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	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Excess_chemical_potential&amp;diff=259583</id>
		<title>Excess chemical potential</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Excess_chemical_potential&amp;diff=259583"/>
		<updated>2014-08-19T14:10:35Z</updated>

		<summary type="html">&lt;p&gt;141.14.232.141: /* References */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;28 yrs old Aeroplane Pilot Augustine Ziolkowski from Saint-Jerome, likes becoming a child, [http://www.gameinformer.com/themes/blogs/generic/post.aspx?WeblogApp=juan30230_blog&amp;amp;y=2014&amp;amp;m=10&amp;amp;d=15&amp;amp;WeblogPostName=dungeon-hunter-4-hack-tool&amp;amp;GroupKeys=blogs/members/ Dungeon Hunter 4 Hack] and video games. In recent time took some time to visit Kakadu National Park.&lt;/div&gt;</summary>
		<author><name>141.14.232.141</name></author>
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	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Identifiability&amp;diff=264263</id>
		<title>Identifiability</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Identifiability&amp;diff=264263"/>
		<updated>2014-02-11T10:12:56Z</updated>

		<summary type="html">&lt;p&gt;141.14.16.190: fix formatting error of latex&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;By investing in a premium Word - Press theme, you&#039;re investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. * A community forum for debate of the product together with some other customers in the comments spot. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. &amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. Which is perfect for building a mobile site for business use. Now, I want to anxiety that not every single query will be answered. Now a days it has since evolved into a fully capable CMS platform which make it,  the best platform in the world for performing online business. &amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. Setting Up Your Business Online Using Free Wordpress Websites. User friendly features and flexibility that Word - Press has to offer is second to none. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. &amp;lt;br&amp;gt;&amp;lt;br&amp;gt;If you have any type of inquiries regarding where and just how to utilize [http://off2.net/wordpressbackupplugin23410 wordpress dropbox backup], you could call us at the internet site. Additionally Word - Press add a default theme named Twenty Fourteen. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL. &amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Instead, you can easily just include it with our bodies integration field in e - Panel. If you operate a website that&#039;s been built on HTML then you might have to witness traffic losses because such a site isn&#039;t competent enough in grabbing the attention of potential consumers. Useful Plugins  Uber - Menu Top Megamenu  Now it is the time of sticky Top navbar. Change the entire appearance of you blog using themes with one click. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.&lt;/div&gt;</summary>
		<author><name>141.14.16.190</name></author>
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	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Static_light_scattering&amp;diff=17041</id>
		<title>Static light scattering</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Static_light_scattering&amp;diff=17041"/>
		<updated>2014-01-23T13:57:38Z</updated>

		<summary type="html">&lt;p&gt;141.14.154.129: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{Cleanup|date=September 2008}}&lt;br /&gt;
&lt;br /&gt;
