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passage: A "Hello, World!" program is usually a simple computer program that emits (or displays) to the screen (often the console) a message similar to "Hello, World!". A small piece of code in most general-purpose programming languages, this program is used to illustrate a language's basic syntax. Such a program is of... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: The example program from the book prints , and was inherited from a 1974 Bell Laboratories internal memorandum by Brian Kernighan, Programming in C: A Tutorial:
```c
main( ) {
printf("hello, world");
}
```
In the above example, the function defines where the program should start executing. The functio... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: Outside computing, use of the exact phrase began over a decade prior; it was the catchphrase of New York radio disc jockey William B. Williams beginning in the 1950s.
## Variations
"Hello, World!" programs vary in complexity between different languages. In some languages, particularly scripting languages, the... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: Other human languages have been used as the output; for example, a tutorial for the Go language emitted both English and Chinese or Japanese characters, demonstrating the language's built-in Unicode support. Another notable example is the Rust language, whose management system automatically inserts a "Hello, W... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: The Debian and Ubuntu Linux distributions provide the "Hello, World!" program through their software package manager systems, which can be invoked with the command . It serves as a sanity check and a simple example of installing a software package. For developers, it provides an example of creating a .deb pack... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: This is one measure of a programming language's ease of use. Since the program is meant as an introduction for people unfamiliar with the language, a more complex "Hello, World!" program may indicate that the programming language is less approachable. For instance, the first publicly known "Hello, World!" prog... | https://en.wikipedia.org/wiki/%22Hello%2C_World%21%22_program |
passage: In mathematics, and more specifically in abstract algebra, a
### *-algebra
(or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the id... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: and so on.
*-rings are unrelated to star semirings in the theory of computation.
-algebra
A *-algebra is a *-ring, with involution * that is an associative algebra over a commutative... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: ## Examples
- Any commutative ring becomes a *-ring with the trivial (identical) involution.
- The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers where * is just complex conjugation.
- More generally, a field extension made by adjunction of a square root (such ... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: - Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
- The polynomial ring over a commutative trivially-*-ring is a *-algebra over with .
- If is simultaneously a *-ring, an algebra over a ring (commutative), and , then is a... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: - The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties)... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra:
$$
\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}
$$
Any nontrivial antiautomorphism necessarily has the form:
$$
\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: Consider the following subalgebra:
$$
\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}
$$
Any nontrivial antiautomorphism necessarily has the form:
$$
\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.
### Skew structures
Given a *-ring, there is also the map .
It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), a... | https://en.wikipedia.org/wiki/%2A-algebra |
passage: The (EGA; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the . In it, Grothendieck established systematic foundations of algebraic ge... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rig... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted t... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: III Étude cohomologique des faisceaux cohérents Cohomologie des Faisceaux algébriques cohérents. Applications. First edition complete except for last four sections, intended for publication after Chapter IV: elementary projective duality, local cohomology and its relation to projective cohomology, and Picard g... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: Treated in detail in SGA III. VIII Étude différentielle des espaces fibrés Le schéma de Picard Did not appear. Material apparently intended for first edition can be found in SGA III, construction and results on Picard scheme are summarised in FGA. IX Le groupe fondamental Le groupe fondamental Did not appear. ... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.
Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHÉS in 1970 and soon a... | https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique |
passage: Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of
### Lie groups
, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity an... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Élie had an elder sister Jeanne-Marie (1867–1931) who became a dressmaker; a younger brother Léon (1872–1956) who became a blacksmith working in his father's smithy; and a younger sister Anna Cartan (1878–1923), who, partly under Élie's influence, entered École Normale Supérieure (as Élie had before) and chose... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: He spent five years (1879–1884) at the College of Vienne and then two years (1884–1886) at the Lycée of Grenoble. In 1886 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years; there he met and befriended his classmate Jean-Baptiste Perrin (1870–1942) who later became a famous physici... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, and met Cartan for the first time.
Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the Universi... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: He would say, “After you left, I thought more about your questions...”—he had some results, and some more questions, and so on. He knew all these papers on simple Lie groups, Lie algebras, all by heart. When you saw him on the street, when a certain issue would come up, he would pull out some old envelope and ... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Riemannian geometry
1. Symmetric spaces
1. Topology of compact groups and their homogeneous spaces
1. Integral invariants and classical mechanics
1. Relativity, spinors
Cartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now consider the central an... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: A very fruitful approach to the study of these groups was opened in 1888 when Wilhelm Killing systematically started to study the group in itself, independent of its possible actions on other manifolds. At that time (and until 1920) only local properties were considered, so the main object of study for Killing... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: It was in the process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important role in quantum mechanics.
