formulasearchengine - User contributions [en]

Regular local ring

2014-03-03T06:23:00Z

117.216.197.134: /* Examples */

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Conservation of energy

2014-01-28T17:14:07Z

117.216.70.17: /* Noether's theorem */

{{Infobox ununennium}}
'''Ununennium''', also known as '''[[Mendeleev's predicted elements|eka]]-[[francium]]''' or element 119, is the temporary name of a [[chemical element]] in the [[periodic table]] that has the temporary symbol '''Uue''' and has the [[atomic number]] 119. To date, attempted syntheses of this element have been unsuccessful. Since it is below the [[alkali metals]] it might have properties similar to those of [[francium]] or [[caesium]] and thus be extremely reactive with water and air (though [[relativistic quantum chemistry|relativistic effects]] might make it less reactive than francium and caesium). A predicted oxidation state is +1; however, unlike all the other alkali metals, it is also predicted to show the +3 oxidation state.

Ununennium would be the first element in the eighth [[periodic table period|period]] of the [[periodic table]].

==Attempts at synthesis==
The synthesis of ununennium was attempted in 1985 by bombarding a target of [[einsteinium]]-254 with [[calcium]]-48 ions at the superHILAC accelerator at Berkeley, California:
:<math>\,^{254}_{99}\mathrm{Es} + \,^{48}_{20}\mathrm{Ca} \to \,^{302}_{119}\mathrm{Uue} ^{*} </math>
No atoms were identified, leading to a limiting yield of 300 [[barn (unit)|nb]].<ref>{{cite journal|title=Search for superheavy elements using 48Ca + 254Esg reaction|journal=Physical Reviews C|year=1985|pages=1760–1763|doi=10.1103/PhysRevC.32.1760|volume=32|issue=5|bibcode = 1985PhRvC..32.1760L|last1=Lougheed|first1=R.|last2=Landrum|first2=J.|last3=Hulet|first3=E.|last4=Wild|first4=J.|last5=Dougan|first5=R.|last6=Dougan|first6=A.|last7=Gäggeler|first7=H.|last8=Schädel|first8=M.|last9=Moody|first9=K. |displayauthors=11}}</ref>

As of May 2012, plans are under way to attempt to synthesize the isotopes 295Uue and 296Uue by bombarding a target of [[berkelium]] with [[titanium]] at the [[GSI Helmholtz Centre for Heavy Ion Research]] in [[Darmstadt]], Germany:<ref name="economist">[http://www.economist.com/node/21554502 Modern alchemy: Turning a line], [[The Economist]], May 12, 2012.</ref><ref>Düllmann, Christoph E. (October 20, 2011). [http://fias.uni-frankfurt.de/kollo/Duellmann_FIAS-Kolloquium.pdf Superheavy Element Research: News from GSI and Mainz]. Johannes Gutenberg University Mainz; GSI Helmholtzzentrum für Schwerionenforschung GmbH; Darmstadt Helmholtz Institute Mainz.</ref>
:<math>\,^{249}_{97}\mathrm{Bk} + \,^{50}_{22}\mathrm{Ti} \to \,^{296}_{119}\mathrm{Uue} \,+3\,^{1}_{0}\mathrm{n}</math>

:<math>\,^{249}_{97}\mathrm{Bk} + \,^{50}_{22}\mathrm{Ti} \to \,^{295}_{119}\mathrm{Uue} \,+4\,^{1}_{0}\mathrm{n}</math>

