SklogWiki - User contributions [en]

LaTeX math markup

2011-12-28T18:03:41Z

70.135.124.111:

Note: this page is a subsection of the Wikipedia page [http://en.wikipedia.org/wiki/Help:Formula Help:Displaying a formula].
== Subscripts, superscripts, integrals ==
{| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"
!rowspan="2"|Feature!!rowspan="2"|Syntax!!colspan="2"|How it looks rendered
|-
!HTML!!PNG
|-
|-
|Superscript||<code>a^2</code>||<math>a^2</math>||<math>a^2 \,\!</math>
|-
|Subscript||<code>a_2</code>||<math>a_2</math>||<math>a_2 \,\!</math>
|-
|rowspan=2|Grouping||<code>a^{2+2}</code>||<math>a^{2+2}</math>||<math>a^{2+2}\,\!</math>
|-
|<code>a_{i,j}</code>||<math>a_{i,j}</math>||<math>a_{i,j}\,\!</math>
|-
|Combining sub & super||<code>x_2^3</code>||colspan=2|<math>x_2^3</math>
|-
|rowspan="2"|Preceding and/or Additional sub & super||<code>\sideset{_1^2}{_3^4}\prod_a^b</code>||colspan=2|<math>\sideset{_1^2}{_3^4}\prod_a^b</math>
|-
|<code>{}_1^2\!\Omega_3^4</code>||colspan=2|<math>{}_1^2\!\Omega_3^4</math>
|-
|rowspan="4"|Stacking
|<code>\overset{\alpha}{\omega}</code>||colspan="2"|<math>\overset{\alpha}{\omega}</math>
|-
|<code>\underset{\alpha}{\omega}</code>||colspan="2"|<math>\underset{\alpha}{\omega}</math>
|-
|<code>\overset{\alpha}{\underset{\gamma}{\omega}}</code>||colspan="2"|<math>\overset{\alpha}{\underset{\gamma}{\omega}}</math>
|-
|<code>\stackrel{\alpha}{\omega}</code>||colspan="2"|<math>\stackrel{\alpha}{\omega}</math>
|-
|Derivative (forced PNG)||<code>x', y'', f', f''\!</code>|| ||<math>x', y'', f', f''\!</math>
|-
|Derivative (f in italics may overlap primes in HTML)||<code>x', y'', f', f''</code>||<math>x', y'', f', f''</math>||<math>x', y'', f', f''\!</math>
|-
|Derivative (wrong in HTML)||<code>x^\prime, y^{\prime\prime}</code>||<math>x^\prime, y^{\prime\prime}</math>||<math>x^\prime, y^{\prime\prime}\,\!</math>
|-
|Derivative (wrong in PNG)||<code>x\prime, y\prime\prime</code>||<math>x\prime, y\prime\prime</math>||<math>x\prime, y\prime\prime\,\!</math>
|-
|Derivative dots||<code>\dot{x}, \ddot{x}</code>||colspan=2|<math>\dot{x}, \ddot{x}</math>
|-
|rowspan="3"|Underlines, overlines, vectors||<code>\hat a \ \bar b \ \vec c</code>||colspan=2|<math>\hat a \ \bar b \ \vec c</math>
|-
|<code>\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}</code>||colspan=2|<math>\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}</math>
|-
|<code>\overline{g h i} \ \underline{j k l}</code>||colspan=2|<math>\overline{g h i} \ \underline{j k l}</math>
|-
|Arrows||<code> A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C</code>||colspan=2|<math> A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C</math>
|-
|Overbraces||<code>\overbrace{ 1+2+\cdots+100 }^{5050}</code>||colspan=2|<math>\overbrace{ 1+2+\cdots+100 }^{5050}</math>
|-
|Underbraces||<code>\underbrace{ a+b+\cdots+z }_{26}</code>||colspan=2|<math>\underbrace{ a+b+\cdots+z }_{26}</math>
|-
|Sum||<code>\sum_{k=1}^N k^2</code>||colspan=2|<math>\sum_{k=1}^N k^2</math>
|-
|Sum (force <code>\textstyle</code>)||<code>\textstyle \sum_{k=1}^N k^2 </code>||colspan=2|<math>\textstyle \sum_{k=1}^N k^2</math>
|-
|Product||<code>\prod_{i=1}^N x_i</code>||colspan=2|<math>\prod_{i=1}^N x_i</math>
|-
|Product (force <code>\textstyle</code>)||<code>\textstyle \prod_{i=1}^N x_i</code>||colspan=2|<math>\textstyle \prod_{i=1}^N x_i</math>
|-
|Coproduct||<code>\coprod_{i=1}^N x_i</code>||colspan=2|<math>\coprod_{i=1}^N x_i</math>
|-
|Coproduct (force <code>\textstyle</code>)||<code>\textstyle \coprod_{i=1}^N x_i</code>||colspan=2|<math>\textstyle \coprod_{i=1}^N x_i</math>
|-
|Limit||<code>\lim_{n \to \infty}x_n</code>||colspan=2|<math>\lim_{n \to \infty}x_n</math>
|-
|Limit (force <code>\textstyle</code>)||<code>\textstyle \lim_{n \to \infty}x_n</code>||colspan=2|<math>\textstyle \lim_{n \to \infty}x_n</math>
|-
|Integral||<code>\int\limits_{-N}^{N} e^x\, dx</code>||colspan=2|<math>\int\limits_{-N}^{N} e^x\, dx</math>
|-
|Integral (force <code>\textstyle</code>)||<code>\textstyle \int\limits_{-N}^{N} e^x\, dx</code>||colspan=2|<math>\textstyle \int\limits_{-N}^{N} e^x\, dx</math>
|-
|Double integral||<code>\iint\limits_{D} \, dx\,dy</code>||colspan=2|<math>\iint\limits_{D} \, dx\,dy</math>
|-
|Triple integral||<code>\iiint\limits_{E} \, dx\,dy\,dz</code>||colspan=2|<math>\iiint\limits_{E} \, dx\,dy\,dz</math>
|-
|Quadruple integral||<code>\iiiint\limits_{F} \, dx\,dy\,dz\,dt</code>||colspan=2|<math>\iiiint\limits_{F} \, dx\,dy\,dz\,dt</math>
|-
|Path integral||<code>\oint\limits_{C} x^3\, dx + 4y^2\, dy</code>||colspan=2|<math>\oint\limits_{C} x^3\, dx + 4y^2\, dy</math>
|-
|Intersections||<code>\bigcap_1^{n} p</code>||colspan=2|<math>\bigcap_1^{n} p</math>
|-
|Unions||<code>\bigcup_1^{k} p</code>||colspan=2|<math>\bigcup_1^{k} p</math>
|}
== Fractions, matrices, multilines ==
<table class="wikitable">

