source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/On%20the%20Cruelty%20of%20Really%20Teaching%20Computer%20Science
|
"On the Cruelty of Really Teaching Computing Science" is a 1988 scholarly article by E. W. Dijkstra which argues that computer programming should be understood as a branch of mathematics, and that the formal provability of a program is a major criterion for correctness.
Despite the title, most of the article is on Dijkstra’s attempt to put computer science into a wider perspective within science, teaching being addressed as a corollary at the end.
Specifically, Dijkstra made a “proposal for an introductory programming course for freshmen” that
consisted of Hoare logic as an uninterpreted formal system.
Debate over feasibility
Since the term "software engineering" was coined, formal verification has almost always been considered too resource-intensive to be feasible. In complex applications, the difficulty of correctly specifying what the program should do in the first place is also a common source of error. Other methods of software testing are generally employed to try to eliminate bugs and many other factors are considered in the measurement of software quality.
Until the end of his life, Dijkstra maintained that the central challenges of computing hadn't been met to his satisfaction, due to an insufficient emphasis on program correctness (though not obviating other requirements, such as maintainability and efficiency).
Pedagogical legacy
Computer science as taught today does not follow of Dijkstra's advice. The curricula generally emphasize techniques for managing complexity and preparing for future changes, following Dijkstra's earlier writings. These include abstraction, programming by contract, and design patterns. Programming techniques to avoid bugs and conventional software testing methods are taught as basic requirements, and students are exposed to certain mathematical tools, but formal verification methods are not included in the curriculum except perhaps as an advanced topic. So in some ways, Dijkstra's ideas have been adhered to; however, the ideas he felt most strongly about have not been.
Newly formed curricula in software engineering have adopted Dijkstra's recommendations. The focus of these programs is the formal specification of software requirements and design in order to facilitate the formal validation of system correctness. In Canada, they are often accredited engineering degrees with similar core competencies in physics-based engineering.
References
1988 documents
Computer science papers
Computer science education
Works by Edsger Dijkstra
|
https://en.wikipedia.org/wiki/List%20of%20partition%20topics
|
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are
partition of a set or an ordered partition of a set,
partition of a graph,
partition of an integer,
partition of an interval,
partition of unity,
partition of a matrix; see block matrix, and
partition of the sum of squares in statistics problems, especially in the analysis of variance,
quotition and partition, two ways of viewing the operation of division of integers.
Integer partitions
Composition (number theory)
Ewens's sampling formula
Ferrers graph
Glaisher's theorem
Landau's function
Partition function (number theory)
Pentagonal number theorem
Plane partition
Quotition and partition
Rank of a partition
Crank of a partition
Solid partition
Young tableau
Young's lattice
Set partitions
Bell number
Bell polynomials
Dobinski's formula
Cumulant
Data clustering
Equivalence relation
Exact cover
Knuth's Algorithm X
Dancing Links
Exponential formula
Faà di Bruno's formula
Feshbach–Fano partitioning
Foliation
Frequency partition
Graph partition
Kernel of a function
Lamination (topology)
Matroid partitioning
Multipartition
Multiplicative partition
Noncrossing partition
Ordered partition of a set
Partition calculus
Partition function (quantum field theory)
Partition function (statistical mechanics)
Derivation of the partition function
Partition of an interval
Partition of a set
Ordered partition
Partition refinement
Disjoint-set data structure
Partition problem
3-partition problem
Partition topology
Quotition and partition
Recursive partitioning
Stirling number
Stirling transform
Stratification (mathematics)
Tverberg partition
Twelvefold way
In probability and stochastic processes
Chinese restaurant process
Dobinski's formula
Ewens's sampling formula
Law of total cumulance
Partition
Partition topics
|
https://en.wikipedia.org/wiki/Persistence%20of%20a%20number
|
In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number.
Usually, this involves additive or multiplicative persistence of a non-negative integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix. In the remainder of this article, base ten is assumed.
The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root.
Examples
The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest number of multiplicative persistence 3.
Smallest numbers of a given multiplicative persistence
In base 10, there is thought to be no number with a multiplicative persistence > 11: this is known to be true for numbers up to 1020,000. The smallest numbers with persistence 0, 1, 2, ... are:
0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899.
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be in increasing order (with the exception of the second number, 10), and – except for the first two digits – all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for n-digit numbers with record-breaking persistence is only proportional to the square of n, a tiny fraction of all possible n-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
Properties of additive persistence
The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero.
For base and natural numbers and the numbers and have the same additive persistence.
More about the additive persistence of a number can be found here.
Smallest numbers of a given additive persistence
The additive persistence of a number, however, can become arbitrarily large (proof: for a given number , the persistence of the number consisting of repetitions of the digit 1 is 1 higher than that of ). The smallest numbers of additive persistence 0, 1, 2, ... are:
0, 10, 19, 199, 19999999999999999999999, ...
The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×
|
https://en.wikipedia.org/wiki/Monte%20Carlo%20algorithm
|
In computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability. Two examples of such algorithms are the Karger–Stein algorithm and the Monte Carlo algorithm for minimum feedback arc set.
The name refers to the Monte Carlo casino in the Principality of Monaco, which is well-known around the world as an icon of gambling. The term "Monte Carlo" was first introduced in 1947 by Nicholas Metropolis.
Las Vegas algorithms are a dual of Monte Carlo algorithms and never return an incorrect answer. However, they may make random choices as part of their work. As a result, the time taken might vary between runs, even with the same input.
If there is a procedure for verifying whether the answer given by a Monte Carlo algorithm is correct, and the probability of a correct answer is bounded above zero, then with probability one, running the algorithm repeatedly while testing the answers will eventually give a correct answer. Whether this process is a Las Vegas algorithm depends on whether halting with probability one is considered to satisfy the definition.
One-sided vs two-sided error
While the answer returned by a deterministic algorithm is always expected to be correct, this is not the case for Monte Carlo algorithms. For decision problems, these algorithms are generally classified as either false-biased or true-biased. A false-biased Monte Carlo algorithm is always correct when it returns false; a true-biased algorithm is always correct when it returns true. While this describes algorithms with one-sided errors, others might have no bias; these are said to have two-sided errors. The answer they provide (either true or false) will be incorrect, or correct, with some bounded probability.
For instance, the Solovay–Strassen primality test is used to determine whether a given number is a prime number. It always answers true for prime number inputs; for composite inputs, it answers false with probability at least and true with probability less than . Thus, false answers from the algorithm are certain to be correct, whereas the true answers remain uncertain; this is said to be a -correct false-biased algorithm.
Amplification
For a Monte Carlo algorithm with one-sided errors, the failure probability can be reduced (and the success probability amplified) by running the algorithm k times. Consider again the Solovay–Strassen algorithm which is -correct false-biased. One may run this algorithm multiple times returning a false answer if it reaches a false response within k iterations, and otherwise returning true. Thus, if the number is prime then the answer is always correct, and if the number is composite then the answer is correct with probability at least 1−(1−)k = 1−2−k.
For Monte Carlo decision algorithms with two-sided error, the failure probability may again be reduced by running the algorithm k times and returning the majority function of the answers.
Complexity classes
The comp
|
https://en.wikipedia.org/wiki/Littlewood%20conjecture
|
In mathematics, the Littlewood conjecture is an open problem () in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,
where is the distance to the nearest integer.
Formulation and explanation
This means the following: take a point (α, β) in the plane, and then consider the sequence of points
(2α, 2β), (3α, 3β), ... .
For each of these, multiply the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will converge; it typically does not, in fact. The conjecture states something about the limit inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
o(1/n)
in the little-o notation.
Connection to further conjectures
It is known that this would follow from a result in the geometry of numbers, about the minimum on a non-zero lattice point of a product of three linear forms in three real variables: the implication was shown in 1955 by Cassels and Swinnerton-Dyer. This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold for n ≥ 3: it is stated in terms of G = SLn(R), Γ = SLn(Z), and the subgroup D of diagonal matrices in G.
Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed.
This in turn is a special case of a general conjecture of Margulis on Lie groups.
Partial results
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero. Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown that it must have Hausdorff dimension zero; and in fact is a union of countably many compact sets of box-counting dimension zero. The result was proved by using a measure classification theorem for diagonalizable actions of higher-rank groups, and an isolation theorem proved by Lindenstrauss and Barak Weiss.
These results imply that non-trivial pairs satisfying the conjecture exist: indeed, given a real number α such that , it is possible to construct an explicit β such that (α,β) satisfies the conjecture.
See also
Littlewood polynomial
References
Further reading
Diophantine approximation
Conjectures
Unsolved problems in mathematics
|
https://en.wikipedia.org/wiki/Double%20Mersenne%20number
|
In mathematics, a double Mersenne number is a Mersenne number of the form
where p is prime.
Examples
The first four terms of the sequence of double Mersenne numbers are :
Double Mersenne primes
A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number can be prime only if Mp is itself a Mersenne prime. For the first values of p for which Mp is prime, is known to be prime for p = 2, 3, 5, 7 while explicit factors of have been found for p = 13, 17, 19, and 31.
Thus, the smallest candidate for the next double Mersenne prime is , or 22305843009213693951 − 1.
Being approximately 1.695,
this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036.
There are probably no other double Mersenne primes than the four known.
Smallest prime factor of (where p is the nth prime) are
7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1 × 1036)
Catalan–Mersenne number conjecture
The recursively defined sequence
is called the sequence of Catalan–Mersenne numbers. The first terms of the sequence are:
Catalan discovered this sequence after the discovery of the primality of by Lucas in 1876. Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if is not prime, there is a chance to discover this by computing modulo some small prime (using recursive modular exponentiation). If the resulting residue is zero, represents a factor of and thus would disprove its primality. Since is a Mersenne number, such a prime factor would have to be of the form . Additionally, because is composite when is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.
In popular culture
In the Futurama movie The Beast with a Billion Backs, the double Mersenne number is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".
See also
Cunningham chain
Double exponential function
Fermat number
Perfect number
Wieferich prime
References
Further reading
.
External links
Tony Forbes, A search for a factor of MM61.
Status of the factorization of double Mersenne numbers
Double Mersennes Prime Search
Operazione Doppi Mersennes
Eponymous numbers in mathematics
Integer sequences
Large integers
Unsolved problems in number theory
Mersenne primes
|
https://en.wikipedia.org/wiki/International%20Congress%20of%20Mathematicians
|
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before 2022 as the Nevanlinna Prize), the Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame".
History
German mathematicians Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.
The University of Chicago, which had opened in 1892, organized an International Mathematical Congress at the Chicago World's Fair in 1893, where Felix Klein participated as the official German representative.
The first official International Congress of Mathematicians was held in Zurich in August 1897. The organizers included such prominent mathematicians as Luigi Cremona, Felix Klein, Gösta Mittag-Leffler, Andrey Markov, and others. The congress was attended by 208 mathematicians from 16 countries, including 12 from Russia and 7 from the US. Only four were women: Iginia Massarini, Vera von Schiff, Charlotte Scott, and Charlotte Wedell.
During the 1900 congress in Paris, France, David Hilbert announced his famous list of 23 unsolved mathematical problems, now termed Hilbert's problems. Moritz Cantor and Vito Volterra gave the two plenary lectures at the start of the congress.
At the 1904 ICM Gyula Kőnig delivered a lecture where he claimed that Cantor's famous continuum hypothesis was false. An error in Kőnig's proof was discovered by Ernst Zermelo soon thereafter. Kőnig's announcement at the congress caused considerable uproar, and Klein had to personally explain to the Grand Duke of Baden (who was a financial sponsor of the congress) what could cause such an unrest among mathematicians.
During the 1912 congress in Cambridge, England, Edmund Landau listed four basic problems about prime numbers, now called Landau's problems. The 1924 congress in Toronto was organized by John Charles Fields, initiator of the Fields Medal; it included a roundtrip railway excursion to Vancouver and ferry to Victoria. The first two Fields Medals were awarded at the 1936 ICM in Oslo.
In the aftermath of World War I, at the insistence of the Allied Powers, the 1920 ICM in Strasbourg and the 1924 ICM in Toronto excluded mathematicians from the countries formerly part of the Central Powers. This resulted in a still unresolved controversy as to whether to count the Strasbourg and Toronto congresses as true ICMs. At the opening of the 1932 ICM in Zürich, Hermann Weyl said: "We attend here to an extraordinary improbable event. For the number of n, corresponding to the
|
https://en.wikipedia.org/wiki/Tsirelson%20space
|
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive.
It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article () where they used the notation T for the dual of Tsirelson's example. Today, the letter T is the standard notation for the dual of the original example, while the original Tsirelson example is denoted by T*. In T* or in T, no subspace is isomorphic, as Banach space, to an ℓ p space, 1 ≤ p < ∞, or to c0.
All classical Banach spaces known to , spaces of continuous functions, of differentiable functions or of integrable functions, and all the Banach spaces used in functional analysis for the next forty years, contain some ℓ p or c0. Also, new attempts in the early '70s to promote a geometric theory of Banach spaces led to ask whether or not every infinite-dimensional Banach space has a subspace isomorphic to some ℓ p or to c0. Moreover, it was shown
by Baudier, Lancien, and Schlumprecht that
ℓ p and c0 do not even coarsely
embed into T*.
The radically new Tsirelson construction is at the root of several further developments in Banach space theory: the arbitrarily distortable space of Schlumprecht (), on which depend Gowers' solution to Banach's hyperplane problem and the Odell–Schlumprecht solution to the distortion problem. Also, several results of Argyros et al. are based on ordinal refinements of the Tsirelson construction, culminating with the solution by Argyros–Haydon of the scalar plus compact problem.
Tsirelson's construction
On the vector space ℓ∞ of bounded scalar sequences , let Pn denote the linear operator which sets to zero all coordinates xj of x for which j ≤ n.
A finite sequence of vectors in ℓ∞ is called block-disjoint if there are natural numbers so that , and so that when or , for each n from 1 to N.
The unit ball B∞ of ℓ∞ is compact and metrizable for the topology of pointwise convergence (the product topology). The crucial step in the Tsirelson construction is to let K be the smallest pointwise closed subset of B∞ satisfying the following two properties:
a. For every integer j in N, the unit vector ej and all multiples , for |λ| ≤ 1, belong to K.
b. For any integer N ≥ 1, if is a block-disjoint sequence in K, then belongs to K.
This set K satisfies the following stability property:
c. Together with every element x of K, the set K contains all vectors y in ℓ∞ such that |y| ≤ |x| (for the pointwise comparison).
It is then shown that K is actually a subset of c0, the Banach subspace of ℓ∞ consisting of scalar sequences tending to zero at infinity. This is done by proving that
d: for every element x in K, there exists an integer n such that 2 Pn(x) belongs to K,
and iterating this fact. Since K is pointwise compact and contained in c0, it is weakly compact in c0. Let V be the closed convex hul
|
https://en.wikipedia.org/wiki/Outer%20automorphism%20group
|
In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a trivial center, then is said to be complete.
An automorphism of a group that is not inner is called an outer automorphism. The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is an outer automorphism of or more precisely "represents the class of the (non-trivial) outer automorphism of ", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.
Structure
The Schreier conjecture asserts that is always a solvable group when is a finite simple group. This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.
As dual of the center
The outer automorphism group is dual to the center in the following sense: conjugation by an element of is an automorphism, yielding a map . The kernel of the conjugation map is the center, while the cokernel is the outer automorphism group (and the image is the inner automorphism group). This can be summarized by the exact sequence
Applications
The outer automorphism group of a group acts on conjugacy classes, and accordingly on the character table. See details at character table: outer automorphisms.
Topology of surfaces
The outer automorphism group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of the surface is the outer automorphism group of its fundamental group.
In finite groups
For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic simple groups and alternating groups (other than the alternating group, ; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for , when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for , when it is the symmetric group on 3 points). These extensions are not always semidirect products, as the case of the alterna
|
https://en.wikipedia.org/wiki/Sergei%20Novikov%20%28mathematician%29
|
Sergei Petrovich Novikov (also Serguei) (Russian: Серге́й Петро́вич Но́виков) (born 20 March 1938) is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. In 1970, he won the Fields Medal.
Early life
Novikov was born on 20 March 1938 in Gorky, Soviet Union (now Nizhny Novgorod, Russia).
He grew up in a family of talented mathematicians. His father was Pyotr Sergeyevich Novikov, who gave a negative solution to the word problem for groups. His mother, Lyudmila Vsevolodovna Keldysh, and maternal uncle, Mstislav Vsevolodovich Keldysh, were also important mathematicians.
In 1955 Novikov entered Moscow State University, from which he graduated in 1960. Four years later he received the Moscow Mathematical Society Award for young mathematicians. In the same year he defended a dissertation for the Candidate of Science in Physics and Mathematics degree (equivalent to the PhD) at Moscow State University. In 1965 he defended a dissertation for the Doctor of Science in Physics and Mathematics degree there. In 1966 he became a Corresponding member of the Academy of Sciences of the Soviet Union.
Research in topology
Novikov's early work was in cobordism theory, in relative isolation. Among other advances he showed how the Adams spectral sequence, a powerful tool for proceeding from homology theory to the calculation of homotopy groups, could be adapted to the new (at that time) cohomology theory typified by cobordism and K-theory. This required the development of the idea of cohomology operations in the general setting, since the basis of the spectral sequence is the initial data of Ext functors taken with respect to a ring of such operations, generalising the Steenrod algebra. The resulting Adams–Novikov spectral sequence is now a basic tool in stable homotopy theory.
