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Multiplication theorem
2013-12-03T19:20:31Z
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<hr />
<div>[[File:Quaternion Plaque on Broom Bridge.jpg|thumb|right|Plaque on Broom bridge (Dublin) commemorating Hamilton's invention of quaternions]]<br />
In [[mathematics]] and [[mechanics]], the set of '''dual quaternions''' is a [[Clifford algebra#Examples:_constructing_quaternions_and_dual_quaternions|Clifford algebra]] that can be used to represent spatial rigid body displacements.<ref>A.T. Yang, ''Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms'', Ph.D thesis, Columbia University, 1963.</ref> A dual quaternion is an ordered pair of [[quaternion]]s {{nowrap|''Â'' {{=}} (''A'', ''B'')}} and therefore is constructed from eight real parameters. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints.<br />
<br />
In [[ring theory]], dual quaternions are a [[ring (mathematics)|ring]] constructed in the same way as the quaternions, except using [[dual number]]s instead of [[real number]]s as coefficients. A dual quaternion can be represented in the form ''p'' + ε ''q'' where ''p'' and ''q'' are ordinary quaternions and ε is the dual unit (εε = 0) and commutes with every element of the algebra. Unlike quaternions they do not form a [[division ring]].<br />
<br />
Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical [[kinematics]] (see McCarthy<ref name="mccarthy"/>), and in applications to 3D [[computer graphics]], [[robot]]ics and [[computer vision]].<ref>[http://www.dsi.unive.it/~rodola/cvpr11-dual.pdf A. Torsello, E. Rodolà and A. Albarelli, ''Multiview Registration via Graph Diffusion of Dual Quaternions'', Proc. of the XXIV IEEE Conference on Computer Vision and Pattern Recognition, pp. 2441-2448, June 2011.]</ref><br />
<br />
==History==<br />
[[W. R. Hamilton]] introduced [[quaternion]]s<ref>W. R. Hamilton, "On quaternions, or on a new system of imaginaries in algebra," Philos. Mag. 18, installments July 1844 – April 1850, ed. by D. E. Wilkins (2000)</ref><ref>[http://books.google.com/books?id=GFYtAAAAYAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false W. R. Hamilton, ''Elements of Quaternions, Longmans, Green & Co., London, 1866]</ref> in 1843, and by 1873 [[William Kingdon Clifford|W. K. Clifford]] obtained a broad generalization of these numbers that he called ''biquaternions'',<ref>W. K. Clifford, "Preliminary sketch of bi-quaternions, Proc. London Math. Soc. Vol. 4 (1873) pp. 381–395</ref><ref>W. K. Clifford, ''Mathematical Papers'', (ed. R. Tucker), London: Macmillan, 1882.</ref> which is an example of what is now called a [[Clifford algebra]].<ref name="mccarthy">[http://books.google.com/books?id=glOqQgAACAAJ&dq=inauthor:%22J.+M.+McCarthy%22&hl=en&ei=_QoMToDvMcfd0QGFh-mvDg&sa=X&oi=book_result&ct=book-thumbnail&resnum=3&ved=0CDsQ6wEwAg J. M. McCarthy, ''An Introduction to Theoretical Kinematics'', pp.&nbsp;62–5, MIT Press 1990.]</ref> At the turn of the 20th century, [[Aleksandr Kotelnikov]]<ref>A. P. Kotelnikov, ''Screw calculus and some applications to geometry and mechanics'', Annal. Imp. Univ. Kazan (1895)</ref> and E. Study<ref>E. Study, ''Geometrie der Dynamen'', Teubner, Leipzig, 1901.</ref> developed dual vectors and dual quaternions for use in the study of mechanics.<br />
<br />
In 1891 [[Eduard Study]] realized that this [[associative algebra]] was ideal for describing the group of motions of [[three-dimensional space]]. He further developed the idea in ''Geometrie der Dynamen'' in 1901. [[B. L. van der Waerden]] called the structure "Study biquaternions", one of three eight-dimensional algebras referred to as [[biquaternion]]s.<br />
<br />
==Formulas==<br />
In order to describe operations with dual quaternions, it is helpful to first consider [[quaternion]]s.<ref>[http://books.google.com/books?id=f8I4yGVi9ocC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false O. Bottema and B. Roth, ''Theoretical Kinematics'', North Holland Publ. Co., 1979]</ref><br />
<br />
A quaternion is a linear combinations of the basis elements 1, ''i'', ''j'', and ''k''. Hamilton's product rule for ''i'', ''j'', and ''k'' is often written as<br />
:<math> i^2 = j^2 = k^2 = ijk = -1.\!</math><br />
Compute {{nowrap|''i'' ( ''i j k'' ) {{=}} −''j k'' {{=}} −''i''}}, to obtain {{nowrap|''j k'' {{=}} ''i''}}, and {{nowrap|( ''i j k'' ) ''k'' {{=}} −''i j'' {{=}} −''k''}} or {{nowrap|''i j'' {{=}} ''k''}}. Now because {{nowrap|''j'' ( ''j k'' ) {{=}} ''j i'' {{=}} −''k''}}, we see that this product yields {{nowrap|''i j'' {{=}} −''j i''}}, which links quaternions to the properties of determinants.<br />
<br />
A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector, that is {{nowrap|''A'' {{=}} ''a''<sub>0</sub> + '''A'''}}, where ''a''<sub>0</sub> is a real number and {{nowrap|'''A''' {{=}} ''A''<sub>1</sub> ''i'' + ''A''<sub>2</sub> ''j'' + ''A''<sub>3</sub> ''k''}} is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of {{nowrap|''A'' {{=}} ''a''<sub>0</sub> + '''A'''}} and {{nowrap|''C'' {{=}} ''c''<sub>0</sub> + '''C'''}} as<br />
<!--reformatted this equation<br />
:G = AC = (a<sub>0</sub>+'''A''')(c<sub>0</sub>+'''C''') = (a<sub>0</sub>c<sub>0</sub> − '''A'''·'''C''') + (c<sub>0</sub>'''A''' + a<sub>0</sub>'''C''' + '''A'''×'''C''').<br />
--><br />
:<math> G = AC = (a_0 + \mathbf{A})(c_0 + \mathbf{C}) = (a_0 c_0 - \mathbf{A}\cdot \mathbf{C}) + (c_0 \mathbf{A} + a_0 \mathbf{C} + \mathbf{A}\times\mathbf{C}).</math><br />
<br />
A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A [[dual number]] is an ordered pair {{nowrap|''â'' {{=}} ( ''a'', ''b'' )}}. Two dual numbers add componentwise and multiply by the rule {{nowrap|''â ĉ'' {{=}} ( ''a'', ''b'' ) ( ''c'', ''d'' ) {{=}} (''a c'', ''a d'' + ''b c'')}}. Dual numbers are often written in the form {{nowrap|''â'' {{=}} ''a'' + ε''b''}}, where ε is the dual unit that commutes with ''i'', ''j'', ''k'' and has the property {{nowrap|ε<sup>2</sup> {{=}} 0}}. <br />
<br />
The result is that a dual quaternion is the ordered pair of quaternions {{nowrap|''Â'' {{=}} ( ''A'', ''B'' )}}. Two dual quaternions add componentwise and multiply by the rule,<br />
<!-- reformatted this equation<br />
:ÂĈ = (A, B)(C, D) = (AC, AD+BC).<br />
--><br />
:<math> \hat{A}\hat{C} = (A, B)(C, D) = (AC, AD+BC).\!</math><br />
It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector, {{nowrap|''Â'' {{=}} ''â''<sub>0</sub> + ''A''}}, where {{nowrap|''â''<sub>0</sub> {{=}} ( ''a'', ''b'' )}} and {{nowrap|''A'' {{=}} ( '''A''', '''B''' )}} is the dual vector that defines a [[screw theory|screw]]. This notation allows us to write the product of two dual quaternions as<br />
<!-- reformatted equation<br />
:Ĝ = ÂĈ =(â<sub>0</sub> + A)(ĉ<sub>0</sub> + C) = (â<sub>0</sub>ĉ<sub>0</sub> - A•C) + (ĉ<sub>0</sub>A + â<sub>0</sub>C + A×C).<br />
--><br />
:<math> \hat{G} = \hat{A}\hat{C} = (\hat{a}_0 + \mathsf{A})(\hat{c}_0 + \mathsf{C}) = (\hat{a}_0 \hat{c}_0 - \mathsf{A}\cdot \mathsf{C}) + (\hat{c}_0 \mathsf{A} + \hat{a}_0 \mathsf{C} + \mathsf{A}\times\mathsf{C}).</math><br />
<br />
=== Addition ===<br />
The addition of dual quaternions is defined componentwise so that given, <br />
<!-- reformatted equation<br />
:Â = (A, B) = a<sub>0</sub> + a<sub>1</sub>i + a<sub>2</sub>j + a<sub>3</sub>k + b<sub>0</sub>ε + b<sub>1</sub>εi + b<sub>2</sub>εj + b<sub>3</sub>εk, and <br />
:Ĉ = (C, D) = c<sub>0</sub> + c<sub>1</sub>i + c<sub>2</sub>j + c<sub>3</sub>k + d<sub>0</sub>ε + d<sub>1</sub>εi + d<sub>2</sub>εj + d<sub>3</sub>εk, <br />
--><br />
:<math> \hat{A} = (A, B) = a_0 + a_1 i + a_2 j + a_3 k + b_0 \epsilon + b_1 \epsilon i + b_2 \epsilon j + b_3 \epsilon k, </math><br />
and<br />
:<math> \hat{C} = (C, D) = c_0 + c_1 i + c_2 j + c_3 k + d_0 \epsilon + d_1 \epsilon i + d_2 \epsilon j + d_3 \epsilon k, </math><br />
then<br />
<!-- reformatted equation<br />
:Â + Ĉ = (a<sub>0</sub>+c<sub>0</sub>) + (a<sub>1</sub>+c<sub>1</sub>)i + (a<sub>2</sub>+c<sub>2</sub>)j + (a<sub>3</sub>+c<sub>3</sub>)k + (b<sub>0</sub>+d<sub>0</sub>)ε + (b<sub>1</sub>+d<sub>1</sub>)εi + (b<sub>2</sub>+d<sub>2</sub>)εj + (b<sub>3</sub>+d<sub>3</sub>)εk.<br />
--><br />
:<math> \hat{A} + \hat{C} = (A+C, B+D) = (a_0+c_0) + (a_1+c_1) i + (a_2+c_2) j + (a_3+c_3) k + (b_0+d_0) \epsilon + (b_1+d_1) \epsilon i + (b_2+d_2) \epsilon j + (b_3+d_3) \epsilon k, </math><br />
<br />
===Multiplication===<br />
Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε. In particular, given<br />
<!-- :Â = (A, B) = A + εB and Ĉ = (C, D) = C + εD, --><br />
:<math> \hat{A} = (A, B) = A + \epsilon B, </math><br />
and<br />
:<math> \hat{C} = (C, D) = C + \epsilon D, </math><br />
then<br />
<!-- :ÂĈ = AC + ε(AD+BC) --><br />
:<math> \hat{A}\hat{C} = (A + \epsilon B)(C + \epsilon D) = AC + \epsilon (AD+BC). \!</math><br />
Notice that there is no ''BD'' term, because the definition of dual numbers requires that {{nowrap|ε<sup>2</sup> {{=}} 0}}.<br />
<br />
This gives us the multiplication table (note the multiplication order is row times column):<br />
<br />
{| class="wikitable" style="text-align: center; width: 400px; height: 200px;"<br />
|+ Multiplication table for dual quaternion units<br />
|-<br />
! scope="col" | ×<br />
! scope="col" | 1 <br />
! scope="col" | ''i'' <br />
! scope="col" | ''j''<br />
! scope="col" | ''k''<br />
! scope="col" | ε<br />
! scope="col" | ε''i''<br />
! scope="col" | ε''j''<br />
! scope="col" | ε''k''<br />
|-<br />
! scope="row" | 1<br />
| 1 || ''i'' || ''j'' || ''k'' || ε || ε''i'' || ε''j'' || ε''k''<br />
|-<br />
! scope="row" | ''i''<br />
| ''i'' || −1 || ''k'' || −''j'' || ε''i'' || −ε || ε''k'' || −ε''j''<br />
|-<br />
! scope="row" | ''j''<br />
| ''j'' || −''k'' || −1 || ''i'' || ε''j'' || −ε''k'' || −ε || ε''i''<br />
|-<br />
! scope="row" | ''k''<br />
| ''k'' || ''j'' || −''i'' || −1 || ε''k'' || ε''j'' || −ε''i'' || −ε<br />
|-<br />
! scope="row" | ε<br />
| ε || ε''i'' || ε''j'' || ε''k'' || 0 || 0 || 0 || 0<br />
|-<br />
! scope="row" | ε''i''<br />
| ε''i'' || −ε || ε''k'' || −ε''j'' || 0 || 0 || 0 || 0<br />
|-<br />
! scope="row" | ε''j''<br />
| ε''j'' || −ε''k'' || −ε || ε''i'' || 0 || 0 || 0 || 0<br />
|-<br />
! scope="row" | ε''k''<br />
| ε''k'' || ε''j'' || −ε''i'' || −ε || 0 || 0 || 0 || 0<br />
|-<br />
|}<br />
<br />
<!-- replaced this table with wiki table format<br />
<table border="1"><br />
<tr><br />
<td><math>Q_1 * Q_2\,\!</math></td><br />
<td><math>Q_2.1\,\!</math></td><br />
<td><math>Q_2.i\,\!</math></td><br />
<td><math>Q_2.j\,\!</math></td><br />
<td><math>Q_2.k\,\!</math></td><br />
<td><math>Q_2.\varepsilon</math></td><br />
<td><math>Q_2.\varepsilon i</math></td><br />
<td><math>Q_2.\varepsilon j</math></td><br />
<td><math>Q_2.\varepsilon k</math></td><br />
</tr><br />
<tr><br />
<td><math>Q_1.1\,\!</math></td><br />
<td>1</td><br />
<td>i</td><br />
<td>j</td><br />
<td>k</td><br />
<td><math>\varepsilon</math></td><br />
<td><math>\varepsilon i</math></td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>\varepsilon k</math></td><br />
</tr><br />
<tr><br />
<td><math>Q_1.i\,\!</math></td><br />
<td>i</td><br />
<td>-1</td><br />
<td>k</td><br />
<td>-j</td><br />
<td><math>\varepsilon i</math></td><br />
<td><math>-\varepsilon</math></td><br />
<td><math>\varepsilon k</math></td><br />
<td><math>-\varepsilon j</math></td><br />
</tr><br />
<tr><br />
<td><math>Q_1.j\,\!</math></td><br />
<td>j</td><br />
<td>-k</td><br />
<td>-1</td><br />
<td>i</td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>-\varepsilon k</math></td><br />
<td><math>-\varepsilon</math></td><br />
<td><math>\varepsilon i</math></td><br />
</tr><br />
<tr><br />
<td><math>Q_1.k\,\!</math></td><br />
<td>k</td><br />
<td>j</td><br />
<td>-i</td><br />
<td>-1</td><br />
<td><math>\varepsilon k</math></td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>-\varepsilon i</math></td><br />
<td><math>-\varepsilon</math></td><br />
</tr><br />
<tr><br />
<td><math>Q_1.\varepsilon</math></td><br />
<td><math>\varepsilon</math></td><br />
<td><math>\varepsilon i</math></td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>\varepsilon k</math></td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
</tr><br />
<tr><br />
<td><math>Q_1.\varepsilon i</math></td><br />
<td><math>\varepsilon i</math></td><br />
<td><math>-\varepsilon</math></td><br />
<td><math>\varepsilon k</math></td><br />
<td><math>-\varepsilon j</math></td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
</tr><br />
<tr><br />
<td><math>Q_1.\varepsilon j</math></td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>-\varepsilon k</math></td><br />
<td><math>-\varepsilon</math></td><br />
<td><math>\varepsilon i</math></td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
</tr><br />
<tr><br />
<td><math>Q_1.\varepsilon k</math></td><br />
<td><math>\varepsilon k</math></td><br />
<td><math>\varepsilon j</math></td><br />
<td><math>-\varepsilon i</math></td><br />
<td><math>-\varepsilon</math></td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
<td>0</td><br />
</tr><br />
</table><br />
--><br />
<br />
===Conjugate===<br />
<!-- This is not a useful definition of the conjugate of a dual quaternion <br />
A dual quaternion has three different definitions of conjugate, which can be expressed as follows.<br />
Where <math>q</math> is the dual quaternion, <math>r</math> is a the 'real' quaternion part, and <math>d</math> is the dual part.<br />
* <math>q^\dagger = r^* + \varepsilon d^*</math><br />
* <math>q_\varepsilon = r - \varepsilon d</math><br />
* <math>q^\dagger_\varepsilon = r^* - \varepsilon d^*</math><br />
--><br />
<br />
The conjugate of a dual quaternion is the extension of the conjugate of a quaternion, that is<br />
:<math> \hat{A}^* = (A^*, B^*) = A^* + \epsilon B^*. \!</math><br />
<br />
As for quaternions, the conjugate of the product of dual quaternions, {{nowrap|''Ĝ'' {{=}} ''ÂĈ''}}, is the product of their conjugates in reverse order,<br />
:<math> \hat{G}^* = (\hat{A}\hat{C})^* = \hat{C}^*\hat{A}^*.\!</math><br />
<br />
It is useful to introduce the functions Sc(∗) and Vec(∗) that select the scalar and vector parts of a quaternion, or the dual scalar and dual vector parts of a dual quaternion. In particular, if {{nowrap|''Â'' {{=}} ''â''<sub>0</sub> + ''A''}}, then<br />
:<math> \mbox{Sc}(\hat{A}) = \hat{a}_0, \mbox{Vec}(\hat{A}) = \mathsf{A}.\!</math><br />
This allows the definition of the conjugate of ''Â'' as<br />
:<math> \hat{A}^* = \mbox{Sc}(\hat{A}) - \mbox{Vec}(\hat{A}).\!</math><br />
or, <br />
:<math> (\hat{a}_0+\mathsf{A})^* = \hat{a}_0 - \mathsf{A}.\!</math><br />
<br />
The product of a dual quaternion with its conjugate yields<br />
:<math>\hat{A}\hat{A}^* = (\hat{a}_0+\mathsf{A})(\hat{a}_0 - \mathsf{A}) = \hat{a}_0^2 + \mathsf{A}\cdot\mathsf{A}.\!</math><br />
This is a dual scalar which is the ''magnitude squared'' of the dual quaternion.<br />
<br />
=== Norm ===<br />
The ''norm'' of a dual quaternion |''Â''| is computed using the conjugate to compute {{nowrap|{{!}}''Â''{{!}} {{=}} √(''Â Â''<sup>*</sup>)}}. This is a dual number called the ''magnitude'' of the dual quaternion. Dual quaternions with {{nowrap|{{!}}''Â''{{!}} {{=}} 1}} are ''unit dual quaternions''.<br />
<br />
Dual quaternions of magnitude 1 are used to represent spatial Euclidean displacements. Notice that the requirement that {{nowrap|''Â Â''<sup>*</sup>}} {{=}} 1, introduces two algebraic constraints on the components of ''Â'', that is<br />
:<math> \hat{A}\hat{A}^* = (A, B)(A^*, B^*) = (AA^*, AB^* + BA^*) = (1, 0).\!</math><br />
<br />
=== Inverse ===<br />
If ''p'' + ε ''q'' is a dual quaternion, and ''p'' is not zero, then the inverse dual quaternion is given by <br />
:''p''<sup>&minus;1</sup> (1 &minus; ε ''q'' ''p''<sup>&minus;1</sup>).<br />
Thus the elements of the subspace { ε q : q ∈ H } do not have inverses. This subspace is called an [[ideal (ring theory)|ideal]] in ring theory. It happens to be the unique [[maximal ideal]] of the ring of dual numbers.<br />
<br />
The [[group of units]] of the dual number ring then consists of numbers not in the ideal. The dual numbers form a [[local ring]] since there is a unique maximal ideal. The group of units is a [[Lie group]] and can be studied using the [[exponential mapping]]. Dual quaternions have been used to exhibit transformations in the [[Euclidean group]]. A typical element can be written as a [[screw theory#Homography|screw transformation]].<br />
<br />
== Dual quaternions and spatial displacements ==<br />
A benefit of the dual quaternion formulation of the composition of two spatial displacements D<sub>B</sub>=([R<sub>B</sub>], '''b''') and D<sub>A</sub>=([R<sub>A</sub>],'''a''') is that the resulting dual quaternion yields directly the [[screw axis]] and dual angle of the composite displacement D<sub>C</sub>=D<sub>B</sub>D<sub>A</sub>.<br />
<br />
In general, the dual quaternion associated with a spatial displacement D = ([A],'''d''') is constructed from its [[screw axis]] S=('''S''', '''V''') and the dual angle (φ, d) where φ is the rotation about and d the slide along this axis, which defines the displacement D. The associated dual quaternion is given by,<br />
:<math> \hat{S} = \cos\frac{\hat{\phi}}{2} + \sin\frac{\hat{\phi}}{2} \mathsf{S}. </math><br />
<br />
Let the composition of the displacement D<sub>B</sub> with D<sub>A</sub> be the displacement D<sub>C</sub>=D<sub>B</sub>D<sub>A</sub>. The screw axis and dual angle of D<sub>C</sub> is obtained from the product of the dual quaternions of D<sub>A</sub> and D<sub>B</sub>, given by<br />
:<math>\hat{A}=\cos(\hat{\alpha}/2)+ \sin(\hat{\alpha}/2)\mathsf{A}\quad<br />
\text{and}\quad \hat{B}=\cos(\hat{\beta}/2)+ \sin(\hat{\beta}/2)\mathsf{B}.</math><br />
That is, the composite displacement D<sub>C</sub>=D<sub>B</sub>D<sub>A</sub> has the associated dual quaternion given by<br />
:<math> \hat{C} = \cos\frac{\hat{\gamma}}{2}+\sin\frac{\hat{\gamma}}{2}\mathsf{C}<br />
=<br />
\Big(\cos\frac{\hat{\beta}}{2}+\sin\frac{\hat{\beta}}{2}\mathsf{B}\Big) \Big(\cos\frac{\hat{\alpha}}{2}+<br />
\sin\frac{\hat{\alpha}}{2}\mathsf{A}\Big).<br />
</math><br />
<br />
Expand this product in order to obtain<br />
:<math><br />
\cos\frac{\hat{\gamma}}{2}+\sin\frac{\hat{\gamma}}{2} \mathsf{C} = <br />
\Big(\cos\frac{\hat{\beta}}{2}\cos\frac{\hat{\alpha}}{2} - <br />
\sin\frac{\hat{\beta}}{2}\sin\frac{\hat{\alpha}}{2} \mathsf{B}\cdot \mathsf{A}\Big) + \Big(\sin\frac{\hat{\beta}}{2}\cos\frac{\hat{\alpha}}{2} \mathsf{B} + <br />
\sin\frac{\hat{\alpha}}{2}\cos\frac{\hat{\beta}}{2} \mathsf{A} + <br />
\sin\frac{\hat{\beta}}{2}\sin\frac{\hat{\alpha}}{2} \mathsf{B}\times \mathsf{A}\Big).<br />
</math><br />
<br />
Divide both sides of this equation by the identity<br />
:<math> \cos\frac{\hat{\gamma}}{2} = \cos\frac{\hat{\beta}}{2}\cos\frac{\hat{\alpha}}{2} - <br />
\sin\frac{\hat{\beta}}{2}\sin\frac{\hat{\alpha}}{2} \mathsf{B}\cdot \mathsf{A}</math><br />
to obtain<br />
:<math> \tan\frac{\hat{\gamma}}{2} \mathsf{C} = \frac{\tan\frac{\hat{\beta}}{2}\mathsf{B} + <br />
\tan\frac{\hat{\alpha}}{2} \mathsf{A} + <br />
\tan\frac{\hat{\beta}}{2}\tan\frac{\hat{\alpha}}{2} \mathsf{B}\times \mathsf{A}}{1 - <br />
\tan\frac{\hat{\beta}}{2}\tan\frac{\hat{\alpha}}{2} \mathsf{B}\cdot \mathsf{A}}.<br />
</math><br />
<br />
This is Rodrigues formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements. He derived this formula in 1840.<ref>Rodrigues, O. (1840), Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et la variation des coordonnées provenant de ses déplacements con- sidérés indépendamment des causes qui peuvent les produire, Journal de Mathématiques Pures et Appliquées de Liouville 5, 380–440.</ref> <br />
<br />
The three screw axes A, B, and C form a [http://books.google.com/books?id=f8I4yGVi9ocC&lpg=PP1&pg=PA64#v=twopage&q&f=false spatial triangle] and the dual angles at these ''vertices'' between the common normals that form the sides of this triangle are directly related to the dual angles of the three spatial displacements.<br />
<br />
== Matrix form of dual quaternion multiplication ==<br />
The matrix representation of the quaternion product is convenient for programming quaternion computations using matrix algebra, which is true for dual quaternion operations as well. <br />
<br />
The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A operating on the vector formed from the components of C.<br />
<br />
Assemble the components of the quaternion C=c<sub>0</sub>+'''C''' into the array C=(C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, c<sub>0</sub>). Notice that the components of the vector part of the quaternion are listed first and the scalar is listed last. This is an arbitrary choice, but once this convention is selected we must abide by it.<br />
<br />
The quaternion product AC can now be represented as the matrix product<br />
:<math> <br />
AC = [A^+] C = <br />
\begin{bmatrix}<br />
a_0 & -A_3 & A_2 & A_1 \\<br />
A_3 & a_0 & -A_1 & A_2 \\<br />
-A_2 & A_1 & a_0 & A_3 \\<br />
-A_1 & -A_2 & -A_3 & a_0<br />
\end{bmatrix}<br />
\begin{Bmatrix} C_1 \\ C_2 \\ C_3 \\ c_0 \end{Bmatrix}.<br />
</math><br />
<br />
The product AC can also be viewed as an operation by C on the components of A, in which case we have<br />
:<math><br />
AC = [C^-] A = \begin{bmatrix}<br />
c_0 & C_3 & -C_2 & C_1 \\<br />
-C_3 & c_0 & C_1 & C_2 \\<br />
C_2 & -C_1 & c_0 & C_3 \\<br />
-C_1 & -C_2 & -C_3 & c_0<br />
\end{bmatrix}<br />
\begin{Bmatrix} A_1 \\ A_2 \\ A_3 \\ a_0 \end{Bmatrix}.<br />
</math><br />
<br />
The dual quaternion product ÂĈ = (A, B)(C, D) = (AC, AD+BC) can be formulated as a matrix operation as follows. Assemble the components of Ĉ into the eight dimensional array Ĉ = (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, c<sub>0</sub>, D<sub>1</sub>, D<sub>2</sub>, D<sub>3</sub>, d<sub>0</sub>), then ÂĈ is given by the 8x8 matrix product<br />
:<math><br />
\hat{A}\hat{C} = [\hat{A}^+]\hat{C} = \begin{bmatrix} A^+ & 0 \\ B^+ & A^+ \end{bmatrix}\begin{Bmatrix} C \\ D\end{Bmatrix}.<br />
</math><br />
<br />
As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as,<br />
:<math><br />
\hat{A}\hat{C} = [\hat{C}^-]\hat{A} = \begin{bmatrix} C^- & 0 \\ D^- & C^- \end{bmatrix}\begin{Bmatrix} A \\ B\end{Bmatrix}.<br />
</math><br />
<br />
== More on spatial displacements ==<br />
The dual quaternion of a displacement D=([A], '''d''') can be constructed from the quaternion S=cos(φ/2) + sin(φ/2)'''S''' that defines the rotation [A] and the vector quaternion constructed from the translation vector '''d''', given by D = d<sub>1</sub>i + d<sub>2</sub>j + d<sub>3</sub>k. Using this notation, the dual quaternion for the displacement D=([A], '''d''') is given by<br />
:<math> \hat{S} = S + \varepsilon \frac{1}{2}DS. </math><br />
<br />
Let the Plücker coordinates of a line in the direction '''x''' through a point '''p''' in a moving body and its coordinates in the fixed frame which is in the direction '''X''' through the point '''P''' be given by,<br />
:<math>\hat{x}=\mathbf{x} + \varepsilon \mathbf{p}\times\mathbf{x}\quad\text{and}\quad\hat{X}=\mathbf{X} + \varepsilon \mathbf{P}\times\mathbf{X}.</math><br />
Then the dual quaternion of the displacement of this body transforms Plücker coordinates in the moving frame to Plücker coordinates in the fixed frame by the formula<br />
:<math>\hat{X} = \hat{S}\hat{x}\hat{S}^*.</math><br />
<br />
Using the matrix form of the dual quaternion product this becomes,<br />
:<math>\hat{X} =[\hat{S}^+][\hat{S}^-]^*\hat{x}.</math><br />
This calculation is easily managed using matrix operations.<br />
<br />
== Dual quaternions and 4×4 homogeneous transforms ==<br />
It might be helpful, especially in rigid body motion, to represent dual quaternions as [[homogeneous transformation matrix|homogeneous matrices]]. As given above a dual quaternion can be written as: <math>\hat q = r + d\varepsilon</math> where ''r'' and ''d'' are both quaternions. The ''r'' quaternion is known as the real or rotational part and the <math>d</math> quaternion is known as the dual or displacement part. A 3 dimensional position vector, <br />
:<math>\vec{v} = (v_0, v_1, v_2)</math><br />
can be transformed by constructing the dual-quaternion representation,<br />
:<math>\hat v := 1 + \varepsilon (v_0 i + v_1 j + v_2 k)</math><br />
then a transformation by <math>\hat q</math> is given by<br />
:<math>\vec{v}' = \hat q \cdot \hat v \cdot \hat q^{-1}</math>.<br />
The rotation part can be given by<br />
:<math>r = r_w + r_xi + r_yj + r_zk = \cos \left( \frac{\theta}{2} \right) + \sin \left( \frac{\theta}{2} \right) \cdot \vec{a}</math><br />
where <math>\theta</math> is the angle of rotation about axis <math>\vec{a}</math>. The rotation part can be expressed as a 3×3 [[orthogonal matrix]] by <br />
: <math>R =\begin{pmatrix}<br />
r_w^2+r_x^2-r_y^2-r_z^2 &2r_xr_y-2r_wr_z &2r_xr_z+2r_wr_y \\<br />
2r_xr_y+2r_wr_z &r_w^2-r_x^2+r_y^2-r_z^2 &2r_yr_z-2r_wr_x \\<br />
2r_xr_z-2r_wr_y &2r_yr_z+2r_wr_x &r_w^2-r_x^2-r_y^2+r_z^2\\<br />
\end{pmatrix}.</math><br />
The displacement can be written as<br />
: <math>d r^* = 0 + \frac{\Delta x}{2}i + \frac{\Delta y}{2}j + \frac{\Delta z}{2}k</math>.<br />
Translation and rotation combined in one transformation matrix is:<br />
: <math> [R|t] = \begin{pmatrix}<br />
& & & \Delta x \\<br />
& R & & \Delta y \\<br />
& & & \Delta z \\<br />
0 & 0 & 0 & 1 \\<br />
\end{pmatrix}</math><br />
Where the left upper 3×3 matrix is the rotation matrix we just calculated.<br />
<br />
==Eponyms==<br />
Since both [[Eduard Study]] and [[William Kingdon Clifford]] used, and wrote upon, the dual quaternions, at times authors refer to dual quaternions as “Study biquaternions” or “Clifford biquaternions”. The latter [[eponym]] has also been used to refer to [[split-biquaternion]]s. Read the article by Joe Rooney linked below for view of a supporter of W.K. Clifford’s claim. Since the claims of Clifford and Study are in contention, it is convenient to use the current designation ''dual quaternion'' to avoid conflict.<br />
<br />
==See also==<br />
* [[Screw theory]]<br />
* [[Screw axis]]<br />
* [[Quaternion]]<br />
* [[Rational motion]]<br />
* [[Quaternions and spatial rotation]]<br />
* [[Biquaternion]]<br />
* [[Conversion between quaternions and Euler angles]]<br />
* [[Clifford algebra]]<br />
* [[Olinde Rodrigues]]<br />
<br />
==References==<br />
{{Reflist}}<br />
* A.T. Yang (1963) ''Application of quaternion algebra and dual numbers to the analysis of spatial mechanisms'', Ph.D thesis, [[Columbia University]].<br />
* A.T. Yang (1974) "Calculus of Screws" in ''Basic Questions of Design Theory'', William R. Spillers, editor, [[Elsevier]], pages 266 to 281.<br />
* J.M. McCarthy (1990) ''An Introduction to Theoretical Kinematics'', pp.&nbsp;62–5, [[MIT]] Press [ISBN 0-262-13252-4].<br />
* L. Kavan, S. Collins, C. O'Sullivan, J. Zara (2006) [https://www.cs.tcd.ie/publications/tech-reports/reports.06/TCD-CS-2006-46.pdf ''Dual Quaternions for Rigid Transformation Blending''], Technical report, Trinity College Dublin.<br />
* Joe Rooney [http://oro.open.ac.uk/8455/01/chapter4(020507).pdf William Kingdon Clifford], Department of Design and Innovation, the Open University, London.<br />
* Joe Rooney (2007) "William Kingdon Clifford", in Marco Ceccarelli, ''Distinguished figures in mechanism and machine science'', Springer.<br />
* [[Eduard Study]] (1891) "Von Bewegungen und Umlegung", [[Mathematische Annalen]] 39:520.<br />
<br />
==External links==<br />
* [[Wilhelm Blaschke]] (1958) D.H. Delphenich translator, [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/blaschke_-_dual_quaternions.pdf "Applications of dual quaternions to kinematics"]<br />
* Wilhelm Blaschke (1960) D.H. Delphenich translator, [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/blaschke_-_kinematics_and_quaternions.pdf Quaternions and Kinematics] from Neo-classical-physics.info.<br />
<br />
{{DEFAULTSORT:Dual Quaternion}}<br />
[[Category:Machines]]<br />
[[Category:Kinematics]]<br />
[[Category:Quaternions]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Quantum_amplifier&diff=16091
Quantum amplifier
2013-12-02T00:28:45Z
<p>99.153.64.179: /* Exposition */ rm misleading statement I added earlier; see talk page</p>
<hr />
<div>{{Starbox begin<br />
| name=IK Pegasi }}<br />
{{Starbox image<br />
| image=[[File:Location of IK Pegasi.png|236px]]<br />
| caption=Location of IK Pegasi. }}<br />
{{Starbox observe<br />
| epoch=J2000<br />
| ra={{RA|21|26|26.6624}}<ref name="simbad"/><br />
| dec={{DEC| +19|22|32.304}}<ref name="simbad" /><br />
| appmag_v=6.078<ref name="simbad" /><br />
| constell=[[Pegasus (constellation)|Pegasus]] }}<br />
{{Starbox character<br />
| class=A8m:<ref name="apj221" />/DA<ref name="mnras270" /><br />
| b-v=0.24<ref name="simbad" />/–<br />
| u-b=0.03<ref name="simbad" />/–<br />
| variable=[[Delta Scuti variable|Delta Scuti]]<ref name="apj221" /> }}<br />
{{Starbox astrometry<br />
| radial_v=-11.4<ref name="simbad" /><br />
| prop_mo_ra=80.23<ref name="simbad" /><br />
| prop_mo_dec=17.28<ref name="simbad" /><br />
| parallax=21.72<br />
| p_error=0.78<br />
| parallax_footnote=<ref name="simbad" /><br />
| absmag_v=2.762<ref name="a" group="nb"/> }}<br />
{{Starbox detail<br />
| mass=1.65<ref name="mnras267" />/1.15<ref name="pasp105" /><br />
| radius=1.6<ref name="mnras267" />/0.006<ref name="mnras270" /><br />
| luminosity=8.0/0.12<ref name="b" group="nb"/><br />
| temperature=7,700<ref name="nras278"/>/35,500<ref name="pasp105" /><br />
| metal=117<ref name="mnras267" /><ref name="nras278" />/– % Sun<br />
| rotation=<&nbsp;32.5<ref name="nras278" />/–&nbsp;km/s<br />
| gravity=4.25<ref name="mnras267" />/8.95<ref name="mnras270" /><br />
| age=5–60&nbsp;×&nbsp;10<sup>7</sup><ref name="mnras267" /> }}<br />
{{Starbox catalog<br />
| names='''AB:''' V*&nbsp;IK&nbsp;Peg, [[Harvard Revised catalogue|HR&nbsp;8210]], [[Bonner Durchmusterung|BD&nbsp;+18°4794]], [[Henry Draper catalogue|HD&nbsp;204188]], [[Smithsonian Astrophysical Observatory|SAO&nbsp;107138]], [[Hipparcos catalogue|HIP&nbsp;105860]].<ref name="simbad" /><br />
<br />'''B:''' WD&nbsp;2124+191, EUVE&nbsp;J2126+193.<ref name="apj502"/><ref name=apj497_2_77/> }}<br />
{{Starbox end}}<br />
<br />
'''IK Pegasi''' (or '''HR 8210''') is a [[binary star]] [[star system|system]] in the [[constellation]] [[Pegasus (constellation)|Pegasus]]. It is just luminous enough to be seen with the unaided eye, at a distance of about 150&nbsp;[[light year]]s from the [[Solar System]].<br />
<br />
The primary (IK Pegasi A) is an [[A-type main-sequence star]] that displays minor pulsations in [[luminosity]]. It is categorized as a [[Delta Scuti variable]] star and it has a periodic cycle of luminosity variation that repeats itself about 22.9 times per day.<ref name="mnras267"/> Its companion (IK Pegasi B) is a massive [[white dwarf]]—a star that has evolved past the main sequence and is no longer generating energy through [[nuclear fusion]]. They orbit each other every 21.7 days with an average separation of about 31&nbsp;million kilometres, or 19&nbsp;million miles, or 0.21&nbsp;[[astronomical unit]]s (AU). This is smaller than the orbit of [[Mercury (planet)|Mercury]] around the [[Sun]].<br />
<br />
IK Pegasi B is the nearest known [[supernova]] progenitor candidate. When the primary begins to evolve into a [[red giant]], it is expected to grow to a radius where the white dwarf can [[Accretion (astrophysics)|accrete]] matter from the expanded gaseous envelope. When the white dwarf approaches the [[Chandrasekhar limit]] of 1.44&nbsp;[[solar mass]]es,<ref>{{cite doi|10.1126/science.1136259 }}</ref> it may explode as a [[Type Ia supernova]].<ref name="mnras262"/><br />
<br />
==Observation==<br />
This star system was catalogued in the 1862 ''[[Durchmusterung|Bonner Durchmusterung]]'' ("Bonn astrometric Survey") as BD&nbsp;+18°4794B. It later appeared in [[Edward Charles Pickering|Pickering's]] 1908 ''[[Bright Star Catalogue|Harvard Revised Photometry Catalogue]]'' as HR 8210.<ref name=Pickering1908/> The designation "IK Pegasi" follows the expanded form of the [[variable star designation|variable star nomenclature]] introduced by [[Friedrich Wilhelm Argelander|Friedrich W. Argelander]].<ref name=rabinowitz_vogel2009/><br />
<br />
Examination of the [[spectrograph]]ic features of this star showed the characteristic [[absorption line]] shift of a binary star system. This shift is created when their orbit carries the member stars toward and then away from the observer, producing a [[doppler effect|doppler shift]] in the wavelength of the line features. The measurement of this shift allows astronomers to determine the relative orbital velocity of at least one of the stars even though they are unable to resolve the individual components.<ref name=sb/><br />
<br />
In 1927, the Canadian astronomer [[William Edmund Harper|William E. Harper]] used this technique to measure the period of this single-line spectroscopic binary and determined it to be 21.724&nbsp;days. He also initially estimated the [[orbital eccentricity]] as 0.027. (Later estimates gave an eccentricity of essentially zero, which is the value for a circular orbit.)<ref name="mnras262" /> The velocity amplitude was measured as 41.5&nbsp;km/s, which is the maximum velocity of the primary component along the line of sight to the Solar System.<ref name=pdao4_161/><br />
<br />
The distance to the IK Pegasi system can be measured directly by observing the tiny [[parallax]] shifts of this system (against the more distant stellar background) as the [[Earth]] orbits around the Sun. This shift was measured to high precision by the [[Hipparcos]] spacecraft, yielding a distance estimate of 150&nbsp;[[light year]]s (with an accuracy of ±5&nbsp;light years).<ref name=aaa323_L49/> The same spacecraft also measured the [[proper motion]] of this system. This is the small angular motion of IK Pegasi across the sky because of its motion through space.<br />
<br />
The combination of the distance and proper motion of this system can be used to compute the transverse velocity of IK Pegasi as 16.9&nbsp;km/s.<ref name="c" group="nb"/> The third component, the heliocentric [[radial velocity]], can be measured by the average [[red-shift]] (or blue-shift) of the stellar spectrum. The ''General Catalogue of Stellar Radial Velocities'' lists a radial velocity of -11.4&nbsp;km/s for this system.<ref name=wilson1953/> The combination of these two motions gives a [[space velocity (astronomy)|space velocity]] of 20.4&nbsp;km/s relative to the Sun.<ref name="d" group="nb"/><br />
<br />
An attempt was made to photograph the individual components of this binary using the [[Hubble Space Telescope]], but the stars proved too close to resolve.<ref name=p12ewwd/> Recent measurements with the [[Extreme Ultraviolet Explorer]] [[space telescope]] gave a more accurate orbital period of {{nowrap|21.72168 ± 0.00009 days}}.<ref name="apj502"/> The [[inclination]] of this system's [[Orbital plane (astronomy)|orbital plane]] is believed to be nearly edge-on (90°) as seen from the Earth. If so it may be possible to observe an [[eclipse]].<ref name="pasp105" /><br />
<br />
==IK Pegasi A==<br />
The [[Hertzsprung-Russell diagram]] (HR diagram) is a plot of [[luminosity]] versus a [[color index]] for a set of stars. IK Pegasi A is currently a [[main sequence]] star—a term that is used to describe a nearly linear grouping of core hydrogen-fusing stars based on their position on the HR diagram. However, IK Pegasi A lies in a narrow, nearly vertical band of the HR diagram that is known as the [[instability strip]]. Stars in this band oscillate in a coherent manner, resulting in periodic pulsations in the star's luminosity.<ref name="araa33"/><br />
<br />
The pulsations result from a process called the [[κ-mechanism]]. A part of the star's outer [[atmosphere]] becomes [[Optical density|optically thick]] due to partial [[ionization]] of certain elements. When these atoms lose an [[electron]], the likelihood that they will absorb energy increases. This results in an increase in temperature that causes the atmosphere to expand. The inflated atmosphere becomes less ionized and loses energy, causing it to cool and shrink back down again. The result of this cycle is a periodic pulsation of the atmosphere and a matching variation of the luminosity.<ref name="araa33" /><br />
<br />
[[File:Size IK Peg.png|right|320px|thumb|The relative dimensions of IK Pegasi A (left), B (lower center) and the Sun (right).<ref>For an explanation of the star colors, see: {{cite web<br />
|date=December 21, 2004<br />
|url=http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_colour.html|title=The Colour of Stars|publisher=Australia Telescope Outreach and Education|accessdate=2007-09-26 }}</ref>]]<br />
Stars within the portion of the instability strip that crosses the main sequence are called [[Delta Scuti variable]]s. These are named after the prototypical star for such variables: [[Delta Scuti]]. Delta Scuti variables typically range from [[spectral class]] A2 to F8, and a stellar luminosity class of III ([[subgiant]]s) to V ([[main sequence]] stars). They are short-period variables that have a regular pulsation rate between 0.025 and 0.25&nbsp;days. Delta Scuti stars have an abundance of elements similar to the Sun's (see [[Population I]] stars) and between 1.5 and 2.5&nbsp;[[solar mass]]es.<ref name=templeton2004/> The pulsation rate of IK Pegasi A has been measured at 22.9 cycles per day, or once every 0.044 days.<ref name="mnras267" /><br />
