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Maxwell material

2014-09-14T21:27:37Z

130.101.20.196: /* Definition */

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2014-02-13T19:33:08Z

130.101.15.36: /* Fanning friction factor formulæ */

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Conditional entropy

2013-12-04T17:27:43Z

130.101.96.122: /* Definition */

{{about|concept in health and disease|game theory|risk dominance|finance|risk factor (finance)|criminology|Risk factor (criminology)}}
In [[epidemiology]], a '''risk factor''' is a variable associated with an increased [[risk]] of [[disease]] or [[infection]]. Sometimes, '''determinant''' is also used, being a variable associated with either increased or decreased risk.

==Correlation vs causation==
Risk factors or determinants are [[correlation]]al and not necessarily [[Causality|causal]], because [[correlation does not prove causation]]. For example, being young cannot be said to cause [[measles]], but young people have a higher rate of measles because they are less likely to have developed [[immunity (medical)|immunity]] during a previous epidemic. [[Statistics|Statistical]] methods are frequently used to assess the strength of an association and to provide causal evidence (for example in the [[British doctors study|study]] of the link between smoking and [[lung cancer]]). Statistical analysis along with the biological sciences can establish that risk factors are causal. Some prefer the term risk factor to mean causal determinants of increased rates of disease, and for unproven links to be called possible risks, associations, etc.

==Terms of description==
Mainly taken from [[risk factors for breast cancer]], risk factors can be described in terms of, for example:
*[[Relative risk]], such as "A woman is more than 100 times more likely to develop breast cancer in her 60s than in her 20s.<ref name=Margolese>{{cite book
|author=Margolese, Richard G, Bernard Fisher, Gabriel N Hortobagyi, and William D Bloomer
|editor=Bast RC, Kufe DW, Pollock RE, et al.
|title=Cancer Medicine
|edition=e.5
|publisher=B.C. Decker
|location=Hamilton, Ontario
|year=2000
|chapter=118
|isbn=1-55009-113-1
|oclc=
|url=http://www.ncbi.nlm.nih.gov/books/NBK20900/#A29677
|accessdate=27 January 2011 }}</ref>"
*Fraction of incidences occurring in the group having the property of or being exposed to the risk factor, such as "99% of breast cancer cases are diagnosed in women<ref name="Giordano">{{cite journal |author=Giordano SH, Cohen DS, Buzdar AU, Perkins G, Hortobagyi GN |title=Breast carcinoma in men: a population-based study |journal=Cancer |volume=101 |issue=1 |pages=51–7 |date=July 2004 |pmid=15221988 |doi=10.1002/cncr.20312}}</ref>"
*Increase in incidence in the exposed group, such as "each daily alcoholic beverage increases the incidence of breast cancer by 11 cases per 1000 women<ref name="pmid19244173">{{cite journal |author=Allen NE, Beral V, Casabonne D, ''et al.'' |title=Moderate alcohol intake and cancer incidence in women |journal=Journal of the National Cancer Institute |volume=101 |issue=5 |pages=296–305 |date=March 2009 |pmid=19244173 |doi=10.1093/jnci/djn514}}</ref>"
*[[Hazard ratio]], such as "an increase in both total and invasive breast cancers in women randomized to receive estrogen and progestin for an average of 5 years, with a hazard ratio of 1.24 compared to controls"<ref>{{cite doi|10.1001/jama.299.9.1036}}</ref>

==Example==
The following example of a risk factor is described in terms of the [[relative risk]] it confers, which is evaluated by comparing the risk of those exposed to the potential risk factor to those not exposed. Let's say that at a wedding, 74 people ate the chicken and 22 of them were ill, while of the 35 people who had the fish or vegetarian meal only 2 were ill. Did the chicken make the people ill?

:<math>
Risk = \frac {\mbox{number of persons experiencing event (food poisoning)}} {\mbox{number of persons exposed to risk factor (food)}}
</math>

So the [[chicken]] eaters' risk = 22/74 = 0.297 
And non-chicken eaters' risk = 2/35 = 0.057.

