formulasearchengine - User contributions [en]

Ramp function

2014-01-23T08:40:11Z

169.229.108.87:

[[File:Contact Process.svg|thumb|350px|The Contact Process (on a 1–D lattice): Active sites are indicated by grey circles and inactive sites by dotted circles. Active sites can activate inactive sites to either side of them at a rate ''r/2'' or become inactive at rate 1.]]
The '''contact process''' is a model of an [[interacting particle system]]. It is a continuous time [[Markov process]] with state space <math>\{0,1\}^S</math>, where <math>S</math> is a finite or countable [[Graph (mathematics)|graph]], usually '''Z'''<math>{}^d</math>. The process is usually interpreted as a model for the spread of an infection: if the state of the process at a given time is <math>\eta</math>, then a site <math>x</math> in <math>S</math> is "infected" if <math>\eta(x)=1</math> and healthy if <math>\eta(x)=0</math>. Infected sites become healthy at a constant rate, while healthy sites become infected at a rate proportional to the number infected neighbors. One can generalize the state space to <math>\{0,\ldots, \kappa\}^S</math>, such is called the '''multitype contact process'''. It represents a model when more than one type of infection is competing for space.

==Dynamics==
{{unreferenced|section|date=June 2012}}

More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site <math>x</math>,
:<math>1\rightarrow0\quad\mbox{at rate }1,</math>
:<math>0\rightarrow1\quad\mbox{at rate }\lambda\sum_{y:y\sim x}\eta(y),</math>
where the sum is over all the neighbors in <math>S</math> of <math>x</math>. This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and viceversa).

For each graph <math>S</math> there exists a critical value <math>\lambda_c</math> for the parameter <math>\lambda</math> so that if <math>\lambda>\lambda_c</math> then the 1's survive (that is, if there is at least one 1 at time zero, then at any time there are ones) with positive probability, while if <math>\lambda<\lambda_c</math> then the process dies out. For contact process on the integer lattice, a major breakthrough{{cn|date=June 2012}} came in 1990 when Bezuidenhout and [[G. R. Grimmett|Grimmett]] showed that the contact process also dies out at the critical value.{{cn|date=June 2012}} Their proof makes use of [[percolation theory]].

==Voter model==
{{Main|Voter model}}

The [[voter model]] (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process <math>\eta(x)</math> is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform.

===Discrete time process===

In the discrete time voter model in one dimension, <math>\xi_t(x): \mathbb{Z} \to \{0,1\}</math> represents the state of particle <math>x</math> at time <math>t</math>. Informally each individual is arranged on a line and can "see" other individuals that are within a radius, <math>r</math>. If more than a certain proportion, <math>\theta</math> of these people disagree then the individual changes her attitude, otherwise she keeps it the same. [[Rick Durrett|Durrett]] and Steif (1993) and Steif (1994) show that for large radii there is a critical value <math>\theta_c</math> such that if <math>\theta > \theta_c</math> most individuals never change, and for <math>\theta \in (1/2, \theta_c)</math> in the limit most sites agree. (Both of these results assume the probability of <math>\xi_0(x) = 1</math> is one half.)

This process has a natural generalization to more dimensions, some results for this are discussed in [[Rick Durrett|Durrett]] and Steif (1993).

===Continuous time process===

The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by [[Thomas M. Liggett|Liggett]] (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and [[Thomas M. Liggett|Liggett]] (1975).

This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are conflicting over territory and the animals are equally matched. A site is selected to be invaded by a neighbor at a given time.