<br />
Astronomers define the [[metallicity]] of a star as the abundance of [[chemical element]]s that have a higher [[atomic number]] than helium. This is measured by a [[spectroscopic]] analysis of the atmosphere, followed by a comparison with the results expected from computed stellar models. In the case of IK Pegasus A, the estimated metal abundance is [M/H]&nbsp;=&nbsp;+0.07&nbsp;±&nbsp;0.20. This notation gives the [[logarithm]] of the ratio of metal elements (M) to hydrogen (H), minus the logarithm of the Sun's metal ratio. (Thus if the star matches the metal abundance of the Sun, this value will be zero.) A logarithmic value of 0.07 is equivalent to an actual metallicity ratio of 1.17, so the star is about 17% richer in metallic elements than the Sun.<ref name="mnras267" /> However the margin of error for this result is relatively large.<br />
<br />
The spectrum of A-class stars such as IK Pegasi A show strong [[Balmer line]]s of hydrogen along with absorption lines of ionized metals, including the K&nbsp;line of ionized [[calcium]] (Ca&nbsp;II) at a wavelength of 393.3&nbsp;[[nanometre|nm]].<ref name=saha2007/> The spectrum of IK Pegasi A is classified as marginal Am (or "Am:"), which means it displays the characteristics of a spectral class A but is marginally metallic-lined. That is, this star's atmosphere displays slightly (but anomalously) higher than normal absorption line strengths for metallic isotopes.<ref name="apj221"/> Stars of spectral type Am are often members of close binaries with a companion of about the same mass, as is the case for IK Pegasi.<ref name=baas26_868/><br />
<br />
Spectral class-A stars are hotter and more massive than the Sun. But, in consequence, their life span on the main sequence is correspondingly shorter. For a star with a mass similar to IK Pegasi A (1.65 solar), the expected lifetime on the main sequence is 2–3{{nowrap| × 10<sup>9</sup> years}}, which is about half the current age of the Sun.<ref name=gsu2005/><br />
<br />
In terms of mass, the relatively young [[Altair]] is the nearest star to the Sun that is a stellar analogue of component A—it has an estimated 1.7 times the [[solar mass]]. The binary system as a whole has some similarities to the nearby system of [[Sirius]], which has a class-A primary and a white dwarf companion. However, Sirius A is more massive than IK Pegasi A and the orbit of its companion is much larger, with a semimajor axis of 20&nbsp;A.U.<br />
<br />
==IK Pegasi B==<br />
The companion star is a dense [[white dwarf]] star. This category of stellar object has reached the end of its evolutionary life span and is no longer generating energy through [[nuclear fusion]]. Instead, under normal circumstances, a white dwarf will steadily radiate away its excess energy, mainly stored heat, growing cooler and dimmer over the course of many billions of years.<ref name=chandra20060829/><br />
<br />
===Evolution===<br />
Nearly all small and intermediate-mass stars (below about nine [[solar mass]]es) will end up as white dwarfs once they have exhausted their supply of [[thermonuclear]] fuel.<ref name=apj591_1_288/> Such stars spend most of their energy-producing life span as a [[main sequence]] star. The amount of time that a star spends on the main sequence depends primarily on its mass, with the lifespan decreasing with increasing mass.<ref name=seligman2007/> Thus, for IK Pegasi B to have become a white dwarf before component A, it must once have been more massive than component A. In fact, the progenitor of IK Pegasi B is thought to have had a mass between 5 and 8&nbsp;[[solar mass]]es.<ref name="mnras262" /><br />
<br />
As the hydrogen fuel at the core of the progenitor of IK Pegasi B was consumed, it evolved into a [[red giant]]. The inner core contracted until hydrogen burning commenced in a shell surrounding the helium core. To compensate for the temperature increase, the outer envelope expanded to many times the radius it possessed as a main sequence star. When the core reached a temperature and density where helium could start to undergo fusion this star contracted and became what is termed a [[horizontal branch]] star. That is, it belonged to a group of stars that fall upon a roughly horizontal line on the H-R diagram. The fusion of helium formed an inert core of carbon and oxygen. When helium was exhausted in the core a helium-burning shell formed in addition to the hydrogen-burning one and the star moved to what astronomers term the [[asymptotic giant branch]], or AGB. (This is a track leading to the upper-right corner of the H-R diagram.) If the star had sufficient mass, in time [[Carbon burning process|carbon fusion]] could begin in the core, producing [[oxygen]], [[neon]] and [[magnesium]].<ref name="evolution"/><ref name=richmond20061005/><ref name=darling/><br />
<br />
The outer envelope of a red giant or AGB star can expand to several hundred times the radius of the Sun, occupying a radius of about {{nowrap|5 × 10<sup>8</sup> km}} (3 A.U.) in the case of the pulsating AGB star [[Mira]].<ref name=hubble19970806/> This is well beyond the current average separation between the two stars in IK Pegasi, so during this time period the two stars shared a common envelope. As a result, the outer atmosphere of IK Pegasi A may have received an isotope enhancement.<ref name="pasp105"/><br />
<br />
[[File:NGC7293 (2004).jpg|right|thumb|The [[Helix Nebula]] is being created by a star evolving into a white dwarf. ''[[NASA]] & [[ESA]] image.'']]<br />
Some time after an inert oxygen-carbon (or oxygen-magnesium-neon) core formed, thermonuclear fusion began to occur along two shells concentric with the core region; hydrogen was burned along the outermost shell, while helium fusion took place around the inert core. However, this double-shell phase is unstable, so it produced thermal pulses that caused large-scale mass ejections from the star's outer envelope.<ref name=science289_5476_88/> This ejected material formed an immense cloud of material called a [[planetary nebula]]. All but a small fraction of the hydrogen envelope was driven away from the star, leaving behind a white dwarf remnant composed primarily of the inert core.<ref name="apjs76"/><br />
<br />
===Composition and structure===<br />
The interior of IK Pegasi B may be composed wholly of carbon and oxygen; alternatively, if its progenitor underwent [[carbon burning]], it may have a core of oxygen and neon, surrounded by a mantle enriched with carbon and oxygen.<ref name=aaa375_1_87/><ref name=rmp74_4_1015/> In either case, the exterior of IK Pegasi B is covered by an atmosphere of almost pure hydrogen, which gives this star its [[stellar classification]] of DA. Due to higher [[atomic mass]], any helium in the envelope will have sunk beneath the hydrogen layer.<ref name="mnras270"/> The entire mass of the star is supported by [[electron degeneracy pressure]]—a [[quantum mechanics|quantum mechanical]] effect that limits the amount of matter that can be squeezed into a given volume.<br />
<br />
[[File:ChandrasekharLimitGraph.svg|left|280px|thumb|This graph shows the theoretical radius of a white dwarf, given its mass. The green curve is for a [[Special relativity|relativistic]] electron gas model.]]<br />
At an estimated 1.15&nbsp;[[solar masses]], IK Pegasi B is considered to be a high-mass white dwarf.<ref name="e" group="nb"/> Although its radius has not been observed directly, it can be estimated from known theoretical relationships between the mass and radius of white dwarfs,<ref name=sb_se/> giving a value of about 0.60% of the [[solar radius|Sun's radius]].<ref name="mnras270" /> (A different source gives a value of 0.72%, so there remains some uncertainty in this result.)<ref name="mnras267" /> Thus this star packs a mass greater than the Sun into a volume roughly the size of the Earth, giving an indication of this object's extreme [[density]].<ref name="f" group="nb"/><br />
<br />
The massive, compact nature of a white dwarf produces a strong surface gravity. Astronomers denote this value by the decimal [[logarithm]] of the [[gravitational force]] in [[Centimeter gram second system of units|cgs units]], or log ''g''. For IK Pegasi B, log ''g'' is 8.95.<ref name="mnras270" /> By comparison, log ''g'' for the Earth is 2.99. Thus the surface gravity on IK Pegasi is over 900,000 times the gravitational force on the Earth.<ref name="g" group="nb"/><br />
<br />
The effective surface temperature of IK Pegasi B is estimated to be about {{nowrap|35,500 ± 1,500 K}},<ref name="pasp105" /> making it a strong source of [[ultraviolet]] radiation.<ref name="mnras270" /><ref name="h" group="nb"/> Under normal conditions this white dwarf would continue to cool for more than a billion years, while its radius would remain essentially unchanged.<ref name=imamura19950224/><br />
<br />
==Future evolution==<br />
In a 1993 paper, David Wonnacott, Barry J. Kellett and David J. Stickland identified this system as a candidate to evolve into a [[Type Ia supernova]] or a [[Cataclysmic variable star|cataclysmic variable]].<ref name="mnras262" /> At a distance of 150&nbsp;light years, this makes it the nearest known candidate supernova progenitor to the [[Earth]]. However, in the time it will take for the system to evolve to a state where a supernova could occur, it will have moved a considerable distance from Earth but may yet pose a threat.<br />
<br />
[[File:Mira 1997 UV.jpg|right|thumb|This [[Hubble Space Telescope]] image shows the pulsating AGB ([[asymptotic giant branch]]) star Mira. ''NASA image.'']]<br />
At some point in the future, IK Pegasi A will consume the hydrogen fuel at its core and start to evolve away from the main sequence to form a red giant. The envelope of a red giant can grow to significant dimensions, extending up to a hundred times its previous radius (or larger). Once IK Pegasi A expands to the point where its outer envelope overflows the [[Roche lobe]] of its companion, a gaseous [[accretion disk]] will form around the white dwarf. This gas, composed primarily of hydrogen and helium, will then accrete onto the surface of the companion. This mass transfer between the stars will also cause their mutual orbit to shrink.<ref name=lrr2006/><br />
<br />
On the surface of the white dwarf, the accreted gas will become compressed and heated. At some point the accumulated gas can reach the conditions necessary for hydrogen fusion to occur, producing a [[thermal runaway|runaway]] reaction that will drive a portion of the gas from the surface. This would result in a (recurrent) [[nova]] explosion—a cataclysmic variable star—and the luminosity of the white dwarf rapidly would increase by several [[Apparent magnitude|magnitudes]] for a period of several days or months.<ref name=aavso200105/> An example of such a star system is [[RS Ophiuchi]], a binary system consisting of a red giant and a white dwarf companion. RS Ophiuchi has flared into a (recurrent) nova on at least six occasions, each time accreting the critical mass of hydrogen needed to produce a runaway explosion.<ref name="vsom0501"/><ref name=hendrix20070720/><br />
<br />
It is possible that IK Pegasi B will follow a similar pattern.<ref name="vsom0501" /> In order to accumulate mass, however, only a portion of the accreted gas can be ejected, so that with each cycle the white dwarf would steadily increase in mass. Thus, even should it behave as a recurring nova, IK Pegasus B could continue to accumulate a growing envelope.<ref name=aaa362_1046/><br />
<br />
An alternate model that allows the white dwarf to steadily accumulate mass without erupting as a nova is called the close-binary [[supersoft x-ray source]] (CBSS). In this scenario, the mass transfer rate to the close white dwarf binary is such that a steady fusion burn can be maintained on the surface as the arriving hydrogen is consumed in thermonuclear fusion to produce helium. This category of super-soft sources consist of high-mass white dwarfs with very high surface temperatures ({{nowrap|0.5 × 10<sup>6</sup>}} to {{nowrap|1 × 10<sup>6</sup> K}}<ref name=asp2002/>).<ref name=di_stefano_greiner1996/><br />
<br />
Should the white dwarf's mass approach the [[Chandrasekhar limit]] of 1.44&nbsp;[[solar mass]]es it will no longer be supported by [[electron degeneracy pressure]] and it will undergo a collapse. For a core primarily composed of oxygen, neon and magnesium, the collapsing white dwarf is likely to form a [[neutron star]]. In this case, only a fraction of star's mass will be ejected as a result.<ref name=lr20060124/> If the core is instead made of carbon-oxygen, however, increasing pressure and temperature will initiate carbon fusion in the center prior to attainment of the Chandrasekhar limit. The dramatic result is a runaway nuclear fusion reaction that consumes a substantial fraction of the star within a short time. This will be sufficient to unbind the star in a cataclysmic, Type Ia supernova explosion.<ref name=chandra20060829/><br />
<br />
Such a supernova event may pose some threat to life on the Earth. It is thought that the primary star, IK Pegasi A, is unlikely to evolve into a red giant in the immediate future. As shown previously, the space velocity of this star relative to the Sun is 20.4&nbsp;km/s. This is equivalent to moving a distance of one light year every 14,700&nbsp;years. After 5&nbsp;million years, for example, this star will be separated from the Sun by more than 500&nbsp;light years. A Type&nbsp;Ia supernova within a thousand parsecs (3300&nbsp;light-years) is thought to be able to affect the Earth,<ref name=richmond20050408/> but it must be closer than about 10 parsecs (around thirty light-years) to cause a major harm to the terrestrial biosphere.<ref name=beech2011/><br />
<br />
Following a supernova explosion, the remnant of the donor star (IK Pegasus A) would continue with the final velocity it possessed when it was a member of a close orbiting binary system. The resulting relative velocity could be as high as 100–200&nbsp;km/s, which would place it among the [[High-velocity star|high-velocity members]] of the [[Milky Way|galaxy]]. The companion will also have lost some mass during the explosion, and its presence may create a gap in the expanding debris. From that point forward it will evolve into a single white dwarf star.<ref name=apj582_2_915/><ref name=apjss128_2_615/> The supernova explosion will create a [[Supernova remnant|remnant]] of expanding material that will eventually merge with the surrounding [[interstellar medium]].<ref name=nasa20060907/><br />
<br />
==Notes==<br />
{{reflist|group="nb"|refs=<br />
<ref name="a">The absolute magnitude ''M<sub>v</sub>'' is given by:<br />
:''M<sub>v</sub>'' = ''V'' + 5(log<sub>10</sub> π + 1) = 2.762<br />
where ''V'' is the visual magnitude and ''π'' is the parallax. See:<br>{{cite book | first=Roger John | last=Tayler | year=1994 | title=The Stars: Their Structure and Evolution | publisher=Cambridge University Press | page=16 | isbn=0-521-45885-4 }}</li><br />
</ref><br />
<br />
<ref name="b">Based upon:<br />
:<math>\begin{smallmatrix} \frac{L}{L_{sun}} = \left ( \frac{R}{R_{sun}} \right )^2 \left ( \frac{T_{eff}}{T_{sun}} \right )^4 \end{smallmatrix}</math><br />
where ''L'' is luminosity, ''R'' is radius and ''T<sub>eff</sub>'' is the effective temperature. See:<br>{{cite web | last=Krimm | first=Hans | date=August 19, 1997 | url=http://ceres.hsc.edu/homepages/classes/astronomy/spring99/Mathematics/sec20.html | title=Luminosity, Radius and Temperature | publisher=Hampden-Sydney College | accessdate=2007-05-16 }}<br />
</ref><br />
<br />
<ref name="c">The net proper motion is given by:<br />
:<math>\begin{smallmatrix} \mu = \sqrt{ {\mu_\delta}^2 + {\mu_\alpha}^2 \cdot \cos^2 \delta } = 77.63\, \end{smallmatrix}</math>&nbsp;mas/y.<br />
where <math>\mu_\alpha</math> and <math>\mu_\delta</math> are the components of proper motion in the RA and Dec., respectively. The resulting transverse velocity is:<br />
:''V<sub>t</sub>'' = μ • 4.74 ''d'' (pc) = 16.9&nbsp;km.<br />
where ''d''(pc) is the distance in parsecs. See:<br>{{cite web | last=Majewski | first=Steven R. | year=2006 | url=http://www.astro.virginia.edu/class/majewski/astr551/lectures/VELOCITIES/velocities.html | title=Stellar Motions | publisher =University of Virginia | accessdate=2007-05-14 }}<br />
</ref><br />
<br />
<ref name="d">By the [[Pythagorean theorem]], the net velocity is given by:<br />
:<math>\begin{smallmatrix} V = \sqrt{{V_r}^2 + {V_t}^2} = \sqrt{11.4^2 + 16.9^2} = 20.4\, \end{smallmatrix}</math>&nbsp;km/s.<br />
where <math>V_r</math> is the radial velocity and <math>V_t</math> is the transverse velocity, respectively.<br />
</ref><br />
<br />
<ref name="e">The white-dwarf population is narrowly distributed around the mean mass of 0.58 solar masses, and only 2%. See:<br>{{cite journal | author=Holberg, J. B.; Barstow, M. A.; Bruhweiler, F. C.; Cruise, A. M.; Penny, A. J. | title=Sirius B: A New, More Accurate View | journal=The Astrophysical Journal | year=1998 | volume=497 | issue=2 | pages=935–942 | doi=10.1086/305489 | bibcode=1998ApJ...497..935H}} of all white dwarfs have at least one solar mass.<br />
</ref><br />
<br />
<ref name="f">R<sub>*</sub> = 0.006 • (6.96 × 10<sup>8</sup>) ≈ 4,200&nbsp;km.</ref><br />
<br />
<ref name="g">The surface gravity of the Earth is 9.780 m/s<sup>2</sup>, or 978.0 cm/s<sup>2</sup> in cgs units. Thus:<br />
:<math>\begin{smallmatrix} \log\ \operatorname{g}=\log\ 978.0=2.99 \end{smallmatrix}</math><br />
The log of the gravitational force ratios is 8.95 - 2.99 = 5.96. So:<br />
:<math>\begin{smallmatrix} 10^{5.96} \approx 912,000 \end{smallmatrix}</math><br />
</ref><br />
<br />
<ref name="h">From [[Wien's displacement law]], the peak emission of a [[black body]] at this temperature would be at a [[wavelength]] of:<br />
:<math>\begin{smallmatrix} \lambda_b = (2.898 \times 10^6 \operatorname{nm\ K})/(35,500\ \operatorname{K}) \approx 82\, \end{smallmatrix}</math>&nbsp;nm<br />
which lies in the far ultraviolet part of the [[electromagnetic spectrum]].<br />
</ref><br />
<br />
}}<br />
<br />
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<ref name=hendrix20070720>{{citation | first=Susan | last=Hendrix | title=Scientists see Storm Before the Storm in Future Supernova | publisher=NASA | date=July 20, 2007 | url=http://www.nasa.gov/vision/universe/starsgalaxies/rxte_supernova.html | accessdate=2007-05-25 }}</ref><br />
<br />
<ref name=aaa362_1046>{{citation | last1=Langer | first1=N. | last2=Deutschmann | first2=A. | last3=Wellstein | first3=S. | last4=Höflich | first4=P. | title=The evolution of main sequence star + white dwarf binary systems towards Type Ia supernovae | journal=Astronomy and Astrophysics | year=2000 | volume=362 | pages=1046–1064 | bibcode=2000astro.ph..8444L | arxiv=astro-ph/0008444 }}</ref><br />
<br />
<ref name=asp2002>{{citation | last1=Langer | first1=N. | last2=Yoon | first2=S.-C. | last3=Wellstein | first3=S. | last4=Scheithauer | first4=S. | editor1-last=Gänsicke | editor1-first=B. T. | editor2-last=Beuermann | editor2-first=K. | editor3-last=Rein | editor3-first=K. | contribution=On the evolution of interacting binaries which contain a white dwarf | title=The Physics of Cataclysmic Variables and Related Objects, ASP Conference Proceedings | pages =252 | publisher=Astronomical Society of the Pacific | year=2002 | location =San Francisco, California | bibcode=2002ASPC..261..252L }}</ref><br />
<br />
<ref name=di_stefano_greiner1996>{{citation | first=Rosanne | last=Di Stefano | editor=J. Greiner | title=Luminous Supersoft X-Ray Sources as Progenitors of Type Ia Supernovae | booktitle=Proceedings of the International Workshop on Supersoft X-Ray Sources | publisher=Springer-Verlag | date=February 28–March 1, 1996 | location=Garching, Germany | url=http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Aastro-ph%2F9701199 | format=PDF | accessdate=2007-05-19 | isbn=3-540-61390-0 }}</ref><br />
<br />
<ref name=lr20060124>{{citation | last1=Fryer | first1=C. L. | last2=New | first2=K. C. B. | date=January 24, 2006 | url=http://relativity.livingreviews.org/Articles/lrr-2011-1/ | title=2.1 Collapse scenario | work=Gravitational Waves from Gravitational Collapse | publisher=Max-Planck-Gesellschaft | accessdate = 2007-06-07 }}</ref><br />
<br />
<ref name=chandra20060829>{{citation | author=Staff | date =August 29, 2006 | url=http://chandra.harvard.edu/edu/formal/stellar_ev/story/index8.html | title=Stellar Evolution - Cycles of Formation and Destruction | publisher=Harvard-Smithsonian Center for Astrophysics | accessdate = 2006-08-10 }}</ref><br />
<br />
<ref name=richmond20050408>{{citation | url=http://www.tass-survey.org/richmond/answers/snrisks.txt | title=Will a Nearby Supernova Endanger Life on Earth? | first=Michael | last=Richmond | date=April 8, 2005 | format=TXT | accessdate=2006-03-30 }}—see section 4.</ref><br />
<br />
<ref name=apj582_2_915>{{citation | last=Hansen | first=Brad M. S. | title=Type Ia Supernovae and High-Velocity White Dwarfs | journal=The Astrophysical Journal | year=2003 | volume=582 | issue=2 | pages=915–918 | bibcode=2002astro.ph..6152H | doi=10.1086/344782 |arxiv = astro-ph/0206152 }}</ref><br />
<br />
<ref name=apjss128_2_615>{{citation | last1=Marietta | first1=E. | last2=Burrows | first2=A. | last3=Fryxell | first3=B. | title=Type Ia Supernova Explosions in Binary Systems: The Impact on the Secondary Star and Its Consequences | journal=The Astrophysical Journal Supplement Series | year=2000 | volume=128 | issue=2 | pages=615–650 | bibcode=2000ApJS..128..615M | doi=10.1086/313392 |arxiv=astro-ph/9908116 }}</ref><br />
<br />
<ref name=nasa20060907>{{citation | author=Staff | date=September 7, 2006 | url=http://heasarc.gsfc.nasa.gov/docs/objects/snrs/snrstext.html | title=Introduction to Supernova Remnants | publisher=NASA/Goddard | accessdate=2007-05-20 }}</ref><br />
<br />
}}<br />
<br />
==External links==<br />
{{commons category|IK Pegasi}}<br />
* {{citation | last=Davies | first=Ben | year=2006 | url=http://ben.davies.net/supernovae2.htm | title=Supernova events | accessdate = 2007-06-01 }}<br />
* {{citation | last=Richmond | first=Michael | date=April 8, 2005 | url=http://www.tass-survey.org/richmond/answers/snrisks.txt | title=Will a Nearby Supernova Endanger Life on Earth? | publisher =The Amateur Sky Survey | accessdate = 2007-06-07 }}<br />
* {{citation | last=Tzekova | first=Svetlana Yordanova | year=2004 | url=http://www.eso.org/public/outreach/eduoff/cas/cas2004/casreports-2004/rep-310/#2.%20The%20main%20star%20-%20IK%20Peg%20A | title=IK Pegasi (HR 8210) | publisher=ESO (European Organisation for Astronomical Research in the Southern Hemisphere) | accessdate=2007-09-30 }}<br />
<br />
{{featured article}}<br />
<br />
{{Stars of Pegasus}}<br />
<br />
[[Category:A-type main-sequence stars]]<br />
[[Category:Pegasus (constellation)]]<br />
[[Category:Spectroscopic binaries]]<br />
[[Category:Supernovae]]<br />
[[Category:White dwarfs]]<br />
[[Category:Objects named with variable star designations|Pegasi, IK]]<br />
[[Category:Henry Draper Catalogue objects|204188]]<br />
[[Category:HR objects|8210]]<br />
<br />
{{Link FA|es}}<br />
{{Link FA|it}}<br />
{{Link FA|ko}}<br />
{{Link FA|pt}}<br />
{{Link FA|vi}}<br />
{{Link GA|zh}}</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Rho_meson&diff=11523
Rho meson
2013-12-01T18:54:16Z
<p>99.153.64.179: punctuation, awkward wording</p>
<hr />
<div>{{More footnotes|date=April 2009}}<br />
'''Fermi acceleration''',<ref>M. A. Lieberman and A. J. Lichtenberg, Phys. Rev. A 5, 1852 (1972)</ref> sometimes referred to as ''diffusive shock acceleration'' (a subclass of Fermi acceleration), is the [[acceleration]] that [[charge (physics)|charged]] [[Subatomic particle|particles]] undergo when being repeatedly reflected, usually by a [[magnetic mirror]]. This is thought to be the primary mechanism by which particles gain non thermal energies in astrophysical [[shock waves]]. It plays a very important role in many astrophysical models, mainly of shocks including [[solar flares]] and [[supernova remnant]]s. It is studied by using the [[Fermi-Ulam model]].<br />
<br />
There are two types of Fermi acceleration: first-order Fermi acceleration (in shocks) and second-order Fermi acceleration (in the environment of moving magnetized gas clouds). In both cases the environment has to be collisionless in order for the mechanism to be effective. This is because Fermi acceleration only applies to particles with energies exceeding the thermal energies, and frequent collisions with surrounding particles will cause severe energy loss and as a result no acceleration will occur.<br />
<br />
==First order Fermi acceleration==<br />
Shock waves typically have moving magnetic inhomogeneities both preceding and following them. Consider the case of a charged particle traveling through the shock wave (from upstream to downstream). If it encounters a moving change in the magnetic field, this can reflect it back through the shock (downstream to upstream) at increased velocity. If a similar process occurs upstream, the particle will again gain energy. These multiple reflections greatly increase its energy. The resulting energy spectrum of many particles undergoing this process (assuming that they do not influence the structure of the shock) turns out to be a power law:<br /><br />
<math>\frac{dN(\epsilon)}{d\epsilon}\propto \epsilon ^{-p}</math> <br /><br />
where the spectral index <math>p\gtrsim2</math> depends, for non-relativistic shocks, only on the compression ratio of the shock. <br /><br />
The term "First order" comes from the fact that the energy gain per shock crossing is proportional to <math>\beta_s</math>, the velocity of the shock divided by the speed of light.<br />
<br />
===The injection problem===<br />
A mystery of first order Fermi processes is the ''injection problem''. In the environment of a shock, only particles with energies that exceed the thermal energy by much (a factor of a few at least) can cross the shock and 'enter the game' of acceleration. It is presently unclear what mechanism causes the particles to initially have energies sufficiently high to do so.<br />
<br />
==Second order Fermi acceleration==<br />
Second order Fermi Acceleration relates to the amount of energy gained during the motion of a charged particle in the presence of randomly moving "magnetic mirrors". So, if the magnetic mirror is moving towards the particle, the particle will end up with increased energy upon reflection. The opposite holds if the mirror is receding. This notion was used by Fermi (1949)<ref>On the Origin of the Cosmic Radiation, E. Fermi, Physical Review 75, pp. 1169-1174, 1949</ref> to explain the mode of formation of cosmic rays. In this case the magnetic mirror is a moving interstellar magnetized cloud. In a random motion environment, Fermi argued, the probability of a head-on collision is greater than a head-tail collision, so particles would, on average, be accelerated. This random process is now called second-order Fermi acceleration, because the mean energy gain per bounce depends on the mirror velocity squared, <math>\beta_m^2</math>.<br />
Surprisingly, the resulting energy spectrum anticipated from this physical setup is very similar to the one found for first order Fermi acceleration.<br />
<br />
==References==<br />
{{reflist}}<br />
{{cite book |url=http://books.google.com/books?id=tNDgb-U9PLMC&printsec=frontcover |title=High Energy Astrophysics, Volume 2 |last=Longair |first = Malcolm S. |publisher=Cambridge University Press |year=1994 |isbn=978-0-521-43584-0}}<br />
<br />
==External links==<br />
*[http://www.daviddarling.info/encyclopedia/F/Fermi_acceleration.html David Darling's article on Fermi acceleration]<br />
*[http://arxiv.org/abs/astro-ph/0610141 Rieger, Bosch-Ramon and Duffy: Fermi acceleration in astrophysical jets.] Astrophys.Space Sci. 309:119-125 (2007)<br />
<br />
[[Category:Fusion power]]<br />
[[Category:Physical quantities]]<br />
[[Category:Dynamics]]<br />
[[Category:Cosmic rays]]<br />
[[Category:Acceleration]]<br />
[[Category:Enrico Fermi|Acceleration]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Hypergeometric_function&diff=9897
Hypergeometric function
2013-12-01T06:21:10Z
<p>99.153.64.179: Hmm missing section on identies and their theory</p>
<hr />
<div>[[File:GlassyCarbon.PNG|thumb|250px|A small rod of glassy carbon.]]<br />
[[File:Vitreos carbon crucible 2.PNG|thumb|250px|vitreous-glassy carbon crucibles]]<br />
'''Glass-like carbon''', often called '''glassy carbon''' or '''vitreous carbon''', is a non-graphitizing [[carbon]] which combines glassy and [[ceramic]] properties with those of [[graphite]]. The most important properties are high temperature resistance, hardness (7 [[Mohs scale of mineral hardness|Mohs]]), low density, low electrical resistance, low friction, low thermal resistance, extreme resistance to chemical attack and impermeability to gases and liquids. Glassy carbon is widely used as an electrode material in [[electrochemistry]], as well as for high temperature [[crucible]]s and as a component of some prosthetic devices, and can be fabricated as different shapes, sizes and sections.<br />
<br />
The names ''glassy carbon'' and ''vitreous carbon'' have been introduced as trademarks; therefore, [[IUPAC]] does not recommend their use as technical terms.<ref name="iupac"/><br />
<br />
==History==<br />
It was first observed in the laboratories of The Carborundum Company, Manchester, UK, in the mid-1950s by Bernard Redfern, the inventor, a materials scientist and diamond technologist. He noticed that [[Sellotape]] he used to hold ceramic (rocket nozzle) samples in a furnace maintained a sort of structural identity after firing in an inert atmosphere.<br />
He searched for a polymer matrix to mirror a diamond structure and discovered a resole resin that would, with special preparation, set without a catalyst. Using this phenolic resin, crucibles were produced. Crucibles were distributed to organisations such as UKAEA Harwell. <br />
<br />
Redfern left The Carborundom Co., which officially wrote off all interests in the glassy carbon invention. Whilst working at [[Plessey|the Plessey Company]] laboratory (in a disused church) in Towcester, UK, Redfern received a glassy carbon crucible for duplication from UKAEA. He identified it as one he had made from markings he had engraved into the uncured precursor prior to carbonisation. (It is almost impossible to engrave the finished product.) The Plessey Company set up a laboratory first in a factory previously used to make briar pipes, in Litchborough, UK, and then a permanent facility at Caswell, near Blakesly, UK. Caswell became the Plessey Research Centre and then the Allen Clark Research Centre. Glassy carbon arrived at the Plessey Company Limited as a fait accompli. Redfern was assigned J.C. Lewis, as a laboratory assistant, for the production of glassy carbon. F.C. Cowlard was assigned to Redfern's department later, as a laboratory administrator. Cowlard was an administrator who previously had some association with Silane (Silane US Patent assignee 3,155,621 3 Nov 1964). Neither he nor Lewis had any previous connection with glassy carbon. <br />
<br />
Refern's contribution to the invention and production of glassy / Vitreous carbon is acknowledged by his co-authorship of early articles,.<ref>{{cite journal |last=Lewis |first=J.C. |coauthors=Redfern, B., Cowlard, F.C.|year=1963 |title=Vitreous carbon as a crucible material for semiconductors |journal=Solid-State Electronics |volume=6 |issue=3 |pages=251–254 |doi=10.1016/0038-1101(63)90081-9 |bibcode = 1963SSEle...6..251L }}</ref> But references to Redfern were not obvious in subsequent publications by Cowlard and Lewis.<ref>{{cite journal |last=Cowlard |first=F.C. |coauthors=Lewis, J.C. |year=1967 |title=Vitreous carbon — A new form of carbon |journal=Journal of Materials Science |volume=2 |issue=6 |pages=507–512 |doi=10.1007/BF00752216 |bibcode = 1967JMatS...2..507C }}</ref> Original boat crucibles and precursor samples exist.<br />
<br />
UK patent application were filed in 11 Jan.1960 and US patent filed 9 Jan. 1961 (finalised as US patent 3,109,712). This prior art is not referenced in US patent 4,668,496, 26 May 1987 for Vitreous Carbon. Patents were filed "Bodies and shapes of carbonaceous materials and processes for their production" and the name "Vitreous Carbon" presented to the product by the son of Redfern.<br />
<br />
Glassy/Vitreous Carbon was under investigation used for components for thermonuclear detonation systems and at least some of the patents surrounding the material were rescinded (in the interests of national security) in the 1960s.<br />
<br />
Large sections of the precursor material were produced as castings, moldings or machined into a predetermined shape. Large crucibles and other forms were manufactured. Carbonisation took place in two stages. Shrinkage during this process is considerable (48.8%) but is absolutely uniform and predictable. A nut and bolt can be made to fit as the polymer, processed separately and subsequently give a perfect fit.<br />
<br />
Some of the first ultrapure samples of Gallium Arsenide were zone refined in these crucibles. (Glassy carbon is extremely pure and unreactive to GaAs).<br />
<br />
Doped/impure glassy carbon exhibited semiconductor phenomena.<br />
<br />
Uranium carbide inclusions were fabricated (using U238 carbide at experimental scale).<br />
<br />
On October 11, 2011, research conducted at the Carnegie Geophysical Laboratory led by Stanford’s Wendy L. Mao and her graduate student Yu Lin described a new form of glassy carbon formed under high pressure with hardness equal to diamond. Unlike diamond, however its structure is that of [[amorphous carbon]] so its hardness may be isotropic. Research is ongoing.<ref>[http://www.physorg.com/news/2011-10-superhard-carbon.html New form of superhard carbon observed]</ref><br />
<br />
==Structure==<br />
The structure of glassy carbon has long been a subject of debate. Early structural models assumed that both sp<sup>2</sup>- and sp<sup>3</sup>-bonded atoms were present, but it is now known that glassy carbon is 100% sp<sup>2</sup>. However, more recent research has suggested that glassy carbon has a [[fullerenes|fullerene]]-related structure.<ref>[http://www.personal.rdg.ac.uk/~scsharip/PM_glassy.pdf Fullerene-related structure of commercial glassy carbons], P.J.F. Harris, 2003.</ref><br />
<br />
Note that glassy carbon should not be confused with [[amorphous carbon]]. This from IUPAC: "Glass-like carbon cannot be described as amorphous carbon because it consists of two-dimensional structural elements and does not exhibit ‘dangling’ bonds."<ref name="iupac">[http://goldbook.iupac.org/G02639.html The entry for "Glass-like carbon" in IUPAC Goldbook.]</ref><br />
<br />
It exhibits a [[conchoidal fracture]].<br />
<br />
==Electrochemical properties==<br />
Glassy carbon electrode (GCE) in aqueous solutions is considered to be an inert electrode for hydronium ion reduction:<ref name="Sawyer">{{cite book |first=D. T. |last=Sawyer |first2=A. |last2=Sobkowiak |first3=J. L., Jr. |last3=Roberts |title=Electrochemistry for Chemists |edition=Second |publisher=John Wiley & Sons |location=New York |year=1995 |isbn=0-471-59468-7 }}</ref><br />
<br />
:H<sub>3</sub>O<sup>+</sup><sub>(aq)</sub> + e<sup>-</sup> <math> \rm \stackrel{GCE} {\rightleftharpoons} </math> H<math>\cdot</math><sub>(aq)</sub> &nbsp; E<sup><s>o</s></sup> = −2.10 V versus [[standard hydrogen electrode|NHE]] at 25 °C<br />
<br />
Comparable reaction on platinum:<br />
:H<sub>3</sub>O<sup>+</sup><sub>(aq)</sub> + Pt<sub>(s)</sub> + e<sup>-</sup> <math> \rightleftharpoons </math> Pt:H<sub>(s)</sub> &nbsp; E<sup><s>o</s></sup> = 0.000 V versus NHE at 25 °C<br />
<br />
The difference of 2.1 V is attributed to the properties of platinum which stabilizes a covalent Pt-H bond.<ref name="Sawyer"/><br />
<br />
== Advantages ==<br />
