formulasearchengine - User contributions [en]

Logit

2014-12-02T02:03:57Z

140.180.245.144: /* Comparison with probit */

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Dynamic mode decomposition

2014-11-21T21:19:47Z

140.180.247.30: Modified the summary of DMD to bring it up to speed with recent advances in the field.

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Elasticity of substitution

2014-10-17T18:58:30Z

140.180.244.150:

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Stochastic dominance

2014-09-12T15:30:20Z

140.180.250.101: "The same is true for non-expected utility maximizers with utility functions that are locally concave." <--- this either needs to be eliminated or made much more precise

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Probability vector

2014-07-15T21:00:39Z

140.180.252.226:

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Marginal distribution

2014-03-02T21:48:51Z

140.180.255.46: /* Continuous variables */

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Metric expansion of space

2014-03-01T20:47:39Z

140.180.252.23: /* Hubble's concerns over the rate of expansion */

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Statistical mechanics

2014-02-03T16:34:50Z

140.180.255.23: /* Applications outside thermodynamics */

In [[genetics]], '''shotgun sequencing''', also known as '''shotgun cloning''', is a method used for [[sequencing]] long [[DNA]] strands. It is named by analogy with the rapidly expanding, quasi-random firing pattern of a [[shotgun]].

Since the [[chain termination method]] of [[DNA sequencing]] can only be used for fairly short strands (100 to 1000 basepairs), longer sequences must be subdivided into smaller fragments, and subsequently re-assembled to give the overall sequence. Two principal methods are used for this: [[chromosome walking]], which progresses through the entire strand, piece by piece, and shotgun sequencing, which is a faster but more complex process, and uses random fragments.

In shotgun sequencing,<ref name="Staden">{{cite journal
| last = Staden
| first = R
| coauthors =
| title = A strategy of DNA sequencing employing computer programs
| journal = Nucleic Acids Research
| volume = 6
| issue = 7
| pages = 2601–10
| year = 1979
| pmid = 461197
| doi = 10.1093/nar/6.7.2601
| pmc = 327874}}</ref><ref>{{cite journal
| last = Anderson
| first = S
| coauthors =
| title = Shotgun DNA sequencing using cloned DNase I-generated fragments
| journal = Nucleic Acids Research
| volume = 9
| issue = 13
| pages = 3015–27
| year = 1981
| pmid = 6269069
| doi = 10.1093/nar/9.13.3015
| pmc = 327328}}</ref>
DNA is broken up randomly into numerous small segments, which are sequenced using the chain termination method to obtain ''reads''. Multiple overlapping reads for the target DNA are obtained by performing several rounds of this fragmentation and sequencing. Computer programs then use the overlapping ends of different reads to assemble them into a continuous sequence.<ref name="Staden" />

Shotgun sequencing was one of the precursor technologies that was responsible for enabling [[full genome sequencing]].

==Example==
For example, consider the following two rounds of shotgun reads:
{| class="wikitable"
|-
! Strand
! Sequence
|-
| Original
| <code><big>AGCATGCTGCAGTCATGCTTAGGCTA</big></code>
|-
| First shotgun sequence
| <code><big>AGCATGCTGCAGTCATGCT------- -------------------TAGGCTA</big></code>
|-
| Second shotgun sequence
| <code><big>AGCATG-------------------- ------CTGCAGTCATGCTTAGGCTA</big></code>
|-
| Reconstruction
| <code><big>AGCATGCTGCAGTCATGCTTAGGCTA</big></code>
|}

In this extremely simplified example, none of the reads cover the full length of the original sequence, but the four reads can be assembled into the original sequence using the overlap of their ends to align and order them. In reality, this process uses enormous amounts of information that are rife with ambiguities and sequencing errors. Assembly of complex genomes is additionally complicated by the great abundance of [[Repeated sequence (DNA)|repetitive sequence]], meaning similar short reads could come from completely different parts of the sequence.

Many overlapping reads for each segment of the original DNA are necessary to overcome these difficulties and accurately assemble the sequence. For example, to complete the [[Human Genome Project]], most of the human genome was sequenced at 12X or greater ''coverage''; that is, each base in the final sequence was present, on average, in 12 reads. Even so, current methods have failed to isolate or assemble reliable sequence for approximately 1% of the ([[Euchromatin|euchromatic]]) human genome.{{Citation needed|date=November 2011}}

==Whole genome shotgun sequencing==
Whole genome shotgun sequencing for small (4000 to 7000 basepair) genomes was already in use in 1979.<ref name=Staden /> Broader application benefited from [[DNA_sequencing_theory#Pairwise_end-sequencing|pairwise end sequencing]], known colloquially as ''double-barrel shotgun sequencing''. As sequencing projects began to take on longer and more complicated DNAs, multiple groups began to realize that useful information could be obtained by sequencing both ends of a fragment of DNA. Although sequencing both ends of the same fragment and keeping track of the paired data was more cumbersome than sequencing a single end of two distinct fragments, the knowledge that the two sequences were oriented in opposite directions and were about the length of a fragment apart from each other was valuable in reconstructing the sequence of the original target fragment. The first published description of the use of paired ends was in 1990<ref>{{cite journal
| last = Edwards
| first = A
| coauthors = Caskey, T
| title = Closure strategies for random DNA sequencing
| journal = Methods: A Companion to Methods in Enzymology
| volume = 3
| issue = 1
| pages = 41–47
| year = 1991
| doi = 10.1016/S1046-2023(05)80162-8 }}</ref>
as part of the sequencing of the human [[Hypoxanthine-guanine phosphoribosyltransferase|HGPRT]] locus, although the use of paired ends was limited to closing gaps after the application of a traditional shotgun sequencing approach. The first theoretical description of a pure pairwise end sequencing strategy, assuming fragments of constant length, was in 1991.<ref>{{cite journal
| last = Edwards
| first = A
| coauthors = Voss, H.; Rice, P.; Civitello, A.; Stegemann, J.; Schwager, C.; Zimmerman, J.; Erfle, H.; Caskey, T.; Ansorge, W.
| title = Automated DNA sequencing of the human HPRT locus
| journal = Genomics
| volume = 6
| pages = 593–608
| year = 1990
| pmid = 2341149
| doi = 10.1016/0888-7543(90)90493-E
| issue = 4 }}
</ref> At the time, there was community consensus that the optimal fragment length for pairwise end sequencing would be three times the sequence read length. In 1995 Roach et al.<ref>{{cite journal
| last = Roach
| first = JC
| coauthors = Boysen, C; Wang, K; Hood, L
| title = Pairwise end sequencing: a unified approach to genomic mapping and sequencing
| journal = Genomics
| volume = 26
| pages = 345–353
| year = 1995
| pmid = 7601461
| doi = 10.1016/0888-7543(95)80219-C
| issue = 2 }}</ref>
introduced the innovation of using fragments of varying sizes, and demonstrated that a pure pairwise end-sequencing strategy would be possible on large targets. The strategy was subsequently adopted by [[The Institute for Genomic Research]] (TIGR) to sequence the genome of the bacterium ''[[Haemophilus influenzae]]'' in 1995,<ref>{{cite journal
| last = Fleischmann
| first = RD
| coauthors = et al.
| title = Whole-genome random sequencing and assembly of Haemophilus influenzae Rd
| journal = Science
| volume = 269
| issue = 5223
| pages = 496–512
| year = 1995
| pmid = 7542800
| doi = 10.1126/science.7542800 |bibcode = 1995Sci...269..496F }}</ref> and then by [[Celera Genomics]] to sequence the ''[[Drosophila melanogaster]]'' (fruit fly) genome in 2000,<ref>{{cite journal
| last = Adams
| first = MD
| coauthors = et al.
| title = The genome sequence of Drosophila melanogaster
| journal = Science
| volume = 287
| issue = 5461
| pages = 2185–95
| year = 2000
| pmid = 10731132 | doi = 10.1126/science.287.5461.2185
| bibcode=2000Sci...287.2185.}}</ref>
and subsequently the human genome.

To apply the strategy, high-molecular-weight DNA is sheared into random fragments, size-selected (usually 2, 10, 50, and 150 kb), and [[clone (genetics)|clone]]d into an appropriate [[vector DNA|vector]]. The clones are then sequenced from both ends using the [[chain termination method]] yielding two short sequences. Each sequence is called an ''end-read'' or ''read'' and two reads from the same clone are referred to as ''[[Paired-end Tags|mate pairs]]''. Since the chain termination method usually can only produce reads between 500 and 1000 bases long, in all but the smallest clones, [[Paired-end Tags|mate pairs]] will rarely overlap.

The original sequence is reconstructed from the reads using sequence assembly [[software]]. First, overlapping reads are collected into longer composite sequences known as ''[[contig]]s''. Contigs can be linked together into ''scaffolds'' by following connections between [[Paired-end Tags|mate pairs]]. The distance between contigs can be inferred from the [[Paired-end Tags|mate pair]] positions if the average fragment length of the library is known and has a narrow window of deviation. Depending on the size of the gap between contigs, different techniques can be used to find the sequence in the gaps. If the gap is small (5-20kb) then the use of PCR to amplify the region is required, followed by sequencing. If the gap is large (>20kb) then the large fragment is cloned in special vectors such as BAC ([[Bacterial artificial chromosome]]s) followed by sequencing of the vector.

Proponents of this approach argue that it is possible to sequence the whole [[genome]] at once using large arrays of sequencers, which makes the whole process much more efficient than more traditional approaches. Detractors argue that although the technique quickly sequences large regions of DNA, its ability to correctly link these regions is suspect, particularly for genomes with repeating regions. As [[sequence assembly]] programs become more sophisticated and computing power becomes cheaper, it may be possible to overcome this limitation.{{Citation needed|date=February 2007}}

===Coverage===

Coverage (read depth or depth) is the average number of reads representing a given [[nucleotide]] in the reconstructed sequence. It can be calculated from the length of the original genome (''G''), the number of reads(''N''), and the average read length(''L'') as <math>N\times L/G</math>. For example, a hypothetical genome with 2,000 base pairs reconstructed from 8 reads with an average length of 500 nucleotides will have 2x redundancy. This parameter also enables one to estimate other quantities, such as the percentage of the genome covered by reads (sometimes also called coverage). A high coverage in shotgun sequencing is desired because it can overcome errors in base calling and assembly. The subject of [[DNA sequencing theory]] addresses the relationships of such quantities.

Sometimes a distinction is made between sequence coverage and physical coverage. Sequence coverage is the average number of times a base is read (as described above). Physical coverage is the average number of times a base is read or spanned by mate paired reads.<ref name="MeyersonFig1">{{cite doi|10.1038/nrg2841}}</ref>

==Hierarchical Shotgun sequencing==

[[File:Whole genome shotgun sequencing versus Hierarchical shotgun sequencing.png|thumb| In whole genome shotgun sequencing (top), the entire genome is sheared randomly into small fragments (appropriately sized for sequencing) and then reassembled. In hierarchical shotgun sequencing (bottom), the genome is first broken into larger segments. After the order of these segments is deduced, they are further sheared into fragments appropriately sized for sequencing.]]
Although shotgun sequencing can in theory be applied to a genome of any size, its direct application to the sequencing of large genomes (for instance, the [[Human Genome]]) was limited until the late 1990s, when technological advances made practical the handling of the vast quantities of complex data involved in the process.<ref name="genome sequencing">Dunham, I. ''Genome Sequencing''. Encyclopedia of Life Sciences, 2005. {{doi|10.1038/npg.els.0005378}}</ref> Historically, full-genome shotgun sequencing was believed to be limited by both the sheer size of large genomes and by the complexity added by the high percentage of repetitive DNA (greater than 50% for the human genome) present in large genomes.<ref name="venter">Venter, J. C. ‘’Shotgunning the Human Genome: A Personal View.’’ Encyclopedia of Life Sciences, 2006.</ref> It was not widely accepted that a full-genome shotgun sequence of a large genome would provide reliable data. For these reasons, other strategies that lowered the computational load of sequence assembly had to be utilized before shotgun sequencing was performed.<ref name="venter" />
In hierarchical sequencing, also known as top-down sequencing, a low-resolution [[Gene mapping#Physical Mapping|physical map]] of the genome is made prior to actual sequencing. From this map, a minimal number of fragments that cover the entire chromosome are selected for sequencing.<ref name="textbook">Gibson, G. and Muse, S. V. ''A Primer of Genome Science''. 3rd ed. P.84</ref> In this way, the minimum amount of high-throughput sequencing and assembly is required.
The amplified genome is first sheared into larger pieces (50-200kb) and cloned into a bacterial host using [[Bacterial artificial chromosome|BACs]] or [[P1-derived artificial chromosome|PACs]]. Because multiple genome copies have been sheared at random, the fragments contained in these clones have different ends, and with enough coverage (see section above) finding a '''scaffold''' of [[Contig#BAC contigs|BAC contigs]] that covers the entire genome is theoretically possible. This scaffold is called a '''tiling path'''.[[File:Tiling path.png|thumb|A BAC contig that covers the entire genomic area of interest makes up the tiling path.]] Once a tiling path has been found, the BACs that form this path are sheared at random into smaller fragments and can be sequenced using the shotgun method on a smaller scale.
Although the full sequences of the BAC contigs is not known, their orientations relative to one another are known. There are several methods for deducing this order and selecting the BACs that make up a tiling path. The general strategy involves identifying the positions of the clones relative to one another and then selecting the least number of clones required to form a contiguous scaffold that covers the entire area of interest. The order of the clones is deduced by determining the way in which they overlap.<ref name="genome map">Dear, P. H. ''Genome Mapping''. Encyclopedia of Life Sciences, 2005. {{doi|10.1038/npg.els.0005353}}.</ref> Overlapping clones can be identified in several ways. A small radioactively or chemically labeled probe containing a [[sequence-tagged site]] (STS) can be hybridized onto a microarray upon which the clones are printed.<ref name="genome map" /> In this way, all the clones that contain a particular sequence in the genome are identified. The end of one of these clones can then be sequenced to yield a new probe and the process repeated in a method called chromosome walking. Alternatively, the BAC [[BAC library#Genomic libraries|library]] can be restriction-digested. Two clones that have several fragment sizes in common are inferred to overlap because they contain multiple similarly spaced restriction sites in common.<ref name="genome map" /> This method of genomic mapping is called restriction fingerprinting because it identifies a set of restriction sites contained in each clone. Once the overlap between the clones has been found and their order relative to the genome known, a scaffold of a minimal subset of these contigs that covers the entire genome is shotgun-sequenced.<ref name="textbook" />
Because it involves first creating a low-resolution map of the genome, hierarchical shotgun sequencing is slower than whole-genome shotgun sequencing but relies less heavily on computer algorithms for genome assembly than whole-genome shotgun sequencing. The process of extensive BAC library creation and tiling path selection, however, make hierarchical shotgun sequencing slow and labor intensive. Now that the technology is available and the reliability of the data demonstrated,<ref name="venter" /> the speed and cost efficiency of whole-genome shotgun sequencing has made it the primary method for genome sequencing.

