s development of sociological and intellectual autonomy.
Kamii studied under Jean Piaget to develop an early childhood curriculum based on his theory. This work can be seen in Physical Knowledge in Preschool Education (1978) and Group Games in Early Education (1980) (which she wrote with Rheta DeVries), and Number in Preschool and Kindergarten (1982). From 1980 to 2000, she developed a primary arithmetic program based on Piaget's theory. She abandoned this effort in 2000 because many parents of fourth graders were teaching "carrying" and "borrowing" at home.
References
External links
Home page
In Loving Memory of Constance Kamii
Year of birth missing (living people)
Living people
Pomona College alumni
University of Michigan School of Education alumni
University of Alabama faculty
Mathematics educators
American educational psychologists
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https://en.wikipedia.org/wiki/Polarization
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Polarization or polarisation may refer to:
Mathematics
Polarization of an Abelian variety, in the mathematics of complex manifolds
Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables
Polarization identity, expresses an inner product in terms of its associated norm
Polarization (Lie algebra)
Physical sciences
Polarization (physics), the ability of waves to oscillate in more than one direction, in particular polarization of light, responsible for example for the glare-reducing effect of polarized sunglasses
Polarization (antenna), the state of polarization (in the above sense) of electromagnetic waves transmitted by or received by a radio antenna
Dielectric polarization, charge separation in insulating materials:
Polarization density, volume dielectric polarization
Dipolar polarization, orientation of permanent dipoles
Ionic polarization, displacement of ions in a crystal
Maxwell–Wagner–Sillars polarization, slow long-distance charge separation in dielectric spectroscopy on inhomogeneous soft matter
Polarization (electrochemistry), a change in the equilibrium potential of an electrochemical reaction
Concentration polarization, the shift of the electrochemical potential difference across an electrochemical cell from its equilibrium value
Spin polarization, the degree by which the spin of elementary particles is aligned to a given direction
Polarizability, an electrical property of atoms or molecules and a separate magnetic property of subatomic particles
Polarization function, a feature of some molecular modelling methods
Photon polarization, the mathematical link between wave polarization and spin polarization
Vacuum polarization, a process in which a background electromagnetic field produces virtual electron-positron pairs
Social sciences
Polarization (economics), faster decrease of moderate-skill jobs relative to low-skill and high-skill jobs
Political polarization, when public opinion divides and becomes oppositional
Social polarization, segregation of society into social groups, from high-income to low-income
Group polarization, tendency of a group to make more extreme decisions than individuals' initial inclinations
Attitude polarization, when disagreement becomes more extreme as different parties consider evidence
Racial polarization, when a population with varying ancestry is divided into distinct racial groups
Others
Polarization (album), an album by American jazz trombonist and composer Julian Priester
Polarization, in many disciplines, is a tendency to be located close to one of the opposite ends of a continuum
See also
Polar (disambiguation)
Polar opposite (disambiguation)
Polarity (disambiguation)
Pole (disambiguation)
Reversal (disambiguation)
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https://en.wikipedia.org/wiki/Plane%20geometry%20%28disambiguation%29
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In mathematics, plane geometry may refer to the geometry of a two-dimensional geometric object called a plane.
Most times it refers to Euclidean plane geometry, the geometry of plane figures,
More specifically it can refer to:
Euclidean plane geometry:
Cartesian geometry, the study of geometry using a coordinate system
Two-dimensional space
synthetic geometry, the study of geometry using a logical deduction and compass and straightedge constructions
geometry of a projective plane,
often the real projective plane,
but possibly the complex projective plane or projective plane defined over some other field,
either infinite or finite, such as the Fano plane, or others;
geometry of an affine plane,
geometry of a non-Euclidean plane,
either the hyperbolic plane,
or the elliptic plane,
or the geometry of the related two-dimensional spherical geometry.
See also
Plane curve
Geometrography
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https://en.wikipedia.org/wiki/Elliptic%20complex
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In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.
Definition
If E0, E1, ..., Ek are vector bundles on a smooth manifold M (usually taken to be compact), then a differential complex is a sequence
of differential operators between the sheaves of sections of the Ei such that Pi+1 ∘ Pi=0. A differential complex with first order operators is elliptic if the sequence of symbols
is exact outside of the zero section. Here π is the projection of the cotangent bundle T*M to M, and π* is the pullback of a vector bundle.
See also
Chain complex
References
Differential geometry
Elliptic partial differential equations
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https://en.wikipedia.org/wiki/Collinearity
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In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
Points on a line
In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being in a row.
A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property.
The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation.
Examples in Euclidean geometry
Triangles
In any triangle the following sets of points are collinear:
The orthocenter, the circumcenter, the centroid, the Exeter point, the de Longchamps point, and the center of the nine-point circle are collinear, all falling on a line called the Euler line.
The de Longchamps point also has other collinearities.
Any vertex, the tangency of the opposite side with an excircle, and the Nagel point are collinear in a line called a splitter of the triangle.
The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points bisect the perimeter), and the center of the Spieker circle are collinear in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the center of mass of the perimeter of the triangle.)
Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear.
From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle.
The lines connecting the feet of the altitudes intersect the opposite sides at collinear points.
A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side a
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https://en.wikipedia.org/wiki/Dolbeault%20cohomology
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In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
Construction of the cohomology groups
Let Ωp,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections
Since
this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space
Dolbeault cohomology of vector bundles
If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the sheaf of holomorphic sections of E, using the Dolbeault operator of E. This is therefore a resolution of the sheaf cohomology of .
In particular associated to the holomorphic structure of is a Dolbeault operator taking sections of to -forms with values in . This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator on differential forms, and is therefore sometimes known as a -connection on , Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative, the Dolbeault operator of can be extended to an operator
which acts on a section by
and is extended linearly to any section in . The Dolbeault operator satisfies the integrability condition and so Dolbeault cohomology with coefficients in can be defined as above:
The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator compatible with the holomorphic structure of , so are typically denoted by dropping the dependence on .
Dolbeault–Grothendieck lemma
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or -Poincaré lemma). First we prove a one-dimensional version of the -Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions:
Proposition: Let the open ball centered in of radius open and , then
Lemma (-Poincaré lemma on the complex plane): Let be as before and a smooth form, then
satisfies on
Proof. Our claim is that defined above is a well-defined smooth function and . To show this we choose a point and an open neighbourhood , then we can find a smooth function whose support is compact and lies in and Then we can write
and define
Since in then is clearly well-defined and smooth; we note that
which is indeed well-defined and smooth, therefore the same is true for . Now we show that on .
since is holomorphic in .
applying the generalised Cauchy formula to we find
since , but then on . Since was arbitrary, the lemma is now proved.
Proof of Dolbeault–Grothendie
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https://en.wikipedia.org/wiki/Set%20%28music%29
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A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
A set by itself does not necessarily possess any additional structure, such as an ordering or permutation. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.
Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.
A time-point set is a duration set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes.
Serial
In the theory of serial music, however, some authors (notably Milton Babbitt) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a twelve-tone row) used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. (See below.) This is a different usage of the term "set" from that described above (and referred to in the term "set theory").
For these authors, a set form (or row form) is a particular arrangement of such an ordered set: the prime form (original order), inverse (upside down), retrograde (backwards), and retrograde inverse (backwards and upside down).
A derived set is one which is generated or derived from consistent operations on a subset, for example Webern's Concerto, Op.24, in which the last three subsets are derived from the first:
This can be represented numerically as the integers 0 to 11:
0 11 3 4 8 7 9 5 6 1 2 10
The first subset (B B D) being:
0 11 3 prime-form, interval-string =
The second subset (E G F) being the retrograde-inverse of the first, transposed up one semitone:
3 11 0 retrograde, interval-string = mod 12
3 7 6 inverse, interval-string = mod 12
+ 1 1 1
------
= 4 8 7
The third subset (G E F) being the retrograde of the first, transposed up (or down) six semitones:
3 11 0 retrograde
+ 6 6 6
------
9 5 6
And the fourth subset (C C A) being the inverse of the first, transposed up one semitone:
0 11 3 prime form, interval-vector = mod 12
0 1 9 inverse, interval-string = mod 12
+ 1 1 1
-------
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https://en.wikipedia.org/wiki/Gerard%20Murphy
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Gerard Murphy may refer to:
Gerard Murphy (politician) (born 1951), Irish Fine Gael politician, TD for Cork North West
Gerard Murphy (mathematician) (1948–2006), Irish mathematics professor
Gerard Murphy (actor) (1948–2013), Irish film, television and theatre actor
See also
Gerry Murphy (disambiguation)
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https://en.wikipedia.org/wiki/Bring%20radical
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In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial
The Bring radical of a complex number a is either any of the five roots of the above polynomial (it is thus multi-valued), or a specific root, which is usually chosen such that the Bring radical is real-valued for real a and is an analytic function in a neighborhood of the real line. Because of the existence of four branch points, the Bring radical cannot be defined as a function that is continuous over the whole complex plane, and its domain of continuity must exclude four branch cuts.