Due to their specific surface orientation glassy carbon is employed as an electrode material for the fabrication of sensors. Glassy carbon paste, glassy carbon, carbon paste etc electrodes when modified are termed as chemically modified electrodes. Chemically modified electrodes have been employed for the analysis of organic molecules (viz., Paracetamol, aspirin, caffeine, phenol, catechol, resorcinol, hydroquinone, dopamine, L-dopa, epinephrine, nor epinephrine, methyl parathion, ethyl parathion, venlafaxine, desvenlafaxine, imipramine, trimipramine, desipramine etc) as well as metal ions (bismuth, antimony etc).<ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors= Srivastava, Ashwini|title=Simultaneous voltammetric determination of acetaminophen, aspirin and caffeine using an in situ surfactant-modified multiwalled carbon nanotube paste electrode|journal=Electrochimica Acta|year=2010|volume=55|pages=8638–8648|doi=10.1016/j.electacta.2010.07.093|url=http://www.sciencedirect.com/science/article/pii/S0013468610010510}}</ref><ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors=Mobin, Shaikh; Mathur, Pradeep; Lahiri, Goutam; Srivastava, Ashwini|title=Biomimetic sensor for certain catecholamines employing copper(II) complex and silver nanoparticle modified glassy carbon paste electrode|journal=Biosensors and Bioelectronics|year=2013|volume=39|pages=124–132|doi=10.1016/j.bios.2012.07.008|url=http://www.sciencedirect.com/science/article/pii/S0956566312004289}}</ref><ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors=Srivastava, Ashwini|title=Simultaneous voltammetric determination of acetaminophen and tramadol using Dowex50wx2 and gold nanoparticles modified glassy carbon paste electrode|year=2011|volume=706|pages=246–254|doi=10.1016/j.aca.2011.08.040|url=http://www.sciencedirect.com/science/article/pii/S0003267011011706}}</ref> <ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors=Srivastava, Ashwini|title=Adsorptive stripping differential pulse voltammetric determination of venlafaxine and desvenlafaxine employing Nafion–carbon nanotube composite glassy carbon electrode|journal=Electrochimica Acta|year=2011|volume=56|pages=4188–4196|doi=10.1016/j.electacta.2011.01.097|url=http://www.sciencedirect.com/science/article/pii/S001346861100171X}}</ref> <ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors=Hirsch, Gary; Karna, Shashi; Srivastava, Ashwini|title=Potentiometric stripping analysis of methyl and ethyl parathion employing carbon nanoparticles and halloysite nanoclay modified carbon paste electrode|journal=Analytica Chimica Acta|year=2012|volume=735|pages=37–45|doi=10.1016/j.aca.2012.05.029|url=http://www.sciencedirect.com/science/article/pii/S0003267012007672}}</ref> <ref>{{cite journal|last=Mobin|first=Shaikh|coauthors=Sanghavi, Bankim; Srivastava, Ashwini; Mathur, Pradeep; Lahiri, Goutam|title=Biomimetic Sensor for Certain Phenols Employing a Copper(II) Complex|journal=Analytical Chemistry|year=2010|volume=82|pages=5983–5992|doi=10.1021/ac1004037|url=http://pubs.acs.org/doi/abs/10.1021/ac1004037}}</ref> <ref>{{cite journal|last=Gadhari|first=Nayan|coauthors=Sanghavi, Bankim; Srivastava, Ashwini|title=Potentiometric stripping analysis of antimony based on carbon paste electrode modified with hexathia crown ether and rice husk|journal=Analytica Chimica Acta|year=2011|volume=703|pages=31–40|doi=10.1016/j.aca.2011.07.017|url=http://www.sciencedirect.com/science/article/pii/S0003267011009585}}</ref> <ref>{{cite journal|last=Gadhari|first=Nayan|coauthors=Sanghavi, Bankim; Karna, Shashi; Srivastava, Ashwini|title=Potentiometric stripping analysis of bismuth based on carbon paste electrode modified with cryptand 2.2.1 and multiwalled carbon nanotubes|journal=Electrochimica Acta|year=2010|volume=56|pages=627–635|doi=10.1016/j.electacta.2010.09.100|url=http://www.sciencedirect.com/science/article/pii/S0013468610013599}}</ref> <ref>{{cite journal|last=Sanghavi|first=Bankim|coauthors=Srivastava, Ashwini|title=Adsorptive stripping voltammetric determination of imipramine, trimipramine and desipramine employing titanium dioxide nanoparticles and an Amberlite XAD-2 modified glassy carbon paste electrode|journal=Analyst|year=2013|doi=10.1039/C2AN36330E|url=http://pubs.rsc.org/en/Content/ArticleLanding/2013/AN/C2AN36330E|bibcode = 2013Ana...138.1395S }}</ref><br />
<br />
== See also ==<br />
* [[Graphite]]<br />
* [[Electrochemistry]]<br />
* [[Fullerenes]]<br />
<br />
==References==<br />
{{reflist}}<br />
<br />
== External links ==<br />
* [http://www.htw-gmbh.de HTW, supplier's website], for Glassy Carbon SIGRADUR<br />
<br />
{{Allotropes of carbon}}<br />
{{Carbon and its allotrophs}}<br />
{{DEFAULTSORT:Glassy Carbon}}<br />
[[Category:Carbon forms]]<br />
[[Category:Amorphous solids]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Theta_representation&diff=13231
Theta representation
2013-11-30T20:25:42Z
<p>99.153.64.179: /* Hilbert space */ typo</p>
<hr />
<div>{{original research|date=May 2013}}<br />
{{orphan|date=December 2008}}<br />
'''2-choice hashing''', also known as '''2-choice chaining''', is a variant of a [[hash table]] in which keys are added by hashing with two [[hash function]]s. The key is put in the array position with the fewer (colliding) keys. Some [[collision resolution scheme]] is needed, unless keys are kept in buckets. The [[average-case cost]] of a successful search is [[Big O notation|O(2 + (m-1)/n)]], where m is the number of keys and n is the size of the array. The most collisions is <math>\log_2 \ln n + \theta(m/n)</math> with high probability.<br />
<br />
==How It Works==<br />
<br />
2-choice hashing utilizes two hash functions h1(x) and h2(x) which work as hash functions are expected to work (i.e. mapping integers from the universe into a specified range). The two hash functions should be independent and have no correlation to each other. Having two hash functions allows any integer x to have up to two potential locations to be stored based on the values of the respective outputs, h1(x) and h2(x). It is important to note that, although there are two hash functions, there is only one table; both hash functions map to locations on that table. <br />
<br />
==Implementation== <br />
<br />
The most important functions of the hashing implementation in this case are insertion and search.<br />
<br />
'''Insertion:''' When inserting the values of both hash functions are computed for the to-be-inserted object. The object is then placed in the bucket which contains fewer objects. If the buckets are equal in size, the default location is the h1(x) value. <br />
<br />
'''Search:''' Effective searches are done by looking in both buckets, that is, the bucket locations which h1(x) and h2(x) mapped to for the desire value. <br />
<br />
==Performance==<br />
<br />
As is true with all hash tables, the performance is based on the largest bucket. Although there are instances where bucket sizes happen to be large based on the values and the hash functions used, this is rare. Having two hash functions and, therefore, two possible locations for any one value, makes the possibility of large buckets even more unlikely to happen. <br />
<br />
The expected bucket size while using 2-choice hashing is: '''θ(log(log(n)))'''. This massive improvement is due to the randomized concept known as The Power of Two Choices.<br />
<br />
'''Note: The idea of multiple hash functions is optimized using 2 hash functions. There is no improvement found if the number of hash functions is increased - that is 3 hash functions does not have better performance than 2. '''<br />
==See also==<br />
*[[2-left hashing]]<br />
<br />
{{DADS|2-choice hashing|twoChoiceHashing}}<br />
<br />
[[Category:Hashing]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Segal%E2%80%93Bargmann_space&diff=30122
Segal–Bargmann space
2013-11-30T20:19:30Z
<p>99.153.64.179: /* See also */ Hardy space</p>
<hr />
<div>In algebraic geometry, a '''derived scheme''' is a pair <math>(X, \mathcal{O})</math> consisting of a [[topological space]] ''X'' and a [[sheaf of spectra|sheaf]] <math>\mathcal{O}</math> of [[commutative ring spectrum|commutative ring spectra]] <ref>also often called <math>E_\infty</math>-ring spectra</ref> on ''X'' such that (1) the pair <math>(X, \pi_0 \mathcal{O})</math> is a [[scheme (mathematics)|scheme]] and (2) <math>\pi_k \mathcal{O}</math> is a [[quasi-coherent sheaf|quasi-coherent]] <math>\pi_0 \mathcal{O}</math>-module. The notion gives a homotopy-theoretic generalization of a scheme.<br />
<br />
Just as affine algebraic geometry is equivalent (in categorical sense) to the theory of [[commutative ring]]s (commonly called [[commutative algebra]]), affine derived algebraic geometry is (roughly in homotopical sense) equivalent to the theory of [[differential graded algebra|commutative differential graded rings]].<br />
<br />
== Notes ==<br />
{{reflist}}<br />
<br />
== References ==<br />
*P. Goerss, [http://www.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf Topological Modular Forms <nowiki>[after Hopkins, Miller, and Lurie]</nowiki>]<br />
* B. Toën, [http://math.berkeley.edu/~aaron/gaelxx/DAG.pdf Introduction to derived algebraic geometry]<br />
<br />
<br />
{{geometry-stub}}<br />
<br />
<br />
[[Category:Algebraic geometry]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Quantum_instrument&diff=17084
Quantum instrument
2013-11-30T05:39:39Z
<p>99.153.64.179: /* Definition */ typo</p>
<hr />
<div>{{Multiple issues|orphan = September 2012|unreferenced = March 2008}}<br />
<br />
The '''illness rate''' is calculated by comparing employee illness-related absences against planned [[working time]], within a specific period. Illness-related absence times and planned working times are calculated in days.<br />
<br />
==Interpretation==<br />
A high illness rate may be interpreted as an indicator of a heavy [[workload]], bad working conditions, dangerous working environment, low [[employee satisfaction]], and so on. As a simple key figure it can be used for planning purposes, for example, to shift resources from one area into an area with a high Illness Rate. An analysis of the illness reasons or causes must include other factors as well. For example, a high [[overtime]] rate combined with a high number of accidents may indicate the reasons for an increase of the illness rate.<br />
<br />
==Calculation Formula==<br />
<math> \textstyle{\mbox{Illness rate } = \frac{\sum{\mbox{Illness-related Absence Times in Days}}}{\sum{\mbox{Planned Working Times in Days}}}} </math><br />
* Unit of Measure: %<br />
<br />
==Direction of Improvement==<br />
One will usually try to minimize the illness rate.<br />
<br />
==Industry and Country Relevance==<br />
The illness rate is generic for all industries and countries.<br />
<br />
[[Category:Human resource management]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Dual_(category_theory)&diff=3876
Dual (category theory)
2013-11-30T05:16:45Z
<p>99.153.64.179: dab, per talk page</p>
<hr />
<div>{{about|coproducts in categories|"coproduct" in the sense of comultiplication|Coalgebra}}<br />
<br />
In [[category theory]], the '''coproduct''', or '''categorical sum''', is a category-theoretic construction which includes as examples the [[disjoint union|disjoint union of sets]] and [[disjoint union (topology)|of topological spaces]], the [[free product|free product of groups]], and the [[direct sum]] of [[Module (mathematics)|modules]] and [[vector space]]s. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a [[morphism]]. It is the category-theoretic [[Dual (category theory)|dual notion]] to the [[product (category theory)|categorical product]], which means the definition is the same as the product but with all [[morphism|arrows]] reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.<br />
<br />
==Definition==<br />
The formal definition is as follows: Let ''C'' be a category and let {''X<sub>j</sub>'' : ''j'' &isin; ''J''} be an [[index set|indexed family]] of objects in ''C''. The coproduct of the family {''X<sub>j</sub>''} is an object ''X'' together with a collection of [[morphism]]s ''i<sub>j</sub>'' : ''X<sub>j</sub>'' &rarr; ''X'' (called ''[[canonical injection]]s'' although they need not be [[injective function|injections]] or even [[monomorphism|monic]]) which satisfy a [[universal property]]: for any object ''Y'' and any collection of morphisms ''f<sub>j</sub>'' : ''X<sub>j</sub>'' &rarr; ''Y'', there exists a unique morphism ''f'' from ''X'' to ''Y'' such that ''f<sub>j</sub>'' = ''f'' ∘ ''i<sub>j</sub>''. That is, the following diagram [[commutative diagram|commutes]] (for each ''j''):<br />
<br />
[[Image:Coproduct-01.svg|160px|center]]<br />
<br />
The coproduct of the family {''X<sub>j</sub>''} is often denoted<br />
:<math> X = \coprod_{j\in J}X_j</math><br />
or<br />
:<math>X = \bigoplus_{j \in J} X_j.</math><br />
<br />
Sometimes the morphism ''f'' may be denoted<br />
:<math>f=\coprod_{j \in J} f_j: \coprod_{j \in J} X_j \to Y</math><br />
to indicate its dependence on the individual ''f''<sub>''j''</sub>.<br />
<br />
If the family of objects consists of only two members the coproduct is usually written ''X''<sub>1</sub> ∐ ''X''<sub>2</sub> or ''X''<sub>1</sub> ⊕ ''X''<sub>2</sub> or sometimes simply ''X''<sub>1</sub> + ''X''<sub>2</sub>, and the diagram takes the form:<br />
<br />
[[Image:Coproduct-03.svg|280px|center]]<br />
<br />
The unique arrow ''f'' making this diagram commute is then correspondingly denoted ''f''<sub>1</sub> ∐ ''f''<sub>2</sub> or ''f''<sub>1</sub> ⊕ ''f''<sub>2</sub> or ''f''<sub>1</sub> + ''f''<sub>2</sub> or [''f''<sub>1</sub>, ''f''<sub>2</sub>].<br />
<br />
== Examples ==<br />
The coproduct in the [[category of sets]] is simply the '''[[disjoint union#Set theory definition|disjoint union]]''' with the maps ''i<sub>j</sub>'' being the [[inclusion map]]s. Unlike [[direct product]]s, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the [[category of groups]], called the '''[[free product]]''', is quite complicated. On the other hand, in the [[category of abelian groups]] (and equally for [[vector spaces]]), the coproduct, called the '''[[direct sum]]''', consists of the elements of the direct product which have only [[finite set|finitely]] many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)<br />
<br />
In the case of [[topological space]]s coproducts are disjoint unions with their [[disjoint union (topology)|disjoint union topologies]]. That is, it is a disjoint union of the underlying sets, and the [[open set]]s are sets ''open in each of the spaces'', in a rather evident sense. In the category of [[pointed space]]s, fundamental in [[homotopy theory]], the coproduct is the [[wedge sum]] (which amounts to joining a collection of spaces with base points at a common base point).<br />
<br />
Despite all this dissimilarity, there is still, at the heart of the whole thing, a disjoint union: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space [[linear span|spanned]] by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.<br />
<br />
== Discussion ==<br />
<br />
The coproduct construction given above is actually a special case of a [[colimit]] in category theory. The coproduct in a category ''C'' can be defined as the colimit of any [[functor]] from a [[discrete category]] ''J'' into ''C''. Not every family {''X''<sub>''j''</sub>} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if ''i''<sub>''j''</sub> : ''X''<sub>''j''</sub> → ''X'' and ''k''<sub>''j''</sub> : ''X''<sub>''j''</sub> → ''Y'' are two coproducts of the family {''X''<sub>''j''</sub>}, then (by the definition of coproducts) there exists a unique [[isomorphism]] ''f'' : ''X'' → ''Y'' such that ''fi''<sub>''j''</sub> = ''k''<sub>''j''</sub>&nbsp; for each ''j'' in ''J''.<br />
<br />
As with any [[universal property]], the coproduct can be understood as a universal morphism. Let Δ: ''C'' → ''C''×''C'' be the [[diagonal functor]] which assigns to each object ''X'' the [[ordered pair]] (''X'',''X'') and to each morphism ''f'':''X'' → ''Y'' the pair (''f'',''f''). Then the coproduct ''X''+''Y'' in ''C'' is given by a universal morphism to the functor Δ from the object (''X'',''Y'') in ''C''×''C''.<br />
<br />
The coproduct indexed by the [[empty set]] (that is, an ''empty coproduct'') is the same as an [[initial object]] in ''C''.<br />
<br />
If ''J'' is a set such that all coproducts for families indexed with ''J'' exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor ''C''<sup>''J''</sup> → ''C''. The coproduct of the family {''X''<sub>''j''</sub>} is then often denoted by ∐<sub>''j''</sub> ''X''<sub>''j''</sub>, and the maps ''i''<sub>''j''</sub> are known as the '''[[inclusion map|natural injections]]'''. <br />
<br />
Letting Hom<sub>''C''</sub>(''U'',''V'') denote the set of all morphisms from ''U'' to ''V'' in ''C'' (that is, a [[hom-set]] in ''C''), we have a [[natural isomorphism]] <br />
:<math>\operatorname{Hom}_C\left(\coprod_{j\in J}X_j,Y\right) \cong \prod_{j\in J}\operatorname{Hom}_C(X_j,Y)</math><br />
given by the [[bijection]] which maps every [[tuple]] of morphisms<br />
:<math>(f_j)_{j\in J} \in \prod_{j \in J}\operatorname{Hom}(X_j,Y)</math><br />
(a product in '''Set''', the [[category of sets]], which is the [[Cartesian product]], so it is a tuple of morphisms) to the morphism<br />
:<math>\coprod_{j\in J} f_j \in \operatorname{Hom}\left(\coprod_{j\in J}X_j,Y\right).</math><br />
That this map is a [[surjection]] follows from the commutativity of the diagram: any morphism ''f'' is the coproduct of the tuple <br />
:<math>(f\circ i_j)_{j \in J}.</math><br />
That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant [[hom-functor]] changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the [[opposite category]] ''C''<sup>opp</sup> to '''Set''' is continuous; it preserves limits (a coproduct in ''C'' is a product in ''C''<sup>opp</sup>).<br />
<br />
If ''J'' is a [[finite set|finite]] set, say ''J'' = {1,...,''n''}, then the coproduct of objects ''X''<sub>1</sub>,...,''X''<sub>''n''</sub> is often denoted by ''X''<sub>1</sub>⊕...⊕''X''<sub>''n''</sub>.<br />
Suppose all finite coproducts exist in ''C'', coproduct functors have been chosen as above, and 0 denotes the [[initial object]] of ''C'' corresponding to the empty coproduct. We then have [[natural isomorphism]]s<br />
:<math>X\oplus (Y \oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z</math><br />
:<math>X\oplus 0 \cong 0\oplus X \cong X</math><br />
:<math>X\oplus Y \cong Y\oplus X.</math><br />
These properties are formally similar to those of a commutative [[monoid]]; a category with finite coproducts is an example of a symmetric [[monoidal category]].<br />
<br />
If the category has a [[zero object]] ''Z'', then we have unique morphism ''X'' → ''Z'' (since ''Z'' is [[terminal object|terminal]]) and thus a morphism ''X'' ⊕ ''Y'' → ''Z'' ⊕ ''Y''. Since ''Z'' is also initial, we have a canonical isomorphism ''Z'' ⊕ ''Y'' ≅ ''Y'' as in the preceding paragraph. We thus have morphisms ''X'' ⊕ ''Y'' → ''X'' and ''X'' ⊕ ''Y'' → ''Y'', by which we infer a canonical morphism ''X'' ⊕ ''Y'' → ''X''×''Y''. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in '''Grp''' it is a proper [[epimorphism]] while in '''Set'''<sub>*</sub> (the category of [[pointed set]]s) it is a proper [[monomorphism]]. In any [[preadditive category]], this morphism is an isomorphism and the corresponding object is known as the [[biproduct]]. A category with all finite biproducts is known as an [[additive category]].<br />
<br />
If all families of objects indexed by ''J'' have coproducts in ''C'', then the coproduct comprises a functor ''C''<sup>''J''</sup> → ''C''. Note that, like the product, this functor is ''covariant''.<br />
<br />
==See also==<br />
*[[Product (category theory)|Product]]<br />
*[[Limit (category theory)|Limits and colimits]]<br />
*[[Coequalizer]]<br />
*[[Direct limit]]<br />
<br />
==References==<br />
* {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 }}<br />
<br />
== External links ==<br />
<br />
*[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of coproducts in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].<br />
<br />
[[Category:Limits (category theory)]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Talk:Quantum_teleportation/Archive/1&diff=307419
Talk:Quantum teleportation/Archive/1
2013-11-28T15:40:43Z
<p>99.153.64.179: {{talk archive}}</p>
<hr />
<div><br><br>[http://www.google.de/url?url=https://www.facebook.com/Hostgator1CentCoupon&rct=j&q=&esrc=s&sa=U&ei=FZAoVYrPLZHoavm2gagM&ved=0CDAQFjAB&usg=AFQjCNFTG8OBlZH3Y4vUeo8aoL5IRVCH1w google.de]Selecting the right web hosting service is essential. The last thing you desire to do is get a website released with a hosting service (after discovering to utilize that hosting service), and afterwards be dissatisfied with the service and have to mess around with moving your internet site to another hosting service.<br><br>I'm a delighted HostGator consumer. I make use of both HostGator and Bluehost.<br><br>I'll start with one of the most important factors to consider: Rate<br><br>HostGator has a number of prices strategies. They are as follows (as of the date this short article was published):.<br><br>Hatchling Strategy: as reduced as $4.95 per month with a 3 year commitment. On this plan you can host one internet site.<br>Baby Plan: as reduced as $7.95 per month with a 3 year commitment. 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99.153.64.179
https://en.formulasearchengine.com/index.php?title=Frobenius_algebra&diff=9367
Frobenius algebra
2013-11-28T06:24:30Z
<p>99.153.64.179: /* Generalization: Frobenius extension */ h2 header not h3</p>
<hr />
<div>[[File:Catalytic Reformer Unit.jpg|thumb|right|250px|A catalytic reformer unit in a petroleum refinery. © BP p.l.c]]<br />
<br />
'''Catalytic reforming''' is a chemical process used to convert [[petroleum refinery]] [[petroleum naphtha|naphtha]]s, typically having low [[octane rating]]s, into high-octane liquid products called '''reformates''' which are components of high-octane [[gasoline]] (also known as high-octane petrol). Basically, the process re-arranges or re-structures the [[hydrocarbon]] [[molecules]] in the naphtha feedstocks as well as breaking some of the molecules into smaller molecules. The overall effect is that the product reformate contains hydrocarbons with more complex molecular shapes having higher octane values than the hydrocarbons in the naphtha feedstock. In so doing, the process separates [[hydrogen]] [[atoms]] from the hydrocarbon molecules and produces very significant amounts of byproduct hydrogen gas for use in a number of the other processes involved in a modern petroleum refinery. Other byproducts are small amounts of [[methane]], [[ethane]], [[propane]], and [[butanes]]. <br />
<br />
This process is quite different from and not to be confused with the catalytic [[steam reforming]] process used industrially to produce various products such as [[hydrogen]], [[ammonia]], and [[methanol]] from [[natural gas]], naphtha or other petroleum-derived feedstocks. Nor is this process to be confused with various other catalytic reforming processes that use methanol or [[biomass|biomass-derived]] feedstocks to produce hydrogen for [[fuel cells]] or other uses.<br />
<br />
==History==<br />
In the 1940s, Vladimir Haensel,<ref>[http://newton.nap.edu/html/biomems/vhaensel.pdf A Biographical Memoir of Vladimir Haensel] written by Stanley Gembiki, published by the National Academy of Sciences in 2006.</ref> a research chemist working for [[Universal Oil Products]] (UOP), developed a [[catalytic]] reforming process using a [[catalyst]] containing [[platinum]]. Haensel's process was subsequently commercialized by UOP in 1949 for producing a high octane gasoline from low octane naphthas and the UOP process become known as the Platforming process.<ref>[http://www.uop.com/refining/1030.html Platforming described on UOP's website]</ref> The first Platforming unit was built in 1949 at the refinery of the Old Dutch Refining Company in [[Muskegon]], [[Michigan]]. <br />
<br />
In the years since then, many other versions of the process have been developed by some of the major oil companies and other organizations. Today, the large majority of gasoline produced worldwide is derived from the catalytic reforming process. <br />
<br />
To name a few of the other catalytic reforming versions that were developed, all of which utilized a platinum and/or a [[rhenium]] catalyst:<br />
<br />
*Rheniforming: Developed by [[Chevron Oil Company]].<br />
*Powerforming: Developed by [[Esso|Esso Oil Company]], currently known as [[ExxonMobil]].<br />
*Magnaforming: Developed by [[Engelhard]] and [[ARCO|Atlantic Richfield Oil Company]].<br />
*Ultraforming: Developed by [[Standard Oil of Indiana]], now a part of the [[British Petroleum|British Petroleum Company]].<br />
*Houdriforming: Developed by the Houdry Process Corporation.<br />
*CCR Platforming: A Platforming version, designed for continuous catalyst regeneration, developed by UOP.<br />
*Octanizing: A catalytic reforming version developed by Axens, a subsidiary of [[Institut francais du petrole]] (IFP), designed for continuous catalyst regeneration.<br />
<br />
==Chemistry==<br />
Before describing the reaction chemistry of the catalytic reforming process as used in petroleum refineries, the typical naphthas used as catalytic reforming feedstocks will be discussed.<br />
<br />
===Typical naphtha feedstocks===<br />
A petroleum refinery includes many [[unit operations]] and [[unit processing|unit processes]]. The first unit operation in a refinery is the [[Continuous distillation#Continuous distillation of crude oil|continuous distillation]] of the [[petroleum|petroleum crude oil]] being refined. The overhead liquid distillate is called naphtha and will become a major component of the refinery's gasoline (petrol) product after it is further processed through a [[Hydrodesulfurization|catalytic hydrodesulfurizer]] to remove [[sulfur]]-containing hydrocarbons and a catalytic reformer to reform its hydrocarbon molecules into more complex molecules with a higher octane rating value. The naphtha is a mixture of very many different hydrocarbon compounds. It has an initial [[boiling point]] of about 35 °C and a final boiling point of about 200 °C, and it contains [[Alkane|paraffin]], [[naphthene]] (cyclic paraffins) and [[aromatic]] hydrocarbons ranging from those containing 4 [[carbon]] atoms to those containing about 10 or 11 carbon atoms.<br />
<br />
The naphtha from the crude oil distillation is often further distilled to produce a "light" naphtha containing most (but not all) of the hydrocarbons with 6 or fewer carbon atoms and a "heavy" naphtha containing most (but not all) of the hydrocarbons with more than 6 carbon atoms. The heavy naphtha has an initial boiling point of about 140 to 150 °C and a final boiling point of about 190 to 205 °C. The naphthas derived from the distillation of crude oils are referred to as "straight-run" naphthas.<br />
<br />
It is the straight-run heavy naphtha that is usually processed in a catalytic reformer because the light naphtha has molecules with 6 or fewer carbon atoms which, when reformed, tend to crack into butane and lower molecular weight hydrocarbons which are not useful as high-octane gasoline blending components. Also, the molecules with 6 carbon atoms tend to form aromatics which is undesirable because governmental environmental regulations in a number of countries limit the amount of aromatics (most particularly [[benzene]]) that gasoline may contain.<ref>[http://www.ec.gc.ca/CEPARegistry/regulations/detailReg.cfm?intReg=1 Canadian regulations on benzene in gasoline]</ref><ref>[http://www.ukpia.com/industry_issues/environment_air_quality_health_safety/benzene_in_petrol.aspx United Kingdom regulations on benzene in gasoline]</ref><ref>[http://www.washingtonpost.com/wp-dyn/content/article/2006/03/01/AR2006030102113.html USA regulations on benzene in gasoline]</ref> <br />
<br />
It should be noted that there are a great many petroleum [[List of oil fields|crude oil sources]] worldwide and each crude oil has its own unique composition or [[Crude oil assay|"assay"]]. Also, not all refineries process the same crude oils and each refinery produces its own straight-run naphthas with their own unique initial and final boiling points. In other words, naphtha is a generic term rather than a specific term. <br />
<br />
The table just below lists some fairly typical straight-run heavy naphtha feedstocks, available for catalytic reforming, derived from various crude oils. It can be seen that they differ significantly in their content of paraffins, naphthenes and aromatics: <br />
<br />
{| class="wikitable"<br />
|+ Typical Heavy Naphtha Feedstocks<br />
|-<br />
! Crude oil name <math>\Rightarrow</math><br>Location <math>\Rightarrow</math> <br />
! Barrow Island<br>Australia<ref>[http://www.santos.com/library/barrow_crude.pdf Barrow Island crude oil assay]</ref><br />
! Mutineer-Exeter<br>Australia<ref>[http://www.santos.com/library/refining_characteristics.pdf Mutineer-Exeter crude oil assay]</ref><br />
! CPC Blend<br>Kazakhstan<ref>[http://crudemarketing.chevron.com/overview.asp?cpc CPC Blend crude oil assay]</ref><br />
! Draugen<br>North Sea<ref>[http://www.statoil.com/STATOILCOM/crude/svg02659.nsf/UNID/C9AC3EF9CE76B0DFC1256B5600528D6D/$FILE/Dra4kv02.pdf Draugen crude oil assay]</ref><br />
|-<br />
| Initial boiling point, °C ||align=center|149||align=center|140||align=center|149||align=center|150<br />
|- <br />
| Final boiling point, °C ||align=center|204||align=center|190||align=center|204||align=center|180 <br />
|-<br />
| Paraffins, liquid volume % ||align=center|46||align=center|62||align=center|57||align=center|38<br />
|-<br />
| Naphthenes, liquid volume % ||align=center|42||align=center|32||align=center|27||align=center|45<br />
|-<br />
| Aromatics, liquid volume % ||align=center|12||align=center|6||align=center|16||align=center|17<br />
|}<br />
<br />
Some refinery naphthas include [[olefins|olefinic hydrocarbons]], such as naphthas derived from the [[fluid catalytic cracking]] and [[Delayed coker|coking]] processes used in many refineries. Some refineries may also [[hydrodesulfurization|desulfurize]] and catalytically reform those naphthas. However, for the most part, catalytic reforming is mainly used on the straight-run heavy naphthas, such as those in the above table, derived from the distillation of crude oils.<br />
<br />
===The reaction chemistry===<br />
There are many chemical reactions that occur in the catalytic reforming process, all of which occur in the presence of a catalyst and a high [[partial pressure]] of hydrogen. Depending upon the type or version of catalytic reforming used as well as the desired reaction severity, the reaction conditions range from temperatures of about 495 to 525 °C and from pressures of about 5 to 45 [[atmosphere|atm]].<ref>[http://www.osha.gov/dts/osta/otm/otm_iv/otm_iv_2.html#3 OSHA Technical Manual, Section IV, Chapter 2, ''Petroleum refining Processes''] (A publication of the [[Occupational Safety and Health Administration]])</ref> <br />
<br />
The commonly used catalytic reforming catalysts contain [[noble metals]] such as platinum and/or rhenium, which are very susceptible to [[Catalyst poisoning|poisoning]] by sulfur and [[nitrogen]] compounds. Therefore, the naphtha feedstock to a catalytic reformer is always pre-processed in a [[hydrodesulfurization]] unit which removes both the sulfur and the nitrogen compounds.<br />
<br />
The four major catalytic reforming reactions are:<ref name=Gary>{{cite book|author=Gary, J.H. and Handwerk, G.E.|title=Petroleum Refining Technology and Economics|edition=2nd Edition|publisher=Marcel Dekker, Inc|year=1984|isbn=0-8247-7150-8}}</ref><br />
<br />
:1: The [[dehydrogenation]] of naphthenes to convert them into aromatics as exemplified in the conversion [[methylcyclohexane]] (a naphthene) to [[toluene]] (an aromatic), as shown below:<br />
<br />
[[File:Methylcyclohexanetotoluene.svg|center]]<br />
<br />
:2: The [[isomerization]] of normal paraffins to [[isoparaffin]]s as exemplified in the conversion of [[Octane|normal octane]] to 2,5-Dimethylhexane (an isoparaffin), as shown below:<br />
<br />
[[File:Paraffintoisoparaffin.svg|center]]<br />
<br />
:3: The dehydrogenation and [[aromatization]] of paraffins to aromatics (commonly called dehydrocyclization) as exemplified in the conversion of [[Heptane|normal heptane]] to toluene, as shown below:<br />
<br />
[[Image:CatReformerEq2.png|center]]<br />
<br />
:4: The [[hydrocracking]] of paraffins into smaller molecules as exemplified by the cracking of normal heptane into [[isopentane]] and ethane, as shown below:<br />
<br />
[[Image:CatReformerEq4.png|center]]<br />
<br />
The hydrocracking of paraffins is the only one of the above four major reforming reactions that consumes hydrogen. The isomerization of normal paraffins does not consume or produce hydrogen. However, both the dehydrogenation of naphthenes and the dehydrocyclization of paraffins produce hydrogen. The overall net production of hydrogen in the catalytic reforming of petroleum naphthas ranges from about 50 to 200 cubic meters of hydrogen gas (at 0 °C and 1 atm) per cubic meter of liquid naphtha feedstock. In the [[United States customary units]], that is equivalent to 300 to 1200 cubic feet of hydrogen gas (at 60 °F and 1 atm) per [[barrel (unit)|barrel]] of liquid naphtha feedstock.<ref>[http://www.freepatentsonline.com/5011805.html US Patent 5011805, ''Dehydrogenation, dehydrocyclization and reforming catalyst''] (Inventor: Ralph Dessau, Assignee: Mobil Oil Corporation)</ref> In many petroleum refineries, the net hydrogen produced in catalytic reforming supplies a significant part of the hydrogen used elsewhere in the refinery (for example, in hydrodesulfurization processes). The hydrogen is also necessary in order to [[Hydrogenolysis|hydrogenolyze]] any polymers that form on the catalyst.<br />
<br />
==Process description==<br />
The most commonly used type of catalytic reforming unit has three [[Chemical reactor|reactors]], each with a fixed bed of catalyst, and all of the catalyst is regenerated [[In situ#Chemistry and chemical engineering|''in situ'']] during routine catalyst regeneration shutdowns which occur approximately once each 6 to 24 months. Such a unit is referred to as a [[semi-regenerative catalytic reformer]] (SRR).<br />
<br />
Some catalytic reforming units have an extra ''spare'' or ''swing'' reactor and each reactor can be individually isolated so that any one reactor can be undergoing in situ regeneration while the other reactors are in operation. When that reactor is regenerated, it replaces another reactor which, in turn, is isolated so that it can then be regenerated. Such units, referred to as ''cyclic'' catalytic reformers, are not very common. Cyclic catalytic reformers serve to extend the period between required shutdowns. <br />
<br />
The latest and most modern type of catalytic reformers are called Continuous Catalyst Regeneration (CCR) reformers. Such units are characterized by continuous in-situ regeneration of part of the catalyst in a special regenerator, and by continuous addition of the regenerated catalyst to the operating reactors. As of 2006, two CCR versions available: UOP's CCR Platformer process<ref>[http://www.uop.com/objects/CCR%20Platforming.pdf CCR Platforming] (UOP website)</ref> and Axens' Octanizing process.<ref>[http://www.axens.net/upload/news/fichier/ptq_q1_06_octanizing_reformer_options.pdf Octanizing Options] (Axens website)</ref> The installation and use of CCR units is rapidly increasing. <br />
<br />
Many of the earliest catalytic reforming units (in the 1950s and 1960s) were non-regenerative in that they did not perform in situ catalyst regeneration. Instead, when needed, the aged catalyst was replaced by fresh catalyst and the aged catalyst was shipped to catalyst manufacturers to be either regenerated or to recover the platinum content of the aged catalyst. Very few, if any, catalytic reformers currently in operation are non-regenerative.<br />
<br />
The [[process flow diagram]] below depicts a typical semi-regenerative catalytic reforming unit.<br />
<br />
[[Image:CatReformer.png|frame|center|Schematic diagram of a typical semi-regenerative catalytic reformer unit in a petroleum refinery]]<br />
<br />
The liquid feed (at the bottom left in the diagram) is [[pump]]ed up to the reaction pressure (5 to 45 atm) and is joined by a stream of hydrogen-rich recycle gas. The resulting liquid-gas mixture is preheated by flowing through a [[heat exchanger]]. The preheated feed mixture is then totally [[vaporized]] and heated to the reaction temperature (495 to 520 °C) before the vaporized reactants enter the first reactor. As the vaporized reactants flow through the fixed bed of catalyst in the reactor, the major reaction is the dehydrogenation of naphthenes to aromatics (as described earlier herein) which is highly [[endothermic]] and results in a large temperature decrease between the inlet and outlet of the reactor. To maintain the required reaction temperature and the rate of reaction, the vaporized stream is reheated in the second fired heater before it flows through the second reactor. The temperature again decreases across the second reactor and the vaporized stream must again be reheated in the third fired heater before it flows through the third reactor. As the vaporized stream proceeds through the three reactors, the reaction rates decrease and the reactors therefore become larger. At the same time, the amount of reheat required between the reactors becomes smaller. Usually, three reactors are all that is required to provide the desired performance of the catalytic reforming unit.<br />
<br />
Some installations use three separate fired heaters as shown in the schematic diagram and some installations use a single fired heater with three separate heating coils.<br />
<br />
The hot reaction products from the third reactor are partially cooled by flowing through the heat exchanger where the feed to the first reactor is preheated and then flow through a water-cooled heat exchanger before flowing through the pressure controller (PC) into the gas separator.<br />
<br />
Most of the hydrogen-rich gas from the gas separator vessel returns to the suction of the recycle hydrogen [[gas compressor]] and the net production of hydrogen-rich gas from the reforming reactions is exported for use in the other refinery processes that consume hydrogen (such as hydrodesulfurization units and/or a [[Hydrocracking|hydrocracker unit]]).<br />
<br />
