formulasearchengine - User contributions [en]

Heston model

2014-10-26T06:02:50Z

207.38.223.67: /* Risk-neutral measure */

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Cauchy's theorem (group theory)

2014-09-19T04:42:48Z

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2014-02-24T14:41:05Z

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Generalized mean

2014-02-11T13:10:22Z

207.38.151.173: /* Geometric mean */

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3SUM

2014-01-31T23:34:27Z

207.38.151.173: /* Quadratic algorithm */

In [[commutative algebra]], a '''regular local ring''' is a [[Noetherian]] [[local ring]] having the property that the minimal number of generators of its [[maximal ideal]] is equal to its [[Krull dimension]]. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by [[Krull's principal ideal theorem]] ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''.

The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an [[algebraic variety]] ''X'' is [[Singular point of an algebraic variety|nonsingular]] if and only if the local ring <math>\mathcal{O}_{X, x}</math> of [[germ (mathematics)|germs]] at ''x'' is regular. Regular local rings are ''not'' related to [[von Neumann regular ring]]s.<ref>A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.</ref>

==Characterizations==
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if <math>A</math> is a Noetherian local ring with maximal ideal <math>\mathfrak{m}</math>, then the following are equivalent definitions
* Let <math>\mathfrak{m} = (a_1, \ldots, a_n)</math> where <math>n</math> is chosen as small as possible. Then <math>A</math> is regular if
::<math>\mbox{dim } A = n\,</math>,
:where the dimension is the Krull dimension. The minimal set of generators of <math>a_1, \ldots, a_n</math> are then called a ''regular system of parameters''.
* Let <math>k = A / \mathfrak{m}</math> be the residue field of <math>A</math>. Then <math>A</math> is regular if
::<math>\dim_k \mathfrak{m} / \mathfrak{m}^2 = \dim A\,</math>,
:where the second dimension is the [[Krull dimension]].
* Let <math>\mbox{gl dim } A := \sup \{ \mbox{pd } M \mbox{ }|\mbox{ } M \mbox{ is an A-module} \}</math> be the [[global dimension]] of <math>A</math> (i.e., the supremum of the [[projective dimension]]s of all <math>A</math>-modules.) Then <math>A</math> is regular if
::<math>\mbox{gl dim } A < \infty\,</math>,
:in which case, <math>\mbox{gl dim } A = \dim A</math>.

==Examples==
# Every [[field (mathematics)|field]] is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
# Any [[discrete valuation ring]] is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Specifically, if ''k'' is a field and ''X'' is an indeterminate, then the ring of [[formal power series]] ''k''[[''X'']] is a regular local ring having (Krull) dimension 1.
# If ''p'' is an ordinary prime number, the ring of [[p-adic integer]]s is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field.
# More generally, if ''k'' is a field and ''X''1, ''X''2, ..., ''X''''d'' are indeterminates, then the ring of formal power series ''k''[[''X''1, ''X''2, ..., ''X''''d'']] is a regular local ring having (Krull) dimension ''d''.
# If ''A'' is a local ring, then it follows that the [[formal power series]] ring ''A''[[''x'']] is regular local.
# If '''Z''' is the ring of integers and ''X'' is an indeterminate, the ring '''Z'''[''X''](2, ''X'') is an example of a 2-dimensional regular local ring which does not contain a field.

==Basic properties==
The [[Auslander–Buchsbaum theorem]] states that every regular local ring is a [[unique factorization domain]].

Every [[localization of a ring|localization]] of a regular local ring is regular.

The [[completion (ring theory)|completion]] of a regular local ring is regular.

If <math>(A, \mathfrak{m})</math> is a complete regular local ring that contains a field, then
:<math>A \cong k[[x_1, \ldots, x_d]]</math>,
where <math>k = A / \mathfrak{m}</math> is the [[residue field]], and <math>d = \dim A</math>, the Krull dimension.

==Origin of basic notions==
Regular local rings were originally defined by [[Wolfgang Krull]] in 1937,<ref>{{Citation | last1=Krull | first1=Wolfgang | author1-link= Wolfgang Krull | title=Beiträge zur Arithmetik kommutativer Integritätsbereiche III | journal=Math. Z. | year=1937 | pages=745–766}}</ref> but they first became prominent in the work of [[Oscar Zariski]] a few years later,<ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic varieties over ground fields of characteristic 0 | journal=Amer. J. Math. | year=1940 | volume=62 | pages=187–221}}</ref><ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=The concept of a simple point of an abstract algebraic variety | journal=Trans. Amer. Math. Soc. | year=1947 | volume=62 | pages=1–52}}</ref> who showed that geometrically, a regular local ring corresponds to a smooth point on an [[algebraic variety]]. Let ''Y'' be an [[algebraic variety]] contained in affine ''n''-space over a perfect field, and suppose that ''Y'' is the vanishing locus of the polynomials ''f1'',...,''fm''. ''Y'' is nonsingular at ''P'' if ''Y'' satisfies a [[Jacobian]] condition: If ''M'' = (∂''fi''/∂''xj'') is the matrix of partial derivatives of the defining equations of the variety, then the rank of the matrix found by evaluating ''M'' at ''P'' is ''n'' − dim ''Y''. Zariski proved that ''Y'' is nonsingular at ''P'' if and only if the local ring of ''Y'' at ''P'' is regular. (Zariski observed that this can fail over non-perfect fields.) This implies that smoothness is an intrinsic property of the variety, in other words it does not depend on where or how the variety is embedded in affine space. It also suggests that regular local rings should have good properties, but before the introduction of techniques from [[homological algebra]] very little was known in this direction. Once such techniques were introduced in the 1950s, Auslander and Buchsbaum proved that every regular local ring is a [[unique factorization domain]].

Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular. Again, this lay unsolved until the introduction of homological techniques. However, [[Jean-Pierre Serre]] found a homological characterization of regular local rings: A local ring ''A'' is regular if and only if ''A'' has finite [[global dimension]]. It is easy to show that the property of having finite global dimension is preserved under localization, and consequently that localizations of regular local rings at prime ideals are again regular. This allows us to define regularity for all rings, not just local ones: A ring ''A'' is said to be a '''[[regular ring]]''' if its localizations at all of its prime ideals are regular local rings. It is equivalent to say that ''A'' has finite global dimension.

==Notes==
<references />

==References==
* [[Jean-Pierre Serre]], ''Local algebra'', [[Springer-Verlag]], 2000, ISBN 3-540-66641-9. Chap.IV.D.

