formulasearchengine - User contributions [en]

Copula (probability theory)

2015-01-09T22:52:48Z

162.93.80.1: /* External links */ Fixed broken link

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Friction stir welding

2014-01-29T18:52:17Z

162.93.65.2: Removed a paragraph of blatant product placement: "The DeltaN FS® system is a tool for friction stir welding developed by EADS Innovation Works"

{{Unreferenced|date=December 2009}}
{{Redirect|Conformable|the topic in [[geology]]|Unconformity}}

In [[mathematics]], a [[matrix (mathematics)|matrix]] is '''conformable''' if its dimensions are suitable for defining some operation (''e.g.'' addition, multiplication, etc.).

==Examples==
* In order to be conformable to addition or subtraction, matrices need to have the same dimensions. Thus ''A'', ''B'' and ''C'' all must have dimensions ''m'' × ''n'' in the equation

::<math>A + B = C</math>

:or

::<math>A - B = C</math>

:for some fixed ''m'' and ''n''.

* For [[matrix multiplication]], consider the equation

::<math>AB = C.</math>

:If ''A'' has dimensions ''m'' × ''n'', then ''B'' has to have dimensions ''n'' × ''p'' for some ''p'', so that ''C'' will have dimensions ''m'' × ''p''. That is, the number of columns in ''A'' must equal the number of rows in ''B'' for ''A'' and ''B'' to be conformable for multiplication in that sequence.

* Since squaring a matrix involves multiplying it by itself (<math>A^2=AA</math>) a matrix must be ''m''×''m'' (that is, it must be a [[square matrix]]) to be conformable for squaring. Thus for example only a square matrix can be [[Idempotent matrix|idempotent]].

*Only a square matrix is conformable for [[matrix inversion]]. However, the [[Moore-Penrose pseudoinverse]] and other [[generalized inverse]]s do not have this requirement.

* Only a square matrix is conformable for [[matrix exponentiation]].

==See also==
* [[Linear algebra]]

{{DEFAULTSORT:Conformable Matrix}}
[[Category:Linear algebra]]
[[Category:Matrices]]

Handicap (golf)

2013-12-30T22:07:10Z

162.93.80.2: /* Specific example */

An '''artificial chemistry''' is a [[computer model]] used to simulate various types of [[system]]s. Artificial chemistry is in some ways similar to a chemical reaction, hence the name. The field of artificial chemistry originated in [[artificial life]] but has shown to be a versatile method with applications in many fields such as [[chemistry]], [[economics]], [[sociology]] and [[linguistics]].

==Formal definition==
An artificial chemistry is defined in general as a triple (S,R,A). In some cases it is sufficient to define it as a tuple (S,I).

*S is the [[Set (mathematics)|set]] of possible molecules S={s1...,sn}, where n is the number of elements in the set, possibly infinite.
*R is a set of [[arity|n-ary]] [[operation (mathematics)|operation]]s on the molecules in S, the reaction rules R={r1...,rn}. Each rule ri is written like a chemical reaction a+b+c->a*+b*+c*. Note here that ri are operators, as opposed to +.
*A is an [[algorithm]] describing how to apply the rules R to a [[subset]] P<math>\subset</math>S.
*I are the interaction rules of the molecules in S.

==Types of artificial chemistries==

* depending on the space of possible molecules
** finite
** infinite

* depending on the type of reactions
** catalytic systems
** reactive systems

* depending on the space topology
** well stirred reactor
** topologically arranged (1,2,3 dimensional)

==Important concepts==

* Organizations: An organization is a set of molecules that is closed and self-maintaining. As such, it is a set that does not create anything outside itself, and such that any molecule inside the set can be generated within the set.
* Closed sets
* Self-maintaining sets
* Hasse diagram of organizations

==History of artificial chemistries==

Artificial chemistries emerged as a sub-field of [[artificial life]], in particular from [[strong artificial life]]. The idea behind this field was that if one wanted to build something alive, it had to be done by a combination of non-living entities. For instance, a cell is itself alive, and yet is a combination of non-living molecules. Artificial chemistry enlists, among others, researchers that believe in an extreme bottom-up approach to artificial life.

==Important contributors==

The first reference about artificial Chemistries come from a Technical paper written by [[John McCaskill]]. [[Walter Fontana]] working with [[Leo Buss]] then took up the work developing the [[AlChemy model]]. The model was presented at the second International Conference of Artificial Life. In his first papers he presented the concept of [[organization]], as a set of molecules that is algebraically closed and self-maintaining.

Two main schools of artificial chemistries have been in Japan and Germany. In Japan the main researchers have been [[Takashi Ikegami]], [[Hideaki Suzuki]] and [[Yasuhiro Suzuki]]. In [[Germany]], it was [[Wolfgang Banzhaf]], who, together with his students [[Peter Dittrich]] and [[Jens Ziegler]], developed various
artificial chemistry models. Their 2001 paper 'Artificial Chemistries - A Review' became a standard in the field. [[Jens Ziegler]], as part of his PhD thesis, proved that an artificial chemistry could be used to control a small Khepera robot. Among other models, [[Peter Dittrich]] developed the [[Seceder model]] which is able to explain group formation in society through some simple rules. Since then he became a professor in [[Jena]] where he investigates artificial chemistries as a way to define a general theory of [[constructive dynamical system]]s.

==Applications of artificial chemistries==

Artificial Chemistries are often used in the study of protobiology, in trying to bridge the gap between [[chemistry]] and [[biology]].
A further motivation to study artificial chemistries is the interest in constructive dynamical systems. [[Yasuhiro Suzuki]] has modeled various systems such as membrane systems, signaling pathways (P53), ecosystems, and enzyme systems by using his method, abstract rewriting system on multisets (ARMS).

==See also==
*[[Cellular automaton|Cellular automata]]
*[[Computational chemistry]] - the use of simplified models to simulate chemical interactions

==External links and references==
*[http://web.cs.mun.ca/~banzhaf/papers/alchemistry_review_MIT.pdf Artificial chemistries—a review (PDF)]
*[http://www.sq3.org.uk/wiki.pl?Papers Tim Hutton's Papers & Talks] - includes several papers on artificial chemistries for artificial life
*[http://www.minet.uni-jena.de/~biosys/twiki/bin/view.pl/ACHEM/WebHome the ARTIFICIAL CHEMISTRY webpage] of [[Peter Dittrich]]'s workgroup.
*[http://protobiology.org the protobiology.org website]

[[Category:Artificial life]]