formulasearchengine - User contributions [en]

Rankine vortex

2014-11-09T10:48:35Z

89.110.14.92:

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Quantum decoherence

2014-01-22T05:10:39Z

89.110.30.224: /* Collective dephasing */

In [[mathematics]], '''complex geometry''' is the study of [[complex manifold]]s and functions of many [[complex variable]]s. Application of transcendental methods to [[algebraic geometry]] falls in this category, together with more geometric chapters of [[complex analysis]].

Throughout this article, "[[Analytic function|analytic]]" is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones. Following the convention in Wikipedia, varieties are assumed to be irreducible.

== Definitions ==
An ''[[Analytic variety|analytic subset]]'' of a complex-analytic manifold ''M'' is locally the zero-locus of some family of holomorphic functions on ''M''. It is called an analytic subvariety if it is irreducible in the Zariski topology.

== Line bundles and divisors ==
Throughout this section, ''X'' denotes a complex manifold.

Let <math>\operatorname{Pic}(X)</math> be the set of all isomorphism classes of line bundles on ''X''. It is called the [[Picard group]] of ''X'' and is naturally isomorphic to <math>H^1(X, \mathcal{O}^*)</math>. Taking the short exact sequence of
:<math>0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0</math>
where the second map is <math>f \mapsto \exp (2\pi i f)</math>
yields a homomorphism of groups:
:<math>\operatorname{Pic}(X) \to H^2(X, \mathbb{Z}).</math>
The image of a line bundle <math>\mathcal{L}</math> under this map is denoted by <math>c_1(\mathcal{L})</math> and is called the first [[Chern class]] of <math>\mathcal{L}</math>.

A [[divisor (algebraic geometry)|divisor]] ''D'' on ''X'' is a [[formal sum]] of hypersurfaces (subvariety of codimension one):
:<math>D = \sum a_i V_i, \quad a_i \in \mathbb{Z}</math>
that is locally a finite sum.<ref>This last condition is automatic for a noetherian scheme or a compact complex manifold.</ref> The set of all divisors on ''X'' is denoted by <math>\operatorname{Div}(X)</math>. It can be canonically identified with <math>H^0(X, \mathcal{M}^*/\mathcal{O}^*)</math>. Taking the long exact sequence of the quotient <math>\mathcal{M}^*/\mathcal{O}^*</math>, one obtains a homomorphism:
:<math>\operatorname{Div}(X) \to \operatorname{Pic}(X).</math>

A line bundle is said to be [[positive line bundle|positive]] if its first Chern class is represented by a closed positive real <math>(1,1)</math>-form. Equivalently, a line bundle is positive if it admits a hermitian structure such that the induced connection has [[Griffiths-positive]] curvature. A complex manifold admitting a positive line bundle is kähler.

The [[Kodaira embedding theorem]] states that a line bundle on a compact kähler manifold is positive if and only if it is [[ample line bundle|ample]].

==Complex vector bundles==
Let ''X'' be a differentiable manifold. The basic invariant of a complex vector bundle <math>\pi: E \to X</math> is the [[Chern class]] of the bundle. By definition, it is a sequence <math>c_1, c_2, \dots</math> such that <math>c_i(E)</math> is an element of <math>H^{2i}(X, \mathbb{Z})</math> and that satisfies the following axioms:<ref>{{harvnb|Kobayashi–Nomizu|1996|Ch XII}}</ref>
# <math>c_i(f^*(E)) = f^*(c_i(E))</math> for any differentiable map <math>f: Z \to X</math>.
# <math>c(E \oplus F) = c(E) \cup c(F)</math> where ''F'' is another bundle and <math>c = 1 + c_1 + c_2 + \dots.</math>
# <math>c_i(E) = 0</math> for <math>i > \operatorname{rk}E</math>.
# <math>-c_1(E_1)</math> generates <math>H^2(\mathbb{C}\mathbf{P}^1, \mathbb{Z})</math> where <math>E_1</math> is the [[canonical line bundle]] over <math>\mathbb{C}\mathbf{P}^1</math>.

If ''L'' is a line bundle, then the [[Chern character]] of ''L'' is given by
:<math>\operatorname{ch}(L) = e^{c_1(L)}</math>.
More generally, if ''E'' is a vector bundle of rank ''r'', then we have the formal factorization:
<math>\sum c_i(E)t^i = \prod_1^r (1+ \eta_i t)</math> and then we set
:<math>\operatorname{ch}(E) = \sum e^{\eta_i}</math>.