{{DEFAULTSORT:Regular Local Ring}}
[[Category:Algebraic geometry]]
[[Category:Ring theory]]

Characteristic equation (calculus)

2014-01-29T21:41:38Z

207.38.151.173:

In [[mathematics]], a '''real closed ring''' is a [[commutative ring]] ''A'' that
is a subring of a product of [[real closed field]]s, which is closed under
continuous [[Semialgebraic set|semi-algebraic]] functions defined over the integers.

== Examples of real closed rings ==
Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first. The following rings are all real closed rings:
* [[real closed field]]s. These are exactly the real closed rings that are fields.
* the ring of all [[Tychonoff_space#Real-valued continuous functions|real valued continuous functions]] on a [[completely regular space]] ''X''. Also, the ring of all bounded real valued continuous functions on ''X'' is real closed.
* convex subrings of real closed fields. These are precisely those real closed rings which are also [[valuation ring]]s and were initially studied by Cherlin and Dickmann (they used the term 'real closed ring' for what is now called 'real closed valuation ring').
* the ring ''A'' of all continuous [[semialgebraic set|semi-algebraic function]]s on a semi-algebraic set of a real closed field (with values in that field). Also, the subring of all bounded (in any sense) functions in ''A'' is real closed.
* (generalizing the previous example) the ring of all (bounded) continuous definable functions on a [[definable set]] ''S'' of an arbitrary first-order [[expansion of a first-order structure|expansion]] ''M'' of a real closed field (with values in ''M''). Also, the ring of all (bounded) definable functions <math>S\to M</math> is real closed.
* Real closed rings are precisely the rings of [[global section]]s of affine real closed spaces (a generalization of [[semialgebraic space]]s) and in this context they were invented by Niels Schwartz in the early 1980s.

==Definition==
A real closed ring is a reduced, commutative unital ring ''A'' which has the following properties:
#The set of squares of ''A'' is the set of nonnegative elements of a partial order ≤ on ''A'' and ''(A,≤)'' is an [[f-ring]].
#Convexity condition: For all a,b from ''A'', if 0≤a≤b then b|a2.
#For every [[prime ideal]] ''p'' of ''A'', the [[residue class ring]] ''A/p'' is [[integrally closed]] and its [[field of fractions]] is a real closed field.
The link to the definition at the beginning of this article is given in the section on algebraic properties below.

==The real closure of a commutative ring==
Every commutative unital ring ''R'' has a so-called '''real closure''' rcl(''R'') and this is unique up to a unique ring
homomorphism over ''R''. This means that rcl(''R'') is a real closed ring and there is a (not necessarily injective) ring homomorphism
<math>r:R\to rcl(R)</math> such that for every ring homomorphism <math>f:R\to A</math> to some other real closed ring ''A'', there is a unique ring homomorphism <math>g:rcl(R)\to A</math> with <math>f=g\circ r</math>.

For example the real closure of the polynomial ring <math>\mathbb{R}[T_1,...,T_n]</math>
is the ring of continuous semi-algbebraic functions <math>\mathbb{R}^n\to \mathbb{R}</math>.

Note that an arbitrary ring ''R'' is semi-real (i.e. -1 is not a sum of squares in ''R'')
if and only if the real closure of ''R'' is not the null ring.

Note also that the real closure of an [[ordered field]] is in general '''not''' the real closure of the underlying field. For example, the real closure of the '''ordered''' subfield <math>\mathbb{Q}(\sqrt 2)</math>
of <math>\mathbb{R}</math> is the field <math>\mathbb{R}_{alg}</math> of real algebraic numbers,
whereas the real closure of the field <math>\mathbb{Q}(\sqrt 2)</math> is the ring
<math>\mathbb{R}_{alg}\times \mathbb{R}_{alg}</math> (corresponding to the two orders of <math>\mathbb{Q}(\sqrt 2)</math>). More generally the real closure of a field ''F''
is a certain subdirect product of the real closures of the ordered fields ''(F,P)'', where ''P'' runs through the orderings of ''F''.

==Algebraic properties==
* The [[category theory|category]] ''RCR'' of real closed rings which has real closed rings as objects and ring homomorphisms as maps has the following properties:
#Arbitrary products, direct limits and inverse limits (in the category of commutative unital rings) of real closed rings are again real closed. The [[Pushout (category theory)|fibre sum]] of two real closed rings ''B,C'' over some real closed ring ''A'' exists in ''RCR'' and is the real closure of the [[tensor product]] of ''B'' and ''C'' over ''A''.
#''RCR'' has arbitrary limits and co-limits.
#''RCR'' is a [[Variety (universal algebra)|variety]] in the sense of [[universal algebra]] (but not a subvariety of commutative rings).
* For a real closed ring ''A'', the natural homomorphism of ''A'' to the product of all its [[residue field]]s is an isomorphism onto a subring of this product that is closed under continuous [[Semialgebraic set|semi-algebraic]] functions defined over the integers. Conversely, every subring of a product of real closed fields with this property is real closed.
* If ''I'' is a [[radical ideal]] of a real closed ring ''A'', then also the [[residue class ring]] ''A''/''I'' is real closed. If ''I'' and ''J'' are radical ideals of a real closed ring then the sum ''I'' + ''J'' is again a radical ideal.
* All classical [[Localization of a ring|localization]]s ''S''−1''A'' of a real closed ring ''A'' are real closed. The epimorphic hull and the complete ring of quotients of a real closed ring are again real closed.
* The (real) holomorphy ring ''H''(''A'') of a real closed ring ''A'' is again real closed. By definition, ''H''(''A'') consists of all elements ''f'' in ''A'' with the property ''−N'' ≤ ''f'' ≤ ''N'' for some natural number ''N''. Applied to the examples above, this means that the rings of bounded (semi-algberaic/definable) continuous functions are all real closed.
* The support map from the [[real spectrum]] of a real closed ring to its [[Spectrum of a ring|Zariski spectrum]], which sends an ordering ''P'' to its support <math>P\cap -P</math> is an [[homeomorphism]]. In particular, the Zariski spectrum of every real closed ring ''A'' is a root system (in the sense of [[graph theory]]) and therefore ''A'' is also a Gel'fand ring (i.e. every [[prime ideal]] of ''A'' is contained in a unique maximal ideal of ''A''). The comparison of the Zariski spectrum of ''A'' with the Zariski spectrum of ''H(A)'' leads to a homeomorphism between the maximal spectra of these rings, generalizing the Gel'fand-Kolmogorov theorem for rings of real valued continuous functions.
* The natural map ''r'' from an arbitrary ring ''R'' to its real closure rcl(''R'') as explained above, induces a homeomorphism from the real spectrum of rcl(''R'') to the real spectrum of ''R''.
* Summarising and significantly strengthening the previous two properties, the following is true: The natural map ''r'' from an arbitrary ring ''R'' to its real closure rcl(''R'') induces an identification of the [[affine scheme]] of rcl(''R'') with the affine real closed space of ''R''.