== Simplicial continuation ==&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&#039;&#039;&#039;Simplicial continuation&#039;&#039;&#039;, or &#039;&#039;&#039;piecewise linear continuation&#039;&#039;&#039; (Allgower and Georg),&amp;lt;ref name=&amp;quot;one&amp;quot;&amp;gt;Eugene L. Allgower, K. Georg, &amp;quot;Introduction to Numerical Continuation Methods&amp;quot;, &#039;&#039;SIAM Classics in Applied Mathematics&#039;&#039; 45, 2003.&amp;lt;/ref&amp;gt;&amp;lt;ref name=&amp;quot;three&amp;quot;&amp;gt;E. L. Allgower, K. Georg, &amp;quot;Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations&amp;quot;, &#039;&#039;SIAM Review&#039;&#039;, Volume 22, 28-85, 1980.&amp;lt;/ref&amp;gt; is a one parameter [[numerical continuation|continuation method]] which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher dimensional manifolds by (Allgower and Gnutzman)&amp;lt;ref name=&amp;quot;two&amp;quot;&amp;gt;Eugene L. Allgower, Stefan Gnutzmann, &amp;quot;An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces&amp;quot;, &#039;&#039;SIAM Journal on Numerical Analysis&#039;&#039;, Volume 24, Number 2, 452-469, 1987.&amp;lt;/ref&amp;gt; and (Allgower and Schmidt).&amp;lt;ref name=&amp;quot;four&amp;quot;&amp;gt;Eugene L. Allgower, Phillip H. Schmidt, &amp;quot;An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold&amp;quot;, &#039;&#039;SIAM Journal on Numerical Analysis&#039;&#039;, Volume 22, Number 2, 322-346, April 1985.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The algorithm for drawing contours is a simplicial continuation algorithm, and since it is easy to visualize, it serves as a good introduction to the algorithm.&lt;br /&gt;
&lt;br /&gt;
== Contour plotting ==&lt;br /&gt;
&lt;br /&gt;
The contour plotting problem is to find the zeros (contours) of &amp;lt;math&amp;gt; f(x,y)=0\,&amp;lt;/math&amp;gt; (&amp;lt;math&amp;gt; f(\cdot)\,&amp;lt;/math&amp;gt; a smooth scalar valued function) in the square &amp;lt;math&amp;gt;0\leq x \leq 1, 0\leq y \leq 1\,&amp;lt;/math&amp;gt;,&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
[[Image:Contours.gif|An example of contours]] [[Image:ContoursA.gif|Contours, three-dimensional view]]&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
The square is divided into small triangles, usually by introducing points at the corners of a regular square mesh &amp;lt;math&amp;gt;ih_x\leq x\leq (i+1)h_x\,&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;jh_y\leq y \leq (j+1)h_y\,&amp;lt;/math&amp;gt;, making a table of the values of &amp;lt;math&amp;gt;f(x_i,y_j)\,&amp;lt;/math&amp;gt; at each corner &amp;lt;math&amp;gt;(i,j)\,&amp;lt;/math&amp;gt;, and then dividing each square into two triangles. The value of &amp;lt;math&amp;gt;f(x_i,y_j)\,&amp;lt;/math&amp;gt; at the corners of the triangle defines a unique Piecewise Linear interpolant &amp;lt;math&amp;gt;lf(x,y)\,&amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;f(\cdot)\,&amp;lt;/math&amp;gt; over each triangle. One way of writing this interpolant on the triangle with corners&lt;br /&gt;
&amp;lt;math&amp;gt;(x_0,y_0),~(x_1,y_1),~(x_2,y_2)\,&amp;lt;/math&amp;gt; is as the set of equations&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt; (x,y) = (x_0,y_0)+(x_1-x_0,y_1-y_0)s+(x_2-x_0,y_2-y_0)t\,&amp;lt;/math&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; 0\leq s\,&amp;lt;/math&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; 0\leq t\,&amp;lt;/math&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; s+t \leq 1\,&amp;lt;/math&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; lf(x,y) = f(x_0,y_0)+(f(x_1,y_1)-f(x_0,y_0))s+(f(x_2,y_2)-f(x_0,y_0))t\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The first four equations can be solved for &amp;lt;math&amp;gt;(s,t)\,&amp;lt;/math&amp;gt; (this maps the original triangle to a right unit triangle), then the remaining equation gives the interpolated value of &amp;lt;math&amp;gt;f(\cdot)\,&amp;lt;/math&amp;gt;. Over the whole mesh of triangles, this piecewise linear interpolant is continuous.&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
[[Image:LinearInterpolant.gif|An example of a triangulation and marked vertices]] [[Image:LinearInterpolantA.gif|Linear interpolant, three-dimensional view]]&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