== Methods from harmonic analysis ==
Some deep results in complex geometry are obtained with the aid of harmonic analysis.

== Vanishing theorem ==
There are several versions of vanishing theorems in complex geometry for both compact and non-compact complex manifolds. They are however all based on the [[Bochner method]].

==See also==
* [[Bivector (complex)]]
* [[Deformation Theory#Deformations of complex manifolds]]
* [[Complex analytic space]]
* [[GAGA]]
* [[Several complex variables]]
* [[Complex projective space]]
* [[List of complex and algebraic surfaces]]
* [[Enriques–Kodaira classification]]
* [[Kähler manifold]]
* [[Stein manifold]]
* [[Pseudoconvexity]]
* [[Kobayashi metric]]
* [[Projective variety]]
* [[Cousin problems]]
* [[Cartan's theorems A and B]]
* [[Hartogs' extension theorem]]
* [[Calabi–Yau manifold]]
* [[Reflection symmetry|Mirror symmetry]]
* [[Hermitian symmetric space]]
* [[Complex Lie group]]
* [[Hopf manifold]]
* [[Hodge decomposition]]
* [[Kobayashi-Hitchin correspondence]]
* [[Holomorphic Higgs pairs]]

==References==
{{Reflist}}

*{{cite book |title=Complex Geometry: An Introduction|first=Daniel|last=Huybrechts
|publisher=Springer|year=2005|isbn=3-540-21290-6}}
* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}
*{{Citation
| last = Hörmander
| first = Lars
| author-link = Lars Hörmander
| title = An Introduction to Complex Analysis in Several Variables
| place = Amsterdam–London–New York–Tokyo
| publisher = [[Elsevier|North-Holland]]
| origyear = 1966
| year = 1990
| series = North–Holland Mathematical Library
| volume = 7
| edition = 3rd (Revised)
| url =
| doi =
| mr = 1045639
| zbl = 0685.32001
| isbn = 0-444-88446-7
}}
*{{Kobayashi-Nomizu}}

[[Category:Complex manifolds]]
[[Category:Several complex variables]]

Spaghettification

2013-12-07T02:07:25Z

89.110.26.70: /* A simple example */

In [[queueing theory]], a discipline within the mathematical [[theory of probability]], '''mean value analysis (MVA)''' is a recursive technique for computing [[expected value|expected]] queue lengths, waiting time at queueing nodes and throughput in equilibrium for a closed separable system of queues. The first approximate techniques were published indepdenently by Schweitzer<ref name="Schweitzer" /> and Bard,<ref>{{cite journal | title = Some Extensions to Multiclass Queueing Network Analysis | first = Yonathan | last = Bard | year = 1979 | journal = Proceedings of the Third International Symposium on Modelling and Performance Evaluation of Computer Systems: Performance of Computer Systems | pages = 51–62 | isbn = 0-444-85332-4 | publisher = North-Holland Publishing Co.}}</ref><ref>{{cite doi|10.1007/978-1-4419-6472-4_13}}</ref> followed later by an exact version by Lavenberg and Reiser published in 1980.<ref>{{cite doi|10.1145/322186.322195}}</ref><ref>{{cite doi|10.1007/3-540-46506-5_22}}</ref>

It is based on the [[arrival theorem]], which states that when one customer in an ''M''-customer closed system arrives at a service facility he/she observes the rest of the system to be in the equilibrium state for a system with ''M'' − 1 customers.

==Problem setup==
Consider a closed queueing network of ''K'' [[M/M/1 queue]]s, with ''M'' customers circulating in the system. To compute the mean queue length and waiting time at each of the nodes and throughput of the system we use an iterative algorithm starting with a network with 0 customers.

Write ''μ''''i'' for the service rate at node ''i'' and ''P'' for the customer routing matrix where element ''p''''ij'' denotes the probability that a customer finishing service at node ''i'' moves to node ''j'' for service. To use the algorithm we first compute the visit ratio row vector '''v''', a vector such that '''v''' = '''v''' P.