==Model theoretic properties==

The class of real closed rings is [[first-order]] [[axiom]]atizable and [[Decidability (logic)|undecidable]]. The class of all real closed valuation rings is [[Decidability (logic)|decidable]] (by Cherlin-Dickmann) and the class of all real closed fields is decidable (by Tarski). After naming a definable radical relation, real closed rings have a [[Model complete theory|model companion]], namely [[von Neumann regular]] real closed rings.

==Comparison with characterizations of real closed fields==

There are many different characterizations of [[real closed field|real closed '''fields''']]. For example
in terms of maximality (with respect to algebraic extensions): a real closed field is a maximally orderable field; or, a real closed field (together with its unique ordering) is a maximally ordered field. Another characterization says that the intermediate value theorem holds for all polynomials in one variable over the (ordered) field. In the case of commutative rings, all these properties can be (and are) analyzed in the literature. They all lead to different classes of rings which are unfortunately also called 'real closed' (because a certain characterization of real closed fields has been extended to rings). '''None''' of them lead to the class of real closed rings and none of them allow a satisfactory notion of a closure operation. A central point in the definition of real closed rings is the globalisation of the notion of a real closed field to rings when these rings are represented as rings of functions on some space (typically, the real spectrum of the ring).

== References ==

{{Reflist}}
* Cherlin, Gregory. Rings of continuous functions: decision problems Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), pp. 44–91, Lecture Notes in Math., 834, Springer, Berlin, 1980.
* Cherlin, Gregory(1-RTG2); Dickmann, Max A. Real closed rings. II. Model theory. Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231.
* A. Prestel, N. Schwartz. Model theory of real closed rings. Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 261–290, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002.
* Schwartz, Niels. The basic theory of real closed spaces. Memoirs of the American Mathematical Society 1989 (ISBN 0821824600 )
* Schwartz, Niels; Madden, James J. Semi-algebraic function rings and reflectors of partially ordered rings. Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin, 1999
* Schwartz, Niels. Real closed rings. Algebra and order (Luminy-Marseille, 1984), 175–194, Res. Exp. Math., 14, Heldermann, Berlin, 1986
* Schwartz, Niels. Rings of continuous functions as real closed rings. Ordered algebraic structures (Curaçao, 1995), 277–313, Kluwer Acad. Publ., Dordrecht, 1997.
* Tressl, Marcus. Super real closed rings. Fundamenta Mathematicae 194 (2007), no. 2, 121–177.


[[Category:Ring theory]]
[[Category:Real algebraic geometry]]
[[Category:Ordered algebraic structures]]
[[Category:Model theory]]
[[Category:Real closed field]]

Argument principle

2014-01-07T18:30:02Z

207.38.151.173:

{{Unreferenced|date=December 2009}}
In [[mathematics]], '''''differential of the first kind''''' is a traditional term used in the theories of [[Riemann surface]]s (more generally, [[complex manifold]]s) and [[algebraic curve]]s (more generally, [[algebraic variety|algebraic varieties]]), for everywhere-regular [[differential form|differential 1-forms]]. Given a complex manifold ''M'', a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere [[holomorphic form|holomorphic]]; on an [[algebraic variety]] ''V'' that is [[non-singular]] it would be a [[global section]] of the [[coherent sheaf]] Ω1 of [[Kähler differential]]s. In either case the definition has its origins in the theory of [[abelian integral]]s.

The dimension of the space of differentials of the first kind, by means of this identification, is the [[Hodge number]]

:''h''0,1.

The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the [[elliptic integral]]s to all curves over the [[complex number]]s. They include for example the '''hyperelliptic integrals''' of type

: <math> \int\frac{x^k \, dx}{\sqrt{Q(x)}} </math>

where ''Q'' is a [[square-free polynomial]] of any given degree > 4. The allowable power ''k'' has to be determined by analysis of the possible pole at the [[point at infinity]] on the corresponding [[hyperelliptic curve]]. When this is done, one finds that the condition is

:''k'' ≤ ''g'' − 1,

or in other words, ''k'' at most 1 for degree of ''Q'' 5 or 6, at most 2 for degree 7 or 8, and so on.

Quite generally, as this example illustrates, for a [[compact Riemann surface]] or [[algebraic curve]], the Hodge number is the [[genus (mathematics)|genus]] ''g''. For the case of [[algebraic surface]]s, this is the quantity known classically as the [[irregularity of a surface|irregularity]] ''q''. It is also, in general, the dimension of the [[Albanese variety]], which takes the place of the [[Jacobian variety]].

==Differentials of the second and third kind==
The traditional terminology also included differentials '''of the second kind''' and '''of the third kind'''. The idea behind this has been supported by modern theories of [[algebraic differential form]]s, both from the side of more [[Hodge theory]], and through the use of morphisms to [[commutative]] [[algebraic group]]s.

The [[Weierstrass zeta function]] was called an ''integral of the second kind'' in [[elliptic function]] theory; it is a [[logarithmic derivative]] of a [[theta function]], and therefore has [[simple pole]]s, with integer residues. The decomposition of a ([[meromorphic]]) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a [[linear combination]] of translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.

The same type of decomposition exists in general, ''mutatis mutandis'', though the terminology is not completely consistent. In the algebraic group ([[generalized Jacobian]]) theory the three kinds are [[abelian varieties]], [[algebraic tori]], and [[affine space]]s, and the decomposition is in terms of a [[composition series]].

On the other hand, a meromorphic abelian differential of the ''second kind'' has traditionally been one with residues at all poles being zero. There is a higher-dimensional analogue available, using the [[Poincaré residue]]

=== See also ===
[[Logarithmic form]]

{{DEFAULTSORT:Differential Of The First Kind}}
[[Category:Complex manifolds]]
[[Category:Algebraic geometry]]

Phragmén–Lindelöf principle

2014-01-07T06:54:28Z

207.38.151.173: /* Applications */

{{about|topological covering group|algebraic covering group|universal perfect central extension}}
In [[mathematics]], a '''covering group''' of a [[topological group]] ''H'' is a [[covering space]] ''G'' of ''H'' such that ''G'' is a topological group and the covering map ''p'' : ''G'' → ''H'' is a [[continuous (topology)|continuous]] [[group homomorphism]]. The map ''p'' is called the '''covering homomorphism'''. A frequently occurring case is a '''double covering group''', a [[double cover (topology)|topological double cover]] in which ''H'' has [[Index of a subgroup|index]] 2 in ''G;'' examples include the [[Spin group]]s, [[Pin group]]s, and [[metaplectic group]]s.