George Jerrard showed that some quintic equations can be solved in closed form using radicals and Bring radicals, which had been introduced by Erland Bring.
In this article, the Bring radical of a is denoted For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior for large .
Normal forms
The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form:
The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using Tschirnhaus transformations to reduce the number of independent coefficients.
Principal quintic form
The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed:
If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation
the coefficients α and β may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities. This leads to a system of equations in α and β consisting of a quadratic and a linear equation, and either of the two sets of solutions may be used to obtain the corresponding three coefficients of the principal quintic form.
This form is used by Felix Klein's solution to the quintic.
Bring–Jerrard normal form
It is possible to simplify the quintic still further and eliminate the quadratic term, producing the Bring–Jerrard normal form:
Using the power-sum formulae again with a cubic transformation as Tschirnhaus tried does not work, since the resulting system of equations results in a sixth-degree equation. But in 1796 Bring found a way around this by using a quartic Tschirnhaus transformation to relate the roots of a principal quintic to those of a Bring–Jerrard quintic:
The extra parameter this fourth-order transformation provides allowed Bring to decrease the degrees of the other parameters. This leads to a system of five equations in six unknowns, which then requires the solution of a cubic and a quadratic equation. This method was also discovered by Jerrard in 1852,
but it is likely that he was unaware of Bring's previous work in this area. The full transformation may readily be accomplished using a computer algebra package such as Mathematica
or Maple.
As might be expected from the complexity of these
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https://en.wikipedia.org/wiki/Tubular%20neighborhood
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In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood.
In general, let S be a submanifold of a manifold M, and let N be the normal bundle of S in M. Here S plays the role of the curve and M the role of the plane containing the curve. Consider the natural map
which establishes a bijective correspondence between the zero section of N and the submanifold S of M. An extension j of this map to the entire normal bundle N with values in M such that is an open set in M and j is a homeomorphism between N and is called a tubular neighbourhood.
Often one calls the open set rather than j itself, a tubular neighbourhood of S, it is assumed implicitly that the homeomorphism j mapping N to T exists.
Normal tube
A normal tube to a smooth curve is a manifold defined as the union of all discs such that
all the discs have the same fixed radius;
the center of each disc lies on the curve; and
each disc lies in a plane normal to the curve where the curve passes through that disc's center.
Formal definition
Let be smooth manifolds. A tubular neighborhood of in is a vector bundle together with a smooth map such that
where is the embedding and the zero section
there exists some and some with and such that is a diffeomorphism.
The normal bundle is a tubular neighborhood and because of the diffeomorphism condition in the second point, all tubular neighborhood have the same dimension, namely (the dimension of the vector bundle considered as a manifold is) that of
Generalizations
Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces.
These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle, which are replacements for the tangent bundle (which does not admit a direct description for these spaces).
See also
(aka offset curve)
References
Manifolds
Geometric topology
Smooth manifolds
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https://en.wikipedia.org/wiki/ACM%20Computing%20Classification%20System
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The ACM Computing Classification System (CCS) is a subject classification system for computing devised by the Association for Computing Machinery (ACM). The system is comparable to the Mathematics Subject Classification (MSC) in scope, aims, and structure, being used by the various ACM journals to organize subjects by area.
History
The system has gone through seven revisions, the first version being published in 1964, and revised versions appearing in 1982, 1983, 1987, 1991, 1998, and the now current version in 2012.
Structure
It is hierarchically structured in four levels. For example, one branch of the hierarchy contains:
Computing methodologies
Artificial intelligence
Knowledge representation and reasoning
Ontology engineering
See also
Computer Science Ontology
Physics and Astronomy Classification Scheme
arXiv, a preprint server allowing submitted papers to be classified using the ACM CCS
Physics Subject Headings
References
.
.
.
External links
dl.acm.org/ccs is the homepage of the system, including links to four complete versions of the system:
the 1964 version
the 1991 version
the 1998 version
the current 2012 version.
The ACM Computing Research Repository uses a classification scheme that is much coarser than the ACM subject classification, and does not cover all areas of CS, but is intended to better cover active areas of research. In addition, papers in this repository are classified according to the ACM subject classification.
Computing Classification System
Classification systems
Computer science literature
1964 in computing
Computer-related introductions in 1964
Computing Classification System
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https://en.wikipedia.org/wiki/Multiple%20line%20segment%20intersection
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In computational geometry, the multiple line segment intersection problem supplies a list of line segments in the Euclidean plane and asks whether any two of them intersect (cross).
Simple algorithms examine each pair of segments. However, if a large number of possibly intersecting segments are to be checked, this becomes increasingly inefficient since most pairs of segments are not close to one another in a typical input sequence. The most common, and more efficient, way to solve this problem for a high number of segments is to use a sweep line algorithm, where we imagine a line sliding across the line segments and we track which line segments it intersects at each point in time using a dynamic data structure based on binary search trees. The Shamos–Hoey algorithm applies this principle to solve the line segment intersection detection problem, as stated above, of determining whether or not a set of line segments has an intersection; the Bentley–Ottmann algorithm works by the same principle to list all intersections in logarithmic time per intersection.
See also
Bentley–Ottmann algorithm
References
Further reading
Chapter 2: Line Segment Intersection, pp. 19–44.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. . Section 33.2: Determining whether any pair of segments intersects, pp. 934–947.
J. L. Bentley and T. Ottmann., Algorithms for reporting and counting geometric intersections, IEEE Trans. Comput. C28 (1979), 643–647.
External links
Intersections of Lines and Planes Algorithms and sample code by Dan Sunday
Robert Pless. Lecture 4 notes. Washington University in St. Louis, CS 506: Computational Geometry (cached copy).
Line segment intersection in CGAL, the Computational Geometry Algorithms Library
"Line Segment Intersection" lecture notes by Jeff Erickson.
Line-Line Intersection Method With C Code Sample Darel Rex Finley
Geometric algorithms
Geometric intersection
Computational geometry
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https://en.wikipedia.org/wiki/Binary%20icosahedral%20group
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In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120.
It is an extension of the icosahedral group I or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism
of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120.
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3).
The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements
Explicitly, the binary icosahedral group is given as the union of all even permutations of the following vectors:
8 even permutations of
16 even permutations of
96 even permutations of
Here is the golden ratio.
In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp(1).
The 120 elements in 4-dimensional space match the 120 vertices the 600-cell, a regular 4-polytope.
Properties
Central extension
The binary icosahedral group, denoted by 2I, is the universal perfect central extension of the icosahedral group, and thus is quasisimple: it is a perfect central extension of a simple group.
Explicitly, it fits into the short exact sequence
This sequence does not split, meaning that 2I is not a semidirect product of { ±1 } by I. In fact, there is no subgroup of 2I isomorphic to I.
The center of 2I is the subgroup { ±1 }, so that the inner automorphism group is isomorphic to I. The full automorphism group is isomorphic to S5 (the symmetric group on 5 letters), just as for - any automorphism of 2I fixes the non-trivial element of the center (), hence descends to an automorphism of I, and conversely, any automorphism of I lifts to an automorphism of 2I, since the lift of generators of I are generators of 2I (different lifts give the same automorphism).
Superperfect
The binary icosahedral group is perfect, meaning that it is equal to its commutator subgroup. In fact, 2I is the unique perfect group of order 120. It follows that 2I is not solvable.
Further, the binary icosahedral group is superperfect, meaning abstractly that its first two group homology groups vanish: Concretely, this means that its abelianization is trivial (it has no non-trivial abelian quotients) and that its Schur multiplier is trivial (it has no non-trivial perfect central extensions). In fact, the binary icosahedral group is the smallest (non-trivial) superperfect group.
The binary icosahedral group is not acyclic, however, as Hn(2I,Z) is cyclic of order 120 for n = 4k+3, and trivial for n > 0 otherwise, .
Isomorphism
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https://en.wikipedia.org/wiki/Fatou%E2%80%93Lebesgue%20theorem
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In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou and Henri Léon Lebesgue.
If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to Lebesgue's dominated convergence theorem.