Those who ate the chicken had a risk over five times as high as those who did not, that is, a relative risk of more than five. This suggests that eating chicken was the cause of the illness, but this is ''not'' proof.

==General determinants==
The probability of an outcome usually depends on an interplay between multiple associated variables. When performing [[epidemiological studies]] to evaluate one or more determinants for a specific outcome, the other determinants may act as [[confounding]] factors, and need to be controlled for, e.g. by [[stratification (statistics)|stratification]]. The potentially confounding determinants varies with what outcome is studied, but the following general confounders are common to most epidemiological associations, and are the determinants most commonly controlled for in epidemiological studies:
*Age
*Sex or gender
*Ethnicity

Other less commonly adjusted for possible confounders include:
*Social status/income
*Geographic location
*Genetic predisposition
*Gender identity
*Occupation
*Sexual orientation
*Level of [[chronic stress]]
*Diet
*Level of [[physical exercise]]
*Alcohol consumption and [[tobacco smoking]]
*Other [[social determinants of health]]

==Risk marker==
A ''risk marker'' is a variable that is quantitatively associated with a disease or other outcome, but direct alteration of the risk marker does not necessarily alter the risk of the outcome.

==History==
The term "risk factor" was first coined by former [[Framingham Heart Study]] Director, Dr. [[William B. Kannel]] in a 1961 article in ''[[Annals of Internal Medicine]]''.<ref>{{cite news| url=http://www.forbes.com/sites/larryhusten/2011/08/23/william-kannel-former-director-of-the-framingham-heart-study-dead-at-87/ | work=Forbes | first=Larry | last=Husten | date=23 August 2011}}</ref>

==References==
{{reflist}}

==Further reading==
* Case, S.P. and Haines, K.R. (2009) Understanding Youth Offending: Risk Factor Research, Policy and Practice. Cullompton: Willan. http://www.willanpublishing.co.uk/cgi-bin/indexer?product=9781843923411

==See also==
*[[protective factor]]

{{DEFAULTSORT:Risk Factor}}
[[Category:Epidemiology]]
[[Category:Risk]]
[[Category:Medical statistics]]
[[Category:Risk factors]]

Trimean

2013-10-17T16:18:21Z

130.101.99.134: /* Efficiency */

In [[mathematics]], in the field of [[abstract algebra]], the '''structure theorem for finitely generated modules over a principal ideal domain''' is a generalization of the [[fundamental theorem of finitely generated abelian groups]] and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a [[prime factorization]]. The result provides a simple framework to understand various canonical form results for square matrices over fields.

==Statement==

When a vector space over a [[Field (mathematics)|field]] ''F'' has a finite generating set, then one may extract from it a [[basis (vector space)|basis]] consisting of a finite number ''n'' of vectors, and the space is therefore isomorphic to ''Fn''. The corresponding statement with the ''F'' generalized to a [[principal ideal domain]] ''R'' is no longer true, as a [[finitely generated module]] over ''R'' need not have any basis. However such a module is still isomorphic to a quotient of some module ''Rn'' with ''n'' finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis ''Rn'' to the generators of the module, and take the quotient by its [[kernel (algebra)|kernel]].) By changing the choice of generating set, one can in fact describe the module as the quotient of some ''Rn'' by a particularly simple submodule, and this is the structure theorem.

The structure theorem for [[finitely generated module]]s over a [[principal ideal domain]] usually appears in the following two forms.

===Invariant factor decomposition===
Every finitely generated module ''M'' over a principal ideal domain ''R'' is isomorphic to a unique one of the form
:<math>\bigoplus_i R/(d_i) = R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_n)</math>
where <math>(d_i) \neq R</math> and <math>d_i \vert d_{i+1}</math>. The order of the nonzero <math>(d_i) \neq R</math> ideals is invariant, and the number of <math>(d_i)=0</math> is invariant.