Novikov also carried out important research in geometric topology, being one of the pioneers with William Browder, Dennis Sullivan, and C. T. C. Wall of the surgery theory method for classifying high-dimensional manifolds. He proved the topological invariance of the rational Pontryagin classes, and posed the Novikov conjecture. This work was recognised by the award in 1970 of the Fields Medal. He was not allowed to travel to Nice to accept his medal, but he received it in 1971 when the International Mathematical Union met in Moscow. From about 1971 he moved to work in the field of isospectral flows, with connections to the theory of theta functions. Novikov's conjecture about the Riemann–Schottky problem (characterizing principally polarized abelian varieties that are the Jacobian of some algebraic curve) stated, essentially, that this was the case if and only if the corresponding theta function provided a solution to the Kadomtsev–Petviashvili equation of soliton theory. This was proved by Takahiro Shiota (1986), following earlier work by Enrico Arbarello and Corrado de Concini (1984), and by Motohico Mulase (1984).
Later career
Since 1971 Novik
|
https://en.wikipedia.org/wiki/Approximation%20property
|
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.
Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955).
Later many other counterexamples were found. The space of bounded operators on does not have the approximation property. The spaces for and (see Sequence space) have closed subspaces that do not have the approximation property.
Definition
A locally convex topological vector space X is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.
For a locally convex space X, the following are equivalent:
X has the approximation property;
the closure of in contains the identity map ;
is dense in ;
for every locally convex space Y, is dense in ;
for every locally convex space Y, is dense in ;
where denotes the space of continuous linear operators from X to Y endowed with the topology of uniform convergence on pre-compact subsets of X.
If X is a Banach space this requirement becomes that for every compact set and every , there is an operator of finite rank so that , for every .
Related definitions
Some other flavours of the AP are studied:
Let be a Banach space and let . We say that X has the -approximation property (-AP), if, for every compact set and every , there is an operator of finite rank so that , for every , and .
A Banach space is said to have bounded approximation property (BAP), if it has the -AP for some .
A Banach space is said to have metric approximation property (MAP), if it is 1-AP.
A Banach space is said to have compact approximation property (CAP), if in the
definition of AP an operator of finite rank is replaced with a compact operator.
Examples
Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property. In particular,
every Hilbert space has the approximation property.
every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.
every nuclear space possesses the approximation property.
Every separable Frechet space that contains a Schauder basis possesses the approximation property.
Every space with a Schauder basis has the AP (we can use the projections associated to the base as the 's in the definition), thus many spaces with the AP can be found. For example, the spaces, or the symmetric Tsirelson space.
References
Bibliography
Enflo, P.: A counterexample to the approximation property in Banach spaces. Acta Math. 130, 309–317(1973).
Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 (1955).
Paul R. Halmos, "Has pr
|
https://en.wikipedia.org/wiki/Ba%20space
|
In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is
If Σ is a sigma-algebra, then the space is defined as the subset of consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive.
If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then is the subspace of consisting of all regular Borel measures on X.
Properties
All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus is a closed subset of , and is a closed set of for Σ the algebra of Borel sets on X. The space of simple functions on is dense in .
The ba space of the power set of the natural numbers, ba(2N), is often denoted as simply and is isomorphic to the dual space of the ℓ∞ space.
Dual of B(Σ)
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm. Then ba(Σ) = B(Σ)* is the continuous dual space of B(Σ). This is due to Hildebrandt and Fichtenholtz & Kantorovich. This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to define the integral with respect to a finitely additive measure (note that the usual Lebesgue integral requires countable additivity). This is due to Dunford & Schwartz, and is often used to define the integral with respect to vector measures, and especially vector-valued Radon measures.
The topological duality ba(Σ) = B(Σ)* is easy to see. There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(Σ)* if it is continuous in the sup-norm.
Dual of L∞(μ)
If Σ is a sigma-algebra and μ is a sigma-additive positive measure on Σ then the Lp space L∞(μ) endowed with the essential supremum norm is by definition the quotient space of B(Σ) by the closed subspace of bounded μ-null functions:
The dual Banach space L∞(μ)* is thus isomorphic to
i.e. the space of finitely additive signed measures on Σ that are absolutely continuous with respect to μ (μ-a.c. for short).
When the measure space is furthermore sigma-finite then L∞(μ) is in turn dual to L1(μ), which by the Radon–Nikodym theorem is identified with the set of all countably additive μ-a.c. measures.
In other words, the inclusion in the bidual
is isomorphic to the inclusion of the space of countably additive μ-a.c. bounded measures inside the space of all finitely additive μ-a.c. bounded measures.
References
Further reading
Measure theory
Ba
|
https://en.wikipedia.org/wiki/Giacomo%20Albanese
|
Giacomo Albanese (11 July 1890 – 8 June 1947) was an Italian mathematician known for his work in algebraic geometry. He took a permanent position in the University of São Paulo, Brazil, in 1936.
Biography
Albanese attended the school in Palermo, Sicily. He graduated from there in 1909. Then he entered the Scuola Normale Superiore di Pisa as a student of mathematics, and received his doctorate in 1913. He was awarded the Ulisse Dini prize for his doctoral essay on the topic Continuous systems of curves on an algebraic surface, written under the direction of Eugenio Bertini.
He taught at Scuola Normale Superiore from 1913 to 1919, with an interruption in 1917–1918, when he was conscripted into the Italian Army to fight in World War I. After spending a year at the University of Padua to work with Francesco Severi, he took up in 1920 a professorship in Analysis and Algebra at the Italian Naval Academy in Livorno. Five years later Albanese moved to the University of Catania, and then went in 1927 to the University of Palermo, where he spent the next two years. From 1929 until 1936 he held the chair of Geometry at the University of Pisa.
See also
Albanese variety
References
External links
1890 births
1947 deaths
20th-century Italian mathematicians
Algebraic geometers
Italian algebraic geometers
Mathematicians from Sicily
Scuola Normale Superiore di Pisa alumni
Academic staff of the University of Pisa
Italian military personnel of World War I
Academic staff of the University of Padua
Academic staff of the University of Catania
Academic staff of the University of Palermo
Academic staff of the Scuola Normale Superiore di Pisa
Academic staff of the University of São Paulo
Italian emigrants to Brazil
|
https://en.wikipedia.org/wiki/Polynomially%20reflexive%20space
|
In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.
Given a multilinear functional Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as
(that is, applying Mn on the diagonal) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous.
We define the space Pn as consisting of all n-homogeneous polynomials.
The P1 is identical to the dual space, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.
Relation to continuity of forms
On a finite-dimensional linear space, a quadratic form x↦f(x) is always a (finite) linear combination of products x↦g(x) h(x) of two linear functionals g and h. Therefore, assuming that the scalars are complex numbers, every sequence xn satisfying g(xn) → 0 for all linear functionals g, satisfies also f(xn) → 0 for all quadratic forms f.
In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence xn satisfies g(xn) → 0 for all linear functionals g, and nevertheless f(xn) = 1 where f is the quadratic form f(x) = ||x||2. In more technical words, this quadratic form fails to be weakly sequentially continuous at the origin.
On a reflexive Banach space with the approximation property the following two conditions are equivalent:
every quadratic form is weakly sequentially continuous at the origin;
the Banach space of all quadratic forms is reflexive.
Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for n-homogeneous polynomials, n=3,4,...
Examples
For the spaces, the Pn is reflexive if and only if < . Thus, no is polynomially reflexive. ( is ruled out because it is not reflexive.)
Thus if a Banach space admits as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.
The Tsirelson space T* is polynomially reflexive.
Notes
References
Alencar, R., Aron, R. and S. Dineen (1984), "A reflexive space of holomorphic functions in infinitely many variables", Proc. Amer. Math. Soc. 90: 407–411.
Farmer, Jeff D. (1994), "Polynomial reflexivity in Banach spaces", Israel Journal of Mathematics 87: 257–273.
Jaramillo, J. and Moraes, L. (2000), "Dualily and reflexivity in spaces of polynomials", Arch. Math. (Basel) 74: 282–293.
Mujica, Jorge (2001), "Reflexive spaces of homogeneous polynomials", Bull. Polish Acad. Sci. Math. 49:3, 211–222.
Banach spaces
|
https://en.wikipedia.org/wiki/Stefan%20Mazurkiewicz
|
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (PAU). His students included Karol Borsuk, Bronisław Knaster, Kazimierz Kuratowski, Stanisław Saks, and Antoni Zygmund. For a time Mazurkiewicz was a professor at the University of Paris; however, he spent most of his career as a professor at the University of Warsaw.
The Hahn–Mazurkiewicz theorem, a basic result on curves prompted by the phenomenon of space-filling curves, is named for Mazurkiewicz and Hans Hahn. His 1935 paper Sur l'existence des continus indécomposables is generally considered the most elegant piece of work in point-set topology.
During the Polish–Soviet War (1919–21), Mazurkiewicz as early as 1919 broke the most common Russian cipher for the Polish General Staff's cryptological agency. Thanks to this, orders issued by Soviet commander Mikhail Tukhachevsky's staff were known to Polish Army leaders. This contributed substantially, perhaps decisively, to Polish victory at the critical Battle of Warsaw and possibly to Poland's survival as an independent country.
See also
Biuro Szyfrów
List of Polish mathematicians
External links
1888 births
1945 deaths
Warsaw School of Mathematics
People from Warsaw Governorate
Polish cryptographers
Topologists
Academic staff of the University of Paris
Academic staff of the University of Warsaw
Mathematical analysts
Cipher Bureau (Poland)
University of Warsaw alumni
|
https://en.wikipedia.org/wiki/Hadwiger%27s%20theorem
|
In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in It was proved by Hugo Hadwiger.
Introduction
Valuations
Let be the collection of all compact convex sets in A valuation is a function such that and for every that satisfy
A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if whenever and is either a translation or a rotation of
Quermassintegrals
The quermassintegrals are defined via Steiner's formula
where is the Euclidean ball. For example, is the volume, is proportional to the surface measure, is proportional to the mean width, and is the constant
is a valuation which is homogeneous of degree that is,
Statement
Any continuous valuation on that is invariant under rigid motions can be represented as
Corollary
Any continuous valuation on that is invariant under rigid motions and homogeneous of degree is a multiple of
See also
References
An account and a proof of Hadwiger's theorem may be found in
An elementary and self-contained proof was given by Beifang Chen in
Integral geometry
Theorems in convex geometry
Probability theorems
|
https://en.wikipedia.org/wiki/Dirichlet%20function
|
In mathematics, the Dirichlet function is the indicator function of the set of rational numbers , i.e. if is a rational number and if is not a rational number (i.e. is an irrational number).
It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of pathological function which provides counterexamples to many situations.
Topological properties
Periodicity
For any real number and any positive rational number , . The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of .
Integration properties
See also
Thomae's function, a variation that is discontinuous only at the rational numbers
References
Dirichlet
Real analysis
|
https://en.wikipedia.org/wiki/Abstract%20nonsense
|
In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them. These terms are mainly used for abstract methods related to category theory and homological algebra. More generally, "abstract nonsense" may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.
Background
Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a non sequitur. Authors sometimes dub these proofs "abstract nonsense" as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually not intended to be derogatory, and is instead used jokingly, in a self-deprecating way, affectionately, or even as a compliment to the generality of the argument.
Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties, use of the Yoneda lemma, natural transformations between functors, and diagram chasing.
When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars. For example, one might say that "By abstract nonsense, products are unique up to isomorphism when they exist", instead of arguing about how these isomorphisms can be derived from the universal property that defines the product. This allows one to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.
History
The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'. The term is often used to describe the application of category theory and its techniques to less abstract domains.
The term is believed to have been coined by the mathematician Norman Steenrod, himself one of the developers of the categorical point of view.
Notes and references
External links
Usage in mathematical exposition from Noam Elkies' class notes
Mathematical terminology
Category theory
|
https://en.wikipedia.org/wiki/Hugo%20Hadwiger
|
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Biography
Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland. He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern.
Mathematical concepts named after Hadwiger
Hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in d-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the area, the perimeter, and the Euler characteristic.
The Hadwiger–Finsler inequality, proven by Hadwiger with Paul Finsler, is an inequality relating the side lengths and area of any triangle in the Euclidean plane. It generalizes Weitzenböck's inequality and was generalized in turn by Pedoe's inequality. In the same 1937 paper in which Hadwiger and Finsler published this inequality, they also published the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex.
Hadwiger's name is also associated with several important unsolved problems in mathematics:
The Hadwiger conjecture in graph theory, posed by Hadwiger in 1943 and called by “one of the deepest unsolved problems in graph theory,” describes a conjectured connection between graph coloring and graph minors. The Hadwiger number of a graph is the number of vertices in the largest clique that can be formed as a minor in the graph; the Hadwiger conjecture states that this is always at least as large as the chromatic number.
The Hadwiger conjecture in combinatorial geometry concerns the minimum number of smaller copies of a convex body needed to cover the body, or equivalently the minimum number of light sources needed to illuminate the surface of the body; for instance, in three dimensions, it is known that any convex body can be illuminated by 16 light sources, but Hadwiger's conjecture implies that only eight light sources are always sufficient.
The Hadwiger–Kneser–Poulsen conjecture states that, if the centers of a system of balls in Euclidean space are moved closer together, then the volume of the union of the balls cannot increase. It has been proven in the plane, but remains open in higher dimensions.
The Hadwiger–Nelson problem concerns the minimum number of colors needed to color the points of the Euclidean plane so that no two points at unit distance from each other are given the same color. It was first proposed by Edward Nelson in 1950. Hadwiger popularized it by including it in a problem collection in 1961;
|
https://en.wikipedia.org/wiki/Algebraic%20geometry%20and%20analytic%20geometry
|
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Main statement
Let X be a projective complex algebraic variety. Because X is a complex variety, its set of complex points X(C) can be given the structure of a compact complex analytic space. This analytic space is denoted Xan. Similarly, if is a sheaf on X, then there is a corresponding sheaf on Xan. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating X and Xan says that for any two coherent sheaves and on X, the natural homomorphism:
is an isomorphism. Here is the structure sheaf of the algebraic variety X and is the structure sheaf of the analytic variety Xan. In other words, the category of coherent sheaves on the algebraic variety X is equivalent to the category of analytic coherent sheaves on the analytic variety Xan, and the equivalence is given on objects by mapping to . (Note in particular that itself is coherent, a result known as the Oka coherence theorem, and also, it was proved in “Faisceaux Algebriques Coherents” () that the structure sheaf of the algebraic variety is coherent.)
Another important statement is as follows: For any coherent sheaf on an algebraic variety X the homomorphisms
are isomorphisms for all qs. This means that the q-th cohomology group on X is isomorphic to the cohomology group on Xan.
The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem, the Lefschetz principle and Kodaira vanishing theorem.
Background
Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.
For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either
the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riem
|
https://en.wikipedia.org/wiki/Chow%27s%20theorem
|
In mathematics, Chow's theorem may refer to a number of theorems due to Wei-Liang Chow:
Chow's theorem: The theorem that asserts that any analytic subvariety in projective space is actually algebraic.
Chow–Rashevskii theorem: In sub-Riemannian geometry, the theorem that asserts that any two points are connected by a horizontal curve.
See also
Chow's lemma
Chow's moving lemma
Zhou, Weiliang
|
https://en.wikipedia.org/wiki/Arithmetic%20group
|
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
History
One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces. The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the discriminant. Arithmetic groups can be thought of as a vast generalisation of the unit groups of number fields to a noncommutative setting.
The same groups also appeared in analytic number theory as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in Langlands' computation of the volume of certain fundamental domains using analytic methods. This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases.
For the modern theory to begin foundational work was needed, and was provided by the work of Armand Borel, André Weil, Jacques Tits and others on algebraic groups. Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra. Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972.
In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group. Some limited results in this direction had been obtained earlier by Selberg, but Margulis' methods (the use of ergodic-theoretical tools for actions on homogeneous spaces) were completely new in this context and were to be extremely influential on later developments, effectively renewing the old subject of geometry of numbers and allowing Margulis himself to prove the Oppenheim conjecture; stronger results (Ratner's theorems) were later obtained by Marina Ratner.
In another direction the classical topic of modular forms has blossomed into the modern theory of automorphic forms. The driving force behind this effort is mainly the Langlands program initiated by Robert Langlands. One of the main tool used there is the trace formula originat
|
https://en.wikipedia.org/wiki/Smith%20number
|
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.
Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
4937775 = 3 · 5 · 5 · 65837
while
4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)
in base 10.
Mathematical definition
Let be a natural number. For base , let the function be the digit sum of in base . A natural number with prime factorisation
is a Smith number if
Here the exponent is the multiplicity of as a prime factor of (also known as the p-adic valuation of ).
For example, in base 10, 378 = 21 · 33 · 71 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 21 · 111 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.
The first few Smith numbers in base 10 are
4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985.
Properties
W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.
The number of Smith numbers in base 10 below 10n for n = 1, 2, ... is given by
1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... .
Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers. It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple (meaning n consecutive Smith numbers) in base 10 for n = 1, 2, ... are
4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... .
Smith numbers can be constructed from factored repunits. , the largest known Smith number in base 10 is
9 × R1031 × (104594 + 3 + 1)1476
where R1031 is the base 10 repunit (101031 − 1)/9.