Statement of the theorem
Let f1, f2, ... denote a sequence of real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a Lebesgue-integrable function g on S which dominates the sequence in absolute value, meaning that |fn| ≤ g for all natural numbers n, then all fn as well as the limit inferior and the limit superior of the fn are integrable and
Here the limit inferior and the limit superior of the fn are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of g.
Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy to remember.
Proof
All fn as well as the limit inferior and the limit superior of the fn are measurable and dominated in absolute value by g, hence integrable.
The first inequality follows by applying Fatou's lemma to the non-negative functions fn + g and using the linearity of the Lebesgue integral. The last inequality is the reverse Fatou lemma.
Since g also dominates the limit superior of the |fn|,
by the monotonicity of the Lebesgue integral. The same estimates hold for the limit superior of the fn.
References
Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.
External links
Theorems in real analysis
Theorems in measure theory
Articles containing proofs
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https://en.wikipedia.org/wiki/Developable
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In mathematics, the term developable may refer to:
A developable space in general topology.
A developable surface in geometry.
A tangent developable surface of a space curve
Mathematics disambiguation pages
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https://en.wikipedia.org/wiki/School%20Mathematics%20Project
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The School Mathematics Project arose in the United Kingdom as part of the new mathematics educational movement of the 1960s. It is a developer of mathematics textbooks for secondary schools, formerly based in Southampton in the UK.
Now generally known as SMP, it began as a research project inspired by a 1961 conference chaired by Bryan Thwaites at the University of Southampton, which itself was precipitated by calls to reform mathematics teaching in the wake of the Sputnik launch by the Soviet Union, the same circumstances that prompted the wider New Math movement. It maintained close ties with the former Collaborative Group for Research in Mathematics Education at the university.
Instead of dwelling on 'traditional' areas such as arithmetic and geometry, SMP dwelt on subjects such as set theory, graph theory and logic, non-cartesian co-ordinate systems, matrix mathematics, affine transforms, Euclidean vectors, and non-decimal number systems.
Course books
SMP, Book 1
This was published in 1965. It was aimed at entry level pupils at secondary school, and was the first book in a series of 4 preparing pupils for Elementary Mathematics Examination at 'O' level.
SMP, Book 3
The computer paper tape motif on early educational material reads "THE SCHOOL MATHEMATICS PROJECT DIRECTED BY BRYAN THWAITES".
O O O O O O OO O O O O OO O O O O O
O O OOOO O O O O OO O O O O O O O
O O O OO O O OO O O O O O O OOO O O O OO O
···································································
O OO OO OO OOO O O O O OO O O O O
O O OO OO OO OOO OOO O OO O OO O O OO OOO OO O
THE SCHOOL MATHEMATICS PROJECT DIRECTED BY BRYAN THWAITES
The code for this tape is introduced in Book 3 as part of the notional computer system now described.
Simpol programming language
The Simpol language was devised by The School Mathematics Project in the 1960s so as to introduce secondary pupils (typically aged 13) to what was then the novel concept of computer programming. It runs on the fictitious Simon computer.
An interpreter for the Simpol language (that will run on a present-day PC) can be downloaded from the University of Southampton, at their SMP 2.0 website.
References
External links
Mathematics education in the United Kingdom
University of Southampton
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https://en.wikipedia.org/wiki/Boolean%20operation
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Boolean operation or Boolean operator may refer to:
Boolean function, a function whose arguments and result assume values from a two-element set
Boolean operation (Boolean algebra), a logical operation in Boolean algebra (AND, OR and NOT)
Boolean operator (computer programming), part of a Boolean expression in a computer programming language
An operation or operator as characterized in the logical truth table
Logical operator, in logic, a logical constant used to connect two or more formulas
Set operation (Boolean), a set-theoretic operation in the algebra of sets (union, intersection, and complementation)
Boolean operations on polygons, an application to polygon sets in computer graphics
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https://en.wikipedia.org/wiki/Overlapping%20interval%20topology
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In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Definition
Given the closed interval of the real number line, the open sets of the topology are generated from the half-open intervals with and with . The topology therefore consists of intervals of the form , , and with , together with itself and the empty set.
Properties
Any two distinct points in are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in , making with the overlapping interval topology an example of a T0 space that is not a T1 space.
The overlapping interval topology is second countable, with a countable basis being given by the intervals , and with and r and s rational.
See also
List of topologies
Particular point topology, a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space
References
(See example 53)
Topological spaces
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https://en.wikipedia.org/wiki/Whitehead%20group
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Whitehead group in mathematics may mean:
A group W with Ext(W, Z)=0; see Whitehead problem
For a ring, the Whitehead group Wh(A) of a ring A, equal to
For a group, the Whitehead group Wh(G) of a group G, equal to K1(Z[G])/{±G}. Note that this is a quotient of the Whitehead group of the group ring.
The Whitehead group Wh(A) of a simplicial complex or PL-manifold A, equal to Wh(π1(A)); see Whitehead torsion.
All named after J. H. C. Whitehead.
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https://en.wikipedia.org/wiki/Blind%20deconvolution
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In electrical engineering and applied mathematics, blind deconvolution is deconvolution without explicit knowledge of the impulse response function used in the convolution. This is usually achieved by making appropriate assumptions of the input to estimate the impulse response by analyzing the output. Blind deconvolution is not solvable without making assumptions on input and impulse response. Most of the algorithms to solve this problem are based on assumption that both input and impulse response live in respective known subspaces. However, blind deconvolution remains a very challenging non-convex optimization problem even with this assumption.
In image processing
In image processing, blind deconvolution is a deconvolution technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown point spread function (PSF). Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.
Most of the work on blind deconvolution started in early 1970s. Blind deconvolution is used in astronomical imaging and medical imaging.
Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF. Iterative methods include maximum a posteriori estimation and expectation-maximization algorithms. A good estimate of the PSF is helpful for quicker convergence but not necessary.
Examples of non-iterative techniques include SeDDaRA, the cepstrum transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the information about the scene is provided in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.
Examples
Any blurred image can be given as input to blind deconvolution algorithm, it can deblur the image, but essential condition for working of this algorithm must not be violated as discussed above. In the first example (picture of shapes), recovered image was very fine, exactly similar to original image because L > K + N. In the second example (picture of a girl), L < K + N, so essential condition is violated, hence recovered image is far different from original image.
In signal processing
Seismic data
In the case of deconvolution of seismic data, the original unknown signal is made of spikes hence is possible to characterize with sparsity constraints or regularizations such as l1 norm/l2 norm norm ratios, suggested by W. C. Gray in 1978.
Audio dec
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https://en.wikipedia.org/wiki/Transcritical%20bifurcation
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In bifurcation theory, a field within mathematics, a transcritical bifurcation is a particular kind of local bifurcation, meaning that it is characterized by an equilibrium having an eigenvalue whose real part passes through zero.
A transcritical bifurcation is one in which a fixed point exists for all values of a parameter and is never destroyed. However, such a fixed point interchanges its stability with another fixed point as the parameter is varied. In other words, both before and after the bifurcation, there is one unstable and one stable fixed point. However, their stability is exchanged when they collide. So the unstable fixed point becomes stable and vice versa.
The normal form of a transcritical bifurcation is
This equation is similar to the logistic equation, but in this case we allow and to be positive or negative (while in the logistic equation and must be non-negative).
The two fixed points are at and . When the parameter is negative, the fixed point at is stable and the fixed point is unstable. But for , the point at is unstable and the point at is stable. So the bifurcation occurs at .
A typical example (in real life) could be the consumer-producer problem where the consumption is proportional to the (quantity of) resource.
For example:
where
is the logistic equation of resource growth; and
is the consumption, proportional to the resource .
References
Bifurcation theory
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https://en.wikipedia.org/wiki/Statistics%20Commission
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The Statistics Commission was a non-departmental public body established in June 2000 by the UK Government to oversee the work of the Office for National Statistics.
Its chairman was Professor David Rhind who succeeded the first chairman, Sir John Kingman, in May 2003. Although it was non-departmental, the commission was funded by grant-in-aid from the Treasury. Following the implementation of the Statistics & Registration Services Act 2007, the commission was abolished. Its functions were to be taken over and considerably enhanced by the UK Statistics Authority (UKSA), whose powers began on 1 April 2008 under the chairmanship of Sir Michael Scholar. Professor Rhind is among the non-executive members of the new authority, to which the ONS is accountable. This contrasts with the duties of the previous Commission which were limited to reporting, observing and criticizing ONS while it, until 2008, has been publicly accountable via a Treasury minister.