Now write ''L''''i''(''n'') for the mean number of customer at queue ''i'' when there are a total of ''n'' customers in the system (this includes the job currently being served at queue ''i'') and ''W''''j''(''n'') for the mean time spent by a customer in queue ''i'' when there are a total of ''n'' customers in the system. Denote the throughput of a system with ''m'' customers by ''λ''''m''.

==Algorithm==
The algorithm<ref>{{cite book |first=Sanjay K.|last=Bose|title=An introduction to queueing systems|publisher=Springer|year=2001|isbn=0-306-46734-8|url=http://books.google.com/books?id=39-jISti_zkC|page=174}}</ref> starts with an empty network (zero customers), then increases the number of customers by 1 at each iteration until there are the required number (''M'') of customers in the system.

To initialise, set ''L''''k''(0) = 0 for ''k'' = 1,...,''K''. (This sets the average queue length in a system with no customers to zero at all nodes.)

Repeat for ''m'' = 1,...,''M'':

:1. For ''k'' = 1, ..., ''K'' compute the waiting time at each node using the arrival theorem
:::<math>W_k(m) = \frac{L_k\left(m-1\right)+1}{\mu_k}.</math>
:2. Then compute the system throughput using Little's law
:::<math>\lambda_m=\frac{m}{\sum_{k=1}^K W_k(m) v_k}.</math>
:3. Finally, use little's law applied to each queue to compute the mean queue lengths for ''k'' = 1, ..., ''K''
:::<math>L_k(m)=v_k \lambda_m W_k(m).</math>

End repeat.

==Schweitzer's approximation==

Schweitzer's approximation estimates the average number of jobs at node ''k'' to be<ref name="Schweitzer">{{cite doi|10.1007/BFb0013865}}</ref><ref>{{cite journal | last = Schweitzer | first = Paul | title = Approximate analysis of multiclass closed networks of queues | journal = Proceedings of International Conference on Stochastic Control and Optimization | year = 1979}}</ref>

:<math>L_k(m-1) \approx \frac{m-1}{m} L_k(m)</math>

which from the above formulas yields [[fixed-point iteration|fixed-point relationships]] which can be solved numerically. This iterative approach is typically faster than the recursive approach of MVA.<ref>{{cite doi|10.2200/S00282ED1V01Y201005CSL002}}</ref>{{rp|38}}

===Pseudo-code===

<tt>
set ''L''''k''(''m'') = ''M''/''K''

repeat until convergence:

::<math>\lambda_m = \frac{m}{\sum_{k=1}^K \frac{\frac{m-1}{m}L_k(m) + 1}{\mu_k} v_k}</math>
::<math>L_k(m) = v_k \lambda_m \frac{\frac{m-1}{m}L_k(m) + 1}{\mu_k}</math>

</tt>

==Multiclass networks==

For networks with a single customer class the MVA algorithm is very fast and time taken grows linearly with the number of customers and number of queues. However, the number of multiplications and additions required for MVA grows exponentially with the number of customer classes. Practically, the algorithm works for 3-4 customer classes.<ref name="casale">{{cite doi|10.1016/j.peva.2010.12.009}}</ref> The ''method of moments'' is an exact method which required log-quadratic time and can solve models with up to 10 classes of customers.<ref name="casale" /> Approximate algorithms have also been proposed with lower complexity.<ref>{{cite doi|10.1145/79147.214074}}</ref>

==Extensions==

The mean value analysis algorithm has been applied to a class of [[PEPA]] models describing [[queueing network]]s and the performance of a [[key distribution center]].<ref>{{cite doi|10.1093/comjnl/bxq064}}</ref>

==Software==

*[http://jmt.sourceforge.net/JMVA.html JMVA], a tool written in [[Java (programming language)|Java]] which implements MVA.<ref>{{cite doi|10.1145/1530873.1530877}}</ref>

==External links==
*[http://www.netlab.tkk.fi/opetus/s383143/kalvot/E_qnets.pdf J. Virtamo: Queuing networks]. Handout from Helsinki Tech gives good overview of Jackson's Theorem and MVA.
*[http://www.cs.utexas.edu/users/lam/Vita/Jpapers/Lam83.pdf Simon Lam: A simple derivation of the MVA algorithm]. Shows relationship between [[Buzen's algorithm]] and MVA.

==References==
{{reflist}}

{{Queueing theory}}

[[Category:Stochastic processes]]
[[Category:Queueing theory]]