Roughly explained, saying that for example the metaplectic group ''Mp''2''n'' is a ''double cover'' of the [[symplectic group]] ''Sp''2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group.

==Properties==

Let ''G'' be a covering group of ''H''. The [[kernel (group theory)|kernel]] ''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a [[discrete group|discrete]] [[normal subgroup]] of ''G''. The kernel ''K'' is [[closed set|closed]] in ''G'' if and only if ''G'' is [[Hausdorff space|Hausdorff]] (and if and only if ''H'' is Hausdorff). Going in the other direction, if ''G'' is any topological group and ''K'' is a discrete normal subgroup of ''G'' then the quotient map ''p'' : ''G'' → ''G''/''K'' is a covering homomorphism.

If ''G'' is [[connected space|connected]] then ''K'', being a discrete normal subgroup, necessarily lies in the [[center (group theory)|center]] of ''G'' and is therefore [[abelian group|abelian]]. In this case, the center of ''H'' = ''G''/''K'' is given by
:<math>Z(H) \cong Z(G)/K.</math>

As with all covering spaces, the [[fundamental group]] of ''G'' injects into the fundamental group of ''H''. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. In particular, if ''G'' is [[path-connected]] then the [[quotient group]] <math>\pi_1(H)/\pi_1(G)</math> is isomorphic to ''K''. The group ''K'' [[group action|acts]] simply transitively on the fibers (which are just left [[coset]]s) by right multiplication. The group ''G'' is then a [[principal bundle|principal ''K''-bundle]] over ''H''.

If ''G'' is a covering group of ''H'' then the groups ''G'' and ''H'' are [[locally isomorphic groups|locally isomorphic]]. Moreover, given any two connected locally isomorphic groups ''H''1 and ''H''2, there exists a topological group ''G'' with discrete normal subgroups ''K''1 and ''K''2 such that ''H''1 is isomorphic to ''G''/''K''1 and ''H''2 is isomorphic to ''G''/''K''2.

==Group structure on a covering space==

Let ''H'' be a topological group and let ''G'' be a covering space of ''H''. If ''G'' and ''H'' are both [[path-connected]] and [[locally path-connected]], then for any choice of element ''e''* in the fiber over ''e'' ∈ ''H'', there exists a unique topological group structure on ''G'', with ''e''* as the identity, for which the covering map ''p'' : ''G'' → ''H'' is a homomorphism.

The construction is as follows. Let ''a'' and ''b'' be elements of ''G'' and let ''f'' and ''g'' be [[path (topology)|path]]s in ''G'' starting at ''e''* and terminating at ''a'' and ''b'' respectively. Define a path ''h'' : ''I'' → ''H'' by ''h''(''t'') = ''p''(''f''(''t''))''p''(''g''(''t'')). By the path-lifting property of covering spaces there is a unique lift of ''h'' to ''G'' with initial point ''e''*. The product ''ab'' is defined as the endpoint of this path. By construction we have ''p''(''ab'') = ''p''(''a'')''p''(''b''). One must show that this definition is independent of the choice of paths ''f'' and ''g'', and also that the group operations are continuous.

The non-connected case is interesting and is studied in the papers by Taylor and by Brown-Mucuk cited below. Essentially there is an obstruction to the existence of a universal cover which is also a topological group such that the covering map is a morphism: this obstruction lies in the third cohomology group of the group of components of ''G'' with coefficients in the fundamental group of ''G'' at the identity.

==Universal covering group==

If ''H'' is a path-connected, locally path-connected, and [[semilocally simply connected]] group then it has a [[universal cover]]. By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism. This group is called the '''universal covering group''' of ''H''. There is also a more direct construction which we give below.

Let ''PH'' be the [[path group]] of ''H''. That is, ''PH'' is the space of [[path (topology)|path]]s in ''H'' based at the identity together with the [[compact-open topology]]. The product of paths is given by pointwise multiplication, i.e. (''fg'')(''t'') = ''f''(''t'')''g''(''t''). This gives ''PH'' the structure of a topological group. There is a natural group homomorphism ''PH'' → ''H'' which sends each path to its endpoint. The universal cover of ''H'' is given as the quotient of ''PH'' by the normal subgroup of [[null-homotopic]] [[loop (topology)|loop]]s. The projection ''PH'' → ''H'' descends to the quotient giving the covering map. One can show that the universal cover is [[simply connected]] and the kernel is just the [[fundamental group]] of ''H''. That is, we have a [[short exact sequence]]
:<math>1\to \pi_1(H) \to \tilde H \to H \to 1</math>
where <math>\tilde H</math> is the universal cover of ''H''. Concretely, the universal covering group of ''H'' is the space of homotopy classes of paths in ''H'' with pointwise multiplication of paths. The covering map sends each path class to its endpoint.

== Lattice of covering groups ==
As the above suggest, if a group has a universal covering group (if it is path-connected, locally path-connected, and semilocally simply connected), with discrete center, then the set of all topological groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of the universal covering group: inclusion of subgroups corresponds to covering of quotient groups. The maximal element is the universal covering group <math>\tilde H,</math> while the minimal element is the universal covering group mod its center, <math>\tilde H/Z(\tilde H)</math>.

This corresponds algebraically to the [[universal perfect central extension]] (called "covering group", by analogy) as the maximal element, and a group mod its center as minimal element.

This is particularly important for Lie groups, as these groups are all the (connected) realizations of a particular Lie algebra. For many Lie groups the center is the group of scalar matrices, and thus the group mod its center is the projectivization of the Lie group. These covers are important in studying [[projective representation]]s of Lie groups, and [[spin representation]]s lead to the discovery of [[spin group]]s: a projective representation of a Lie group need not come from a linear representation of the group, but does come from a linear representation of some covering group, in particular the universal covering group. The finite analog led to the covering group or Schur cover, as discussed above.