==Predicted decay characteristics==
The alpha-decay half-lives of 1700 nuclei with 100 ≤ Z ≤ 130 have been calculated in a quantum tunneling model with alpha-decay Q-values from different mass estimates.<ref name=npa07>{{cite journal|journal=Nucl. Phys. A|volume=789|pages=142–154|year=2007|title=Predictions of alpha decay half lives of heavy and superheavy elements|author=Chowdhury, P. Roy; Samanta, C. and Basu, D. N. |doi=10.1016/j.nuclphysa.2007.04.001|bibcode=2007NuPhA.789..142S|arxiv = nucl-th/0703086 }}</ref><ref>{{cite journal|journal=Phys. Rev. C|volume=77|pages=044603|year=2008|title=Search for long lived heaviest nuclei beyond the valley of stability|author=Chowdhury, P. Roy; Samanta, C. and Basu, D. N. |doi=10.1103/PhysRevC.77.044603|issue=4|bibcode=2008PhRvC..77d4603C|arxiv = 0802.3837 }}</ref><ref>{{cite journal|journal=At. Data & Nucl. Data Tables|volume=94|pages=781–806|year=2008|title=Nuclear half-lives for α -radioactivity of elements with 100 ≤ Z ≤ 130|author=Chowdhury, P. Roy; Samanta, C. and Basu, D. N. |doi=10.1016/j.adt.2008.01.003|issue=6|bibcode=2008ADNDT..94..781C|arxiv = 0802.4161 }}</ref> The alpha-decay half-lives predicted for 291–307119 are of the order of micro-seconds. The highest value of the alpha-decay half-life predicted in the quantum tunneling model with the mass estimates from a macroscopic-microscopic model is ~485 microseconds for the isotope 294119. For 302119 it is ~163 microseconds.

==Target-projectile combinations leading to Z=119 compound nuclei==
The below table contains various combinations of targets and projectiles which could be used to form compound nuclei with an atomic number of 119.

{|class="wikitable" style="text-align:center"

!Target!!Projectile!!CN!!Attempt result
|-
!254Es
|48Ca||302Uue||{{no|Failure to date}}
|-
!249Bk
|50Ti||299Uue||{{unk|Planned reaction}}
|}

==Theoretical calculations on evaporation cross sections==
The below table contains various targets-projectile combinations for which calculations have provided estimates for cross section yields from various neutron evaporation channels. The channel with the highest expected yield is given.

DNS = Di-nuclear system; σ = cross section

{|class="wikitable" style="text-align:center"
|-
!Target!!Projectile!!CN!!Channel (product)!!σ max!!Model!!Ref
|-
!254Es
|48Ca||302Uue||3n (299Uue)||0.5 pb||DNS||<ref>{{cite journal|arxiv=0803.1117|doi=10.1016/j.nuclphysa.2008.11.003|title=Production of heavy and superheavy nuclei in massive fusion reactions|year=2009|author=Feng, Z|journal=Nuclear Physics A|volume=816|page=33|last2=Jin|first2=G|last3=Li|first3=J|last4=Scheid|first4=W|bibcode=2009NuPhA.816...33F}}</ref>
|}

==Extrapolated chemical properties==
Ununennium is expected to behave normally for an alkali metal and exhibit a strong +1 [[oxidation state]]. However, the energetic properties of its [[valence electron]] would increase its first [[ionization energy]], making it less reactive than expected and more like [[potassium]] than [[caesium]] chemically. This would also decrease the [[metallic radius|metallic]] and [[ionic radius|ionic radii]] of ununennium.<ref name=EB>{{cite web|author=Seaborg|url=http://www.britannica.com/EBchecked/topic/603220/transuranium-element|title=transuranium element (chemical element)|publisher=Encyclopædia Britannica|date=c. 2006|accessdate=2010-03-16}}</ref> Ununennium is also predicted to be the first alkali metal to display the +3 oxidation state, due to the ionization energy of the 7p3/2 electrons, which is predicted to be very low.<ref name=Haire/>

==See also==
*[[Extended periodic table]]

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

{{Compact extended periodic table}}

[[Category:Chemical elements]]
[[Category:Alkali metals]]
[[Category:Hypothetical chemical elements]]

Gross enrolment ratio

2013-12-08T06:31:01Z

117.216.176.13: /* Example */

{{Unreferenced|date=December 2009}}

In [[probability theory]], it is possible to approximate the [[moment (mathematics)|moments]] of a function ''f'' of a [[random variable]] ''X'' using [[Taylor expansion]]s, provided that ''f'' is sufficiently differentiable and that the moments of ''X'' are finite. This technique is often used by [[statistics|statisticians]].