== References ==
*{{cite journal
| last = Clifford
| first = Peter
| coauthors = Aidan Sudbury
| year = 1973
| title = A Model for Spatial Conflict
| journal = Biometrika
| volume = 60
| issue = 3
| pages = 581–588
| doi = 10.1093/biomet/60.3.581 }}
* {{cite journal
| last = Durrett
| first = Richard
|authorlink=Rick Durrett
| coauthors = Jeffrey E. Steif
| year = 1993
| title = Fixation Results for Threshold Voter Systems
| journal = The Annals of Probability
| volume = 21
| issue = 1
| pages = 232–247
| doi = 10.1214/aop/1176989403 }}
* {{cite journal
| last = Holley
| first = Richard A.
| coauthors = [[Thomas M. Liggett]]
| year = 1975
| title = Ergodic Theorems for Weakly Interacting Infinite Systems and The Voter Model
| journal = The Annals of Probability
| volume = 3
| issue = 4
| pages = 643–663
| doi = 10.1214/aop/1176996306 }}
* {{cite journal
| last = Steif
| first = Jeffrey E.
| year = 1994
| title = The Threshold Voter Automaton at a Critical Point
| journal = The Annals of Probability
| volume = 22
| issue = 3
| pages = 1121–1139
| doi = 10.1214/aop/1176988597 }}
*{{cite book
|last=Liggett
|first=Thomas M.
|authorlink=Thomas M. Liggett
|title=Interacting Particle Systems
|year=1985
|publisher=Springer Verlag
|location=New York
|isbn=0-387-96069-4 }}
* [[Thomas M. Liggett]], "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999.
* C. Bezuidenhout and [[G. R. Grimmett]], ''The critical contact process dies out'', Ann. Probab. '''18''' (1990), 1462 – 1482.

{{Stochastic processes}}

[[Category:Stochastic processes]]
[[Category:Lattice models]]

Euler's laws of motion

2013-12-21T03:46:41Z

169.229.108.121: /* Euler's second law */

A '''quorum''' is the minimum number of votes that a distributed transaction has to obtain in order to be allowed to perform an operation in a [[distributed system]]. A '''quorum'''-based technique is implemented to enforce consistent operation in a distributed system.

== Quorum-based techniques in distributed database systems ==
Quorum-based voting can be used as a [[Replication (computer science)#Database replication|replica]] control method<ref name="ozsu">
{{cite book |last1= Ozsu
|first1= Tamer M
|last2= Valduriez
|first2= Patrick
|title= Principles of distributed database systems
|edition= 2nd
|isbn = 0-13-691643-0
|year= 1991
|publisher= Prentice-Hall, Inc.
|location= Upper Saddle River, NJ
|chapter= 12
}}</ref>
, as well as a commit method to ensure [[Database transaction|transaction]] [[Atomicity (database systems)|atomicity]] in the presence of [[network partitioning]].<ref name="ozsu"/>

=== Quorum-based voting in commit protocols ===
In a distributed database system, a transaction could be executing its operations at multiple sites. Since atomicity requires every distributed transaction to be atomic, the transaction must have the same fate ([[Commit (data management)|commit]] or [[Rollback (data management)|abort]]) at every site. In case of network partitioning, sites are partitioned and the partitions may not be able to communicate with each other. This is where a quorum-based technique comes in. The fundamental idea is that a transaction is executed if the majority of sites vote to execute it.

Every site in the system is assigned a vote Vi. Let us assume that the total number of votes in the system is V and the abort and commit quorums are Va and Vc, respectively. Then the following rules must be obeyed in the implementation of the commit protocol:
# Va + Vc > V, where 0 < Vc, Va <math>\le</math> V.
# Before a transaction commits, it must obtain a commit quorum Vc. The total of at least one site that is prepared to commit and zero or more sites waiting <math>\ge</math> Vc.<ref>{{cite web|last=Skeen|first=Dale|title=A Quorum-based Commit Protocol|url=https://ecommons.library.cornell.edu/handle/1813/6323|publisher=Cornell University ECommons Library|accessdate=10 February 2013}}</ref>
# Before a transaction aborts, it must obtain an abort quorum Va The total of zero or more sites that are prepared to abort or any sites waiting <math>\ge</math> Va.

The first rule ensures that a transaction cannot be committed and aborted at the same time. The next two rules indicate the votes that a transaction has to obtain before it can terminate one way or the other.