The contour of the interpolant on an individual triangle is a line segment (it is an interval on the intersection of two planes). The equation for the line can be found, however the points where the line crosses the edges of the triangle are the endpoints of the line segment.&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
[[Image:Contour.gif|The unique linear interpolant on a simplex and its zero set]][[Image:TriangleContour.gif|The contour of the linear interpolant over a triangle]]&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
The contour of the piecewise linear interpolant is a set of curves made up of these line segments. Any point on the edge connecting &amp;lt;math&amp;gt;(x_0,y_0)\,&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;(x_1,y_1)\,&amp;lt;/math&amp;gt; can be written as&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt;(x,y) = (x_0,y_0) + t (x_1-x_0,y_1-y_0),\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
with &amp;lt;math&amp;gt;t\,&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;(0,1)\,&amp;lt;/math&amp;gt;, and the linear interpolant over the edge is&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt; f \sim f_0 + t (f_1-f_0)\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
So setting &amp;lt;math&amp;gt; f = 0\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt;t = -f_0/(f_1-f_0)\,&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; (x,y) = (x_0,y_0)-f_0*(x_1-x_0,y_1-y_0)/(f_1-f_0)\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Since this only depends on values on the edge, every triangle which shares this edge will produce the same point, so the contour will be continuous. Each triangle can be tested independently, and if all are checked the entire set of contour curves can be found.&lt;br /&gt;
&lt;br /&gt;
== Piecewise linear continuation ==&lt;br /&gt;
&lt;br /&gt;
Piecewise linear continuation is similar to contour plotting (Dobkin, Silvio, Thurston and Wilks),&amp;lt;ref name=&amp;quot;five&amp;quot;&amp;gt;[[David P. Dobkin]], Silvio V. F. Levy, [[William Thurston|William P. Thurston]] and Allan R. Wilks, &amp;quot;Contour Tracing by Piecewise Linear Approximations&amp;quot;, &#039;&#039;ACM Transactions on Graphics&#039;&#039;, 9(4) 389-423, 1990.&amp;lt;/ref&amp;gt; but in higher dimensions. The algorithm is based on the following results:&lt;br /&gt;
&lt;br /&gt;
=== Lemma 1 ===&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
| If F(x) maps &amp;lt;math&amp;gt;\mathbb{R}^n&amp;lt;/math&amp;gt; into &amp;lt;math&amp;gt;\mathbb{R}^{n-1}&amp;lt;/math&amp;gt;, there is a unique linear interpolant on an &#039;(n-1)&#039;-dimensional [[simplex]] which agrees with the function values at the vertices of the simplex.&lt;br /&gt;
|}&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
An &#039;(n-1)&#039;-dimensional simplex has n vertices, and the function F assigns an &#039;n&#039;-vector to each. The simplex is [[convex set|convex]], and any point within the simplex is a [[convex combination]] of the vertices. That is:&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
If x is in the interior of an (n-1)-dimensional simplex with n vertices &amp;lt;math&amp;gt; v_i &amp;lt;/math&amp;gt;, then there are positive scalars &amp;lt;math&amp;gt;0&amp;lt;\alpha_i&amp;lt;/math&amp;gt; such that&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; \mathbf{x} = \sum_i \alpha_i \mathbf{v}_i &amp;lt;/math&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt; \sum_i \alpha_i = 1.\,&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
If the vertices of the simplex are [[Linear independence|linearly independent]] the non-negative scalars &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; are unique for each point x, and are called the [[Barycentric coordinates (mathematics)|barycentric coordinates]] of x. They determine the value of the unique [[interpolation|interpolant]] by the formula:&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