See also
Equidigital number
Notes
References
External links
Base-dependent integer sequences
Eponymous numbers in mathematics
Lehigh University
|
https://en.wikipedia.org/wiki/Halton%20sequence
|
In statistics, Halton sequences are sequences used to generate points in space for numerical methods such as Monte Carlo simulations. Although these sequences are deterministic, they are of low discrepancy, that is, appear to be random for many purposes. They were first introduced in 1960 and are an example of a quasi-random number sequence. They generalize the one-dimensional van der Corput sequences.
Example of Halton sequence used to generate points in (0, 1) × (0, 1) in R2
The Halton sequence is constructed according to a deterministic method that uses coprime numbers as its bases. As a simple example, let's take one dimension of the Halton sequence to be based on 2 and the other on 3. To generate the sequence for 2, we start by dividing the interval (0,1) in half, then in fourths, eighths, etc., which generates
,
, ,
, , , ,
, ,...
Equivalently, the nth number of this sequence is the number n written in binary representation, inverted, and written after the decimal point. This is true for any base. As an example, to find the sixth element of the above sequence, we'd write 6 = 1*2 + 1*2 + 0*2 = 110, which can be inverted and placed after the decimal point to give 0.011 = 0*2 + 1*2 + 1*2 = . So the sequence above is the same as
0.1, 0.01, 0.11, 0.001, 0.101, 0.011, 0.111, 0.0001, 0.1001,...
To generate the sequence for 3, we divide the interval (0,1) in thirds, then ninths, twenty-sevenths, etc., which generates
, , , , , , , , ,...
When we pair them up, we get a sequence of points in a unit square:
(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ).
Even though standard Halton sequences perform very well in low dimensions, correlation problems have been noted between sequences generated from higher primes. For example, if we started with the primes 17 and 19, the first 16 pairs of points: (, ), (, ), (, ) ... (, ) would have perfect linear correlation. To avoid this, it is common to drop the first 20 entries, or some other predetermined quantity depending on the primes chosen. Several other methods have also been proposed. One of the most prominent solutions is the scrambled Halton sequence, which uses permutations of the coefficients used in the construction of the standard sequence. Another solution is the leaped Halton, which skips points in the standard sequence. Using, e.g., only each 409th point (also other prime numbers not used in the Halton core sequence are possible), can achieve significant improvements.
Implementation
In pseudocode:
algorithm Halton-Sequence is
inputs: index
base
output: result
while do
return
An alternative implementation that produces subsequent numbers of a Halton sequence for base b is given in the following generator function (in Python). This algorithm uses only integer numbers internally, which makes it robust against round-off errors.
def halton_sequence(b):
"""Generator function for H
|
https://en.wikipedia.org/wiki/Lerp
|
Lerp or LERP may refer to:
Lerp (biology), a structure produced by larvae of psyllid insects as a protective cover
Linear interpolation (Lerp), a method of curve fitting in mathematics
Emil Lerp (1886-1966), German inventor of first gasoline transportable chainsaw
Liberia Equal Rights Party
Lyari Expressway Resettlement Project
|
https://en.wikipedia.org/wiki/Coimage
|
In algebra, the coimage of a homomorphism
is the quotient
of the domain by the kernel.
The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies.
More generally, in category theory, the coimage of a morphism is the dual notion of the image of a morphism. If , then a coimage of (if it exists) is an epimorphism such that
there is a map with ,
for any epimorphism for which there is a map with , there is a unique map such that both and
See also
Quotient object
Cokernel
References
Abstract algebra
Isomorphism theorems
Category theory
pl:Twierdzenie o izomorfizmie#Pierwsze twierdzenie
|
https://en.wikipedia.org/wiki/Haboush%27s%20theorem
|
In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that
F(v) ≠ 0.
The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V, and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p.
When K has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of G implies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p was proved by W. J. , about a decade after the problem had been posed by David Mumford, in the introduction to the first edition of his book Geometric Invariant Theory.
Applications
Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where they were already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated.
Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1 on Y).
C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.
It follows from the work of , Haboush, and Popov that the following conditions are equivalent for an affine algebraic group G over a field K:
G is reductive (its unipotent radical is trivial).
For any non-zero invariant vector in a rational representation of G, there is an invariant homogeneous polynomial that does not vanish on it.
For any finitely generated K algebra on which G act rationally, the algebra of fixed elements is finitely generated.
Proof
The theorem is proved in several steps as follows:
We can assume that the group is defined over an algebraically closed field K of characteristic p>0.
Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the case of connected reductive groups (as the connected component has finite index). By taking a central extension which is harmless one can also assume the group G is simply connected.
Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick an element of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V to the element a∈A(G) with a(g) = (g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.
The structure of the representation A(G) is given as follows. Pick a ma
|
https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener%20theorem
|
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration).
Holomorphic Fourier transforms
The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform
and allow to be a complex number in the upper half-plane. One may then expect to differentiate under the integral in order to verify that the Cauchy–Riemann equations hold, and thus that defines an analytic function. However, this integral may not be well-defined, even for in ; indeed, since is in the upper half plane, the modulus of grows exponentially as ; so differentiation under the integral sign is out of the question. One must impose further restrictions on in order to ensure that this integral is well-defined.
The first such restriction is that be supported on : that is, . The Paley–Wiener theorem now asserts the following: The holomorphic Fourier transform of , defined by
for in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has
and by dominated convergence,
Conversely, if is a holomorphic function in the upper half-plane satisfying
then there exists such that is the holomorphic Fourier transform of .
In abstract terms, this version of the theorem explicitly describes the Hardy space . The theorem states that
This is a very useful result as it enables one to pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space
of square-integrable functions supported on the positive axis.
By imposing the alternative restriction that be compactly supported, one obtains another Paley–Wiener theorem. Suppose that is supported in , so that . Then the holomorphic Fourier transform
is an entire function of exponential type , meaning that there is a constant such that
and moreover, is square-integrable over horizontal lines:
Conversely, any entire function of exponential type which is square-integrable over horizontal lines is the holomorphic Fourier transform of an
function supported in .
Schwartz's Paley–Wiener theorem
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on is an entire function on and gives estimates on its growth at infinity. It was proven by Laurent Schwartz (1952). The formulation presented here is from .
Generally, the Fourier transform ca
|
https://en.wikipedia.org/wiki/Aggregate
|
Aggregate or aggregates may refer to:
Computing and mathematics
Aggregate (data warehouse), a part of the dimensional model that is used to speed up query time by summarizing tables
Aggregate analysis, a technique used in amortized analysis in computer science, especially in analysis of algorithms
Aggregate class, a type of class supported by C++
Aggregate data, in statistics, data combined from several measurements
Aggregate function, aggregation function, in database management is a function wherein the values of multiple rows are grouped together to form a single summary value
Aggregate Level Simulation Protocol (ALSP), a protocol and supporting software that enables simulations to interoperate with one another
Aggregate root, a concept in the Domain-driven Design software development process
Aggregate Server Access Protocol, used by the Reliable server pooling (RSerPool) framework
Aggregate throughput, total throughput measured over all links and in all directions in a communication network
Economics
Aggregate demand, the total demand for final goods and services during a specific time period in an economy
Aggregate income, the total of all incomes in an economy without adjustments for inflation, taxation, or types of double counting
Aggregate expenditure, a measure of national income
Aggregate Spend (US), a process to monitor the total amount spent by healthcare manufacturers on individual healthcare professionals and organizations through payments and gifts of various kinds
Aggregate supply, the total supply of goods and services produced during a specific time period in an economy
Religion
Aggregate (Sanskrit, skandha; Pāli, khandha), in Buddhism, a category of sensory experiences
Aggregates, in some Christian churches, are combinations of groupings of multiples of canonical hours (i.e., offices) that form a single religious service
Science
Biology
Aggregate fruit, in botanical terminology, fruit that develops from the merging of ovaries originating from a single flower
Aggregate species (Wiktionary) or Species aggregate, a named species representing a range of very closely related organisms
Materials science
Aggregate (composite), in materials science, a component of a composite material that resists compressive stress
Construction aggregate, materials used in construction, including sand, gravel, crushed stone, slag, or recycled crushed concrete
Other uses in science
Aggregate (geology), a mass of crystals, rock particles, or soil particles
Aggregate (rocket family), in rocketry, a set of experimental rocket designs developed in Nazi Germany
Arts, entertainment, and media
Aggregate, in music, is a set of all twelve pitch classes, also known as the total chromatic
The Aggregate, a 1988 album by Anthony Braxton and the Rova saxophone Quartet
Other uses
Aggregate, in the social sciences, a gathering of people into a cluster or a crowd that does not form a true social group
Aggregate Industries, a manufactu
|
https://en.wikipedia.org/wiki/Alabama%20School%20of%20Mathematics%20and%20Science
|
The Alabama School of Mathematics and Science (ASMS) is a public residential high school in the Midtown neighborhood of Mobile, Alabama. ASMS is a member of the National Consortium of Secondary STEM Schools (NCSSS). It graduated its first class in 1993.
The school was founded in 1989 as a unique public-private partnership. The Alabama School of Mathematics and Science is part of the state government, while the Alabama School of Mathematics and Science Foundation coordinates private support. It was modeled after the North Carolina School of Science and Mathematics and the Louisiana School for Math, Science, and the Arts where students complete their final two or three years of high school focusing on advanced studies in mathematics and the sciences. Although a boarding school, it does not charge for tuition, books, room, or board. The only fees include an annual student activity fee, which covers class trips and other day-to-day activities, along with an enrollment fee (for new students only), a PSAT fee, and a graduation fee. The annual student activity fee was $1,575 for the 2019 - 2020 school year. The school's focus is preparing its students for higher education, and residency is a requirement for all students.
ASMS's mascot is a dragon.
Academics
All courses are taught at the Advanced Placement or Honors level. ASMS offers Advanced Placement courses in Biology, Chemistry, Computer Science A, Environmental Science, Physics B, Physics C, Studio Art, English Literature and United States History. More than 40% of ASMS instructors hold terminal degrees, all have earned at least a master's degree, and nearly 100% have taught at the college level. According to Newsweek, the school was the third most effective in turning out college-ready graduates in the State of Alabama and 182nd nationally in 2012. The academic program, which is comprehensive in the sciences as well as the humanities, is complemented by varsity and intramural sports, residential life activities, and college counseling.
ASMS is accredited by the Southern Association of Colleges and Schools and is a member of the National Consortium of Secondary STEM Schools, Science and Technology, or NCSSS.
Admission is open to all Alabama high school students via a process akin to college admissions. Initially, students could enter only as high school juniors, but in September 1998 the school also allowed sophomores to apply for entry. Students have been admitted as seniors, but this is extremely rare.
For the class of 2016, the average composite ACT score for the class of 2016 was 29.0 and 100% of the graduates matriculated on to institutions of higher education
Summer program
ASMS also offers an academic summer camp for students entering the 6th through 9th grades. The Adventures In Math and Science summer program is typically held in June and enrolls roughly 1,000 students over the course of three weeks. Students from across Alabama can enroll in classes that are taught by ASMS instruct
|
https://en.wikipedia.org/wiki/Analytical%20hierarchy
|
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, , and over functions from to . The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy.
The analytical hierarchy of formulas
The notation
indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarchy for details.
A formula in the language of second-order arithmetic is defined to be if it is logically equivalent to a formula of the form where is . A formula is defined to be if it is logically equivalent to a formula of the form where is . This inductive definition defines the classes and for every natural number .
Kuratowski and Tarski showed in 1931 that every formula in the language of second-order arithmetic has a prenex normal form, and therefore or for some . Because meaningless quantifiers can be added to any formula, once a formula is given the classification or for some it will be given the classifications and for all greater than .
The analytical hierarchy of sets of natural numbers
A set of natural numbers is assigned the classification if it is definable by a formula. The set is assigned the classification if it is definable by a formula. If the set is both and then it is given the additional classification .
The sets are called hyperarithmetical. An alternate classification of these sets by way of iterated computable functionals is provided by the hyperarithmetical theory.
The analytical hierarchy on subsets of Cantor and Baire space
The analytical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor and Baire space because they fit with the language of ordinary second-order arithmetic. Cantor space is the set of all infinite sequences of 0s and 1s; Baire space is the set of all infinite sequences of natural numbers. These are both Polish spaces.
The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification if it is definable by a formula. The set is assigned the classification if it is definable by a formula. If the set is both and then it is given the additional classification .
A subset of Baire space has a corresponding subset of Cantor space under the map
|
https://en.wikipedia.org/wiki/Wheel%20theory
|
A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring.
The term wheel is inspired by the topological picture of the real projective line together with an extra point ⊥ (bottom element) such as .
A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.
Definition
A wheel is an algebraic structure , in which
is a set,
and are elements of that set,
and are binary operations,
is a unary operation,
and satisfying the following properties:
and are each commutative and associative, and have and as their respective identities.
( is an involution)
( is multiplicative)
Algebra of wheels
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument similar (but not identical) to the multiplicative inverse , such that becomes shorthand for , but neither nor in general, and modifies the rules of algebra such that
in the general case
in the general case, as is not the same as the multiplicative inverse of .
Other identities that may be derived are
where the negation is defined by and if there is an element such that (thus in the general case ).
However, for values of satisfying and , we get the usual
If negation can be defined as below then the subset is a commutative ring, and every commutative ring is such a subset of a wheel. If is an invertible element of the commutative ring then . Thus, whenever makes sense, it is equal to , but the latter is always defined, even when .
Examples
Wheel of fractions
Let be a commutative ring, and let be a multiplicative submonoid of . Define the congruence relation on via
means that there exist such that .
Define the wheel of fractions of with respect to as the quotient (and denoting the equivalence class containing as ) with the operations
(additive identity)
(multiplicative identity)
(reciprocal operation)
(addition operation)
(multiplication operation)
Projective line and Riemann sphere
The special case of the above starting with a field produces a projective line extended to a wheel by adjoining a bottom element noted ⊥, where . The projective line is itself an extension of the original field by an element , where for any element in the field. However, is still undefined on the projective line, but is defined in its extension to a wheel.
Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives
|
https://en.wikipedia.org/wiki/Analytic%20set
|
In the mathematical field of descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .
Definition
There are several equivalent definitions of analytic set. The following conditions on a subspace A of a Polish space X are equivalent:
A is analytic.
A is empty or a continuous image of the Baire space ωω.
A is a Suslin space, in other words A is the image of a Polish space under a continuous mapping.
A is the continuous image of a Borel set in a Polish space.
A is a Suslin set, the image of the Suslin operation.
There is a Polish space and a Borel set such that is the projection of onto ; that is,
A is the projection of a closed set in the cartesian product of X with the Baire space.
A is the projection of a Gδ set in the cartesian product of X with the Cantor space 2ω.
An alternative characterization, in the specific, important, case that is Baire space ωω, is that the analytic sets are precisely the projections of trees on . Similarly, the analytic subsets of Cantor space 2ω are precisely the projections of trees on .
Properties
Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images.
The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analytic set is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set including one and disjoint from the other. This is sometimes called the "Luzin separability principle" (though it was implicit in the proof of Suslin's theorem).
Analytic sets are always Lebesgue measurable (indeed, universally measurable) and have the property of Baire and the perfect set property.
Projective hierarchy
Analytic sets are also called (see projective hierarchy). Note that the bold font in this symbol is not the Wikipedia convention, but rather is used distinctively from its lightface counterpart (see analytical hierarchy). The complements of analytic sets are called coanalytic sets, and the set of coanalytic sets is denoted by .
The intersection is the set of Borel sets.
See also
Projection (measure theory)
References
N.N. Lusin, "Leçons sur les ensembles analytiques et leurs applications", Gauthier-Villars (1930)
Martin, Donald A.: Measurable cardinals and analytic games. Fundamenta Mathematicae 66 (1969/1970), p. 287-291.
Descriptive set theory
|
https://en.wikipedia.org/wiki/Bendixson%E2%80%93Dulac%20theorem
|
In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression
has the same sign () almost everywhere in a simply connected region of the plane, then the plane autonomous system
has no nonconstant periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or line.
The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac in 1923 using Green's theorem.
Proof
Without loss of generality, let there exist a function such that
in simply connected region . Let be a closed trajectory of the plane autonomous system in . Let be the interior of . Then by Green's theorem,
Because of the constant sign, the left-hand integral in the previous line must evaluate to a positive number. But on , and , so the bottom integrand is in fact 0 everywhere and for this reason the right-hand integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory .
References
Henri Dulac (1870-1955) was a French mathematician from Fayence
Differential equations
Theorems in dynamical systems
|
https://en.wikipedia.org/wiki/Hausdorff%20paradox
|
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (a 3-dimensional sphere in ). It states that if a certain countable subset is removed from , then the remainder can be divided into three disjoint subsets and such that and are all congruent. In particular, it follows that on there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of is simultaneously , , and of the non-zero measure of the whole sphere).
The paradox was published in Mathematische Annalen in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the axiom of choice.
This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no countably additive measure defined on all subsets.) The structure of the group of rotations on the sphere plays a crucial role here the statement is not true on the plane or the line. In fact, as was later shown by Banach, it is possible to define an "area" for all bounded subsets in the Euclidean plane (as well as "length" on the real line) in such a way that congruent sets will have equal "area". (This Banach measure, however, is only finitely additive, so it is not a measure in the full sense, but it equals the Lebesgue measure on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal area.