After 1925 Cartan grew more and more interested in topological questions. Spurred by Weyl's brilliant results on ... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: The Lie pseudogroup considered by Cartan is a set of transformations between subsets of a space that contains the identical transformation and possesses the property that the result of composition of two transformations in this set (whenever this is possible) belongs to the same set. Since the composition of t... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Such pseudogroups of transformations are called primitive. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations belongs to one of the six classes: 1) the pseudogroup of all analytic transformations of n complex variables; 2) the pseudogroup of all analytic tra... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Breaking with tradition, he sought from the start to formulate and solve the problems in a completely invariant fashion, independent of any particular choice of variables and unknown functions. He thus was able for the first time to give a precise definition of what is a "general" solution of an arbitrary diff... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight.
Differential geometry
Cartan's contributions to differential geometry are no less impressive, and it may be said that he revitalized the whole subje... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Cartan's ability to handle many other types of fibers and groups allows one to credit him with the first general idea of a fiber bundle, although he never defined it explicitly. This concept has become one of the most important in all fields of modern mathematics, chiefly in global differential geometry and in... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: Symmetric Riemann spaces may be defined in various ways, the simplest of which postulates the existence around each point of the space of a "symmetry" that is involutive, leaves the point fixed, and preserves distances. The unexpected fact discovered by Cartan is that it is possible to give a complete descript... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: ## Publications
Cartan's papers have been collected in his Oeuvres complètes, 6 vols. (Paris, 1952–1955). Two excellent obituary notices are S. S. Chern and C. Chevalley, in Bulletin of the American Mathematical Society, 58 (1952); and J. H. C. Whitehead, in Obituary Notices of the Royal Society (1952).
-
-
... | https://en.wikipedia.org/wiki/%C3%89lie_Cartan |
passage: In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomo... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results s... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in m... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology. However, for co... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space. He defined étale cohomology by replacing the category o... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: ## Definitions
For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: The idea of derived functor here is that the functor of sections doesn't respect exact sequences as it is not right exact; according to general principles of homological algebra there will be a sequence of functors H 0, H 1, ... that represent the 'compensations' that must be made in order to restore some meas... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.
## ℓ-adic cohomology groups
In applications to algebraic geometry o... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Étale cohomology works fine for coefficients Z/nZ for n co-prime to p, but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-ad... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-adic cohomology, where ℓ stands for any prime number different from p. One considers, for schemes V, the cohomology groups
$$
H^i(V, \ma... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: An ℓ-adic sheaf is a special sort of inverse system of étale sheaves Fi, where i runs through positive integers, and Fi is a module over Z/ℓi Z and the map from Fi+1 to Fi is just reduction mod Z/ℓi Z.
When V is a non-singular algebraic curve of genus g, H1 is a free Zℓ-module of rank 2g, dual to the Tate mo... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: It also shows one reason why the condition ℓ ≠ p is required: when ℓ = p the rank of the Tate module is at most g.
Torsion subgroups can occur, and were applied by Michael Artin and David Mumford to geometric questions. To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology group... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds.
For example, the first cohomology group of a complex elliptic c... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: This phenomenon of Galois representations is related to the fact that the fundamental group of a topological space acts on the singular cohomology groups, because Grothendieck showed that the Galois group can be regarded as a sort of fundamental group. (See also Grothendieck's Galois theory.)
## Calculation o... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: For curves the calculation takes several steps, as follows . Let Gm denote the sheaf of non-vanishing functions.
### Calculation of H1(X, Gm)
The exact sequence of étale sheaves
$$
1\to \mathbf{G}_m\to j_*\mathbf{G}_{m, K}\to \bigoplus_{x\in |X|}i_{x*}\mathbf{Z}\to 1
$$
gives a long exact sequence of cohom... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Let Gm denote the sheaf of non-vanishing functions.
### Calculation of H1(X, Gm)
The exact sequence of étale sheaves
$$
1\to \mathbf{G}_m\to j_*\mathbf{G}_{m, K}\to \bigoplus_{x\in |X|}i_{x*}\mathbf{Z}\to 1
$$
gives a long exact sequence of cohomology groups
$$
\begin{align}
0 &\to H^0(\mathbf{G}_m)\to H... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: so their sum is just the divisor group of X. Moreover, the first cohomology group H 1(X, j∗Gm,K) is isomorphic to the Galois cohomology group H 1(K, K*) which vanishes by Hilbert's theorem 90. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence
$$
K\to \operatorname{Div}(X)\to... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Tsen's theorem implies that the Brauer group of a function field K in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1, so all the cohomology groups H i(X, Gm) vanish if i ≥ 2.