==Shotgun and Next-generation sequencing==
The classical shotgun sequencing was based on the Sanger sequencing method: this was the most advanced technique for sequencing genomes from about 1995–2005. The shotgun strategy is still applied today, however using other sequencing technologies, called [[DNA_sequencing#Next-generation_methods|next-generation sequencing]]. These technologies produce shorter reads (anywhere from 25–500bp) but many hundreds of thousands or millions of reads in a relatively short time (on the order of a day).<ref>{{cite journal
| last = Karl
| first = V
| coauthors = et al
| title = Next Generation Sequencing: From Basic Research to Diagnostics
| journal = Clinical Chemistry
| volume = 55
| issue = 4
| pages = 41–47
| year = 2009
| pmid = 19246620
| doi = 10.1373/clinchem.2008.112789 }}</ref>
This results in high coverage, but the assembly process is much more computationally expensive. These technologies are vastly superior to Sanger sequencing due to the high volume of data and the relatively short time it takes to sequence a whole genome.<ref>{{cite journal
| last = Metzker
| first = Michael L.
| title = Sequencing technologies - the next generation
| journal = Nat Rev Genet
| volume = 11
| issue = 1
| pages = 31–46
| year = 2010
| pmid = 19997069
| doi = 10.1038/nrg2626 }}</ref>

==See also==
*[[DNA sequencing theory]]

==References==
{{Reflist|30em}}

===Further reading===
{{Refbegin}}
*{{cite web | title=Shotgun sequencing comes of age | work=The Scientist | url=http://www.the-scientist.com/news/20021231/06 | accessdate=December 31, 2002}}
*{{cite web | title=Shotgun sequencing finds nanoorganisms - Probe of acid mine drainage turns up unsuspected virus-sized Archaea
| work=SpaceRef.com| url=http://www.spaceref.com/news/viewpr.html?pid=21532
| accessdate=December 23, 2006}}
*{{cite web | title=Genomic shotgun sequencing | work=biology science | url=http://www.cd-genomics.com/gene/shotgun.htm | accessdate=April 11, 2009}}
{{Refend}}

==External links==
{{NCBI-handbook}}

{{DEFAULTSORT:Shotgun Sequencing}}
[[Category:Molecular biology]]
[[Category:DNA sequencing]]

Knudsen diffusion

2014-02-01T20:41:37Z

140.180.243.156: removed nonstandard notation

'''Fiducial inference''' is one of a number of different types of [[statistical inference]]. These are rules, intended for general application, by which conclusions can be drawn from [[Sample (statistics)|samples]] of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of [[frequentist inference]], [[Bayesian inference]] and [[decision theory]]. However, fiducial inference is important in the [[history of statistics]] since its development led to the parallel development of concepts and tools in [[theoretical statistics]] that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it.

==Background==

The general approach of fiducial inference was proposed by [[Ronald Fisher|R A Fisher]].{{Citation needed|date=June 2011}} Here "fiducial" comes from the Latin for faith. Fiducial inference can be interpreted as an attempt to perform [[inverse probability]] without calling on [[prior probability distribution]]s.<ref>Quenouille (1958), Chapter 6</ref> Fiducial inference quickly attracted controversy and was never widely accepted.{{Citation needed|date=June 2011}} Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published.{{Citation needed|date=June 2011}} These counter-examples cast doubt on the coherence of "fiducial inference" as a system of [[statistical inference]] or [[inductive logic]]. Other studies showed that, where the steps of fiducial inference are said to lead to "fiducial probabilities" (or "fiducial distributions"), these probabilities lack the property of additivity, and so cannot constitute a [[probability measure]].{{Citation needed|date=June 2011}}

The concept of fiducial inference can be outlined by comparing its treatment of the problem of [[interval estimation]] in relation to other modes of statistical inference.
*A [[confidence interval]], in [[frequentist inference]], with [[coverage probability]] ''γ'' has the interpretation that among all confidence intervals computed by the same method, a proportion ''γ'' will contain the true value that needs to be estimated. This has either a repeated sampling (or [[frequency probability|frequentist]]) interpretation, or is the probability that an interval calculated from yet-to-be-sampled data will cover the true value. However, in either case, the probability concerned is not the probability that the true value is in the particular interval that has been calculated since at that stage both the true value and the calculated are fixed and are not random.

*[[Credible interval]]s, in [[Bayesian inference]], do allow a probability to be given for the event that an interval, once it has been calculated does include the true value, since it proceeds on the basis that a probability distribution can be associated with the state of knowledge about the true value, both before and after the sample of data has been obtained.

Fisher’s fiducial method was designed to meet perceived problems with the Bayesian approach, at a time when the frequentist approach had yet to be fully developed. Such problems related to the need to assign a [[prior distribution]] to the unknown values. The aim was to have a procedure whose results could still be given the interpretation that a probability could be assigned to whether or not a calculated interval includes the true value. The method proceeds by attempting to derive a "fiducial distribution", which is a measure of the degree of faith that can be put on any given value of the unknown parameter.

Unfortunately Fisher did not give a general definition of the fiducial method and he denied that the method could always be applied.{{Citation needed|date=June 2011}} His only examples were for a single parameter; different generalisations have been given when there are several parameters. A relatively complete presentation of the fiducial approach to inference is given by Quenouille (1958), while Williams (1959) describes the application of fiducial analysis to the [[Calibration (statistics)|calibration]] problem (also known as "inverse regression") in [[regression analysis]].<ref>Williams (1959, Chapter 6)</ref> Further discussion of fiducial inference is given by Kendall & Stuart (1973).<ref name=KS>Kendall, M.G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition'', Griffin. ISBN 0-85264-215-6 (Chapter 21)</ref>

==The fiducial distribution==
Fisher required the existence of a [[sufficient statistic]] for the fiducial method to apply. Suppose there is a single sufficient statistic for a single parameter. That is, suppose that the [[conditional distribution]] of the data given the statistic does not depend on the value of the parameter. For example suppose that ''n'' independent observations are uniformly distributed on the interval <math>[0,\omega]</math>. The maximum, ''X'', of the ''n'' observations is a [[sufficient statistic]] for ω. If only ''X'' is recorded and the values of the remaining observations are forgotten, these remaining observations are equally likely to have had any values in the interval <math>[0,X]</math>. This statement does not depend on the value of ω. Then ''X'' contains all the available information about ω and the other observations could have given no further information.

The [[cumulative distribution function]] of ''X'' is
:<math>F(x) = P(X \leq x) = P\left(\mathrm{all\ observations} \leq x\right) = \left(\frac{x}{\omega}\right)^n .</math>

Probability statements about ''X''/ω may be made. For example, given α, a value of ''a'' can be chosen with 0  <  ''a''  <  1 such that
:<math>P\left(a < \frac{X}{\omega}\right) = 1-a^n = \alpha.</math>
Thus
:<math>a = (1-\alpha)^{\frac{1}{n}} .</math>
Then Fisher says{{Citation needed|date=October 2010}} that this statement may be inverted into the form
:<math>P\left(\omega < \frac{X}{a}\right) = \alpha .</math>
In this latter statement, ω is now regarded as a [[random variable]] and ''X'' is fixed, whereas previously it was the other way round. This distribution of ω is the ''fiducial distribution'' which may be used to form fiducial intervals.

The calculation is identical to the [[pivotal quantity|pivotal method]] for finding a confidence interval, but the interpretation is different. In fact older books use the terms ''confidence interval'' and ''fiducial interval'' interchangeably.{{Citation needed|date=October 2010}} Notice that the fiducial distribution is uniquely defined when a single sufficient statistic exists.

The pivotal method is based on a random variable that is a function of both the observations and the parameters but whose distribution does not depend on the parameter. Such random variables are called [[pivotal quantity|pivotal quantities]]. By using these, probability statements about the observations and parameters may be made in which the probabilities do not depend on the parameters and these may be inverted by solving for the parameters in much the same way as in the example above. However, this is only equivalent to the fiducial method if the pivotal quantity is uniquely defined based on a sufficient statistic.

A fiducial interval could be taken to be just a different name for a confidence interval and give it the fiducial interpretation. But the definition might not then be unique.{{Citation needed|date=October 2010}} Fisher would have denied that this interpretation is correct: for him, the fiducial distribution had to be defined uniquely and it had to use all the information in the sample.{{Citation needed|date=October 2010}}

==Status of the approach==
After its formulation by Fisher, fiducial inference quickly attracted controversy and was never widely accepted. Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published.

Fisher admitted that "fiducial inference" had problems. Fisher wrote to [[George A. Barnard]] that he was "not clear in the head" about one problem on fiducial inference,<ref name=Z>{{cite journal|doi=10.1214/ss/1177011233|title=R. A. Fisher and Fiducial Argument
|first=S. L.
|last=Zabell 
|journal=Statistical Science
|volume=7
|issue=3
|date=Aug 1992
|pages=369–387
|jstor=2246073
}} (page 381)
</ref> and, also writing to Barnard, Fisher complained that his theory seemed to have only "an asymptotic approach to intelligibility".<ref name=Z/> Later Fisher confessed that "I don't understand yet what fiducial probability does. We shall have to live with it a long time before we know what it's doing for us. But it should not be ignored just because we don't yet have a clear interpretation".<ref name=Z/>

Lindley{{Citation needed|date=August 2011}}<ref>Sharon Bertsch McGrayne (2011) The Theory That Would Not Die. p. 133 {{full|date=November 2012}}</ref> showed that fiducial probability lacked additivity, and so was not a [[probability measure]]. Cox points out<ref>Cox (2006) p. 66</ref> that the same argument applies to the so-called "[[Confidence Distribution|confidence distribution]]" associated with [[confidence intervals]], so the conclusion to be drawn from this is moot. Fisher sketched "proofs" of results using fiducial probability. When the conclusions of Fisher's fiducial arguments are not false, many have been shown to also follow from Bayesian inference.{{Citation needed|date=February 2010}}<ref name=KS/>

In 1978, JG Pederson wrote that "the fiducial argument has had very limited success and is now essentially dead."<ref>{{Cite journal|doi=10.2307/1402811|first=JG| last=Pederson| title=Fiducial Inference |journal=International Statistical Review | volume= 46 | year= 1978 | pages= 147–170 | mr=0514060 | issue= 2|postscript=|jstor=1402811 }}</ref> Davison<ref>Davison, A.C. (2001) "''Biometrika'' Centenary: Theory and general methodology" ''[[Biometrika]]'' 2001 (page 12 in the republication edited by D. M. Titterton and [[David R. Cox]])</ref> wrote "A few subsequent attempts have been made to resurrect fiducialism, but it now seems largely of historical importance, particularly in view of its restricted range of applicability when set alongside models of current interest."