<tr>
<th>Feature</th>
<th>Syntax</th>
<th>How it looks rendered</th>
</tr>

<tr>
<td>Fractions</td>
<td><code>\frac{2}{4}=0.5</code></td>
<td><math>\frac{2}{4}=0.5</math></td>
</tr>

<tr>
<td>Small Fractions</td>
<td><code>\tfrac{2}{4} = 0.5</code></td>
<td><math>\tfrac{2}{4} = 0.5</math></td>
</tr>

<tr>
<td>Large (normal) Fractions</td>
<td><code>\dfrac{2}{4} = 0.5</code></td>
<td><math>\dfrac{2}{4} = 0.5</math></td>
</tr>

<tr>
<td>Large (nested) Fractions</td>
<td><code>\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a</code></td>
<td><math>\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a</math></td>
</tr>

<tr>
<td>Binomial coefficients</td>
<td><code>\binom{n}{k}</code></td>
<td><math>\binom{n}{k}</math></td>
</tr>

<tr>
<td>Small Binomial coefficients</td>
<td><code>\tbinom{n}{k}</code></td>
<td><math>\tbinom{n}{k}</math></td>
</tr>

<tr>
<td>Large (normal) Binomial coefficients</td>
<td><code>\dbinom{n}{k}</code></td>
<td><math>\dbinom{n}{k}</math></td>
</tr>

<tr>
<td rowspan="7">Matrices</td>
<td><pre>\begin{matrix}
x & y \\
z & v
\end{matrix}</pre></td>
<td><math>\begin{matrix} x & y \\ z & v
\end{matrix}</math></td>
</tr>

<tr>
<td><pre>\begin{vmatrix}
x & y \\
z & v
\end{vmatrix}</pre></td>
<td><math>\begin{vmatrix} x & y \\ z & v
\end{vmatrix}</math></td>
</tr>