: &amp;lt;math&amp;gt;LF = \sum_i \alpha_i F(\mathbf{v}_i)&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
=== Lemma 2 ===&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
| An (n-1)-dimensional simplex can be tested to determine if it contains the origin.&lt;br /&gt;
|}&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
There are basically two tests. The one which was first used labels the vertices of the simplex with a vector of signs (+/-) of the coordinates of the vertex. For example the vertex (.5,-.2,1.) would be labelled (+,-,+). A simplex is called &#039;&#039;completely labelled&#039;&#039; if there is a vertex whose label begins with a string of &amp;quot;+&amp;quot; signs of length 0,1,2,3,4,...n. A completely labelled simplex contains a neighborhood of the origin. This may be surprising, but what underlies this result is that for each coordinate of a completely labelled simplex there is a vector with &amp;quot;+&amp;quot; and another with a &amp;quot;-&amp;quot;. Put another way, the smallest cube with edges parallel to the coordinate axes and which covers the simplex has pairs of faces on opposite sides of 0. (i.e. a &amp;quot;+&amp;quot; and a &amp;quot;-&amp;quot; for each coordinate).&lt;br /&gt;
&lt;br /&gt;
The second approach is called &#039;&#039;vector labelling&#039;&#039;. It is based on the barycentric coordindates of the vertices of the simplex. The first step is to find the barycentric coordinates of the origin, and then the test that the simplex contains the origin is simply that all the barycentric coordinates are positive and the sum is less than 1.&lt;br /&gt;
&lt;br /&gt;
=== Lemma 3 ===&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
| There is a triangulation (the Coxeter-Freudenthal-Kuhn triangulation [1]) which is invariant under the pivot operation&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;P_{v_i} (v_0,v_1,...,v_n) := (v_0,v_1,...,v_{i-1},\tilde v_i,v_{i+1},...,v_n)&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\tilde v_i = \left\{ \begin{array}{lcl}&lt;br /&gt;
v_1+v_n-v_0 &amp;amp; &amp;amp;i=0 \\&lt;br /&gt;
v_{i+1}+v_{i-1}-v_i&amp;amp;\qquad\qquad&amp;amp;0\\&lt;br /&gt;
v_{n-1}+v_0-v_n &amp;amp; &amp;amp;i=n \\ \end{array}\right.&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
|}&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
[[Image:Simplical3dOne.gif|The first step of three-dimensional simplicial continuation]] [[Image:Simplical3dTwo.gif|The second step of three-dimensional simplicial continuation]]&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
[[Image:Simplicial.gif]]&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Numerical analysis]]&lt;/div&gt;</summary>
		<author><name>141.14.154.129</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Wulff_construction&amp;diff=23429</id>
		<title>Wulff construction</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Wulff_construction&amp;diff=23429"/>
		<updated>2014-01-09T12:50:19Z</updated>

		<summary type="html">&lt;p&gt;141.14.162.129: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;A &#039;&#039;&#039;Peierls transition&#039;&#039;&#039; or &#039;&#039;&#039;Peierls distortion&#039;&#039;&#039; is a distortion of the periodic lattice of a one-dimensional crystal.  Atomic positions oscillate so that the perfect order of the 1-D crystal is broken.&lt;br /&gt;
&lt;br /&gt;
== Peierls’ Theorem&amp;lt;ref&amp;gt;{{cite web | last=Fowler | first=Michael | title=Electrons in One Dimension: the Peierls Transition | date=28 Feb 2007 | url=http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/PeierlsTrans.htm}}&amp;lt;/ref&amp;gt; ==&lt;br /&gt;
&lt;br /&gt;