However, fiducial inference is still being studied<ref>Hannig, J. (2009) "Generalized fiducial inference for wavelet regression" ''[[Biometrika]]'', 96(4),847–860.</ref><ref>Hannig, J. (2009) "On generalized fiducial inference", ''Statistica Sinica'', 19, 491–544</ref> and other current work is ongoing under the name of [[confidence distribution]]s.

{{More footnotes|date=February 2010}}

==Notes==
{{Reflist}}

==References==
*[[David R. Cox|Cox, D. R.]] (2006). ''Principles of Statistical Inference'', CUP. ISBN 0-521-68567-2.
*{{cite book |last=Fisher |first=R A |coauthors= |title=Statistical Methods and Scientific Inference|year=1956 |publisher=Hafner |location=New York |isbn=0-02-844740-9}}
* [[Ronald A. Fisher|Fisher, Ronald]] "Statistical methods and scientific induction" ''Journal of the Royal Statistical Society, Series B'' 17 (1955), 69—78. (criticism of statistical theories of [[Jerzy Neyman]] and [[Abraham Wald]] from a fiducial perspective)
* {{cite journal|
title=Note on an Article by Sir Ronald Fisher
|authorlink=Jerzy Neyman
|first=Jerzy
|last=Neyman
|journal=[[Journal of the Royal Statistical Society, Series B]]
|volume=18
|issue=2
|year=1956
|pages=288–294
|jstor=2983716
}} (reply to Fisher 1955, which diagnoses a fallacy of "fiducial inference")
*{{cite book |editor=Tukey, J W, ed.|last=|first= |coauthors= |title=R.A. Fisher's Contributions to Mathematical Statistics|year=1950|publisher=Wiley |location=New York }}
*Quenouille, M.H. (1958) ''Fundamentals of Statistical Reasoning''. Griffin, London
*Williams, E.J. (1959) ''Regression Analysis'', Wiley {{LCCN|59011815}}
*Young, G.A., Smith, R.L. (2005) ''Essentials of Statistical Inference'', CUP. ISBN 0-521-83971-8
* Fraser, D.A.S. (1961) "The fiducial method and invariance." ''Biometrika'', '''48''', 261-80.
* Fraser, D.A.S. (1961) "On fiducial inference." ''Annals of Mathematical Statistics,'' '''32''', 661-676.

{{DEFAULTSORT:Fiducial Inference}}
[[Category:Statistical theory]]
[[Category:Statistical inference]]

Darboux's theorem (analysis)

2014-01-15T13:20:31Z

140.180.246.196: /* Proof */

{{merge|Cat state|date=February 2013|discuss=talk: Cat state #Proposed merge}}
In [[physics]], in the area of [[quantum information theory]], a '''Greenberger–Horne–Zeilinger state''' is a certain type of [[quantum entanglement|entangled]] [[quantum state]] which involves at least three subsystems (particles). It was first studied by D. Greenberger, M.A. Horne and [[Anton Zeilinger]] in 1989.<ref>{{citation |author=Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger |year=2007 |title=Going beyond Bell's Theorem |arxiv=0712.0921|bibcode = 2007arXiv0712.0921G }}</ref> They have noticed the extremely non-classical properties of the state.

==Definition==
The '''GHZ state''' is an [[quantum entanglement|entangled]] [[quantum state]] of {{math|''M'' > 2}} subsystems. In the case of each of the subsystems being two-dimensional, that is for [[qubit]]s, it reads
:<math>|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}.</math>
In simple words it is a quantum superposition of all subsystems being in state 0 with all of them being in state 1 (states 0 and 1 of a single subsystem are fully distinguishable).

The simplest one is the 3-qubit GHZ state:
<math>|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math>

==Properties==
There is no standard measure of multi-partite entanglement because different types of multi-partite entanglement exist which are not mutually convertible. Nonetheless, many measures define the GHZ to be [[Maximally entangled state|maximally entangled]].

Another important property of the GHZ state is that when we [[partial trace|trace]] over one of the three systems
we get
:<math>\mathrm{Tr}_3\big((|000\rangle + |111\rangle)(\langle 000|+\langle 111|) \big) = |00\rangle \langle 00| + |11\rangle \langle 11|</math>
which is an unentangled [[mixed state (physics)|mixed state]]. It has certain two-particle (qubit) correlations, but these are [[covariation|of a classical nature]].

On the other hand, if we were to measure one of the subsystems, in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either <math>|00\rangle</math> or <math>|11\rangle</math> which are unentangled pure states. This is unlike the [[W state]] which leaves bipartite entanglements even when we measure one of its subsystems.

The GHZ state leads to striking non-classical correlations (1989). Particles prepared in this state lead to a version of [[Bell's theorem]], which shows the internal inconsistency of the notion of elements-of-reality introduced in the famous [[Einstein–Podolsky–Rosen paradox|Einstein–Podolsky–Rosen]] paper. The first laboratory observation of GHZ correlations was by the group of [[Anton Zeilinger]] (1998). Many, more accurate observations followed. The correlations can be utilized in some [[quantum information]] tasks. These include multipartner [[quantum cryptography]] (1998) and [[communication complexity]] tasks (1997, 2004).

==See also==
* [[Bell's theorem]]
* [[Bell state]]
* [[GHZ experiment]]
* [[Local hidden variable theory]]
* [[Quantum entanglement]]
* [[Qubit]]
* [[Measurement in quantum mechanics]]

==References==
<references/>

{{DEFAULTSORT:Greenberger-Horne-Zeilinger state}}
[[Category:Quantum information theory]]

{{Physics-stub}}

[[de:GHZ-Experiment|Greenberger-Horne-Zeilinger]]
[[ja:Greenberger-Horne-Zeilinger 状態]]

Force field (chemistry)

2014-01-06T20:41:32Z

140.180.243.65: /* Post-translational modifications */ Added Unnatural Amino Acids to this section since both Post-translational modifications and unnatural amino acids are non-canonical

This page provides supplementary chemical data on [[ethylene]].

== Structure and properties == 

{| border="1" cellspacing="0" cellpadding="3" style="margin: 0 0 0 0.5em; background: #FFFFFF; border-collapse: collapse; border-color: #C0C090;"
! {{chembox header}} | Structure and properties
|-
| [[Index of refraction]]
| ? 
|-
| [[Dielectric constant]]
| ? ε0 at ? °C 
|-
| [[Bond strength (chemistry)|Bond strength]]
| ? 
|-
| [[Bond length]]
| C-C 133pm C-H 108 pm
|-
| [[Bond angle]]
| 121.7 
|-
| [[Magnetic susceptibility]]
| ? 
|-
|}

== Thermodynamic properties == 

{| border="1" cellspacing="0" cellpadding="6" style="margin: 0 0 0 0.5em; background: #FFFFFF; border-collapse: collapse; border-color: #C0C090;"
! {{chembox header}} | Phase behavior
|-
| [[Triple point]]
| 104 K (−169 °C), 120 Pa
|-
| [[Critical point (thermodynamics)|Critical point]]
| 282.5 K (9.4 °C), 50.6 bar
|-
| [[Standard enthalpy change of fusion|Std enthalpy change of fusion]], Δfus''H''<s>o</s>
| +3.35 kJ/mol
|-
| [[Standard entropy change of fusion|Std entropy change of fusion]], Δfus''S''<s>o</s>
| +32.2 J/(mol·K)
|-
| [[Standard enthalpy change of vaporization|Std enthalpy change of vaporization]], Δvap''H''<s>o</s>
| +13.5 kJ/mol
|-
| [[Standard entropy change of vaporization|Std entropy change of vaporization]], Δvap''S''<s>o</s>
| ? J/(mol·K)
|-
! {{chembox header}} | Solid properties
|-
| [[Standard enthalpy change of formation|Std enthalpy change of formation]], Δf''H''<s>o</s>solid
| ? kJ/mol
|-
| [[Standard molar entropy]], ''S''<s>o</s>solid
| ? J/(mol K)
|-
| [[Heat capacity]], ''cp''
| ? J/(mol K)
|-
! {{chembox header}} | Liquid properties
|-
| [[Standard enthalpy change of formation|Std enthalpy change of formation]], Δf''H''<s>o</s>liquid
| ? kJ/mol
|-
| [[Standard molar entropy]], ''S''<s>o</s>liquid
| 117.8 J/(mol K)
|-
| [[Heat capacity]], ''cp''
| 67.4 J/(mol K)
|-
! {{chembox header}} | Gas properties
|-
| [[Standard enthalpy change of formation|Std enthalpy change of formation]], Δf''H''<s>o</s>gas
| +52.47 kJ/mol
|-
| [[Standard molar entropy]], ''S''<s>o</s>gas
| 219.32 J/(mol K)
|-
| [[Enthalpy of combustion]], Δc''H''<s>o</s>
| –1387.4 kJ/mol
|-
| [[Heat capacity]], ''cp''
| 42.9 J/(mol K)
|-
| [[van der Waals equation|van der Waals' constants]]<ref name="lange1522">''Lange's Handbook of Chemistry'' 10th ed, pp 1522-1524</ref>
| a = 453.02 L2 kPa/mol2 b = 0.05714 liter per mole
|-
|}

==Vapor pressure of liquid==
{| border="1" cellspacing="0" cellpadding="6" style="margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;"
|-
| {{chembox header}} | '''P in mm of hg''' || 1 || 10 || 40 || 100 || 400 || 760 || 1520 || 3800 || 7600 || 15200 || 30400 ||
|-
| {{chembox header}} | '''T in °C''' || –168.3 || –153.2 || –141.3 || –131.8 || –113.9 || –103.7 || –90.8 || –71.1 || –52.8 || –29.1 || –1.5 ||  —
|} 
Table data obtained from ''CRC Handbook of Chemistry and Physics'', 44th ed.

[[Image:LogEthyleneVaporPressure.png|thumb|475px|left|'''log10 of ethylene vapor pressure.''' Uses formula: <math>\scriptstyle \log_{10} P_{mmHg} = 6.74756 - \frac {585.00} {255.00+T}</math>, obtained from ''Lange's Handbook of Chemistry'', 10th ed.]]{{Clear}}

== Spectral data == 

{| border="1" cellspacing="0" cellpadding="3" style="margin: 0 0 0 0.5em; background: #FFFFFF; border-collapse: collapse; border-color: #C0C090;"
! {{chembox header}} | [[UV/VIS spectroscopy|UV-Vis]]
|-
| [[Lambda-max|λmax]]
| ? [[Nanometre|nm]]
|-
| [[molar absorptivity|Extinction coefficient]], ε
| ?
|-
! {{chembox header}} | [[Infrared|IR]]
|-
| Major absorption bands
| 974 cm−1
|-
! {{chembox header}} | [[NMR Spectroscopy|NMR]]
|-
| [[Proton NMR]] 
|  
|-
| [[Carbon-13 NMR]] 
|  
|-
| Other NMR data 
|  
|-
! {{chembox header}} | [[Mass Spectrometry|MS]]
|-
| Masses of main fragments
|   
|-
|}

== Material Safety Data Sheet == 

The handling of this chemical may incur notable safety precautions. It is highly recommend that you seek the Material Safety Datasheet ([[Material safety data sheet|MSDS]]) for this chemical from a reliable source such as [http://www.siri.org/msds/index.php SIRI], and follow its directions.

==References==
<references/>
*[http://webbook.nist.gov/chemistry/ NIST Standard Reference Database]

Except where noted otherwise, data relate to [[standard ambient temperature and pressure]].

[[wikipedia:Chemical infobox|Disclaimer]] applies.

{{DEFAULTSORT:Ethylene (Data Page)}}
[[Category:Chemical data pages]]

Abelian group

2014-01-05T20:45:23Z

140.180.251.122:

{{other uses|Abelian (disambiguation)}}
{{dablink|"Abelian group" is also an archaic name for the [[Symplectic group]]}}
{{Group theory sidebar |Basics}}

In [[abstract algebra]], an '''abelian group''', also called a '''commutative group''', is a [[group (mathematics)|group]] in which the result of applying the group [[Operation (mathematics)|operation]] to two group elements does not depend on their order (the axiom of [[commutativity]]). Abelian groups [[generalization|generalize]] the [[arithmetic]] of addition of [[integer]]s. They are named after [[Niels Henrik Abel]].<ref>Jacobson (2009), p. 41</ref>

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a [[module (mathematics)|module]] and a [[vector space]], being its refinements. The theory of abelian groups is generally simpler than that of their [[nonabelian group|non-abelian]] counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.