History
The Commission arose from an election manifesto commitment by the Labour Government whilst in Opposition to provide independent national statistics. The commitment was implemented by the Government first publishing a Green Paper in 1998 inviting consultation which offered four options for overseeing the production of statistics for ministers. The subsequent white paper revealed that, of those four options, the one which received significantly more support than the others was the establishment of a Commission Consequently, in drawing up the new framework for national statistics, the Statistics Commission was established. Its main function is to
"...give independent, reliable and relevant advice on National Statistics to Ministers and, by so doing, to provide an additional safeguard on the quality and integrity of National Statistics."
The white paper charged the commission with four principal aims:
To consider and comment to government on National Statistics's programme and scope of work
To comment on National Statistics's quality assurance processes and to arrange audits where it finds concern
To comment on the application of the code of practice for official statistics
To prepare for the UK Parliament an annual report on National Statistics and the Commission
Commissioners
The last commissioners were:
Professor David Rhind, formerly Vice-Chancellor and President of City University, London
Ian Beesley, a retired senior partner at PricewaterhouseCoopers and Fellow of the Royal Statistical Society
Sir Kenneth Calman, Vice-Chancellor and Warden of the University of Durham
Janet Trewsdale, chairman of the Northern Ireland Economic Council
Sir Derek Wanless, formerly a director of Northern Rock plc
Colette Bowe, chairman of the Ofcom Consumer Panel
Patricia Hodgson, governor of the Wellcome Foundation and non-executive director of the Competition Commission
Martin Weale, director of the National Institute of Economic and Social Research
Joly Dixon, a retired director at the European Commission an
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https://en.wikipedia.org/wiki/F.%20and%20M.%20Riesz%20theorem
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In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure μ on the circle, any part of μ that is not absolutely continuous with respect to the Lebesgue measure dθ can be detected by means of Fourier coefficients.
More precisely, it states that if the Fourier–Stieltjes coefficients of
satisfy
for all ,
then μ is absolutely continuous with respect to dθ.
The original statements are rather different (see Zygmund, Trigonometric Series, VII.8). The formulation here is as in Walter Rudin, Real and Complex Analysis, p.335. The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H1.
Expansions to this theorem were made by James E. Weatherbee in his 1968 dissertation: Some Extensions Of The F. And M. Riesz Theorem On Absolutely Continuous Measures.
References
F. and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, (1916), pp. 27-44.
Theorems in measure theory
Fourier series
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https://en.wikipedia.org/wiki/Family%20Resources%20Survey
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The Family Resources Survey (FRS) is one of the United Kingdom's largest household surveys. It is carried out by the Office for National Statistics (ONS) with the National Centre for Social Research (NatCen), and Northern Ireland Statistics and Research Agency (NISRA) on an annual basis, by collecting information on the incomes and characteristics of private households in the United Kingdom. It is sponsored by the Department for Work and Pensions (DWP).
History
The survey was launched in 1992 to supply DWP with the information it required for policy analysis and it has been conducted annually since then. Before 1992, the department had to rely on other government surveys, for example the Family Expenditure Survey (now established as Living Cost and Food Survey) and the General Household Survey. However their sample sizes proved insufficient for the needs of DWP.
Beginning with a sample size of about 26,000 households, the number was reduced in 1997 to 24,000 households. After Northern Ireland was included in the sample and a 100% boost was introduced for Scotland, the sample size rose to 29,000 households in 2002 across the UK. In the most recent survey, the sample size was nearly 20,000 households.
Further changes occurred in 1998 when certain parts of the questionnaire were dropped in order to reduce the length, and in 1999 a system of rotating blocks of questions was introduced to reduce respondent fatigue. As a result, in recent years, the mean time of an interview is approximately 1 hour long, and the questions asked deal with topics such as the receipt of Social Security benefits, assets and savings, housing costs and income.
In addition, households who take part receive a £10 Post Office voucher, per household, as a token of the FRS's appreciation. Note that the survey sample excludes those living in hostels, communal areas and care homes, as they are not included in the FRS's definition of a "private" household.
At present, the FRS is designated by the UK Statistics Authority as National Statistics. The FRS provides the data for a number of other DWP National Statistics publications: Households Below Average Income, Pensioners' Incomes Series, and Income-Related Benefits: Estimates of Take-up.
Methodology and scope
The interviews are carried out on a face-to-face basis and all adult residents in a household aged 16 and older are interviewed. The reference period is based on the financial year (April to March) and data is released annually.
Survey results
The DWP reviews the survey results and uses the data from the FRS in its Policy Simulation Model (PSM) in order to evaluate existing policies and costing policy options. The FRS data is released annually, around February/ March, and is covered by a published report which has different chapters to highlight the breadth of information the FRS collects, as well as the change that may have occurred since the previous year's publication. For 2023, the release date is 23 March. The chapt
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https://en.wikipedia.org/wiki/Wedderburn%27s%20theorem
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Wedderburn's theorem may refer to:
Artin–Wedderburn theorem, classifying semisimple rings and semisimple algebras
Wedderburn's theorem on simple rings with a unit and a minimal left ideal
Wedderburn's little theorem, that a finite domain is a commutative field
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https://en.wikipedia.org/wiki/Fekete%20polynomial
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In mathematics, a Fekete polynomial is a polynomial
where is the Legendre symbol modulo some integer p > 1.
These polynomials were known in nineteenth-century studies of Dirichlet L-functions, and indeed to Dirichlet himself. They have acquired the name of Michael Fekete, who observed that the absence of real zeroes t of the Fekete polynomial with 0 < t < 1 implies an absence of the same kind for the L-function
This is of considerable potential interest in number theory, in connection with the hypothetical Siegel zero near s = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect.
References
Peter Borwein: Computational excursions in analysis and number theory. Springer, 2002, , Chap.5.
External links
Brian Conrey, Andrew Granville, Bjorn Poonen and Kannan Soundararajan, Zeros of Fekete polynomials, arXiv e-print math.NT/9906214, June 16, 1999.
Polynomials
Zeta and L-functions
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https://en.wikipedia.org/wiki/Decahedron
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In geometry, a decahedron is a polyhedron with ten faces. There are 32300 topologically distinct decahedra, and none are regular, so this name does not identify a specific type of polyhedron except for the number of faces.
Some decahedra have regular faces:
Octagonal prism (uniform 8-prism)
Square antiprism (uniform 4-antiprism)
Square cupola (Johnson solid 4)
Pentagonal bipyramid (Johnson solid 13, 5-bipyramid)
Augmented pentagonal prism (Johnson solid 52)
The decahedra with irregular faces include:
Pentagonal trapezohedron (5-trapezohedron, antiprism dual) often used as a die in role playing games, known as d10
Truncated square trapezohedron
Enneagonal pyramid (9-pyramid)
Ten of diamonds decahedron - a space-filling polyhedron with D2d symmetry.
References
External links
Polyhedra
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https://en.wikipedia.org/wiki/Dmitri%20Egorov
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Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Society (1923–1930).
Life
Egorov held spiritual beliefs to be of great importance, and openly defended the Church against Marxist supporters after the Russian Revolution. He was elected president of the Moscow Mathematical Society in 1921, and became director of the Institute for Mechanics and Mathematics at Moscow State University in 1923. He also edited the journal Matematicheskii Sbornik of the Moscow Mathematical Society. However, because of Egorov's stance against the repression of the Russian Orthodox Church, he was dismissed from the Institute in 1929 and publicly rebuked. In 1930 he was arrested and imprisoned as a "religious sectarian", and soon after was expelled from the Moscow Mathematical Society. Upon imprisonment, Egorov began a hunger strike until he was taken to the prison hospital, and eventually to the house of fellow mathematician Nikolai Chebotaryov where he died. He was buried in Arskoe Cemetery in Kazan.
Research work
Egorov studied potential surfaces and triply orthogonal systems, and made contributions to the broader areas of differential geometry and integral equations. His work was influenced by that of Jean Gaston Darboux on differential geometry and by Henri Lebesgue in mathematical analysis. A theorem in real analysis and integration theory, Egorov's Theorem, is named after him.
Works
, available at Gallica.
Notes
Bibliography
.
.
External links
Mathematicians from the Russian Empire
Differential geometers
Academic staff of Moscow State University
Corresponding Members of the Russian Academy of Sciences (1917–1925)
Corresponding Members of the USSR Academy of Sciences
Honorary Members of the USSR Academy of Sciences
Mathematicians from Moscow
Egorov, Dmitri
1869 births
1931 deaths
Professorships at the Imperial Moscow University
Imperial Moscow University alumni
Burials at Arskoe Cemetery
Russian scientists
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https://en.wikipedia.org/wiki/Local%20cohomology
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In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module (where is the coordinate ring of ) detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class in the local cohomology module .