==First moment==
: <math>
\begin{align}
\operatorname{E}\left[f(X)\right] & {} = \operatorname{E}\left[f(\mu_X + \left(X - \mu_X\right))\right] \\
& {} \approx \operatorname{E}\left[f(\mu_X) + f'(\mu_X)\left(X-\mu_X\right) + \frac{1}{2}f''(\mu_X) \left(X - \mu_X\right)^2 \right].
\end{align}
</math>

Noting that <math>E[X-\mu_X]=0</math>, the 2nd term disappears. Also <math>E[(X-\mu_X)^2]</math> is <math>\sigma_X^2</math>. Therefore,

:<math>\operatorname{E}\left[f(X)\right]\approx f(\mu_X) +\frac{f''(\mu_X)}{2}\sigma_X^2</math>
where <math>\mu_X</math> and <math>\sigma^2_X</math> are the mean and variance of X respectively.

It is possible to generalize this to functions of more than one variable using [[Taylor expansion#Taylor series in several variables|multivariate Taylor expansions]]. For example,

:<math>\operatorname{E}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]} -\frac{\operatorname{cov}\left[X,Y\right]}{\operatorname{E}\left[Y\right]^2}+\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{var}\left[Y\right]</math>

==Second moment==
Analogously,

:<math>\operatorname{var}\left[f(X)\right]\approx \left(f'(\operatorname{E}\left[X\right])\right)^2\operatorname{var}\left[X\right] = \left(f'(\mu_X)\right)^2\sigma^2_X.</math>

The above is using a first order approximation unlike for the method used in estimating the first moment. It will be a poor approximation in cases where <math>f(X)</math> is highly non-linear. This is a special case of the [[delta method]]. For example,

:<math>\operatorname{var}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{var}\left[X\right]}{\operatorname{E}\left[Y\right]^2}-\frac{2\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{cov}\left[X,Y\right]+\frac{\operatorname{E}\left[X\right]^2}{\operatorname{E}\left[Y\right]^4}\operatorname{var}\left[Y\right].</math>

==See also==
*[[Propagation of uncertainty]]
*[[WKB approximation]]
*http://www.stanford.edu/class/cme308/notes/TaylorAppDeltaMethod.pdf

{{DEFAULTSORT:Taylor Expansions For The Moments Of Functions Of Random Variables}}
[[Category:Statistical approximations]]
[[Category:Algebra of random variables]]

Skew-Hermitian matrix

2013-06-30T05:59:12Z

117.216.34.162:

'''LIX''' is a [[readability test|readability measure]] indicating the difficulty of reading a text<ref>http://reap.cs.cmu.edu/Papers/IASTED-HCI-05-jonbrown.pdf</ref> developed by [[Swedes|Swedish]] scholar [[Carl-Hugo Björnsson]]. It is computed as follows:

<math>\text{LIX} = \frac{A}{B} + \frac{C \cdot 100}{A}</math>, where

<math>A</math> is the number of words,

<math>B</math> is the number of periods (defined by period, colon or capital first letter), and

<math>C</math> is the number of long words (more than 6 letters).<ref>[http://www.iva.dk/bh/core%20concepts%20in%20lis/articles%20a-z/readability.htm ]</ref>

==References==
<references/>

==Further reading==
* Björnsson, C. H. (1968). Läsbarhet. Stockholm: Liber.
* Björnsson, C. H. (1971). Læsbarhed. København: Gad.

==External links==
* {{Official website|http://www.lix.se/ }}
* [http://www.standards-schmandards.com/exhibits/rix/ Calculator for LIX and other readability indices]

[[Category:Readability tests]]