{{Algebraic structures |Group}}

== Definition ==
An abelian group is a [[set (mathematics)|set]], ''A'', together with an [[Binary operation|operation]] "•" that combines any two [[element (mathematics)|elements]] ''a'' and ''b'' to form another element denoted {{nowrap|''a'' • ''b''}}. The symbol "•" is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, {{nowrap|(''A'', •)}}, must satisfy five requirements known as the ''abelian group axioms'':

;Closure: For all ''a'', ''b'' in ''A'', the result of the operation ''a'' • ''b'' is also in ''A''.
;Associativity: For all ''a'', ''b'' and ''c'' in ''A'', the equation (''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'') holds.
;Identity element: There exists an element ''e'' in ''A'', such that for all elements ''a'' in ''A'', the equation {{nowrap|''e'' • ''a'' {{=}} ''a'' • ''e'' {{=}} ''a''}} holds.
;Inverse element: For each ''a'' in ''A'', there exists an element ''b'' in ''A'' such that ''a'' • ''b'' = ''b'' • ''a'' = ''e'', where ''e'' is the identity element.
;Commutativity: For all ''a'', ''b'' in ''A'', ''a'' • ''b'' = ''b'' • ''a''.

More compactly, an abelian group is a [[commutative]] [[group (mathematics)|group]]. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".

== Facts ==

=== Notation ===
{{see also|Additive group|Multiplicative group}}
There are two main notational conventions for abelian groups – additive and multiplicative.

{| class="wikitable" style="margin: 1em auto 1em auto"
|-
! Convention
! Operation
! Identity
! Powers
! Inverse
|-
! Addition
| ''x'' + ''y'' || 0 || ''nx'' || −''x''
|-
! Multiplication
| ''x'' ⋅ ''y'' or ''xy'' || ''e'' or 1
| ''x''''n''
| ''x''−1
|}

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for [[module (mathematics)|module]]s and [[ring (mathematics)|ring]]s. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered.

=== Multiplication table ===
To verify that a [[finite group]] is abelian, a table (matrix) – known as a [[Cayley table]] – can be constructed in a similar fashion to a [[multiplication table]]. If the group is ''G'' = {''g''1 = ''e'', ''g''2, ..., ''g''''n''} under the operation ⋅, the (''i'', ''j'')'th entry of this table contains the product ''g''''i'' ⋅ ''g''''j''. The group is abelian [[if and only if]] this table is symmetric about the main diagonal.

This is true since if the group is abelian, then ''g''''i'' ⋅ ''g''''j'' = ''g''''j'' ⋅ ''g''''i''. This implies that the (''i'', ''j'')'th entry of the table equals the (''j'', ''i'')'th entry, thus the table is symmetric about the main diagonal.

== Examples ==
* For the [[integer]]s and the operation [[addition]] "+", denoted ('''Z''',+), the operation + combines any two integers to form a third integer, addition is associative, zero is the [[additive identity]], every integer ''n'' has an [[additive inverse]], −''n'', and the addition operation is commutative since ''m'' + ''n'' = ''n'' + ''m'' for any two integers ''m'' and ''n''.

* Every [[cyclic group]] ''G'' is abelian, because if ''x'', ''y'' are in ''G'', then ''xy'' = ''a''''m''''a''''n'' = ''a''''m'' + ''n'' = ''a''''n'' + ''m'' = ''a''''n''''a''''m'' = ''yx''. Thus the [[integer]]s, '''Z''', form an abelian group under addition, as do the [[modular arithmetic|integers modulo ''n'']], '''Z'''/''n'''''Z'''.

* Every [[Ring theory|ring]] is an abelian group with respect to its addition operation. In a [[commutative ring]] the invertible elements, or [[unit (ring theory)|units]], form an abelian [[multiplicative group]]. In particular, the [[real number]]s are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.

* Every [[subgroup]] of an abelian group is [[normal subgroup|normal]], so each subgroup gives rise to a [[quotient group]]. Subgroups, quotients, and [[Direct sum of groups|direct sums]] of abelian groups are again abelian.

In general, [[matrix (mathematics)|matrices]], even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of 2×2 [[rotation matrix|rotation matrices]].

== Historical remarks ==

Abelian groups were named for [[Norway|Norwegian]] [[mathematician]] [[Niels Henrik Abel]] by [[Camille Jordan]] because Abel found that the commutativity of the group of a [[polynomial]] implies that the roots of the polynomial can be [[solvability by radicals|calculated by using radicals]]. See Section 6.5 of Cox (2004) for more information on the historical background.

== Properties ==
If ''n'' is a [[natural number]] and ''x'' is an element of an abelian group ''G'' written additively, then ''nx'' can be defined as ''x'' + ''x'' + ... + ''x'' (''n'' summands) and (−''n'')''x'' = −(''nx''). In this way, ''G'' becomes a [[module (mathematics)|module]] over the [[ring (mathematics)|ring]] '''Z''' of integers. In fact, the modules over '''Z''' can be identified with the abelian groups.

Theorems about abelian groups (i.e. [[module (mathematics)|module]]s over the [[principal ideal domain]] '''Z''') can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of [[finitely generated abelian group]]s which is a specialization of the [[structure theorem for finitely generated modules over a principal ideal domain]]. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form '''Z'''/''p''k'''Z''' for ''p'' prime, and the latter is a direct sum of finitely many copies of '''Z'''.

If ''f'', ''g'' : ''G''  →  ''H'' are two [[group homomorphism]]s between abelian groups, then their sum ''f'' + ''g'', defined by (''f'' + ''g'') (''x'') = ''f''(''x'') + ''g''(''x''), is again a homomorphism. (This is not true if ''H'' is a non-abelian group.) The set Hom (''G'', ''H'') of all group homomorphisms from ''G'' to ''H'' thus turns into an abelian group in its own right.

Somewhat akin to the [[dimension]] of [[vector space]]s, every abelian group has a ''[[rank of an abelian group|rank]]''. It is defined as the [[cardinal number|cardinality]] of the largest set of [[linearly independent]] elements of the group. The integers and the [[rational number]]s have rank one, as well as every subgroup of the rationals.

== Finite abelian groups ==
Cyclic groups of [[modular arithmetic|integers modulo ''n'']], '''Z'''/''n'''''Z''', were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The [[automorphism group]] of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of [[Georg Frobenius]] and [[Ludwig Stickelberger]] and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of [[linear algebra]].

=== Classification ===
The '''fundamental theorem of finite abelian groups''' states that every finite abelian group ''G'' can be expressed as the direct sum of cyclic subgroups of [[prime number|prime]]-power order. This is a special case of the [[fundamental theorem of finitely generated abelian groups]] when ''G'' has zero [[rank of an abelian group|rank]].

The cyclic group '''Z'''''mn'' of order ''mn'' is isomorphic to the direct sum of '''Z'''''m'' and '''Z'''''n'' if and only if ''m'' and ''n'' are [[coprime]]. It follows that any finite abelian group ''G'' is isomorphic to a direct sum of the form

:<math>\mathbf{Z}_{k_1} \oplus \cdots \oplus \mathbf{Z}_{k_u}</math>

in either of the following canonical ways:
* the numbers ''k''1,...,''k''''u'' are powers of primes
* ''k''1 [[divisor|divides]] ''k''2, which divides ''k''3, and so on up to ''k''''u''.

For example, '''Z'''15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: '''Z'''15 ≅ {0, 5, 10} ⊕ {0, 3, 6, 9, 12}. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are [[group isomorphism|isomorphic]].

For another example, every abelian group of order 8 is isomorphic to either '''Z'''8 (the integers 0 to 7 under addition modulo 8), '''Z'''4 ⊕ '''Z'''2 (the odd integers 1 to 15 under multiplication modulo 16), or '''Z'''2 ⊕ '''Z'''2 ⊕ '''Z'''2.

See also [[list of small groups]] for finite abelian groups of order 16 or less.

=== Automorphisms ===
One can apply the [[#Classification|fundamental theorem]] to count (and sometimes determine) the [[Group isomorphism#Automorphisms|automorphisms]] of a given finite abelian group ''G''. To do this, one uses the fact (which will not be proved here) that if ''G'' splits as a direct sum ''H'' ⊕ ''K'' of subgroups of [[coprime]] order, then Aut(''H'' ⊕ ''K'') ≅ Aut(''H'') ⊕ Aut(''K'').

Given this, the fundamental theorem shows that to compute the automorphism group of ''G'' it suffices to compute the automorphism groups of the [[Sylow theorems|Sylow]] ''p''-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of ''p''). Fix a prime ''p'' and suppose the exponents ''e''''i'' of the cyclic factors of the Sylow ''p''-subgroup are arranged in increasing order:

:<math>e_1\leq e_2 \leq\cdots\leq e_n</math>

for some ''n'' > 0. One needs to find the automorphisms of

:<math>\mathbf{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbf{Z}_{p^{e_n}}.</math>

One special case is when ''n'' = 1, so that there is only one cyclic prime-power factor in the Sylow ''p''-subgroup ''P''. In this case the theory of automorphisms of a finite [[cyclic group]] can be used. Another special case is when ''n'' is arbitrary but ''e''''i'' = 1 for 1 ≤ ''i'' ≤ ''n''. Here, one is considering ''P'' to be of the form

:<math>\mathbf{Z}_p \oplus \cdots \oplus \mathbf{Z}_p,</math>

so elements of this subgroup can be viewed as comprising a vector space of dimension ''n'' over the finite field of ''p'' elements '''F'''''p''. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

:<math>\mathrm{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_p),</math>

where GL is the appropriate [[general linear group]]. This is easily shown to have order

:<math>|\mathrm{Aut}(P)|=(p^n-1)\cdots(p^n-p^{n-1}).</math>

In the most general case, where the ''e''''i'' and ''n'' are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

:<math>d_k=\mathrm{max}\{r|e_r = e_k^{\,}\}</math>

and

:<math>c_k=\mathrm{min}\{r|e_r=e_k^{\,}\}</math>

then one has in particular ''d''''k'' ≥ ''k'', ''c''''k'' ≤ ''k'', and

:<math>|\mathrm{Aut}(P)| = \left(\prod_{k=1}^n{p^{d_k} - p^{k-1}}\right)\left(\prod_{j=1}^n{(p^{e_j})^{n-d_j}}\right)\left(\prod_{i=1}^n{(p^{e_i-1})^{n-c_i+1}}\right).</math>

One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).

== Infinite abelian groups ==
Тhe simplest infinite abelian group is the [[infinite cyclic group]] '''Z'''. Any [[finitely generated abelian group]] ''A'' is isomorphic to the direct sum of ''r'' copies of '''Z''' and a finite abelian group, which in turn is decomposable into a direct sum of finitely many [[cyclic group]]s of primary orders. Even though the decomposition is not unique, the number ''r'', called the '''[[Rank of an abelian group|rank]]''' of ''A'', and the prime powers giving the orders of finite cyclic summands are uniquely determined.

By contrast, classification of general infinitely generated abelian groups is far from complete. [[Divisible group]]s, i.e. abelian groups ''A'' in which the equation ''nx'' = ''a'' admits a solution ''x'' ∈ ''A'' for any natural number ''n'' and element ''a'' of ''A'', constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to '''Q''' and [[Prüfer group]]s '''Q'''''p''/'''Z'''''p'' for various prime numbers ''p'', and the cardinality of the set of summands of each type is uniquely determined.<ref>For example, '''Q'''/'''Z''' ≅ ∑''p'' '''Q'''''p''/'''Z'''''p''.</ref> Moreover, if a divisible group ''A'' is a subgroup of an abelian group ''G'' then ''A'' admits a direct complement: a subgroup ''C'' of ''G'' such that ''G'' = ''A'' ⊕ ''C''. Thus divisible groups are [[injective module]]s in the category of abelian groups, and conversely, every injective abelian group is divisible ([[Baer's criterion]]). An abelian group without non-zero divisible subgroups is called '''reduced'''.

Two important special classes of infinite abelian groups with diametrically opposite properties are ''torsion groups'' and ''torsion-free groups'', exemplified by the groups '''Q'''/'''Z''' (periodic) and '''Q''' (torsion-free).

=== Torsion groups ===

An abelian group is called '''[[periodic group|periodic]]''' or '''[[torsion (algebra)|torsion]]''' if every element has finite [[order (group theory)|order]]. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second [[Prüfer theorems]] state that if ''A'' is a periodic group and either it has '''bounded exponent''', i.e. ''nA'' = 0 for some natural number ''n'', or if ''A'' is countable and the [[height (abelian group)|''p''-heights]] of the elements of ''A'' are finite for each ''p'', then ''A'' is isomorphic to a direct sum of finite cyclic groups.<ref>Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the [[direct product]] of the cyclic groups '''Z'''/''p''''m'''''Z''' for all natural ''m'' is not a direct sum of cyclic groups.</ref> The cardinality of the set of direct summands isomorphic to '''Z'''/''p''''m'''''Z''' in such a decomposition is an invariant of ''A''. These theorems were later subsumed in the '''Kulikov criterion'''. In a different direction, [[Helmut Ulm]] found an extension of the second Prüfer theorem to countable abelian ''p''-groups with elements of infinite height: those groups are completely classified by means of their [[Ulm invariant]]s.