The liquid from the gas separator vessel is routed into a [[fractionating column]] commonly called a ''stabilizer''. The overhead offgas product from the stabilizer contains the byproduct methane, ethane, propane and butane gases produced by the hydrocracking reactions as explained in the above discussion of the reaction chemistry of a catalytic reformer, and it may also contain some small amount of hydrogen. That offgas is routed to the refinery's central gas processing plant for removal and recovery of propane and butane. The residual gas after such processing becomes part of the refinery's fuel gas system.<br />
<br />
The bottoms product from the stabilizer is the high-octane liquid reformate that will become a component of the refinery's product gasoline.<br />
<br />
==Catalysts and mechanisms==<br />
<br />
Most catalytic reforming catalysts contain platinum or rhenium on a [[Silicon dioxide|silica]] or [[amorphous silica-alumina|silica-alumina]] support base, and some contain both platinum and rhenium. Fresh catalyst is [[chloride]]d (chlorinated) prior to use.<br />
<br />
The noble metals (platinum and rhenium) are considered to be catalytic sites for the dehydrogenation reactions and the chlorinated alumina provides the [[acid]] sites needed for isomerization, cyclization and hydrocracking reactions.<ref name=Gary/><br />
<br />
The activity (i.e., effectiveness) of the catalyst in a semi-regenerative catalytic reformer is reduced over time during operation by [[Carbon|carbonaceous coke]] deposition and chloride loss. The activity of the catalyst can be periodically regenerated or restored by in situ high temperature oxidation of the coke followed by chlorination. As stated earlier herein, semi-regenerative catalytic reformers are regenerated about once per 6 to 24 months.<br />
<br />
Normally, the catalyst can be regenerated perhaps 3 or 4 times before it must be returned to the manufacturer for reclamation of the valuable platinum and/or rhenium content.<ref name=Gary/><br />
<br />
==References==<br />
{{reflist}}<br />
<br />
==External links==<br />
*[http://www.processengr.com/ppt_presentations/oil_refinery_processes.pdf Oil Refinery Processes, A Brief Overview]<br />
*[http://home.comcast.net/~jjechura/CHEN409/09_Gasoline_Upgrading.pdf Colorado School of Mines, Lecture Notes] (''Chapter 10, Refining Processes, Catalytic Refinery'' by John Jechura, Adjunct Professor)<br />
*[http://www.cheresources.com/refining3.shtml Students' Guide to Refining] (scroll down to ''Platforming'')<br />
*[http://www.dct.tudelft.nl/race/education/smst/smst200303.pdf Modern Refinery] Website of [[Delft University of Technology]], [[Netherlands]] (use search function for ''Reforming'')<br />
*[http://www.ifp.fr/IFP/fr/IFP02OGS.nsf/(VNoticesOGST)/AD4A1392D20E5AAEC1256CDE0055399E/$file/decroocq_52n5.pdf?openelement Major scientific and technical challenges about development of new refining processes] (IFP website)<br />
<br />
[[Category:Oil refining]]<br />
[[Category:Chemical engineering]]<br />
[[Category:Chemical processes]]<br />
[[Category:Unit processes]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Double_pushout_graph_rewriting&diff=27753
Double pushout graph rewriting
2013-11-28T06:14:59Z
<p>99.153.64.179: /* Generalization */ misc wikilinks</p>
<hr />
<div>In [[general relativity]] the '''Kantowski-Sachs metric''' describes a [[homogeneous]] but [[anisotropic]] universe whose spatial section has the topology of <math> \mathbb{R} \times S^{2}</math>. The metric is:<br />
:<math><br />
ds^{2} = -dt^{2} + e^{2\sqrt{\Lambda}t} dz^{2} + \frac{1}{\Lambda}(d\theta^{2} + \sin^{2}\theta d\phi^{2})<br />
</math><br />
The [[isometry]] group of this spacetime is <math> \mathbb{R} \times SO(3)</math>. Remarkably, the isometry group does not act simply transitively on spacetime, nor does it possess a subgroup with simple transitive action.<br />
<br />
==See also==<br />
*[[Bianchi classification]]<br />
*[[Ronald Kantowski]]<br />
*[[Rainer Kurt Sachs]]<br />
<br />
[[Category:Physical cosmology]]<br />
<br />
<br />
{{astronomy-stub}}<br />
{{physics-stub}}</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Category_of_relations&diff=14448
Category of relations
2013-11-27T20:15:06Z
<p>99.153.64.179: /* See also */ recat to monoidal cats</p>
<hr />
<div>{| class="wikitable" align="right" style="margin-left:10px" width="320"<br />
!bgcolor=#e7dcc3 colspan=2|{{PAGENAME}}<br />
|-<br />
|bgcolor=#ffffff align=center colspan=2|[[File:Truncated cubic tiling.png|200px]][[File:HC A2-P3.png|110px]]<br />
|-<br />
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]<br />
|-<br />
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{4,3,4}<BR>t<sub>0,1</sub>{4,3,4}<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node_1|3|node|4|node}}<br />
|-<br />
|bgcolor=#e7dcc3|Cell type||[[truncated cube|3.8.8]], [[octahedron|{3,4}]]<br />
|-<br />
|bgcolor=#e7dcc3|Face type||[[triangle|{3}]], [[square (geometry)|{4}]], [[octagon|{8}]]<br />
|-<br />
|bgcolor=#e7dcc3|Cells/edge||(3.8.8)<sup>4</sup><BR>{3,4}.(3.8.8)<sup>2</sup><br />
|-<br />
|bgcolor=#e7dcc3|Faces/edge||{8}<sup>4</sup><BR>{3}<sup>2</sup>.{8}<br />
|-<br />
|bgcolor=#e7dcc3|Cells/vertex||[[truncated cube|3.8.8]] (4)<BR>[[octahedron|{3,4}]] (1)<br />
|-<br />
|bgcolor=#e7dcc3|Faces/vertex||{8}<sup>4</sup>+{3}<sup>4</sup><br />
|-<br />
|bgcolor=#e7dcc3|Edges/vertex||5<br />
|-<br />
|bgcolor=#e7dcc3|[[Euler characteristic]]||0<br />
|-<br />
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated cubic honeycomb verf.png|80px]]<BR>[[square pyramid]]<br />
|-<br />
|bgcolor=#e7dcc3|[[Space group]]<BR>[[Fibrifold notation]]||Pm{{overline|3}}m (221)<BR>4<sup>−</sup>:2<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]<br />
|-<br />
|bgcolor=#e7dcc3|Dual||[[Pyramidille]]<BR>(Hexakis cubic honeycomb)<br />
|-<br />
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]<br />
|}<br />
The '''truncated cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. It is composed of [[truncated cube]]s and [[octahedron|octahedra]] in a ratio of 1:1.<br />
<br />
[[John Horton Conway]] calls this honeycomb a '''truncated cubille''', and its dual [[pyramidille]].<br />
<br />
== Symmetry ==<br />
<br />
There is a second [[uniform coloring]]s by reflectional symmetry of the [[Coxeter group]]s, the second seen with alternately colored truncated cubic cells.<br />
{| class="wikitable" width=280<br />
!Construction<br />
!Bicantellated alternate cubic<br />
!Truncated cubic honeycomb<br />
|- valign=top<br />
![[Coxeter group]]<br />
![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math><br />
![4,3,4], <math>{\tilde{C}}_3</math><BR>=<[4,3<sup>1,1</sup>]><br />
|-<br />
![[Space group]]||Fm{{overline|3}}m||Pm{{overline|3}}m<br />
|- align=center<br />
!Coloring<br />
|[[Image:Truncated cubic honeycomb2.png|120px]]<br />
|[[Image:Truncated cubic honeycomb.png|120px]]<br />
|- align=center<br />
![[Coxeter-Dynkin diagram]]<br />
!{{CDD|node_1|4|node_1|split1|nodes}}<br />
!{{CDD|node_1|4|node_1|3|node|4|node}}<br />
|- align=center<br />
![[Vertex figure]]<br />
|[[File:Bicantellated alternate cubic honeycomb verf.png|80px]]<br />
|[[Image:Truncated cubic honeycomb verf.png|80px]]<br />
|}<br />
== Related honeycombs==<br />
The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, [[Coxeter group]] generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The [[Expansion (geometry)|expanded]] cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.<br />
{{C3 honeycombs}}<br />
<br />
The [4,3<sup>1,1</sup>], {{CDD|node|4|node|split1|nodes}}, [[Coxeter group]] generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.<br />
{{B3 honeycombs}}<br />
<br />
== See also==<br />
*[[Architectonic and catoptric tessellation]]<br />
<br />
== References ==<br />
{{Commons category|Truncated cubic honeycomb}}<br />
{{reflist}}<br />
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, (2008) ''The Symmetries of Things'', ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)<br />
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''<br />
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.<br />
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, [[Peter McMullen]], Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]<br />
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)<br />
* [[Alfredo Andreini|A. Andreini]], ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.<br />
* {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x4x3o4o - tich - O14}}<br />
* [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 03-Tich]<br />
<br />
{{polychora-stub}}<br />
<br />
[[Category:Honeycombs (geometry)]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Direct_integral&diff=8050
Direct integral
2013-11-23T17:13:26Z
<p>99.153.64.179: /* Direct integrals of Hilbert spaces */ clarify remark</p>
<hr />
<div>{{Unreferenced|date=December 2009}}<br />
<br />
If <math>L_1</math> and <math>L_2</math> are [[formal languages]], then the '''left quotient''' of <math>L_1</math> with <math>L_2</math> is the language consisting of strings ''w'' such that ''xw'' is in <math>L_2</math> for some string ''x'' in <math>L_1</math>. In symbols, we write:<br />
<br />
<math>L_1 \backslash L_2 = \{w \ | \ \exists x ((x \in L_1) \land (xw \in L_2))\}</math><br />
<br />
You can regard the '''left quotient''' as the set of postfixes that complete words from <math>L_1</math>, such that the resulting word is in <math>L_2</math>.<br />
<br />
For more details, see [[right quotient]].<br />
<br />
{{DEFAULTSORT:Left Quotient}}<br />
[[Category:Formal languages]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Choi%27s_theorem_on_completely_positive_maps&diff=13869
Choi's theorem on completely positive maps
2013-11-23T15:57:05Z
<p>99.153.64.179: wikilink</p>
<hr />
<div>{{confusing|date=October 2013}}<br />
{{Refimprove|date=May 2010}}<br />
In [[theoretical physics]], '''scalar field theory''' can refer to a [[Classical field theory|classical]] or [[Quantum field theory|quantum theory]] of [[scalar field]]s. A field which is invariant under any [[Lorentz transformation]] is called a "scalar", in contrast to a [[vector field|vector]] or [[tensor field]]. The quanta of the quantized scalar field are spin-zero particles, and as such are [[boson]]s.<br />
<br />
No fundamental scalar fields have been observed in nature, though the [[Higgs boson]] may yet prove the first example. However, scalar fields appear in the [[effective field theory]] descriptions of many physical phenomena. An example is the [[pion]], which is actually a "pseudoscalar", which means it is not invariant under parity transformations which invert the spatial directions, distinguishing it from a true scalar, which is parity-invariant. Because of the relative simplicity of the mathematics involved, scalar fields are often the first field introduced to a student of classical or quantum field theory.<br />
<br />
In this article, the repeated index notation indicates the [[Einstein summation convention]] for summation over repeated indices. The theories described are defined in flat, D-dimensional [[Minkowski space]], with (D-1) spatial dimension and one time dimension and are, by construction, relativistically [[Covariance|covariant]]. The Minkowski space [[Metric (mathematics)|metric]], <math>\eta_{\mu\nu}</math>, has a particularly simple form: it is [[diagonal]], and here we use the [[metric signature|+ − − −]] [[sign convention]].<br />
<br />
==Classical scalar field theory==<br />
<br />
===Linear (free) theory===<br />
<br />
The most basic scalar field theory is the [[linear]] theory. The [[Action (physics)|action]] for the free [[theory of relativity|relativistic]] scalar field theory is<br />
<br />
:<math>\mathcal{S}=\int \mathrm{d}^{D-1}x \mathrm{d}t \mathcal{L} = \int \mathrm{d}^{D-1}x \mathrm{d}t<br />
\left[ \frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -\frac{1}{2} m^2\phi^2 \right]</math><br />
:<math>=\int \mathrm{d}^{D-1}x \mathrm{d}t \left[\frac{1}{2}(\partial_t\phi)^2- \frac{1}{2}\delta^{ij}\partial_i\phi \partial_j\phi -\frac{1}{2} m^2\phi^2 \right],</math><br />
<br />
where <math>\mathcal{L}</math> is known as a [[Lagrangian]] density. This is an example of a quadratic action, since each of the terms is quadratic in the field, <math>\phi</math>. The term proportional to <math>m^2</math> is sometimes known as a mass term, due to its interpretation in the quantized version of this theory in terms of particle mass.<br />
<br />
The equation of motion for this theory is obtained by [[Euler–Lagrange|extremizing]] the action above. It takes the following form, linear in <math>\phi</math>:<br />
<br />
:<math>\eta^{\mu\nu}\partial_\mu\partial_\nu\phi+m^2\phi=\partial^2_t\phi-\nabla^2\phi+m^2\phi=0</math><br />
<br />
Note that this is the same as the [[Klein–Gordon equation]], but that here the interpretation is as a classical field equation, rather than as a quantum mechanical wave equation.<br />
<br />
===Nonlinear (interacting) theory===<br />
<br />
The most common generalization of the linear theory above is to add a [[scalar potential]]<br />
<math>V(\phi)</math> to the equations of motion, where typically, ''V'' is a polynomial in ''φ'' of order 3 or more (often a monomial). Such a theory is sometimes said to be [[interaction|interacting]], because the Euler-Lagrange equation is now nonlinear, implying a [[Self-energy|self-interaction]]. The action for the most general such theory is<br />
<br />
:<math>\mathcal{S}=\int \mathrm{d}^{D-1}x \, \mathrm{d}t \mathcal{L} = \int<br />
\mathrm{d}^{D-1}x \mathrm{d}t \left[\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - V(\phi) \right]</math><br />
<br />
:<math>=\int \mathrm{d}^{D-1}x \, \mathrm{d}t \left[\frac{1}{2}(\partial_t\phi)^2- \frac{1}{2}\delta^{ij}\partial_i\phi\partial_j\phi -<br />
\frac{1}{2}m^2\phi^2-\sum_{n=3}^\infty \frac{1}{n!} g_n\phi^n \right]</math><br />
<br />
The ''n''! factors in the expansion are introduced because they are useful in the Feynman diagram expansion of the quantum theory, as described below. The corresponding Euler-Lagrange equation of motion is<br />
<br />
:<math>\eta^{\mu\nu}\partial_\mu\partial_\nu\phi+V'(\phi)=\partial^2_t\phi-\nabla^2\phi<br />
+V'(\phi)=0</math>.<br />
<br />
===Dimensional analysis and scaling===<br />
<br />
Physical quantities in these scalar field theories may have dimensions of length, time or mass, or some combination of the three. However, in a relativistic theory, any quantity ''t'', with dimensions of time, can be 'converted' into a length, <math>l=ct</math>, by using the [[velocity of light]], ''c''.<br />
<br />
Similarly, any length ''l'' is equivalent to an inverse mass, <math>l=\frac{\hbar}{mc}</math>, using [[Planck's constant]], <math>\hbar</math>. Heuristically, one can think of a time as a length, or either time or length as an inverse mass. In short, one can think of the dimensions of any physical quantity as defined in terms of just one independent dimension, rather than in terms of all three. This is most often termed the [[Classical scaling dimension|mass dimension]] of the quantity.<br />
<br />
One objection is that this theory is classical, and therefore it is not obvious that Planck's constant should be a part of the theory at all. In a sense this is a valid objection, and if desired one can indeed recast the theory without mass dimensions at all. However, this would be at the expense of making the connection with the quantum scalar field slightly more obscure. Given that one has dimensions of mass, Planck's constant is thought of here as an essentially arbitrary fixed quantity with dimensions appropriate to convert between mass and [[inverse length]].<br />
<br />
====Scaling Dimension====<br />
The [[classical scaling dimension]], or mass dimension, <math>\Delta</math>, of <math>\phi</math> describes the transformation of the field under a rescaling of coordinates:<br />
<br />
:<math>x\rightarrow\lambda x</math><br />
<br />
:<math>\phi\rightarrow\lambda^{-\Delta}\phi</math><br />
<br />
The units of action are the same as the units of <math>\hbar</math>, and so the action itself has zero mass dimension. This fixes the scaling dimension of <math>\phi</math> to be<br />
<br />
:<math>\Delta =\frac{D-2}{2}</math>.<br />
<br />
====Scale invariance====<br />
There is a specific sense in which some scalar field theories are [[Scale invariance|scale-invariant]]. While the actions above are all constructed to have zero mass dimension, not all actions are invariant under the scaling transformation<br />
<br />
:<math>x\rightarrow\lambda x </math><br />
<br />
:<math>\phi\rightarrow\lambda^{-\Delta}\phi</math><br />
<br />
The reason that not all actions are invariant is that one usually thinks of the parameters ''m'' and <math>g_n</math> as fixed quantities, which are not rescaled under the transformation above. The condition for a scalar field theory to be scale invariant is then quite obvious: all of the parameters appearing in the action should be dimensionless quantities. In other words, a scale invariant theory is one without any fixed length scale (or equivalently, mass scale) in the theory.<br />
<br />
For a scalar field theory with D spacetime dimensions, the only dimensionless parameter <math>g_n</math> satisfies <math>n=\frac{2D}{D-2}</math>. For example, in D=4 only <math>g_4</math> is classically dimensionless, and so the only classically scale-invariant scalar field theory in <math>D=4</math> is the massless <math>\phi^4</math> theory. Classical scale invariance normally does not imply quantum scale invariance. See the discussion of the beta function below.<br />
<br />
====Conformal invariance====<br />
<br />
A transformation<br />
<br />
:<math>x\rightarrow \tilde{x}(x)</math><br />
<br />
is said to be [[Conformal symmetry|conformal]] if the transformation satisfies<br />
<br />
:<math>\frac{\partial\tilde{x^\mu}}{\partial x^\rho}\frac{\partial\tilde{x^\nu}}{\partial<br />
x^\sigma}\eta_{\mu\nu}=\lambda^2(x)\eta_{\rho\sigma}</math><br />
<br />
for some function <math>\lambda^2(x)</math>. The conformal group contains as subgroups the [[isometry|isometries]] of the metric <math>\eta_{\mu\nu}</math> (the [[Poincaré group]]) and also the scaling transformations (or [[Scale invariance|dilatation]]s) considered above. In fact, the scale-invariant theories in the previous section are also conformally-invariant.<br />
<br />
===φ<sup>4</sup> theory===<br />
{{See also|Quartic interaction}}<br />
Massive <math>\phi^4</math> theory illustrates a number of interesting phenomena in scalar field theory.<br />
<br />
The Lagrangian density is<br />
<br />
:<math>\mathcal{L}=\frac{1}{2}(\partial_t\phi)^2 -\frac{1}{2}\delta^{ij}\partial_i\phi\partial_j\phi - \frac{1}{2}m^2\phi^2-\frac{g}{4!}\phi^4.</math><br />
<br />
====Spontaneous symmetry breaking====<br />
This Lagrangian has a <math>Z_2</math> symmetry under the transformation <math>\phi\rightarrow-\phi</math><br />
<br />
This is an example of an [[internal symmetry]], in contrast to a [[Spacetime symmetries|space-time symmetry]].<br />
<br />
If <math>m^2</math> is positive, the potential <math>V(\phi)=\frac{1}{2}m^2\phi^2 +\frac{g}{4!}\phi^4</math> has a single minimum, at the origin. The solution <math>\phi=0</math> is clearly invariant under the <math>Z_2</math> symmetry. Conversely, if <math>m^2</math> is negative, then one can readily see that the potential <math>\, V(\phi)=\frac{1}{2}m^2\phi^2+\frac{g}{4!}\phi^4\!</math> has two minima. This is known as a ''double well potential'', and the lowest energy states (known as the vacua, in quantum field theoretical language) in such a theory are '''not''' invariant under the <math>Z_2</math> symmetry of the action (in fact it maps each of the two vacua into the other). In this case, the <math>Z_2</math> symmetry is said to be [[Spontaneous symmetry breaking|spontaneously broken]].<br />
<br />
====Kink solutions====<br />
The <math>\phi^4</math> theory with a negative <math>m^2</math> also has a kink solution, which is a canonical example of a [[soliton]]. Such a solution is of the form<br />
<br />
:<math>\phi(\vec{x},t)=\pm\frac{m}{2\sqrt{\frac{g}{4!}}}\tanh\left(\frac{m(x-x_0)}{\sqrt{2}}\right)</math><br />
<br />
where x is one of the spatial variables (<math>\phi</math> is taken to be independent of t, and the remaining spatial variables). The solution interpolates between the two different vacua of the double well potential. It is not possible to deform the kink into a constant solution without passing through a solution of infinite energy, and for this reason the kink is said to be stable. For <math>D>2</math>, i.e. theories with more than one spatial dimension, this solution is called a [[Domain wall (string theory)|domain wall]].<br />
<br />
Another well-known example of a scalar field theory with kink solutions is the [[Sine-Gordon|sine-Gordon theory]].<br />
<br />
===Complex scalar field theory===<br />
<br />
In a complex scalar field theory, the scalar field takes values in the complex numbers,<br />
rather than the real numbers. The action considered normally takes the form<br />
:<math>\mathcal{S}=\int \mathrm{d}^{D-1}x \, \mathrm{d}t<br />
\mathcal{L} = \int \mathrm{d}^{D-1}x \, \mathrm{d}t \left[\eta^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi<br />
-V(|\phi|^2)\right]</math><br />
<br />
This has a [[U(1)]] symmetry, whose action on the space of fields rotates <math>\phi\rightarrow e^{i\alpha}\phi</math>, for some real phase angle <math>\alpha</math>.<br />
<br />
As for the real scalar field, spontaneous symmetry breaking is found if ''m''<sup>2</sup> is negative. This gives rise to a [[Mexican hat potential]] which is analogous to the double-well potential in real scalar<br />
field theory, but now the choice of vacuum breaks a continuous ''U''(1) symmetry instead of a discrete one.<br />
This leads to a [[Goldstone boson]].<br />
<br />
===''O''(''N'') theory===<br />
<br />
One can express the complex scalar field theory in terms of two real fields, <math>\phi^1=Re{\phi}</math> and <math>\phi^2=Im{\phi}</math> which transform in the vector representation of the <math>U(1)=O(2)</math> internal symmetry. Although such fields transform as a vector under the internal symmetry, they are still Lorentz scalars. This can be generalised to a theory of N scalar fields transforming in the vector representation of the [[Orthogonal group|''O''(''N'')]] symmetry. The Lagrangian for an ''O''(''N'')-invariant scalar field theory is typically of the form<br />
<br />
:<math>\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi\cdot\partial_\nu\phi -V(\phi\cdot\phi)</math><br />
<br />
using an appropriate <math>O(N)</math>-invariant [[inner product]].<br />
<br />
==Quantum scalar field theory==<br />
<br />
In [[quantum field theory]], the fields, and all observables constructed from them, are replaced by quantum operators on a [[Hilbert space]]. This Hilbert space is built on a [[vacuum state]], and dynamics are governed by a [[Hamiltonian (quantum mechanics)|Hamiltonian]], a positive operator which annihilates the vacuum. A construction of a quantum scalar field theory may be found in the [[canonical quantization]] article, which uses canonical commutation relations among the fields as a basis for the construction. In brief, the basic variables are the field φ and its canonical momentum π. Both fields are [[Hermitian operator|Hermitian]]. At spatial points <math>\vec{x}, \vec{y}</math> at equal times, the [[canonical commutation relations]] are given by<br />
<br />
:<math>[\phi(\vec{x}),\phi(\vec{y})]=[\pi(\vec{x}),\pi(\vec{y})]=0, </math><br />
<br />
:<math> [\phi(\vec{x}),\pi(\vec{y})]=i \delta(\vec{x}-\vec{y}),</math><br />
<br />
and the free [[Hamiltonian (quantum theory)|Hamiltonian]] is<br />
<br />
:<math>H=\int d^3x \left[{1\over 2}\pi^2+{1\over 2}(\nabla \phi)^2+{m^2\over 2}\phi^2\right].</math><br />
<br />
A spatial [[Fourier transform]] leads to [[momentum space]] fields<br />
<br />
:<math>\tilde{\phi}(\vec{k})=\int d^3x e^{-i\vec{k}\cdot\vec{x}}\phi(\vec{x}), </math><br />
<br />
:<math>\tilde{\pi}(\vec{k})=\int d^3x e^{-i\vec{k}\cdot\vec{x}}\pi(\vec{x})</math><br />
<br />
which are used to define annihilation and creation operators<br />
<br />
:<math>a(\vec{k})=\left(E\tilde{\phi}(\vec{k})+i\tilde{\pi}(\vec{k})\right),</math><br />
<br />
:<math> a^\dagger(\vec{k})=\left(E\tilde{\phi}(\vec{k})-i\tilde{\pi}(\vec{k})\right),</math><br />
<br />
where <math>E=\sqrt{k^2+m^2}</math>. These operators satisfy the commutation relations<br />
<br />
:<math>[a(\vec{k}_1),a(\vec{k}_2)]=[a^\dagger(\vec{k}_1),a^\dagger(\vec{k}_2)]=0,</math><br />
<br />
:<math>[a(\vec{k}_1),a^\dagger(\vec{k}_2)]=(2\pi)^3 2E \delta(\vec{k}_1-\vec{k}_2).</math><br />
<br />
The state <math>| 0\rangle</math> annihilated by all of the operators ''a'' is identified as the ''bare vacuum'', and a particle with momentum <math>\vec{k}</math> is created by applying <math>a^\dagger(\vec{k})</math> to the vacuum. Applying all possible combinations of creation operators to the vacuum constructs the Hilbert space. This construction is called [[Fock space]]. The vacuum is annihilated by the Hamiltonian<br />
:<math>H=\int {d^3k\over (2\pi)^3}\frac{1}{2} a^\dagger(\vec{k}) a(\vec{k}) , </math><br />
<br />
where the [[zero-point energy]] has been removed by [[Wick ordering]]. (See [[canonical quantization]].)<br />
<br />
Interactions can be included by adding an interaction Hamiltonian. For a ''φ''<sup>4</sup> theory, this corresponds to adding a Wick ordered term ''g'':''φ''<sup>4</sup>:/4! to the Hamiltonian, and integrating over ''x''. Scattering amplitudes may be calculated from this Hamiltonian in the [[interaction picture]]. These are constructed in [[Perturbation theory (quantum mechanics)|perturbation theory]] by means of the [[Dyson series]], which gives the time-ordered products, or ''n''-particle Green's functions <math>\langle 0|\mathcal{T}\{{\phi}(x_1)\cdots {\phi}(x_n)\}|0\rangle</math> as described in the [[Dyson series]] article. The Green's functions may also be obtained from a generating function that is constructed as a solution to the [[Schwinger-Dyson equation]].<br />
<br />
===Feynman Path Integral===<br />
The [[Feynman diagram]] expansion may be obtained also from the Feynman [[path integral formulation]].<ref>A general reference for this section is {{cite book|last=Ramond|first=Pierre|title=Field Theory: A Modern Primer (Second Edition)|publisher=Westview Press|location=USA|date=2001-12-21|isbn=0-201-30450-3}}</ref> The [[time ordered]] [[vacuum expectation value]]s of polynomials in φ, known as the ''n''-particle Green's functions, are constructed by integrating over all possible fields, normalized by the [[vacuum expectation value]] with no external fields,<br />
<br />
:<math>\langle 0|\mathcal{T}\{{\phi}(x_1)\cdots {\phi}(x_n)\}|0\rangle=\frac{\int \mathcal{D}\phi \phi(x_1)\cdots \phi(x_n) e^{i\int d^4x \left({1\over 2}\partial^\mu \phi \partial_\mu \phi -{m^2 \over 2}\phi^2-{g\over 4!}\phi^4\right)}}{\int \mathcal{D}\phi e^{i\int d^4x \left({1\over 2}\partial^\mu \phi \partial_\mu \phi -{m^2 \over 2}\phi^2-{g\over 4!}\phi^4\right)}}.</math><br />
<br />
All of these Green's functions may be obtained by expanding the exponential in ''J''(''x'')φ(''x'') in the generating function<br />
:<math>Z[J] =\int \mathcal{D}\phi e^{i\int d^4x \left({1\over 2}\partial^\mu \phi \partial_\mu \phi -{m^2 \over 2}\phi^2-{g\over 4!}\phi^4+J\phi\right)} = Z[0] \sum_{n=0}^{\infty} \frac{i^n J(x_1) \cdots J(x_n)}{n!} \langle 0|\mathcal{T}\{{\phi}(x_1)\cdots {\phi}(x_n)\}|0\rangle.</math><br />
<br />
A [[Wick rotation]] may be applied to make time imaginary. Changing the signature to (++++) then turns the Feynman integral into a [[partition function (statistical mechanics)|statistical mechanics partition function]] in [[Euclidean space]],<br />
<br />
:<math>Z[J]=\int \mathcal{D}\phi e^{-\int d^4x \left({1\over 2}(\nabla\phi)^2+{m^2 \over 2}\phi^2+{g\over 4!}\phi^4+J\phi\right)}.</math><br />
<br />
Normally, this is applied to the scattering of particles with fixed momenta, in which case, a [[Fourier transform]] is useful, giving instead<br />
:<math>\tilde{Z}[\tilde{J}]=\int \mathcal{D}\tilde\phi e^{-\int d^4p \left({1\over 2}(p^2+m^2)\tilde\phi^2+{\lambda\over 4!}\tilde\phi^4-\tilde{J}\tilde\phi\right)}.</math><br />
<br />
The standard trick to evaluate this [[functional integral]] is to write it as a product of exponential factors, schematically,<br />
:<math>\tilde{Z}[\tilde{J}]\sim\int \mathcal{D}\tilde\phi \prod_p \left[e^{-(p^2+m^2)\tilde\phi^2/2} e^{-g\tilde\phi^4/4!} e^{\tilde{J}\tilde\phi}\right].</math><br />
The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically. The integral with λ = 0 can be treated as a product of infinitely many elementary Gaussian integrals, and the result may be expressed as a sum of [[Feynman diagrams]], calculated using the following Feynman rules:<br />
<br />
* Each field <math>\tilde{\phi}(p)</math> in the ''n''-point Euclidean Green's function is represented by an external line (half-edge) in the graph, and associated with momentum ''p''.<br />
* Each vertex is represented by a factor ''-g''.<br />
* At a given order ''g''<sup>''k''</sup>, all diagrams with ''n'' external lines and ''k'' vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a propagator 1/(''q''<sup>2</sup> + ''m''<sup>2</sup>), where ''q'' is the momentum flowing through that line.<br />
* Any unconstrained momenta are integrated over all values.<br />
* The result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity.<br />
* Do not include graphs containing "vacuum bubbles", connected subgraphs with no external lines.<br />
<br />
The last rule takes into account the effect of dividing by <math>\tilde{Z}[0]</math>. The Minkowski-space Feynman rules are similar, except that each vertex is represented by ''-ig'', while each internal line is represented by a propagator ''i''/(''q''<sup>2</sup>-''m''<sup>2</sup> + ''i''ε), where the ''ε'' term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.<br />
<br />
===Renormalization===<br />
<br />
The integrals over unconstrained momenta, called "loop integrals", in the Feynman graphs typically diverge. This is normally handled by [[renormalization]], which is a procedure of adding divergent counter-terms to the Lagrangian in such a way that the diagrams constructed from the original Lagrangian and counter-terms is finite.<ref>See the previous reference, or for more detail, {{cite book|last1=Itzykson|first1=Zuber|last2=Zuber|first2=Jean-Bernard|title=Quantum Field Theory|publisher=Dover|date=2006-02-24|isbn=0-07-032071-3}}</ref> A renormalization scale must be introduced in the process, and the coupling constant and mass become dependent upon it.<br />
<br />
The dependence of a coupling constant ''g'' on the scale λ is encoded by<br />
a [[Beta function (physics)|beta function]], β(''g''), defined by the relation<br />
<br />
:<math>\beta(g) = \lambda\,\frac{\partial g}{\partial \lambda}</math><br />
<br />
This dependence on the energy scale is known as the running of the coupling parameter, and theory of this kind of scale-dependence in quantum field theory is described by the [[renormalization group]].<br />
<br />
Beta-functions are usually computed in an approximation scheme, most commonly [[Perturbation theory (quantum mechanics)|perturbation theory]], where one assumes that the coupling constant is small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher [[Feynman graph|loop]] contributions, due to the number of loops in the corresponding [[Feynman graph]]s).<br />
<br />
The beta-function at one loop (the first perturbative contribution) for the <math>\phi^4</math> theory is<br />
<br />
:<math>\beta(g)=\frac{3}{16\pi^2}g^2+O(g^3)</math><br />
<br />
The fact that the sign in front of the lowest-order term is positive suggests that the coupling constant increases with energy. If this behavior persists at large couplings, this would indicate the presence of a [[Landau pole]] at finite energy, or [[quantum triviality]]. The question can only be answered non-perturbatively, since it involves strong coupling.<br />
<br />
A quantum field theory is trivial when the running coupling, computed through its [[Beta function (physics)|beta function]], goes to zero when the cutoff is removed. Consequently, the [[propagator]] becomes that of a free particle and the field is no longer interacting. Alternatively, the field theory may be interpreted as an effective theory, in which the cutoff is not removed, giving finite interactions but leading to a [[Landau pole]] at some energy scale. For a φ<sup>4</sup> interaction, [[Michael Aizenman]] proved that the theory is indeed trivial for space-time dimension <math>D\ge 5</math>.<ref name="Aiz81"><br />
{{cite journal<br />
|last=Aizenman|first=M. |authorlink=Michael Aizenman<br />
|year=1981<br />
|title=Proof of the Triviality of ϕ{{su|b=d|p=4}} Field Theory and Some Mean-Field Features of Ising Models for d>4<br />
|journal=[[Physical Review Letters]]<br />
|volume=47<br />
|issue=1 |pages=1–4 |doi=10.1103/PhysRevLett.47.1 |bibcode=1981PhRvL..47....1A<br />
}}</ref> For <math>D=4</math> the triviality has yet to be proven rigorously, but lattice computations have confirmed this. (See [[Landau pole]] for details and references.) This fact is relevant as the [[Higgs mechanism|Higgs field]], for which triviality bounds are used to set limits on the Higgs mass, based on the new physics must enter at a higher scale (perhaps the [[Planck scale]]) to prevent the Landau pole from being reached.<br />
<br />
==See also==<br />
<br />
*[[Renormalization]]<br />
*[[Quantum triviality]]<br />
*[[Landau pole]]<br />
<br />
==References==<br />
{{Reflist}}<br />
<br />
==Further reading==<br />
* Peskin, M and Schroeder, D. ;''An Introduction to Quantum Field Theory,'' Westview Press (1995)<br />
* Weinberg, Steven ; ''The Quantum Theory of Fields,'' (3 volumes) Cambridge University Press (1995)<br />
* Srednicki, Mark; ''Quantum Field Theory,'' Cambridge University Press (2007)<br />
* Zinn-Justin, Jean ; ''Quantum Field Theory and Critical Phenomena,'' Oxford University Press (2002)<br />
<br />
==External links==<br />
*[http://www.quantumfieldtheory.info Pedagogic Aides to Quantum Field Theory] Click on the link for Chap. 3 to find an extensive, simplified introduction to scalars in relativistic quantum mechanics and quantum field theory.<br />
* 't Hooft, G., "The Conceptual Basis of Quantum Field Theory" ([http://www.phys.uu.nl/~thooft/lectures/basisqft.pdf ''online version'']).<br />
<br />
{{DEFAULTSORT:Scalar Field Theory}}<br />
[[Category:Quantum field theory]]<br />
<br />
[[ko:사승 상호작용]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Symmetric_monoidal_category&diff=12644
Symmetric monoidal category
2013-11-19T17:44:45Z
<p>99.153.64.179: fix ref</p>
<hr />
<div>A '''lead–lag compensator''' is a component in a [[control system]] that improves an undesirable [[frequency response]] in a feedback and [[control system]]. It is a fundamental building block in classical [[control theory]].<br />
<br />
== Applications ==<br />
Lead–lag compensators influence disciplines as varied as [[robotics]],<br />
[[satellite]] control, automobile diagnostics, [[laser]] frequency stabilization, and many more.<br />
They are an important building block in analog control systems, and<br />
can also be used in digital control.<br />
<br />
Given the control plant, desired specifications can be achieved using compensators. I, D, [[PI controller|PI]], [[PD controller|PD]], and [[PID controller|PID]], are optimizing controllers which are used to improve system parameters (such as reducing steady state error, reducing resonant peak, improving system response by reducing rise time). All these operations can be done by compensators as well.<br />
<br />
== Theory ==<br />
Both lead compensators and lag compensators introduce a [[Pole (complex analysis)|pole]]–[[Zero (complex analysis)|zero]] pair into the open loop [[transfer function]]. The transfer function can be written in the Laplace domain as<br />
<br />
:<math>\frac{Y}{X} = \frac{s+z}{s+p} </math><br />
<br />
where ''X'' is the input to the compensator, ''Y'' is the output, ''s'' is the complex [[Laplace transform]] variable, ''z'' is the zero frequency and ''p'' is the pole frequency. The pole and zero are both typically negative. In a lead compensator, the pole is left of the zero in the [[complex plane]], <math>|z| < |p|</math>,<br />
while in a lag compensator <math> |z| > |p| </math>.<br />
<br />
A lead-lag compensator consists of a lead compensator cascaded with a lag compensator. The overall transfer function can be written as<br />
<br />
:<math> \frac{Y}{X} = \frac{(s+z_1)(s+z_2)}{(s+p_1)(s+p_2)}. </math><br />
<br />
Typically <math> |p_1| > |z_1| > |z_2| > |p_2| </math>, where ''z''<sub>1</sub> and ''p''<sub>1</sub> are the zero and pole of the lead compensator and ''z''<sub>2</sub> and ''p''<sub>2</sub> are the zero and pole of the lag compensator. The lead compensator provides phase lead at high frequencies. This shifts the poles to the left, which enhances the responsiveness and stability of the system. The lag compensator provides phase lag at low frequencies which reduces<br />
the steady state error.<br />
<br />
The precise locations of the poles and zeros depend on both the desired characteristics of the closed loop response and the characteristics of the system being controlled. However, the pole and zero of the lag compensator should be close together so as not to cause the poles to shift right, which could cause instability or slow convergence.<br />
Since their purpose is to affect the low frequency behaviour, they should be near the origin.<br />