The nonzero <math>d_i</math> elements, together with the number of <math>d_i</math> which are zero, form a [[complete set of invariants]] for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic. The <math>R/(d_i)</math> themselves are called [[invariant factor]]s of ''M''.

The ideals <math>(d_i)</math> are unique. In terms of the <math>d_i</math> elements, this means that the <math>d_i</math> are unique up to multiplication by a [[unit (ring theory)|unit]].

The free part is visible in the part of the decomposition corresponding to the <math>d_i = 0</math> factors. These occur at the end of the sequence of <math>d_i</math>'s, as everything divides zero.

Some prefer to write the free part of ''M'' separately:
:<math>R^f \oplus \bigoplus_i R/(d_i) = R^f \oplus R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_{n-f})</math>
where the visible <math>d_i</math> are nonzero, and ''f'' is the number of <math>d_i</math>'s which are 0.

===Primary decomposition===
:Every [[finitely generated module]] ''M'' over a [[principal ideal domain]] ''R'' is isomorphic to one of the form
::<math>\bigoplus_i R/(q_i)</math>
:where <math>(q_i) \neq R</math> and the <math>(q_i)</math> are [[primary ideal]]s. The <math>q_i</math> are unique (up to multiplication by units).

The elements <math>q_i</math> are called the ''elementary divisors'' of ''M''. In a PID, primary ideals are powers of primes, and so <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>.

The summands <math>R/(q_i)</math> are [[indecomposable module|indecomposable]], so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a [[indecomposable module|completely decomposable module]]. Since PID's are Noetherian rings, this can be seen as a manifestation of the [[Lasker-Noether theorem]].

As before, it is possible to write the free part (where <math>q_i=0</math>) separately and express ''M'' as:
:<math>R^f \oplus(\bigoplus_i R/(q_i))</math>
where the visible <math>q_i </math> are nonzero.

==Proofs==
One proof proceeds as follows:
* Every [[finitely generated module]] over a PID is also [[finitely presented module|finitely presented]] because a PID is [[noetherian ring|Noetherian]], an even stronger condition than [[coherent ring|coherence]].
* Take a presentation, which is a map <math>R^r \to R^g</math> (relations to generators), and put it in [[Smith normal form]].
This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.

Another outline of a proof:
* Denote by ''tM'' the [[torsion submodule]] of M. Then ''M''/''tM'' is a finitely generated [[torsion-free module|torsion free]] module, and such a module over a commutative PID is a [[free module]] of finite rank, so it is isomorphic to <math>R^n</math> for a positive integer ''n''. This free module can be embedded as a submodule ''F'' of ''M'', such that the embedding splits (is a right inverse of) the projection map; it suffices to lift each of the generators of ''F'' into ''M''. As a consequence <math>M= tM\oplus F</math>.
* For a prime ''p'' in ''R'' we can then speak of <math>N_p= \{m\in tM\mid \exists i, mp^i=0\}</math> for each prime ''p''. This is a submodule of ''tM'', and it turns out that each ''N''''p'' is a direct sum of cyclic modules, and that ''tM'' is a direct sum of ''N''''p'' for a finite number of distinct primes ''p''.
* Putting the previous two steps together, ''M'' is decomposed into cyclic modules of the indicated types.



==Corollaries==
This includes the classification of [[finite-dimensional vector space]]s as a special case, where <math>R = K</math>. Since fields have no non-trivial ideals, every finitely generated vector space is free.

Taking <math>R=\mathbb{Z}</math> yields the [[fundamental theorem of finitely generated abelian groups]].