=== Torsion-free and mixed groups ===

An abelian group is called '''torsion-free''' if every non-zero element has infinite order. Several classes of [[torsion-free abelian group]]s have been studied extensively:

* [[Free abelian group]]s, i.e. arbitrary direct sums of '''Z'''
* [[Cotorsion group|Cotorsion]] and [[algebraically compact module|algebraically compact]] torsion-free groups such as the [[p-adic integer|''p''-adic integers]]
* [[Slender group]]s

An abelian group that is neither periodic nor torsion-free is called '''mixed'''. If ''A'' is an abelian group and ''T''(''A'') is its [[torsion subgroup]] then the factor group ''A''/''T''(''A'') is torsion-free. However, in general the torsion subgroup is not a direct summand of ''A'', so ''A'' is ''not'' isomorphic to ''T''(''A'') ⊕ ''A''/''T''(''A''). Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups.

=== Invariants and classification ===
One of the most basic invariants of an infinite abelian group ''A'' is its [[rank of an abelian group|rank]]: the cardinality of the maximal [[linearly independent]] subset of ''A''. Abelian groups of rank 0 are precisely the periodic groups, while [[torsion-free abelian groups of rank 1]] are necessarily subgroups of '''Q''' and can be completely described. More generally, a torsion-free abelian group of finite rank ''r'' is a subgroup of '''Q'''''r''. On the other hand, the group of [[p-adic integer|''p''-adic integers]] '''Z'''''p'' is a torsion-free abelian group of infinite '''Z'''-rank and the groups '''Z'''''p''''n'' with different ''n'' are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are [[pure subgroup|pure]] and [[basic subgroup|basic]] subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by [[Irving Kaplansky]], László Fuchs, [[Phillip Griffith]], and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent results.

=== Additive groups of rings ===
The additive group of a [[ring (mathematics)|ring]] is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:

* [[Tensor product]]
* Corner's results on countable torsion-free groups
* Shelah's work to remove cardinality restrictions

== Relation to other mathematical topics ==
Many large abelian groups possess a natural [[topology]], which turns them into [[topological group]]s.

The collection of all abelian groups, together with the [[Group homomorphism|homomorphisms]] between them, forms the [[category of abelian groups|category]] '''Ab''', the prototype of an [[abelian category]].

Nearly all well-known [[algebraic structure]]s other than [[Boolean algebra (structure)|Boolean algebras]], are [[Decidability (logic)|undecidable]]. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research:
*Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the [[torsion-free abelian groups of rank 1|rank 1]] case are well understood;
*There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
*While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
*Many mild extensions of the first order theory of abelian groups are known to be undecidable.
*Finite abelian groups remain a topic of research in computational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the [[set theory]] commonly assumed to underlie all of mathematics. Take the [[Whitehead problem]]: are all Whitehead groups of infinite order also [[free abelian group]]s? In the 1970s, [[Saharon Shelah]] proved that the Whitehead problem is:
* [[list of statements undecidable in ZFC|Undecidable in ZFC]], the conventional [[axiomatic set theory]] from which nearly all of present day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
* Undecidable even if [[ZFC]] is augmented by taking the [[generalized continuum hypothesis]] as an axiom;
* Positively answered if ZFC is augmented with the axiom of [[Constructible universe|constructibility]] (see [[statements true in L]]).

== A note on the typography ==
Among mathematical [[adjective]]s derived from the [[proper name]] of a [[mathematician]], the word "abelian" is rare in that it is often spelled with a lowercase '''a''', rather than an uppercase '''A''', indicating how ubiquitous the concept is in modern mathematics.<ref>[http://www.maa.org/devlin/devlin_04_04.html Abel Prize Awarded: The Mathematicians' Nobel]</ref>

== See also ==

*[[Abelianization]]
*[[Class field theory]]
*[[Commutator subgroup]]
*[[Dihedral group of order 6]], the smallest non-Abelian group
*[[Elementary abelian group]]
*[[Pontryagin duality]]
*[[Pure injective module]]
*[[Pure projective module]]

== Notes ==
<references />

== References ==
* {{cite book |last=Cox |first=David |year=2004 |title=Galois Theory |publisher=[[Wiley-Interscience]] |mr=2119052 }}
* {{cite book |last=Fuchs |first=László |year=1970 |title=Infinite Abelian Groups, Vol. I |series=Pure and Applied Mathematics |volume=36-I |publisher=[[Academic Press]] |mr=0255673 }}
* {{cite book |last=Fuchs |first=László |year=1973 |title=Infinite Abelian Groups, Vol. II |series=Pure and Applied Mathematics |volume=36-II |publisher=[[Academic Press]] |mr=0349869 }}
* {{cite book |first=Phillip A. |last=Griffith |year=1970 |title=Infinite Abelian group theory |series=Chicago Lectures in Mathematics |publisher=[[University of Chicago Press]] |isbn=0-226-30870-7}}
* {{cite book |last=Herstein |first=I. N. |year=1975 |title=Topics in Algebra |edition=2nd |publisher=[[John Wiley & Sons]] |isbn=0-471-02371-X}}
* {{cite journal |last1=Hillar |first1=Christopher |last2=Rhea |first2=Darren |year=2007 |title=Automorphisms of finite abelian groups |journal=[[American Mathematical Monthly]] |volume=114 |issue=10 |pages=917–923 |arxiv=math/0605185}}
* {{Cite book| last=Jacobson| first=Nathan | year=2009| title=Basic Algebra I | edition=2nd | publisher=[[Dover Publications]] | isbn = 978-0-486-47189-1}}
* {{cite journal |last=Szmielew |first=Wanda |year=1955 |title=Elementary properties of abelian groups |journal=[[Fundamenta Mathematicae]] |volume=41 |pages=203–271}}

{{DEFAULTSORT:Abelian Group}}
[[Category:Abelian group theory]]
[[Category:Properties of groups]]
[[Category:Niels Henrik Abel]]

Lattice theorem

2013-11-07T14:06:23Z

140.180.243.169: Undid revision 580600729 by 140.180.243.169 (talk)

In [[field theory (physics)|field theory]], the '''Stueckelberg action''' (named after [[Ernst Stueckelberg]], (1938), "Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kräfte", ''Helv. Phys. Acta.'' '''11:''' 225) describes a massive spin-1 field as an '''R''' (the [[real number]]s are the [[Lie algebra]] of [[U(1)]]) [[Yang-Mills theory]] coupled to a real [[scalar field]] φ. This scalar field takes on values in a real 1D [[affine representation]] of '''R''' with'' m'' as the [[coupling constant|coupling strength]].

:<math>\mathcal{L}=-\frac{1}{4}(\partial^\mu A^\nu-\partial^\nu A^\mu)(\partial_\mu A_\nu-\partial_\nu A_\mu)+\frac{1}{2}(\partial^\mu \phi+m A^\mu)(\partial_\mu \phi+m A_\mu) </math>

This is a special case of the [[Higgs mechanism]], where, in effect, the mass of the Higgs scalar excitation has been taken to infinity, so the Higgs has decoupled and is ignorable, resulting in a nonlinear, affine representation of the field, instead of a [[linear representation]]—— in contemporary terminology, a ''U(1)'' nonlinear σ-model.

Gauge-fixing φ=0, yields the [[Proca action]].

This explains why, unlike the case for non-abelian vector fields, [[quantum electrodynamics]] with a massive photon '''''is''''', in fact, [[renormalizable]], even though it is not manifestly [[gauge invariant]] (after the Stückelberg scalar has been eliminated in the Proca action).

==The Stueckelberg Extension of the Standard Model==
The Stueckelberg Lagrangian of the ''StSM'' (Stueckelberg extension of the Standard Model) consists of a [[gauge invariant]] kinetic term for a massive [[U(1)]] gauge field. Such a term can be implemented into the Lagrangian of the [[Standard Model]]
without destroying the renormalizability of the theory and further provides a mechanism for
mass generation that is distinct from the [[Higgs mechanism|Higgs]] mechanism in the context of [[Abelian group|Abelian]] gauge theories.

The model involves a non-trivial
mixing of the Stueckelberg and the Standard Model sectors by including an additional term in the effective Lagrangian of the Standard Model given by
:<math>\mathcal{L}_{St}=-\frac{1}{4}C_{\mu \nu }C^{\mu\nu }+g_XC_{\mu }\mathcal{J}_X^{\mu }-\frac{1}{2}\left(\partial _{\mu }\sigma +M_1C_{\mu}+M_2B_{\mu }\right)^2.</math>

The first term above is the Stueckelberg field strength, <math>M_1</math> and <math>M_2</math> are topological mass parameters and <math>\sigma </math> is the axion.
After symmetry breaking in the electroweak sector the photon remains massless. The model predicts a new type of gauge boson dubbed <math>Z'_{St}</math> which inherits a very distinct narrow [[decay width]] in this model. The St sector of the StSM decouples from the SM in limit <math>M_2/M_1 \to 0</math>.

Stueckelberg type couplings arise quite naturally in theories involving [[compactification (physics)|compactification]]s of higher dimensional [[string theory]], in particular, these couplings appear in the dimensional reduction of the ten dimensional N = 1 [[supergravity]] coupled to [[supersymmetric]] Yang-Mills gauge fields in the presence of internal gauge fluxes. In the context of intersecting [[D-brane]] model building, products of U(N) gauge groups are broken to their [[SU(N)]] subgroups via the Stueckelberg couplings and thus the Abelian gauge fields become massive. Further, in a much simpler fashion one may consider a model with only one extra dimension (a type of [[Kaluza–Klein model]]) and compactify down to a four dimensional theory. The resulting Lagrangian will contain massive vector gauge bosons that acquire masses through the Stueckelberg mechanism.

==See also==
* [[Higgs mechanism#Affine Higgs mechanism]]

==References==
{{reflist}}
*[http://www.stuckelberg.org/ The edited PDF files of the physics course of Professor Stueckelberg, openly accessible, with commentary and complete biographical documents.]
*[http://arxiv.org/abs/hep-ph/0503208 Review:Stueckelberg Extension of the Standard Model and the MSSM]
* Boris Kors, Pran Nath: [http://arxiv.org/abs/hep-ph/0402047], [http://arxiv.org/abs/hep-ph/0406167], [http://arxiv.org/abs/hep-ph/0503208]
* Daniel Feldman, Zuowei Liu, Pran Nath: [http://arxiv.org/abs/hep-ph/0603039], [http://arxiv.org/abs/hep-ph/0606294]

[[Category:Quantum field theory]]
[[Category:Particle physics]]
[[Category:Symmetry]]
[[Category:Theoretical physics]]

Thermal desorption spectroscopy

2013-11-05T22:29:21Z

140.180.253.70: /* Desorption */ Physisorption occurs too.

:''Not to be confused with s-p mixing in Molecular Orbital theory. See [[Molecular orbital diagram]].

In [[chemistry]], '''hybridisation''' (or '''[[American and British English spelling differences#-ise, -ize (-isation, -ization)|hybridization]]''') is the concept of mixing [[atomic orbital]]s into new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form [[chemical bond]]s in [[valence bond theory]]. Hybrid orbitals are very useful in the explanation of [[molecular geometry]] and atomic bonding properties. Although sometimes taught together with the [[VSEPR|valence shell electron-pair repulsion (VSEPR) theory]], valence bond and hybridisation are in fact not related to the VSEPR model.<ref>{{citation | last= Gillespie | first=R.J. | year=2004 | title=Teaching molecular geometry with the VSEPR model | journal=Journal of Chemical Education | volume=81 | issue=3 | pages=298–304 | doi=10.1021/ed081p298 |bibcode = 2004JChEd..81..298G }}</ref>

== Historical development ==

[[Chemist]] [[Linus Pauling]] first developed the hybridisation theory in order to explain the structure of [[molecule]]s such as [[methane]] (CH4).<ref>{{citation | last=Pauling | first=L. | year=1931 | title=The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules | journal=[[Journal of the American Chemical Society]] | volume=53 |issue=4 | pages=1367–1400 | doi=10.1021/ja01355a027 }}</ref> This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective [[heuristic]] for rationalising the structures of [[organic compounds]].

Orbitals are a model representation of the behaviour of electrons within molecules. In the case of simple hybridisation, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the [[Schrödinger equation]] can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen. Hybrid orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. It provides a [[quantum mechanics|quantum mechanical]] insight to [[Lewis structure]]s. Hybridisation theory finds its use mainly in organic chemistry.

== spx and sdx terminology ==
This terminology describes the weight of the respective components of a hybrid orbital. For example, in methane, the C hybrid orbital which forms each [[carbon]]–[[hydrogen]] bond consists of 25% s character and 75% p character and is thus described as sp3 (read as ''s-p-three'') hybridised. [[Quantum mechanics]] describes this hybrid as an sp3 [[wavefunction]] of the form N(s + {{sqrt|3}}pσ), where N is a [[normalization constant]] (here 1/2) and pσ is a p orbital directed along the C-H axis to form a [[sigma bond]]. The ratio of coefficients (denoted λ in general) is [[square root of 3|{{sqrt|3}}]] in this example. Since the [[electron density]] associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ2 = 3. The p character or the weight of the p component is N2λ2 = 3/4.