<br />
== Implementation ==<br />
Both analog and digital control systems use lead-lag compensators. The technology used for the implementation is different in each case, but the underlying principles are the same. The transfer function is rearranged so that the output is expressed in terms of sums of terms involving the input, and integrals of the input and output. For example,<br />
<br />
:<math><br />
Y = X - (z_1 + z_2) \frac{X}{s} + z_1 z_2 \frac{X}{s^2}+ (p_1+p_2)\frac{Y}{s} - p_1 p_2 \frac{Y}{s^2}.<br />
</math><br />
<br />
In analog control systems, where integrators are expensive, it is common to group terms<br />
together to minimize the number of integrators required:<br />
<br />
:<math><br />
Y = X + \frac{1}{s}\left((p_1+p_2)Y - (z_1+z_2)X<br />
+ \frac{1}{s}(z_1 z_2 X - p_1 p_2 Y)\right).<br />
</math><br />
<br />
In analog control, the control signal is typically an electrical [[voltage]] or [[electric current|current]]<br />
(although other signals such as [[hydraulic]] pressure can be used).<br />
In this case a lead-lag compensator will consist of <br />
a network of [[operational amplifier]]s ("op-amps") connected as [[Operational amplifier applications#Integrator|integrators]] and<br />
[[Operational amplifier applications#Summing amplifier|weighted adders]]. In digital control, the operations are performed numerically.<br />
<br />
The reason for expressing the transfer function as an [[integral equation]] is that<br />
differentiating signals amplify the [[noise]] on the signal, since even very small<br />
amplitude noise has a high derivative if its frequency is high, while integrating a<br />
signal averages out the noise. This makes implementations in terms of integrators<br />
the most numerically stable.<br />
<br />
== Intuitive explanation ==<br />
<br />
To begin designing a lead-lag compensator, an engineer must consider whether the system<br />
needing correction can be classified as a lead-network, a lag-network, or a combination<br />
of the two: a lead-lag network (hence the name "lead-lag compensator"). The electrical<br />
response of this network to an input signal is expressed by the network's [[Laplace transform|Laplace-domain]]<br />
transfer function, a [[Complex number|complex]] mathematical function which itself can be expressed as one<br />
of two ways: as the current-gain ratio transfer function or as the voltage-gain ratio<br />
transfer function. Remember that a complex function can be in general written as<br />
<math>F(x) = A(x) + i B(x)</math>, where <math>A(x)</math> is the "Real Part" and <math>B(x)</math> is the "Imaginary Part" of<br />
the single-variable function, <math>F(x)</math>.<br />
<br />
The "phase angle" of the network is the [[Complex number#Complex plane|argument]] of <math>F(x)</math>; in the left half plane this is <math>\tan^{-1}(B(x)/A(x))</math>. If the phase angle<br />
is negative for all signal frequencies in the network then the network is classified<br />
as a "lag network". If the phase angle is positive for all signal frequencies<br />
in the network then the network is classified as a "lead network". If the total network<br />
phase angle has a combination of positive and negative phase as a function of frequency<br />
then it is a "lead-lag network".<br />
<br />
Depending upon the nominal operation design parameters of a system under an active<br />
feedback control, a lag or lead network can cause [[Stability theory|instability]] and poor speed and<br />
response times.<br />
<br />
==See also==<br />
<br />
* [[Proportional control]]<br />
* [[PID controller]]<br />
<br />
==References==<br />
#Nise, Norman S. (2004); ''Control Systems Engineering'' (4 ed.); Wiley & Sons; ISBN 0-471-44577-0<br />
#Horowitz, P. & Hill, W. (2001); ''The Art of Electronics'' (2 ed.); Cambridge University Press; ISBN 0-521-37095-7<br />
#Cathey, J.J. (1988); ''Electronic Devices and Circuits (Schaum's Outlines Series)''; McGraw-Hill ISBN 0-07-010274-0<br />
<br />
==External links==<br />
* [http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Leadlag Matlab Control Tutorials: lead and lag compensators]<br />
* [http://www.sccs.swarthmore.edu/users/06/adem/engin/e58/lab6/index.php lead controller using Matlab]<br />
* [http://www.mathpages.com/home/kmath249/kmath249.htm Lead-Lag Frequency Response] at MathPages<br />
* [http://www.mathpages.com/home/kmath198/kmath198.htm Lead-Lag Algorithms] at MathPages<br />
<br />
{{DEFAULTSORT:Lead-lag compensator}}<br />
[[Category:Control theory]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Cocycle_category&diff=30155
Cocycle category
2013-11-19T05:49:48Z
<p>99.153.64.179: wikilink</p>
<hr />
<div>{| class="wikitable" align="right" style="margin-left:10px" width="320"<br />
!bgcolor=#e7dcc3 colspan=2|Stericated 24-cell honeycomb<BR>Stericated 16-cell honeycomb<br />
|-<br />
|bgcolor=#ffffff align=center colspan=2|(No image)<br />
|-<br />
|bgcolor=#e7dcc3|Type||[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]<br />
|-<br />
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t04{3,4,3,3}<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node|4|node|3|node_1}}<br />
|-<br />
|bgcolor=#e7dcc3|4-face type||[[24-cell|{3,4,3}]] [[File:Schlegel_wireframe_24-cell.png|40px]]<BR>[[16-cell|{3,3,4}]] [[File:Schlegel_wireframe_16-cell.png|40px]]<BR>[[octahedral prism|{3,4}x{}]] [[File:Octahedral prism.png|40px]]<BR>[[tetrahedral prism|{3,3}x{}]] [[File:Tetrahedral prism.png|40px]]<BR>[[duoprism|{3}x{3}]] [[File:3-3 duoprism.png|40px]]<br />
|-<br />
|bgcolor=#e7dcc3|Cell type||[[octahedron|{3,4}]]<BR>[[tetrahedron|{3,3}]]<BR>[[triangular prism|{3}x{}]]<br />
|-<br />
|bgcolor=#e7dcc3|Face type||{3}, {4}<br />
|-<br />
|bgcolor=#e7dcc3|[[Vertex figure]]||<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{F}}_4</math>, [3,4,3,3]<br />
|-<br />
|bgcolor=#e7dcc3|Properties||[[Vertex transitive]]<br />
|}<br />
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''stericated 24-cell honeycomb''' (or '''stericated 16-cell honeycomb''') is a uniform space-filling [[honeycomb (geometry)|honeycomb]]. It can be seen as a [[Sterication (geometry)|sterication]] of the regular [[24-cell honeycomb]], containing [[24-cell]], [[16-cell]], [[octahedral prism]], [[tetrahedral prism]], and 3-3 [[duoprism]] cells. <br />
<br />
== Alternate names==<br />
* Stericated icositetrachoric/hexadecaachoric tetracomb/honeycomb<br />
* Small cellated demitesseractic tetracomb (scicot)<br />
<br />
== Related honeycombs==<br />
{{F4 honeycombs}}<br />
<br />
== See also ==<br />
Regular and uniform honeycombs in 4-space:<br />
*[[Tesseractic honeycomb]]<br />
*[[16-cell honeycomb]]<br />
*[[24-cell honeycomb]]<br />
*[[Rectified 24-cell honeycomb]]<br />
*[[Snub 24-cell honeycomb]]<br />
*[[5-cell honeycomb]]<br />
*[[Truncated 5-cell honeycomb]]<br />
*[[Omnitruncated 5-cell honeycomb]]<br />
<br />
== References ==<br />
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.&nbsp;296, Table II: Regular honeycombs<br />
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]<br />
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]<br />
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' Model 121 (Wrongly named ''runcinated'' icositetrachoric honeycomb)<br />
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3o4o3x - scicot - O121<br />
<br />
{{Honeycombs}}<br />
[[Category:5-polytopes]]<br />
[[Category:Honeycombs (geometry)]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Dagger_symmetric_monoidal_category&diff=15116
Dagger symmetric monoidal category
2013-11-19T02:53:50Z
<p>99.153.64.179: /* Examples */ ... are dagger compact</p>
<hr />
<div>{{refimprove|date=June 2013}}<br />
{{no footnotes|date=June 2013}}<br />
[[File:DirectSolarRadiationNiamey26December2006.png|thumb|Direct solar radiation, at the various wavelengths indicated in [[nanometers]], as measured at Niamey Niger on 24 December 2006, with a Multi-Filter Rotating Shadowband Radiometer (MFRSR). Measurements are plotted as a function of time in UTC.]]<br />
'''Langley extrapolation''' is a method for measuring the Sun's [[radiance]] with ground-based instrumentation,<br />
thereby removing the effect of the [[atmosphere]]. It is based on repeated measurements with a [[sun photometer]] operated at a given location for a cloudless morning or afternoon, as the [[Sun]] moves across the [[sky]]. It is named for American astronomer and physicist [[Samuel Pierpont Langley]].<br />
<br />
It is known from [[Beer's law]] that, for every instantaneous measurement, the ''direct-Sun radiance'' ''I'' is linked to the ''solar extraterrestrial radiance'' ''I''<sub>0</sub> and the atmospheric [[optical depth]] &tau; by the following equation:<br />
<br />
:<math>I / I_0 = e^{-m \tau},\, </math><br />
<br />
where ''m'' is a geometrical factor accounting for the slant path through the atmosphere, known as the [[airmass|airmass factor]].<br />
For a plane-parallel atmosphere, the airmass factor is simple to determine if one knows the solar [[celestial coordinate system | zenith angle]] &theta;: ''m'' = 1/cos(&theta;). As time passes, the Sun moves across the sky, and therefore <math>\theta</math> and <math>m</math> vary according to known [[astronomy | astronomical]] laws.<br />
<br />
[[File:LangleyPlotNiamey26December2006.png|thumb|Direct solar radiation as a function of secant of solar zenith angle at Niamey, Niger. December 24, 2006. From ARM data, from an MFRSR instrument. Wavelength in units of nanometers is indicated. Log is base 10.]]<br />
By taking the logarithm of the above equation, one obtains<br />
<br />
:<math>\ln I = \ln I_0 - m \tau,</math><br />
<br />
and if one assumes that the atmospheric disturbance &tau; does not change during the observations (which last for a morning or an afternoon), the plot of ln&nbsp;''I'' versus ''m'' is a straight line. Then, by [[extrapolation | linear extrapolation]] to ''m'' = 0, one obtains ''I''<sub>0</sub>, i.e. the Sun's radiance that would be observed by an instrument placed above the atmosphere.<br />
<br />
[[File:LangleyExtraoplationToTopOfAtmosphere.png|thumb|Points are Langley extrapolation to top of atmosphere of direct solar radiation measured at Niamey, Niger 24 December 2006. Compared with Planck functions with the wavelength in [[micrometers]].]]<br />
The requirement for good ''Langley plots'' is a constant atmosphere (constant &tau;). This requirement can be fulfilled only under particular conditions, since the atmosphere is continuously changing. Needed conditions are in particular: the absence of clouds along the optical path, and the absence of variations in the [[particulate | atmospheric aerosol]] layer. Since aerosols tend to be more concentrated at low altitude, Langley extrapolation is often performed at high mountain sites. Data from [[NASA]] [[Glenn Research Center]] indicates that the Langley plot accuracy is improved if the data is taken above the [[tropopause]].<br />
<br />
==Solar cell calibration==<br />
<br />
A Langley plot can also be used as a method to calculate the performance of [[solar cell]]s outside the Earth's atmosphere. At the [[Glenn Research Center]], the performance of solar cells is measured as a function of altitude. By extrapolation, researchers determine their performance under space conditions [http://rt.grc.nasa.gov/power-in-space-propulsion/photovoltaics-power-technologies/technology-thrusts/solar-cell-measurement-and-calibration/].<br />
<br />
==References==<br />
<br />
* Glenn E. Shaw, "Sun photometry", ''Bulletin of the American Meteorological Society'' '''64''', 4-10, 1983.<br />
[[Category:Radiometry]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=No-broadcast_theorem&diff=14721
No-broadcast theorem
2013-11-18T20:20:36Z
<p>99.153.64.179: argh another typo</p>
<hr />
<div>In [[metallurgy]], the '''Scheil-Gulliver equation''' (or '''Scheil equation''') describes [[solution|solute]] redistribution during [[solidification]] of an [[alloy]].<ref>{{cite journal|last=Xiong|first=M.|coauthors=Kuznetsov, A.V.|journal=Flow, Turbulence and Combustion|date=1 January 2001|volume=67|issue=4|pages=305–323|doi=10.1023/A:1015291706970}}</ref> <br />
<br />
== Assumptions ==<br />
Four key assumptions in Scheil analysis enable determination of phases present in a cast part. These assumptions are:<br />
<br />
# No diffusion occurs in solid phases once they are formed (<math>\ D_S = 0 </math>)<br />
# Infinitely fast diffusion occurs in the liquid at all temperatures by virtue of a high diffusion coefficient, thermal [[convection]], [[Marangoni effect|Marangoni convection]], etc. (<math>\ D_L = \infty</math>)<br />
# [[Thermodynamic equilibrium|Equilibrium]] exists at the solid-liquid interface, and so compositions from the phase diagram are valid<br />
# Solidus and liquidus lines are straight segments<br />
<br />
It should be noted that the fourth condition (straight solidus/liquidus segments) may be relaxed when numerical techniques are used, such as those used in [[CALPHAD]] software packages, though these calculations rely on calculated equilibrium phase diagrams. Calculated diagrams may include odd artifacts (i.e. retrograde solubility) that influence Scheil calculations.<br />
<br />
== Derivation ==<br />
<br />
[[Image:Scheil solidification.svg]]<br />
<br />
The hatched areas in the figure represent the amount of solute in the solid and liquid. Considering that the total amount of solute in the system must be conserved, the areas are set equal as follows:<br />
<br />
:<math>(C_L-C_S) \ df_S = (f_L) \ dC_L</math>.<br />
<br />
Since the [[partition coefficient]] (related to solute distribution) is <br />
<br />
:<math>k = \frac{C_S}{C_L}</math> (determined from the phase diagram)<br />
<br />
and mass must be [[Conservation of mass|conserved]]<br />
<br />
:<math>\ f_S + f_L = 1</math><br />
<br />
the mass balance may be rewritten as<br />
<br />
:<math>C_L(1-k) \ df_S = (1-f_S) \ dC_L</math>.<br />
<br />
Using the boundary condition<br />
<br />
:<math>\ C_L = C_o </math> at <math>\ f_S = 0</math><br />
<br />
the following integration may be performed:<br />
<br />
:<math>\displaystyle\int^{f_S}_0 \frac{df_S}{1-f_S} = \frac{1}{1-k} \displaystyle\int^{C_L}_{C_o} \frac{dC_L}{C_L}</math>.<br />
<br />
Integrating results in the Scheil-Gulliver equation for composition of the liquid during solidification:<br />
<br />
:<math>\ C_L = C_o(f_L)^{k - 1}</math><br />
<br />
or for the composition of the solid:<br />
<br />
:<math>\ C_S = kC_o(1-f_S)^{k - 1}</math>.<br />
<br />
== References ==<br />
<references /><br />
*Gulliver, G.H., ''J. Inst. Met.'', 9:120, 1913.<br />
*Kou, S., ''Welding Metallurgy'', 2nd Edition, Wiley -Interscience, 2003.<br />
*Porter, D. A., and Easterling, K. E., ''Phase Transformations in Metals and Alloys'' (2nd Edition), Chapman & Hall, 1992.<br />
*Scheil, E., ''Z. Metallk.'', 34:70, 1942.<br />
* Karl B. Rundman Principles of Metal Casting Textbook - Michigan Technological University<br />
* H. Fredriksson, Y. Akerlind, Materials Processing during Casting, Chapter 7, Wiley:Hoboken, 2006.<br />
[[Category:Metallurgy]]<br />
[[Category:Differential equations]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Separable_state&diff=11173
Separable state
2013-11-17T20:14:24Z
<p>99.153.64.179: /* Separable pure states */ reorder sequence of sentences</p>
<hr />
<div>The '''Constructive Systems Engineering Cost Model (COSYSMO)''' was created by Ricardo Valerdi while at the [[University of Southern California]] Center for Software Engineering. It gives an estimate of the number of person-months it will take to staff systems engineering resources on hardware and software projects. Initially developed in 2002, the model now contains a calibration data set of more than 50 projects provided by major aerospace and defense companies such as [[Raytheon]], [[Northrop Grumman]], [[Lockheed Martin]], [[Science Applications International Corporation|SAIC]], [[General Dynamics]], and [[BAE Systems]].<br />
<br />
Similar to its predecessor [[COCOMO]], COSYSMO computes effort (and cost) as a function of system functional size and adjusts it based on a number of environmental factors related to systems engineering.<br />
<br />
COSYSMO's central cost estimating relationship, or CER is of the form:<br />
<math>PM_{NS}=A \times Size^E \times \prod^n_{i=1} EM_i</math><br />
<br />
where "Size" is one of four size additive size drivers, and EM represents one of fourteen multiplicative effort multipliers.<br />
<br />
==See also==<br />
* [[Comparison of development estimation software]]<br />
* [[Software development effort estimation]]<br />
<br />
==Further reading==<br />
* Valerdi, R., Boehm, B., Reifer, D., COSYSMO: A Constructive Systems Engineering Cost Model Coming Age, 13th INCOSE Symposium, July 2003, Crystal City, VA.<br />
* Valerdi, R., The Constructive Systems Engineering Cost Estimation Model (COSYSMO), University of Southern California, May 2005.<br />
* Valerdi, R., The Constructive Systems Engineering Cost Model (COSYSMO): Quantifying the Costs of Systems Engineering Effort in Complex Systems, [[VDM Verlag]], 2008.<br />
<br />
==External links==<br />
* [http://cosysmo.mit.edu COSYSMO]<br />
* [http://www.softstarsystems.com/COSYSMO.htm SystemStar Tool]<br />
* [http://www.pricesystems.com/news/INSight/INSightMar2006.html True COSYSMO]<br />
{{comp-sci-stub}}<br />
<br />
[[Category:Software engineering costs]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=De_Sitter_space&diff=3797
De Sitter space
2013-11-03T04:37:12Z
<p>99.153.64.179: /* See also */ fix link</p>
<hr />
<div>[[File:Ohm's Law with Voltage source TeX.svg|right|thumb|Representation of a lumped model made up of a voltage source and a resistor.]]<br />
<br />
The '''lumped element model''' (also called '''lumped parameter model''', or '''lumped component model''') simplifies the description of the behaviour of spatially distributed physical systems into a [[topology]] consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in [[electrical network|electrical systems]] (including [[electronics]]), mechanical [[multibody system]]s, [[heat transfer]], [[acoustics]], etc.<br />
<br />
Mathematically speaking, the simplification reduces the [[State space (controls)|state space]] of the system to a [[finite set|finite]] dimension, and the [[partial differential equation]]s (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into [[ordinary differential equation]]s (ODEs) with a finite number of parameters.<br />
<br />
== Examples ==<br />
<br />
=== Lumped element model in electrical systems ===<br />
The lumped element model of electronic [[Electrical network|circuit]]s makes the simplifying assumption that the attributes of the circuit, [[Electrical resistance|resistance]], [[capacitance]], [[inductance]], and [[gain]], are concentrated into idealized [[electrical component]]s; [[resistor]]s, [[capacitor]]s, and [[inductor]]s, etc. joined by a network of perfectly [[Electrical conduction|conducting]] wires.<br />
<br />
The lumped element model is valid whenever <math>L_c \ll \lambda</math>, where <math>L_c</math> denotes the circuit's characteristic length, and <math>\lambda</math> denotes the circuit's operating [[wavelength]].<br />
Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the [[distributed element model]] (including [[transmission line]]s), whose dynamic behaviour is described by the [[Maxwell's equations]]. Another way of viewing the validity of the lumped element model is to note that this model ignores the finite time it takes signals to propagate around a circuit. Whenever this propagation time is not significant to the application the lumped element model can be used. This is the case when the propagation time is much less than the [[period (physics)|period]] of the signal involved. However, with increasing propagation time there will be an increasing error between the assumed and actual phase of the signal which in turn results in an error in the assumed amplitude of the signal. The exact point at which the lumped element model can no longer be used depends to a certain extent on how accurately the signal needs to be known in a given application.<br />
<br />
Real-world components exhibit non-ideal characteristics which are, in reality, distributed elements but are often represented to a [[first-order approximation]] by lumped elements. To account for leakage in [[capacitor]]s for example, we can model the non-ideal capacitor as having a large lumped [[resistor]] connected in-parallel even though the leakage is, in reality distributed throughout the dielectric. Similarly a [[wire-wound resistor]] has significant [[inductance]] as well as [[Electrical resistance|resistance]] distributed along its length but we can model this as a lumped [[inductor]] in series with the ideal resistor.<br />
<br />
=== Lumped element model in mechanical systems ===<br />
The simplifying assumptions in this domain are:<br />
* all objects are [[rigid body|rigid bodies]];<br />
* all interactions between rigid bodies take place via [[kinematic pair]]s (''joints''), [[spring (device)|spring]]s and [[dashpot|dampers]].<br />
<br />
=== Lumped element model in acoustics ===<br />
In this context, the lumped component model extends the distributed concepts of [[Acoustic theory]] subject to approximation. In the acoustical lumped component model, certain physical components with acoustical properties may be approximated as behaving similarly to standard electronic components or simple combinations of components.<br />
<br />
*A rigid-walled cavity containing air (or similar compressible fluid) may be approximated as a [[capacitor]] whose value is proportional to the volume of the cavity. The validity of this approximation relies on the shortest wavelength of interest being significantly (much) larger than the longest dimension of the cavity.<br />
<br />
*A [[reflex port]] may be approximated as an [[inductor]] whose value is proportional to the effective length of the port divided by its cross-sectional area. The effective length is the actual length plus an [[end correction]]. This approximation relies on the shortest wavelength of interest being significantly larger than the longest dimension of the port.<br />
<br />
*Certain types of damping material can be approximated as a [[resistor]]. The value depends on the properties and dimensions of the material. The approximation relies in the wavelengths being long enough and on the properties of the material itself.<br />
<br />
*A [[loudspeaker]] drive unit (typically a [[woofer]] or [[subwoofer]] drive unit) may be approximated as a series connection of a zero-[[Electrical impedance|impedance]] [[voltage]] source, a [[resistor]], a [[capacitor]] and an [[inductor]]. The values depend on the specifications of the unit and the wavelength of interest.<br />
<br />
==See also==<br />
* [[Lumped matter discipline]]<br />
<br />
==External links==<br />
* [http://www.jat.co.kr/eda/saber/mpp.pdf Advanced modelling and simulation techniques for magnetic components]<br />
*[http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS)], the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling - '''broken link'''<br />
<br />
[[Category:Mechanics]]<br />
[[Category:Acoustics]]<br />
[[Category:Components]]<br />
[[Category:Electronic circuits]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Vakhitov%E2%80%93Kolokolov_stability_criterion&diff=25962
Vakhitov–Kolokolov stability criterion
2013-11-02T20:25:09Z
<p>99.153.64.179: wikilink</p>
<hr />
<div>[[File:Lomonosovportrait.jpg|thumb|220px|[[Mikhail Lomonosov]], Russian [[polymath]] scientist, inventor, poet and artist, the founder of [[Moscow State University]].]]<br />
This '''list of [[Russia]]n and [[Soviet Union|Soviet]] [[scientist]]s''' includes scientists associated with modern [[Russian Federation]], the [[Soviet Union]], [[Imperial Russia]], [[Russian Tsardom]], and the [[Grand Duchy of Moscow]].<br />
<br />
Regardless of ethnicity or emigration, the list includes famous natives of Russia and its predecessor states as well as the people who were born elsewhere but spent most of their active life in Russia.<br />
<br />
==Polymaths==<br />
[[File:Baer Karl von 1792-1876.jpg|100px|thumb|'''[[Karl Baer|Baer]]''']]<br />
* '''[[Karl Ernst von Baer]]''', polymath naturalist, formulated the geological [[Baer's law]] on [[river erosion]] and [[embryological]] [[Baer's laws]], founder of the [[Russian Entomological Society]], co-founder of the [[Russian Geographical Society]]<br />
* '''[[Alexander Borodin]]''', chemist and composer, author of the famous opera ''[[Prince Igor]]'', discovered [[Borodin reaction]], co-discovered [[Aldol reaction]]<br />
* '''[[Alexander Chizhevsky]]''', interdisciplinary scientist, biophysicist, philosopher and artist, founder of [[heliobiology]] and modern [[air ionification]], [[Russian cosmist]]<br />
* '''[[Johann Gottlieb Georgi]]''', naturalist, chemist, mineralogist, ethnographer and explorer, the first to describe [[omul]] fish of [[Lake Baikal|Baikal]], published the first full-scale work on ethnography of [[indigenous peoples of Russia]][[File:Levitzky Lvov 1770.jpg|thumb|95px|'''[[Nikolay Lvov|Lvov]]''']]<br />
* '''[[Mikhail Lomonosov]]''', polymath scientist, artist and inventor; founder of the [[Moscow State University]]; proposed the law of [[conservation of matter]]; disproved the [[phlogiston theory]]; invented [[coaxial rotor]] and the first [[helicopter]]; invented the [[night vision telescope]] and [[off-axis reflecting telescope]]; discovered the [[atmosphere of Venus]]; suggested the [[organic matter|organic]] origin of [[soil]], [[peat]], [[coal]], [[petroleum]] and [[amber]]; pioneered the research of [[atmospheric electricity]]; coined the term ''[[physical chemistry]]''; the first to record the [[freezing]] of [[mercury (element)|mercury]]; co-developed [[Russian porcelain]], re-discovered [[smalt]] and created a number of [[mosaic]]s dedicated to [[Peter the Great|Petrine era]]; author of an early account of [[Russian history]] and the first opponent of the [[Normanist theory]]; reformed Russian literary language by combining [[Old Church Slavonic]] with vernacular tongue in his early grammar; influenced [[Russian poetry]] through his [[ode]]s<br />
* '''[[Nikolay Lvov]]''', polymath artist, geologist, philologist and ethnographer, compiled the first major collection of [[Russian folk songs]], adapted [[rammed earth]] technology for northern climate and built the [[Priory Palace]] in [[Gatchina]], pioneered [[HVAC]] technology, invented [[carton-pierre]]<br />
* '''[[Alexander Middendorf]]''', zoologist and explorer, discoverer of [[Putorana Plateau]], founder of [[permafrost]] science, studied the influence of [[permafrost]] on living beings, coined the term ''[[radula]]'', prominent hippologist and [[horse breeder]][[File:Pallas Peter Simon 1741-1811.png|thumb|100px|'''[[Peter Simon Pallas|Pallas]]''']]<br />
* '''[[Vladimir Obruchev]]''', geologist, paleontologist, geographer and explorer of [[Siberia]] and [[Central Asia]], author of the comprehensive ''Geology of [[Siberia]]'' and two popular [[science fiction]] novels, ''[[Hollow Earth|Plutonia]]'' and ''[[Sannikov Land]]''<br />
* '''[[Peter Simon Pallas]]''', polymath naturalist, geographer, ethnographer, philologist, explorer of [[European Russia]] and [[Siberia]], discoverer of the first [[pallasite]] meteorite ([[Krasnojarsk (meteorite)|Krasnojarsk]]) and multiple animals, including the [[Pallas's cat]], [[Pallas's Squirrel]], and [[Pallas's Gull]]<br />
* '''[[Yakov Perelman]]''', a founder of [[popular science]], author of many popular books, including the ''Physics Can Be Fun'' and ''Mathematics Can Be Fun''<br />
* '''[[Nicholas Roerich]]''', artist, writer, philosopher, archeologist, explorer of [[Central Asia]], public figure, initiator of the international [[Roerich’s Pact]] on the defense of cultural objects, author of over 7000 paintings[[File:Vernadsky.jpg|thumb|95px|'''[[Vladimir Vernadsky|Vernadsky]]''']]<br />
* '''[[Pyotr Semyonov-Tyan-Shansky]]''', geographer, geologist, entomologist, explorer of the [[Tian Shan Mountains]], discoverer of the Peak [[Khan Tengri]], for 40 years the head of the [[Russian Geographical Society]], statistician, organiser of the first [[Russian Empire Census]]<br />
* '''[[Vasily Tatishchev]]''', statesman, economist, geographer, ethnographer, philologist and historian, supervisor of the first instrumental mapping of Russia, coloniser of the [[Urals]] and Siberia, founder of [[Perm]] and [[Yekaterinburg]], discovered and published ''[[Russkaya Pravda]]'', ''[[Sudebnik]]'' of 1550 and the controversial ''[[Ioachim Chronicle]]'', wrote the first full-scale account of Russian history, compiled the first [[encyclopedic dictionary]] of [[Russian language]]<br />
* '''[[Vladimir Vernadsky]]''', philosopher and geologist, a founder of [[geochemistry]], [[biogeochemistry]] and [[radiogeology]], creator of [[noosphere]] theory, popularized the term ''[[biosphere]]'', major [[Russian cosmist]]<br />
* '''[[Ivan Yefremov]]''', paleontologist, philosopher, sci-fi and historical novelist, founder of [[taphonomy]], author of ''[[The Land of Foam]] '', ''[[Andromeda: A Space-Age Tale]]'' and ''[[Thais of Athens]]''<br />
<br />
==Earth scientists==<br />
{{main|Russian Earth scientists}}<br />
[[File:Jan Czerski.jpg|thumb|95px|'''[[Ivan Chersky|Chersky]]''']]<br />
* '''[[Dmitry Anuchin]]''', anthropologist and geographer, coined the term ''[[anthroposphere]]'', determined the location of the [[Volga]] [[river source]]<br />
* '''[[Karl Baer]]''', naturalist, formulated the geological [[Baer's law]] on [[river erosion]], co-founder of the [[Russian Geographical Society]]<br />
* '''[[Lev Berg]]''', determined the depth of [[Central Asia]]n lakes, including [[Balkhash Lake|Balkhash]] and [[Issyk Kul]], a head of the [[Soviet Geographical Society]]<br />
* '''[[Leonid Brekhovskikh]]''', founder of modern [[acoustical oceanography]], discovered the [[deep sound channel]], the first to observe [[mesoscale ocean eddies]]<br />
* '''[[Ivan Chersky]]''', paleontologist, geologist and explorer of [[Siberia]], explained the origin of [[Lake Baikal]], pioneered the [[geomorphological]] evolution theory<br />
* '''[[Pyotr Chikhachyov]]''', early geographer and geologist of [[Central Asia]], discovered [[Kuznetsk Coal Basin]][[File:Dokuchaev2.jpg|thumb|90px|'''[[Dokuchaev]]''']]<br />
* '''[[Vasily Dokuchaev]]''', founder of [[soil science]], created the first [[soil classification]], determined the [[Clorpt|five factors for soil formation]]<br />
* '''[[Alexander Fersman]]''', a founder of [[geochemistry]], discovered [[copper]] in [[Monchegorsk]], [[apatite]]s in [[Khibiny]], [[sulfur]] in [[Central Asia]]<br />
* '''[[Boris Borisovich Galitzine|Boris Golitsyn]]''', inventor of electromagnetic [[seismograph]], the president of ''International Association of [[Seismology]]''<br />
* '''[[Grigory Gamburtsev]]''', major Soviet [[seismologist]], invented a number of seismological methods and devices<br />
* '''[[Ivan Gubkin]]''', founder of the [[Gubkin Russian State University of Oil and Gas]][[File:Wladimir Peter Köppen.jpg|thumb|100px|'''[[Wladimir Köppen|Köppen]]''']]<br />
* '''[[Alexander Karpinsky]]''', geologist and mineralogist, the first President of the [[Soviet Academy of Sciences]]<br />
* '''[[Alexander Keyserling]]''', naturalist, a founder of Russian [[geology]]<br />
* '''[[Maria Klenova]]''', a founder of [[marine geology]], polar explorer<br />
* '''[[Wladimir Köppen]]''', [[meteorologist]], author of the commonly used [[Köppen climate classification]]<br />
* '''[[Stepan Krasheninnikov]]''', geographer, the first Russian naturalist, made the first scientific description of [[Kamchatka]]<br />
* '''[[Alexander Kruber]]''', founder of Russian [[karstology]]<br />
* '''[[Nikolai Kudryavtsev]]''', author of modern [[abiogenic theory]] for origin of [[petroleum]], coordinated oil and gas exploration in [[Siberia]][[File:Middendorff1.jpg|thumb|95px|'''[[Alexander von Middendorff|Middendorf]]''']]<br />
* '''[[Leonid Kulik]]''', meteorite researcher, the first to study the [[Tunguska event]]<br />
* '''[[Mikhail Lomonosov]]''', polymath, suggested the [[organic matter|organic]] origin of [[soil]], [[peat]], [[coal]], [[petroleum]] and [[amber]]; forerrunner of the [[continental drift]] theory, pioneer researcher of [[atmospheric electricity]]<br />
* '''[[Alexander Middendorf]]''', zoologist and explorer, founder of [[permafrost]] science, determined the southern border of permafrost<br />
* '''[[Pavel Molchanov]]''', meteorologist, inventor of [[radiosonde]]<br />
* '''[[Ivan Mushketov]]''', made the first geological map of [[Turkestan]]<br />
* '''[[Vladimir Obruchev]]''', geologist and explorer, author of the comprehensive ''Geology of [[Siberia]]'' and two popular [[science fiction]] novels,[[File:Obruchev young.jpeg|thumb|95px|'''[[Obruchev]]''']] ''[[Hollow Earth|Plutonia]]'' and ''[[Sannikov Land]]''<br />
* '''[[Mikhail Pomortsev]]''', meteorologist, inventor of [[nephoscope]]<br />
* '''[[Farman Salmanov]]''', discoverer of giant [[oil fields]] in [[West Siberian Plain|West Siberia]]<br />
* '''[[Pyotr Semyonov-Tyan-Shansky]]''', explorer of the [[Tian Shan Mountains]], for 40 years the head of the [[Russian Geographical Society]], prominent statistician and organiser of the first [[Russian Empire Census]]<br />
* '''[[Nikolay Shatsky]]''', made a comprehensive [[tectonic]] map of [[North Eurasia]], introduced [[Riphean (stage)|Riphean]] and [[Baikalian]] geological [[Stage (stratigraphy)|stages]]<br />
* '''[[Pyotr Shirshov]]''', polar explorer, founder of the [[Shirshov Institute of Oceanology]], proved that there is life in high latitudes of the [[Arctic Ocean]]<br />
* '''[[Yuly Shokalsky]]''', the first head of the [[Soviet Geographical Society]], coined the term ''[[World Ocean]]''<br />
* '''[[Aleksey Tillo]]''', made the first correct [[wikt:hypsometry|hypsometric]] map of [[European Russia]], coined the term ''[[Central Russian Upland]]'', measured the lengths of main [[Russian rivers]]<br />
* '''[[Andrey Nikolayevich Tychonoff|Andrey Tikhonov]]''', mathematician and inventor of [[magnetotellurics]] in geology<br />
* '''[[Vladimir Vernadsky]]''', philosopher and geologist, a founder of [[geochemistry]], [[biogeochemistry]] and [[radiogeology]], creator of [[noosphere]] theory, popularized the term ''[[biosphere]]''<br />
<br />
==Biologists and paleontologists==<br />
{{main|Russian biologists}}<br />
[[File:Andrei Bolotov.jpg|thumb|90px|'''[[Andrei Bolotov|Bolotov]]''']]<br />
* "'[[M. Blagorazumova]]", original discoverer of [[LDL]] and [[High-density lipoprotein|HDL]]'s roles in atherosclerotic plaque formation<ref>M.A. Blagorazumova, "Collected Scientific Papers of the Theoretical and Clinical Departments of Volgograd Medical Institute [in Russian] (17). 1956. <br />
<br />
M . A . B l a g o r a z u m o v a , I n t h e b o o k : C o l l e c t e d S c i e n t i f i c Pape r s o f t h e T h e o r e t i c a l and C l i n i c a l De p a r t a n e n t s <br />
o f V o l g o g r a d M e d i c a l I n s t i t u t e [ i n R u s s i a n ] , p . 17~(1956)</ref><br />
* '''[[Johann Friedrich Adam]]''', discoverer of the [[Adams mammoth]], the first complete [[woolly mammoth]] skeleton<br />
* '''[[Karl Baer]]''', naturalist, founder of the [[Russian Entomological Society]], formulated [[embryological]] [[Baer's laws]]<br />
* '''[[Jacques von Bedriaga]]''', prominent [[herpetologist]], described [[Bedriaga's Rock Lizard]] and [[Bedriaga's Skink]]<br />
* '''[[Dmitry Konstantinovich Belyaev|Dmitry Belyaev]]''', [[domesticated silver fox]]<br />
* '''[[Lev Berg]]''', [[ichthyologist]] of [[Central Asia]] and [[European Russia]]<br />
* '''[[Nikolai Bernstein]]''', neurophysiologist, coined the term [[biomechanics]]<br />
* '''[[Andrey Bolotov]]''', major 18th century [[agriculturist]], discovered [[dichogamy]], pioneered [[cross-pollination]][[File:Ivanovsky.jpg|thumb|90px|'''[[Dmitry Ivanovsky|Ivanovsky]]''']]<br />
* '''[[August von Bongard]]''', [[botanist]] of [[Alaska]], discoverer of [[Sitka Spruce]] and [[Alnus rubra|Red Alder]]<br />
* '''[[Alexander Bunge]]''', major [[botanist]] of [[Siberia]] (especially [[Altai Mountains|Altai]])<br />
* '''[[Mikhail Chailakhyan]]''', researcher of [[flowering]], described the [[florigen]] hormone<br />
* '''[[Sergei Chetverikov]]''', pioneer of [[modern evolutionary synthesis]]<br />
* '''[[Alexander Chizhevsky]]''', founder of [[heliobiology]] and modern [[air ionification]]<br />
* '''[[Eduard Eversmann]]''', biologist and explorer, pioneer researcher of flora and fauna of southern Russia<br />
* '''[[Andrey Famintsyn]]''', [[plant physiologist]], inventor of [[grow lamp]], developer of [[symbiogenesis]] theory<br />
* '''[[Yuri Filipchenko]]''', entomologist, coined the terms [[microevolution]] and [[macroevolution]]<br />
* '''[[Nikolay Gamaleya]]''', [[microbiologist]] and pioneer of Russian [[vaccine]] research<br />
* '''[[Johann Georg Gmelin]]''', the first researcher of [[Siberia]]n flora[[File:Merezhkovsky K S.jpg|thumb|90px|'''[[Konstantin Merezhkovsky|Merezhkovsky]]''']]<br />
* '''[[Grigory Grumm-Grzhimaylo]]''', zoologist and geographer, obtained two [[Przewalski's Horse]]s and more than 1000 bird specimens from his travels in Central Asia<br />
* '''[[Alexander Gurwitsch]]''', originated the [[morphogenetic field]] theory and discovered the [[biophoton]]<br />