<tr>
<td><pre>\begin{Vmatrix}
x & y \\
z & v
\end{Vmatrix}</pre></td>
<td><math>\begin{Vmatrix} x & y \\ z & v
\end{Vmatrix}</math></td>
</tr>

<tr>
<td><pre>\begin{bmatrix}
0 & \cdots & 0 \\
\vdots & \ddots & \vdots \\
0 & \cdots & 0
\end{bmatrix}</pre></td>
<td><math>\begin{bmatrix} 0 & \cdots & 0 \\ \vdots
& \ddots & \vdots \\ 0 & \cdots &
0\end{bmatrix} </math></td>
</tr>

<tr>
<td><pre>\begin{Bmatrix}
x & y \\
z & v
\end{Bmatrix}</pre></td>
<td><math>\begin{Bmatrix} x & y \\ z & v
\end{Bmatrix}</math></td>
</tr>

<tr>
<td><pre>\begin{pmatrix}
x & y \\
z & v
\end{pmatrix}</pre></td>
<td><math>\begin{pmatrix} x & y \\ z & v
\end{pmatrix}</math></td>
</tr>

<tr>
<td><pre>
\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
</pre></td>
<td><math>
\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
</math></td>
</tr>

<tr>
<td>Case distinctions</td>
<td><pre>
f(n) =
\begin{cases}
n/2, & \mbox{if }n\mbox{ is even} \\
3n+1, & \mbox{if }n\mbox{ is odd}
\end{cases}</pre></td>
<td><math>f(n) =
\begin{cases}
n/2, & \mbox{if }n\mbox{ is even} \\
3n+1, & \mbox{if }n\mbox{ is odd}
\end{cases} </math></td>
</tr>

<tr>
<td rowspan="2">Multiline equations</td>
<td><pre>
\begin{align}
f(x) & = (a+b)^2 \\
& = a^2+2ab+b^2 \\
\end{align}
</pre></td>
<td><math>
\begin{align}
f(x) & = (a+b)^2 \\
& = a^2+2ab+b^2 \\
\end{align}
</math></td>
</tr>

<tr>
<td><pre>
\begin{alignat}{2}
f(x) & = (a-b)^2 \\
& = a^2-2ab+b^2 \\
\end{alignat}
</pre></td>
<td><math>
\begin{alignat}{2}
f(x) & = (a-b)^2 \\
& = a^2-2ab+b^2 \\
\end{alignat}
</math></td>
</tr>
<tr>
<td>Multiline equations <small>(must define number of colums used ({lcr}) <small>(should not be used unless needed)</small></small></td>
<td><pre>
\begin{array}{lcl}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}</pre></td>
<td><math>\begin{array}{lcl}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}</math></td>
</tr>

<tr>
<td>Multiline equations (more)</td>
<td><pre>
\begin{array}{lcr}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}</pre></td>
<td><math>\begin{array}{lcr}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}</math></td>
</tr>

<tr>
<td>Breaking up a long expression so that it wraps when necessary</td>
<td><pre>
<nowiki>
<math>f(x) \,\!</math>
<math>= \sum_{n=0}^\infty a_n x^n </math>
<math>= a_0+a_1x+a_2x^2+\cdots</math>
</nowiki>
</pre>
</td>
<td>
<math>f(x) \,\!</math><math>= \sum_{n=0}^\infty a_n x^n </math><math>= a_0 +a_1x+a_2x^2+\cdots</math>
</td>
</tr>

<tr>
<td>Simultaneous equations</td>
<td><pre>\begin{cases}
3x + 5y + z \\
7x - 2y + 4z \\
-6x + 3y + 2z
\end{cases}</pre></td>
<td><math>\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}</math></td>
</tr>

</table>
[[category: help]]

== Other ==
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Buckingham potential

2011-12-28T18:01:15Z

70.135.124.111:

The '''Buckingham potential''' is given by <ref>[http://dx.doi.org/10.1098/rspa.1938.0173 R. A. Buckingham "The Classical Equation of State of Gaseous Helium, Neon and Argon", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''168''' pp. 264-283 (1938)]</ref>

:<math>\Phi_{12}(r) = A \exp \left(-Br\right) - \frac{C}{r^6}</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math>A</math>, <math>B</math> and <math>C</math> are constants.