### Calculation of Hi(X, μn)
If μn i... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: ### Calculation of Hi(X, μn)
If μn is the sheaf of n-th roots of unity and n and the characteristic of the field k are coprime integers, then:
$$
H^i (X, \mu_n) = \begin{cases} \mu_n(k) & i =0 \\ \operatorname{Pic}_n(X) & i = 1 \\ \mathbf{Z}/n\mathbf{Z} & i =2 \\ 0 & i \geqslant 3 \end{cases}
$$
where Picn(X)... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: &\to \cdots
\end{align}
$$
of the Kummer exact sequence of étale sheaves
$$
1 \to \mu_n \to \mathbf{G}_m \xrightarrow{(\cdot)^n} \mathbf{G}_m \to 1.
$$
and inserting the known values
$$
H^i (X, \mathbf{G}_m) = \begin{cases} k^* & i = 0 \\ \operatorname{Pic}(X) & i =1 \\ 0 &i \geqslant 2 \end{cases}
$$
In ... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an n-th root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.
### Calculation of H i(X, Z/nZ)
By fi... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: This follows from the previous result, using the fact that the Picard group of a curve is the points of its Jacobian variety, an abelian variety of dimension g, and if n is coprime to the characteristic then the points of order dividing n in an abelian variety of dimension g over an algebraically closed field ... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: (For coefficients in Z/pnZ there is a similar sequence involving Witt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.
## Examples of étale cohomology groups
- If X is the spectrum of a field K with absolute Galois group G, then éta... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers. For curves the first cohomology group is the first cohomology group of its Jacobian. This explains why Weil was able to give a more elementary proof of the ... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: ## Poincaré duality and cohomology with compact support
The étale cohomology groups with compact support of a variety X are defined to be
$$
H_c^q(X, F) = H^q(Y, j_!F)
$$
where j is an open immersion of X into a proper variety Y and j! is the extension by 0 of the étale sheaf F to Y. This is independent of the... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: More generally if f is a separated morphism of finite type from X to S (with X and S Noetherian) then the higher direct images with compact support Rqf! are defined by
$$
R^qf_!(F)=R^qg_*(j_!F)
$$
for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to S (with f... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.
If X is a smooth algebraic variety of dimension N and ... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: Let be a curve of genus defined over , the finite field with elements. Then for
$$
\#X \left (\mathbf F_{p^n} \right ) = p^n + 1 -\sum_{i=1}^{2g} \alpha_i^n,
$$
where are certain algebraic numbers satisfying .
This agrees with being a curve of genus with points. It also shows that the number of poin... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: The points of that are defined over are those fixed by , where is the Frobenius automorphism in characteristic .
The étale cohomology Betti numbers of in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.
According to all of these,
$$
\#X \left (\mathbf F_{p^n} \right ) = \operatorname{Tr} \left (F^n|_{H^... | https://en.wikipedia.org/wiki/%C3%89tale_cohomology |
passage: In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
## Motivation
Let X be a topological space, and let
$$
\mathcal{U}
$$
be an open cover of X. Let ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: This is the approach adopted below.
## Construction
Let X be a topological space, and let
$$
\mathcal{F}
$$
be a presheaf of abelian groups on X. Let
$$
\mathcal{U}
$$
be an open cover of X.
### Simplex
A q-simplex σ of
$$
\mathcal{U}
$$
is an ordered collection of q+1 sets chosen from
$$
\mathcal{U}
... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Now let
$$
\sigma = (U_i)_{i \in \{ 0 , \ldots , q \}}
$$
be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$$
\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.
$$
The boundary of σ is defined as t... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$$
\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.
$$
The boundary of σ is defined as the alternating sum of the partial boundaries:
$$
\partial \sigma := \sum_{j=0}^q ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: ### Cochain
A q-cochain of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
is a map which associates with each q-simplex σ an element of
$$
\mathcal{F}(|\sigma|)
$$
, and we denote the set of all q-cochains of
$$
\mathcal{U}
$$
with coefficients in
$$
\mathcal{F}
$$
by
$$
C^q(\mathcal U, \ma... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: A calculation shows that
$$
\delta_{q+1} \circ \delta_q = 0.
$$
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called
the differential of the cochain complex.
#### Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of
$$
\delta
$$
, henc... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: #### Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of
$$
\delta
$$
, hence
$$
Z^q(\mathcal{U}, \mathcal{F}) := \ker ( \delta_q) \subseteq C^q(\mathcal U, \mathcal F)
$$
is the set of all q-cocycles.