* '''[[Ilya Ivanovich Ivanov (biologist)|Ilya Ivanov]]''', researcher of [[artificial insemination]] and the [[interspecific hybrid]]ization of animals, involved in controversial attempts to create a [[human-ape hybrid]]<br />
* '''[[Dmitry Ivanovsky]]''', discoverer of [[virus]]es<br />
* '''[[Georgii Karpechenko]]''', inventor of [[rabbage]] (the first ever non-sterile [[Hybrid (biology)|hybrid]] obtained through [[crossbreeding]])<br />
* '''[[Nikolai Koltsov]]''', discoverer of [[cytoskeleton]][[File:Michurin 1936.jpg|thumb|95px|'''[[Ivan Vladimirovich Michurin|Michurin]]''']]<br />
* '''[[Vladimir Leontyevich Komarov|Vladimir Komarov]]''', [[plant geographer]], President of the [[Soviet Academy of Sciences]], founder of the [[Komarov Botanical Institute]]<br />
* '''[[Alexander Kovalevsky]]''', embryologist, major researcher of [[gastrulation]]<br />
* '''[[Trofim Lysenko]]''', agronomist, developer of [[yarovization]], infamous for [[lysenkoism]]<br />
* '''[[Evgeny Maleev]]''', discoverer of [[Talarurus]], [[Tarbosaurus]], and [[Therizinosaurus]]<br />
* '''[[Carl Maximowicz]]''', pioneer researcher of the [[Far East]]ern [[flora]]<br />
* '''[[Ilya Mechnikov]]''', pioneer researcher of [[immune system]], [[probiotics]] and [[phagocytosis]], coined the term ''[[gerontology]]'', [[Nobel Prize in Medicine]] winner<br />
* '''[[Mikhail Aleksandrovich Menzbier|Mikhail Menzbier]]''', major [[ornithologist]], discoverer of the [[Menzbier's Marmot]]<br />
* '''[[Konstantin Merezhkovsky]]''', major [[lichenologist]], developer of [[symbiogenesis]] theory, a founder of [[endosymbiosis]] theory<br />
* '''[[Ivan Vladimirovich Michurin|Ivan Michurin]]''', [[pomologist]], [[selection]]ist and [[geneticist]], practiced [[hybrid (biology)|crossing]] of geographically distant plants, created hundreds of fruit [[cultivar]]s[[File:Leopold von Schrenck.jpg|thumb|90px|'''[[Leopold von Schrenck|Schrenck]]''']]<br />
* '''[[Alexander Middendorf]]''', zoologist and explorer, studied the influence of [[permafrost]] on living beings, coined the term ''[[radula]]'', prominent [[horse breeder]]<br />
* '''[[Victor Motschulsky]]''', prominent [[coleopterologist]] (researcher of [[beetle]]s)<br />
* '''[[Sergei Navashin]]''', discovered [[double fertilization]]<br />
* '''[[Alexey Olovnikov]]''', predicted existence of [[Telomerase]], suggested the [[Telomere hypothesis of aging]] and the [[Telomere relations to cancer]]<br />
* '''[[Aleksandr Oparin]]''', biologist and [[biochemist]], proposed the famous "[[Primordial soup]]" theory of [[life origin]], showed that many [[food production]] processes are based on [[biocatalysis]]<br />
* '''[[Heinz Christian Pander]]''', embryologist, discoverer of [[germ layers]]<br />
* '''[[Peter Simon Pallas]]''', polymath naturalist and explorer, discoverer of multiple animals, including the [[Pallas's cat]], [[Pallas's Squirrel]], and [[Pallas's Gull]]<br />
* '''[[Ivan Pavlov]]''', founder of modern [[physiology]], the first to research [[classical conditioning]], Nobel Prize in Medicine winner<br />
* '''[[Vladimir Pravdich-Neminsky]]''', published the first [[EEG]] and the [[evoked potential]] of the mammalian brain<br />
* '''[[Nikolai Przhevalsky]]''', explorer and naturalist, brought vast collections from Central Asia, discovered [[Przewalski's Horse|the only extant species of wild horse]][[File:Kliment Timiryazev.jpg|thumb|85px|'''[[Kliment Timiryazev|Timiryazev]]''']]<br />
* '''[[Anatoly Konstantinovich Rozhdestvensky|Anatoly Rozhdestvensky]]''', discoverer of [[Aralosaurus]] and [[Probactrosaurus]]<br />
* '''[[Ivan Schmalhausen]]''', developer of [[modern evolutionary synthesis]]<br />
* '''[[Leopold von Schrenck]]''', ethnographer, zoologist, discovered the [[Amur sturgeon]], [[Manchurian Black Water Snake]] and [[Schrenck's Bittern]]<br />
* '''[[Boris Schwanwitsch]]''', entomologist, applied colour patterns of insect wings to [[military camouflage]] during [[World War II]]<br />
* '''[[Ivan Sechenov]]''', founder of [[electrophysiology]] and [[neurophysiology]]<br />
* '''[[Georg Wilhelm Steller]]''', naturalist, participant of [[Vitus Bering]]'s voyages, discoverer of [[Steller's Jay]], [[Steller's Eider]],[[File:Nikolai Vavilov NYWTS.jpg|thumb|100px|'''[[Nikolai Vavilov|Vavilov]]''']] extinct [[Steller's Sea Cow]] and multiple other animals<br />
* '''[[Lina Stern]]''', pioneer researcher of [[blood–brain barrier]]<br />
* '''[[Armen Takhtajan]]''', developer of [[Takhtajan system]] of [[flowering plant]] classification, major [[biogeographer]]<br />
* '''[[Kliment Timiryazev]]''', plant physiologist and evolutionist, major researcher of [[chlorophyll]]<br />
* '''[[Nikolai Timofeeff-Ressovsky]]''', major researcher of [[radiation genetics]], [[population genetics]], and [[microevolution]]<br />
* '''[[Lev Tsenkovsky]]''', pioneer researcher of the [[ontogenesis]] of [[lower plants]] and animals<br />
* '''[[Mikhail Tsvet]]''', inventor of [[chromatography]]<br />
* '''[[Nikolai Vavilov]]''', botanist and geneticist,[[File:Vinogradsky.jpg|thumb|100px|'''[[Sergei Winogradsky|Vinogradsky]]''']] gathered the world's largest collection of plant [[seed]]s, identified the [[Vavilov Center|centres of origin]] of main cultivated plants<br />
* '''[[Mikhail Stepanovich Voronin|Mikhail Voronin]]''', major researcher of [[fungi]] and [[plant pathology]]<br />
* '''[[Sergey Vinogradsky]]''', microbiologist, ecologist, and soil scientist, pioneered the [[biogeochemical cycle]] concept, discovered [[lithotrophy]] and [[chemosynthesis]], invented the [[Winogradsky column]] for breeding of [[microorganism]]s<br />
* '''[[Ivan Yefremov]]''', paleontologist, sci-fi author, founded [[taphonomy]]<br />
* '''[[Sergey Zimov]]''', creator of the [[Pleistocene Park]]<br />
<br />
==Physicians and psychologists==<br />
{{main|Russian physicians}}<br />
[[File:Bekhterev by Repin.jpg|thumb|90px|'''[[Vladimir Bekhterev|Bekhterev]]''']]<br />
* '''[[Aleksandr Bakulev]]''', prominent [[cardiovascular surgery]] developer<br />
* '''[[Vladimir Bekhterev]]''', neuropathologist, founder of [[objective psychology]], noted the role of the [[hippocampus]] in memory, major contributor to [[reflexology]], studied the [[Bekhterev’s Disease]]<br />
* '''[[Vladimir Betz]]''', discovered [[Betz cells]] of [[primary motor cortex]]<br />
* '''[[Peter Borovsky]]''', described the causative agent of [[Oriental sore]]<br />
* '''[[Sergey Botkin]]''', major [[therapist]] and court physician<br />
* '''[[Nikolay Burdenko]]''', major developer of [[neurosurgery]]<br />
* '''[[Konstantin Buteyko]]''', developed the [[Buteyko method]] for the treatment of asthma and other breathing disorders<br />
* '''[[Mikhail Chumakov]]''', co-discovered [[tick-borne encephalitis]], co-developed oral [[polio vaccine]]<br />
* '''[[Livery Darkshevich]]''', [[neurologist]], described the nucleus of [[posterior commissure]][[File:Sergey Korsakov.jpg|thumb|90px|'''[[Sergey Korsakov|Korsakov]]''']]<br />
* '''[[Vladimir Demikhov]]''', major pioneer of [[transplantology]]<br />
* '''[[Vladimir Filatov]]''', [[ophthalmologist]], [[corneal transplantation]] pioneer<br />
* '''[[Svyatoslav Fyodorov]]''', inventor of [[radial keratotomy]]<br />
* '''[[Georgy Gause]]''', inventor of [[gramicidin S]] and other [[antibiotics]]<br />
* '''[[Oleg Gazenko]]''', founder of [[space medicine]], selected and trained [[Laika]], the first [[space dog]]<br />
* '''[[Vera Gedroitz]]''', the first female Professor of [[Surgery]] in the world<br />
* '''[[Ilya Gruzinov]]''', found that [[vocal folds]] are the source of [[phonation]]<br />
* '''[[Waldemar Haffkine]]''', invented the first vaccines against [[cholera]] and [[bubonic plague]]<br />
* '''[[Gavriil Ilizarov]]''', invented [[Ilizarov apparatus]], developed [[distraction osteogenesis]]<br />
* '''[[Nikolai Korotkov]]''', invented [[auscultatory blood pressure measurement]], pioneer of [[vascular surgery]][[File:Ilya Mechnikov nobel.jpg|thumb|90px|'''[[Ilya Mechnikov|Mechnikov]]''']]<br />
* '''[[Sergey Korsakov]]''', studied the effects of [[alcoholism]] on the [[nervous system]], described [[Korsakoff's syndrome]], introduced [[paranoia]] concept<br />
* '''[[Aleksei Kozhevnikov]]''', neurologist and psychiatrist, described the [[epilepsia partialis continua]]<br />
* '''[[Aleksey Leontyev]]''', founder of [[activity theory]] in psychology<br />
* '''[[Peter Lesgaft]]''', founder of the modern system of [[physical education]] in Russia<br />
* '''[[Alexander Luria]]''', co-developer of activity theory and [[cultural-historical psychology]], major researcher of [[aphasia]]<br />
* '''[[Ilya Mechnikov]]''', pioneer researcher of [[immune system]], [[probiotics]] and [[phagocytosis]]; coined the term ''[[gerontology]]'', [[Nobel Prize in Medicine]] winner[[File:Ivan Pavlov (Nobel).png|thumb|90px|'''[[Ivan Pavlov|Pavlov]]''']]<br />
* '''[[Lazar Minor]]''', neurologist, described [[Minor's disease]]<br />
* '''[[Pyotr Nikolsky]]''', [[dermatologist]], discoveror of [[Nikolsky's sign]]<br />
* '''[[Alexey Olovnikov]]''', predicted existence of [[Telomerase]], suggested the [[Telomere hypothesis of aging]] and the [[Telomere relations to cancer]]<br />
* '''[[Ivan Pavlov]]''', founder of modern [[physiology]], the first to research [[classical conditioning]], influenced [[comparative psychology]] and [[behaviorism]] by his works on [[reflex]]es, Nobel Prize in Medicine winner<br />
* '''[[Nikolay Ivanovich Pirogov|Nikolay Pirogov]]''', pioneer of [[ether]] [[anaesthesia]] and modern [[field surgery]], the first to perform anaesthesia in the field conditions, invented a number of surgical operations<br />
* '''[[Leonid Rogozov]]''', performed an [[appendectomy]] on himself during the 6th [[Soviet Antarctic Expedition]], a famous case of [[self-surgery]][[File:Sechenov.jpg|thumb|90px|'''[[Sechenov]]''']]<br />
* '''[[Grigory Rossolimo]]''', pioneer of child [[neuropsychology]]<br />
* '''[[Vladimir Roth]]''', [[neuropathologist]], described [[meralgia paraesthetica]]<br />
* '''[[Ivan Sechenov]]''', founder of [[electrophysiology]] and [[neurophysiology]], author of the classic work ''[[Reflex]]es of the [[Brain]]''<br />
* '''[[Vladimir Serbsky]]''', founder of [[forensic psychiatry]] in Russia<br />
* '''[[Nikolay Sklifosovskiy]]''', prominent 19th century [[field surgeon]]<br />
* '''[[Lina Stern]]''', pioneer researcher of [[blood–brain barrier]]<br />
* '''[[Fyodor Uglov]]''', the oldest practicing surgeon in history<br />
* '''[[Alexander Varshavsky]]''', researched [[ubiquitin]]ation, [[Wolf Prize in Medicine]] winner<br />
* '''[[Luka Voyno-Yasenetsky]]''', founder of [[purulent]] [[surgery]], saint<br />
* '''[[Lev Vygotsky]]''', founder of [[cultural-historical psychology]], major contributor to [[child development]] and [[psycholinguistics]], introduced [[zone of proximal development]] and [[cultural mediation]] concepts<br />
* '''[[Josias Weitbrecht]]''', the first to describe the construction and function of [[intervertebral disc]]s<br />
* '''[[Sergei Yudin]]''', inventor of [[cadaveric blood transfusion]]<br />
* '''[[Alexander Zalmanov]]''', developer of [[turpentine bath]] therapy<br />
* '''[[Bluma Zeigarnik]]''', psychiatrist, discovered the [[Zeigarnik effect]], founded experimental [[psychopathology]]<br />
<br />
==Economists and sociologists==<br />
[[File:Leonid Kantorovich 1975.jpg|thumb|90px|'''[[Kantorovich]]''']]<br />
* '''[[Alexander Chayanov]]''', developed the [[consumption-labour-balance principle]]<br />
* '''[[Georges Gurvitch]]''', major developer of [[sociology of knowledge]] and [[sociology of law]]<br />
* '''[[Leonid Kantorovich]]''', mathematician and economist, founded [[linear programming]], developed the theory of [[Optimization (mathematics)|optimal allocation]] of resources, [[Nobel Prize in Economics]] winner<br />
* '''[[Nikolai Kondratiev]]''', discoverer of the [[Kondratiev wave]]s<br />
* '''[[Andrey Korotayev]]''', historian and anthropologist, a founder of [[cliodynamics]], a prominent developer of [[social cycle theory]]<br />
* '''[[Gleb Krzhizhanovsky]]''', developer of [[GOELRO plan]], the first Chief of [[Gosplan]]<br />
* '''[[Simon Kuznets]]''', discovered the [[Kuznets swing]]s, built the [[Kuznets curve]], disproved the [[Absolute Income Hypothesis]], Nobel Prize in Economics winner[[File:Питирим Сорокин.jpg|thumb|90px|'''[[Pitirim Sorokin|Sorokin]]''']]<br />
* '''[[Vladimir Lenin]]''', leader of the [[October Revolution]] and founder of the [[Soviet Union]], introduced [[planned economy]] and [[Leninism]]<br />
* '''[[Evsei Liberman]]''', laid the scientific support for the Soviet [[1965 Soviet economic reform|Kosygin reform]] (iniatied by [[Alexei Kosygin]]) in economy<br />
* '''[[Wassily Leontief]]''', developed [[input-output analysis]] and the [[Leontief paradox]], Nobel Prize in Economics winner<br />
* '''[[Vasily Nemchinov]]''', created the mathematical basis for the Soviet [[central planning]]<br />
* '''[[Grigory Orlov]]''', founder of the [[Free Economic Society]]<br />
* '''[[Pitirim Sorokin]]''', sociologist, a prominent developer of the [[social cycle theory]]<br />
* '''[[Eugen Slutsky]]''', statistician and economist,developed the [[Slutsky equation]]<br />
* '''[[Stanislav Strumilin]]''', pioneer of the [[planned economy]], developed the [[First Five-Year Plan (Soviet Union)|first]] [[Five-Year Plans for the National Economy of the Soviet Union|five-year plans]]<br />
<br />
==Historians and archeologists==<br />
{{main|Russian historians}}<br />
[[File:Artamonov student.jpg|thumb|90px|'''[[Mikhail Artamonov|Artamonov]]''']]<br />
* '''[[Friedrich von Adelung]]''', historian and [[museologist]], researched the European accounts of the [[Time of Troubles]]<br />
* '''[[Valeri Pavlovich Alekseyev|Valery Alekseyev]]''', anthropologist, proposed ''[[Homo rudolfensis]]''<br />
* '''[[Mikhail Artamonov]]''', historian and archaeologist, founder of modern [[Khazar]] studies, excavated a great number of [[Scythian]] and Khazar [[kurgan]]s and settlements, including the fortress of [[Sarkel]]<br />
* '''[[Artemiy Artsikhovsky]]''', archaeologist, discoverer of [[birch bark document]]s in [[Novgorod]]<br />
* '''[[Vasily Bartold]]''', [[turkologist]], the ''"[[Edward Gibbon|Gibbon]] of [[Turkestan]]"'', an archaeologist of [[Samarcand]]<br />
* '''[[Konstantin Bestuzhev-Ryumin]]''', 19th-century historian and [[paleographer]], founder of the [[Bestuzhev Courses]] for women[[File:Nikolay Danilevski.jpg|thumb|95px|'''[[Nikolay Danilevsky|Danilevsky]]''']]<br />
* '''[[Nikita Bichurin]]''', a founder of [[Sinology]], published many documents on [[Chinese history|Chinese]] and [[Mongolian history]], opened the first Chinese-language school in Russia<br />
* '''[[Nikolay Danilevsky]]''', ethnologist, philosopher and historian, a founder of [[Eurasianism]], the first to present an account of history as a series of distinct [[civilisations]]<br />
* '''[[Igor Diakonov]]''', historian and linguist, a prominent researcher of [[Sumer]] and [[Assyria]]<br />
* '''[[Boris Farmakovsky]]''', archaeologist of [[Ancient Greek]] colony [[Olbia, Ukraine|Olbia]]<br />
* '''[[Vladimir Golenishchev]]''', egyptologist, excavated [[Wadi Hammamat]], discovered over 6,000 antiquities, including the ''[[Moscow Mathematical Papyrus]]'', the ''[[Story of Wenamun]]'', and various [[Fayum portraits]][[File:Tropinin karamzin.JPG|thumb|90px|'''[[Nikolai Karamzin|Karamzin]]''']]<br />
* '''[[Timofey Granovsky]]''', a founder of [[mediaeval studies]] in Russia, disproved the historicity of [[Vineta]]<br />
* '''[[Boris Grekov]]''', prominent researcher of [[Kievan Rus']] and [[Golden Horde]]<br />
* '''[[Lev Gumilev]]''', historian and ethnologist, prominent researcher of ancient [[Central Asia]]n peoples, related [[ethnogenesis]] and [[biosphere]], influenced the rise of [[Neo-Eurasianism]]<br />
* '''[[Boris Hessen]]''', physicist who brought [[externalism]] into modern [[historiography of science]]<br />
* '''[[Dmitry Ilovaysky]]''', major 19th century anti-[[Normanist]]<br />
* '''[[Pyotr Kafarov]]''', prominent [[sinologist]], discovered many invaluable manuscripts, including ''[[The Secret History of the Mongols]]''<br />
* '''[[Nikolai Karamzin]]''', [[sentimentalist]] writer and historian, author of the 12-volume ''History of the Russian State'', the principal early 19th-century account of national history<br />
* '''[[Vasily Klyuchevsky]]''', dominated Russian historiography at the turn of the 20th century, shifted focus from politics and society to geography and economy<br />
* '''[[Alexander Kazhdan]]''', [[Byzantinist]], editor of the ''[[Oxford Dictionary of Byzantium]]''<br />
* '''[[Nikodim Kondakov]]''', prominent researcher of [[Byzantine art]][[File:Gerhard Friedrich Müller.jpg|thumb|90px|'''[[Gerhardt Friedrich Müller|Müller]]''']]<br />
* '''[[Andrey Korotayev]]''', historian and anthropologist, a founder of [[cliodynamics]], a prominent developer of [[social cycle theory]]<br />
* '''[[Nikolay Kostomarov]]''', historian, folklorist and romantic writer, researched the differences between [[Great Russia]] and [[Little Russia]] and the history of [[Ukraine]]<br />
* '''[[Pyotr Kozlov]]''', explorer of [[Central Asia]], discoverer of the ancient [[Tangut people|Tangut]] city of [[Khara-Khoto]] and [[Xiongnu]] royal burials at [[Noin-Ula]]<br />
* '''[[Platon Levshin]]''', president of the [[Most Holy Synod]] during the [[Age of Enlightenment]], author of the first systematic course of the history of [[Russian Orthodox Church]]<br />
* '''[[Nikolay Likhachyov]]''', the first and foremost Russian [[sigillographer]], prominent also in a number of other auxiliary historical disciplines<br />
* '''[[Aleksey Lobanov-Rostovsky]]''', statesman, published the major ''Russian [[Genealogical]] Book''[[File:MAntokolsky Nestor.JPG|thumb|100px|'''[[Nestor the Chronicler|Nestor]]''']]<br />
* '''[[Mikhail Lomonosov]]''', polymath scientist and artist, the first opponent of the [[Normanist theory]], published an early account of Russian history<br />
* '''[[Boris Marshak]]''', excavated the [[Sogdian people|Sogdian]] ruins at [[Panjakent]]<br />
* '''[[Friedrich Martens]]''', legal historian, drafted the [[Martens Clause]] of the [[Hague Peace Conference]]<br />
* '''[[Vladimir Minorsky]]''', prominent historian of [[Persia]]<br />
* '''[[Gerhardt Friedrich Müller]]''', co-founder of the [[Russian Academy of Sciences]], explorer and the first academic historian of [[Siberia]], a founder of [[ethnography]], author of the first academic account of [[Russian history]], put forth the [[Normanist theory]]<br />
* '''[[Aleksei Musin-Pushkin]]''', prominent collector of ancient Russian manuscripts, discovered ''[[The Tale of Igor's Campaign]]''<br />
* '''[[Nestor the Chronicler]]''', author of the ''[[Primary Chronicle]]'' (the first [[East Slavic languages|East Slavic]] chronicle) and several [[hagiographies]], saint<br />
* '''[[Dimitri Obolensky]]''', [[Byzantine commonwealth]] researcher[[File:Charsolomon TProskouriakoff.jpg|thumb|90px|'''[[Tatyana Proskuryakova|Proskuryakova]]''']]<br />
* '''[[Alexey Okladnikov]]''', prominent historian and archaeologist of [[Siberia]] and [[Mongolia]]<br />
* '''[[Sergey Oldenburg]]''', a founder of Russian [[Indology]] and the [[Academic Institute of Oriental Studies]]<br />
* '''[[George Ostrogorsky]]''', preeminent 20th-century [[Byzantinist]]<br />
* '''[[Avraamy Palitsyn]]''', 17th-century historian of the [[Time of Troubles]]<br />
* '''[[Evgeny Pashukanis]]''', legal historian, wrote ''The General Theory of Law and [[Marxism]]''<br />
* '''[[Boris Piotrovsky]]''', prominent researcher of [[Urartu]], [[Scythia]], and [[Nubia]], long-term director of the [[Hermitage Museum]]<br />
* '''[[Mikhail Piotrovsky]]''', orientalist, current director of the [[Hermitage Museum]]<br />
* '''[[Mikhail Pogodin]]''', leading mid-19th-century Russian historian, proponent of the [[Normanist theory]]<br />
* '''[[Boris Petrovich Polevoy|Boris Polevoy]]''', major historian of the [[Russian Far East]][[File:Rybakov BA.jpg|thumb|90px|'''[[Boris Rybakov|Rybakov]]''']]<br />
* '''[[Mikhail Pokrovsky]]''', [[Marxist]] historian prominent in 1920s<br />
* '''[[Natalia Polosmak]]''', archaeologist of [[Pazyryk burials]], discoverer of [[Pazyryk Ice Maiden|Ice Maiden]] [[mummy]]<br />
* '''[[Alexander Polovtsov]]''', statesman, historian and [[Maecenas]], founder of the Russian Historian Society<br />
* '''[[Tatyana Proskuryakova]]''', [[Mayanist]] scholar and archaeologist, deciphered the ancient [[Maya script]]<br />
* '''[[Semyon Remezov]]''', cartographer and the first historian of [[Siberia]], author of the ''[[Remezov Chronicle]]''<br />
* '''[[Mikhail Rostovtsev]]''', archeologist and economist, the first to thoroughly examine the social and economic systems of the [[Ancient World]], excavated [[Dura-Europos]][[File:Solovjev S. M..jpg|thumb|100px|'''[[Sergey Solovyov|Solovyov]]''']]<br />
* '''[[Nicholas Roerich]]''', painter, archeologist, and public figure, explorer of [[Central Asia]], initiator of the international [[Roerich’s Pact]] on protection of historical monuments<br />
* '''[[Sergei Rudenko]]''', discoverer of [[Scythian]] [[Pazyryk burials]]<br />
* '''[[Boris Rybakov]]''', historian and chief Soviet archaeologist for 40 years, primary opponent of the [[Normanist theory]]<br />
* '''[[Dmitry Samokvasov]]''', [[Black Grave]] discoverer<br />
* '''[[Viktor Sarianidi]]''', discoverer of the [[Bactria-Margiana Archaeological Complex]] and the ''[[Bactrian Gold]]'' in [[Central Asia]]<br />
* '''[[Mikhail Shcherbatov]]''', a man of [[Russian Enlightenment]], conservative historian<br />
* '''[[Sergey Solovyov]]''', principal Russian 19th-century historian, author of the 29-volume ''History of Russia''<br />
* '''[[Vasily Vasilievich Struve|Vasily Struve]]''', orientalist and historian of the [[Ancient World]], put forth the [[Marxist]] theory of five socio-economic formations that dominated the Soviet education[[File:Tatishchev.png|thumb|105px|'''[[Vasily Tatishchev|Tatishchev]]''']]<br />
* '''[[Yevgeny Tarle]]''', author of the famous studies on [[Napoleon's invasion of Russia]] and on the [[Crimean War]]<br />
* '''[[Vasily Tatischev]]''', statesman, geographer and historian, discovered and published ''[[Russkaya Pravda]]'', ''[[Sudebnik]]'' of 1550 and the controversial ''[[Ioachim Chronicle]]'', wrote the first full-scale account of Russian history<br />
* '''[[Mikhail Tikhomirov]]''', leading specialist in medieval Russian [[paleography]], published the ''[[Complete Collection of Russian Chronicles]]''<br />
* '''[[Boris Turayev]]''', author of the first full-scale ''History of [[Ancient World|Ancient East]]''<br />
* '''[[Peter Turchin]]''', population biologist and historian, coined the term ''[[cliodynamics]]''<br />
* '''[[Fyodor Uspensky]]''', Byzantinist, researcher of the [[Trapezuntine Empire]]<br />
* '''[[Aleksey Uvarov]]''', founder of the first Russian archaeological society, discovered over 750 ancient [[kurgan]]s<br />
* '''[[Vasily Vasilievsky]]''', prominent 19th century [[Byzantinist]]<br />
* '''[[Alexander Vasiliev (historian)|Alexander Vasiliev]]''', author of a comprehensive ''History of the [[Byzantine Empire]]''<br />
* '''[[Nikolay Veselovsky]]''', the first to excavate [[Afrasiab]] (the oldest part of [[Samarkand]]), as well as the [[Solokha]] and [[Maikop kurgan]]s in Southern Russia<br />
* '''[[Nikolai Yadrintsev]]''', discoverer of [[Genghis Khan]]'s capital [[Karakorum]] and the [[Orkhon script]] of ancient [[Türks]]<br />
* '''[[Valentin Yanin]]''', primary researcher of ancient [[birch bark document]]s<br />
* '''[[Gennady Zdanovich]]''', discoverer of [[Sintashta culture]] settlement [[Arkaim]]<br />
<br />
==Linguists and ethnographers==<br />
{{main|Russian linguists}}<br />
[[File:Baudouin1.jpg|thumb|90px|'''[[Baudouin de Courtenay]]''']]<br />
* '''[[Vasily Abaev]]''', a prominent researcher of [[Iranian languages]]<br />
* '''[[Alexander Afanasyev]]''', leading Russian [[folklorist]], recorded and published over 600 [[Russian fairy tales]], by far the largest folktale collection by any one man in the world<br />
* '''[[Ivan Baudouin de Courtenay]]''', co-inventor of the concept of [[phoneme]] and the systematic treatment of [[alternation (linguistics)|alternations]], pioneer of [[synchronic analysis]] and [[mathematical linguistics]]<br />
* '''[[Vladimir Bogoraz]]''', researcher of [[Chukchi people]], founder of the [[Institute of the Peoples of the North]]<br />
* '''[[Otto von Böhtlingk]]''', prominent [[Indologist]] and [[Sanskrit grammarian]]<br />
* '''[[Fyodor Buslaev]]''', philologist and folklorist, representative of the [[Mythological school]] of [[comparative literature]][[File:Портрет писателя Владимира Ивановича Даля.jpg|thumb|100px|'''[[Vladimir Dahl|Dahl]]''']]<br />
* '''[[Vladimir Dahl]]''', the greatest [[Russian language]] [[lexicographer]] of the 19th century, folklorist and [[turkologist]], author of the ''[[Explanatory Dictionary of the Live Great Russian language]]''<br />
* '''[[Johann Gottlieb Georgi]]''', explorer, published the first full-scale work on ethnography of [[indigenous peoples of Russia]]<br />
* '''[[Dmitry Gerasimov]]''', medieval translator, diplomat and philologist, correspondent of European [[Renaissance]] scholars<br />
* '''[[Vladislav Illich-Svitych]]''', founder of [[Nostratic]] linguistics<br />
* '''[[Vyacheslav Vsevolodovich Ivanov|Vyacheslav Ivanov]]''', founder of [[glottalic theory]] of [[Indo-European]] [[consonant]]ism<br />
* '''[[Roman Jakobson]]''', literary theorist and preeminent linguist of the 20th century, a founder of [[phonology]], made numerous contributions to [[Slavic linguistics]], author of [[Jackobson's Communication Model]]<br />
* '''[[Wilhelm Junker]]''', explorer and ethnographer of [[Equatorial Africa]], studied the [[Zande people]] from Niam-Niam [[File:Архимандрит Палладий.jpg|thumb|100px|'''[[Pyotr Kafarov|Kafarov]]''']]<br />
* '''[[Pyotr Kafarov]]''', prominent [[sinologist]], developed the [[cyrillization of Chinese]], discovered and published many invaluable manuscripts, including ''[[The Secret History of the Mongols]]''<br />
* '''[[Yuri Knorozov]]''', linguist, [[epigrapher]] and ethnographer, deciphered the ancient [[Maya script]], proposed a decipherment for the [[Indus script]]<br />
* '''[[Nikolay Krushevsky]]''', co-inventor of the concept of [[phoneme]] and the systematic treatment of [[alternation (linguistics)|alternations]]<br />
* '''[[Gerasim Lebedev]]''', pioneer of [[Indology]], introduced [[Bengali script]] typing to Europe, founded the first European-style drama [[theater]] in India<br />
* '''[[Dmitry Likhachov]]''', major 20th century expert on [[Old Russian language]] and literature[[File:Miklukho-Maklai.jpg|thumb|90px|'''[[Nicholai Miklukho-Maklai|Miklukho-Maklai]]''']]<br />
* '''[[Mikhail Lomonosov]]''', polymath scientist and artist, wrote a [[grammar]] that reformed Russian literary language by combining [[Old Church Slavonic]] with vernacular tongue<br />
* '''[[Yuri Lotman]]''', prominent literary scholar and semiotician<br />
* '''[[Nikolay Lvov]]''', polymath artist and scientist, compiled the first significant collection of [[Russian folk songs]], published epic [[bylina]]s<br />
* '''[[Richard Maack]]''', naturalist and ethographer of [[Siberia]]<br />
* '''[[Sergey Malov]]''', turkologist, classified the [[Turkic alphabet (disambiguation)|Turkic alphabet]]s, deciphered ancient [[Orkhon script]]<br />
* '''[[Nicholas Marr]]''', put forth a pseudo-linguistic ''[[Japhetic theory]]'' on the [[origin of language]]<br />
* '''[[Igor Melchuk]]''', structural linguist, author of [[Meaning-Text Theory]]<br />
* '''[[Nicholai Miklukho-Maklai]]''', anthropologist who lived and traveled among the natives of [[Papua New Guinea]] and [[Pacific islands]], prominent [[anti-racist]]<br />
* '''[[Gerhardt Friedrich Müller]]''', explorer and historian, a founder of [[ethnography]]<br />
* '''[[Semyon Novgorodov]]''', [[Yakuts|Yakut]] politician and linguist, creator of written [[Yakut language]] ([[Sakha scripts]])<br />
* '''[[Sergei Ozhegov]]''', author of the most widely used explanatory dictionary of [[Russian language]][[File:Nikolai Trubetzkoy.jpg|thumb|95px|'''[[Nikolai Trubetzkoy|Trubetzkoy]]''']]<br />
* '''[[Stephan of Perm]]''', 14th century missionary, converted [[Komi Permyaks]] to Christianity and invented the [[Old Permic script]]<br />
* '''[[Yevgeny Polivanov]]''', linguist, orientalist and [[polyglot (person)|polyglot]], developed the [[cyrillization of Japanese]]<br />
* '''[[Nicholas Poppe]]''', prominent [[Altaic languages]] researcher<br />
* '''[[Grigory Potanin]]''', explorer of [[Central Asia]], the first to research [[Salar people]]<br />
* '''[[Vladimir Propp]]''', [[Formalism (literature)|formalist]] scholar, major researcher of [[Folklore|folk tale]]s and [[mythology]]<br />
* '''[[Tatyana Proskuryakova]]''', [[Mayanist]] scholar and archaeologist, deciphered the ancient [[Maya script]]<br />
* '''[[George de Roerich]]''', major 20th century [[Tibetologist]]<br />
* '''[[Franz Anton Schiefner]]''', prominent [[tibetologist]], [[Finnic languages|Finnic]] and [[Caucasus]] languages researcher<br />
* '''[[Isaac Jacob Schmidt]]''', the first researcher of [[Mongolian language|Mongolian]][[File:Zalizn 5.jpg|thumb|90px|'''[[Andrey Zaliznyak|Zaliznyak]]''']]<br />
* '''[[Leopold von Schrenck]]''', naturalist and ethnographer, coined the term ''[[Paleo-Asiatic]] peoples'', the first director of the [[Peter the Great Museum of Anthropology and Ethnography]]<br />
* '''[[Aleksey Shakhmatov]]''', a founder of [[textology]], prepared major 20th century [[reforms of Russian orthography]], pioneered the systematic research of [[Old Russian]] and medieval [[Russian literature]]<br />
* '''[[Lev Shcherba]]''', [[phonetics|phonetist]] and [[phonologist]], author of the ''[[glokaya kuzdra]]'' phrase<br />
* '''[[Fyodor Shcherbatskoy]]''', [[Indologist]], initiated the scholarly study of [[Buddhist]] philosophy in the West<br />
* '''[[Ivan Snegiryov]]''', an early collector of [[Russian proverbs]] and researcher of [[lubok]] prints<br />
* '''[[Izmail Sreznevsky]]''', leading 19th century [[Slavist]], published ''[[Codex Zographensis]]'', ''[[Codex Marianus]]'' and ''[[Kiev Fragments]]''[[File:1908-kl-t-zamenhof.jpg|thumb|90px|'''[[Ludwik Zamenhof|Zamenhof]]''']]<br />
* '''[[Sergei Starostin]]''', prominent supporter of [[Altaic languages]] theory, proposed [[Dené–Caucasian languages]] [[macrofamily]], reconstructed a number of Eurasian proto-languages<br />
* '''[[Vasily Tatischev]]''', geographer, ethnographer and historian, compiled the first encyclopedic [[dictionary]] of [[Russian language|Russian]]<br />
* '''[[Tenevil]]''', [[Chukchi people|Chukchi]] [[reindeer herder]] who created a writing system for the [[Chukchi language]]<br />
* '''[[Nikolai Trubetzkoy]]''', principal developer of [[phonology]] and inventor of [[morphophonology]], defined [[phoneme]], a founder of the [[Prague School]] of [[structural linguistics]]<br />
* '''[[Dmitry Ushakov]]''', author of the academic ''[[Explanatory Dictionary of the Russian Language (Ushakov)|Explanatory Dictionary of the Russian Language]]''<br />
* '''[[Max Vasmer]]''', leading [[Indo-European languages|Indo-European]], [[Finno-Ugric languages|Finno-Ugric]] and [[Turkic languages|Turkic]] [[etymologist]], author of the ''Etymological dictionary of the [[Russian language]]''<br />
* '''[[Viktor Vinogradov]]''', linguist and philologist, founder of the [[Russian Language Institute]]<br />
* '''[[Alexander Vostokov]]''', coined the term ''[[Old Church Slavonic]]'', discovered ''[[Ostromir Gospel]]'' (the most ancient [[East Slavic languages|East Slavic]] book), pioneer researcher of the [[Russian grammar]]<br />
* '''[[Andrey Zaliznyak]]''', author of the comprehensive systematic description of Russian [[inflection]], prominent researcher of the [[Old Novgorod dialect]] and [[birch bark document]]s, proved the authentity of the ''[[Tale of Igor's Campaign]]''<br />
* '''[[Ludwik Zamenhof]]''', inventor of [[Esperanto]], the most widely spoken [[Constructed language|constructed]] [[international auxiliary language]]<br />
<br />
==Mathematicians==<br />
{{main|Russian mathematicians}}<br />
[[File:Vladimir Arnold-1.jpg|thumb|90px|'''[[Vladimir Arnold|Arnold]]''']]<br />
* '''[[Georgy Adelson-Velsky]]''', inventor of [[AVL tree]] algorithm, developer of [[Kaissa]], the first world computer chess champion<br />
* '''[[Aleksandr Danilovich Aleksandrov|Aleksandr Aleksandrov]]''', developer of [[CAT(k) space]] and [[Alexandrov's uniqueness theorem]] in geometry<br />
* '''[[Pavel Alexandrov]]''', author of the [[Alexandroff compactification]] and the [[Alexandrov topology]]<br />
* '''[[Dmitri Anosov]]''', developed [[Anosov diffeomorphism]]<br />
* '''[[Vladimir Arnold]]''', an author of the [[Kolmogorov–Arnold–Moser theorem]] in [[dynamical system]]s, solved [[Hilbert's 13th problem]], raised the [[ADE classification]] and [[Arnold's rouble problem]]s<br />
* '''[[Sergey Bernstein]]''', developed the [[Bernstein polynomial]], [[Bernstein's theorem on monotone functions]] and [[Bernstein inequalities in probability theory]]<br />
* '''[[Nikolay Bogolyubov]]''', mathematician and theoretical physicist, author of the [[edge-of-the-wedge theorem]], [[Krylov–Bogolyubov theorem]], [[describing function]] and multiple important contributions to [[quantum mechanics]][[File:Chebyshev.jpg|thumb|90px|'''[[Chebyshev]]''']]<br />
* '''[[Sergey Alexeyevich Chaplygin|Sergey Chaplygin]]''', author of [[Chaplygin's equation]] important in [[aerodynamics]] and notion of [[Chaplygin gas]].<br />
* '''[[Nikolai Chebotaryov]]''', author of [[Chebotarev's density theorem]]<br />
* '''[[Pafnuti Chebyshev]]''', prominent tutor and founding father of Russian mathematics, contributed to [[probability]], [[statistics]] and [[number theory]], author of the [[Chebyshev's inequality]], [[Chebyshev distance]], [[Chebyshev function]], [[Chebyshev equation]] etc.<br />
* '''[[Boris Delaunay]]''', inventor of [[Delaunay triangulation]], organised the first [[Soviet Student Olympiad]] in mathematics<br />
* '''[[Vladimir Drinfeld]]''', mathematician and theoretical physicist, introduced [[quantum group]]s and [[ADHM construction]], [[Fields Medal]] winner<br />
* '''[[Eugene Dynkin]]''', developed [[Dynkin diagram]], [[Doob–Dynkin lemma]] and [[Dynkin system]] in [[algebra]] and [[probability]]<br />
* '''[[Leonhard Euler]]''', preeminent 18th century mathematician, arguably the greatest of all time, made important discoveries in [[mathematical analysis]],[[File:Leonhard Euler 2.jpg|thumb|100px|'''[[Euler]]''']] [[graph theory]] and [[number theory]], introduced much of the modern mathematical terminology and notation ([[mathematical function]], [[Euler's number]], [[Euler circles]] etc.)<br />
* '''[[Anatoly Fomenko]]''', topologist and chronologist, put forth a controversial theory of the [[New Chronology (Fomenko)|New Chronology]]<br />
* '''[[Evgraf Fedorov]]''', mathematician and crystallographer, identified [[Periodic graph (geometry)|Periodic graph]] in geometry, the first to catalogue all of the 230 [[space groups]] of crystals<br />
* '''[[Boris Galerkin]]''', developed the [[Galerkin method]] in [[numerical analysis]]<br />
* '''[[Israel Gelfand]]''', major contributor to numerous areas of mathematics, including [[group theory]], [[representation theory]] and [[linear algebra]], author of the [[Gelfand representation]], [[Gelfand pair]], [[Gelfand triple]], [[integral geometry]] etc.<br />
* '''[[Alexander Gelfond]]''', author of [[Gelfond's theorem]], provided means to obtain infinite number of [[transcendentals]], including [[Gelfond–Schneider constant]] and [[Gelfond's constant]], [[Wolf Prize in Mathematics]] winner<br />
* '''[[Sergei Godunov]]''', developed [[Godunov's theorem]] and [[Godunov's scheme]] in [[differential equations]]<br />
* '''[[Valery Goppa]]''', inventor of [[Goppa code]]s in [[algebraic geometry]][[File:Gromov Mikhail Leonidovich.jpg|thumb|105px|'''[[Mikhail Gromov (mathematician)|Gromov]]''']]<br />
* '''[[Mikhail Gromov (mathematician)|Mikhail Gromov]]''', a prominent developer of [[geometric group theory]], inventor of [[homotopy principle]], introduced [[Gromov's compactness theorem (disambiguation)|Gromov's compactness theorems]], [[Gromov norm]], [[Gromov product]] etc., Wolf Prize winner<br />
* '''[[Leonid Kantorovich]]''', mathematician and economist, founded [[linear programming]], introduced the [[Kantorovich inequality]] and [[Kantorovich metric]], developed the theory of [[Optimization (mathematics)|optimal allocation]] of resources, [[Nobel Prize in Economics]] winner<br />
* '''[[Anatoly Karatsuba]]''', developed the [[Karatsuba algorithm]] (the first fast [[multiplication algorithm]])<br />
* '''[[Leonid Khachiyan]]''', developed the [[Ellipsoid algorithm]] for [[linear programming]]<br />
* '''[[Aleksandr Khinchin]]''', developed the [[Pollaczek-Khinchine formula]], [[Wiener–Khinchin theorem]] and [[Khinchin inequality]] in [[probability theory]][[File:Sofja Wassiljewna Kowalewskaja 1.jpg|thumb|95px|'''[[Sofia Kovalevskaya|Kovalevskaya]]''']]<br />
* '''[[Andrey Kolmogorov]]''', a preeminent 20th century mathematician, Wolf Prize winner; multiple contributions to mathematics include: [[probability axioms]], [[Chapman–Kolmogorov equation]] and [[Kolmogorov extension theorem]] in [[probability]]; [[Kolmogorov complexity]] etc.<br />