A key example arises from [[SL2(R)|SL2('''R''')]], which has center {±1} and fundamental group '''Z'''. It is a double cover of the centerless [[projective special linear group]] PSL2('''R'''), which is obtained by taking the quotient by the center. By [[Iwasawa decomposition]], both groups are circle bundles over the complex upper half-plane, and their universal cover <math>{\mathrm{S}\widetilde{\mathrm{L}_2(}\mathbf{R})}</math> is a real line bundle over the half-plane that forms one of [[Geometrization conjecture|Thurston's eight geometries]]. Since the half-plane is contractible, all bundle structures are trivial. The preimage of SL2('''Z''') in the universal cover is isomorphic to the [[braid group]] on three strands.

== Lie groups ==
{{see also|Group extension#Central_extension}}

The above definitions and constructions all apply to the special case of [[Lie group]]s. In particular, every covering of a [[manifold]] is a manifold, and the covering homomorphism becomes a [[smooth map]]. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism.

Two Lie groups are locally isomorphic if and only if their [[Lie algebras]] are isomorphic. This implies that a homomorphism φ : ''G'' → ''H'' of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras
:<math>\phi_* : \mathfrak g \to \mathfrak h</math>
is an isomorphism.

Since for every Lie algebra <math>\mathfrak g</math> there is a unique simply connected Lie group ''G'' with Lie algebra <math>\mathfrak g</math>, from this follows that the universal convering group of a connected Lie group ''H'' is the (unique) simply connected Lie group ''G'' having the same Lie algebra as ''H''.

== Examples ==

* The universal covering group of the [[circle group]] '''T''' is the additive group of [[real number]]s '''R''' with the covering homomorphism given by the [[exponential function]] exp: '''R''' → '''T'''. The kernel of the exponential map is isomorphic to '''Z'''.
* For any integer ''n'' we have a covering group of the circle by itself '''T''' → '''T''' which sends ''z'' to ''z''''n''. The kernel of this homomorphism is the [[cyclic group]] consisting of the ''n''th [[roots of unity]].
* The rotation group [[SO(3)]] has as a universal cover the group [[SU(2)]] which is isomorphic to the group of unit [[quaternion]]s [[Sp(1)]]. This is a double cover since the kernel has order 2.
* The [[unitary group]] U(''n'') is covered by the compact group '''T''' × SU(''n'') with the covering homomorphism given by ''p''(''z'', ''A'') = ''zA''. The universal cover is just '''R''' × SU(''n'').
* The [[special orthogonal group]] SO(''n'') has a double cover called the [[spin group]] Spin(''n''). For ''n'' ≥ 3, the spin group is the universal cover of SO(''n'').
* For ''n'' ≥ 2, the universal cover of the [[special linear group]] SL(''n'', '''R''') is ''not'' a [[matrix group]] (i.e. it has no faithful finite-dimensional [[group representation|representation]]s).

== References ==

*{{cite book | last = Pontryagin | first = Lev S. | authorlink = Lev Semenovich Pontryagin | title = Topological Groups | year = 1986 | edition = 3rd ed. | others = trans. from Russian by Arlen Brown and P.S.V. Naidu | publisher = Gordon and Breach Science Publishers | location = New York | isbn = 2-88124-133-6}}

* Taylor, R.L. ''Covering groups of nonconnected topological groups'', Proc. Amer. Math. Soc. 5 (1954) 753–768.

* Brown, R. and Mucuk, O. ''Covering groups of nonconnected topological groups revisited'', Math. Proc. Cambridge Philos. Soc. 115~(1) (1994) 97–110.

[[Category:Topological groups]]
[[Category:Lie groups]]

Arc length

2014-01-07T06:52:52Z

207.38.151.173:

{{expert|date=August 2012}}
{{citations|date=August 2012}}

In a [[Plasma (physics)|plasma]], the '''Boltzmann relation''' describes the [[number density]] of an [[isothermal]] [[charged particle]] [[fluid]] when the thermal and the electrostatic forces acting on the fluid have reached [[Mechanical equilibrium|equilibrium]].

In many situations, the electron density of a plasma is assumed to behave according to the Boltzmann relation, due to their small mass and high mobility.<ref name="Chen">{{cite book |title=Introduction to Plasma Physics and Controlled Fusion |last=Chen |first=Francis F. |year=2006 |publisher=Springer |edition=2nd |page=75 |isbn=978-0-306-41332-2}}</ref>

==Equation==

If the local [[electrostatic potential]]s at two nearby locations are φ1 and φ2, the Boltzmann relation for the electrons takes the form:

:<math>n_e (\phi_2) = n_e(\phi_1) e^{- (\phi_2-\phi_1)/k_B T_e}</math>

where ''n''e is the electron [[number density]], ''T''e the [[temperature]] of the plasma, and ''k''B is [[Boltzmann constant]].

==Derivation==

A simple derivation of the Boltzmann relation for the electrons can be obtained using the momentum fluid equation of the two-fluid model of [[plasma physics]] in absence of a [[magnetic field]]. When the electrons reach [[dynamic equilibrium]], the inertial and the collisional terms of the momentum equations are zero, and the only terms left in the equation are the pressure and electric terms. For an [[Isothermal flow|isothermal fluid]], the [[pressure]] force takes the form
:<math>F_{\rm fluid}=-k_BT_e\nabla n_e,</math>
while the electric term is
:<math>F_{\rm electric}=e n_e \nabla\phi </math>.
[[Integral|Integration]] leads to the expression given above.

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the [[Poisson equation]] because the electron and ion densities are not known ''a priori'', and if they were, because of [[Plasma (physics)#Potentials|quasineutrality]] the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the ion density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation.

==Inaccurate situations==

Discrepancies with the Boltzmann relation can occur, for example, when oscillations occur so fast that the electrons cannot find a new equilibrium (see e.g. [[plasma oscillation]]s) or when the electrons are prevented from moving by a magnetic field (see e.g. [[lower hybrid oscillation]]s).