See also
References
Further reading
(Original article; in German)
External links
Hausdorff Paradox on ProofWiki
Mathematical paradoxes
Theorems in analysis
Measure theory
|
https://en.wikipedia.org/wiki/Bohr%20compactification
|
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the study of almost periodic functions, on the real line.
Definitions and basic properties
Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr(G) and a continuous homomorphism
b: G → Bohr(G)
which is universal with respect to homomorphisms into compact Hausdorff groups; this means that if K is another compact Hausdorff topological group and
f: G → K
is a continuous homomorphism, then there is a unique continuous homomorphism
Bohr(f): Bohr(G) → K
such that f = Bohr(f) ∘ b.
Theorem. The Bohr compactification exists and is unique up to isomorphism.
We will denote the Bohr compactification of G by Bohr(G) and the canonical map by
The correspondence G ↦ Bohr(G) defines a covariant functor on the category of topological groups and continuous homomorphisms.
The Bohr compactification is intimately connected to the finite-dimensional unitary representation theory of a topological group. The kernel of b consists exactly of those elements of G which cannot be separated from the identity of G by finite-dimensional unitary representations.
The Bohr compactification also reduces many problems in the theory of almost periodic functions on topological groups to that of functions on compact groups.
A bounded continuous complex-valued function f on a topological group G is uniformly almost periodic if and only if the set of right translates gf where
is relatively compact in the uniform topology as g varies through G.
Theorem. A bounded continuous complex-valued function f on G is uniformly almost periodic if and only if there is a continuous function f1 on Bohr(G) (which is uniquely determined) such that
Maximally almost periodic groups
Topological groups for which the Bohr compactification mapping is injective are called maximally almost periodic (or MAP groups). In the case G is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with vector groups
of finite dimension.
See also
References
Topological groups
Harmonic analysis
Compactification (mathematics)
|
https://en.wikipedia.org/wiki/Giuseppe%20Vitali
|
Giuseppe Vitali (26 August 1875 – 29 February 1932) was an Italian mathematician who worked in several branches of mathematical analysis. He gives his name to several entities in mathematics, most notably the Vitali set with which he was the first to give an example of a non-measurable subset of real numbers.
Biography
Giuseppe Vitali was the eldest of five children. His father, Domenico Vitali, worked for a railway company in Ravenna while his mother, Zenobia Casadio, was able to stay at home and look after her children.
He completed his elementary education in Ravenna in 1886, and then spent three years at the Ginnasio Comunale in Ravenna where his performance in the final examinations of 1889 was average.
He continued his secondary education in Ravenna at the Dante Alighieri High School. There his mathematics teacher was Giuseppe Nonni who quickly realised the young Giuseppe had great potential. He wrote to Giuseppe's father, in a letter dated 28 June 1895, asking that he allow his son to pursue further studies in mathematics.
He became a student of the Scuola Normale Superiore in Pisa and graduated to the University of Pisa in 1899. He spent two years as assistant before leaving the academic world. From 1901 to 1922 he taught in secondary schools, first in Sassari, then Voghera and then from 1904 at the Classical High School Christopher Columbus in Genoa. In those years he was involved in politics as a member of the Italian Socialist Party until it was forcibly disbanded by the fascists in 1922. His pursuit of mathematical analysis then led him to almost total social isolation. In 1923 he won a position as professor of calculus at the University of Modena and Reggio Emilia . He also taught at the Universities of Padua (1924 to 1925) and Bologna (from 1930). He was an invited speaker at the International Congress of Mathematicians held in Bologna in September 1928, giving the lecture Rapporti inattesi su alcuni rami della matematica (Unexpected relationships of some branches of mathematics).
From 1926 Vitali developed a serious illness and suffered a paralysed arm, meaning he could no longer write. Despite this about half his research papers were written in the last four years of his life.
On 29 February 1932 he delivered a lecture at the University of Bologna and was walking in conversation with fellow mathematician Ettore Bortolotti when he collapsed and died in the street. He was aged 56.
Vitali published a remarkable volume of mathematics over his career with his most significant output taking place in the first eight years of the twentieth century.
He was honoured with election to the Academy of Sciences of Turin in 1928, to the Accademia Nazionale dei Lincei in 1930, and to the Academy of Bologna in 1931.
Mathematical contributions
In 1905 Vitali was the first to give an example of a non-measurable subset of real numbers, see Vitali set. His covering theorem is a fundamental result in measure theory. He also proved seve
|
https://en.wikipedia.org/wiki/Thomas%20Kurtz
|
Thomas Kurtz may refer to:
Thomas E. Kurtz (born 1928), professor of mathematics and computer scientist
Thomas G. Kurtz (born 1941), professor of mathematics and statistics
Tom Kurtz, rhythm guitarist for the band Starstruck that recorded the hit song Black Betty#Ram Jam version
|
https://en.wikipedia.org/wiki/Pietro%20Cataldi
|
Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclid's fifth postulate.
Cataldi discovered the sixth and seventh perfect numbers by 1588. His discovery of the 6th, that corresponding to p=17 in the formula Mp=2p-1, exploded a many-times repeated number-theoretical myth that the perfect numbers had units digits that invariably alternated between 6 and 8. (Until Cataldi, 19 authors going back to Nicomachus are reported to have made the claim, with a few more repeating this afterward, according to L.E.Dickson's History of the Theory of Numbers). Cataldi's discovery of the 7th (for p=19) held the record for the largest known prime for almost two centuries, until Leonhard Euler discovered that 231 - 1 was the eighth Mersenne prime. Although Cataldi incorrectly claimed that p=23, 29, 31 and 37 all also generate Mersenne primes (and perfect numbers), his text's clear demonstration shows that he had genuinely established primality through p=19.
References
External links
Galileo Project
1548 births
1626 deaths
16th-century Italian mathematicians
17th-century Italian mathematicians
|
https://en.wikipedia.org/wiki/Hyperbolic%20orthogonality
|
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.
Geometry
Two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola.
Two particular hyperbolas are frequently used in the plane:
The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote A, a pair of lines (a, b) are hyperbolic orthogonal if there is a pair (c, d) such that , and c is the reflection of d across A.
Similar to the perpendularity of a circle radius to the tangent, a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.
A bilinear form is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of complex numbers , the bilinear form is , while in the plane of hyperbolic numbers the bilinear form is
The vectors z1 and z2 in the complex number plane, and w1 and w2 in the hyperbolic number plane are said to be respectively Euclidean orthogonal or hyperbolic orthogonal if their respective inner products [bilinear forms] are zero.
The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
entails perpendicularity in the complex plane, while
implies the ws are hyperbolic orthogonal.
The notion of hyperbolic orthogonality arose in analytic geometry in consideration of conjugate diameters of ellipses and hyperbolas. If g and g′ represent the slopes of the conjugate diameters, then in the case of an ellipse and in the case of a hyperbola. When a = b the ellipse is a circle and the conjugate diameters are perpendicular while the hyperbola is rectangular and the conjugate diameters are hyperbolic-orthogonal.
In the terminology of projective geometry, the operation of taking the hyperbolic orthogonal line is an involution. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the projectively extended real line. Then whichever hyperbola (A) or (B) is used, the operation is an example of a hyperbolic involution where the asymptote is invariant. Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a heterogeneous relation on sets of lines in the plane.
Simultaneity
Since Hermann Minkowski's foundation for spacetime study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a world lin
|
https://en.wikipedia.org/wiki/Flux%20%28disambiguation%29
|
Flux is a rate of flow through a surface or substance in physics, and has a related meaning in applied mathematics.
Flux may also refer to:
Science and technology
Biology and healthcare
Flux (biology), movement of a substance between compartments
Flux (metabolism), the rate of turnover of molecules through a metabolic pathway
4-Fluoroamphetamine (4-FA; PAL-303; "Flux"), a central nervous system stimulant with quasi-amphetamine effects
Dysentery, or other diseases called "flux", which cause the loss of fluid by diarrhea or hemorrhage
Rheumatism (historically), or "flux", thought to be caused by an excessive flow of rheum or fluid into a joint
Slime flux, a bacterial disease that occurs on certain trees
Computing
Flux (machine-learning framework)
Flux (graphics software), a suite of VRML/X3D viewing/authoring software
Flux (software company), a developer of workflow software
f.lux, a program that adjusts the color temperature of a computer display
Fast flux, a DNS technique used by botnets to hide phishing and malware delivery sites
Physics and engineering
Flux (metallurgy), a chemical cleaning agent, flowing agent, or purifying agent enhancing success in soldering and like joining of metals
Ceramic flux, a substance which lowers the melting point and promotes glass formation in ceramic materials and glasses
Secondary flux, a substance which acts as a ceramic flux in combination with other materials or at higher temperatures
Electric flux, a measure of quantity of electricity
Luminous flux, in photometry
Magnetic flux, a measure of quantity of magnetism
Radiant flux, in radiometry
People with the name
Alfred William Flux (1867–1942), British economist and statistician
Robert Flux, guitarist of the band Oomph!
Maarten van der Vleuten (born 1967), Dutch composer and producer, alias Flux
Art and entertainment
Fictional characters
Flux (comics), a Marvel Comics antagonist opposing the Hulk
Flux (Dennis Sykes), a character in the Marvel Comics series 1 Month 2 Live
Flux Wildly, a character in the computer game Toonstruck
Literature
Flux (novel), a book by Stephen Baxter
"Flux", a short story by Michael Moorcock
Music
Flux (Caveman Shoestore album), 1994
Flux (Poppy album), 2021
Flux (Rich Robinson album), 2016
Flux, an album by Love Spirals Downwards, 1998
"Flux" (Bloc Party song), 2007
"Flux" (Ellie Goulding song), 2019
"Flux", a song by Lamb of God from New American Gospel, 2000
Other uses in art and entertainment
Flux (magazine), a music, comic book and video game magazine of the 1990s
MTV Flux, a former television channel in the United Kingdom and Japan
Doctor Who: Flux, the thirteenth series of the British science fiction television programme
Organizations
Flux (political party), a political party in Australia
The Flux Foundation, an arts non-profit group
Other uses
Flux, Utah, a ghost town and former mining community in Tooele County, Utah
Flow (psychology), often referred to as flux
"Flux", t
|
https://en.wikipedia.org/wiki/Ernst%20Leonard%20Lindel%C3%B6f
|
Ernst Leonard Lindelöf (; 7 March 1870 – 4 June 1946) was a Finnish mathematician, who made contributions in real analysis, complex analysis and topology. Lindelöf spaces are named after him. He was the son of mathematician Lorenz Leonard Lindelöf and brother of the philologist .
Biography
Lindelöf studied at the University of Helsinki, where he completed his PhD in 1893, became a docent in 1895 and professor of Mathematics in 1903. He was a member of the Finnish Society of Sciences and Letters.
In addition to working in a number of different mathematical domains including complex analysis, conformal mappings, topology, ordinary differential equations and the gamma function, Lindelöf promoted the study of the history of Finnish mathematics. He is known for the Picard–Lindelöf theorem on differential equations and the Phragmén–Lindelöf principle, one of several refinements of the maximum modulus principle that he proved in complex function theory. He was the PhD supervisor for Lars Ahlfors at the University of Helsinki.
Selected bibliography
Le calcul des résidus et ses applications à la théorie des fonctions (Paris, 1905)Mémoire sur la théorie des fonctions entières d'ordre fini ("Acta societatis scientiarum fennicae" 31, 1903)With Lars Edvard Phragmén: "Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier", in: Acta Mathematica'' 31, 1908.
External links
1870 births
1946 deaths
Scientists from Helsinki
20th-century Finnish mathematicians
Topologists
19th-century Finnish mathematicians
Academic staff of the University of Helsinki
Members of the Royal Society of Sciences in Uppsala
Mathematicians from the Russian Empire
|
https://en.wikipedia.org/wiki/196%20%28number%29
|
196 (one hundred [and] ninety-six) is the natural number following 195 and preceding 197.
In mathematics
196 is a square number, the square of 14. As the square of a Catalan number, it counts the number of walks of length 8 in the positive quadrant of the integer grid that start and end at the origin, moving diagonally at each step. It is part of a sequence of square numbers beginning 0, 1, 4, 25, 196, ... in which each number is the smallest square that differs from the previous number by a triangular number.
There are 196 one-sided heptominoes, the polyominoes made from 7 squares. Here, one-sided means that asymmetric polyominoes are considered to be distinct from their mirror images.
A Lychrel number is a natural number which cannot form a palindromic number through the iterative process of repeatedly reversing its digits and adding the resulting numbers. 196 is the smallest number conjectured to be a Lychrel number in base 10; the process has been carried out for over a billion iterations without finding a palindrome, but no one has ever proven that it will never produce one.
See also
196 (disambiguation)
References
Arithmetic dynamics
Integers
|
https://en.wikipedia.org/wiki/151%20%28number%29
|
151 (one hundred [and] fifty-one) is a natural number. It follows 150 and precedes 152.
In mathematics
151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime, a centered decagonal number, and a lucky number.
151 appears in the Padovan sequence, preceded by the terms 65, 86, 114; it is the sum of the first two of these.
151 is a unique prime in base 2, since it is the only prime with period 15 in base 2.
There are 151 4-uniform tilings, such that the symmetry of tilings with regular polygons have four orbits of vertices.
151 is the number of uniform paracompact honeycombs with infinite facets and vertex figures in the third dimension, which stem from 23 different Coxeter groups. Split into two whole numbers, 151 is the sum of 75 and 76, both relevant numbers in Euclidean and hyperbolic 3-space:
75 is the total number of non-prismatic uniform polyhedra, which incorporate regular polyhedra, semiregular polyhedra, and star polyhedra,
75 uniform compound polyhedra, inclusive of seven types of families of prisms and antiprisms,
76 is the number of unique uniform compact hyperbolic honeycombs that are solely generated from Wythoff constructions.
While 151 is the 36th indexed prime, its twin prime 149 has a reciprocal whose repeating decimal expansion has a digit sum of 666, which is the magic constant in a prime reciprocal magic square equal to the sum of the first 36 non-zero integers, or equivalently the 36th triangular number.
Furthermore, the sum between twin primes (149, 151) is 300, which in turn is the 24th triangular number.
In music
The song 151 by Slick Shoes
The song Cell 151 from Steve Hackett's Highly Strung album
The song Psalm 151 by the Polish band Kult
The song Oddfellows Local 151 by R.E.M. of the album Document
The song "151" by Armando from A Bugged Out Mix
The song "151" by Jedi Mind Tricks
The song "151 Rum (J.I.D song)" by J.I.D
The song "Hyakugojyuuichi" (151 in Japanese), the original ending theme of the Pokemon anime
In sports
The De La Salle High School football team's national-record 151-game winning streak from 1992 to 2004.
UFC 151 was cancelled due to a medial collateral ligament injury sustained by challenger Dan Henderson who was scheduled to face champion Jon Jones. Jones declined to fight a new challenger on an 8-day notice. This became the first UFC event in history to have been completely cancelled.
In other fields
151 is also:
The year AD 151 or 151 BC
151 AH is a year in the Islamic calendar that corresponds to 759 – 760 CE
151 is the name of a Manchester-based company that sells a variety of household products
The proof number of Bacardi 151 rum
The total number of Pokémon in the first Generation, counting Mewtwo and Mew
151 Abundantia is a main belt asteroid
The atomic number of an element temporarily called unquintunum
Samarium-151 is a radioisotope of samarium
The IQ 151 personal computer
MG 151 cannon was a 15 mm
|
https://en.wikipedia.org/wiki/Algebra%20of%20sets
|
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being and the top being the universe set under consideration.
Fundamentals
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
The fundamental properties of set algebra
The binary operations of set union () and intersection () satisfy many identities. Several of these identities or "laws" have well established names.
Commutative property:
Associative property:
Distributive property:
The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over union. However, unlike addition and multiplication, union also distributes over intersection.
Two additional pairs of properties involve the special sets called the empty set Ø and the universe set ; together with the complement operator ( denotes the complement of . This can also be written as , read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context).
Identity :
Complement :
The identity expressions (together with the commutative expressions) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively.
Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.
The preceding five pairs of formulae—the commutative, associative, distributive, identity and complement formulae—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.
Note that if the complement f
|
https://en.wikipedia.org/wiki/Near
|
NEAR or Near may refer to:
People
Thomas J. Near, US evolutionary ichthyologist
Near, a developer who created the higan emulator
Science, mathematics, technology, biology, and medicine
National Emergency Alarm Repeater (NEAR), a former alarm device to warn civilians of a foreign nuclear attack on the United States
National Emergency Airway Registry (NEAR), a patient registry for intubations in the United States
Nicking enzyme amplification reaction (NEAR), a method of DNA amplification
NEAR Shoemaker, a spacecraft that studied the near-Earth asteroid Eros
Nearness or proximity space
"Near", a city browser by NearGlobal
Television, film, music, and books
Near (Death Note), Nate River, a character
Other uses
Near v. Minnesota, a U.S. press freedom Supreme Court decision
New England Auto Racers Hall of Fame
|
https://en.wikipedia.org/wiki/Probably
|
Probably may refer to:
Probability, the chance that something is likely to happen or be the case
"Probably" (South Park), an episode of the TV series South Park
"Probably" (song), a song by Fool's Garden
See also
Probability (disambiguation)
Problem (disambiguation)
Pro (disambiguation)
|
https://en.wikipedia.org/wiki/Finitely%20generated%20group
|
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
Examples
Every quotient of a finitely generated group G is finitely generated; the quotient group is generated by the images of the generators of G under the canonical projection.