=== Quorum-based voting for replica control ===
In replicated databases, a data object has copies present at several sites. To ensure [[serializability]], no two transactions should be allowed to read or write a data item concurrently. In case of replicated databases, a quorum-based replica control protocol can be used to ensure that no two copies of a data item are read or written by two transactions concurrently.

The quorum-based voting for replica control is due to [Gifford, 1979]<ref>
{{Cite document
| first = David K.
| last = Gifford
| contribution = Weighted voting for replicated data
| title = SOSP '79: Proceedings of the seventh ACM symposium on Operating systems principles
| year = 1979
| pages = 150–162
| place = Pacific Grove, California, United States
| url = http://doi.acm.org/10.1145/800215.806583
| publisher = ACM
| doi = 10.1145/800215.806583
| postscript = 
}}</ref>
. Each copy of a replicated data item is assigned a vote. Each operation then has to obtain a ''read quorum'' (Vr) or a ''write quorum'' (Vw) to read or write a data item, respectively. If a given data item has a total of V votes, the quorums have to obey the following rules:
# Vr + Vw > V
# Vw > V/2

The first rule ensures that a data item is not read and written by two transactions concurrently. The second rule ensures that two write operations from two transactions cannot occur concurrently on the same data item. The two rules ensure that one-copy serializability is maintained.

==See also==
* [[CAP theorem]]
* [[Database transaction]]
* [[Replication (computer science)]]
* [[Atomicity (database systems)]]

==References==
<references/>


[[Category:Database management systems]]
[[Category:Transaction processing]]

Ethanol (data page)

2013-12-06T09:49:04Z

169.229.108.77: /* Density of ethanol at various temperatures (kg/l or g/cm3) */

In [[mathematics]], a '''Bézout matrix''' (or '''Bézoutian''' or '''Bezoutiant''') is a special [[matrix (mathematics)#Square matrices and related definitions|square matrix]] associated with two [[polynomial]]s, introduced by {{harvs|txt|last=Sylvester|authorlink=James Joseph Sylvester|year=1853}} and {{harvs|txt|last=Cayley|year=1857|authorlink=Arthur Cayley}} and named after [[Étienne Bézout]]. Such matrices are sometimes used to test the [[stable polynomial|stability]] of a given polynomial.

==Definition==
Let ''f''(''z'') and ''g''(''z'') be two complex polynomials of degree at most ''n'' with coefficients (note that any coefficient could be zero):
:<math>f(z)=\sum_{i=0}^n u_i z^i,\quad\quad g(z)=\sum_{i=0}^n v_i z^i.</math>

The '''Bézout matrix''' of order ''n'' associated with the polynomials ''f'' and ''g'' is
:<math>B_n(f,g)=\left(b_{ij}\right)_{i,j=1,\dots,n}</math>
where the coefficients result from the identity
:<math>
\frac{f(x)g(y)-f(y)g(x)}{x-y}
=\sum_{i,j=1}^n b_{ij}\,x^{i-1}\,y^{j-1}.
</math>

It is in <math>\C^{n\times n}</math> and the entries of that matrix are such that if we note for each ''i'',''j''=1,...,n, <math>m_{ij}=\min\{i,n+1-j\}</math>, then:
:<math>b_{ij}=\sum_{k=1}^{m_{ij}}u_{j+k-1}v_{i-k}-u_{i-k}v_{j+k-1}.</math>

To each Bézout matrix, one can associate the following [[bilinear form]], called the Bézoutian:
:<math>\operatorname{Bez}:\C^n\times\C^n\to \C:(x,y)\mapsto \operatorname{Bez}(x,y)=x^*B_n(f,g)y.</math>

==Examples==
* For ''n''=3, we have for any polynomials ''f'' and ''g'' of degree (at most) 3:
:<math>B_3(f,g)=\left[\begin{matrix}u_1v_0-u_0 v_1 & u_2 v_0-u_0 v_2 & u_3 v_0-u_0 v_3\\u_2 v_0-u_0 v_2 & u_2v_1-u_1v_2+u_3v_0-u_0v_3 & u_3 v_1-u_1v_3\\u_3v_0-u_0v_3 & u_3v_1-u_1v_3 & u_3v_2-u_2v_3\end{matrix}\right].</math>