The Buckingham potential describes the exchange repulsion, which originates from the Pauli exclusion principle, by a more realistic exponential function of distance, in contrast to the inverse twelfth power used by the [[Lennard-Jones model |Lennard-Jones potential]]. However, since the Buckingham potential remains finite even at very small distances, it runs the risk of an un-physical "Buckingham catastrophe" at short range when used in simulations of charged systems. This occurs when the electrostatic attraction artificially overcomes the repulsive barrier. The Lennard-Jones potential is also about 4 times quicker to compute <ref>[http://dx.doi.org/10.1023/A:1007911511862 David N. J. White "A computationally efficient alternative to the Buckingham potential for molecular mechanics calculations", Journal of Computer-Aided Molecular Design '''11''' pp.517-521 (1997)]</ref> and so is more frequently used in [[Computer simulation techniques | computer simulations]].
==See also==
*[[Exp-6 potential]]
==References==
<references/>
[[category: models]]

== Other ==
[http://thetvtopc.com/Reverse_Cell_Phone_Lookup_Number reverse phone lookup]

Ben-Naim models of water

2011-12-28T17:54:11Z

70.135.124.111:

{{Stub-water}}
The page treats the models for [[water]] proposed over the years by Arieh Ben-Naim and co-workers.
==BNS model==
The '''BNS''' model was proposed by Ben-Naim and Stillinger (Ref. 1).
====References====
#A. Ben-Naim and F.H. Stillinger "Aspects of the Statistical-Mechanical Theory of Water", in "Structure and Transport of Processes in Water and Aqueous Solutions", Wiley-Interscience, New York pp. 295-330 (1972)
==Mercedes-Benz model==
The so called '''Mercedes-Benz''' model of water is a two dimensional model proposed in 1971 (Refs. 1 and 2). Recently a three dimensional version of the model has been proposed (Ref. 3) and studied (Ref. 4).
====References====
#[http://dx.doi.org/10.1063/1.1675414 A. Ben-Naim "Statistical Mechanics of "Waterlike" Particles in Two Dimensions. I. Physical Model and Application of the Percus–Yevick Equation", Journal of Chemical Physics '''54''' pp. 3682-3695 (1971)]
#[http://dx.doi.org/10.1080/00268977200101851 A. Ben-Naim "Statistical mechanics of water-like particles in two-dimensions II. One component system", Molecular Physics '''24''' pp. 705-721 (1972)]
#[http://dx.doi.org/10.1063/1.3183935 Cristiano L. Dias, Tapio Ala-Nissila, Martin Grant, and Mikko Karttunen "Three-dimensional “Mercedes-Benz” model for water", Journal of Chemical Physics '''131''' 054505 (2009)]
#[http://dx.doi.org/10.1063/1.3259970 Alan Bizjak, Tomaz Urbic, Vojko Vlachy, and Ken A. Dill "Theory for the three-dimensional Mercedes-Benz model of water", Journal of Chemical Physics '''131''' 194504 (2009)]

==Primitive model==
====References====
#Ben-Naim "Statistical Thermodynamics for Chemists and Biochemists", Plenum, New York (1992)
#[http://dx.doi.org/10.1063/1.2818051 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. I. Pure liquid water", Journal of Chemical Physics '''128''' 024505 (2008)]
#[http://dx.doi.org/10.1063/1.2818067 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. II. Solvation of inert solutes in water", Journal of Chemical Physics '''128''' 024506 (2008)]
#[http://dx.doi.org/10.1063/1.2899730 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. III. Solvation of hard rods in aqueous mixtures", Journal of Chemical Physics '''128''' 164507 (2008)]
#[http://dx.doi.org/10.1063/1.2976442 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. IV. A study of “hydrophobic interactions”", Journal of Chemical Physics '''129''' 104506 (2008)]

==Primitive cluster model==
====References====
#[http://dx.doi.org/10.1063/1.1672463 Ronald A. Lovett and A. Ben-Naim "One-Dimensional Model for Aqueous Solutions of Inert Gases", Journal of Chemical Physics '''51''' pp. 3108-3119 (1969)]
#[http://dx.doi.org/10.1063/1.2818051 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. I. Pure liquid water", Journal of Chemical Physics '''128''' 024505 (2008)]
#[http://dx.doi.org/10.1063/1.2818067 Arieh Ben-Naim "One-dimensional model for water and aqueous solutions. II. Solvation of inert solutes in water", Journal of Chemical Physics '''128''' 024506 (2008)]
[[category:water]]
[[category:models]]

== Other ==
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