A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z.
A locally cyclic group is a group in which every finitely generated subgroup is cyclic.
The free group on a finite set is finitely generated by the elements of that set (§Examples).
A fortiori, every finitely presented group (§Examples) is finitely generated.
Finitely generated abelian groups
Every abelian group can be seen as a module over the ring of integers Z, and in a finitely generated abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators,
x = α1⋅x1 + α2⋅x2 + ... + αn⋅xn
with integers α1, ..., αn.
Subgroups of a finitely generated abelian group are themselves finitely generated.
The fundamental theorem of finitely generated abelian groups states that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of which are unique up to isomorphism.
Subgroups
A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.
On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
A subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required.
In 1954, Albert G. Howson showed that the intersection of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if and are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most generators. This upper bound was then significantly improved by Hanna Neumann to ; see Hanna Neumann conjecture.
The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called Noetherian.
A group such that every finitely generated subgroup is finite is called locally finit
|
https://en.wikipedia.org/wiki/Screw%20%28disambiguation%29
|
A screw is an externally threaded fastener. "Screw" or "screws" may also refer to:
Engineering and mathematics
Devices with a helical thread:
Screw (simple machine)
Screw thread, screw thread principles and standards
Archimedes' screw, a simple machine for transporting water to a higher elevation
Leadscrew, a type of screw used to provide controlled and quantifiable movement in machine tools
Screw (motion), a description of spiral motion used in rigid body dynamics
Screw propeller
Screw, some specific pair of vectors (e.g., force+moment or linear+angular velocity); see Screw theory
Screw axis, the axis of rotation in 3D geometry
People with the name
Homer Screws (born 1966), American soccer defender
Kattie B. Screws (born 1930), matriarch of the Jackson family of American singers
William W. Screws (1839–1913), American politician in Alabama
Arts, entertainment, and media
Music
Screw (band), a Japanese rock band
"Screw" (song), a 2009 song by Japanese singer Kotoko
A Screw, an EP by Swans
Chopped and screwed music, a technique of remixing hip hop music by slowing the tempo
"(Let's Dance) The Screw", a 1963 song by The Crystals
Other uses in arts, entertainment, and media
Screw (magazine), a pornographic tabloid published and edited by Al Goldstein
Screw (TV series), a 2022 British prison-drama series from Channel 4.
Other
Screws v. United States, 1945 US Supreme Court case
See also
Screwed (disambiguation)
Thumbscrew (disambiguation)
|
https://en.wikipedia.org/wiki/Maurer%E2%80%93Cartan%20form
|
In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.
As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group . The Lie algebra is identified with the tangent space of at the identity, denoted . The Maurer–Cartan form is thus a one-form defined globally on which is a linear mapping of the tangent space at each into . It is given as the pushforward of a vector in along the left-translation in the group:
Motivation and interpretation
A Lie group acts on itself by multiplication under the mapping
A question of importance to Cartan and his contemporaries was how to identify a principal homogeneous space of . That is, a manifold identical to the group , but without a fixed choice of unit element. This motivation came, in part, from Felix Klein's Erlangen programme where one was interested in a notion of symmetry on a space, where the symmetries of the space were transformations forming a Lie group. The geometries of interest were homogeneous spaces , but usually without a fixed choice of origin corresponding to the coset .
A principal homogeneous space of is a manifold abstractly characterized by having a free and transitive action of on . The Maurer–Cartan form gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on satisfying an integrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain, locally, a group action on .
Construction
Intrinsic construction
Let be the tangent space of a Lie group at the identity (its Lie algebra). acts on itself by left translation
such that for a given we have
and this induces a map of the tangent bundle to itself:
A left-invariant vector field is a section of such that
The Maurer–Cartan form is a -valued one-form on defined on vectors by the formula
Extrinsic construction
If is embedded in by a matrix valued mapping , then one can write explicitly as
In this sense, the Maurer–Cartan form is always the left logarithmic derivative of the identity map of .
Characterization as a connection
If we regard the Lie group as a principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique principal connection on the principal bundle . Indeed, it is the unique valued -form on satisfying
where is the pullback of forms along the right-translation in the group and is the adjoint action on the Lie algebra.
Properties
If is a left-invariant vector field on , then is constant on . Furthermore,
|
https://en.wikipedia.org/wiki/Wolfgang%20Krull
|
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. He attended the Universities of Freiburg, Rostock and finally Göttingen from 1919–1921, where he earned his doctorate under Alfred Loewy. He worked as an instructor and professor at Freiburg, then spent a decade at the University of Erlangen. In 1939, Krull moved to become chair at the University of Bonn, where he remained for the rest of his life. Wolfgang Krull was not a member of the Nazi Party.
His 35 doctoral students include Wilfried Brauer, Karl-Otto Stöhr and Jürgen Neukirch.
See also
Cohen structure theorem
Jacobson ring
Local ring
Prime ideal
Real algebraic geometry
Regular local ring
Valuation ring
Krull dimension
Krull ring
Krull topology
Krull–Azumaya theorem
Krull–Schmidt category
Krull–Schmidt theorem
Krull's intersection theorem
Krull's principal ideal theorem
Krull's separation lemma
Krull's theorem
Publications
References
External links
1899 births
1971 deaths
20th-century German mathematicians
Nazi Party members
Algebraists
|
https://en.wikipedia.org/wiki/Indecomposability
|
Indecomposability or indecomposable may refer to any of several subjects in mathematics:
Indecomposable module, in algebra
Indecomposable distribution, in probability
Indecomposable continuum, in topology
Indecomposability (intuitionistic logic), a principle in constructive analysis and in computable analysis
Indecomposability of a polynomial in polynomial decomposition
A property of certain ordinals; see additively indecomposable ordinal
|
https://en.wikipedia.org/wiki/Functional%20calculus
|
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.)
If is a function, say a numerical function of a real number, and is an operator, there is no particular reason why the expression should make sense. If it does, then we are no longer using on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of and an matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation.
The most immediate case is to apply polynomial functions to a square matrix, extending what has just been discussed. In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator . This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let be the finite dimension of the algebra of matrices, then is linearly dependent. So for some scalars , not all equal to 0. This implies that the polynomial lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial . Multiplying by a unit if necessary, we can choose to be monic. When this is done, the polynomial is precisely the minimal polynomial of . This polynomial gives deep information about . For instance, a scalar is an eigenvalue of if and only if is a root of . Also, sometimes can be used to calculate the exponential of efficiently.
The polynomial calculus is not as informative in the infinite-dimensional case. Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitions should be.
See also
References
External links
|
https://en.wikipedia.org/wiki/Minimal%20counterexample
|
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true.
If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof.
The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind.
Examples
The minimal counterexample method has been much used in the classification of finite simple groups. The Feit–Thompson theorem, that finite simple groups that are not cyclic groups have even order, was based on the hypothesis of some, and therefore some minimal, simple group G of odd order. Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.
Euclid's proof of the fundamental theorem of arithmetic is a simple proof which uses a minimal counterexample.
Courant and Robbins used the term minimal criminal for a minimal counter-example in the context of the four color theorem.
References
Mathematical proofs
Mathematical terminology
|
https://en.wikipedia.org/wiki/Valuation%20ring
|
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D.
Given a field F, if D is a subring of F such that either x or x−1 belongs to
D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring.
The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement, where
dominates if and .
Every local ring in a field K is dominated by some valuation ring of K.
An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.
Definitions
There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For an integral domain D and its field of fractions K, the following are equivalent:
For every nonzero x in K, either x is in D or x−1 is in D.
The ideals of D are totally ordered by inclusion.
The principal ideals of D are totally ordered by inclusion (i.e. the elements in D are, up to units, totally ordered by divisibility.)
There is a totally ordered abelian group Γ (called the value group) and a valuation ν: K → Γ ∪ {∞} with D = { x ∈ K | ν(x) ≥ 0 }.
The equivalence of the first three definitions follows easily. A theorem of states that any ring satisfying the first three conditions satisfies the fourth: take Γ to be the quotient K×/D× of the unit group of K by the unit group of D, and take ν to be the natural projection. We can turn Γ into a totally ordered group by declaring the residue classes of elements of D as "positive".
Even further, given any totally ordered abelian group Γ, there is a valuation ring D with value group Γ (see Hahn series).
From the fact that the ideals of a valuation ring are totally ordered, one can conclude that a valuation ring is a local domain, and that every finitely generated ideal of a valuation ring is principal (i.e., a valuation ring is a Bézout domain). In fact, it is a theorem of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain. It also follows from this that a valuation ring is Noetherian if and only if it is a principal ideal domain. In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a discrete valuation ring. (By convention, a field is not a discrete valuation ring.)
A value group is called discrete if it is isomorphic to the additive group of the integers, and a valuation r
|
https://en.wikipedia.org/wiki/Algebraic
|
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
Algebraic data type, a datatype in computer programming each of whose values is data from other datatypes wrapped in one of the constructors of the datatype
Algebraic numbers, a complex number that is a root of a non-zero polynomial in one variable with integer coefficients
Algebraic functions, functions satisfying certain polynomials
Algebraic element, an element of a field extension which is a root of some polynomial over the base field
Algebraic extension, a field extension such that every element is an algebraic element over the base field
Algebraic definition, a definition in mathematical logic which is given using only equalities between terms
Algebraic structure, a set with one or more finitary operations defined on it
Algebraic, the order of entering operations when using a calculator (contrast reverse Polish notation)
Algebraic sum, a summation of quantities that takes into account their signs; e.g. the algebraic sum of 4, 3, and -8 is -1.
See also
Algebra (disambiguation)
Algebraic notation (disambiguation)
|
https://en.wikipedia.org/wiki/R.%20H.%20Bing
|
R. H. Bing (October 20, 1914 – April 28, 1986) was an American mathematician who worked mainly in the areas of geometric topology and continuum theory. His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. She compromised by abbreviating it to R. H. Consequently, R. H. does not stand for a first or middle name.
Mathematical contributions
Bing's mathematical research was almost exclusively in 3-manifold theory and in particular, the geometric topology of . The term Bing-type topology was coined to describe the style of methods used by Bing.
Bing established his reputation early on in 1946, soon after completing his Ph.D. dissertation, by solving the Kline sphere characterization problem. In 1948 he proved that the pseudo-arc is homogeneous, contradicting a published but erroneous 'proof' to the contrary.
In 1951 he proved results regarding the metrizability of topological spaces, including what would later be called the Bing–Nagata–Smirnov metrization theorem.
In 1952, Bing showed that the double of a solid Alexander horned sphere was the 3-sphere. This showed the existence of an involution on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere, which meant that the original Smith conjecture needed to be phrased in a suitable category. This result also jump-started research into crumpled cubes. The proof involved a method later developed by Bing and others into set of techniques called Bing shrinking. Proofs of the generalized Schoenflies conjecture and the double suspension theorem relied on Bing-type shrinking.
Bing was fascinated by the Poincaré conjecture and made several major attacks which ended unsuccessfully, contributing to the reputation of the conjecture as a very difficult one. He did show that a simply connected, closed 3-manifold with the property that every loop was contained in a 3-ball is homeomorphic to the 3-sphere. Bing was responsible for initiating research into the Property P conjecture, as well as its name, as a potentially more tractable version of the Poincaré conjecture. It was proven in 2004 as a culmination of work from several areas of mathematics. With some irony, this proof was announced some time after Grigori Perelman announced his proof of the Poincaré conjecture.
The side-approximation theorem was considered by Bing to be one of his key discoveries. It has many applications, including a simplified proof of Moise's theorem, which states that every 3-manifold can be triangulated in an essentially unique way.
Notable examples
The house with two rooms
The house with two rooms is a contractible 2-complex that is not collapsible. Another such example, popularized by E.C. Zeeman, is the dunce hat.
The house with two rooms can also be thickened and then triangulated to be unshellable, despite the thickened house topologically being a 3-ball. The house with two rooms shows up in various ways in topology. For example, it is used i
|
https://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin%20theorem
|
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.
This theorem bounds the norms of linear maps acting between spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.
Motivation
First we need the following definition:
Definition. Let be two numbers such that . Then for define by: .
By splitting up the function in as the product and applying Hölder's inequality to its power, we obtain the following result, foundational in the study of -spaces:
This result, whose name derives from the convexity of the map on , implies that .
On the other hand, if we take the layer-cake decomposition , then we see that and , whence we obtain the following result:
In particular, the above result implies that is included in , the sumset of and in the space of all measurable functions. Therefore, we have the following chain of inclusions:
In practice, we often encounter operators defined on the sumset . For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps boundedly into , and Plancherel's theorem shows that the Fourier transform maps boundedly into itself, hence the Fourier transform extends to by setting
for all and . It is therefore natural to investigate the behavior of such operators on the intermediate subspaces .
To this end, we go back to our example and note that the Fourier transform on the sumset was obtained by taking the sum of two instantiations of the same operator, namely
These really are the same operator, in the sense that they agree on the subspace . Since the intersection contains simple functions, it is dense in both and . Densely defined continuous operators admit unique extensions, and so we are justified in considering and to be the same.
Therefore, the problem of studying operators on the sumset essentially reduces to the study of operators that map two natural domain spaces, and , boundedly to two target spaces: and , respectively. Since such operators map the sumset space to , it is natural to expect that these operators map the intermediate space to the corresponding intermediate space .
Statement of the theorem
There are several ways to state the Riesz–Thorin interpolation theorem; to be consistent with the notations in the previous section, we shall use the sumset formulation.
In other words, if is simultaneously of type and of type , then is of type for all . In this manner, the i
|
https://en.wikipedia.org/wiki/Nobuo%20Yoneda
|
was a Japanese mathematician and computer scientist.
In 1952, he graduated the Department of Mathematics, the Faculty of Science, the University of Tokyo, and obtained his Bachelor of Science. That same year, he was appointed Assistant Professor in the Department of Mathematics of the University of Tokyo. He obtained his Doctor of Science (DSc) degree from the University of Tokyo in 1961, under the direction of Shokichi Iyanaga. In 1962, he was appointed Associate Professor in the Faculty of Science at Gakushuin University, and was promoted in 1966 to the rank of Professor. He became a professor of Theoretical Foundation of Information Science in 1972. After retiring from the University of Tokyo in 1990, he moved to Tokyo Denki University.
The Yoneda lemma in category theory and the Yoneda product in homological algebra are named after him.
In computer science, he is known for his work on dialects of the programming language ALGOL. He became involved with developing international standards in programming and informatics, as a Japanese representative on the International Federation for Information Processing (IFIP) IFIP Working Group 2.1 on Algorithmic Languages and Calculi, which specified, supports, and maintains the languages ALGOL 60 and ALGOL 68.
References
External links
1930 births
1996 deaths
Japanese computer scientists
20th-century Japanese mathematicians
University of Tokyo alumni
Academic staff of the University of Tokyo
Academic staff of Gakushuin University
Academic staff of Tokyo Denki University
|
https://en.wikipedia.org/wiki/Integrability%20conditions%20for%20differential%20systems
|
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system).
Given a collection of differential 1-forms on an -dimensional manifold , an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point is annihilated by (the pullback of) each .
A maximal integral manifold is an immersed (not necessarily embedded) submanifold
such that the kernel of the restriction map on forms
is spanned by the at every point of . If in addition the are linearly independent, then is ()-dimensional.
A Pfaffian system is said to be completely integrable if admits a foliation by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.)
An integrability condition is a condition on the to guarantee that there will be integral submanifolds of sufficiently high dimension.
Necessary and sufficient conditions
The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Example of a non-integrable system
Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form :
If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product
But a direct calculation gives
which is a nonzero multiple of the standard volume form on R3. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.
On the other hand, for the curve defined by
then θ defined as above is 0, and hence the curve is easily verified to be a solution (i.e. an integral curve) for the above Pfaffian system for any nonzero constant c.
Examples of applications
In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi, i.e., a collection of 1-forms forming a basis of the cotangent space at every point with which are closed (dθi = 0, i = 1, 2, ..., n). By the Poincaré lemma, the θi locally will have the form dxi for so
|
https://en.wikipedia.org/wiki/European%20Mathematical%20Society
|
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current president is Jan Philip Solovej, professor at the Department of Mathematics at the University of Copenhagen.
Goals
The Society seeks to serve all kinds of mathematicians in universities, research institutes and other forms of higher education. Its aims are to
Promote mathematical research, both pure and applied,
Assist and advise on problems of mathematical education,
Concern itself with the broader relations of mathematics to society,
Foster interaction between mathematicians of different countries,
Establish a sense of identity amongst European mathematicians,
Represent the mathematical community in supra-national institutions.
The EMS is itself an Affiliate Member of the International Mathematical Union and an Associate Member of the International Council for Industrial and Applied Mathematics.
History
The precursor to the EMS, the European Mathematical Council was founded in 1978 at the International Congress of Mathematicians in Helsinki. This informal federation of mathematical societies was chaired by Sir Michael Atiyah. The European Mathematical Society was founded on 28 October 1990 in Mądralin near Warsaw, Poland, with Friedrich Hirzebruch as founding President. Initially, the EMS had 27 member societies. The first European Congress of Mathematics (ECM) was held at the Sorbonne and Panthéon-Sorbonne universities in Paris in 1992, and is now held every 4 years at different locations around Europe, organised by the EMS. The last ECM (postponed for a year due to the covid pandemic) was in 2021 in Portorož in Slovenia.