==See also==
*[[List of plasma (physics) articles]]

==References==
*{{cite book
| last = Wesson
| first = John
| coauthors = et al.
| year = 2004
| title = Tokamaks
| publisher = Oxford University Press
| isbn = 0-19-850922-7
}}
{{Reflist}}

==External links==

[[Category:Plasma physics]]

{{physics-stub}}

Egorov's theorem

2013-12-14T17:39:15Z

207.38.151.173: /* Statement of the theorem */

In economics, '''Aggregate Expenditure''' is a [[measures of national income|measure of national income]].<ref>{{cite book|last=Trosten|first=Jochem|title=Macro-recitation|year=2009|pages=2–7|url=http://www.torstenjochem.de/macro101/07_Measuring_GDP.pdf}}</ref> Aggregate Expenditure is defined as the current value of all the finished goods and services in the economy.<ref>{{cite web|last=Haworth|first=Barry|title=The Aggregate Expenditure Model|url=http://econpage.com/202/handouts/AEmodel/index.html|work=The Aggregate expenditure model|publisher=University of Louisville|accessdate=13 November 2011}}</ref> The aggregate expenditure is thus the sum total of all the expenditures undertaken in the economy by the factors during a given time period. It refers to the expenditure incurred on consumer goods, planned investment (or savings) and in the Keynesian model also includes the expenditure made by the government in the economy. In an open economy scenario, the aggregate expenditure also includes the difference between the exports and the imports.

Aggregate Expenditures is defined as
: <math>AE = C+I+G+Xn</math>,
here,
* C = Household Consumption
* Ip = Planned Investment
* Iu = Unplanned Investment
* G = Government spending
* Xn = Net exports (Exports-Imports)

Aggregate Expenditure is one of the methods to calculate the sum total of all economic activities in an economy which is referred to as the Gross Domestic product of an economy. The gross domestic product which is an important measure of the growth of the economy is calculated through the [[Keynesian cross|Aggregate expenditure model]] also known as the [[Keynesian cross]].AE is also used in the [[Aggregate demand|Aggregate Demand]]-[[Aggregate supply|Aggregate Supply]] Model which advances the Aggregate Expenditures Model with the inclusion of Price changes.

Components of '''Aggregate Expenditure''' (AE) - defined as the total amount that firms and households plan to spend on goods and services at each level of income. Also, it can be seen that the aggregate expenditure is the sum of expenditures on consumption, investment, government expenses and net exports. It is normally derived from all the components of the [[Aggregate Demand]]. Aggregate demand (AD) refers to the sum total of goods that are demanded in an economy over a period and thus AD is defined by the planned total expenditure in an economy for a given price level.

==Components of AE==
Various school of thoughts use various components to come up with the Aggregate Expenditure. The major school of economic thoughts which are the classical and Keynesian economists use the following components:

*Consumption expenditure (C)'''
Consumption refers to the household consumption over a given period of time. The total household consumption can be divided into two parts, they are: Autonomous Consumption and Induced consumption. [[Autonomous consumption|Autonomous Consumption]] refers to the amount of consumption regardless of the amount of income, hence, even if the income is zero, the autonomous consumption would be the total consumption. [[Induced consumption]] refers to the level of consumption dependent on the level of income.<ref>{{cite book|last=Rittenberg, Tregarthen|first=Libby,Timothy|title=Principles of Macroeconomics|url=http://www.flatworldknowledge.com/node/30043#web-30043}}</ref>

C = Ca + MPC (Y)
* Ca = Autonomous Consumption
* MPC = Marginal Propensity to Consume
* Y = Income

*Investment (I)'''
Investment is the amount of expenditure towards the capital goods. Investment refers to the expenditure towards goods that are expected to yield a return or increase their own value over time. The investment expenditure can be further divided into two parts, planned investment and unplanned investment. Over the long run the sum of differences in the unplanned investment would equal to zero as economy approaches equilibrium.
'''

*Government Expenditure:(G)'''
The Keynesian model propagates an active state to control and regulate the economy. The government can make expenditure in terms of infrastructure or through transfers and thus increase the total expenditure in the economy as advocated by Keynes.

*Net Exports:(NX)'''
In an open economy, the total expenditure of the economy also includes the components of the net exports which is the total exports minus the total imports.<ref>{{cite web|title=Components of Aggregate Expenditure|url=http://su-bc.org/PDF/tutorials/ECN112.pdf|work=Stirling University|accessdate=13 November 2011}}</ref>

==Classical Economics==

Classical economists relied on the [[Say's law]] which states that supply creates its own demand, which stemmed from the belief that wages,prices and interest rates are all flexible.<ref>{{cite book|last=Byrns|title=Student Guide for Learning Contemporary Economics|pages=127|url=http://www.unc.edu/depts/econ/byrns_web/PrinEcon/SG/SVE_SG-27.pdf}}</ref> This comes from the classical thought that the factor payments which are made to the various [[factors of production]] during the production process, would create enough income in the economy to create a demand for the products produced.[[File:Classical aggregate expenditure.png|thumb|Classical Aggregate Expenditure]]
This supports the classical thought which revolves around Adam Smith's [[invisible hand]] which states that the markets would achieve equilibrium via the market forces that impact economic activity and thus there is no need for government intervention.Moreover, the classical economists believed that the economy was operating at a [[full employment]] Hence the classical Aggregate expenditure model is:<ref>{{cite book|last=Burnette|first=Jeffery|title=Classical Economics- Principles of Micro|publisher=Rochester Institute of Technology|pages=9|url=http://people.rit.edu/jdbgse/Documents%20402/CN_intromacro_3.pdf}}</ref>

Aggregate Expenditure = Aggregate Consumption + Planned Investment

Therefore,

<math> --~~~~AE = C + I </math>

Where,
*C= Consumption Expenditure
*I = Aggregate investment ( Savings = Investment<ref>{{cite book|last=Burnette|first=Jeffery|title=Classical Economics- Principles of Micro|publisher=Rochester Institute of Technology|pages=9|url=http://people.rit.edu/jdbgse/Documents%20402/CN_intromacro_3.pdf}}</ref>)

Classical economics has been criticized for its assumptions that the economy works on a full-employment equilibrium which is a false assumption as in reality, the economy operates at an under-employment equilibrium which provides the foundation for the Keynesian model of Aggregate Expenditure.

==Keynesian Economics==
[[Keynesian Economics]] believes, contrary to the classical thought that the Wages, Prices and interest rates are not flexible and hence violating Say's Law, which provided the foundation for the maxim that "supply creates its own demand". Keynes believed that the economy was subject to [[Sticky Prices]] and thus the economy was not in a state of perpetual equilibrium and also operated at an under-employment equilibrium. [[File:Keynesian aggregate expenditure.png|thumb|Keynesian aggregate expenditure]] Keynesian economics calls for a government intervention and is called demand side economics as it believes that aggregate demand and not the aggregate supply determines the GDP because of the difference between the Aggregate Supply and Planned expenditure in an economy.Hence Keynes believed that the government played an important role in the determination on the Aggregate Expenditure in an economy and was thus included Government Expenditure in the Aggregate Expenditure Function.