Peierls&#039; Theorem states that &#039;&#039;a one-dimensional equally spaced chain with one electron per ion is unstable&#039;&#039;. It was asserted in the 1930s by [[Rudolf Peierls]]. It can be proven using a simple model of the potential for an electron in a 1-D crystal with lattice spacing a.  The periodicity of the crystal creates energy band gaps in the E–k diagram at multiples of the value k = π/a (similar to the result of the [[Particle_in_a_one-dimensional_lattice_(periodic_potential)#Kronig–Penney_model|Kronig–Penney model]], which helps to explain the origin of band gaps in semiconductors). If the ions each contribute one electron, then the band will be half-filled, up to values of k = ±π/2a in the ground state.&lt;br /&gt;
&lt;br /&gt;
Imagine a lattice distortion where every other ion moves closer to one neighbor and further away from the other, the unfavourable energy of the long bond between ions is outweighed by the energy gain of the short bond.  The period has just doubled from a to 2a.  In essence, the proof relies on the fact that doubling the period would introduce new band gaps located at multiples of k = π/2a.  This would cause small energy savings, based on the distortion of the bands in the vicinity of the new gaps.  Approaching k = π/2a from the left, the distortion due to the introduction of the new band gap will cause the electrons to be at a lower energy than they would be in the perfect crystal.  Therefore, this lattice distortion becomes energetically favorable when the energy savings due to the new band gaps outweighs the elastic energy cost of rearranging the ions.  Of course, this effect will be noticeable only when the electrons are arranged close to their ground state – in other words, thermal excitation should be minimized.  Therefore, the Peierls transition should be seen at low temperature.  This is the basic argument for the occurrence of the Peierls transition, sometimes called dimerization.&lt;br /&gt;
&lt;br /&gt;
== Historical background ==&lt;br /&gt;
&lt;br /&gt;
Peierls’ discovery gained experimental backing during the effort to find new superconducting materials.  In 1964, Dr. William Little of the Stanford University Department of Physics theorized that a certain class of polymer chains may experience a high T&amp;lt;sub&amp;gt;C&amp;lt;/sub&amp;gt; superconducting transition.&amp;lt;ref&amp;gt;{{cite journal | author = W. A. Little | journal = [[Physical Review]] | volume = 134 | year = 1964 | pages = A1416–A1424 | doi = 10.1103/PhysRev.134.A1416 | title = Possibility of Synthesizing an Organic Superconductor | issue = 6A|bibcode = 1964PhRv..134.1416L }}&amp;lt;/ref&amp;gt;  The basis for his assertion was that the lattice distortions which lead to pairing of electrons in the [[BCS theory]] of [[superconductivity]] could be replaced instead by rearranging the electron density in a series of side chains.  This means that now electrons would be responsible for creating the Cooper pairs instead of ions.  Because the transition temperature is inversely proportional to the square root of the mass of the charged particle responsible for the distortions, the T&amp;lt;sub&amp;gt;C&amp;lt;/sub&amp;gt; should be improved by a corresponding factor:&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;\frac{T}{T_i} = \sqrt{\frac{M_i}{m_e}}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The subscript &#039;&#039;i&#039;&#039; represents &amp;quot;ion,&amp;quot; while &#039;&#039;e&#039;&#039; represents &amp;quot;electron.&amp;quot;  The predicted benefit in superconducting transition temperature was therefore a factor of about 300.&lt;br /&gt;
&lt;br /&gt;
In the 1970s, various organic materials such as [[Charge-transfer_complex#Electrical_conductivity|TTF-TCNQ]] were synthesized.&amp;lt;ref&amp;gt;{{cite journal | author = P. W. Anderson, P. A. Lee, M. Saitoh | journal = [[Solid State Communications]] | volume = 13 | year = 1973 | pages = 595–598 | doi = 10.1016/S0038-1098(73)80020-1 | title = Remarks on giant conductivity in TTF-TCNQ | bibcode=1973SSCom..13..595A | issue = 5}}&amp;lt;/ref&amp;gt;  What was found is that these materials underwent an insulating transition rather than a superconducting one.  Eventually it was realized that these were the first experimental observations of the Peierls transition.  With the introduction of new band gaps after the lattice becomes distorted, electrons must overcome this new energy barrier in order to become free to conduct.  The simple model of the Peierls distortion as a rearrangement of ions in a 1-D chain could describe why these materials became insulators rather than superconductors.&lt;br /&gt;