Outside of algebraic geometry, local cohomology has found applications in commutative algebra, combinatorics, and certain kinds of partial differential equations.
Definition
In the most general geometric form of the theory, sections are considered of a sheaf of abelian groups, on a topological space , with support in a closed subset , The derived functors of form local cohomology groups
In the theory's algebraic form, the space X is the spectrum Spec(R) of a commutative ring R (assumed to be Noetherian throughout this article) and the sheaf F is the quasicoherent sheaf associated to an R-module M, denoted by . The closed subscheme Y is defined by an ideal I. In this situation, the functor ΓY(F) corresponds to the I-torsion functor, a union of annihilators
i.e., the elements of M which are annihilated by some power of I. As a right derived functor, the ith local cohomology module with respect to I is the ith cohomology group of the chain complex obtained from taking the I-torsion part of an injective resolution of the module . Because consists of R-modules and R-module homomorphisms, the local cohomology groups each have the natural structure of an R-module.
The I-torsion part may alternatively be described as
and for this reason, the local cohomology of an R-module M agrees with a direct limit of Ext modules,
It follows from either of these definitions that would be unchanged if were replaced by another ideal having the same radical. It also follows that local cohomology does not depend on any choice of generators for I, a fact which becomes relevant in the following definition involving the Čech complex.
Using Koszul and Čech complexes
The derived functor definition of local cohomology requires an injective resolution of the module , which can make it inaccessible for use in explicit computations. The Čech complex is seen as more practical in certain contexts. , for example,
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https://en.wikipedia.org/wiki/Hirzebruch%20surface
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In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .
Definition
The Hirzebruch surface is the -bundle, called a Projective bundle, over associated to the sheafThe notation here means: is the -th tensor power of the Serre twist sheaf , the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface is isomorphic to , and is isomorphic to blown up at a point so is not minimal.
GIT quotient
One method for constructing the Hirzebruch surface is by using a GIT quotientwhere the action of is given byThis action can be interpreted as the action of on the first two factors comes from the action of on defining , and the second action is a combination of the construction of a direct sum of line bundles on and their projectivization. For the direct sum this can be given by the quotient varietywhere the action of is given byThen, the projectivization is given by another -action sending an equivalence class toCombining these two actions gives the original quotient up top.
Transition maps
One way to construct this -bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts of defined by there is the local model of the bundleThen, the transition maps, induced from the transition maps of give the mapsendingwhere is the affine coordinate function on .
Properties
Projective rank 2 bundles over P1
Note that by Grothendieck's theorem, for any rank 2 vector bundle on there are numbers such thatAs taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to is the Hirzebruch surface since this bundle can be tensored by .
Isomorphisms of Hirzebruch surfaces
In particular, the above observation gives an isomorphism between and since there is the isomorphism vector bundles
Analysis of associated symmetric algebra
Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebrasThe first few symmetric modules are special since there is a non-trivial anti-symmetric -module . These sheaves are summarized in the tableFor the symmetric sheaves are given by
Intersection theory
Hirzebruch surfaces for have a special rational curve on them: The surface is the projective bundle of and the curve is the zero section. This curve has self-intersection number , and is the only irreducible curve with negative self intersection number. The only irreducible curves with zero self intersection number are the fibers of the Hirzebruch surface (considered as a fiber bundle over ). The Picard group is generated by the curve and one of the fibers, and these generators have intersection matrixso the bilinear form is two dimensional unimodular, and is even or odd depending on whether is even or odd.
The Hirzebruch surface () blown up at a point on the special curve is isomorphic to blown up at a point not on the
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https://en.wikipedia.org/wiki/Schwarz%20reflection%20principle
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In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane. In notation, if is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula,
That is, we make the definition that agrees along the real axis.
The result proved by Hermann Schwarz is as follows. Suppose that F is a continuous function on the closed upper half plane , holomorphic on the upper half plane , which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane.
In practice it would be better to have a theorem that allows F certain singularities, for example F a meromorphic function. To understand such extensions, one needs a proof method that can be weakened. In fact Morera's theorem is well adapted to proving such statements. Contour integrals involving the extension of F clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results.
The principle also adapts to apply to harmonic functions.
See also
Kelvin transform
Method of image charges
Schwarz function
References
External links
Harmonic functions
Theorems in complex analysis
Mathematical principles
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https://en.wikipedia.org/wiki/B%C3%B4cher%27s%20theorem
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In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.
Bôcher's theorem in complex analysis
In complex analysis, the theorem states that the finite zeros of the derivative of a non-constant rational function that are not multiple zeros are also the positions of equilibrium in the field of force due to particles of positive mass at the zeros of and particles of negative mass at the poles of , with masses numerically equal to the respective multiplicities, where each particle repels with a force equal to the mass times the inverse distance.
Furthermore, if C1 and C2 are two disjoint circular
regions which contain respectively all the zeros and all the poles of , then C1 and C2 also contain all the critical
points of .
Bôcher's theorem for harmonic functions
In the theory of harmonic functions, Bôcher's theorem states that a positive harmonic function in a punctured domain (an open domain minus one point in the interior) is a linear combination of a harmonic function in the unpunctured domain with a scaled fundamental solution for the Laplacian in that domain.
See also
Marden's theorem
External links
(Review of Joseph L. Walsh's book.)
Theorems in complex analysis
Harmonic functions
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https://en.wikipedia.org/wiki/Hausdorff%20moment%20problem
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In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments
of some Borel measure supported on the closed unit interval . In the case , this is equivalent to the existence of a random variable supported on , such that .
The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line , and in the Hamburger moment problem one considers the whole line . The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinite measures corresponding to the same prescribed moments and they consist of a convex set. The set of polynomials may or may not be dense in the associated Hilbert spaces if the moment problem is indeterminate, and it depends on whether measure is extremal or not. But in the determinate moment problem case, the set of polynomials is dense in the associated Hilbert space.
Completely monotonic sequences
In 1921, Hausdorff showed that is such a moment sequence if and only if the sequence is completely monotonic, that is, its difference sequences satisfy the equation
for all . Here, is the difference operator given by
The necessity of this condition is easily seen by the identity
which is non-negative since it is the integral of a non-negative function. For example, it is necessary to have
See also
Total monotonicity
References
Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Mathematische Zeitschrift 9, 74–109, 1921.
Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Mathematische Zeitschrift 9, 280–299, 1921.
Feller, W. "An Introduction to Probability Theory and Its Applications", volume II, John Wiley & Sons, 1971.
Shohat, J.A.; Tamarkin, J. D. The Problem of Moments, American mathematical society, New York, 1943.
Probability problems
Moment (mathematics)
Mathematical problems
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https://en.wikipedia.org/wiki/Hamburger%20moment%20problem
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In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m0, m1, m2, ...), does there exist a positive Borel measure μ (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that
In other words, an affirmative answer to the problem means that (m0, m1, m2, ...) is the sequence of moments of some positive Borel measure μ.
The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff).
Characterization
The Hamburger moment problem is solvable (that is, (mn) is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers
is positive definite, i.e.,
for every arbitrary sequence (cj)j ≥ 0 of complex numbers that are finitary (i.e. cj = 0 except for finitely many values of j).
For the "only if" part of the claims simply note that
which is non-negative if is non-negative.
We sketch an argument for the converse. Let Z+ be the nonnegative integers and F0(Z+) denote the family of complex valued sequences with finitary support. The positive Hankel kernel A induces a (possibly degenerate) sesquilinear product on the family of complex-valued sequences with finite support. This in turn gives a Hilbert space
whose typical element is an equivalence class denoted by [f].
Let en be the element in F0(Z+) defined by en(m) = δnm. One notices that
Therefore, the "shift" operator T on , with T[en] = [en + 1], is symmetric.
On the other hand, the desired expression
suggests that μ is the spectral measure of a self-adjoint operator. (More precisely stated, μ is the spectral measure for an operator defined below and the vector [1], ). If we can find a "function model" such that the symmetric operator T is multiplication by x, then the spectral resolution of a self-adjoint extension of T proves the claim.
A function model is given by the natural isomorphism from F0(Z+) to the family of polynomials, in one single real variable and complex coefficients: for n ≥ 0, identify en with xn. In the model, the operator T is multiplication by x and a densely defined symmetric operator. It can be shown that T always has self-adjoint extensions. Let be one of them and μ be its spectral measure. So
On the other hand,
For an alternative proof of the existence that only uses Stieltjes integrals, see also, in particular theorem 3.2.