For atoms forming equivalent hybrids with no lone pairs, there is a correspondence to the number and type of orbitals used. For example, sp3 hybrids are formed from one s and three p orbitals. However, in all other cases, there is no such correspondence. The two bond-forming hybrid orbitals of oxygen in water can be described as sp4, which means that they have 20% s character and 80% p character, but does ''not'' imply that they are formed from one s and four p orbitals. As a result, the amount of ''p''-character is not restricted to integer values; i.e., hybridisations like ''sp''2.5 are also readily described. For more information see [[variable hybridization]].

An analogous notation is used to describe sdx hybrids. For example, the [[permanganate ion]] (MnO4–) has sd3 hybridisation with orbitals that are 25% s and 75% d.

== Types of hybridisation ==

=== sp3 hybrids ===
[[Image:AE4h.svg|thumb|150px|Four sp3 orbitals.]]
Hybridisation describes the bonding atoms from an atom's point of view. That is, for a tetrahedrally coordinated carbon (e.g., [[methane]] CH4), the carbon should have 4 orbitals with the correct symmetry to bond to the 4 hydrogen atoms.

Carbon's [[ground state]] configuration is 1s2 2s2 2p''x''1 2p''y''1 or more easily read:

{| cellpadding=4px align=center
| rowspan=2 |C||↑↓||↑↓||↑||↑|| 
|-
|{{overline|1s}}
|{{overline|2s}}
|{{overline|2p''x''}}
|{{overline|2p''y''}}
|{{overline|2p''z''}}
|}

The carbon atom can utilize its two singly occupied p-type orbitals (the designations p''x'' p''y'' or p''z'' are meaningless at this point, as they do not fill in any particular order), to form two [[covalent bond]]s with two hydrogen atoms, yielding the "free radical" [[methylene radical|methylene]] CH2, the simplest of the [[carbene]]s. The carbon atom can also bond to four hydrogen atoms by an excitation of an electron from the doubly occupied 2s orbital to the empty 2p orbital, so that there are four singly occupied orbitals.


{| cellpadding=4px align=center
| rowspan=2 |C*||↑↓||↑||↑||↑||↑
|-
|{{overline|1s}}
|{{overline|2s}}
|{{overline|2p''x''}}
|{{overline|2p''y''}}
|{{overline|2p''z''}}
|}

As the energy released by formation of two additional bonds more than compensates for the excitation energy required, the formation of four C-H bonds is energetically favoured.

Quantum mechanically, the lowest energy is obtained if the four bonds are equivalent which requires that they be formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained which are linear combinations of the valence-shell (core orbitals are almost never involved in bonding) s and p wave functions<ref>McMurray, J. (1995). Chemistry Annotated Instructors Edition (4th ed.). Prentice Hall. p. 272. ISBN 978-0-131-40221-8</ref> which are the four '''sp3 hybrids'''.

{| cellpadding=4px align=center
| rowspan=2 |C*||↑↓||↑||↑||↑||↑
|-
|{{overline|1s}}
|{{overline|sp3}}
|{{overline|sp3}}
|{{overline|sp3}}
|{{overline|sp3}}
|}

In CH4, four sp3 hybrid orbitals are overlapped by [[hydrogen]] 1s orbitals, yielding four [[sigma bond|σ (sigma) bonds]] (that is, four single covalent bonds) of equal length and strength.

[[Image:Ch4 hybridization.svg|A schematic presentation of hybrid orbitals overlapping hydrogen orbitals]] translates into [[Image:Ch4-structure.png|Methane's tetrahedral shape]]

=== sp2 hybrids ===
[[Image:AE3h.svg|thumb|150px|Three sp2 orbitals.]]
[[Image:Ethene-2D-flat.png|thumb|120px|Ethene structure]]

Other carbon based compounds and other molecules may be explained in a similar way as methane. For example, [[ethene]] (C2H4) has a double bond between the carbons.

For this molecule, carbon will sp2 hybridise, because one [[pi bond|π (pi) bond]] is required for the [[covalent bond|double bond]] between the carbons, and only three σ bonds are formed per carbon atom. In '''sp2 hybridisation''' the 2s orbital is mixed with only two of the three available 2p orbitals:

{| cellpadding=4px align=center
| rowspan=2 |C*||↑↓||↑||↑||↑||↑
|-
|{{overline|1s}}
|{{overline|sp2}}
|{{overline|sp2}}
|{{overline|sp2}}
|{{overline|2p}}
|}

forming a total of three sp2 orbitals with one p orbital remaining. In ethylene ([[ethene]]) the two carbon atoms form a σ bond by overlapping two sp2 orbitals and each carbon atom forms two covalent bonds with hydrogen by s–sp2 overlap all with 120° angles. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. The hydrogen–carbon bonds are all of equal strength and length, which agrees with experimental data.

=== sp hybrids ===
[[Image:AE2h.svg|thumb|150px|Two sp orbitals]]
The chemical bonding in compounds such as [[alkyne]]s with triple bonds is explained by '''sp hybridisation'''.

{| cellpadding=4px align=center
| rowspan=2 |C*||↑↓||↑||↑||↑||↑
|-
|{{overline|1s}}
|{{overline|sp}}
|{{overline|sp}}
|{{overline|2p}}
|{{overline|2p}}
|}

In this model, the 2s orbital mixes with only one of the three p orbitals resulting in two sp orbitals and two remaining unchanged p orbitals. The chemical bonding in [[acetylene]] (ethyne) (C2H2) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional [[pi bonds|π bonds]] formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.

== Hybridisation and molecule shape ==
Hybridisation helps to explain [[molecular geometry|molecule shape]] since the angles between bonds are (approximately) equal to the angles between hybrid orbitals, as explained above for the tetrahedral geometry of methane. As another example, the three sp2 hybrid orbitals are at angles of 120° to each other, so this hybridisation favours [[trigonal planar molecular geometry]] with bond angles of 120°. Other examples are given in the table below.

{| class="wikitable"
! Classification
! Main group
! Transition metal<ref>{{cite book |last1=Weinhold |first1= Frank |last2= Landis |first2= Clark R. |title=Valency and bonding: A Natural Bond Orbital Donor-Acceptor Perspective |location=Cambridge |publisher=Cambridge University Press |year=2005 |pages=381–383 |isbn=978-0-521-83128-4}}</ref>
|-----
! AX2
|
* [[linear molecular geometry|Linear]] (180°)
* sp hybridisation
* E.g., CO2
|
* [[bent molecular geometry|Bent]] (90°)
* sd hybridisation
* E.g., VO2+
|-----
! AX3
|
* [[trigonal planar molecular geometry|Trigonal planar]] (120°)
* sp2 hybridisation
* E.g., BCl3
|
* [[trigonal pyramidal molecular geometry|Trigonal pyramidal]] (90°)
* sd2 hybridisation
* E.g., CrO3
|-----
! AX4
|
*[[tetrahedral molecular geometry|Tetrahedral]] (109.5°)
* sp3 hybridisation
* E.g., CCl4
|
*[[tetrahedral molecular geometry|Tetrahedral]] (109.5°)
* sd3 hybridisation
* E.g., MnO4−
|-----
! AX5
| align="center" | -
|
* [[Square pyramidal molecular geometry|Square pyramidal]] (66°, 114°)<ref name=Kaupp>{{cite journal
| title = "Non-VSEPR" Structures and Bonding in d(0) Systems
| first = Martin | last = Kaupp
| journal = Angew Chem Int Ed Engl.
| year = 2001
| volume = 40
| issue = 1
| pages = 3534–3565
| doi = 10.1002/1521-3773(20011001)40:19<3534::AID-ANIE3534>3.0.CO;2-#
}}</ref><ref name=RBKing>{{cite journal |journal= Coordination Chemistry Reviews |volume= 197 |year= 2000 |pages= 141–168 |title= Atomic orbitals, symmetry, and coordination polyhedra |first= R. Bruce |last= King }}</ref>
* sd4 hybridisation
* E.g., Ta(CH3)5
|-----
! AX6
| align="center" | -
|
* [[trigonal prismatic molecular geometry|Trigonal prismatic]] (63°, 117°)<ref name=Kaupp/><ref name=RBKing/>
* sd5 hybridisation
* E.g., W(CH3)6
|}

===Main group compounds with lone pairs===
For main group compounds with lone electron pairs, the s orbital lone pair can be hybridised to a certain extent with the bond pairs.<ref name=Weinhold>{{cite web |url=http://isites.harvard.edu/fs/docs/icb.topic818673.files/Lecture%202%20-%20Weinhold%20et%20al%20-%20Shape%20of%20Oxygen%20Lone%20Pairs.pdf |title=Rabbit Ears Hybrids, VSEPR Sterics, and Other Orbital Absurdities |first=Frank |last=Weinhold |year= |work= |publisher= |location=University of Wisconsin |accessdate=2012-11-11}}</ref> This is analogous to s-p mixing in [[molecular orbital theory]], and maximizes energetic stability according to the [[Walsh diagram]] for the molecule. This rationalisation is applied to explain deviations from ideal bond angles, such as when only p orbitals are used for bonding, most commonly in second and third period elements.

* [[trigonal pyramidal molecular geometry|Trigonal pyramidal]] (AX3E1)
** The s-orbital can be hybridised with the three p-orbital bonds to give bond angles greater than 90°.
** Ex. NH3
* [[bent molecular geometry|Bent]] (AX2E1-2)
** The s-orbital lone pair can be hybridised with the two p-orbital bonds to give bond angles greater than 90°. The out-of-plane p-orbital can either be a lone pair or pi bond. If it is a lone pair, the in-plane and out-of-plane lone pairs are inequivalent, contrary to the common picture depicted by VSEPR theory (see below).
** Exs. SO2, H2O
* Monocoordinate (AX1E1-3)
** The s-orbital lone pair can be hybridised with the p-orbital bond. The two out-of-line p-orbitals can either be lone pairs or pi bonds. The p-orbital lone pairs are not equivalent to the s-rich lone pair.
** Exs. CO, SO, HF

== Hybridisation of [[hypervalent molecule]]s ==

=== Traditional description ===
In general chemistry courses and mainstream textbooks, hybridisation is often presented for main group AX5 and above, as well as for transition metal complexes, using the hybridisation scheme first proposed by Pauling.
{| class="wikitable"
! Classification
! Main group
! Transition metal
|-----
! AX2
| align="center" | -
|
* [[linear molecular geometry|Linear]] (180°)
* sp hybridisation
* E.g., Ag(NH3)2+
|-----
! AX3
| align="center" | -
|
* [[trigonal planar molecular geometry|Trigonal planar]] (120°)
* sp2 hybridisation
* E.g., Cu(CN)32−
|-----
! AX4
| align="center" | -
|
* [[square planar molecular geometry|Square planar]] (90°)
* dsp2 hybridisation
* E.g., PtCl42−
|-----
! AX5
|
*[[trigonal bipyramidal molecular geometry|Trigonal bipyramidal]] (90°, 120°)
* sp3d hybridisation
* E.g., PCl5
|
*[[trigonal bipyramidal molecular geometry|Trigonal bipyramidal]] (90°, 120°)
* dsp3 hybridisation
* E.g., Fe(CO)5
|-----
! AX6
|
*[[octahedral molecular geometry|Octahedral]] (90°)
* sp3d2 hybridisation
* E.g., SF6
|
*[[octahedral molecular geometry|Octahedral]] (90°)
* d2sp3 hybridisation
* E.g., Mo(CO)6
|-----
! AX7
|
*[[pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]] (90°, 72°)
* sp3d3 hybridisation
* E.g., IF7
|
*[[pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]] (90°, 72°)
* d3sp3 hybridisation
* E.g., V(CN)74−
|-----
! AX8
|
*[[square antiprismatic molecular geometry|Square antiprismatic]]
* sp3d4 hybridisation
* E.g., IF8−
|
*[[square antiprismatic molecular geometry|Square antiprismatic]]
* d4sp3 hybridisation
* E.g., Re(CN)83−
|-----
! AX9
| align="center" | -
|
*[[Tricapped trigonal prismatic molecular geometry|Tricapped trigonal prismatic]]
* d5sp3 hybridisation
* E.g., ReH92−
|}

In this notation, d orbitals of main group atoms are listed after the s and p orbitals since they have the same principal quantum number (''n''), while d orbitals of transition metals are listed first since the s and p orbitals have a higher n. Thus for AX5 molecules, sp3d hybridisation in the P atom involves 3s, 3p and 3d orbitals, while dsp3 for Fe involves 3d, 4s and 4p orbitals.

However, hybridisation of s, p and d orbitals together is no longer accepted, as more recent calculations based on molecular orbital theory have shown that in main-group molecules the d component is insignificant, while in transition metal complexes the p component is insignificant (see below).