Presidents of the EMS
Friedrich Hirzebruch, 1990–1994
Jean-Pierre Bourguignon, 1995–1998
Rolf Jeltsch, 1999–2002
John Kingman, 2003–2006
Ari Laptev, 2007–2010
Marta Sanz-Solé, 2011–2014
Pavel Exner, 2015–2018
Volker Mehrmann, 2019–2022
Jan Philip Solovej, 2023–2026
Structure and Governance
The governing body of the EMS is its Council, which comprises delegates representing all of the societies which are themselves members of the EMS, along with delegates representing the institutional and individual EMS members. The Council meets every 2 years, and appoints the President and Executive Committee who are responsible for the running of the society.
Besides the Executive Committee, the EMS has standing committees on: Applications and Interdisciplinary Relations, Developing Countries, Mathematical Education, ERCOM (Directors of European Research Centres in the Mathematical Sciences), Ethics, European Solidarity, Meetings, Publications and Electronic Dissemination, Raising Public Awareness of Mathematics, Women in Mathematics.
The EMS's rules are set down in its Statutes and Bylaws. The EMS is headquartered at the University of Helsinki.
Prizes
The European Congress of Mathemati
|
https://en.wikipedia.org/wiki/Chern%E2%80%93Weil%20homomorphism
|
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.
Let G be a real or complex Lie group with Lie algebra and let denote the algebra of -valued polynomials on (exactly the same argument works if we used instead of Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G-bundle P on M, there is an associated homomorphism of -algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Definition of the homomorphism
Choose any connection form ω in P, and let Ω be the associated curvature form; i.e., the exterior covariant derivative of ω. If is a homogeneous polynomial function of degree k; i.e., for any complex number a and x in then, viewing f as a symmetric multilinear functional on (see the ring of polynomial functions), let
be the (scalar-valued) 2k-form on P given by
where vi are tangent vectors to P, is the sign of the permutation in the symmetric group on 2k numbers (see Lie algebra-valued forms#Operations as well as Pfaffian).
If, moreover, f is invariant; i.e., , then one can show that is a closed form, it descends to a unique form on M and that the de Rham cohomology class of the form is independent of . First, that is a closed form follows from the next two lemmas:
Lemma 1: The form on P descends to a (unique) form on M; i.e., there is a form on M that pulls-back to .
Lemma 2: If a form of on P descends to a form on M, then .
Indeed, Bianchi's second identity says and, since D is a graded derivation, Finally, Lemma 1 says satisfies the hypothesis of Lemma 2.
To see Lemma 2, let be the projection and h be the projection of onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that (the kernel of is precisely the vertical subspace.) As for Lemma 1, first note
which is because and f is invariant. Thus, one can define by the formula:
where are any lifts of : .
Next, we show that the de Rham cohomology class
|
https://en.wikipedia.org/wiki/Total%20least%20squares
|
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.
The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix.
Linear model
Background
In the least squares method of data modeling, the objective function, S,
is minimized, where r is the vector of residuals and W is a weighting matrix. In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector , so the residuals are given by
There are m observations in y and n parameters in β with m>n. X is a m×n matrix whose elements are either constants or functions of the independent variables, x. The weight matrix W is, ideally, the inverse of the variance-covariance matrix of the observations y. The independent variables are assumed to be error-free. The parameter estimates are found by setting the gradient equations to zero, which results in the normal equations
Allowing observation errors in all variables
Now, suppose that both x and y are observed subject to error, with variance-covariance matrices and respectively. In this case the objective function can be written as
where and are the residuals in x and y respectively. Clearly these residuals cannot be independent of each other, but they must be constrained by some kind of relationship. Writing the model function as , the constraints are expressed by m condition equations.
Thus, the problem is to minimize the objective function subject to the m constraints. It is solved by the use of Lagrange multipliers. After some algebraic manipulations, the result is obtained.
or alternatively
where M is the variance-covariance matrix relative to both independent and dependent variables.
Example
When the data errors are uncorrelated, all matrices M and W are diagonal. Then, take the example of straight line fitting.
in this case
showing how the variance at the ith point is determined by the variances of both independent and dependent variables and by the model being used to fit the data. The expression may be generalized by noting that the parameter is the slope of the line.
An expression of this type is used in fitting pH titration data where a small error on x translates to a large error on y when the slope is large.
Algebraic point of view
As was shown in 1980 by Golub and Van Loan, the TLS problem does not have a solution in general. The following considers the simple case where a unique solution exists without making any particular assumptions.
The computation of the TLS using singular value decomposition (SVD) is described in standard texts. We can solve the equation
for B
|
https://en.wikipedia.org/wiki/Plane%20curve
|
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions.
Symbolic representation
A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as or for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form for specific functions and
Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.
Smooth plane curve
A smooth plane curve is a curve in a real Euclidean plane and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function.
Equivalently, a smooth plane curve can be given locally by an equation where is a smooth function, and the partial derivatives and are never both 0 at a point of the curve.
Algebraic plane curve
An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (or where is a homogeneous polynomial, in the projective case.)
Algebraic curves have been studied extensively since the 18th century.
Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation has degree 2.
The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle (that is the projective curve of equation The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.
Examples
Numerous examples of plane curves are shown in Gallery of curves and listed at List of curves. The algebraic curves of degree 1 or 2 are shown here (an algebraic curve of degree less than 3 is always contained in a plane):
See also
Algebraic geometry
Convex curve
Differential geometry
Osgood curve
Plane curve fitting
Projective varieties
Skew curve
References
.
.
.
External links
Euclidean geometry
es:Curva plana
|
https://en.wikipedia.org/wiki/Principal%20branch
|
In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that
or that
.
Exponentiation to fractional powers
A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of .
For example, take the relation , where is any positive real number.
This relation can be satisfied by any value of equal to a square root of (either positive or negative). By convention, is used to denote the positive square root of .
In this instance, the positive square root function is taken as the principal branch of the multi-valued relation .
Complex logarithms
One way to view a principal branch is to look specifically at the exponential function, and the logarithm, as it is defined in complex analysis.
The exponential function is single-valued, where is defined as:
where .
However, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined. One way to see this is to look at the following:
and
where is any integer and continues the values of the -function from their principal value range , corresponding to into the principal value range of the -function , covering all four quadrants in the complex plane.
Any number defined by such criteria has the property that .
In this manner log function is a multi-valued function (often referred to as a "multifunction" in the context of complex analysis). A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between and . These are the chosen principal values.
This is the principal branch of the log function. Often it is defined using a capital letter, .
See also
Branch point
Branch cut
Complex logarithm
Riemann surface
External links
Branches of Complex Functions Module by John H. Mathews
Complex analysis
|
https://en.wikipedia.org/wiki/Hyperfinite%20type%20II%20factor
|
In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor.
There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.
Constructions
The von Neumann group algebra of a discrete group with the infinite conjugacy class property is a factor of type II1, and if the group is amenable and countable the factor is hyperfinite. There are many groups with these properties, as any locally finite group is amenable. For example, the von Neumann group algebra of the infinite symmetric group of all permutations of a countable infinite set that fix all but a finite number of elements gives the hyperfinite type II1 factor.
The hyperfinite type II1 factor also arises from the group-measure space construction for ergodic free measure-preserving actions of countable amenable groups on probability spaces.
The infinite tensor product of a countable number of factors of type In with respect to their tracial states is the hyperfinite type II1 factor. When n=2, this is also sometimes called the Clifford algebra of an infinite separable Hilbert space.
If p is any non-zero finite projection in a hyperfinite von Neumann algebra A of type II, then pAp is the hyperfinite type II1 factor. Equivalently the fundamental group of A is the group of positive real numbers. This can often be hard to see directly. It is, however, obvious when A is the infinite tensor product of factors of type In, where n runs over all integers greater than 1 infinitely many times: just take p equivalent to an infinite tensor product of projections pn on which the tracial state is either or .
Properties
The hyperfinite II1 factor R is the unique smallest infinite
dimensional factor in the following sense: it is contained in any other infinite dimensional factor, and any infinite dimensional factor contained in R is isomorphic to R.
The outer automorphism group of R is an infinite simple group with countable many conjugacy classes, indexed by pairs consisting of a positive integer p and a complex pth root of 1.
The projections of the hyperfinite II1 factor form a continuous geometry.
The infinite hyperfinite type II factor
While there are other factors of type II∞, there is a unique hyperfinite one, up to isomorphism. It consists of those infinite square matrices with entries in the hyperfinite type II1 factor that define bounded operators.
See also
Subfactors
References
A. Connes, Classification of Injective Factors The Annals of Mathematics 2nd Ser., Vol. 104, No. 1 (Jul., 1976), pp. 73–115
F.J. Murray, J. von Neumann, On rings of operators IV Ann. of Math. (2), 44 (1943) pp. 716–808. This shows that all approximately finite factors of type II1 are isomorphic.
Von Neum
|
https://en.wikipedia.org/wiki/Q1
|
Q1 or Q-1 may refer to:
Quarter 1, as in the first quarter of a calendar year or fiscal year
first quartile in descriptive statistics
The first quarto, usually meaning the earliest published version, of one of William Shakespeare's works
Q1 Tower, a residential apartment building in Surfers Paradise, Australia
DIGITAL Q1, a digital camera model (Fujifilm)
Q-1 visa, allows individuals traveling to the U.S. to participate in a cultural exchange program
Quake (video game) 1, a 1996 video game
Samsung Q1, an Ultra Mobile Personal Computer (UMPC)
Qualifying 1, or first qualifying in Formula 1
Quran 1, al fātiḥah the 1st chapter of the Islamic Holy book
Transport
Air
Radioplane Q-1, an American experimental unmanned aircraft of the 1950s
The primary United States Air Force designation for a series of unmanned aerial vehicles built by General Atomics, which includes the MQ-1 Predator and the MQ-1C Warrior
Road
Q1 (New York City bus)
Rossion Q1, a sports car from US car maker 1g Racing/Rossion Automotive
Rail
LNER Thompson Class Q1, a class of steam locomotives of the London and North Eastern Railway, UK
PRR Q1, a steam locomotive of the Pennsylvania Railroad, USA
SECR Q1 class, a steam locomotive of the South Eastern and Chatham Railway, UK
SR Q1 class, a steam locomotive of the Southern Railway, UK
See also
1Q (disambiguation)
QI, a British comedy television quiz
|
https://en.wikipedia.org/wiki/Q2
|
Q2 or Q-2 may refer to:
The second quarter of a calendar year (April, May, June) or fiscal year
The second quarto of William Shakespeare's works
Q2 (statistics), the second quartile in descriptive statistics (i.e. the median)
Q2 Stadium, a sports stadium in Austin, Texas
Quake II, first person shooter game developed by id Software and distributed by Activision
Quran 2, al-baqarah, the 2nd chapter of the Islamic Holy book
Leica Q2, a digital camera
Media and broadcasting
KZJO, a television station in Seattle, Washington, United States, which formerly held the call sign KMYQ and was known on air as "MyQ²"
KTVQ, a television station in Billings, Montana, United States known on air as "Q2"
Q2 (TV channel), a Belgian television station
Q2, a short-lived spinoff network of QVC in the 1990s
Q2, a Star Trek character played by Corbin Bernsen
"Q2" (Star Trek: Voyager), an episode of Star Trek: Voyager
Q2, a contemporary classical music internet radio station run by New York's WQXR
Transportation
Air Cargo Carriers, a cargo airline with the IATA code 2Q
PRR Q2, a class of steam locomotive
Metrobus route Q2, a bus route in Washington, D.C.
Q2 (New York City bus)
Alfa Romeo's limited slip differential
Audi Q2, a mini SUV by German manufacturer
See also
02 (disambiguation)
2Q (disambiguation)
|
https://en.wikipedia.org/wiki/Crelle%27s%20Journal
|
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).
History
The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as Borchardt's Journal. The current editor-in-chief is Daniel Huybrechts (Rheinische Friedrich-Wilhelms-Universität Bonn).
Past editors
1826–1856 August Leopold Crelle
1856–1880 Carl Wilhelm Borchardt
1881–1888 Leopold Kronecker, Karl Weierstrass
1889–1892 Leopold Kronecker
1892–1902 Lazarus Fuchs
1903–1928 Kurt Hensel
1929–1933 Kurt Hensel, Helmut Hasse, Ludwig Schlesinger
1934–1936 Kurt Hensel, Helmut Hasse
1937–1952 Helmut Hasse
1952–1977 Helmut Hasse, Hans Rohrbach
1977–1980 Helmut Hasse
References
External links
Journal archive at the Göttinger Digitalisierungszentrum
Journal index at The European Digital Mathematics Library
Multilingual journals
English-language journals
French-language journals
German-language journals
Mathematics journals
Publications established in 1826
Monthly journals
De Gruyter academic journals
1826 establishments in Prussia
|
https://en.wikipedia.org/wiki/Infinite%20conjugacy%20class%20property
|
In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.
The von Neumann group algebra of a group is a factor if and only if the group has the infinite conjugacy class property. It will then be, provided the group is nontrivial, of type II1, i.e. it will possess a unique, faithful, tracial state.
Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed, and free groups on two generators.
In abelian groups, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible.
References
Infinite group theory
Properties of groups
|
https://en.wikipedia.org/wiki/Addition%20theorem
|
In mathematics, an addition theorem is a formula such as that for the exponential function:
ex + y = ex · ey,
that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions and , several functions may be involved; this is more apparent than real, in that case, since there is an algebraic function of (in other words, we usually take their functions both as defined on the unit circle).
The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To "classify" addition theorems it is necessary to put some restriction on the type of function G admitted, such that
F(x + y) = G(F(x), F(y)).
In this identity one can assume that F and G are vector-valued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution.
In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups. The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of formal groups.
See also
Timeline of abelian varieties
Addition theorem for spherical harmonics
Mordell–Weil theorem
References
Theorems in algebraic geometry
Theorems in algebra
|
https://en.wikipedia.org/wiki/List%20of%20manifolds
|
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see :Category:Manifolds and its subcategories.
Generic families of manifolds
Euclidean space, Rn
n-sphere, Sn
n-torus, Tn
Real projective space, RPn
Complex projective space, CPn
Quaternionic projective space, HPn
Flag manifold
Grassmann manifold
Stiefel manifold
Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
Manifolds of a specific dimension
1-manifolds
Circle, S1
Long line
Real line, R
Real projective line, RP1 ≅ S1
2-manifolds
Cylinder, S1 × R
Klein bottle, RP2 # RP2
Klein quartic (a genus 3 surface)
Möbius strip
Real projective plane, RP2
Sphere, S2
Surface of genus g
Torus
Double torus
3-manifolds
3-sphere, S3
3-torus, T3
Poincaré homology sphere
SO(3) ≅ RP3
Solid Klein bottle
Solid torus
Whitehead manifold
Meyerhoff manifold
Weeks manifold
For more examples see 3-manifold.
4-manifolds
Complex projective plane
Del Pezzo surface
E8 manifold
Enriques surface
Exotic R4
Hirzebruch surface
K3 surface
For more examples see 4-manifold.
Special types of manifolds
Manifolds related to spheres
Brieskorn manifold
Exotic sphere
Homology sphere
Homotopy sphere
Lens space
Spherical 3-manifold
Special classes of Riemannian manifolds
Einstein manifold
Ricci-flat manifold
G2 manifold
Kähler manifold
Calabi–Yau manifold
Hyperkähler manifold
Quaternionic Kähler manifold
Riemannian symmetric space
Spin(7) manifold
Categories of manifolds
Manifolds definable by a particular choice of atlas
Affine manifold
Analytic manifold
Complex manifold
Differentiable (smooth) manifold
Piecewise linear manifold
Lipschitz manifold
Topological manifold
Manifolds with additional structure
Almost complex manifold
Almost symplectic manifold
Calibrated manifold
Complex manifold
Contact manifold
CR manifold
Finsler manifold
Hermitian manifold
Hyperkähler manifold
Kähler manifold
Lie group
Pseudo-Riemannian manifold
Riemannian manifold
Sasakian manifold
Spin manifold
Symplectic manifold
Infinite-dimensional manifolds
Banach manifold
Fréchet manifold
Hilbert manifold
See also
References
Manifolds
|
https://en.wikipedia.org/wiki/Automorphic%20function
|
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group.
Factor of automorphy
In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group acts on a complex-analytic manifold . Then, also acts on the space of holomorphic functions from to the complex numbers. A function is termed an automorphic form if the following holds:
where is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of .
The factor of automorphy for the automorphic form is the function . An automorphic function is an automorphic form for which is the identity.
Some facts about factors of automorphy:
Every factor of automorphy is a cocycle for the action of on the multiplicative group of everywhere nonzero holomorphic functions.
The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form.
For a given factor of automorphy, the space of automorphic forms is a vector space.
The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding factors of automorphy.
Relation between factors of automorphy and other notions:
Let be a lattice in a Lie group . Then, a factor of automorphy for corresponds to a line bundle on the quotient group . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.
The specific case of a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.
Examples
Kleinian group
Elliptic modular function
Modular function
Complex torus
References
Automorphic forms
Discrete groups
Types of functions
Complex manifolds
|
https://en.wikipedia.org/wiki/Algebraic%20function
|
In mathematics, an algebraic function is a function that can be defined
as the root of an irreducible polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:
Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by
.