Let ''T'' be a linear operator on a [[finite-dimensional vector space]] ''V'' over ''K''. Taking <math>R=K[T]</math>, the algebra of polynomials with coefficients in ''K'' evaluated at ''T'', yields structure information about ''T''. ''V'' can be viewed as a finitely generated module over <math>K[T]</math>. The last invariant factor is the [[Minimal polynomial (field theory)|minimal polynomial]], and the product of invariant factors is the [[characteristic polynomial]]. Combined with a standard matrix form for <math>K[T]/p(T)</math>, this yields various [[canonical form]]s:
* [[invariant factors]] + [[companion matrix]] yields [[Frobenius normal form]] (aka, [[rational canonical form]])
* [[primary decomposition]] + [[companion matrix]] yields [[primary rational canonical form]]
* [[primary decomposition]] + [[Jordan block]]s yields [[Jordan canonical form]] (this latter only holds over an [[algebraically closed field]])

==Uniqueness==
While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between ''M'' and its [[canonical form]] is not unique, and does not even preserve the [[direct sum of modules|direct sum]] decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.

However, one has a canonical torsion submodule ''T'', and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:
:<math>0 < \cdots < T < M.</math>
Compare [[composition series]] in [[Jordan–Hölder theorem]].

For instance, if <math>M \approx \mathbf{Z} \oplus \mathbf{Z}/2</math>, and <math>(1,0), (0,1)</math> is one basis, then
<math>(1,1), (0,1)</math> is another basis, and the change of basis matrix <math>\begin{bmatrix}1 & 1 \\0 & 1\end{bmatrix}</math> does not preserve the summand <math>\mathbf{Z}</math>. However, it does preserve the <math>\mathbf{Z}/2</math> summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).

==Generalizations==
===Groups===
The [[Jordan–Hölder theorem]] is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a [[composition series]], rather than a [[direct sum of modules|direct sum]].

The [[Krull–Schmidt theorem]] and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of [[indecomposable module]]s in which the summands are unique up to order.

===Primary decomposition===
The primary decomposition generalizes to finitely generated modules over commutative [[Noetherian ring]]s, and this result is called the [[Lasker–Noether theorem]].

===Indecomposable modules===
By contrast, unique decomposition into ''indecomposable'' submodules does not generalize as far, and the failure is measured by the [[ideal class group]], which vanishes for PIDs.

For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring R = Z[√−5], both the module R and its submodule M generated by 2 and 1 + √−5 are indecomposable. While R is not isomorphic to M, R ⊕ R is isomorphic to M ⊕ M; thus the images of the M summands give indecomposable submodules L1, L2 < R ⊕ R which give a different decomposition of R ⊕ R. The failure of uniquely factorizing R ⊕ R into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R.

===Non-finitely generated modules===
Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are '''Z'''-submodules of '''Q'''4 which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, '''Z'''.

Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring '''Z''' of integers. Then '''Q''' is a torsion-free '''Z'''-module which is not free. Another classical example of such a module is the [[Baer–Specker group]], the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which [[large cardinal]]s exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axioms and may be invalid under a different choice.

==References==
{{reflist}}
{{refbegin}}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}
*{{Citation | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract algebra | publisher=Wiley | location=New York | edition=3rd | isbn=978-0-471-43334-7 | id={{MathSciNet | id = 2286236}} | year=2004}}
*{{Citation | last=Hungerford | first1=Thomas W. | author1-link=Thomas W. Hungerford | title=Algebra | publisher=Springer | location=New York | isbn=978-0-387-90518-1 | year=1980 | pages=218–226, Section IV.6: Modules over a Principal Ideal Domain }}
*{{Citation |author=Jacobson, Nathan |author1-link=Nathan Jacobson|title=Basic algebra. I |edition=2 |publisher=W. H. Freeman and Company |place=New York |date=1985 |pages=xviii+499 |isbn=0-7167-1480-9 |mr=780184}}
*{{Citation | last1=Lam | first1=T. Y. | title=Lectures on modules and rings | publisher=Springer-Verlag | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
{{refend}}

{{DEFAULTSORT:Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain}}
[[Category:Theorems in abstract algebra]]
[[Category:Module theory]]

[[de:Hauptidealring#Moduln über Hauptidealringen]]

ICE table

2012-06-26T14:19:44Z

130.101.8.66:

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