&lt;br /&gt;
== Related physical consequences ==&lt;br /&gt;
&lt;br /&gt;
Peierls predicted that the rearrangement of the ion cores in a Peierls transition would produce periodic fluctuations in the electron density.  These are commonly called [[charge density wave]]s, and they are an example of collective charge transport.  Several materials systems have verified the existence of these waves.  Good candidates are weakly coupled molecular chains, where electrons can move freely along the direction of the chains but motion is restricted perpendicular to the chains.  NbSe&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; and K&amp;lt;sub&amp;gt;0.3&amp;lt;/sub&amp;gt;MoO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; are two examples in which charge density waves have been observed at relatively high temperatures of 145K and 180K, respectively.&amp;lt;ref&amp;gt;{{Cite news | last=Thorne | first=Robert | title=Charge-Density-Wave Conductors | magazine=[[Physics Today]] | date = May 1996 | url=http://pages.physics.cornell.edu/~rthorne/thorne_phystod_1996.pdf}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Furthermore, the 1-D nature of the material causes a breakdown of the [[Fermi liquid]] theory for electron behavior.  Therefore, a 1-D conductor should behave as a [[Luttinger liquid]] instead.  A Luttinger liquid is a [[Paramagnetism|paramagnetic]] one-dimensional metal without Landau [[Quasiparticle|quasi-particle]] excitations.&lt;br /&gt;
&lt;br /&gt;
== Research topics ==&lt;br /&gt;
&lt;br /&gt;
1-D metals have been the subject of much research.  Here are a few examples of both theoretical and experimental research efforts to illustrate the broad range of topics:&lt;br /&gt;
&lt;br /&gt;
* Theory has shown that polymer chains that have been looped and formed into rings undergo a Peierls transition.  These rings demonstrate a persistent current and the Peierls distortion can be modified by modulating the magnetic flux through the loop.&amp;lt;ref&amp;gt;{{cite journal | author = S. D. Liang, Y. H. Bai, B. Beng | journal = [[Physical Review B]] | volume = 74 | year = 2006 | pages = 113304 | doi = 10.1103/PhysRevB.74.113304 | title = Peierls instability and persistent current in mesoscopic conducting polymer rings | issue = 11 |bibcode = 2006PhRvB..74k3304L }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* Density functional theory has been used to calculate the bond length alterations predicted in increasingly long chains of organic oligomers.  The selection of which hybrid functional to use is paramount in obtaining an accurate estimate of the bond length alteration caused by Peierls distortions, as some functionals have been shown to overestimate the oscillation while others underestimate it.&amp;lt;ref&amp;gt;{{cite journal | author = D. Jacquemin, A. Femenias, H. Chermette, I. Ciofini, C. Adamo, J. M. Andr, E. A. Perpte | journal = [[Journal of Physical Chemistry A]] | volume = 110 | year = 2006 | pages = 5952–5959 | doi = 10.1021/jp060541w | title = Assessment of Several Hybrid DFT Functionals for the Evaluation of Bond Length Alternation of Increasingly Long Oligomers | pmid = 16640395 | issue = 17 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* Gold deposited on a stepped Si(553) surface has shown evidence of two simultaneous Peierls transitions.  The lattice period is distorted by factors of 2 and 3, and energy gaps open for nearly 1/2-filled and 1/3-1/4 filled bands.  