Uniqueness of solutions
The solutions form a convex set, so the problem has either infinitely many solutions or a unique solution.
Consider the (n + 1) × (n + 1) Hankel matrix
Positivity of A means that for each n, det(Δn) ≥ 0. If det(Δn) = 0, for some n, then
is finite-dimensional and T is self-adjoint. So in this case the solution to the Hamburger moment problem is uniq
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https://en.wikipedia.org/wiki/Statistics%20Online%20Computational%20Resource
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The Statistics Online Computational Resource (SOCR) is an online multi-institutional research and education organization. SOCR designs, validates and broadly shares a suite of online tools for statistical computing, and interactive materials for hands-on learning and teaching concepts in data science, statistical analysis and probability theory. The SOCR resources are platform agnostic based on HTML, XML and Java, and all materials, tools and services are freely available over the Internet.
The core SOCR components include interactive distribution calculators, statistical analysis modules, tools for data modeling, graphics visualization, instructional resources, learning activities and other resources.
All SOCR resources are licensed under either the Lesser GNU Public License or CC BY; peer-reviewed, integrated internally and interoperate with independent digital libraries developed by other professional societies and scientific organizations like NSDL, Open Educational Resources, Mathematical Association of America, California Digital Library, LONI Pipeline, etc.
See also
List of statistical packages
Comparison of statistical packages
External links
SOCR University of Michigan site
SOCR UCLA site
References
Educational math software
Research institutes in the United States
Statistical software
University of Michigan
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https://en.wikipedia.org/wiki/Steven%20Pemberton
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Steven Pemberton is a researcher affiliated with the Distributed and Interactive Systems group at the Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands.
He was one of the designers of ABC, a programming language released in 1987, and editor-in-chief of the Special Interest Group on Computer–Human Interaction (SIGCHI)'s Bulletin from 1993-1999 and the Association for Computing Machinery (ACM)'s Interactions from 1998-2004.
Contributions to web standards
Pemberton was a contributing author of HyperText Markup Language (HTML) 4.0 and HTML 4.01, and chair of the World Wide Web Consortium (W3C) HTML Working Group. He was a contributing author of the Extensible HyperText Markup Language (XHTML) specifications 1.0 in 2000 and 1.1 in 2001, and chair of the XHTML 2 Working Group from 2006-9.
He chaired the first W3C workshop on style sheets in 1995, and was a contributing author of the Cascading Style Sheets (CSS) Level 1 specification in 1996, Level 2 in 1998, and CSS Color Module Level 3 in 2002.
Pemberton was co-chair of the W3C XForms Working Group from 2000-2007, and in 2003 co-authored the XForms 1.0 specification. In 2009 he co-authored the XForms 1.1 and XML Events specifications. He was co-chair of the W3C Forms Working Group from 2010-2012.
Awards
2009: SIGCHI Lifetime Service Award.
2022: SIGCHI Lifetime Practice Award.
References
External links
Personal website
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/K-group
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K-group or K group may refer to:
A group in algebraic K-theory
A group in topological K-theory
A complemented group
K-gruppen (K-groups), small Communist groups in 1970s Germany
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https://en.wikipedia.org/wiki/Landscape%20of%20Geometry
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Landscape of Geometry was an educational television show that illustrated the principles and applications of geometry. The series was produced and broadcast by TVOntario in 1982–83 and was hosted by David Stringer. A videotape edition of the show was produced in 1992 by Films for the Humanities.
Episode list
Eight episodes were produced. They were:
"The Shape of Things"
"It's Rude to Point"
"Lines That Cross"
"Lines That Don't Cross"
"Up, Down, and Sideways"
"Trussworthy"
"Cracked Up"
"The Range of Change"
All episodes were 15 minutes in length.
References
TVO original programming
1980s Canadian children's television series
Mathematics education television series
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https://en.wikipedia.org/wiki/Symmetric%20derivative
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In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as
The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.
If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function , which is not differentiable at , but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.
The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist.
Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
Examples
The absolute value function
For the absolute value function , using the notation for the symmetric derivative, we have at that
Hence the symmetric derivative of the absolute value function exists at and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at ).
Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected.
The function x−2
For the function , at we have
Again, for this function the symmetric derivative exists at , while its ordinary derivative does not exist at due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an essential discontinuity.
The Dirichlet function
The Dirichlet function, defined as
has a symmetric derivative at every , but is not symmetrically differentiable at any ; i.e. the symmetric derivative exists at rational numbers but not at irrational numbers.
Quasi-mean-value theorem
The symmetric derivative does not obey the usual mean-value theorem (of Lagrange). As a counterexample, the symmetric derivative of has the image , but secants for f can have a wider range of slopes; for instance, on the interval , the mean-value theorem would mandate that there exist a point where the (symmetric) derivative takes the value
A theorem somewhat analogous to Rolle's theorem but for the symmetric derivative was established in 1967 by C. E. Aull, who named it quasi-Rolle theorem. If is continuous on the closed interval and symmetrically differentiable on the open interval , and , then there exist two points , in such that , and . A lemma also established by Aull as a stepping stone to this theorem states that if is continuous on the closed interval and symmetrically differentiable on the open interval ,
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https://en.wikipedia.org/wiki/Napier-Hastings%20Urban%20Area
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The Napier-Hastings Urban Area was defined by Statistics New Zealand (Stats NZ) as a main urban area of New Zealand that was based around the twin cities of Napier and Hastings in the Hawke's Bay Region. It was defined under the New Zealand Standard Areas Classification 1992 (NZSAC92), which has since been superseded by the Statistical Standard for Geographic Areas 2018 (SSGA18).
The urban area lay mostly on the Heretaunga Plains, with part on surrounding hills. It was a city cluster consisting of the cities of Napier and Hastings, the town of Havelock North and some smaller settlements. It was the sixth-most-populous urban area in the country under the NZSAC92, with residents, fewer than Tauranga () and more than Dunedin ().
While the two cities are separated by of rural land from city edge to edge (20 km from one city centre to the next), there is sufficient economic and social integration between the cities that Stats NZ treated them as a single urban area. Stats NZ also subdivided the urban area into urban zones, as they did for the urban areas of Auckland, Hamilton and Wellington. The Napier-Hastings Urban Area contains about three-quarters of the population of the entire Hawke's Bay.
Under SSGA18, Stats NZ split up the Napier-Hastings Urban Area for statistical purposes. It was split into two "large urban areas" (30,000 to 99,999 population) of Napier and Hastings, the "medium urban area" (10,000 to 29,999) of Havelock North, the "small urban area" (1,000 to 9,999) of Clive, and the "rural settlements" (300 to 999) of Haumoana, Te Awanga, Whakatu and Whirinaki. Some settlements of less than 300 that were included in the Napier-Hastings Urban Area were left out entirely, including Eskdale, Omahu, Pakipaki, Pakowhai and Waiohiki. Stats NZ similarly split Wellington into Lower Hutt, Porirua, Upper Hutt and the urban part of Wellington City under SSGA18.
Urban zones
Hastings Urban Zone
Hastings Urban Zone lay within Hastings District and included Hastings, Havelock North and localities from Omahu and Pakowhai in the north, Pakipaki in the west, to settlements near the coast from Clive to Te Awanga. The main urban zone of Hastings contained around 68,000 people. Hastings hosts the regional hospital, showgrounds, racecourse, newspaper print, sports park and a water park.
Hastings on its own would have been the 8th largest urban area in New Zealand.
Napier Urban Zone
Napier Urban Zone consisted of Napier city and two parts of Hastings District that adjoin it: the Eskdale–Whirinaki area to the north and the Waiohiki area to the south. Napier hosts the region's port (the majority of export coming from Hastings' industry and food processing), airport, High Court and polytechnic college, and the regional council headquarters.
Napier on its own would have been the 10th largest urban area in New Zealand.
Amalgamation
Amalgamation is an ongoing debate for Napier and Hastings residents. Both cities have previously had smaller amalgamations (Nap
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https://en.wikipedia.org/wiki/2E6
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2E6 may refer to:
EIA Class 2 dielectric
2E6 group in mathematics
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https://en.wikipedia.org/wiki/Statistical%20literacy
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Statistical literacy is the ability to understand and reason with statistics and data. The abilities to understand and reason with data, or arguments that use data, are necessary for citizens to understand material presented in publications such as newspapers, television, and the Internet. However, scientists also need to develop statistical literacy so that they can both produce rigorous and reproducible research and consume it. Numeracy is an element of being statistically literate and in some models of statistical literacy, or for some populations (e.g., students in kindergarten through 12th grade/end of secondary school), it is a prerequisite skill. Being statistically literate is sometimes taken to include having the abilities to both critically evaluate statistical material and appreciate the relevance of statistically-based approaches to all aspects of life in general or to the evaluating, design, and/or production of scientific work.