Hence,

* <math> --~~~~AE = C+I+G+NX </math>

Where,

* C = Household Consumption Expenditure
* I = Investment (Planned)
* G = Government Expenditure
* NX= Net Exports ( Exports - Imports )

Keynesian economics preaches that in times of a recession, the government must undertake the expenditure to compensate for the lack in the components of Household expenditure (C) and private investment (I) so as to ensure that the demand is maintained in the markets. This also leads to the [[Multiplier (economics)|Keynesian Multiplier]] which suggests that every dollar spent on investment creates a multiplier effect and leads to an increased expenditure of more than one dollar.

==Aggregate Expenditure and Aggregate Supply==
[[File:Keynesian cross and growth in expenditure.png|thumb|effect of increase in expenditure]]
An economy is said to be in an equilibrium when aggregate expenditure is equal to the aggregate supply (production) in the economy. According to Keynes, the economy does not stay in a perpetual state of equilibrium but the Aggregate expenditure and Aggregate Supply adjust each other towards equilibrium. When there is an excess supply over the expenditure and hence the demand there is an inventory leftover with the producers, which leads to a reduction in either the prices or the quantity of output and hence reducing the total output (GDP) of the economy. On the other hand, if there is an excess of expenditure over supply, then there is excess demand leading to an increase in prices or output. Hence the economy constantly keeps shifting between excess supply ( inventory ) and excess demand. Thus, the economy is constantly moving towards an equilibrium between the aggregate expenditure and aggregate supply.<ref>{{cite book|last=Branson|first=William|title=Macroeconomic theory and policy|year=1979}}</ref> In an under-employment equilibrium the Keynesian Cross refers to the point of intersection of the Aggregate Supply and the Aggregate Expenditure curve. The rise in the expenditure by either Consumption (C), Investment (I) or the Government (G) or an increase in the exports or a decrease in the imports leads to a rise in the aggregate expenditure and thus pushes the economy towards a higher equilibrium and thus reaching a higher level towards the potential of the GDP.<ref>{{cite web|title=Aggregate Expenditure|url=http://www.murraylax.org/eco120/notes/keynesian.pdf|work=Keynesian Model|accessdate=13 November 2011}}</ref>

==See also==
* [[Aggregate income]]
* [[Aggregate demand]]

==References==
{{Reflist}}
{{Refbegin}}
*Parry G., and Kemp S., (2009) Discovering Economics Tactic Publications, South Perth.
{{Refend}}

{{DEFAULTSORT:Aggregate Expenditure}}
[[Category:Income]]
[[Category:Macroeconomic aggregates]]

Skew-symmetric matrix

2013-12-06T19:16:49Z

207.38.151.173: /* Properties */

In [[physics]], the '''Planck time''' (''tP'') is the [[unit of time]] in the system of [[natural units]] known as [[Planck units]]. It is the time required for [[light]] to travel, in a [[vacuum]], a distance of 1 [[Planck length]].<ref name="gsu_hbase">{{cite web | url = http://hyperphysics.phy-astr.gsu.edu/hbase/astro/planck.html | title = Big Bang models back to Planck time | publisher = [[Georgia State University]] | date = 19 June 2005}}</ref> The unit is named after [[Max Planck]], who was the first to propose it.

The Planck time is defined as:<ref>[http://physics.nist.gov/cgi-bin/cuu/Value?plkt CODATA Value: Planck Time] – The [[NIST]] Reference on Constants, Units, and Uncertainty.</ref>
:<math>t_P \equiv \sqrt{\frac{\hbar G}{c^5}}</math> ≈ 5.39106(32) × 10−44 s

where:
:<math>\hbar = h / 2 \pi</math> is the [[reduced Planck constant]] (sometimes <math>h</math> is used instead of <math>\hbar</math> in the definition<ref name="gsu_hbase" />)
:''G'' = [[gravitational constant]]
:''c'' = [[speed of light]] in a [[vacuum]]
:s is the [[SI]] unit of time, the [[second]].

The two digits between [[parentheses]] denote the [[Standard error (statistics)|standard error]] of the estimated value.

==Physical significance==
One Planck time is the time it would take a photon traveling at the speed of [[light]] to cross a distance equal to one [[Planck length]]. Theoretically, this is the smallest time measurement that will ever be possible,<ref>{{cite encyclopedia |url=http://astronomy.swin.edu.au/cosmos/P/Planck+Time |title=Planck Time |encyclopedia =COSMOS - The SAO Encyclopedia of Astronomy |publisher= Swinburne University}}</ref> roughly 10−43 seconds. Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change.
{{As of| May 2010}}, the smallest time interval uncertainty in direct measurements is on the order of 12 [[attoseconds]] (1.2 × 10−17 seconds), about 3.7 × 1026 Planck times.<ref>
{{cite web | url=http://www.physorg.com/news192909576.html | title=12 attoseconds is the world record for shortest controllable time | date=2010-05-12 | accessdate=2012-04-19}}</ref>

The Planck time comes from a field of [[mathematical physics]] known as [[dimensional analysis]], which studies [[units of measurement]] and [[physical constants]]. The Planck time is the unique combination of the [[gravitational constant]] ''G'', the [[speed of light|relativity constant]] ''c'', and the [[Planck constant|quantum constant]] ''h'', to produce a constant with units of [[time]]. For processes that occur in a time ''t'' less than one Planck time, the [[dimensionless quantity]] ''tP / t'' is greater than one. Dimensional analysis suggests that the effects of both [[quantum mechanics]] and [[gravity]] will be important under these circumstances, requiring a theory of [[quantum gravity]]. All scientific experiments and human experiences happen over billions of billions of billions of Planck times, making any events happening at the Planck scale hard to detect.