The distortions have been studied and imaged using [[Low-energy electron diffraction|LEED]] and [[Scanning tunneling microscope|STM]], while the energy bands were studied with [[ARPES|ARP]].&amp;lt;ref&amp;gt;{{cite journal | author = J. R. Ahn, P. G. Kang, K. D. Ryang, H.W. Yeom | journal = [[Physical Review Letters]] | volume = 95 | year = 2005 | pages = 196402 | doi = 10.1103/PhysRevLett.95.196402 | title = Coexistence of Two Different Peierls Distortions within an Atomic Scale Wire: Si(553)-Au | pmid=16384001 | bibcode=2005PhRvL..95s6402A | issue = 19}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* [[Luttinger liquid]]s have a power law dependence of resistance on temperature.  This has been shown for purple bronze (Li&amp;lt;sub&amp;gt;0.9&amp;lt;/sub&amp;gt;Mo&amp;lt;sub&amp;gt;6&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;17&amp;lt;/sub&amp;gt;).&amp;lt;ref&amp;gt;{{cite journal | author = C. A. M. dos Santos, M. S. da Luz, Yi-Kuo Yu, J. J. Neumeier, J. Moreno, B. D. White | journal = [[Physical Review B]] | volume = 77 | year = 2008 | pages = 193106 | doi = 10.1103/PhysRevB.77.193106 | title = Electrical transport in single-crystalline Li&amp;lt;sub&amp;gt;0.9&amp;lt;/sub&amp;gt;Mo&amp;lt;sub&amp;gt;6&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;17&amp;lt;/sub&amp;gt;: A two-band Luttinger liquid exhibiting Bose metal behavior | issue = 19|bibcode = 2008PhRvB..77s3106D }}&amp;lt;/ref&amp;gt;  Purple bronze may prove to be a very interesting material, since it has shown renormalization of the Luttinger-liquid density of states anomalous exponent,&amp;lt;ref&amp;gt;{{cite journal | author = F. Wang, J.V. Alvarez, S.-K. Mo, J. W. Allen, G.-H. Gweon, J. He, R. Jin, D. Mandrus, H. Höchst | journal = [[Physical Review Letters]] | volume = 96 | year = 2006 | pages = 196403 | doi = 10.1103/PhysRevLett.96.196403 | title = New Luttinger-Liquid Physics from Photoemission on Li&amp;lt;sub&amp;gt;0.9&amp;lt;/sub&amp;gt;Mo&amp;lt;sub&amp;gt;6&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;17&amp;lt;/sub&amp;gt; | bibcode=2006PhRvL..96s6403W | pmid=16803117 | issue = 19|arxiv = cond-mat/0604503 }}&amp;lt;/ref&amp;gt; which is one of the parameters that are used to describe Luttinger liquid behavior.&amp;lt;ref&amp;gt;{{cite arXiv | last=Voit | first=Johannes | title=AIP Conference Proceedings | date=5 May 2000 | eprint=cond-mat/0005114 | class=cond-mat.str-el | doi=10.1063/1.1342524 | chapter=A brief introduction to Luttinger liquids | volume=544 | pages=309}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
* The dependence of resonant tunneling through island barriers in a 1-D wire has been studied, and is also found to be a power law dependence.  This offers additional evidence of Luttinger liquid behavior.&amp;lt;ref&amp;gt;{{cite journal | author = O. M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L. N. Pfeiffer, K.W. West | journal = [[Physical Review Letters]] | volume = 84 | year = 2000 | pages = 1764–1767 | doi = 10.1103/PhysRevLett.84.1764 | title = Experimental evidence for resonant tunneling in a Luttinger liquid | pmid=11017620 | bibcode=2000PhRvL..84.1764A | issue = 8|arxiv = cond-mat/9909138 }}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
== See also ==&lt;br /&gt;
* [[Charge density wave]]&lt;br /&gt;
* [[Luttinger liquid]]&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
[[Category:Superconductivity]]&lt;br /&gt;
[[Category:Phase transitions]]&lt;/div&gt;</summary>
		<author><name>141.14.162.129</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Substitution_model&amp;diff=240601</id>
		<title>Substitution model</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Substitution_model&amp;diff=240601"/>
		<updated>2012-03-16T12:58:48Z</updated>