Promoting statistical literacy
Each day people are inundated with statistical information from advertisements ("4 out of 5 dentists recommend"), news reports ("opinion poll show the incumbent leading by four points"), and even general conversation ("half the time I don't know what you're talking about"). Experts and advocates often use numerical claims to bolster their arguments, and statistical literacy is a necessary skill to help one decide what experts mean and which advocates to believe. This is important because statistics can be made to produce misrepresentations of data that may seem valid. The aim of statistical literacy proponents is to improve the public understanding of numbers and figures.
Health decisions are often manifest as statistical decision problems but few doctors or patients are well equipped to engage with these data.
Results of opinion polling are often cited by news organizations, but the quality of such polls varies considerably. Some understanding of the statistical technique of sampling is necessary in order to be able to correctly interpret polling results. Sample sizes may be too small to draw meaningful conclusions, and samples may be biased. The wording of a poll question may introduce a bias, and thus can even be used intentionally to produce a biased result. Good polls use unbiased techniques, with much time and effort being spent in the design of the questions and polling strategy. Statistical literacy is necessary to understand what makes a poll trustworthy and to properly weigh the value of poll results and conclusions.
For these reasons, and others, many programs around the world have been created to promote or improve statistical literacy. For example, many official statistical agencies such as Statistics Canada and the Australian Bureau of Statistics have programs to educate students in schools about the nature of statistics. A project of the International Statistical Institute is the only international organization whose focus is to promote national progra
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https://en.wikipedia.org/wiki/List%20of%20lay%20Catholic%20scientists
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Many Catholics have made significant contributions to the development of science and mathematics from the Middle Ages to today. These scientists include Galileo Galilei, René Descartes, Louis Pasteur, Blaise Pascal, André-Marie Ampère, Charles-Augustin de Coulomb, Pierre de Fermat, Antoine Laurent Lavoisier, Alessandro Volta, Augustin-Louis Cauchy, Pierre Duhem, Jean-Baptiste Dumas, Alois Alzheimer, Georgius Agricola and Christian Doppler.
Lay Catholic scientists
A
Maria Gaetana Agnesi (1718–1799) – mathematician who wrote on differential and integral calculus
Georgius Agricola (1494–1555) – father of mineralogy
Ulisse Aldrovandi (1522–1605) – father of natural history
Rudolf Allers (1883–1963) – Austrian psychiatrist; the only Catholic member of Sigmund Freud's first group, later a critic of Freudian psychoanalysis
Alois Alzheimer (1864–1915) – credited with identifying the first published case of presenile dementia, which is now known as Alzheimer's disease
André-Marie Ampère (1775–1836) – one of the main discoverers of electromagnetism
Leopold Auenbrugger (1722–1809) – first to use percussion as a diagnostic technique in medicine
Adrien Auzout (1622–1691) – astronomer who contributed to the development of the telescopic micrometer
Amedeo Avogadro (1776–1856) – Italian scientist noted for contributions to molecular theory and Avogadro's Law
B
Jacques Babinet (1794–1872) – French physicist, mathematician, and astronomer who is best known for his contributions to optics
Stefan Banach (1892–1945) – Polish mathematician, founder of modern functional analysis
Stephen M. Barr (1953–) – professor emeritus in the Department of Physics and Astronomy at the University of Delaware and a member of its Bartol Research Institute; founding president of the Society of Catholic Scientists
Joachim Barrande (1799–1883) – French geologist and paleontologist who studied fossils from the Lower Palaeozoic rocks of Bohemia
Laura Bassi (1711–1778) – physicist at the University of Bologna and Chair in experimental physics at the Bologna Institute of Sciences, the first woman to be offered a professorship at a European university
Antoine César Becquerel (1788–1878) – pioneer in the study of electric and luminescent phenomena
Henri Becquerel (1852–1908) – awarded the Nobel Prize in physics for his co-discovery of radioactivity
Carlo Beenakker (1960–) – professor at Leiden University and leader of the university's mesoscopic physics group, established in 1992.
Giovanni Battista Belzoni (1778–1823) – prolific Italian explorer and pioneer archaeologist of Egyptian antiquities
Pierre-Joseph van Beneden (1809–1894) – Belgian zoologist and paleontologist who established one of the world's first marine laboratories and aquariums
Claude Bernard (1813–1878) – physiologist who helped to apply scientific methodology to medicine
Jacques Philippe Marie Binet (1786–1856) – mathematician known for Binet's formula and his contributions to number theory
Jean-Bap
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https://en.wikipedia.org/wiki/Small%20Latin%20squares%20and%20quasigroups
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Latin squares and quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of some very small orders, which is the side length of the square, or the number of elements in the equivalent quasigroup.
The equivalence
Given a quasigroup with elements, its Cayley table (almost universally called its multiplication table) is an table that includes borders; a top row of column headers and a left column of row headers. Removing the borders leaves an array that is a Latin square. This process can be reversed, starting with a Latin square, introduce a bordering row and column to obtain the multiplication table of a quasigroup. While there is complete arbitrariness in how this bordering is done, the quasigroups obtained by different choices are sometimes equivalent in the sense given below.
Isotopy and isomorphism
Two Latin squares, 1 and 2 of size are isotopic if there are three bijections from the rows, columns and symbols of 1 onto the rows, columns and symbols of 2, respectively, that map 1 to 2. Isotopy is an equivalence relation and the equivalence classes are called isotopy classes.
A stronger form of equivalence exists. Two Latin squares, 1 and 2 of side with common symbol set that is also the index set for the rows and columns of each square, are isomorphic if there is a bijection such that for all , in . An alternate way to define isomorphic Latin squares is to say that a pair of isotopic Latin squares are isomorphic if the three bijections used to show that they are isotopic are, in fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes.
An alternate representation of a Latin square is given by an orthogonal array. For a Latin square of order this is an 2 × 3 matrix with columns labeled , and and whose rows correspond to a single position of the Latin square, namely, the row of the position, the column of the position and the symbol in the position. Thus for the order three Latin square,
the orthogonal array is given by:
The condition for an appropriately sized matrix to represent a Latin square is that for any two columns the ordered pairs determined by the rows in those columns are all the pairs () with 1 ≤ , once each.
This property is not lost by permuting the three columns (but not the labels), so another orthogonal array (and thus, another Latin square) is obtained. For example, by permuting the first two columns, which corresponds to transposing the square (reflecting about its main diagonal) gives another Latin square, which may or may not be isotopic to the original. In this case, if the quasigroup corresponding to this Latin square satisfies the commutative law, the new Latin square is the same as the original one. Altogether there are six possibilities including "do nothing", giving at most six Latin squares called the conjugates (also
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https://en.wikipedia.org/wiki/Symmetrically%20continuous%20function
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In mathematics, a function is symmetrically continuous at a point x if
The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function is symmetrically continuous at , but not continuous.
Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability.
The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over , similarly to the usually continuous functions, which form a linear subspace within it.
References
Differential calculus
Theory of continuous functions
Types of functions
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https://en.wikipedia.org/wiki/Second%20derivative
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In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:
where is acceleration, is velocity, is time, is position, and d is the instantaneous "delta" or change. The last expression is the second derivative of position () with respect to time.
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
Second derivative power rule
The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:
Notation
The second derivative of a function is usually denoted . That is:
When using Leibniz's notation for derivatives, the second derivative of a dependent variable with respect to an independent variable is written
This notation is derived from the following formula:
Example
Given the function
the derivative of is the function
The second derivative of is the derivative of , namely
Relation to the graph
Concavity
The second derivative of a function can be used to determine the concavity of the graph of . A function whose second derivative is positive will be concave up (also referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (also simply called concave), and its tangent lines will lie above the graph of the function.
Inflection points
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.
Second derivative test
The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e., a point where ) is a local maximum or a local minimum. Specifically,
If , then has a local maximum at .
If , then has a local minimum at .
If , the second derivative test says nothing about the point , a possible inflection point.
The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly, the position of the vehicle a
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https://en.wikipedia.org/wiki/Reflection%20principle
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In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory.
The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set.