=== Resonance description ===
As shown by computational chemistry, [[hypervalent molecule]]s can only be stable given strongly polar (and weakened) bonds with electronegative ligands such as fluorine or oxygen to reduce the valence electron occupancy of the central atom to a maximum of 8<ref>{{cite journal
| title = Chemical Bonding to Hypercoordinate Second-Row Atoms: d Orbital Participation versus Democracy
| author = David L. Cooper , Terry P. Cunningham , Joseph Gerratt , Peter B. Karadakov , Mario Raimondi
| journal = [[Journal of the American Chemical Society]]
| year = 1994
| volume = 116
| issue = 10
| pages = 4414–4426
| doi = 10.1021/ja00089a033
}}</ref> (or 12 for transition metals). This requires an explanation that invokes [[sigma bond|sigma]] [[resonance (chemistry)|resonance]] in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. As a guideline, all resonance structures have to obey the octet rule for main group compounds and the dodectet (12) rule for transition metal complexes.

{| class="wikitable"
! Classification
! Main group
! Transition metal
|-----
! rowspan="2"| AX2
| align="center" rowspan="2"| -
| align="center"| [[linear molecular geometry|Linear]] (180°)
|-----
| align="center"|[[File:Di silv.svg|300px]]
|-----
! rowspan="2"| AX3
| align="center" rowspan="2"| -
| align="center"| [[trigonal planar molecular geometry|Trigonal planar]] (120°)
|-----
| align="center"|[[File:Tri copp.svg|320px]]
|-----
! rowspan="2"| AX4
| align="center" rowspan="2"| -
| align="center"| [[square planar molecular geometry|Square planar]] (90°)
|-----
| align="center"|[[File:Tetra plat.svg|320px]]
|-----
! rowspan="2"| AX5
| align="center"|[[trigonal bipyramidal molecular geometry|Trigonal bipyramidal]] (90°, 120°)
| align="center"|[[trigonal bipyramidal molecular geometry|Trigonal bipyramidal]] (90°, 120°)
|-----
| align="center"|[[File:Penta phos.png|400px]]
|
* Fractional hybridisation (s and d orbitals)
* E.g., Fe(CO)5
|-----
! rowspan="2"| AX6
| align="center"|[[octahedral molecular geometry|Octahedral]] (90°)
| align="center"|[[octahedral molecular geometry|Octahedral]] (90°)
|-----
| align="center"|[[File:Hexa sulf.png|300px]]
| align="center"|[[File:Hexa moly.svg|300px]]
|-----
! rowspan="2"| AX7
| align="center"|[[pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]] (90°, 72°)
| align="center"|[[pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]] (90°, 72°)
|-----
| align="center"|[[File:Hepta iodi.svg|420px]]
|
* Fractional hybridisation (s and three d orbitals)
* E.g., V(CN)74−
|-----
! rowspan="2"| AX8
| align="center"| [[square antiprismatic molecular geometry|Square antiprismatic]]
| align="center"|[[square antiprismatic molecular geometry|Square antiprismatic]]
|-----
|
* Fractional hybridisation (s and three p orbitals)
* E.g., IF8−
|
* Fractional hybridisation (s and four d orbitals)
* E.g., Re(CN)83−
|-----
! rowspan="2"| AX9
| align="center" rowspan="2"| -
| align="center"|[[Tricapped trigonal prismatic molecular geometry|Tricapped trigonal prismatic]]
|-----
|
* Fractional hybridisation (s and five d orbitals)
* E.g., ReH92−
|}

==== Main group compounds with lone pairs ====
For hypervalent main group compounds with lone electron pairs, the bonding scheme can be split into two components: the "resonant bonding" component and the "regular bonding" component. The "regular bonding" component has the same description (see above), while the "resonant bonding" component consists of resonating bonds utilizing p orbitals. The table below shows how each shape is related to the two components and their respective descriptions.

{| class="wikitable"
| colspan="2" rowspan="2"|
! colspan="3"| Regular bonding component (marked in red)
|-----
! Bent
! Monocoordinate
! align="center" | -
|-----
! rowspan="6"| Resonant bonding component
! rowspan="2"|Linear axis
| align="center" | [[Seesaw molecular geometry|Seesaw]] (AX4E1) (90°, 180°, >90°)
| align="center" | [[T-shaped molecular geometry|T-shaped]] (AX3E2) (90°, 180°)
| align="center" | [[Linear molecular geometry|Linear]] (AX2E3) (180°)
|-----
| align="center"|[[File:Tetra sulf.svg|160px]]
| align="center"|[[File:Tri chlo.svg|160px]]
| align="center"|[[File:Di xeno.svg|160px]]
|-----
! rowspan="2"|Square planar equator
| align="center" rowspan="2"| -
| align="center" | [[Square pyramidal molecular geometry|Square pyramidal]] (AX5E1) (90°, 90°)
| align="center" | [[Square planar molecular geometry|Square planar]] (AX4E2) (90°)
|-----
| align="center"|[[File:Penta chlo.svg|240px]]
| align="center"|[[File:Tetra xeno.svg|240px]]
|-----
! rowspan="2"|Pentagonal planar equator
| align="center" rowspan="2"| -
| align="center" | [[Pentagonal pyramidal molecular geometry|Pentagonal pyramidal]] (AX6E1) (90°, 72°)
| align="center" | [[Pentagonal planar molecular geometry|Pentagonal planar]] (AX5E2) (72°)
|-----
| align="center"|[[File:Hexa xeno.svg|240px]]
| align="center"|[[File:Penta xeno.svg|240px]]
|}

==Clarifying misconceptions==

===VSEPR electron domains and hybrid orbitals are different===
The simplistic picture of hybridisation taught in conjunction with VSEPR theory does not agree with high-level theoretical calculations<ref name=Weinhold/> despite its widespread usage in many textbooks. For example, following the guidelines of VSEPR, the hybridization of the oxygen in water is described with two equivalent lone electron-pairs.<ref>Petrucci R.H., Harwood W.S. and Herring F.G. "General Chemistry. Principles and Modern Applications" (Prentice-Hall 8th edn 2002) p. 441</ref> However, [[molecular orbital]] calculations give orbitals that reflect the [[Molecular symmetry#Common point groups|C2v symmetry]] of the molecule.<ref name=Levine470>Levine I.N. “Quantum chemistry” (4th edn, Prentice-Hall) p. 470–2</ref> One of the two lone pairs is in a pure p-type orbital, with its electron density perpendicular to the H–O–H framework.<ref name=Laing>[http://dx.doi.org/10.1021/ed064p124 Laing, Michael ''J. Chem. Educ.'' (1987) '''64''', 124–128] "No rabbit ears on water. The structure of the water molecule: What should we tell the students?"</ref> The other lone pair is in an approximately sp0.8 orbital that is in the same plane as the H–O–H bonding.<ref name=Laing/> [[Ultraviolet photoelectron spectroscopy|Photoelectron spectra]] confirm the presence of two different energies for the nonbonded electrons.<ref>Levine p. 475</ref>

=== Non-inclusion of d orbitals in main group compounds ===
{{main|Hypervalent molecule}}

In 1990, Magnusson published a seminal work definitively excluding the role of d-orbital hybridization in bonding in hypervalent compounds of second-row elements. This had long been a point of contention and confusion in describing these molecules using molecular orbital theory. Part of the confusion here originates from the fact that one must include d-functions in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result), and the contribution of the d-function to the molecular wavefunction is large. These facts were historically interpreted to mean that d-orbitals must be involved in bonding. However, Magnusson concludes in his work that d-orbital involvement is not implicated in hypervalency.<ref name="ReferenceA">E. Magnusson. Hypercoordinate molecules of second-row elements: d functions or d orbitals? ''J. Am. Chem. Soc.'' '''1990''', ''112'', 7940-7951. {{doi|10.1021/ja00178a014}}</ref>

=== Non-inclusion of p orbitals in transition metal complexes ===
Similarly, p orbitals have long been thought to be utilized by transition metal centers in bonding with ligands, hence the [[18-electron rule|18-electron]] description; however, recent [[molecular orbital]] calculations have found that such p orbital participation in bonding is insignificant,<ref>{{cite journal
| title = Valence and extra-valence orbitals in main group and transition metal bonding
| author = C. R. Landis, F. Weinhold
| journal = [[Journal of Computational Chemistry]]
| year = 2007
| volume = 28
| issue = 1
| pages = 198–203
| doi = 10.1002/jcc.20492
}}</ref><ref>{{cite web |url=http://www.bowdoin.edu/student-fellowships/pdf/summer-2012/ODonnell%20SRR.pdf |title=Investigating P-Orbital Character In Transition Metal-to-Ligand Bonding |first=Mark |last=O’Donnell |year=2012 |work= |publisher=Bowdoin College |location= Brunswick, ME |accessdate=2012-09-16}}</ref> even though the contribution of the p-function to the molecular wavefunction is calculated to be somewhat larger than that of the d-function in main group compounds.

==Hybridization theory vs. Molecular Orbital theory==

Hybridisation theory is an integral part of [[organic chemistry]] and in general discussed together with [[molecular orbital theory]] in advanced organic chemistry textbooks although for different reasons. One textbook notes that for drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons.<ref>{{Clayden|page=105}}</ref> It also comments that predicting bond angles in methane with MO theory is not straightforward. Another textbook treats hybridisation theory when explaining bonding in alkenes<ref>''Organic Chemistry'', Third Edition Marye Anne Fox James K. Whitesell '''2003''' ISBN 978-0-7637-3586-9</ref> and a third<ref>''Organic Chemistry'' 3rd Ed. '''2001''' Paula Yurkanis Bruice ISBN 978-0-130-17858-9</ref> uses MO theory to explain bonding in hydrogen but hybridisation theory for methane.

Bonding orbitals formed from hybrid atomic orbitals may be considered as [[localized molecular orbitals]], which can be formed from the delocalized orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules with a closed electron shell in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore ''equivalent'' to the delocalized orbital description for explaining the ground state total energy and electron density, as well as the molecular geometry which corresponds to the minimum value of the total energy.

There is no such equivalence, however, for ionized or excited states with open electron shells. Hybrid orbitals cannot therefore be used to interpret photoelectron spectra, which measure the energies of ionized states, identified with delocalized orbital energies using [[Koopmans' theorem]]. Nor can they be used to interpret UV-visible spectra which correspond to electronic transitions between delocalized orbitals. From a pedagogical perspective, the hybridisation approach tends to over-emphasize localisation of bonding electrons and does not effectively embrace [[molecular symmetry]] as does MO theory.

== See also ==
* [[Linear combination of atomic orbitals molecular orbital method]]
* [[MO diagram]]s
* [[Ligand field theory]]
* [[Crystal field theory]]

== References ==
{{reflist}}

== External links ==
* [http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_07.html Covalent Bonds and Molecular Structure]
* [http://www.mhhe.com/physsci/chemistry/essentialchemistry/flash/hybrv18.swf Hybridisation flash movie]
* [http://adomas.org/hopv/ Hybrid orbital 3D preview program in OpenGL]
* [http://college.hmco.com/chemistry/shared/media/zumdahl/dswmedia/undr_dcr/Ch14_u14a.dcr Understanding Concepts: Molecular Orbitals]

{{DEFAULTSORT:Orbital Hybridisation}}
[[Category:Chemical bonding]]
[[Category:Quantum chemistry]]
[[Category:Stereochemistry]]

Smoothed octagon

2013-07-03T02:23:08Z

140.180.245.116: The packing shown is best, not just best known

The '''Sethi model''' was developed by [[Suresh P. Sethi]] and describes the process of how sales evolve over time in response to [[advertising]].<ref name="sethi83">{{Cite journal |last=Sethi |first=S. P. |year=1983 |title=Deterministic and Stochastic Optimization of a Dynamic Advertising Model |journal=Optimal Control Application and Methods |volume=4 |issue=2 |pages=179–184 |doi=10.1002/oca.4660040207 }}</ref> The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market, and a random effect that can go either way.