* '''[[Maxim Kontsevich]]''', author of the [[Kontsevich integral]] and [[Kontsevich quantization formula]], Fields Medal winner<br />
* '''[[Vladimir Aleksandrovich Kotelnikov|Vladimir Kotelnikov]]''', a pioneer in [[information theory]], an author of fundamental [[sampling theorem]]<br />
* '''[[Sofia Kovalevskaya]]''', the first woman professor in Northern Europe and Russia, the first female professor of mathematics, discovered the [[Kovalevskaya Top]]<br />
* '''[[Mikhail Kravchuk]]''', developed the [[Kravchuk polynomials]] and [[Kravchuk matrix]]<br />
* '''[[Mark Krein]]''', developed the [[Tannaka-Krein duality]], [[Krein–Milman theorem]] and [[Krein space]], Wolf Prize winner<br />
* '''[[Alexander Kronrod]]''', developer of [[Gauss–Kronrod quadrature formula]] and [[Kaissa]], the first world computer chess champion<br />
* '''[[Nikolay Mitrofanovich Krylov|Nikolay Krylov]]''', author of the [[edge-of-the-wedge theorem]], [[Krylov–Bogolyubov theorem]] and [[describing function]][[File:Lobachevsky cropped.jpg|thumb|90px|'''[[Lobachevsky]]''']]<br />
* '''[[Aleksandr Kurosh]]''', author of the [[Kurosh subgroup theorem]] and [[Kurosh problem]] in [[group theory]]<br />
* '''[[Olga Aleksandrovna Ladyzhenskaya|Olga Ladyzhenskaya]]''', made major contributions to solution of [[Hilbert's nineteenth problem|Hilbert's 19th problem]] and important [[Navier-Stokes equations]]<br />
* '''[[Evgeny Landis]]''', inventor of [[AVL tree]] algorithm<br />
* '''[[Vladimir Levenshtein]]''', developed the [[Levenshtein automaton]], [[Levenshtein coding]] and [[Levenshtein distance]]<br />
* '''[[Leonid Levin]]''', IT scientist, developed the [[Cook-Levin theorem]]<br />
* '''[[Yuri Linnik]]''', developed [[Linnik's theorem]] in [[analytic number theory]]<br />
* '''[[Nikolai Lobachevsky]]''', a ''[[Copernicus]] of [[Geometry]]'' who created the first [[non-Euclidean]] geometry ([[Lobachevskian]] or [[hyperbolic geometry]])<br />
* '''[[Nikolai Lusin]]''', developed [[Luzin's theorem]], [[Polish space#Lusin spaces|Luzin spaces]] and [[Luzin set]]s in [[descriptive set theory]]<br />
* '''[[Aleksandr Lyapunov]]''', founder of [[stability theory]], author of the [[Lyapunov's central limit theorem]], [[Lyapunov equation]], [[Lyapunov fractal]], [[Lyapunov time]] etc.<br />
* '''[[Leonty Magnitsky]]''', a director of the [[Moscow School of Mathematics and Navigation]], author of the principal Russian 18th-century textbook in mathematics<br />
* '''[[Anatoly Maltsev]]''', researched [[decidability (logic)|decidability]] of various [[algebraic group]]s,[[File:Alexander Ljapunow jung.jpg|thumb|90px|'''[[Aleksandr Lyapunov|Lyapunov]]''']] developed the [[Malcev algebra]]<br />
* '''[[Yuri Manin]]''', author of the [[Gauss–Manin connection]] in [[algebraic geometry]], [[Manin-Mumford conjecture]] and [[Manin obstruction]] in [[diophantine geometry]]<br />
* '''[[Grigory Margulis]]''', worked on [[lattice (discrete subgroup)|lattice]]s in [[Lie groups]], Wolf Prize and [[Fields Medal]] winner<br />
* '''[[Andrey Markov|Andrey Markov Sr.]]''', invented the [[Markov chain]]s, proved [[Markov brothers' inequality]], author of the [[hidden Markov model]], [[Markov number]], [[Markov property]], [[Markov's inequality]], [[Markov process]]es, [[Markov random field]], [[Markov algorithm]] etc.<br />
* '''[[Andrey Markov (Soviet mathematician)|Andrey Markov Jr.]]''', author of [[Markov's principle]] and [[Markov's principle#Markov's rule|Markov's rule]] in logics<br />
* '''[[Yuri Matiyasevich]]''', author of [[Matiyasevich's theorem]] in [[set theory]], provided negative solution for [[Hilbert's tenth problem]]<br />
* '''[[Alexander Ivanovich Mikhailov|Alexander Mikhailov]]''', coined the term ''[[Informatics (academic field)|Informatics]]''[[File:AAMarkov.jpg|thumb|100px|'''[[Andrey Markov|Markov Sr.]]''']]<br />
* '''[[Mark Naimark]]''', author of the [[Gelfand–Naimark theorem]] and [[Naimark's problem]]<br />
* '''[[Pyotr Novikov]]''', solved the [[word problem for groups]] and [[Burnside's problem]]<br />
* '''[[Sergei Novikov (mathematician)|Sergei Novikov]]''', worked on [[algebraic topology]] and [[soliton theory]], developed [[Adams–Novikov spectral sequence]] and [[Novikov conjecture]], Wolf Prize and Fields Medal winner<br />
* '''[[Andrei Okounkov]]''', [[infinite symmetric group]]s and [[Hilbert scheme]] researcher, Fields Medal winner<br />
* '''[[Mikhail Vasilievich Ostrogradsky|Mikhail Ostrogradsky]]''', mathematician and physicist, author of [[divergence theorem]] and [[partial fractions in integration]]<br />
* '''[[Grigori Perelman]]''', made landmark contributions to [[Riemannian geometry]] and [[topology]], proved [[Geometrization conjecture]][[File:Sobolev S L.jpeg|thumb|95px|'''[[Sergei Sobolev|Sobolev]]''']] and [[Poincaré conjecture]], won a [[Fields medal]] and the first Clay [[Millennium Prize Problems]] Award (declined both)<br />
* '''[[Lev Pontryagin]]''', blind mathematician, developed [[Pontryagin duality]] and [[Pontryagin class]]es in topology, and [[Pontryagin's minimum principle]] in [[optimal control]]<br />
* '''[[Yury Prokhorov]]''', author of the [[Lévy–Prokhorov metric]] and [[Prokhorov's theorem]] in [[probability]]<br />
* '''[[Alexander Razborov]]''', [[mathematician]] and [[computational theorist]] who won the [[Nevanlinna Prize]] in 1990 and the [[Gödel Prize]] for contributions to [[Theory of computation|computer sciences]]<br />
* '''[[Lev Schnirelmann]]''', developed the [[Lusternik–Schnirelmann category]] in topology and [[Schnirelmann density]] of numbers<br />
* '''[[Moses Schönfinkel]]''', inventor of [[combinatory logic]]<br />
* '''[[Yakov Sinai]]''', developed the [[Kolmogorov–Sinai entropy]] and [[Sinai billiard]], Wolf Prize winner<br />
* '''[[Eugen Slutsky]]''', statistician and economist, developed the [[Slutsky equation]] and [[Slutsky's theorem]]<br />
* '''[[Stanislav Smirnov]]''', prominent researcher of [[triangular lattice]], Fields Medalist<br />
* '''[[Sergei Sobolev]]''', introduced the [[Sobolev space]]s and [[mathematical distribution]]s, co-developer of the first [[ternary computer]] ''[[Setun]]''<br />
* '''[[Vladimir Steklov]]''', mathematician and physicist, founder of [[Steklov Institute of Mathematics]], proved theorems on [[generalized Fourier series]]<br />
* '''[[Jakow Trachtenberg]]''', developed the [[Trachtenberg system]] of [[mental calculation]]<br />
* '''[[Boris Trakhtenbrot]]''', proved the [[Gap theorem]], developed [[Trakhtenbrot's theorem]]<br />
* '''[[Valentin Turchin]]''', inventor of [[Refal programming language]], introduced [[metasystem transition]] and [[supercompilation]]<br />
* '''[[Andrey Tychonoff|Andrey Tikhonov]]''', author of [[Tikhonov space]] and [[Tikhonov's theorem]] (central in [[general topology]]), the [[Tikhonov regularization]] of [[ill-posed problem]]s, invented [[magnetotellurics]]<br />
* '''[[Pavel Urysohn]]''', developed the [[metrization theorems]], [[Urysohn's Lemma]] and [[Fréchet–Urysohn space]] in [[topology]]<br />
* '''[[Nicolai A. Vasiliev|Nicolay Vasilyev]]''', inventor of [[non-Aristotelian logic]], the forerunner of [[Paraconsistent logic|paraconsistent]] and [[multi-valued logic]]s<br />
* '''[[Ivan Matveevich Vinogradov|Ivan Vinogradov]]''', developed [[Vinogradov's theorem]] and [[Pólya–Vinogradov inequality]] in [[analytic number theory]]<br />
* '''[[Vladimir Voevodsky]]''', introduced a [[homotopy theory]] for schemes and modern [[motivic cohomology]], Fields Medalist<br />
* '''[[Georgy Voronoy]]''', invented the [[Voronoi diagram]]<br />
* '''[[Dmitry Yegorov]]''', author of [[Egorov's Theorem]] in [[mathematical analysis]]<br />
* '''[[Efim Zelmanov]]''', solved the [[restricted Burnside problem]], Fields Medal winner<br />
<br />
==Astronomers and cosmologists==<br />
{{main|Russian astronomers}}<br />
[[File:Fyodor Aleksandrovich Bredikhin.jpg|thumb|90px|'''[[Fyodor Bredikhin|Bredikhin]]''']]<br />
* '''[[Viktor Ambartsumian]]''', one of the founders of [[theoretical astrophysics]], discoverer of [[stellar associations]], founder of [[Byurakan Observatory]] in [[Armenia]]<br />
* '''[[Vladimir Belinski]]''', an author of the [[BKL singularity]] model of the Universe evolution<br />
* '''[[Aristarkh Belopolsky]]''', invented a [[spectrograph]] based on the [[Doppler effect]], among the first photographers of [[stellar spectra]]<br />
* '''[[Fyodor Bredikhin]]''', developed the theory of [[comet]] tails, [[meteor]]s and [[meteor shower]]s, a director of the [[Pulkovo Observatory]]<br />
* '''[[Jacob Bruce]]''', statesman, naturalist and astronomer, founder of the first [[observatory]] in Russia (in the [[Sukharev Tower]])<br />
* '''[[Lyudmila Chernykh]]''', astronomer, discovered 268 [[asteroid]]s[[File:Aleksandr Fridman cropped.png|thumb|100px|'''[[Alexander Friedmann|Friedmann]]''']]<br />
* '''[[Nikolai Chernykh]]''', astronomer, discovered 537 [[asteroid]]s and 2 [[comets]]<br />
* '''[[Aleksandr Chudakov]]''', co-discoverer of the Earth's [[radiation belt]]<br />
* '''[[Alexander Friedmann]]''', mathematician and cosmologist, discovered the [[metric expansion of space|expanding-universe]] [[Friedmann equations|solution]] to the [[general relativity]] [[Einstein field equations|field equations]], an author of the [[FLRW]] metric of [[Universe]]<br />
* '''[[George Gamow]]''', theoretical physicist and cosmologist, discovered [[quantum tunneling|alpha decay via quantum tunneling]] and [[Gamow factor]] in [[stellar nucleosynthesis]], introduced the [[big bang nucleosynthesis]] theory, predicted [[cosmic microwave background]][[File:GamovGA 1930.jpg|thumb|95px|'''[[George Gamow|Gamow]]''']]<br />
* '''[[Matvey Gusev]]''', the first to prove the non-sphericity of the [[Moon]], pioneer of photography in astronomy<br />
* '''[[Nikolai Kardashev]]''', astrophysicist, inventor of [[Kardashev scale]] for ranking the space civilizations<br />
* '''[[Isaak Khalatnikov]]''', an author of the [[BKL singularity]] model of the Universe evolution<br />
* '''[[Marian Kowalski]]''', the first to measure the rotation of the [[Milky Way]]<br />
* '''[[Feodosy Krasovsky]]''', astronomer and geodesist, measured the [[Krasovsky ellipsoid]], a coordinate system used in the [[USSR]] and the post-Soviet states<br />
* '''[[Anders Johan Lexell]]''', astronomer and mathematician, researcher of [[celestial mechanics]] and [[comet]] astronomy, proved that [[Uranus]] is a planet rather than a [[comet]]<br />
* '''[[Andrei Linde]]''', created the Universe [[chaotic inflation theory]][[File:Struve.jpg|thumb|100px|'''[[Friedrich Wilhelm Struve|F.W. Struve]]''']]<br />
* '''[[Evgeny Lifshitz]]''', an author of the [[BKL singularity]] model of the Universe evolution<br />
* '''[[Mikhail Lomonosov]]''' polymath, inventor of the [[Reflecting telescope#Off-axis designs|off-axis reflecting telescope]], discoverer of the [[atmosphere of Venus]]<br />
* '''[[Dmitri Dmitrievich Maksutov]]''', inventor of the [[Maksutov telescope]]<br />
* '''[[Igor Dmitriyevich Novikov|Igor Novikov]]''', theoretical astrophysicist and cosmologist, formulated the [[Novikov self-consistency principle]] in the theory of [[time travel]]<br />
* '''[[Viktor Safronov]]''', astronomer and cosmologist, author of the [[planetesimal]] hypothesis of [[planet formation]]<br />
* '''[[Grigory Shayn]]''', astronomer and astrophysicist, the first director of the [[Crimean Astrophysical Observatory]],[[File:Sunyaev.jpg|thumb|90px|'''[[Rashid Sunyaev|Sunyaev]]''']] co-developed a method for measurement of [[stellar rotation]]<br />
* '''[[Iosif Shklovsky]]''', astronomer and astrophysicist, author of several discoveries in the fields of [[radio astronomy]] and [[cosmic rays]], [[extraterrestrial life]] researcher<br />
* '''[[Friedrich Wilhelm Struve]]''', astronomer and geodesist, founder and the first director of the [[Pulkovo Observatory]], prominent researcher and discoverer of new [[double stars]], initiated the construction of 2,820&nbsp;km long [[Struve Geodetic Arc]], progenitor of the [[Struve family]] of astronomers<br />
* '''[[Otto Lyudvigovich Struve]]''', astronomer and astrophysicist, co-developed a method for measurement of [[stellar rotation]], directed several observatories in the [[U.S.]]<br />
* '''[[Otto Wilhelm von Struve]]''', astronomer, director of the [[Pulkovo Observatory]], discovered over 500 [[double stars]]<br />
* '''[[Rashid Sunyaev]]''', astrophysicist, co-predicted the [[Sunyaev–Zel'dovich effect]] of CMB distortion<br />
* '''[[George Volkoff]]''', predicted the existence of [[neutron stars]]<br />
* '''[[Boris Vorontsov-Velyaminov]]''', discovered the absorption of light by [[interstellar dust]], author of the ''[[Morphological Catalogue of Galaxies]]''<br />
* '''[[Ivan Yarkovsky]]''', discovered the [[YORP]] and [[Yarkovsky effect]]s of [[meteoroid]]s or [[asteroid]]s<br />
*'''[[A.L. Zaitsev|Aleksandr Zaitsev]]''', coined the term ''[[Messaging to Extra-Terrestrial Intelligence]]'', conducted the first intercontinental [[radar astronomy]] experiment, transmitted the [[Cosmic Call]]s<br />
*'''[[Felix Ziegel]]''', Soviet researcher, Doctor of Science and docent of Cosmology at the [[Moscow Aviation Institute]], author of more than 40 popular books on astronomy and space exploration, generally regarded as a founder of Russian [[ufology]]<br />
* '''[[Yakov Zel'dovich]]''', physicist, astrophysicist and cosmologist, the first to suggest that [[accretion disc]]s around massive [[black hole]]s are responsible for the [[quasar]] radiation, co-predicted the [[Sunyaev–Zel'dovich effect]] of CMB distortion<br />
<br />
==Physicists==<br />
{{main|Russian physicists}}<br />
[[File:Zhores Alferov.jpg|thumb|95px|'''[[Zhores Alferov|Alferov]]''']]<br />
* '''[[Alexei Alexeyevich Abrikosov|Alexei Abrikosov]]''', discovered how [[magnetic flux]] can penetrate a [[superconductor]] (the [[Abrikosov vortex]]), [[Nobel Prize]] winner<br />
* '''[[Artem Alikhanian]]''', a prominent researcher of [[cosmic rays]], inventor of wide-gap track [[spark chamber]]<br />
* '''[[Franz Aepinus]]''', related [[electricity]] and [[magnetism]], proved the electric nature of [[pyroelectricity]], explained [[electric polarization]] and [[electrostatic induction]], invented [[Achromatic lens|achromatic]] [[microscope]]<br />
* '''[[Abraham Alikhanov]]''', a prominent researcher of [[cosmic rays]], built the first [[nuclear reactor]]s in the [[USSR]]<br />
* '''[[Zhores Alferov]]''', inventor of modern [[heterotransistor]], Nobel Prize winner<br />
* '''[[Semen Altshuler]]''', researched [[Electron paramagnetic resonance|EPR]] and [[NMR]], predicted [[acoustic paramagnetic resonance]][[File:Basov.jpg|thumb|85px|'''[[Nikolay Basov|Basov]]''']]<br />
* '''[[Lev Artsimovich]]''', builder of the first [[tokamak]], researcher of high temperature [[Plasma (physics)|plasma]]<br />
* '''[[Gurgen Askaryan]]''', predicted [[self focusing]] of light, discovered [[Askaryan effect]] in the [[particle physics]]<br />
* '''[[Nikolay Basov]]''', physicist, co-inventor of [[laser]] and [[maser]], Nobel Prize winner<br />
* '''[[Nikolay Bogolyubov]]''', mathematician and theoretical physicist,co-developed the [[BBGKY hierarchy]], formulated a microscopic theory of [[superconductivity]], suggested a triplet [[quark]] model, introduced a new quantum degree of freedom ([[color charge]])<br />
* '''[[Gersh Budker]]''', inventor of [[electron cooling]], co-inventor of [[collider]]<br />
* '''[[Sergey Chaplygin]]''', a founder of [[aerodynamics|aero-]] and [[hydrodynamics]], formulated the [[Chaplygin's equation]]s and [[Chaplygin gas]] concept<br />
* '''[[Pavel Cherenkov]]''', discoverer of [[Cherenkov radiation]], Nobel Prize winner<br />
* '''[[Yuri Denisyuk]]''', inventor of [[Holography|3D holography]]<br />
* '''[[Ludvig Faddeev]]''', discoverer of [[Faddeev–Popov ghosts]] and [[Faddeev equations]] in [[quantum physics]][[File:Cerenkov.jpg|thumb|85px|'''[[Pavel Cherenkov|Cherenkov]]''']]<br />
* '''[[Georgy Flyorov]]''', [[nuclear physicist]], one of the initiators of the [[Soviet atomic bomb project]], co-discoverer of [[seaborgium]] and [[bohrium]], founder of the [[Joint Institute for Nuclear Research]]<br />
* '''[[Vladimir Fock]]''', developed the [[Fock space]], [[Fock state]] and the [[Hartree–Fock method]] in [[quantum mechanics]]<br />
* '''[[Ilya Frank]]''', explained the phenomenonof [[Cherenkov radiation]], Nobel Prize winner<br />
*'''[[Vsevolod Frederiks|Vsevolod Frederiks (Fréedericksz)]]''', discovered the [[Fréedericksz transition]], the Fréedericksz critical field in [[liquid crystal]]s<br />
* '''[[Yakov Frenkel]]''', introduced the notion of [[electron hole]], discovered the [[Frenkel defect]] of a [[crystal lattice]], described the [[Poole–Frenkel effect]] in [[solid-state physics]]<br />
* '''[[Andre Geim]]''', inventor of [[graphene]], developer of [[gecko tape]], Nobel Prize winner and at the same time [[Ig Nobel Prize]] winner for [[diamagnetic levitation]] of a living [[frog]]<br />
* '''[[Vitaly Ginzburg]]''', co-author of the [[Ginzburg–Landau theory]] of [[superconductivity]], a developer of [[hydrogen bomb]], Nobel Prize winner[[File:Pyotr L Kapitsa Russian physicist 1964.jpg|thumb|95px|'''[[Pyotr Kapitsa|Kapitsa]]''']]<br />
* '''[[Vladimir Gribov]]''', introduced [[pomeron]], [[DGLAP]] equations and [[Gribov ambiguity]]<br />
* '''[[Aleksandr Gurevich]]''', author of the [[runaway breakdown]] theory of [[lightning]]<br />
* '''[[Abram Ioffe]]''', founder of the Soviet physics school, tutor of many prominent scientists<br />
* '''[[Dmitri Ivanenko]]''', proposed the first models of [[Nuclear shell model|nuclear shell]] and exchange of [[Atomic nucleus|nuclear forces]], predicted the [[synchrotron radiation]], the author of the hypothesis of [[Color superconductivity|quark stars]]<br />
* '''[[Boris Jacobi]]''', formulated the [[Maximum power theorem]] in [[electrical engineering]], invented [[electroplating]], [[electrotyping]], [[galvanoplastic|galvanoplastic sculpture]] and [[electric boat]][[File:Landau.jpg|thumb|85px|'''[[Lev Landau|Landau]]''']]<br />
* '''[[Pyotr Kapitsa]]''', originated the techniques for creating ultrastrong [[magnetic field]]s, co-discovered a way to measure the magnetic field of an [[atomic nucleus]] discovered [[superfluidity]], Nobel Prize winner<br />
* '''[[Yuly Khariton]]''', chief designer of the [[Soviet atomic bomb]], co-developer of the [[Tsar Bomb]]<br />
* '''[[Orest Khvolson]]''', the first to study the [[Chwolson ring]] effect of [[gravitational lensing]]<br />
* '''[[Sergey Krasnikov]]''', developer of the [[Krasnikov tube]], a speculative mechanism for space travel<br />
* '''[[Igor Kurchatov]]''', builder of the first [[nuclear power plant]], developer of the first [[nuclear reactor]]s for [[Nuclear marine propulsion|surface ships]]<br />
* '''[[Dmitry Lachinov]]''', physicist, electrical engineer, inventor, meteorologist and climatologist<br />
* '''[[Lev Landau]]''', theoretical physicist, developed the [[Ginzburg–Landau theory]] of [[superconductivity]], explained the [[Landau damping]] in [[plasma physics]], pointed out the [[Landau pole]] in [[quantum electrodynamics]], co-author of the famous ''[[Course of Theoretical Physics]]'', Nobel Prize winner<br />
* '''[[Grigory Landsberg]]''', co-discoverer of [[Raman scattering]] of light[[File:Emil Lenz.jpg|thumb|100px|'''[[Heinrich Lenz|Lenz]]''']]<br />
* '''[[Mikhail Lavrentyev]]''', physicist and mathematician, founded the Siberian Division of the [[Soviet Academy of Sciences]] and [[Akademgorodok]] in [[Novosibirsk]]<br />
* '''[[Petr Nikolaevich Lebedev|Pyotr Lebedev]]''', the first to measure the [[radiation pressure]] on a solid body, thus privoving the [[Maxwell's theory of electromagnetism]]<br />
* '''[[Heinrich Lenz]]''', discovered the [[Lenz's law]] of [[electromagnetism]]<br />
* '''[[Evgeny Lifshitz]]''', an author of the [[BKL singularity]] model of the Universe evolution, co-author of the famous ''[[Course of Theoretical Physics]]''<br />
* '''[[Mikhail Lomonosov]]''', polymath scientist, artist and inventor, proposed the law of [[conservation of matter]], disproved the [[phlogiston theory]]<br />
* '''[[Oleg Losev]]''', inventor of [[light-emitting diode]] and [[crystadine]]<br />
* '''[[Alexander Alexeyevich Makarov|Alexander Makarov]]''', inventor of [[orbitrap]]<br />
* '''[[Boris Mamyrin]]''', inventor of [[reflectron]][[File:Vasily petrov.jpg|thumb|95px|'''[[Vasily Vladimirovich Petrov|Petrov]]''']]<br />
* '''[[Leonid Mandelshtam]]''', co-discoverer of [[Raman effect]]<br />
* '''[[Stanislav Mikheyev]]''', co-discoverer of [[Mikheyev–Smirnov–Wolfenstein effect]] of [[neutrino oscillations]]<br />
* '''[[Konstantin Novoselov]]''', inventor of [[graphene]], developer of [[gecko tape]], Nobel Prize winner<br />
* '''[[Vasily Vladimirovich Petrov|Vasily Petrov]]''', discoverer of [[electric arc]], proposed [[arc lamp]] and [[arc welding]]<br />
* '''[[Boris Podolsky]]''', an author of [[EPR Paradox]] in [[quantum physics]]<br />
* '''[[Alexander Markovich Polyakov|Alexander Polyakov]]''', developed the concepts of [[Polyakov action]], [['t Hooft–Polyakov monopole]] and [[BPST instanton]]<br />
* '''[[Isaak Pomeranchuk]]''', predicted [[synchrotron radiation]]<br />
* '''[[Bruno Pontecorvo]]''', a founder of [[neutrino]] [[high energy physics]], his work led to the discovery of [[PMNS matrix]][[File:Aleksandr Prokhorov.jpg|thumb|85px|'''[[Alexander Prokhorov|Prokhorov]]''']]<br />
* '''[[Alexander Stepanovich Popov|Alexander Popov]]''', inventor of [[lightning detector]], one of the [[Invention of radio|inventors of radio]],recorded the first experimental [[radiolocation]] at sea<br />
* '''[[Victor Popov]]''', co-discoverer of [[Faddeev–Popov ghost]]s in [[quantum field theory]]<br />
* '''[[Alexander Prokhorov]]''', co-inventor of [[laser]] and [[maser]], Nobel Prize winner<br />
* '''[[Georg Wilhelm Richmann]]''', inventor of [[electrometer]], pioneer researcher of [[atmospheric electricity]], killed by a [[ball lightning]] in experiment<br />
* '''[[Andrei Sakharov]]''', co-developer of [[tokamak]] and the [[Tsar Bomb]], inventor of [[explosively pumped flux compression generator]], [[Nobel Peace Prize]] winner<br />
* '''[[Nikolay Semyonov]]''', physical chemist, co-discovered a way to measure the magnetic field of an [[atomic nucleus]], Nobel Prize in Chemistry winner<br />
* '''[[Lev Shubnikov]]''', discoverer of [[Shubnikov–de Haas effect]], one of the first researchers of [[solid hydrogen]] and [[liquid helium]]<br />
* '''[[Dmitri Skobeltsyn]]''', the first to use [[cloud chamber]] for studying [[cosmic rays]], the first to observe [[positron]]s<br />
* '''[[Alexei Yuryevich Smirnov|Alexei Smirnov]]''', co-discoverer of [[Mikheyev–Smirnov–Wolfenstein effect]] of [[neutrino oscillations]]<br />
* '''[[Arseny Sokolov]]''', co-discoverer of [[Sokolov–Ternov effect]], a developer of [[synchrotron radiation]] theory[[File:Alexander stoletov.jpg|thumb|90px|'''[[Aleksandr Stoletov|Stoletov]]''']]<br />
* '''[[Igor Tamm]]''', explained the phenomenon of [[Cherenkov radiation]], co-developer of [[tokamak]], Nobel Prize winner<br />
* '''[[Aleksandr Stoletov]]''', inventor of [[photoelectric cell]], built the [[Stoletov curve]] and pioneered the research of [[ferromagnetism]]<br />
* '''[[Igor Ternov]]''', co-discoverer of [[Sokolov–Ternov effect]] of [[synchrotron radiation]]<br />
* '''[[Nikolay Umov]]''', discovered the [[Umov–Poynting vector]] and [[Umov effect]], the first to propose the formula [[Mass–energy equivalence|<small><math>E=kmc^2</math></small>]]<br />
* '''[[Petr Ufimtsev]]''', developed the theory that led to modern [[stealth technology]]<br />
* '''[[Sergey Vavilov]]''', co-discoverer of [[Cherenkov radiation]], formulated the [[Kasha–Vavilov rule]] of [[quantum yield]]s[[File:NikolaiYegorovichZhukovsky.jpg|thumb|90px|'''[[Nikolay Yegorovich Zhukovsky|Zhukovsky]]''']]<br />
* '''[[Vladimir Veksler]]''', inventor of [[synchrophasotron]], co-inventor of [[synchrotron]]<br />
* '''[[Evgeny Velikhov]]''', leader of the international program [[ITER]] (thermonuclear experimental [[tokamak]])<br />
* '''[[Anatoly Vlasov]]''', developed the [[Vlasov equation]] in [[plasma physics]]<br />
* '''[[Alexey Yekimov]]''', discoverer of [[quantum dot]]s<br />
* '''[[Yevgeny Zavoisky]]''', inventor of [[EPR spectroscopy]], co-developer of [[NMR spectroscopy]]<br />
* '''[[Yakov Zel'dovich]]''', physicist and cosmologist, predicted the [[beta decay]] of a [[pi meson]] and the [[muon]] [[catalysis]], co-predicted the [[Sunyaev–Zel'dovich effect]] of CMB distortion<br />
* '''[[Nikolay Yegorovich Zhukovsky|Nikolai Zhukovsky]]''', a founder of [[aerodynamics|aero-]] and [[hydrodynamics]], the first to study airflow, author of [[Joukowsky transform]] and [[Kutta–Joukowski theorem]], founder of [[TsAGI]] and pioneer of aviation<br />
<br />
==Chemists and material scientists==<br />
{{main|Russian chemists}}<br />
[[File:Butlerov A.png|thumb|95px|'''[[Aleksandr Butlerov|Butlerov]]''']]<br />
* '''[[Ernest Beaux]]''', inventor of [[Chanel No. 5]], ''"the world's most legendary fragrance"''<br />
* '''[[Nikolay Beketov]]''', inventor of [[aluminothermy]], a founder of [[physical chemistry]]<br />
* '''[[Friedrich Konrad Beilstein]]''', proposed the [[Beilstein test]] for the detection of [[halogens]], author of the [[Beilstein database]] in [[organic chemistry]]<br />
* '''[[Boris Pavlovich Belousov|Boris Belousov]]''', chemist and biophysicist, discoverer of [[Belousov–Zhabotinsky reaction]], a classical example of [[non-equilibrium thermodynamics]]<br />
* '''[[Alexander Borodin]]''', chemist and composer, the author of the famous opera ''[[Prince Igor]]'', discovered [[Borodin reaction]], co-discovered [[Aldol reaction]]<br />
* '''[[Aleksandr Butlerov]]''', discovered [[hexamine]], [[formaldehyde]] and [[formose reaction]] (the first synthesis of [[sugar]]), the first to incorporate [[double bond]]s into structural formulae, a founder of [[organic chemistry]] and the theory of [[chemical structure]]<br />
* '''[[Dmitry Chernov]]''', founder of modern [[metallography]], discovered [[Polymorphism (materials science)|polymorphism]] in metals, built the [[iron]]-[[carbon]] [[phase diagram]][[File:ClausKE.jpg|thumb|95px|'''[[Karl Ernst Claus|Claus]]''']]<br />
* '''[[Aleksei Chichibabin]]''', discovered [[Chichibabin pyridine synthesis]], [[Bodroux-Chichibabin aldehyde synthesis]] and [[Chichibabin reaction]]<br />
* '''[[Lev Chugaev]]''', discoverer of [[Chugaev elimination]] in [[organic chemistry]]<br />
* '''[[Karl Ernst Claus]]''', chemist and botanist, discoverer of [[ruthenium]]<br />
* '''[[Nikolay Demyanov]]''', discoverer of [[Demjanov rearrangement]] in [[organic chemistry]]<br />
* '''[[Aleksandr Dianin]]''', discoverer of [[Bisphenol A]] and [[Dianin's compound]]<br />
* '''[[Constantin Fahlberg]]''', inventor of [[saccharin]], the first [[artificial sweetener]]<br />
* '''[[Alexey Favorsky]]''', discoverer of [[Favorskii rearrangement]] and [[Favorskii reaction]] in [[organic chemistry]]<br />
* '''[[Alexander Frumkin]]''', a founder of modern [[electrochemistry]], author of the theory of [[electrode]] reactions<br />
* '''[[Evgraf Fedorov]]''', the first to enumerate all of the 230 [[space group]]s of [[crystal]]s, thus founding the modern [[crystallography]][[File:Jewgraf Stepanowitsch Fjodorow.jpg|thumb|95px|'''[[Evgraf Fedorov|Fedorov]]''']]<br />
* '''[[Andre Geim]]''', inventor of [[graphene]], developer of [[gecko tape]], Nobel Prize in Physics winner<br />
* '''[[Igor Gorynin]]''', inventor of [[weldable]] [[titanium]] alloys, high strength [[aluminium]] alloys, and many [[radiation-hardened]] steels<br />
* '''[[Vladimir Ipatieff]]''', inventor of [[Ipatieff bomb]], a founder of [[petrochemistry]]<br />
* '''[[Isidore (inventor)|Isidore]]''', legendary inventor of the [[vodka|Russian vodka]]<br />
* '''[[Boris Jacobi]]''', re-discovered [[electroplating]] and initiated its practical usage<br />
* '''[[Pyotr Kapitsa]]''', discovered [[superfluidity]] while studying [[liquid helium]], Nobel Prize in Physics winner<br />
* '''[[Morris Kharasch]]''', inventor of anti-microbial compound [[thimerosal]][[File:MarkovnikovVV 1870.jpg|thumb|90px|'''[[Vladimir Markovnikov|Markovnikov]]''']]<br />
* '''[[Gottlieb Kirchhoff]]''', discoverer of [[glucose]]<br />
* '''[[Ivan Knunyants]]''', inventor of [[Nylon 6|poly-caprolactam]], founder of Soviet school of [[fluorocarbon]]'s chemistry, a developer of Soviet [[chemical weapons]]<br />
* '''[[Sergei Vasilyevich Lebedev|Sergei Lebedev]]''', inventor of [[polybutadiene]], the first commercially viable [[synthetic rubber]]<br />
* '''[[Mikhail Lomonosov]]''', polymath, coined the term ''[[physical chemistry]]'', re-discovered [[smalt]], proved that the [[phlogiston theory]] was false, the first to record the [[freezing]] of [[mercury (element)|mercury]]<br />
* '''[[Aleksandr Loran]]''', inventor of [[fire fighting foam]]<br />
* '''[[Konstantin Novoselov]]''', inventor of [[graphene]], developer of [[gecko tape]], Nobel Prize in Physics winner[[File:DIMendeleevCab.jpg|thumb|95px|'''[[Dmitri Mendeleev|Mendeleev]]''']]<br />
* '''[[Vladimir Markovnikov]]''', author of the [[Markovnikov's rule]] in [[organic chemistry]], discoverer of [[naphthenes]]<br />
* '''[[Dmitri Mendeleyev]]''', invented the [[Periodic table]] of [[chemical elements]], the first to predict the properties of elements yet to be discovered, invented [[pyrocollodion]], developer of [[pipeline transport|pipelines]] and a prominent researcher of [[vodka]]<br />
* '''[[Nikolai Menshutkin]]''', discoverer of [[Menshutkin reaction]] in [[organic chemistry]]<br />
* '''[[Sergey Namyotkin]]''', a prominent researcher of [[terpenes]], discoverer of [[Nametkin rearrangement]]<br />
* '''[[Ilya Prigogine]]''', researcher of [[dissipative system|dissipative structures]],[[File:Tswett 01.jpg|thumb|95px|'''[[Mikhail Tsvet|Tsvet]]''']] [[complex systems]] and [[irreversibility]], Nobel Prize winner<br />
* '''[[Sergey Reformatsky]]''', discoverer of [[Reformatsky reaction]] in [[organic chemistry]]<br />
* '''[[Nikolay Semyonov]]''', physical chemist, author of the [[chain reaction]] theory, Nobel Prize winner<br />
* '''[[Carl Schmidt (chemist)|Carl Schmidt]]''', analyzed the crystal structure of many biochemicals, proved that [[animal]] and [[plant]] cells are chemically similar<br />
* '''[[Vladimir Shukhov]]''', polymath, inventor of [[chemical cracking]]<br />
*'''[[Mikhail Shultz]]''', physical chemist and artist; one of the creators the [[glass electrode]] theory; author of several thermodynamic methods.<br />
* '''[[Mikhail Tsvet]]''', botanist, inventor of [[chromatography]]<br />
* '''[[Victor Veselago]]''', the first researcher of materials with negative [[permittivity]] and [[Permeability (electromagnetism)|permeability]]<br />
* '''[[Paul Walden]]''', discovered the [[Walden inversion]] and [[ethylammonium nitrate]], the first room temperature [[ionic liquid]]<br />
* '''[[Alexander Mikhaylovich Zaytsev|Alexander Zaytsev]]''', author of the [[Zaitsev's rule]] in [[organic chemistry]]<br />
* '''[[Nikolay Zelinsky]]''', inventor of [[activated charcoal]] [[gas mask]] in [[Europe]] during [[World War I]], co-discoverer of [[Hell-Volhard-Zelinsky halogenation]], a founder of [[petrochemistry]]<br />
* '''[[Nikolai Zinin]]''', discovered [[benzidine]], co-discovered [[aniline]], the first President of the Russian Physical-Chemical Society<br />
* '''[[Anatol Zhabotinsky]]''', discoverer of [[Belousov–Zhabotinsky reaction]], a classical example of [[non-equilibrium thermodynamics]]<br />
<br />
==Structural engineers==<br />
[[File:Augustin de Betancourt in Russian attire, 1810s.jpg|thumb|100px|'''[[Agustín de Betancourt|Betancourt]]''']]<br />
* '''[[Nikolai Belelyubsky]]''', major bridge designer, invented a number of construction schemes<br />
* '''[[Agustín de Betancourt]]''', polymath-engineer and urban planner, designed the [[Moscow Manege]] and the giant cast iron dome of [[St. Isaac's Cathedral]], founded [[Goznak]]<br />
* '''[[Vladimir Barmin]]''', designer of the world's first rocket [[launch complex]] ([[Baikonur Cosmodrome]])<br />
* '''[[Akinfiy Demidov]]''', builder of the [[Leaning Tower of Nevyansk]],the first structure to employ [[rebar]]s and [[cast iron]] [[cupola]], as well as the first [[lightning rod]] in the Western world<br />
* '''[[Alexey Dushkin]]''', designer of the first [[deep column station]], ''[[Mayakovskaya (Moscow Metro)|Mayakovskaya]]''<br />
* '''[[Alexander Hrennikoff]]''', founder of the [[Finite Element Method]]<br />
[[File:Vladimir Grigoryevich Shukhov 1891.jpg|thumb|90px|'''[[Vladimir Shukhov|Shukhov]]''']]<br />
* '''[[Nikolai Nikitin]]''', engineer of the largest Soviet structures: [[Moscow State University]], [[Luzhniki Stadium]], [[The Motherland Calls]] and [[Ostankino Tower]] (once the world's [[tallest freestanding structure]])<br />
* '''[[Lavr Proskuryakov]]''', builder of multiple bridges along the [[Trans-Siberian Railway]], inventor and tutor<br />
* '''[[Vladimir Shukhov]]''', engineer-polymath, inventor of breakthrough [[industrial design]]s ([[hyperboloid structure]], [[thin-shell structure]], [[tensile structure]], [[gridshell]]), builder of [[Shukhov Tower]]s and multiple other structures<br />
<br />
==Aerospace engineers==<br />
{{main|Russian aerospace engineers}}<br />
[[File:Olegantonov.jpg|thumb|90px|'''[[Oleg Antonov|Antonov]]''']]<br />
* '''[[Rostislav Alexeyev]]''', designer of high-speed [[Raketa hydrofoil]]s and [[ekranoplan]]s, including the ''[[Caspian Sea Monster]]''<br />
* '''[[Oleg Antonov]]''', designer of the [[Antonov|An]]-series aircraft, including ''[[Antonov A-40|A-40]]'' [[winged tank]] and ''[[An-124]]'' (the largest serial [[cargo aircraft]], later modified to world's largest [[fixed-wing aircraft]] ''[[An-225]]'')<br />
* '''[[Georgy Babakin]]''', designed the first [[soft lander spacecraft]] ''[[Luna 9]]''<br />
* '''[[Vladimir Barmin]]''', designer of the first rocket [[launch complex]] ([[Baikonur Cosmodrome]])<br />
* '''[[Robert Bartini]]''', developer of [[ekranoplan]]s and [[VTOL]] [[amphibious aircraft]]s, physicist, tutor to many other aerospace designers<br />
* '''[[Alexander Bereznyak]]''', designer of the first fighter [[rocket-powered aircraft]], ''[[BI-1]]''<br />
* '''[[Georgy Beriev]]''', designer of the [[Beriev|Be]]-series [[amphibious aircraft]][[File:Valentin Glushko on a 2008 Russian coin; RR5110-0084R.png|thumb|105px|'''[[Valentyn Glushko|Glushko]]''']]<br />
* '''[[Georgy Bothezat]]''', inventor of [[quadcopter]] helicopter (''[[The Flying Octopus]]'')<br />
* '''[[Vladimir Chelomey]]''', designer of the first [[space station]] ''[[Salyut 1]]'', creator of [[Proton rocket]] ([[Comparison of heavy lift launch systems|the most used heavy lift launch system]])<br />
* '''[[Evgeniy Chertovsky]]''', inventor of [[pressure suit]]<br />
* '''[[Nicolas Florine]]''', builder of the first successful [[tandem rotor]] helicopter<br />
* '''[[Valentyn Glushko]]''', inventor of [[hypergolic propellant]] and [[electrically powered spacecraft propulsion]], designer of the world's most powerful [[liquid-fuel rocket]] engine [[RD-170]][[File:Sergey Korolyov 140-190 for collage.jpg|thumb|90px|'''[[Sergei Korolyov|Korolyov]]''']]<br />
* '''[[Pyotr Grushin]]''', inventor of [[anti-ballistic missile]]<br />
* '''[[Mikhail Gurevich (aircraft designer)|Mikhail Gurevich]]''', designer of the [[MiG]]-series fighter aircraft, including [[most produced aircraft|world's most produced]] [[jet aircraft]] ''[[MiG-15]]'' and most produced [[supersonic aircraft]] ''[[MiG-21]]''<br />
* '''[[Sergey Ilyushin]]''', designed the [[Ilyushin|Il]]-series fighter aircraft, including ''[[Ilyushin Il-2|Il-2]]'' bomber (the [[most produced military aircraft]] in history)<br />
* '''[[Aleksei Isaev]]''', designer of the first [[rocket-powered aircraft|rocket-powered]] fighter aircraft, ''[[Bereznyak-Isayev BI-1|BI-1]]''<br />
* '''[[Mstislav Keldysh]]''', co-developer of the first [[satellite]] (''[[Sputnik]]'') and [[Keldysh bomber]]<br />
* '''[[Kerim Kerimov]]''', the secret figure behind the Soviet space program<br />
* '''[[Nikolay Kamov]]''', designed the [[Kamov|Ka]]-series [[coaxial rotor]] helicopters[[File:Gleb Kotelnikov.jpg|thumb|90px|'''[[Gleb Kotelnikov|Kotelnikov]]''']]<br />
* '''[[Alexander Kemurdzhian]]''', inventor of [[space rover]] (''[[Lunokhod]]'')<br />
* '''[[Sergei Korolyov]]''', ''the Farther of the [[Soviet space program]]'', inventor of the first [[intercontinental ballistic missile]] and the first [[space rocket]] (''[[R-7 Semyorka]]''), creator of the first [[satellite]] (''[[Sputnik]]''), supervisor of the [[first human spaceflight]]<br />
* '''[[Gleb Kotelnikov]]''', inventor of [[knapsack parachute]] and [[drogue parachute]]<br />
* '''[[Semyon Lavochkin]]''', designer of the [[Lavochkin|La]]-series aircraft and the first operational [[surface-to-air missile]] ''[[S-25 Berkut]]''<br />
* '''[[Mikhail Lomonosov]]''', polymath, inventor of [[coaxial rotor]] and the first [[helicopter]]<br />
* '''[[Gleb Lozino-Lozinskiy]]''', designer of the [[Buran space shuttle]] and [[Spiral project]]<br />
* '''[[Arkhip Lyulka]]''', designer of the [[Lyulka]]-series [[aircraft engine]]s, including the first double jet [[turbofan]]<br />
* '''[[Victor Makeev]]''', developer of the first intercontinental [[SLBM]]<br />
* '''[[Artem Mikoyan]]''', designer of the [[MiG]]-series fighter aircraft, including world's most produced jet ''[[MiG-15]]'' and most produced [[supersonic aircraft]] ''[[MiG-21]]''<br />