Motivation
A naive version of the reflection principle states that "for any property of the universe of all sets we can find a set with the same property". This leads to an immediate contradiction: the universe of all sets contains all sets, but there is no set with the property that it contains all sets. To get useful (and non-contradictory) reflection principles we need to be more careful about what we mean by "property" and what properties we allow.
Reflection principles are associated with attempts to formulate the idea that no one notion, idea, or statement can capture our whole view of the universe of sets. Kurt Gödel described it as follows:
Georg Cantor expressed similar views on Absolute Infinity: All cardinality properties are satisfied in this number, in which held by a smaller cardinal.
To find non-contradictory reflection principles we might argue informally as follows. Suppose that we have some collection A of methods for forming sets (for example, taking powersets, subsets, the axiom of replacement, and so on). We can imagine taking all sets obtained by repeatedly applying all these methods, and form these sets into a class X, which can be thought of as a model of some set theory. But in light of this view, V is not be exhaustible by a handful of operations, otherwise it would be easily describable from below, this principle is known as inexhaustibility (of V). As a result, V is larger than X. Applying the methods in A to the set X itself would also result in a collection smaller than V, as V is not exhaustible from the image of X under the operations in A. Then we can introduce the following new principle for forming sets: "the collection of all sets obtained from some set by repeatedly applying all methods in the collection A is also a set". After adding this principle to A, V is still not exhaustible by the operations in this new A. This process may be repeated further and further, adding more and more operations to the set A and obtaining larger and larger models X. Each X resembles V in the sense that it shares the property with V of being closed under the operations in A.
We can use this informal argument in two ways. We can try to formalize it in (say) ZF set theory; by doing this we obtain some theorems of ZF set theory, called reflection theorems. Alternatively we can use this argument to motivate introducing new axioms for set theory, such as some axioms
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https://en.wikipedia.org/wiki/Reflection%20theorem
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In algebraic number theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field , with p a prime number, will be divisible by p if the class number of the maximal real subfield is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field , then 3 also divides the class number of the imaginary quadratic field .
Leopoldt's Spiegelungssatz
Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension.
Let L/K be a finite Galois extension of number fields, with group G, degree prime to p and L containing the p-th roots of unity. Let A be the p-Sylow subgroup of the class group of L. Let φ run over the irreducible characters of the group ring Qp[G] and let Aφ denote the corresponding direct summands of A. For any φ let q = pφ(1) and let the G-rank eφ be the exponent in the index
Let ω be the character of G
The reflection (Spiegelung) φ* is defined by
Let E be the unit group of K. We say that ε is "primary" if is unramified, and let E0 denote the group of primary units modulo Ep. Let δφ denote the G-rank of the φ component of E0.
The Spiegelungssatz states that
Extensions
Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of K/k, but rather by ideals in a group ring over the Galois group of K/k. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about ray class groups. This was further developed into the very general "T-S reflection theorem" of Georges Gras. Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.
References
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/Proth%27s%20theorem
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In number theory, Proth's theorem is a primality test for Proth numbers.
It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which
then p is prime. In this case p is called a Proth prime. This is a practical test because if p is prime, any chosen a has about a 50 percent chance of working, furthermore, since the calculation is mod p, only values of a smaller than p have to be taken into consideration.
In practice, however, a quadratic nonresidue of p is found via a modified Euclid's algorithm and taken as the value of a, since if a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. For such an a the Legendre symbol is
Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly. Note that if a is chosen to be a quadratic nonresidue as described above, the runtime is constant, safe for the time spent on finding such a quadratic nonresidue. Finding such a value is very fast compared to the actual test.
Numerical examples
Examples of the theorem include:
for p = 3 = 1(21) + 1, we have that 2(3-1)/2 + 1 = 3 is divisible by 3, so 3 is prime.
for p = 5 = 1(22) + 1, we have that 3(5-1)/2 + 1 = 10 is divisible by 5, so 5 is prime.
for p = 13 = 3(22) + 1, we have that 5(13-1)/2 + 1 = 15626 is divisible by 13, so 13 is prime.
for p = 9, which is not prime, there is no a such that a(9-1)/2 + 1 is divisible by 9.
The first Proth primes are :
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153 ….
The largest known Proth prime is , and is 9,383,761 digits long. It was found by Peter Szabolcs in the PrimeGrid volunteer computing project which announced it on 6 November 2016. It is also the largest known non-Mersenne prime and largest Colbert number. The second largest known Proth prime is , found by PrimeGrid.
Proof
The proof for this theorem uses the Pocklington-Lehmer primality test, and closely resembles the proof of Pépin's test. The proof can be found on page 52 of the book by Ribenboim in the references.
History
François Proth (1852–1879) published the theorem in 1878.
See also
Pépin's test (the special case k = 1, where one chooses a = 3)
Sierpinski number
References
External links
Primality tests
Theorems about prime numbers
de:Prothsche Primzahl
nl:Prothgetal
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https://en.wikipedia.org/wiki/Coadjoint%20representation
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In mathematics, the coadjoint representation of a Lie group is the dual of the adjoint representation. If denotes the Lie algebra of , the corresponding action of on , the dual space to , is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on .
The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups a basic role in their representation theory is played by coadjoint orbits.
In the Kirillov method of orbits, representations of are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of , which again may be complicated, while the orbits are relatively tractable.
Formal definition
Let be a Lie group and be its Lie algebra. Let denote the adjoint representation of . Then the coadjoint representation is defined by
for
where denotes the value of the linear functional on the vector .
Let denote the representation of the Lie algebra on induced by the coadjoint representation of the Lie group . Then the infinitesimal version of the defining equation for reads:
for
where is the adjoint representation of the Lie algebra .
Coadjoint orbit
A coadjoint orbit for in the dual space of may be defined either extrinsically, as the actual orbit inside , or intrinsically as the homogeneous space where is the stabilizer of with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated.
The coadjoint orbits are submanifolds of and carry a natural symplectic structure. On each orbit , there is a closed non-degenerate -invariant 2-form inherited from in the following manner:
.
The well-definedness, non-degeneracy, and -invariance of follow from the following facts:
(i) The tangent space may be identified with , where is the Lie algebra of .
(ii) The kernel of the map is exactly .
(iii) The bilinear form on is invariant under .
is also closed. The canonical 2-form is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.
Properties of coadjoint orbits
The coadjoint action on a coadjoint orbit is a Hamiltonian -action with momentum map given by the inclusion .
Examples
See also
Borel–Bott–Weil theorem, for a compact group
Kirillov character formula
Kirillov orbit theory
References
Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, ,
External links
Representation theory of Lie groups
Symplectic geometry
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https://en.wikipedia.org/wiki/Sum%20of%20normally%20distributed%20random%20variables
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In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables.
This is not to be confused with the sum of normal distributions which forms a mixture distribution.
Independent random variables
Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if
then
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).
In order for this result to hold, the assumption that X and Y are independent cannot be dropped, although it can be weakened to the assumption that X and Y are jointly, rather than separately, normally distributed. (See here for an example.)
The result about the mean holds in all cases, while the result for the variance requires uncorrelatedness, but not independence.
Proofs
Proof using characteristic functions
The characteristic function
of the sum of two independent random variables X and Y is just the product of the two separate characteristic functions:
of X and Y.
The characteristic function of the normal distribution with expected value μ and variance σ2 is
So
This is the characteristic function of the normal distribution with expected value and variance
Finally, recall that no two distinct distributions can both have the same characteristic function, so the distribution of X + Y must be just this normal distribution.
Proof using convolutions
For independent random variables X and Y, the distribution fZ of Z = X + Y equals the convolution of fX and fY:
Given that fX and fY are normal densities,
Substituting into the convolution:
Defining , and completing the square:
The expression in the integral is a normal density distribution on x, and so the integral evaluates to 1. The desired result follows:
Using the convolution theorem
It can be shown that the Fourier transform of a Gaussian, , is
By the convolution theorem:
Geometric proof
First consider the normalized case when X, Y ~ N(0, 1), so that their PDFs are
and
Let Z = X + Y. Then the CDF for Z will be
This integral is over the half-plane which lies under the line x+y = z.
The key observation is that the function
is radially symmetric. So we rotate the coordinate plane about the origin, choosing new coordinates such that the line x+y = z is described by the equation where is determined geometrically. Because of the radial symmetry, we have , and the CDF for Z is
This is easy to integrate; we find that the CDF for Z is
To determine the value , note that we rotated the plane so that the line x+y = z now runs vertically with x-intercept equal to c. So c is just the distance from the origin to the line x+y = z along the perpendicular bisector,
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