In more precise terms, an algebraic function of degree in one variable is a function that is continuous in its domain and satisfies a polynomial equation
where the coefficients are polynomial functions of , with integer coefficients. It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the 's. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the field generated by these coefficients.
The value of an algebraic function at a rational number, and more generally, at an algebraic number is always an algebraic number.
Sometimes, coefficients that are polynomial over a ring are considered, and one then talks about "functions algebraic over ".
A function which is not algebraic is called a transcendental function, as it is for example the case of . A composition of transcendental functions can give an algebraic function: .
As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field, such as the complex numbers), a polynomial equation does not implicitly define a single function, but up to n
functions, sometimes also called branches. Consider for example the equation of the unit circle:
This determines y, except only up to an overall sign; accordingly, it has two branches:
An algebraic function in m variables is similarly defined as a function which solves a polynomial equation in m + 1 variables:
It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem.
Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1, ..., xm).
Algebraic functions in one variable
Introduction and overview
The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals.
First, note that any polynomial funct
|
https://en.wikipedia.org/wiki/Subfield
|
Subfield may refer to:
an area of research and study within an academic discipline
Field extension, used in field theory (mathematics)
a Division (heraldry)
a division in MARC standards
|
https://en.wikipedia.org/wiki/Abelian%20surface
|
In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety. This subvariety contains all of the points whose period matrices correspond to a period matrix of an abelian variety.
The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century.
Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4.
Hodge diamond:
Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.
See also
Hodge theory
Complex torus
References
Algebraic surfaces
Complex surfaces
|
https://en.wikipedia.org/wiki/Estimation%20of%20covariance%20matrices
|
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data, heteroscedasticity, or autocorrelated residuals require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.
Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles:
to provide initial estimates that can be used to study the inter-relationships;
to provide sample estimates that can be used for model checking.
Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample.
Estimation in a general context
Given a sample consisting of n independent observations x1,..., xn of a p-dimensional random vector X ∈ Rp×1 (a p×1 column-vector), an unbiased estimator of the (p×p) covariance matrix
is the sample covariance matrix
where is the i-th observation of the p-dimensional random vector, and the vector
is the sample mean.
This is true regardless of the distribution of the random variable X, provided of course that the theoretical means and covariances exist. The reason for the factor n − 1 rather than n is essentially the same as the reason for the same factor appearing in unbiased estimates of sample variances and sample covariances, which relates to the fact that the mean is not known and is replaced by the sample mean (see Bessel's correction).
In cases where the distribution of the random variable X is known to be within a certain family of distributions, other estimates may be derived on the basis of that assumption. A well-known instance is when the random variable X is normally distributed: in this case the maximum likelihood es
|
https://en.wikipedia.org/wiki/Monadic
|
Monadic may refer to:
Monadic, a relation or function having an arity of one in logic, mathematics, and computer science
Monadic, an adjunction if and only if it is equivalent to the adjunction given by the Eilenberg–Moore algebras of its associated monad, in category theory
Monadic, in computer programming, a feature, type, or function related to a monad (functional programming)
Monadic or univalent, a chemical valence
Monadic, in theology, a religion or philosophy possessing a concept of a divine Monad
See also
Monadic predicate calculus, in logic
Monad (disambiguation)
|
https://en.wikipedia.org/wiki/Simon%20Donaldson
|
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London.
Biography
Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge, and his mother earned a science degree there. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge, in 1979, and in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and later under Michael Atiyah's supervision. Still a postgraduate student, Donaldson proved in 1982 a result that would establish his fame. He published the result in a paper "Self-dual connections and the topology of smooth 4-manifolds" which appeared in 1983. In the words of Atiyah, the paper "stunned the mathematical world."
Whereas Michael Freedman classified topological four-manifolds, Donaldson's work focused on four-manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang–Mills gauge theory which has its origin in quantum field theory. One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all. Donaldson also derived polynomial invariants from gauge theory. These were new topological invariants sensitive to the underlying smooth structure of the four-manifold. They made it possible to deduce the existence of "exotic" smooth structures—certain topological four-manifolds could carry an infinite family of different smooth structures.
After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998 as Professor of Pure Mathematics.
In 2014, he joined the Simons Center for Geometry and Physics at Stony Brook University in New York, United States.
Awards
Donaldson was an invited speaker of the International Congress of Mathematicians (ICM) in 1983, and a plenary speaker at the ICM in 1986, 1998, and 2018.
In 1985, Donaldson received the Junior Whitehead Prize from the London Mathematical Society. In 1994, he was awarded the Crafoord Prize in Mathematics. In February 2006, Donaldson was awarded the King Faisal International Prize for science for his work in pure mathematical theories linked to physics, which have helped in forming an understanding of the laws
|
https://en.wikipedia.org/wiki/Divided%20differences
|
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.
Divided differences is a recursive division process. Given a sequence of data points , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.
Definition
Given n + 1 data points
where the are assumed to be pairwise distinct, the forward divided differences are defined as:
To make the recursive process of computation clearer, the divided differences can be put in tabular form, where the columns correspond to the value of j above, and each entry in the table is computed from the difference of the entries to its immediate lower left and to its immediate upper left, divided by a difference of corresponding x-values:
Notation
Note that the divided difference depends on the values and , but the notation hides the dependency on the x-values. If the data points are given by a function f,
one sometimes writes the divided difference in the notation
Other notations for the divided difference of the function ƒ on the nodes x0, ..., xn are:
Example
Divided differences for and the first few values of :
Properties
Linearity
Leibniz rule
Divided differences are symmetric: If is a permutation then
Polynomial interpolation in the Newton form: if is a polynomial function of degree , and is the divided difference, then
If is a polynomial function of degree , then
Mean value theorem for divided differences: if is n times differentiable, then for a number in the open interval determined by the smallest and largest of the 's.
Matrix form
The divided difference scheme can be put into an upper triangular matrix:
Then it holds
if is a scalar
This follows from the Leibniz rule. It means that multiplication of such matrices is commutative. Summarised, the matrices of divided difference schemes with respect to the same set of nodes x form a commutative ring.
Since is a triangular matrix, its eigenvalues are obviously .
Let be a Kronecker delta-like function, that is Obviously , thus is an eigenfunction of the pointwise function multiplication. That is is somehow an "eigenmatrix" of : . However, all columns of are multiples of each other, the matrix rank of is 1. So you can compose the matrix of all eigenvectors of from the -th column of each . Denote the matrix of eigenvectors with . Example The diagonalization of can be written as
Polynomials and power series
The matrix
contains the divided difference scheme for the identity function with respect to the nodes , thus contains the divided differences for the power function with exponent .
Consequently, you can obtain the divided differences for a polynomial function by applying to the matrix : If
and
then
This is known as Opitz' formula.
Now consider increasing
|
https://en.wikipedia.org/wiki/Monomial%20basis
|
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
One indeterminate
The polynomial ring of univariate polynomials over a field is a -vector space, which has
as an (infinite) basis. More generally, if is a ring then is a free module which has the same basis.
The polynomials of degree at most form also a vector space (or a free module in the case of a ring of coefficients), which has as a basis.
The canonical form of a polynomial is its expression on this basis:
or, using the shorter sigma notation:
The monomial basis is naturally totally ordered, either by increasing degrees
or by decreasing degrees
Several indeterminates
In the case of several indeterminates a monomial is a product
where the are non-negative integers. As an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular is a monomial.
Similar to the case of univariate polynomials, the polynomials in form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the monomial basis.
The homogeneous polynomials of degree form a subspace which has the monomials of degree as a basis. The dimension of this subspace is the number of monomials of degree , which is
where is a binomial coefficient.
The polynomials of degree at most form also a subspace, which has the monomials of degree at most as a basis. The number of these monomials is the dimension of this subspace, equal to
In contrast to the univariate case, there is no natural total order of the monomial basis in the multivariate case. For problems which require choosing a total order, such as Gröbner basis computations, one generally chooses an admissible monomial order – that is, a total order on the set of monomials such that
and for every monomial
See also
Horner's method
Polynomial sequence
Newton polynomial
Lagrange polynomial
Legendre polynomial
Bernstein form
Chebyshev form
Algebra
Polynomials
|
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt%20operator
|
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator that acts on a Hilbert space and has finite Hilbert–Schmidt norm
where is an orthonormal basis. The index set need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm is identical to the Frobenius norm.
||·|| is well defined
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if and are such bases, then
If then As for any bounded operator, Replacing with in the first formula, obtain The independence follows.
Examples
An important class of examples is provided by Hilbert–Schmidt integral operators.
Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator.
The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional.
Given any and in , define by , which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator;
moreover, for any bounded linear operator on (and into ), .
If is a bounded compact operator with eigenvalues of , where each eigenvalue is repeated as often as its multiplicity, then is Hilbert–Schmidt if and only if , in which case the Hilbert–Schmidt norm of is .
If , where is a measure space, then the integral operator with kernel is a Hilbert–Schmidt operator and .
Space of Hilbert–Schmidt operators
The product of two Hilbert–Schmidt operators has finite trace-class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on .
They also form a Hilbert space, denoted by or , which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces
where is the dual space of .
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).
The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).
The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, is finite-dimensional.
Properties
Every Hilbert–Schmidt operator is a compact operator.
A bounded linear operator is Hilbert–Schmidt if and only if the same is true of the operator , in which case the Hilbert–Schmidt norms of T and |T| are equal.
Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators.
If and are Hilbert–Schmidt operators between Hilb
|
https://en.wikipedia.org/wiki/Whitehead%20theorem
|
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.
Statement
In more detail, let X and Y be topological spaces. Given a continuous mapping
and a point x in X, consider for any n ≥ 1 the induced homomorphism
where πn(X,x) denotes the n-th homotopy group of X with base point x. (For n = 0, π0(X) just means the set of path components of X.) A map f is a weak homotopy equivalence if the function
is bijective, and the homomorphisms f* are bijective for all x in X and all n ≥ 1. (For X and Y path-connected, the first condition is automatic, and it suffices to state the second condition for a single point x in X.) The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.
Combining this with the Hurewicz theorem yields a useful corollary: a continuous map between simply connected CW complexes that induces an isomorphism on all integral homology groups is a homotopy equivalence.
Spaces with isomorphic homotopy groups may not be homotopy equivalent
A word of caution: it is not enough to assume πn(X) is isomorphic to πn(Y) for each n in order to conclude that X and Y are homotopy equivalent. One really needs a map f : X → Y inducing an isomorphism on homotopy groups. For instance, take X= S2 × RP3 and Y= RP2 × S3. Then X and Y have the same fundamental group, namely the cyclic group Z/2, and the same universal cover, namely S2 × S3; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the Künneth formula); thus, X and Y are not homotopy equivalent.
The Whitehead theorem does not hold for general topological spaces or even for all subspaces of Rn. For example, the Warsaw circle, a compact subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence. The study of possible generalizations of Whitehead's theorem to more general spaces is part of the subject of shape theory.
Generalization to model categories
In any model category, a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence.
References
J. H. C. Whitehead, Combinatorial homotopy. I., Bull. Amer. Math. Soc., 55 (1949), 213–245
J. H. C. Whitehead, Co
|
https://en.wikipedia.org/wiki/Cupola%20%28disambiguation%29
|
A cupola is a relatively small, most often dome-like, tall structure on top of a building.
Cupola may also refer to:
Science, mathematics, and technology
Cupola (cave formation), a recess in the ceiling of a lava tube
Cupola (geology), a type of igneous rock intrusion
Cupola (geometry), a geometric solid
Cupola (ISS module), an observation dome on the International Space Station
Cupola (military), a small gun turret mounted on a larger one
Cupola gecko, a species of gecko
Cupola sign, in medicine, a radiologic sign
Cupola furnace, a variety of small blast furnace
Reverberatory furnace, for smelting some non-ferrous metals
Cupola, an observation area on top of a railway caboose
Other uses
Sicilian Mafia Commission or Cupola, a body of Sicilian Mafia leaders
The Cupola (mountain), Tasmania, Australia
The Cupola, the yearbook of Western New England University, Springfield, Massachusetts, US
"Cupola", a 2001 song by Zeromancer from Eurotrash
See also
Cupola House (disambiguation)
Copala (disambiguation)
Coppola (disambiguation)
Copula (disambiguation)
Cupula (disambiguation)
|
https://en.wikipedia.org/wiki/Truncated%20power%20function
|
In mathematics, the truncated power function with exponent is defined as
In particular,
and interpret the exponent as conventional power.
Relations
Truncated power functions can be used for construction of B-splines.
is the Heaviside function.
where is the indicator function.
Truncated power functions are refinable.
See also
Macaulay brackets
External links
Truncated Power Function on MathWorld
References
Numerical analysis
|
https://en.wikipedia.org/wiki/Semi-local%20ring
|
In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R.
The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".
Some literature refers to a commutative semi-local ring in general as a
quasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals.
A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.
Examples
Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
The quotient is a semi-local ring. In particular, if is a prime power, then is a local ring.
A finite direct sum of fields is a semi-local ring.
In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m1, ..., mn
.
(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=J(R), and we see that R/J(R) is indeed a semisimple ring.
The classical ring of quotients for any commutative Noetherian ring is a semilocal ring.
The endomorphism ring of an Artinian module is a semilocal ring.
Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ pi), where the pi are finitely many prime ideals.
Textbooks
Ring theory
Localization (mathematics)
|
https://en.wikipedia.org/wiki/Triple%20product
|
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
Scalar triple product
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
Geometric interpretation
Geometrically, the scalar triple product
is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.
Properties
The scalar triple product is unchanged under a circular shift of its three operands (a, b, c):
Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product:
Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product:
The scalar triple product can also be understood as the determinant of the matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose):
If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume.
If any two vectors in the scalar triple product are equal, then its value is zero:
Also:
The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a Gram determinant.
The ratio of the triple product and the product of the three vector norms is known as a polar sine:which ranges between -1 and 1.
Scalar or pseudoscalar
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the orientation of the frame or the parity of the permutation of the vectors. This means the product is negated if the orientation is reversed, for example by a parity transformation, and so is more properly described as a pseudoscalar if the orientation can change.
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product (of vectors)
|
https://en.wikipedia.org/wiki/Concepts%20of%20Modern%20Mathematics
|
Concepts of Modern Mathematics is a book by mathematician and science popularizer Ian Stewart about then-recent developments in mathematics. It was originally published by Penguin Books in 1975, updated in 1981, and reprinted by Dover publications in 1995 and 2015.
Overview
The book arose out of an extramural class that Ian Stewart taught at the University of Warwick about "Modern mathematics". In the 1995 Dover edition Stewart wrote that the aim of the class was:
to explain why the underlying abstract point of view had gained currency among research mathematicians, and to examine how it opened up entirely new realms of mathematical thought.
The book is aimed at non-mathematicians. However, there are frequent equations and diagrams and the level of presentation is more technical than some of Stewart's other popular books such as Flatterland. Topics covered include analytic geometry, set theory, abstract algebra, group theory, topology, and probability.
References
Books by Ian Stewart (mathematician)
1975 non-fiction books
Mathematics books
|
https://en.wikipedia.org/wiki/Enrico%20Betti
|
Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations, giving early expositions of Galois theory. He also discovered Betti's theorem, a result in the theory of elasticity.
Biography
Betti was born in Pistoia, Tuscany. He graduated from the University of Pisa in 1846 under (1792–1857). In Pisa, he was also a student of Ottaviano-Fabrizio Mossotti and Carlo Matteucci. After a time teaching, he held an appointment there from 1857. In 1858 he toured Europe with Francesco Brioschi and Felice Casorati, meeting Bernhard Riemann. Later he worked in the area of theoretical physics opened up by Riemann's work. He was also closely involved in academic politics, and the politics of the new Italian state.
Works
E. Betti, Sopra gli spazi di un numero qualunque di dimensioni, Ann. Mat. Pura Appl. 2/4 (1871), 140–158. (Betti's most well known paper).
Opere matematiche di Enrico Betti, pubblicate per cura della R. Accademia de' lincei (2vols.) (U. Hoepli, Milano, 1903–1913)
See also
Betti cohomology
Betti group
Betti numbers
Notes
Further reading
External links
An Italian short biography of Enrico Betti in Edizione Nazionale Mathematica Italiana online.
1823 births
1892 deaths
People from Pistoia
19th-century Italian mathematicians
Topologists
University of Pisa alumni
Academic staff of the University of Pisa
Members of the Göttingen Academy of Sciences and Humanities
|
https://en.wikipedia.org/wiki/Nuisance%20parameter
|
In statistics, a nuisance parameter is any parameter which is unspecified but which must be accounted for in the hypothesis testing of the parameters which are of interest.
The classic example of a nuisance parameter comes from the normal distribution, a member of the location–scale family. For at least one normal distribution, the variance(s), σ2 is often not specified or known, but one desires to hypothesis test on the mean(s). Another example might be linear regression with unknown variance in the explanatory variable (the independent variable): its variance is a nuisance parameter that must be accounted for to derive an accurate interval estimate of the regression slope, calculate p-values, hypothesis test on the slope's value; see regression dilution.
Nuisance parameters are often scale parameters, but not always; for example in errors-in-variables models, the unknown true location of each observation is a nuisance parameter. A parameter may also cease to be a "nuisance" if it becomes the object of study, is estimated from data, or known.