Analysis of [[Hubble Space Telescope]]s [[Hubble Ultra-Deep Field|deep field]] images in 2003 led to a debate about the physical implications of the Planck time as a physical minimum time interval. According to Lieu and Hillman,<ref>{{cite journal |last=Lieu |first=Richard |coauthors=Hillman, Lloyd W. |date=2003-03-10 |title=The Phase Coherence of Light from Extragalactic Sources: Direct Evidence against First-Order Planck-Scale Fluctuations in Time and Space |journal=The Astrophysical Journal |volume=585 |issue=2 |pages=L77–L80 |doi=10.1086/374350 |bibcode=2003ApJ...585L..77L|arxiv = astro-ph/0301184 |url=http://iopscience.iop.org/1538-4357/585/2/L77/pdf/1538-4357_585_2_L77.pdf }}
</ref>
speculative theories of quantum gravity "[[quantum foam|foam]]" where there are space–time fluctuations on the Planck scale predict that images of extremely distant objects should be blurry. However, blurring was not seen in the Hubble images, which was claimed to be problematic for such theories.<ref name="space.com">{{cite web|url=http://www.space.com/scienceastronomy/quantum_bits_030402.html|title=Hubble Pictures Too Crisp, Challenging Theories of Time and Space|date=2003-04-02|publisher=Space.com|accessdate=2008-05-30}}
</ref>
Other authors have disputed this, in particular Ng ''et al.'',<ref>{{cite journal|last=Ng|first=Y. Jack|coauthors=Christiansen, W. A.; van Dam H.|date=2003-07-10|title=Probing Planck-Scale Physics with Extragalactic Sources?|journal=The Astrophysical Journal Letters|publisher=The American Astronomical Society|volume=591|issue=2|pages=L87–L89 |doi=10.1086/377121|bibcode=2003ApJ...591L..87N|arxiv = astro-ph/0302372 |url=http://iopscience.iop.org/1538-4357/591/2/L87/pdf/1538-4357_591_2_L87.pdf}}
</ref>
who stated that the blurring effect was overestimated by Lieu and Hillman by factors of between 1015 and 1030, and thus the observations are very much less effective in constraining theory: "the cumulative effects of spacetime ﬂuctuations on the phase coherence of light [in certain theories of 'foamy' spacetime] are too small to be observable".

==See also==
* [[Chronon]]
* [[Orders of magnitude (time)]]
* [[Planck energy]]
* [[Planck length]]
* [[Planck units]]
* [[Quantum clock]]

== Notes and references==
{{Reflist|2}}

{{Planck's natural units}}
{{Time measurement and standards}}
{{Time topics}}
{{Orders of magnitude seconds}}
{{Portal bar|Physics}}

{{Use dmy dates|date=May 2011}}

[[Category:Natural units|Time]]
[[Category:Units of time]]
[[Category:Physical constants]]

Parametric surface

2013-09-24T21:39:49Z

207.38.151.173: /* First fundamental form */

In [[algebraic geometry]], a [[line bundle]] on a [[Complete variety|complete]]
[[algebraic variety]] over a field is said to be '''nef''' if the degree of its restriction to every [[algebraic curve]] in the variety is non-negative. The term "nef" was introduced by [[Miles Reid]]<ref>M. Reid. Minimal models of canonical 3-folds. ''Algebraic Varieties and Analytic Varieties'', 131-180. North-Holland (1983). Section 0.12f.</ref> as a replacement for the older terms "arithmetically effective" {{harv|Zariski|1962|loc=definition 7.6}} and "numerically effective", as well as for the phrase "numerically eventually free". (A line bundle is called '''semi-ample''' or "eventually free" if some positive power is basepoint-free.) The older terminology was confusing because nef divisors are not the same as divisors numerically equivalent to effective divisors. For example, a curve with negative self-intersection number on a surface is effective but not nef.

Every semi-ample divisor is nef, but not every nef divisor is numerically equivalent to a semi-ample divisor, or even to an effective divisor. For example, [[David Mumford|Mumford]] constructed a line bundle ''L'' on a suitable [[ruled surface]] ''X'' such that ''L'' has positive degree on all curves, but the intersection number ''c''1(''L'')2 is zero. It follows that ''L'' is nef, but no positive multiple of the first [[Chern class]] ''c''1(''L'') is numerically equivalent to an effective divisor.<ref>R. Lazarsfeld. ''Positivity in Algebraic Geometry I.'' Springer-Verlag (2004). Example 1.5.2.</ref> (The first Chern class is an isomorphism from the [[Picard group]] of line bundles on a variety ''X'' to the group of [[Cartier divisor]]s modulo [[linear equivalence]].)

A Cartier divisor ''D'' on an algebraic variety ''X'' is said to be nef if the corresponding line bundle ''O''(''D'') is nef on ''X''. Equivalently, ''D'' is nef if

:<math>D\cdot C \ge 0 \, </math>

for any algebraic curve ''C'' in ''X'', in the sense of [[intersection theory]].

To work with inequalities, it is convenient to consider '''R'''-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The '''R'''-divisors modulo [[Adequate equivalence relation|numerical equivalence]] form a real vector space ''N''1(''X'') of finite dimension, the [[Néron–Severi group]] tensored with the real numbers. The nef '''R'''-divisors form a closed convex cone in this vector space, called the '''nef cone'''. The interior of this cone is called the '''ample cone'''. For any projective variety ''X'', [[Steven Kleiman|Kleiman]] showed that a divisor is ample if and only if its numerical equivalence class lies in the interior of the nef cone.<ref>R. Lazarsfeld. ''Positivity in Algebraic Geometry I.'' Springer-Verlag (2004). Theorem 1.4.23.</ref> In particular, every [[ample line bundle]] is nef.

The '''[[cone of curves]]''' is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space ''N''1(X) of 1-cycles modulo numerical equivalence. The vector spaces ''N''1(''X'') and ''N''1(''X'') are dual to each other by the intersection pairing, and the nef cone is the dual of the closure of the cone of curves. (The cone of curves need not be closed. For example, the class of the line bundle ''L'' on Mumford's surface is a 1-cycle which is not in the cone of curves, but is in its closure.)

==Notes==
{{reflist}}

==References==
*{{Citation | authorlink=Robert Lazarsfeld | last1=Lazarsfeld | first1=Robert | mr=2095471 | title=Positivity in algebraic geometry | volume=1 | publisher=Springer-Verlag | location=Berlin | year=2004 | ISBN=3-540-22533-1}}
*{{Citation | authorlink=Miles Reid | last=Reid | first1=Miles | mr=0715649 | title=Algebraic Varieties and Analytic Varieties (Tokyo, 1981)| chapter=Minimal models of canonical 3-folds | pages=131–180 | series=Advanced Studies in Pure Mathematics | volume=1 | publisher=North-Holland | year=1983 | ISBN=0-444-86612-4}}
*{{citation| authorlink=Oscar Zariski | mr=0141668|last=Zariski|first= Oscar|title=The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface|journal=Ann. of Math. (2) |volume=76 |year=1962|pages= 560–615 | url=http://www.jstor.org/stable/1970376}}

[[Category:Algebraic varieties]]

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