		<summary type="html">&lt;p&gt;141.14.31.21: /* Mechanistic vs. empirical models */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;28 years old Medical Laboratory Technician Catanzaro from Saint John, enjoys to spend some time physical exercise (aerobics weights), new launch property singapore and educational courses. In the previous year has completed a journey to Archaeological Ruins at Moenjodaro.&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Feel free to surf to my web site - [http://zoom2x.net/x2/?document_srl=138281 the Skywoods developer]&lt;/div&gt;</summary>
		<author><name>141.14.31.21</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Glucose-1-phosphate_adenylyltransferase&amp;diff=21205</id>
		<title>Glucose-1-phosphate adenylyltransferase</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Glucose-1-phosphate_adenylyltransferase&amp;diff=21205"/>
		<updated>2010-10-29T10:43:26Z</updated>

		<summary type="html">&lt;p&gt;141.14.245.16: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{enzyme&lt;br /&gt;
| Name = glycerate kinase&lt;br /&gt;
| EC_number = 2.7.1.31&lt;br /&gt;
| CAS_number = 9026-61-3&lt;br /&gt;
| IUBMB_EC_number = 2/7/1/31&lt;br /&gt;
| GO_code = 0008887&lt;br /&gt;
| image = &lt;br /&gt;
| width = &lt;br /&gt;
| caption = &lt;br /&gt;
}}&lt;br /&gt;
{{Pfam box |Symbol = Glyc_kinase |Name = Glycerate kinase |Pfam = PF02595 |InterPro = IPR004381 |PROSITE =  |PDB = {{PDB|1to6}} }}&lt;br /&gt;
&lt;br /&gt;
In [[enzymology]], a &#039;&#039;&#039;glycerate kinase&#039;&#039;&#039; ({{EC number|2.7.1.31}}) is an [[enzyme]] that [[catalysis|catalyzes]] the [[chemical reaction]]&lt;br /&gt;
&lt;br /&gt;
:ATP + (R)-glycerate &amp;lt;math&amp;gt;\rightleftharpoons&amp;lt;/math&amp;gt; ADP + 3-phospho-(R)-glycerate&lt;br /&gt;
&lt;br /&gt;
Thus, the two [[substrate (biochemistry)|substrates]] of this enzyme are [[adenosine triphosphate|ATP]] and [[(R)-glycerate]], whereas its two [[product (chemistry)|products]] are [[adenosine diphosphate|ADP]] and [[3-phospho-(R)-glycerate]].&lt;br /&gt;
&lt;br /&gt;
This enzyme belongs to the family of [[transferase]]s, specifically those transferring phosphorus-containing groups ([[phosphotransferase]]s) with an alcohol group as acceptor.  The systematic name of this enzyme class is &#039;&#039;&#039;ATP:(R)-glycerate 3-phosphotransferase&#039;&#039;&#039;. Other names in common use include &#039;&#039;&#039;glycerate kinase (phosphorylating)&#039;&#039;&#039;, &#039;&#039;&#039;D-glycerate 3-kinase&#039;&#039;&#039;, &#039;&#039;&#039;D-glycerate kinase&#039;&#039;&#039;, &#039;&#039;&#039;glycerate-3-kinase&#039;&#039;&#039;, &#039;&#039;&#039;GK&#039;&#039;&#039;, &#039;&#039;&#039;D-glyceric acid kinase&#039;&#039;&#039;, and &#039;&#039;&#039;ATP:D-glycerate 2-phosphotransferase&#039;&#039;&#039;.  This enzyme participates in 3 [[metabolism|metabolic pathways]]: [[serine]]/[[glycine]]/[[threonine]] metabolism, [[glycerolipid]] metabolism, and [[glyoxylate]]-[[dicarboxylate]] metabolism.  &lt;br /&gt;
&lt;br /&gt;
==Structural studies==&lt;br /&gt;
&lt;br /&gt;
As of late 2007, 3 [[tertiary structure|structures]] have been solved for this class of enzymes, with [[Protein Data Bank|PDB]] accession codes {{PDB link|1TO6}}, {{PDB link|1X3L}}, and {{PDB link|2B8N}}.&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
{{reflist|1}}&lt;br /&gt;
* {{cite journal | author = Doughty CC, Hayashi JA, Guenther HL | date = 1966 | title = Purification and properties of D-glycerate 3-kinase from Escherichia coli | journal = J. Biol. Chem.  | volume = 241 | pages = 568&amp;amp;ndash;72  | pmid = 5325263 | issue = 3 }}&lt;br /&gt;
* {{cite journal | author = ICHIHARA A, GREENBERG DM | date = 1957 | title = Studies on the purification and properties of D-glyceric acid kinase of liver | journal = J. Biol. Chem.  | volume = 225 | pages = 949&amp;amp;ndash;58  | pmid = 13416296 | issue = 2 }}&lt;br /&gt;
&lt;br /&gt;
{{enzyme-stub}}&lt;br /&gt;
&lt;br /&gt;
[[Category:EC 2.7.1]]&lt;br /&gt;
[[Category:Enzymes of known structure]]&lt;/div&gt;</summary>
		<author><name>141.14.245.16</name></author>
	</entry>
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