Theoretical statistics
The general treatment of nuisance parameters can be broadly similar between frequentist and Bayesian approaches to theoretical statistics. It relies on an attempt to partition the likelihood function into components representing information about the parameters of interest and information about the other (nuisance) parameters. This can involve ideas about sufficient statistics and ancillary statistics. When this partition can be achieved it may be possible to complete a Bayesian analysis for the parameters of interest by determining their joint posterior distribution algebraically. The partition allows frequentist theory to develop general estimation approaches in the presence of nuisance parameters. If the partition cannot be achieved it may still be possible to make use of an approximate partition.
In some special cases, it is possible to formulate methods that circumvent the presences of nuisance parameters. The t-test provides a practically useful test because the test statistic does not depend on the unknown variance but only the sample variance. It is a case where use can be made of a pivotal quantity. However, in other cases no such circumvention is known.
Practical statistics
Practical approaches to statistical analysis treat nuisance parameters somewhat differently in frequentist and Bayesian methodologies.
A general approach in a frequentist analysis can be based on maximum likelihood-ratio tests. These provide both significance tests and confidence intervals for the parameters of interest which are approximately valid for moderate to large sample sizes and which take account of the presence of nuisance parameters. See Basu (1977) for some general discussion and Spall and Garner (1990) for some discussion relative to the identification of parameters in linear dynamic (i.e., state space representation) models.
In Bayesian analysis, a generally applicable approach cre
|
https://en.wikipedia.org/wiki/Marginal%20likelihood
|
A marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample from a prior and is therefore often referred to as model evidence or simply evidence.
Concept
Given a set of independent identically distributed data points where according to some probability distribution parameterized by , where itself is a random variable described by a distribution, i.e. the marginal likelihood in general asks what the probability is, where has been marginalized out (integrated out):
The above definition is phrased in the context of Bayesian statistics in which case is called prior density and is the likelihood. The marginal likelihood quantifies the agreement between data and prior in a geometric sense made precise in de Carvalho et al. (2019). In classical (frequentist) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter , where is the actual parameter of interest, and is a non-interesting nuisance parameter. If there exists a probability distribution for , it is often desirable to consider the likelihood function only in terms of , by marginalizing out :
Unfortunately, marginal likelihoods are generally difficult to compute. Exact solutions are known for a small class of distributions, particularly when the marginalized-out parameter is the conjugate prior of the distribution of the data. In other cases, some kind of numerical integration method is needed, either a general method such as Gaussian integration or a Monte Carlo method, or a method specialized to statistical problems such as the Laplace approximation, Gibbs/Metropolis sampling, or the EM algorithm.
It is also possible to apply the above considerations to a single random variable (data point) , rather than a set of observations. In a Bayesian context, this is equivalent to the prior predictive distribution of a data point.
Applications
Bayesian model comparison
In Bayesian model comparison, the marginalized variables are parameters for a particular type of model, and the remaining variable is the identity of the model itself. In this case, the marginalized likelihood is the probability of the data given the model type, not assuming any particular model parameters. Writing for the model parameters, the marginal likelihood for the model M is
It is in this context that the term model evidence is normally used. This quantity is important because the posterior odds ratio for a model M1 against another model M2 involves a ratio of marginal likelihoods, the so-called Bayes factor:
which can be stated schematically as
posterior odds = prior odds × Bayes factor
See also
Empirical Bayes methods
Lindley's paradox
Marginal probability
Bayesian information criterion
References
Charles S. Bos. "A comparison of marginal likelihood computation methods". In W. Härdle and B. Ronz, editors, COMPSTAT 2002: Proceedings
|
https://en.wikipedia.org/wiki/Combinatorial%20group%20theory
|
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation.
A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.
It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.
History
See for a detailed history of combinatorial group theory.
A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron.
The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein, in the early 1880s, who gave the first systematic study of groups by generators and relations.
References
|
https://en.wikipedia.org/wiki/Cauchy%20space
|
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.
Definition
Throughout, is a set, denotes the power set of and all filters are assumed to be proper/non-degenerate (i.e. a filter may not contain the empty set).
A Cauchy space is a pair consisting of a set together a family of (proper) filters on having all of the following properties:
For each the discrete ultrafilter at denoted by is in
If is a proper filter, and is a subset of then
If and if each member of intersects each member of then
An element of is called a Cauchy filter, and a map between Cauchy spaces and is Cauchy continuous if ; that is, the image of each Cauchy filter in is a Cauchy filter base in
Properties and definitions
Any Cauchy space is also a convergence space, where a filter converges to if is Cauchy. In particular, a Cauchy space carries a natural topology.
Examples
Any uniform space (hence any metric space, topological vector space, or topological group) is a Cauchy space; see Cauchy filter for definitions.
A lattice-ordered group carries a natural Cauchy structure.
Any directed set may be made into a Cauchy space by declaring a filter to be Cauchy if, given any element there is an element such that is either a singleton or a subset of the tail Then given any other Cauchy space the Cauchy-continuous functions from to are the same as the Cauchy nets in indexed by If is complete, then such a function may be extended to the completion of which may be written the value of the extension at will be the limit of the net. In the case where is the set of natural numbers (so that a Cauchy net indexed by is the same as a Cauchy sequence), then receives the same Cauchy structure as the metric space
Category of Cauchy spaces
The natural notion of morphism between Cauchy spaces is that of a Cauchy-continuous function, a concept that had earlier been studied for uniform spaces.
See also
References
Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.
General topology
|
https://en.wikipedia.org/wiki/Ideal%20quotient
|
In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set
Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).
(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.
Properties
The ideal quotient satisfies the following properties:
as -modules, where denotes the annihilator of as an -module.
(in particular, )
(as long as R is an integral domain)
Calculating the quotient
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f1, f2, f3) and J = (g1, g2) are ideals in k[x1, ..., xn], then
Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
Calculate a Gröbner basis for with respect to lexicographic order. Then the basis functions which have no t in them generate .
Geometric interpretation
The ideal quotient corresponds to set difference in algebraic geometry. More precisely,
If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then
where denotes the taking of the ideal associated to a subset.
If I and J are ideals in k[x1, ..., xn], with k an algebraically closed field and I radical then
where denotes the Zariski closure, and denotes the taking of the variety defined by an ideal. If I is not radical, then the same property holds if we saturate the ideal J:
where .
Examples
In ,
In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal of an integral domain is given by the ideal quotient .
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let in be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in . Then, the ideal quotient is the ideal of the z-plane in . This shows how the ideal quotient can be used to "delete" irreducible subschemes.
A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient , showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal the saturation of is defined as the ideal quotient where . It is a theorem that the set of saturated ideals of contained in is in bijection with the set of projective subschemes in . This shows us that defines th
|
https://en.wikipedia.org/wiki/Barry%20Greenstein
|
Barry Greenstein (born December 30, 1954, in Chicago, Illinois) is an American professional poker player and former mathematics postgraduate student. He has won a number of major events, including three at the World Series of Poker and two on the World Poker Tour. Greenstein donates his profit from tournament winnings to charities, primarily Children Incorporated, earning him the nickname "the Robin Hood of poker". He was elected into the Poker Hall of Fame in 2011.
Personal life
After graduating from Bogan High School, he earned a bachelor's degree in computer science from the University of Illinois at Urbana-Champaign. He studied for a PhD in mathematics without ever defending his completed dissertation.
According to his book, Ace on the River, Greenstein was doing well playing poker, but figured a more conventional job would improve his chances of adopting his stepchildren, so he went to work for the new startup company Symantec, where he worked on their first product Q&A. He left the company in 1991 at age 36.
Greenstein has two children and four stepchildren, and he resides in Rancho Palos Verdes, California. His stepson, Joe Sebok, with whom he started PokerRoad—a poker strategy and entertainment website—also played professionally for a few years.
Poker career
Greenstein has appeared in each of the first three series of Poker Superstars Invitational Tournament and all seven seasons of the GSN series High Stakes Poker. He has also appeared in the first three seasons of the late night series Poker After Dark on NBC. Greenstein has also won two World Poker Tour (WPT) titles: the Fifth Annual Jack Binion World Poker Open ($1,278,370), and 2006 WPT Invitational at the Commerce Casino ($100,000). He has also won two other WPT titles in special events: WPT Father and Sons Tournament, 2006 World Poker Finals; and WPT Poker by the Book: Chapter 2. As of 2008, he began donating his net winnings to charity. At the 2008 World Series of Poker (WSOP), Greenstein cashed in six events for a total of $768,461. He also finished second in the 2008 WSOP Player of the Year standings, behind Erick Lindgren. In 2011, Greenstein cashed five times at the World Series of Poker, including a 3rd place in the $10,000 Limit Hold'em Championship for $169,512 and a 15th place in the $50,000 Player's Championship.
In 2005, Greenstein wrote a book titled Ace on the River. Tuan Le, a WPT Season 3 champion, claims that book showed him how to succeed on the poker circuit. Greenstein is known for giving away a free copy of his book to the player who eliminates him in a tournament, including his autograph and details of the hand. Greenstein taught former girlfriend Mimi Tran how to play poker in exchange for her teaching him how to speak Vietnamese. He also played online at PokerStars, under the alias "barryg1" and was a member of the cardroom's Team PokerStars.
He accepted a bet of $10,000 from 2+2 Poker Forums to say "lol donkaments" on an episode of High Stakes Poker
|
https://en.wikipedia.org/wiki/Integer%20square%20root
|
In number theory, the integer square root (isqrt) of a non-negative integer is the non-negative integer which is the greatest integer less than or equal to the square root of ,
For example,
Introductory remark
Let and be non-negative integers.
Algorithms that compute (the decimal representation of) run forever on each input which is not a perfect square.
Algorithms that compute do not run forever. They are nevertheless capable of computing up to any desired accuracy .
Choose any and compute .
For example (setting ):
Compare the results with
It appears that the multiplication of the input by gives an accuracy of decimal digits.
To compute the (entire) decimal representation of , one can execute an infinite number of times, increasing by a factor at each pass.
Assume that in the next program () the procedure is already defined and — for the sake of the argument — that all variables can hold integers of unlimited magnitude.
Then will print the entire decimal representation of .
// Print sqrt(y), without halting
void sqrtForever(unsigned int y)
{
unsigned int result = isqrt(y);
printf("%d.", result); // print result, followed by a decimal point
while (true) // repeat forever ...
{
y = y * 100; // theoretical example: overflow is ignored
result = isqrt(y);
printf("%d", result % 10); // print last digit of result
}
}
The conclusion is that algorithms which compute are computationally equivalent to algorithms which compute .
Basic algorithms
The integer square root of a non-negative integer can be defined as
For example, because .
Algorithm using linear search
The following C-programs are straightforward implementations.
Linear search using addition
In the program above (linear search, ascending) one can replace multiplication by addition, using the equivalence
// Integer square root
// (linear search, ascending) using addition
unsigned int isqrt(unsigned int y)
{
unsigned int L = 0;
unsigned int a = 1;
unsigned int d = 3;
while (a <= y)
{
a = a + d; // (a + 1) ^ 2
d = d + 2;
L = L + 1;
}
return L;
}
Algorithm using binary search
Linear search sequentially checks every value until it hits the smallest where .
A speed-up is achieved by using binary search instead. The following C-program is an implementation.
// Integer square root (using binary search)
unsigned int isqrt(unsigned int y)
{
unsigned int L = 0;
unsigned int M;
unsigned int R = y + 1;
while (L != R - 1)
{
M = (L + R) / 2;
if (M * M <= y)
L = M;
else
R = M;
}
return L;
}
Numerical example
For example, if one computes using binary search, one obtains the sequence
This computation takes 21 iteration steps, whereas linear search (ascending, starting from ) needs steps.
Algorithm using Newton's method
One way of calculating and is to use Heron's method, which is a special case of Newton's method, to find a solution for the equation , giving the iterative formula
The sequence conv
|
https://en.wikipedia.org/wiki/Selberg%20trace%20formula
|
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given by the trace of certain functions on .
The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the Poisson summation formula.
The case when is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the Arthur–Selberg trace formula.
When is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
Early history
Cases of particular interest include those for which the space is a compact Riemann surface . The initial publication in 1956 of Atle Selberg dealt with this case, its Laplacian differential operator and its powers. The traces of powers of a Laplacian can be used to define the Selberg zeta function. The interest of this case was the analogy between the formula obtained, and the explicit formulae of prime number theory. Here the closed geodesics on play the role of prime numbers.
At the same time, interest in the traces of Hecke operators was linked to the Eichler–Selberg trace formula, of Selberg and Martin Eichler, for a Hecke operator acting on a vector space of cusp forms of a given weight, for a given congruence subgroup of the modular group. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the Riemann–Roch theorem.
Applications
The trace formula has applications to arithmetic geometry and number theory. For instance, using the trace theorem, Eichler and Shimura calculated the Hasse–Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula. The development of parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps c
|
https://en.wikipedia.org/wiki/Ehresmann%27s%20lemma
|
In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping , where and are smooth manifolds, is
a surjective submersion, and
a proper map (in particular, this condition is always satisfied if M is compact),
then it is a locally trivial fibration. This is a foundational result in differential topology due to Charles Ehresmann, and has many variants.
See also
Thom's first isotopy lemma
References
Theorems in differential topology
|
https://en.wikipedia.org/wiki/Singular%20point%20of%20an%20algebraic%20variety
|
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
Definition
A plane curve defined by an implicit equation
,
where is a smooth function is said to be singular at a point if the Taylor series of has order at least at this point.
The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation
whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exist or a special definition must be provided.
In general for a hypersurface
the singular points are those at which all the partial derivatives simultaneously vanish. A general algebraic variety being defined as the common zeros of several polynomials, the condition on a point of to be a singular point is that the Jacobian matrix of the first order partial derivatives of the polynomials has a rank at that is lower than the rank at other points of the variety.
Points of that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers).
In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a manifold near every regular point. But it is important to note that a real variety may be a manifold and have singular points. For example the equation defines a real analytic manifold but has a singular point at the origin. This may be explained by saying that the curve has two complex conjugate branches that cut the real branch at the origin.
Singular points of smooth mappings
As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of smooth mappings (functions from to where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the jets of the mapping. The th jet is the Taylor series of the mapping truncated at degree and deleting the constant term.
Nodes
In classical algebraic geometry, certain special singular points were also called nodes. A node is a singular point where the Hessian matrix is non-singular; this implies that the singular point has multiplicity two and the
|
https://en.wikipedia.org/wiki/L1
|
L1, L01, L.1, L 1 or L-1 may refer to:
Mathematics, science and technology
Math
L1 distance in mathematics, used in taxicab geometry
L1, the space of Lebesgue integrable functions
ℓ1, the space of absolutely convergent sequences
Science
L1 family, a protein family of cell adhesion molecules
L1 (protein), a cell adhesion molecule
L1 or LINE1; transposable elements in the DNA
, Lagrangian point 1, the most intuitive position for an object to be gravitationally stationary relative to two larger objects (such as a satellite with respect to the Earth and Moon)
Anthranilic acid, also called vitamin L1
The first lumbar vertebra of the vertebral column in human anatomy
The first larval stage in the Caenorhabditis elegans worm development
Technology
L1, one of the frequencies used by GPS systems (see GPS frequencies)
L1, the common name for the Soviet space effort known formally as Soyuz 7K-L1, designed to launch men from the Earth to circle the Moon without going into lunar orbit
ISO/IEC 8859-1 (Latin-1), an 8-bit character encoding
An L-carrier cable system developed by AT&T
The level-1 CPU cache in a computer
Sony Xperia L1, an Android smartphone
A class of FM broadcast station in North America
Transportation and military
Lehrgeschwader 1, from its historic Geschwaderkennung code with the Luftwaffe in World War II
Lufthansa Systems' IATA code
Lawrance L-1, a predecessor of the 1920s American Lawrance J-1 aircraft engine
Inner West Light Rail, a light rail service in Sydney, Australia, numbered L1
L1 (New York City bus), a temporary bus route in New York City
Volkswagen L1, a Volkswagen concept hybrid car
L1A1 Self-Loading Rifle
Locomotives
Erie L-1, an American 0-8-8-0 steam locomotive class
GCR Class 1B, latterly known as LNER Class L1, a class of British 2-6-4T steam locomotives
GNR Class L1, latterly known as LNER Class R1, a class of British 0-8-2T steam locomotives
LNER Thompson Class L1, a class of British 2-6-4T steam locomotives
NCC Class L1, a Northern Counties Committee 0-6-0 class steam locomotive
Pennsylvania Railroad class L1s, an American 2-8-2 steam locomotive class
Soo Line L-1 class, an American 2-8-2 steam locomotive class
SP&S Class L-1, an American 4-4-0 steam locomotive class
SR L1 class, a class of 4-4-0 steam locomotives of Great Britain
VR Class Vr1 (originally L1), a Finnish steam locomotive class
Aircraft
Arado L 1, a 1929 German two-seat parasol-wing sporting monoplane
Macchi L.1, a 1915 Austro-Hungarian reconnaissance flying boat
Stinson L-1 Vigilant, the US Army Air Forces designation for the Stinson Model 74 observation aircraft
Submarines
USS L-1 (SS-40), a 1915 United States Navy L-class submarine
HMS L1, a Royal Navy submarine
Other
Ligue 1, the top division of French football
LINE1 or L1, a transposable element in DNA
Lowest unique bid
L1, in linguistics, a subject's first language or mother tongue
L-1 Identity Solutions, a US face-recognition corporation
L-1 visa, a
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.