Suresh Sethi published his paper "Deterministic and Stochastic Optimization of a Dynamic Advertising Model" in 1983.<ref name="sethi83" /> The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.<ref>{{Cite journal |last=Vidale |first=M. L. |last2=Wolfe |first2=H. B. |year=1957 |title=An Operations-Research Study of Sales Response to Advertising |journal=Operations Research |volume=5 |issue=3 |pages=370–381 |doi=10.1287/opre.5.3.370 }}</ref> The model and its competitive extensions have been used extensively in the literature.<ref name="Sorger89">{{Cite journal |last=Sorger |first=G. |year=1989 |title=Competitive Dynamic Advertising: A Modification of the Case Game |journal=Journal of Economic Dynamics and Control |volume=13 |issue=1 |pages=55–80 |doi=10.1016/0165-1889(89)90011-0 }}</ref><ref name="CV92">{{Cite journal |last=Chintagunta |first=P. K. |last2=Vilcassim |first2=N. J. |year=1992 |title=An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly |journal=Management Science |volume=38 |issue=9 |pages=1230–1244 |doi=10.1287/mnsc.38.9.1230 }}</ref><ref name="CJ95">{{Cite journal |last=Chintagunta |first=P. K. |authorlink2=Dipak C. Jain |last2=Jain |first2=D. C. |year=1995 |title=Empirical Analysis of a Dynamic Duopoly Model of Competition |journal=Journal of Economics & Management Strategy |volume=4 |issue=1 |pages=109–131 |doi=10.1111/j.1430-9134.1995.00109.x }}</ref><ref name="PS04">{{Cite journal |last=Prasad |first=A. |last2=Sethi |first2=S. P. |year=2004 |title=Competitive Advertising under Uncertainty: Stochastic Differential Game Approach |journal=Journal of Optimization Theory and Applications |volume=123 |issue=1 |pages=163–185 |doi=10.1023/B:JOTA.0000043996.62867.20 }}</ref><ref name="BKPS05">{{Cite journal |last=Bass |first=F. M. |last2=Krishamoorthy |first2=A. |last3=Prasad |first3=A. |last4=Sethi |first4=S. P. |year=2005 |title=Generic and Brand Advertising Strategies in a Dynamic Duopoly |journal=Marketing Science |volume=24 |issue=4 |pages=556–568 |doi=10.1287/mksc.1050.0119 }}</ref><ref name="NPS08">{{Cite journal |last=Naik |first=P. A. |last2=Prasad |first2=A. |last3=Sethi |first3=S. P. |year=2008 |title=Building Brand Awareness in Dynamic Oligopoly Markets |journal=Management Science |volume=54 |issue=1 |pages=129–138 |doi=10.1287/mnsc.1070.0755 }}</ref><ref name="EricEJOR">{{Cite journal |last=Erickson |first=G. M. |year=2009 |title=An Oligopoly Model of Dynamic Advertising Competition |journal=European Journal of Operations Research |volume= |issue= |pages= |doi= }}</ref><ref>{{Cite journal |last=Prasad |first=A. |last2=Sethi |first2=S. P. |year=2009 |title=Integrated Marketing Communications in Markets with Uncertainty and Competition |journal=Automatica |volume=45 |issue= 3|pages=601–610 |doi=10.1016/j.automatica.2008.09.018 }}</ref><ref name="EricOR">{{Cite journal |last=Erickson |first=G. M. |year=2009 |title=Advertising Competition in a Dynamic Oligopoly with Multiple Brands |journal=Operations Research |volume=57 |issue=5 |pages=1106–1113 |doi=10.1287/opre.1080.0663 }}</ref> Moreover, some of these extensions have been also tested empirically.<ref name="CV92" /><ref name="CJ95" /><ref name="NPS08" /><ref name="EricOR" />

==Model==

The Sethi advertising model or simply the Sethi model provides a sales-advertising dynamics in the form of the following [[stochastic differential equation]]:

: <math> dX_t =\left(rU_t\sqrt{1-X_t} - \delta X_t\right)\,dt+\sigma(X_t)\,dz_t, \qquad X_0=x</math>.

Where:
* <math>X_t</math> is the market share at time <math>t</math>
* <math>U_t</math> is the rate of advertising at time <math>t</math>
* <math>r</math> is the coefficient of the effectiveness of advertising
* <math>\delta</math> is the decay constant
* <math>\sigma(X_t)</math> is the diffusion coefficient
* <math>z_t</math> is the [[Wiener process]] (Standard [[Brownian motion]]); <math>dz_t</math> is known as [[White noise]].

===Explanation===

The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market via <math>r</math>, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market via <math>\delta</math>, and a random effect using a diffusion or White noise term that can go either way.

* The coefficient <math>r</math> is the coefficient of the effectiveness of advertising innovation.
* The coefficient <math>\delta</math> is the decay constant.
* The square-root term brings in the so-called word-of-mouth effect at least at low sales levels.<ref name="sethi83" /><ref name="Sorger89" />

* The diffusion term <math>\sigma(X_t)dz_t</math> brings in the random effect.

===Example of an optimal advertising problem===

Subject to the Sethi model above with the initial market share <math>x</math>, consider the following objective function:

: <math>V(x) = \max_{U_t \geq 0} \;E\left[ \int_0^\infty e^{-\rho t}(\pi X_t-U_t^2)\,dt\right],</math>

where <math>\pi</math> denotes the sales revenue corresponding to the total market, i.e., when <math>x = 1</math>, and <math>\rho > 0</math> denotes the discount rate.

The function <math>V(x)</math> is known as the value function for this problem, and it is shown to be<ref name="Sethi00">Sethi, S.P., Thompson, G.L. (2000). ''Optimal Control Theory: Applications to Management Science and Economics''. Second Edition. Springer. ISBN 0-387-28092-8 and ISBN 0-7923-8608-6, pp. 352-355. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html</ref>

: <math>
V(x)=\bar\lambda x+ \frac{\bar\lambda^2 r^2}{4
\rho},
</math>

where

: <math>
\bar\lambda=\frac{\sqrt{(\rho+\delta)^2+r^2
\pi}-(\rho+\delta)}{r^2/2}.
</math>

The [[optimal control]] for this problem is<ref name="Sethi00" />

: <math> U^*_t = u^*(X_t)=\frac{r\bar\lambda \sqrt{1-\ X_t}}{2} = \begin{cases}
{} > \bar{u} & \text{if } X_t < \bar{x}, \\
{} = \bar{u} & \text{if } X_t = \bar{x}, \\
{} < \bar{u} & \text{if } X_t > \bar{x},
\end{cases} </math>

where

: <math>
\bar x= \frac{r^2 \bar\lambda /2}{r^2 \bar\lambda /2+\delta}
</math>

and

: <math>
\bar u=\frac{r\bar\lambda \sqrt{1-\bar x}}{2}.
</math>

==Extensions of the Sethi model==

* Competitive extensions-Nash differential games<ref name="Sorger89" /><ref name="PS04" /><ref name="BKPS05" /><ref name="NPS08" /><ref name="EricEJOR" /><ref name="EricOR" />
* Empirical testing of the Sethi model and extensions<ref name="CV92" /><ref name="CJ95" /><ref name="NPS08" /><ref name="EricOR" />
* Stackelberg differential games <ref>{{Cite journal | last1 = He | first1 = X. | last2 = Prasad | first2 = A. | last3 = Sethi | first3 = S.P. | year = 2009 | title = Cooperative Advertising and Pricing in a Stochastic Supply Chain: Feedback Stackelberg Strategies | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1069063 | journal = Production and Operations Management | volume = 18 | issue = 1| pages = 78–94 }}</ref><ref>{{Cite journal | doi = 10.1007/s11518-007-5058-2 | last1 = He | first1 = X. | last2 = Prasad | first2 = A. | last3 = Sethi | first3 = S.P. | last4 = Gutierrez | first4 = G. | year = 2007| title = A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1069162 | journal = Journal of Systems Science and Systems Engineering | volume = 16 | issue = 4| pages = 385–413 }}</ref>

* The Sethi durable goods model<ref>{{Cite journal | doi = 10.1007/s10957-008-9472-5 | last1 = Sethi | first1 = S.P. | last2 = Prasad | first2 = A. | last3 = He | first3 = X. | year = 2008 | title = Optimal Advertising and Pricing in a New-Product Adoption Model | url = | journal = Journal of Optimization Theory and Applications | volume = 139 | issue = 2| pages = 351–360 }}</ref><ref>Krishnamoorthy, A., Prasad, A., Sethi, S.P. (2009). Optimal Pricing and Advertising in a Durable-Good Duopoly. ''European Journal of Operations Research''.</ref>

==See also==

* [[Bass diffusion model]]
* [[differential game]]s
* [[Stochastic differential equation]]
* [[Diffusion of innovations]]
* [[Stackleberg competition]]
* [[Nash equilibrium]]

==References==
{{Reflist|colwidth=30em}}

[[Category:Advertising]]
[[Category:Mathematical economics]]
[[Category:Optimal control]]
[[Category:Stochastic processes]]

Template:Elementary

2013-06-13T17:59:55Z

140.180.254.122:

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2013-06-12T15:12:39Z

140.180.244.55: /* The Dean Equations */

<math>Insert formula here</math>{{Use dmy dates|date=January 2011}}
{{Infobox album | 
Name = The Final Wave|
Type = live album |
Artist = [[Australian Crawl]] |
Cover = The Final Wave.jpg|
Released = October 1986 |
Recorded = 27 January 1986|
Genre = [[Rock music|Rock]] |
Length = |
Label = Freestyle Records / [[EMI]] |
Producer = John French|
Last album = ''[[Between a Rock and a Hard Place (Australian Crawl album)|Between A Rock And A Hard Place]]'' (1985) |
This album = '''''The Final Wave''''' (1986) |
Next album = ''[[Lost & Found (Australian Crawl album)|Lost & Found]]'' (1996)
}}
{{Album ratings
|rev1 = [[Allmusic]]
|rev1score = {{Rating|2.5|5}}<ref>[{{Allmusic|class=album|id=r240453|pure_url=yes}} Allmusic review]</ref>
}}
'''''The Final Wave''''' is the second live album released by Australian rock band [[Australian Crawl]]. It is a recording of the band's final Melbourne concert on 27 January 1986. The album reached #16 on the Australian album charts upon its release.<ref name="Kent">{{cite book|title=[[Kent Music Report|Australian Chart Book 1970-1992]]|last=Kent|first=David|authorlink=David Kent (historian)|publisher=Australian Chart Book|location=[[St Ives, New South Wales|St Ives]], N.S.W.|year=1993|isbn=0-646-11917-6}} Note: Used for Australian Singles and Albums charting from 1970 until [[Australian Recording Industry Association|ARIA]] created their own [[ARIA Charts|charts]] in mid-1988.</ref>

The album cover features a copy of [[Japan]]ese artist [[Katsushika Hokusai]]'s best known [[woodblock printing in Japan|woodblock print]], [[The Great Wave off Kanagawa]]. It was first published in 1832 ([[Edo Period]]) and is Hokusai's most famous work. It depicts an enormous [[Ocean surface wave|wave]] threatening boats near the [[Prefectures of Japan|Japanese prefecture]] of [[Kanagawa]]; [[Mount Fuji]] can be seen in the background. The wave is probably not intended to be a [[tsunami]], but a normal ocean wave created by the wind.

==Track listing==
# "[[Beautiful People (Australian Crawl song)|Beautiful People]]" ([[James Reyne]], Mark Hudson) - 3:37
# "Unpublished Critics" (Reyne, Paul Williams) - 5:46
# "Lakeside" (Reyne) - 4:25
# "Love (Beats Me Up)" (Reyne) - 4:39
# "White Limbo" ([[Simon Binks]])- 3:27
# "Two Can Play" ([[Simon Hussey]], Reyne) - 2:35
# "[[Errol (song)|Errol]]" ([[Guy McDonough]], Reyne) - 3:18
# "[[Downhearted]]" (Sean Higgins, G McDonough, William 'Bill' McDonough) - 4:44
# "Daughters of the Northern Coast" (G McDonough, Reyne) - 3:31
# "[[The Boys Light Up (song)|The Boys Light Up]]" (Reyne) - 4:13
# "Indisposed" ([[Brad Robinson (Australian musician)|Brad Robinson]], James Robinson, James Reyne, W McDonough) - 3:04
# "Things Don't Seem" (G McDonough, Higgins) - 3:01
# "Reckless (Don't Be So)" (Reyne) - 5:14
# "(The Last) [[Louie Louie]]" ([[Richard Berry (musician)|Richard Berry]]) - 5:17

Songwriting credits from [[Australasian Performing Right Association]] (APRA).<ref name="APRA">{{cite web |url=http://www.apra.com.au/site/public/searchworksresult.stm |title=Australasian Performing Right Association |publisher=[[Australasian Performing Right Association|APRA]] |accessdate=18 April 2008 }} {{Dead link|date=October 2010|bot=H3llBot}}</ref>

==Personnel==
* Mark Greig - guitar, vocals
* John Watson - drums
* [[Simon Binks]] - guitar, vocals
* Harry Brus - bass, vocals
* [[Brad Robinson (Australian musician)|Brad Robinson]] - guitars, keyboards
* [[James Reyne]] - vocals

==References==
{{Reflist}}

==Releases==

{| class="wikitable"
|colspan="1"|'''Format'''
|colspan="1"| '''Country'''
|colspan="1"| '''Label'''
|colspan="1"| '''Catalogue No.'''
|colspan="1"| '''Year'''
|-
|LP
|AUS
|Freestyle
|SFLI 0142
|1986
|-
|CD
|AUS
|EMI
|8544442
|1997
|-
|}

{{Australian Crawl}}

{{DEFAULTSORT:Final Wave, The}}
[[Category:Australian Crawl albums]]
[[Category:1986 live albums]]

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2012-07-27T01:53:30Z

140.180.3.191:

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2011-10-12T21:16:18Z

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