Thus a (q−1)-cochain
$$
f
$$
is a cocycle if for all q-simplices
$$
\sigma
$$
the coc... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Thus a (q−1)-cochain
$$
f
$$
is a cocycle if for all q-simplices
$$
\sigma
$$
the cocycle condition
$$
\sum_{j=0}^{q} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0
$$
holds.
A 0-cocycle
$$
f
$$
is a collection of local sections of
$$
\mathcal{F}
$$
satisfying a compa... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: For example, a 1-cochain
$$
f
$$
is a 1-coboundary if there exists a 0-cochain
$$
h
$$
such that for every intersecting
$$
A,B\in \mathcal{U}
$$
$$
f(A \cap B) = h(A)|_{A \cap B} - h(B)|_{A \cap B}
$$
### Cohomology
The Čech cohomology of
$$
\mathcal{U}
$$
with values in
$$
\mathcal{F}
$$
is defined ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Thus the qth Čech cohomology is given by
$$
\check{H}^q(\mathcal{U}, \mathcal{F}) := H^q((C^{\bullet}(\mathcal U, \mathcal F), \delta)) = Z^q(\mathcal{U}, \mathcal{F}) / B^q(\mathcal{U}, \mathcal{F})
$$
.
The Čech cohomology of X is defined by considering refinements of open covers. If
$$
\mathcal{V}
$$
is ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in is defined as the direct limit
$$
\check{H}(X,\mathcal F) := \varinjlim_{\mathcal U} \check{H}(\mathcal U,\mathcal F)
$$
of this system.
The Čech c... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: ## Relation to other cohomology theories
If X is homotopy equivalent to a CW complex, then the Čech cohomology
$$
\check{H}^{*}(X;A)
$$
is naturally isomorphic to the singular cohomology
$$
H^*(X;A) \,
$$
. If X is a differentiable manifold, then
$$
\check{H}^*(X;\R)
$$
is also naturally isomorphic to th... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: For example if X is the closed topologist's sine curve, then
$$
\check{H}^1(X;\Z)=\Z,
$$
whereas
$$
H^1(X;\Z)=0.
$$
If X is a differentiable manifold and the cover
$$
\mathcal{U}
$$
of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in
$$
\mathc... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: If X is paracompact Hausdorff, then
$$
\chi
$$
is an isomorphism. More generally,
$$
\chi
$$
is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.
## In algebraic geometry
Čech cohomology can be defined more generally for objects in a site C endowed with ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: ## In algebraic geometry
Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf
$$
\mathcal{F}
$$
is defined as
$$
\check H^n (X, \mathcal{F}) ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite se... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve
$$
N_X \mathcal U : \dots \to \mathcal U \times_X \mathcal U \times_X \mathcal U \to \mathcal U \times_X \mathcal U \to \mathcal U.
$$
A hyperco... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: (This group is the same as
$$
\check H^n(\mathcal U, \mathcal F)
$$
in case K∗ equals
$$
N_X \mathcal U
$$
.) Then, it can be shown that there is a canonical isomorphism
$$
H^n (X, \mathcal F) \cong \varinjlim_{K_*} H^n(\mathcal F(K_*)),
$$
where the colimit now runs over all hypercoverings.
### Examples
T... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: For example, we calculate the first Čech cohomology with values in
$$
\mathbb{R}
$$
of the unit circle
$$
X=S^1
$$
. Dividing
$$
X
$$
into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover
$$
\mathcal{U}=\{U_0,U_1,U_2\}
$$
where
$$
U_i \cap U_j \ne \emptyset
$$
but... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Dividing
$$
X
$$
into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover
$$
\mathcal{U}=\{U_0,U_1,U_2\}
$$
where
$$
U_i \cap U_j \ne \emptyset
$$
but
$$
U_0 \cap U_1 \cap U_2 = \emptyset
$$
.