* Let <math>f(x)=3x^3-x</math> and <math>g(x)=5x^2+1</math> be two polynomials. Then:
:<math>B_4(f,g)=\left[\begin{matrix}-1 & 0 & 3 & 0\\0 &8 &0 &0 \\3&0&15&0\\0&0&0&0\end{matrix}\right].</math>
The last row and column are all zero as ''f'' and ''g'' have degree strictly less than ''n'' (equal 4). The other zero entries are because for each ''i''=0,...,n, either <math>u_i</math> or <math>v_i</math> is zero.

==Properties==
* <math>B_n(f,g)</math> is symmetric (as a matrix);
* <math>B_n(f,g)=-B_n(g,f)</math>;
* <math>B_n(f,f)=0</math>;
* <math>B_n(f,g)</math> is [[bilinear]]{{dn|date=December 2013}} in (''f'',''g'');
* <math>B_n(f,g)</math> is in <math>\mathbb{R}^{n\times n}</math> if ''f'' and ''g'' have real coefficients;
* <math>B_n(f,g)</math> is nonsingular with <math>n=max(deg(f),deg(g))</math> if and only if ''f'' and ''g'' have no common roots.
* <math>B_n(f,g)</math> with <math>n=max(deg(f),deg(g))</math> has [[determinant]] which is the [[resultant]] of ''f'' and ''g''.

==Applications==
An important application of Bézout matrices can be found in [[control theory]]. To see this, let ''f''(''z'') be a complex polynomial of degree ''n'' and denote by ''q'' and ''p'' the real polynomials such that ''f''(i''y'')=''q''(''y'')+i''p''(''y'') (where ''y'' is real). We also note ''r'' for the rank and ''σ'' for the signature of <math>B_n(p,q)</math>. Then, we have the following statements:
* ''f''(''z'') has ''n''-''r'' roots in common with its conjugate;
* the left ''r'' roots of ''f''(''z'') are located in such a way that:
** (''r''+''σ'')/2 of them lie in the open left half-plane, and
** (''r''-''σ'')/2 lie in the open right half-plane;
* ''f'' is [[stable polynomial|Hurwitz stable]] [[if and only if]] <math>B_n(p,q)</math> is [[Positive-definite matrix|positive definite]].

The third statement gives a necessary and sufficient condition concerning stability. Besides, the first statement exhibits some similarities with a result concerning [[Sylvester matrix|Sylvester matrices]] while the second one can be related to [[Routh-Hurwitz theorem]].

==References==

*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=Note sur la methode d’elimination de Bezout | year=1857 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002149818 | journal= J. Reine Angew. Math. | volume=53 | pages=366–367}}
*{{Citation | last2=Naĭmark | first2=M. A. | last1=Kreĭn | first1=M. G. | title=The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations | origyear=1936 | doi=10.1080/03081088108817420 | mr=638124 | year=1981 | journal=Linear and Multilinear Algebra | issn=0308-1087 | volume=10 | issue=4 | pages=265–308}}
* {{cite book |last1=Pan |first1=Victor |last2=Bini |first2=Dario |title=Polynomial and matrix computations |publisher=Birkhäuser |location=Basel, Switzerland |year=1994 |pages= |isbn=0-8176-3786-9 |oclc= |doi= }}
* {{cite book |last1=Pritchard |first1=Anthony J.|first2=Diederich |last2=Hinrichsen |title=Mathematical systems theory I: modelling, state space analysis, stability and robustness |publisher=Springer |location=Berlin |year=2005 |pages= |isbn=3-540-44125-5 |oclc= |doi= }}
*{{Citation | last1=Sylvester | first1=James Joseph | title=On a Theory of the Syzygetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and That of the Greatest Algebraical Common Measure | jstor=108572 | publisher=The Royal Society | year=1853 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=143 | pages= 407–548}}

{{DEFAULTSORT:Bezout matrix}}
[[Category:Polynomials]]
[[Category:Matrices]]