Given any 1-cocycle
$$
f
$$
,
$$
\delta f
$$
is a 2-cochain which takes ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Thus,
$$
Z^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j)\} \cong \mathbb{R}^3.
$$
On the other hand, given any 1-coboundary
$$
f = \delta g
$$
, we have
$$
\begin{cases}
f(U_i,U_i)=g(U_i)-g(U_i)=0 & (i=0,1,2); \\
f(U_i,U_j)=g(U_j)-g(U_i)=-f(U_j,U_i) & (i ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Thus,
$$
Z^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j)\} \cong \mathbb{R}^3.
$$
On the other hand, given any 1-coboundary
$$
f = \delta g
$$
, we have
$$
\begin{cases}
f(U_i,U_i)=g(U_i)-g(U_i)=0 & (i=0,1,2); \\
f(U_i,U_j)=g(U_j)-g(U_i)=-f(U_j,U_i) & (i ... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: $$
and
$$
f(U_1,U_2)
$$
. This gives the set of 1-coboundaries:
$$
\begin{align}
B^1(\mathcal{U},\mathbb{R})=\{f \in C^1(\mathcal{U},\mathbb{R}) : \ & f(U_i,U_i)=0, f(U_j,U_i)=-f(U_i,U_j), \\
&f(U_0,U_2)=f(U_0,U_1)+f(U_1,U_2)\} \cong \mathbb{R}^2.
\end{align}
$$
Therefore,
$$
\check{H}^1(\mathcal{U},\mathbb... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Since
$$
\mathcal{U}
$$
is a good cover of
$$
X
$$
, we have
$$
\check{H}^1(X,\mathbb{R}) \cong \mathbb{R}
$$
by Leray's theorem.
We may also compute the coherent sheaf cohomology of
$$
\Omega^1
$$
on the projective line
$$
\mathbb{P}^1_\mathbb{C}
$$
using the Čech complex. Using the cover
$$
\mathca... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: We may also compute the coherent sheaf cohomology of
$$
\Omega^1
$$
on the projective line
$$
\mathbb{P}^1_\mathbb{C}
$$
using the Čech complex. Using the cover
$$
\mathcal{U} = \{ U_1 = \text{Spec}(\Complex[y]), U_2 = \text{Spec}(\Complex[y^{-1}]) \}
$$
we have the following modules from the cotangent she... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: Using the cover
$$
\mathcal{U} = \{ U_1 = \text{Spec}(\Complex[y]), U_2 = \text{Spec}(\Complex[y^{-1}]) \}
$$
we have the following modules from the cotangent sheaf
$$
\begin{align}
&\Omega^1(U_1) = \Complex[y]dy \\
&\Omega^1(U_2) = \Complex \left [y^{-1} \right ]dy^{-1}
\end{align}
$$
If we take the conventio... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: then we get the Čech complex
$$
0 \to \Complex[y]dy \oplus \Complex \left [y^{-1} \right ]dy^{-1} \xrightarrow{d^0} \Complex \left [y,y^{-1} \right ]dy \to 0
$$
Since
$$
d^0
$$
is injective and the only element not in the image of
$$
d^0
$$
is
$$
y^{-1}dy
$$
we get that
$$
\begin{align}
&H^1(\mathbb{P}_{... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
passage: The μ-law algorithm (sometimes written mu-law, often abbreviated as u-law) is a companding algorithm, primarily used in 8-bit PCM digital telecommunications systems in North America and Japan. It is one of the two companding algorithms in the G.711 standard from ITU-T, the other being the similar A-law. A-law ... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: At the cost of a reduced peak SNR, it can be mathematically shown that μ-law's non-linear quantization effectively increases dynamic range by 33 dB or bits over a linearly-quantized signal, hence 13.5 bits (which rounds up to 14 bits) is the most resolution required for an input digital signal to be compresse... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: ### Discrete
The discrete form is defined in ITU-T Recommendation G.711.
G.711 is unclear about how to code the values at the limit of a range (e.g. whether +31 codes to 0xEF or 0xF0).
However, G.191 provides example code in the C language for a μ-law encoder. The difference between the positive and negative... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: The difference between the positive and negative ranges, e.g. the negative range corresponding to +30 to +1 is −31 to −2. This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
+ Quantized μ-law algor... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
+ Quantized μ-law algorithm 14-bit binary linear input code 8-bit compressed code +8158 to +4063 in 16 intervals of 256 0x80 + interval number ... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: Non-linear ADC Use an analog-to-digital converter with quantization levels which are unequally spaced to match the μ-law algorithm.
Digital Use the quantized digital version of the μ-law algorithm to convert data once it is in the digital domain.
Software/DSP
Use the continuous version of the μ-law algori... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: This pre-existing algorithm had the effect of significantly lowering the amount of bits required to encode a recognizable human voice in digital systems. A sample could be effectively encoded using μ-law in as little as 8 bits, which conveniently matched the symbol size of the majority of common computers.
μ-... | https://en.wikipedia.org/wiki/%CE%9C-law_algorithm |
passage: In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions.
## Definition
Suppose that R(y, x1, ..., xk... | https://en.wikipedia.org/wiki/%CE%9C_operator |
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