* '''[[Mikhail Mil]]''', designer of the [[Mil Helicopters|Mi]]-series helicopters, including ''[[Mil Mi-8]]'' (the world's [[most produced helicopter]]) and ''[[Mil Mi-12]]'' (the world's largest helicopter)<br />
* '''[[Alexander Mozhaysky]]''', author of the first attempt to create [[heavier-than-air craft]] in [[Russia]], designed the largest of 19th century [[airplane]]s<br />
* '''[[Alexander Nadiradze]]''', designer of the first mobile [[ICBM]] ''[[RT-21 Temp 2S]]'' and the first reliable mobile ICBM ''[[RT-2PM Topol]]''<br />
* '''[[Nikolai Polikarpov]]''', designer of the [[Polikarpov|Po]]-series aircraft, including ''[[Po-2]]'' ''[[Polikarpov Po-2|Kukuruznik]]'' (world's [[most produced biplane]])<br />
* '''[[Alexander Procofieff de Seversky]]''', inventor of [[ionocraft]] and [[gyroscopic]]ally stabilized [[bombsight]]<br />
* '''[[Guy Severin]]''', designed the first [[spacewalk]] supporting system<br />
* '''[[Igor Sikorsky]]''', inventor of [[airliner]] and [[strategic bomber]] (''[[Sikorsky Ilya Muromets]]''), father of modern [[helicopter]], founder of the [[Sikorsky Aircraft]]<br />
* '''[[Boris Shavyrin]]''', inventor of [[air-augmented rocket]]<br />
* '''[[Pavel Sukhoi]]''', designer of the [[Sukhoi|Su]]-series fighter aircraft<br />
* '''[[Vladimir Syromyatnikov]]''', designer of the ''[[Androgynous Peripheral Attach System]]''<br />
* '''[[Mikhail Tikhonravov]]''', designer of [[Sputnik]]s, including the first artificial satellite ''[[Sputnik 1]]''[[File:Tsiolkovsky.jpg|thumb|100px|'''[[Tsiolkovsky]]''']]<br />
* '''[[Konstantin Tsiolkovsky]]''', principal pioneer of [[astronautics]]<br />
* '''[[Alexei Tupolev]]''', designer of the [[Tupolev|Tu]]-series aircraft, including the first [[supersonic transport]] ''[[Tu-144]]''<br />
* '''[[Andrey Tupolev]]''', designer of the Tu-series aircraft, including the [[turboprop]] long-range airliner ''[[Tu-114]]'' and [[turboprop]] strategic bomber ''[[Tu-95]]''<br />
* '''[[Vladimir Vakhmistrov]]''', supervisor of [[Zveno project]] (the first bomber with [[parasite aircraft]]s)<br />
* '''[[Alexander Sergeyevich Yakovlev|Alexander Yakovlev]]''', designer of the [[Yakovlev|Yak]]-series aircraft, including the first [[regional jet]] ''[[Yak-40]]''<br />
* '''[[Friedrich Zander]]''', designed the first [[liquid-fuel rocket]] in the Soviet Union, [[GIRD]]-X, pioneer of astronautics<br />
* '''[[Nikolay Yegorovich Zhukovsky|Nikolai Zhukovsky]]''', founder of modern [[aerodynamics|aero-]] and [[hydrodynamics]], pioneer of [[aviation]]<br />
<br />
==Naval engineers==<br />
{{main|Russian naval engineers}}<br />
[[File:Alexey Krylov 1910s.JPG|thumb|90px|'''[[Alexey Krylov|Krylov]]''']]<br />
* '''[[Rostislav Alexeyev]]''', designer of high-speed [[Raketa hydrofoil]]s and [[ekranoplan]]s, including the ''[[Caspian Sea Monster]]''<br />
* '''[[Anatoly Petrovich Alexandrov|Anatoly Alexandrov]]''', inventor of [[degaussing]], developer of [[naval nuclear reactor]]s (including one for the first [[nuclear icebreaker]])<br />
* '''[[Mikhail Britnev]]''', designer of the first [[metal]]-[[ship hull|hull]] [[icebreaker]] ''[[Pilot (icebreaker)|Pilot]]''<br />
* '''[[Stefan Drzewiecki]]''', inventor of electric-powered submarine and [[midget submarine]], designed the first serial submarine, developed the [[blade element theory]]<br />
* '''[[Boris Jacobi]]''', inventor of [[electric boat]], developer of modern [[naval mining]]<br />
* '''[[Konstantin Khrenov]]''', inventor of [[underwater welding]]<br />
* '''[[Alexei Krylov]]''', inventor of [[gyroscopic]] [[damping]] of ships, author of the [[insubmersibility]] theory[[File:SO Makarov 01.jpg|thumb|90px|'''[[Stepan Makarov|Makarov]]''']]<br />
* '''[[Fyodor Litke]]''', explorer, inventor of recording [[tide measurer]]<br />
* '''[[Stepan Makarov]]''', Admiral, war hero, oceanographer, inventor of [[torpedo boat tender]], builder of [[Icebreaker Yermak|the first polar icebreaker]], author of the insubmersibility theory<br />
* '''[[Victor Makeev]]''', developer of the first intercontinental [[submarine-launched ballistic missile]]<br />
* '''[[Ludvig Nobel]]''', designer of the modern [[oil tanker]]<br />
* '''[[Peter the Great]]''', monarch and craftsman, inventor of [[yacht club]] and [[sounding line]] with separating [[Plumb-bob|plummet]], founder of the [[Russian Navy]]<br />
* '''[[Pavel Schilling]]''', inventor of [[electric]] [[naval mine]]<br />
* '''[[Igor Spassky]]''', designer of the [[Sea Launch]] platform and over 200 [[nuclear submarine]]s, including the world's largest submarines (''[[Typhoon class]]'')<br />
* '''[[Vladimir Yourkevitch]]''', designer of ''[[SS Normandie]]'', developer of modern [[ship hull]] design<br />
<br />
==Electrical engineers==<br />
{{main|Russian electrical engineers}}<br />
[[File:Pavel Shilling.jpg|thumb|95px|[[Pavel Schilling|Schilling]]]]<br />
* '''[[Zhores Alferov]]''', physicist, inventor of [[heterotransistor]], [[Nobel Prize]] winner<br />
* '''[[Nikolay Benardos]]''', inventor of [[carbon arc welding]] (the first practical [[arc welding]] method)<br />
* '''[[Mikhail Dolivo-Dobrovolsky]]''', inventor of [[three-phase electric power]]<br />
* '''[[Boris Jacobi]]''', inventor of [[electroplating]], [[electrotyping]], [[galvanoplastic sculpture]] and [[electric boat]]<br />
* '''[[Konstantin Khrenov]]''', inventor of [[underwater welding]]<br />
* '''[[Dmitry Lachinov]]''', inventor ofelectricity [[economizer]], [[electrical insulation]] tester, pioneer of long-distance [[electricity transmission]]<br />
* '''[[Alexander Lodygin]]''', one of the inventors of [[incandescent light bulb]], inventor of [[electric]] [[streetlight]] and [[tungsten filament]]<br />
* '''[[Oleg Losev]]''', inventor of [[light-emitting diode]] and [[crystadine]][[File:Yablochkov 1.jpg|thumb|95px|'''[[Yablochkov]]''']]<br />
* '''[[Vasily Vladimirovich Petrov|Vasily Petrov]]''', inventor of [[electric arc]] and [[arc welding]]<br />
* '''[[Fyodor Pirotsky]]''', inventor of [[railway electrification system]] and [[electric tram]]<br />
* '''[[Alexander Poniatoff]]''', inventor of [[videotape recorder]]<br />
* '''[[Georg Wilhelm Richmann]]''', inventor of [[electrometer]], died from [[ball lightning]] during an experiment<br />
* '''[[Pavel Schilling]]''', inventor of [[shielded cable]], [[naval mine|electric mine]] and [[electromagnetic telegraph]]<br />
* '''[[Nikolay Slavyanov]]''', inventor of [[shielded metal arc welding]]<br />
* '''[[Aleksandr Stoletov]]''', physicist, inventor of [[photoelectric cell]]<br />
* '''[[Pavel Yablochkov]]''', inventor of [[Yablochkov candle]] (the first commercially viable [[electric lamp]]), [[AC transformer]] and [[headlamp]]<br />
<br />
==Computer scientists==<br />
{{main|Russian IT developers}}<br />
[[File:Victor Glushkov envelope-cropped.jpg|thumb|95px|'''[[Victor Glushkov|Glushkov]]''']]<br />
* '''[[Georgy Adelson-Velsky]]''', inventor of [[AVL tree]] algorithm, developer of [[Kaissa]] (the first [[World Computer Chess Champion]])<br />
* '''[[Boris Babaian]]''', developer of the [[Elbrus supercomputer]]s<br />
* '''[[Sergey Brin]]''', inventor of the [[Google web search engine]]<br />
* '''[[Nikolay Brusentsov]]''', inventor of [[ternary computer]] (''[[Setun]]'')<br />
* '''[[Mikhail Donskoy]]''', a leading developer of [[Kaissa]], the first computer chess champion<br />
* '''[[Victor Glushkov]]''', a founder of [[cybernetics]], inventor of the first [[personal computer]] ''[[MIR (computer)|MIR]]''<br />
* '''[[Yevgeny Kaspersky]]''', developer of [[Kaspersky]] [[anti-virus product]]s[[File:Eugene Kaspersky 2007.jpg|thumb|85px|'''[[Yevgeny Kaspersky|Kaspersky]]''']]<br />
* '''[[Semen Korsakov]]''', the first to use [[punched card]]s for information storage and search<br />
* '''[[Evgeny Landis]]''', inventor of [[AVL tree]] algorithm<br />
* '''[[Sergey Alexeyevich Lebedev|Sergey Lebedev]]''', developer of the first Soviet and European [[electronic computer]]s, [[MESM]] and [[BESM]]<br />
* '''[[Leonid Levin]]''', IT scientist, developed the [[Cook-Levin theorem]]<br />
* '''[[Willgodt Theophil Odhner]]''', inventor of the [[Odhner Arithmometer]], the most popular mechanical calculator in the 20th century<br />
* '''[[Alexey Pajitnov]]''', inventor of ''[[Tetris]]''<br />
* '''[[Alexander Razborov]]''', [[mathematician]] and [[computational theorist]] who won the [[Nevanlinna Prize]] in 1990 and the [[Gödel Prize]] for contributions to [[Theory of computation|computer sciences]]<br />
* '''[[Eugene Roshal]]''', developer of the [[FAR file manager]], [[RAR file format]], [[WinRAR]] [[file archiver]]<br />
* '''[[Valentin Turchin]]''', inventor of [[Refal programming language]], introduced [[metasystem transition]] and [[supercompilation]][[File:Alexey Pajitnov - 2575833305 (crop).jpg|thumb|90px|'''[[Pajitnov]]''']]<br />
* '''[[David Yang]]''', developer of [[Cybiko]], founder of [[ABBYY]] company<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
== See also ==<br />
* [[List of scientists]]<br />
* [[List of Russian inventors]]<br />
* [[Science and technology in Russia]]<br />
* [[Science and technology in the Soviet Union]]<br />
* [[Timeline of Russian inventions]]<br />
{{Lists of Russians|state=uncollapsed}}<br />
<br />
{{DEFAULTSORT:List Of Russian Scientists}}<br />
[[Category:Russian scientists| ]]<br />
[[Category:Lists of scientists by nationality|Russian]]<br />
[[Category:Science and technology in Russia| ]]<br />
[[Category:Soviet scientists| ]]<br />
[[Category:Lists of Russian people by occupation|Scientists]]<br />
[[Category:Science and technology in the Soviet Union]]<br />
[[Category:Ukrainian scientists]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Stable_polynomial&diff=9904
Stable polynomial
2013-11-02T20:01:40Z
<p>99.153.64.179: oops typo criterium->criterion</p>
<hr />
<div>In [[mathematics]], the '''Barnes G-function''' ''G''(''z'') is a [[function (mathematics)|function]] that is an extension of [[superfactorial]]s to the [[complex number]]s. It is related to the [[Gamma function]], the [[K-function]] and the [[Glaisher–Kinkelin constant]], and was named after [[mathematician]] [[Ernest William Barnes]].<ref>E.W.Barnes, "The theory of the G-function", ''Quarterly Journ. Pure and Appl. Math.'' '''31''' (1900), 264–314.</ref> Up to elementary factors, it is a special case of the [[double gamma function]].<br />
<br />
<br />
Formally, the Barnes ''G''-function is defined in the following [[Weierstrass product]] form:<br />
<br />
<br />
:<math> G(1+z)=(2\pi)^{z/2} \text{exp}\left(- \frac{z+z^2(1+\gamma)}{2} \right) \, \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)^k \text{exp}\left(\frac{z^2}{2k}-z\right) </math><br />
<br />
<br />
where <math>\, \gamma \, </math> is the [[Euler–Mascheroni constant]], exp(''x'') = ''e''<sup>''x''</sup>, and ∏ is [[capital pi notation]].<br />
<br />
<br />
==Functional equation and integer arguments==<br />
<br />
<br />
The Barnes ''G''-function satisfies the [[functional equation]]<br />
<br />
<br />
:<math> G(z+1)=\Gamma(z)\, G(z) </math><br />
<br />
<br />
with normalisation ''G''(1)&nbsp;=&nbsp;1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler [[Gamma function]]: <br />
<br />
<br />
:<math> \Gamma(z+1)=z \, \Gamma(z) </math><br />
<br />
<br />
The functional equation implies that G takes the following values at [[integer]] arguments:<br />
<br />
<br />
:<math>G(n)=\begin{cases} 0&\text{if }n=0,-1,-2,\dots\\ \prod_{i=0}^{n-2} i!&\text{if }n=1,2,\dots\end{cases}</math><br />
<br />
<br />
and thus<br />
<br />
:<math>G(n)=\frac{(\Gamma(n))^{n-1}}{K(n)}</math><br />
<br />
<br />
where <math>\,\Gamma(x)\,</math> denotes the [[Gamma function]] and ''K'' denotes the [[K-function]]. The functional equation uniquely defines the G function if the convexity condition: <math>\, \frac{d^3}{dx^3}G(x)\geq 0\, </math> is added.<ref>M. F. Vignéras, ''L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL<math>(2,\mathbb{Z})</math>'', Astérisque '''61''', 235–249 (1979).</ref><br />
<br />
==Reflection formula 1.0==<br />
<br />
The [[difference equation]] for the G function, in conjunction with the [[functional equation]] for the [[Gamma function]], can be used to obtain the following [[reflection formula]] for the Barnes G function (originally proved by [[Hermann Kinkelin]]):<br />
<br />
<br />
:<math> \log G(1-z) = \log G(1+z)-z\log 2\pi+ \int_0^z \pi x \cot \pi x \, dx.</math><br />
<br />
<br />
The logtangent integral on the right-hand side can be evaluated in terms of the [[Clausen function]] (of order 2), as is shown below:<br />
<br />
<br />
:<math>2\pi \log\left( \frac{G(1-z)}{G(1+z)} \right)= 2\pi z\log\left(\frac{\sin\pi z}{\pi} \right)+\text{Cl}_2(2\pi z)</math><br />
<br />
<br />
The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation <math>\, Lc(z)\, </math> for the logtangent integral, and using the fact that <math>\,(d/dx) \log(\sin\pi x)=\pi\cot\pi x\,</math>, an integration by parts gives<br />
<br />
<br />
:<math>Lc(z)=\int_0^z\pi x\log(\sin \pi x)\,dx=z\log(\sin \pi z)-\int_0^z\log(\sin \pi x)\,dx=</math><br />
<br />
<br />
:<math>z\log(\sin \pi z)-\int_0^z\Bigg[\log(2\sin \pi x)-\log 2\Bigg]\,dx=</math><br />
<br />
<br />
:<math>z\log(2\sin \pi z)-\int_0^z\log(2\sin \pi x)\,dx</math><br />
<br />
<br />
Performing the integral substitution <math>\, y=2\pi x \Rightarrow dx=dy/(2\pi)\,</math> gives<br />
<br />
<br />
:<math>z\log(2\sin \pi z)-\frac{1}{2\pi}\int_0^{2\pi z}\log\left(2\sin \pi \frac{y}{2} \right)\,dy</math><br />
<br />
<br />
The [[Clausen function]] - of second order - has the integral representation<br />
<br />
<br />
:<math>\text{Cl}_2(\theta) = -\int_0^{\theta}\log\Bigg|2\sin \frac{x}{2} \Bigg|\,dx</math><br />
<br />
<br />
However, within the interval <math>\, 0 < \theta < 2\pi \,</math>, the [[absolute value]] sign within the [[integrand]] can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent itegral, the following relation clearly holds:<br />
<br />
<br />
:<math>Lc(z)=z\log(2\sin \pi z)+\frac{1}{2\pi}\, \text{Cl}_2(2\pi z)</math><br />
<br />
<br />
Thus, after a slight rearrangement of terms, the proof is complete:<br />
<br />
<br />
:<math>2\pi \log\left( \frac{G(1-z)}{G(1+z)} \right)= 2\pi z\log\left(\frac{\sin\pi z}{\pi} \right)+\text{Cl}_2(2\pi z)\, . \, \Box </math><br />
<br />
<br />
Using the relation <math>\, G(1+z)=\Gamma(z)\, G(z) \,</math> and dividing the reflection formula by a factor of <math>\, 2\pi \,</math> gives the equivalent form:<br />
<br />
<br />
:<math> \log\left( \frac{G(1-z)}{G(z)} \right)= z\log\left(\frac{\sin\pi z}{\pi} <br />
\right)+\log\Gamma(z)+\frac{1}{2\pi}\text{Cl}_2(2\pi z) </math><br />
<br />
<br />
Ref: see '''Adamchik''' below for an equivalent form of the [[reflection formula]], but with a different proof.<br />
<br />
==Reflection formula 2.0==<br />
<br />
<br />
Replacing '''z''' with '''(1/2)-z''' in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving [[Bernoulli polynomials]]):<br />
<br />
<br />
:<math>\log\left( \frac{ G\left(\frac{1}{2}+z\right) }{ G\left(\frac{1}{2}-z\right) } \right) =</math><br />
<br />
<br />
:<math><br />
\log \Gamma \left(\frac{1}{2}-z \right) + B_1(z) \log 2\pi-\frac{1}{2}\log 2+\pi \int_0^z B_1(x) \tan \pi x \,dx</math><br />
<br />
<br />
<br />
==Taylor series expansion==<br />
<br />
<br />
By [[Taylor's theorem]], and considering the logarithmic [[derivatives]] of the Barnes function, the following series expansion can be obtained:<br />
<br />
<br />
:<math>\log G(1+z)= \frac{z}{2}\log 2\pi -\left( \frac{z+(1+\gamma)z^2}{2} \right) + \sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1}</math><br />
<br />
<br />
It is valid for <math>\, 0 < z < 1 \,</math>. Here, <math>\, \zeta(x) \,</math> is the [[Riemann Zeta function]]:<br />
<br />
<br />
:<math> \zeta(x)=\sum_{k=1}^{\infty}\frac{1}{k^x} </math><br />
<br />
<br />
Exponentiating both sides of the Taylor expansion gives:<br />
<br />
<br />
:<math> G(1+z)=\exp \left[ \frac{z}{2}\log 2\pi -\left( \frac{z+(1+\gamma)z^2}{2} \right) + \sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right]=</math><br />
<br />
<br />
:<math>(2\pi)^{z/2}\text{exp}\left( -\frac{z+(1+\gamma)z^2}{2} \right) \exp \left[\sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right]</math><br />
<br />
<br />
Comparing this with the [[Weierstrass product]] form of the Barnes function gives the following relation:<br />
<br />
<br />
:<math>\exp \left[\sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right] = \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)^k\text{exp}\left(\frac{z^2}{2k}-z\right)</math><br />
<br />
==Multiplication formula==<br />
<br />
<br />
Like the Gamma function, the G-function also has a multiplication formula:<ref>I. Vardi, ''Determinants of Laplacians and multiple gamma functions'', SIAM J. Math. Anal. '''19''', 493–507 (1988).</ref><br />
<br />
<br />
:<math><br />
G(nz)= K(n) n^{n^{2}z^{2}/2-nz} (2\pi)^{-\frac{n^2-n}{2}z}\prod_{i=0}^{n-1}\prod_{j=0}^{n-1}G\left(z+\frac{i+j}{n}\right)<br />
</math><br />
<br />
<br />
where <math>K(n)</math> is a constant given by:<br />
<br />
<br />
:<math> K(n)= e^{-(n^2-1)\zeta^\prime(-1)} \cdot<br />
n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/2}\,=\,<br />
(Ae^{-\frac{1}{12}})^{n^2-1}\cdot n^{\frac{5}{12}}\cdot (2\pi)^{(n-1)/2}.</math><br />
<br />
<br />
Here <math>\zeta^\prime</math> is the derivative of the [[Riemann zeta function]] and <math>A</math> is the [[Glaisher–Kinkelin constant]].<br />
<br />
== Asymptotic expansion ==<br />
<br />
<br />
The [[logarithm]] of ''G''(''z'' + 1) has the following asymptotic expansion, as established by Barnes:<br />
<br />
<br />
:<math> \log G(z+1)=</math><br />
<br />
<br />
:<math>\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~<br />
\sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right).</math><br />
<br />
<br />
Here the <math>B_{k}</math> are the [[Bernoulli numbers]] and <math>A</math> is the [[Glaisher–Kinkelin constant]]. (Note that somewhat confusingly at the time of Barnes <ref>E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.</ref> the [[Bernoulli number]] <math>B_{2k}</math> would have been written as <math>(-1)^{k+1} B_k </math>, but this convention is no longer current.) This expansion is valid for <math>z </math> in any sector not containing the negative real axis with <math>|z|</math> large.<br />
<br />
==Relation to the Loggamma integral==<br />
<br />
<br />
The parametric Loggamma can be evaluated in terms of the Barnes G-function (Ref: this result is found in '''Adamchik''' below, but stated without proof):<br />
<br />
<br />
:<math> \int_0^z \log \Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log\Gamma(z) -\log G(1+z) </math><br />
<br />
<br />
The proof is somewhat indirect, and involves first considering the logarithmic difference of the [[Gamma function]] and Barnes G-function:<br />
<br />
:<math>z\log \Gamma(z)-\log G(1+z)</math><br />
<br />
<br />
Where<br />
<br />
<br />
:<math>\frac{1}{\Gamma(z)}= z e^{\gamma} \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</math><br />
<br />
<br />
and <math>\,\gamma\,</math> is the [[Euler-Mascheroni constant]].<br />
<br />
<br />
Taking the logarithm of the [[Weierstrass product]] forms of the Barnes function and Gamma function gives:<br />
<br />
<br />
:<math>z\log \Gamma(z)-\log G(1+z)=-z \log\left(\frac{1}{\Gamma (z)}\right)-\log G(1+z)=</math><br />
<br />
<br />
:<math>-z \left[ \log z+\gamma z +\sum_{k=1}^{\infty} \Bigg\{ \log\left(1+\frac{z}{k} \right) -\frac{z}{k} \Bigg\} \right]</math><br />
<br />
<br />
:<math>-\left[ \frac{z}{2}\log 2\pi -\frac{z}{2}-\frac{z^2}{2} -\frac{z^2 \gamma}{2} + \sum_{k=1}^{\infty} \Bigg\{k\log\left(1+\frac{z}{k}\right) +\frac{z^2}{2k} -z \Bigg\} \right]</math><br />
<br />
<br />
A little simplification and re-ordering of terms gives the series expansion:<br />
<br />
<br />
:<math> \sum_{k=1}^{\infty} \Bigg\{ (k+z)\log \left(1+\frac{z}{k}\right)-\frac{z^2}{2k}-z \Bigg\}=</math><br />
<br />
<br />
:<math>-z\log z-\frac{z}{2}\log 2\pi +\frac{z}{2} +\frac{z^2}{2}- \frac{z^2 \gamma}{2}- z\log\Gamma(z) +\log G(1+z)</math><br />
<br />
<br />
Finally, take the logarithm of the [[Weierstrass product]] form of the [[Gamma function]], and integrate over the interval <math>\, [0,\,z]\, </math> to obtain:<br />
<br />
<br />
:<math>\int_0^z\log\Gamma(x)\,dx=-\int_0^z \log\left(\frac{1}{\Gamma(x)}\right)\,dx=</math><br />
<br />
<br />
:<math>-(z\log z-z)-\frac{z^2 \gamma}{2}- \sum_{k=1}^{\infty} \Bigg\{ (k+z)\log \left(1+\frac{z}{k}\right)-\frac{z^2}{2k}-z \Bigg\}</math><br />
<br />
<br />
Equating the two evaluations completes the proof:<br />
<br />
<br />
:<math> \int_0^z \log \Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log\Gamma(z) -\log G(1+z)\, . \, \Box</math><br />
<br />
==References==<br />
<references/><br />
<br />
*{{dlmf|first=R.A. |last=Askey|first2=R.|last2=Roy|id=5.17}}<br />
{{DEFAULTSORT:Barnes G-Function}}<br />
[[Category:Number theory]]<br />
[[Category:Special functions]]<br />
<br />
*{{cite web|last=Adamchik|first=Viktor S.|title=Contributions to the Theory of the Barnes function|url=http://arxiv.org/pdf/math/0308086v1.pdf|accessdate=2003}}</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Jury_stability_criterion&diff=17861
Jury stability criterion
2013-11-02T19:04:26Z
<p>99.153.64.179: wikilink</p>
<hr />
<div>{{Trigonometry}}<br />
{|class="wikitable" style="background-color:#FFFFFF; margin:1em 0 1em 1em" align="right" hspace="3" <br />
! Function<br />
! Derivative<br />
|-<br />
| <math>\sin(x)</math><br />
| <math>\cos(x)</math><br />
|-<br />
|<math>\cos(x)</math><br />
|<math>-\sin(x)</math><br />
|-<br />
|<math>\tan(x)</math><br />
|<math>\sec^2(x)</math><br />
|-<br />
|<math>\cot(x)</math><br />
|<math>-\csc^2(x)</math><br />
|-<br />
|<math>\sec(x)</math><br />
|<math>\sec(x)\tan(x)</math><br />
|-<br />
|<math>\csc(x)</math><br />
|<math>-\csc(x)\cot(x)</math><br />
|-<br />
|<math>\arcsin(x)</math><br />
|<math>\frac{1}{\sqrt{1-x^2}}</math><br />
|-<br />
|<math>\arccos(x)</math><br />
|<math>\frac{-1}{\sqrt{1-x^2}}</math><br />
|-<br />
|<math>\arctan(x)</math><br />
|<math>\frac{1}{x^2+1}</math><br />
|}<br />
The '''differentiation of trigonometric functions''' is the mathematical process of finding the rate at which a [[trigonometric function]] changes with respect to a variable—the [[derivative]] of the trigonometric function. Commonplace trigonometric functions include sin(''x''), cos(''x'') and tan(''x''). For example, in differentiating ''f''(''x'') = sin(''x''), one is calculating a function ''f''&nbsp;′(''x'') which computes the rate of change of sin(''x'') at a particular point ''a''. The value of the rate of change at ''a'' is thus given by ''f''&nbsp;′(''a''). Knowledge of differentiation from first principles is required, along with competence in the use of [[trigonometric identities]] and limits. All functions involve the arbitrary variable ''x'', with all differentiation performed with respect to ''x''.<br />
<br />
It turns out that once one knows the derivatives of sin(''x'') and cos(''x''), one can easily compute the derivatives of the other circular trigonometric functions because they can all be expressed in terms of sine or cosine; the [[quotient rule]] is then implemented to differentiate this expression. Proofs of the derivatives of sin(''x'') and cos(''x'') are given in the proofs section; the results are quoted in order to give proofs of the derivatives of the other circular trigonometric functions. Finding the derivatives of the [[inverse trigonometric functions]] involves using [[implicit differentiation]] and the derivatives of regular trigonometric functions also given in the proofs section.<br />
<br />
==Derivatives of trigonometric functions and their inverses==<br />
:<math> \left(\sin(x)\right)' = \cos(x)</math><br />
:<math> \left(\cos(x)\right)' = -\sin(x)</math><br />
:<math> \left(\tan(x)\right)' = \left(\frac{\sin(x)}{\cos(x)}\right)' = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} = \sec^2(x)</math><br />
:<math> \left(\cot(x)\right)' = \left(\frac{\cos(x)}{\sin(x)}\right)' = \frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)} = -(1+\cot^2(x)) = -\csc^2(x)</math><br />
:<math> \left(\sec(x)\right)' = \left(\frac{1}{\cos(x)}\right)' = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)}.\frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x)</math><br />
:<math> \left(\csc(x)\right)' = \left(\frac{1}{\sin(x)}\right)' = -\frac{\cos(x)}{\sin^2(x)} = -\frac{1}{\sin(x)}.\frac{\cos(x)}{\sin(x)} = -\csc(x)\cot(x)</math><br />
:<math> \left(\arcsin(x)\right)' = \frac{1}{\sqrt{1-x^2}}</math><br />
:<math> \left(\arccos(x)\right)' = \frac{-1}{\sqrt{1-x^2}}</math><br />
:<math> \left(\arctan(x)\right)' = \frac{1}{x^2+1}</math><br />
<br />
==Proofs of derivatives of trigonometric functions==<br />
<br />
===Limit of <math>\frac{\sin\theta}{\theta}</math> as <math>\theta \to 0</math>===<br />
[[File:limit_circle_FbN.jpeg|thumb|<center>Circle, centre ''O'', radius ''r''</center>]]<br />
<br />
The diagram on the right shows a circle, centre ''O'' and radius ''r''. Let θ be the angle at ''O'' made by the two radii ''OA'' and ''OB''. Since we are considering the limit as θ tends to zero, we may assume that θ is a very small positive number: {{nowrap|1=0 < θ ≪ 1}}.<br />
<br />
Consider the following three regions of the diagram: ''R''<sub>1</sub> is the triangle ''OAB'', ''R''<sub>2</sub> is the [[circular sector]] ''OAB'', and ''R''<sub>3</sub> is the triangle ''OAC''. Clearly:<br />
:<math>\text{Area}(R_1) < \text{Area}(R_2) < \text{Area}(R_3) \, . </math><br />
Using basic trigonometric formulae, the area of the triangle ''OAB'' is <br />
:<math> \frac{1}{2} \times ||OA|| \times ||OB|| \times \sin\theta = \frac{1}{2}r^2\sin\theta \, . </math><br />
The [[Circular_sector#Area|area of the circular sector]] ''OAB'' is <math>\frac{1}{2}r^2\theta</math>, while the area of the triangle ''OAC'' is given by<br />
:<math> \frac{1}{2} \times ||OA|| \times ||AC|| = \frac{1}{2}\times r \times r \tan\theta = \frac{1}{2}r^2\tan\theta \, . </math><br />
Collecting together these three areas gives:<br />
:<math>\text{Area}(R_1) < \text{Area}(R_2) < \text{Area}(R_3) \iff <br />
\frac{1}{2}r^2\sin\theta < \frac{1}{2}r^2\theta < \frac{1}{2}r^2\tan\theta \, . </math><br />
Since {{nowrap|1=''r'' > 0}} we can divide through by ½·r<sup>2</sup>; this means that the construction and calculations are all independent of the circle's radius. Moreover, since {{nowrap|1=0 < θ ≪ 1}} it follows that {{nowrap|1=sin(θ) > 0}} and we may divide through by a factor of sin(θ), giving:<br />
:<math>1 < \frac{\theta}{\sin\theta} < \frac{1}{\cos\theta} \implies 1 > \frac{\sin\theta}{\theta} > \cos\theta \, . </math><br />
In the last step we simply took the reciprocal of each of the three terms. Since all three terms are positive this has the effect of reversing the inequities, e.g. if {{nowrap|1=2 < 3}} then {{nowrap|1=½ > ⅓}}. <br />
<br />
[[File:Squeeze FbN.png|thumb|<center>Squeeze: The curves {{nowrap|1=''y'' = 1}} and {{nowrap|1=''y'' = cos(θ)}} shown in red, the curve {{nowrap|1=''y'' = sin(θ)/θ}} shown in blue.</center>]]<br />
<br />
We have seen that if {{nowrap|1=0 < θ ≪ 1}} then sin(θ)/θ is ''always'' less than 1 and, in addition, is ''always'' greater than cos(θ). Notice that as θ gets closer to 0, so cos(θ) gets closer to 1. Informally: as θ gets smaller, sin(θ)/θ is "[[Squeeze theorem|squeezed]]" between 1 and cos(θ), which itself it heading towards 1. It follows that sin(θ)/θ tends to 1 as θ tends to 0 from the positive side. <br />
<br />
For the case where θ is a very small negative number: {{nowrap|1=–1 ≪ θ < 0}}, we use the fact that sine is an [[odd function]]:<br />
:<math>\lim_{\theta \to 0^-} \frac{\sin\theta}{\theta} = \lim_{\theta\to 0^+}\frac{\sin(-\theta)}{-\theta} = \lim_{\theta \to 0^+}\frac{-\sin\theta}{-\theta} = \lim_{\theta\to 0^+}\frac{\sin\theta}{\theta} = 1 \, . </math><br />
<br />
===Limit of <math>\frac{\cos\theta - 1}{\theta}</math> as <math>\theta \to 0</math>===<br />
The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of θ is unimportant. <br />
:<math> \lim_{\theta \to 0} \left(\frac{\cos\theta - 1}{\theta}\right) = \lim_{\theta \to 0} \left[ \left( \frac{\cos\theta - 1}{\theta} \right) \left( \frac{\cos\theta + 1}{\cos\theta + 1} \right) \right] = \lim_{\theta \to 0} \left( \frac{\cos^2\theta - 1}{\theta(\cos\theta + 1)} \right) . </math><br />
The well-known identity {{nowrap|1=sin<sup>2</sup>θ + cos<sup>2</sup>θ = 1}} tells us that {{nowrap|1=cos<sup>2</sup>θ – 1 = –sin<sup>2</sup>θ.}}<br />
Using this, the fact that the limit of a product is the product of the limits, and the result from the last section, we find that:<br />
:<math> \lim_{\theta \to 0} \left(\frac{\cos\theta - 1}{\theta}\right) = \lim_{\theta \to 0} \left( \frac{-\sin^2\theta}{\theta(\cos\theta+1)} \right) = \lim_{\theta \to 0} \left( \frac{-\sin\theta}{\theta}\right) \times \lim_{\theta \to 0} \left( \frac{\sin\theta}{\cos\theta + 1} \right) = (-1) \times \frac{0}{2} = 0 \, . </math><br />
<br />
===Limit of <math>\frac{\tan\theta}{\theta}</math> as <math>\theta \to 0</math>===<br />
Using the limit for the [[#Limit_of_as|sine]] function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of the limits, we find:<br />
:<math><br />
\lim_{\theta \to 0^-} \frac{\tan\theta}{\theta}<br />
= \lim_{\theta\to 0^+} \frac{\tan\theta}{\theta}<br />
= \lim_{\theta \to 0} \frac{\tan\theta}{\theta}<br />
= \lim_{\theta\to 0} \frac{\sin\theta}{\theta} \times \lim_{\theta\to 0} \frac{1}{\cos\theta}<br />
= 1 \times 1<br />
= 1 \, . </math><br />
<br />
===Derivative of the sine function===<br />
To calculate the derivative of the [[sine function]] sin ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = \lim_{\delta \to 0} \left( \frac{\sin(\theta + \delta) - \sin \theta}{\delta} \right) . </math><br />
Using the well-known angle formula {{nowrap|1=sin(α+β) = sin α cos β + sin β cos α}}, we have:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = \lim_{\delta \to 0} \left( \frac{\sin\theta\cos\delta + \sin\delta\cos\theta-\sin\theta}{\delta} \right) = \lim_{\delta \to 0} \left[ \left(\frac{\sin\delta}{\delta} \cos\theta\right) + \left(\frac{\cos\delta -1}{\delta}\sin\theta\right) \right] . </math><br />
Using the limits for the [[#Limit_of_as|sine]] and [[#Limit_of_as_2|cosine]] functions:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = (1\times\cos\theta) + (0\times\sin\theta) = \cos\theta \, . </math><br />
<br />
===Derivative of the cosine function===<br />
To calculate the derivative of the [[cosine function]] cos ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = \lim_{\delta \to 0} \left( \frac{\cos(\theta+\delta)-\cos\theta}{\delta} \right) . </math><br />
Using the well-known angle formula {{nowrap|1=cos(α+β) = cos α cos β – sin α sin β}}, we have:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = \lim_{\delta \to 0} \left( \frac{\cos\theta\cos\delta - \sin\theta\sin\delta-\cos\theta}{\delta} \right) = \lim_{\delta \to 0} \left[ \left(\frac{\cos\delta -1}{\delta}\cos\theta\right) - \left(\frac{\sin\delta}{\delta} \sin\theta\right) \right] . </math><br />
Using the limits for the [[#Limit_of_as|sine]] and [[#Limit_of_as_2|cosine]] functions:<br />
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = (0 \times \cos\theta) - (1 \times \sin\theta) = -\sin\theta \, . </math><br />
<br />
===Derivative of the tangent function===<br />
To calculate the derivative of the [[tangent function]] tan ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:<br />
:<math><br />
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta<br />
= \lim_{\delta \to 0} \left( \frac{\tan(\theta+\delta)-\tan\theta}{\delta} \right) .<br />
</math><br />
Using the well-known angle formula {{nowrap|1=tan(α+β) = (tan α + tan β) / (1 - tan α tan β)}}, we have:<br />
:<math><br />
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta<br />
= \lim_{\delta \to 0} \left[ \frac{\frac{\tan\theta + \tan\delta}{1 - \tan\theta\tan\delta} - \tan\theta}{\delta} \right]<br />
= \lim_{\delta \to 0} \left[ \frac{\tan\theta + \tan\delta - \tan\theta + \tan^2\theta\tan\delta}{\delta \left( 1 - \tan\theta\tan\delta \right)} \right] .<br />
</math><br />
Using the fact that the limit of a product is the product of the limits:<br />
:<math><br />
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta<br />
= \lim_{\delta \to 0} \frac{\tan\delta}{\delta} \times \lim_{\delta \to 0} \left( \frac{1 + \tan^2\theta}{1 - \tan\theta\tan\delta} \right) .<br />
</math><br />
Using the limit for the [[#Limit_of_as_3|tangent]] function, and the fact that tan ''&delta;'' tends to 0 as &delta; tends to 0:<br />
:<math><br />
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta<br />
= 1 \times \frac{1 + \tan^2\theta}{1 - 0} = 1 + \tan^2\theta .<br />
</math><br />
We see immediately that:<br />
:<math><br />
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta<br />
= 1 + \frac{\sin^2\theta}{\cos^2\theta}<br />
= \frac{\cos^2\theta + \sin^2\theta}{\cos^2\theta}<br />
= \frac{1}{\cos^2\theta}<br />
= \sec^2\theta \, .<br />
</math><br />
<br />
==Proofs of derivatives of inverse trigonometric functions==<br />
The following derivatives are found by setting a [[Variable (mathematics)|variable]] ''y'' equal to the [[inverse trigonometric function]] that we wish to take the derivative of. Using [[implicit differentiation]] and then solving for ''dy''/''dx'', the derivative of the inverse function is found in terms of ''y''. To convert ''dy''/''dx'' back into being in terms of ''x'', we can draw a reference triangle on the unit circle, letting ''θ'' be y. Using the [[Pythagorean theorem]] and the definition of the regular trigonometric functions, we can finally express ''dy''/''dx'' in terms of ''x''.<br />
<br />
===Differentiating the inverse sine function===<br />
We let<br />
<br />
:<math>y=\arcsin x\,\!</math><br />
<br />
Where<br />
<br />
:<math>-\frac{\pi}{2}\le y \le \frac{\pi}{2}</math><br />
<br />
Then<br />
<br />
:<math>\sin y=x\,\!</math><br />
<br />
Using [[implicit differentiation]] and solving for dy/dx:<br />
<br />
:<math>{d \over dx}\sin y={d \over dx}x</math><br />
<br />
:<math>{dy \over dx}\cos y=1\,\!</math><br />
<br />
Substituting <math> \cos y = \sqrt{1-\sin^2 y}</math> in from above,<br />
<br />
:<math>{dy \over dx}\sqrt{1-\sin^2 y}=1</math><br />
<br />
Substituting <math>x=\sin y</math> in from above,<br />
<br />
:<math>{dy \over dx}\sqrt{1-x^2}=1</math><br />
<br />
:<math>{dy \over dx}=\frac{1}{\sqrt{1-x^2}}</math><br />
<br />
===Differentiating the inverse cosine function===<br />
We let<br />
<br />
:<math>y=\arccos x\,\!</math><br />
<br />
Where<br />
<br />
:<math>0 \le y \le \pi</math><br />
<br />
Then<br />
<br />
:<math>\cos y=x\,\!</math><br />
<br />
Using [[implicit differentiation]] and solving for dy/dx:<br />
<br />
:<math>{d \over dx}\cos y={d \over dx}x</math><br />
<br />
:<math>-{dy \over dx}\sin y=1</math><br />
<br />
Substituting <math>\sin y = \sqrt{1-\cos^2 y}\,\!</math> in from above, we get<br />
<br />
:<math>-{dy \over dx}\sqrt{1-\cos^2 y} =1</math><br />
<br />
Substituting <math>x=\cos y\,\!</math> in from above, we get<br />
<br />
:<math>-{dy \over dx}\sqrt{1-x^2} =1</math><br />
<br />
:<math>{dy \over dx} = -\frac{1}{\sqrt{1-x^2}}</math><br />
<br />
===Differentiating the inverse tangent function===<br />
<br />
We let<br />
<br />
:<math>y=\arctan x\,\!</math><br />
<br />
Where<br />
<br />
:<math>-\frac{\pi}{2} < y < \frac{\pi}{2}</math><br />
<br />
Then<br />
<br />
:<math>\tan y=x\,\!</math><br />
<br />
Using [[implicit differentiation]] and solving for dy/dx:<br />
<br />
:<math>{d \over dx}\tan y={d \over dx}x</math><br />
<br />
Left side:<br />
<br />
:<math><br />
{d \over dx}\tan y<br />
= {d \over dx}\frac{\sin y}{\cos y}<br />
= \frac{{dy \over dx} \cos^2 y + \sin^2 y {dy \over dx}}{\cos^2 y}<br />
= {dy \over dx} \left (1 + \tan^2 y \right)<br />
</math><br />
<br />
Right side:<br />
<br />
:<math>{d \over dx}x = 1</math><br />
<br />
Therefore<br />
<br />
:<math>{dy \over dx}(1+\tan^2 y)=1</math><br />
<br />
Substituting <math>x=\tan y\,\!</math> in from above, we get<br />
<br />
:<math>{dy \over dx}(1+x^2)=1</math><br />
<br />
:<math>{dy \over dx}=\frac{1}{1+x^2}</math><br />
<br />
===Differentiating the inverse cotangent function===<br />
<br />
We let<br />
<br />
:<math>y=\arccot x\,\!</math><br />
<br />
Where<br />
<br />
:<math> 0 < y < \pi</math><br />
<br />
Then<br />
<br />
:<math>\cot y=x\,\!</math><br />
<br />
Using implicit differentiation and solving for dy/dx:<br />
<br />
:<math>{d \over dx}\cot y={d \over dx}x</math><br />
<br />
:<math>{dy \over dx}-\csc^2 y=1</math><br />
<br />
Substituting <math>1+\cot^2 y = \csc^2 y\,\!</math> into the above,<br />
<br />
:<math>{dy \over dx}-(1+\cot^2 y)=1</math><br />
<br />
Substituting <math>x=\cot y\,\!</math> in from above, we get<br />
<br />
:<math>{dy \over dx}-(1+x^2)=1</math><br />
<br />
:<math>{dy \over dx}=\frac{-1}{1+x^2}</math><br />
<br />
==See also==<br />
*[[Trigonometry]]<br />
*[[Calculus]]<br />
*[[Derivative]]<br />
*[[Table of derivatives]]<br />
<br />
==References==<br />
{{Reflist}}<br />
<br />
==Bibliography==<br />
*''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', Edited by Abramowitz and Stegun, National Bureau of Standards, Applied Mathematics Series, 55 (1964)<br />
<br />
{{DEFAULTSORT:Differentiation Of Trigonometric Functions}}<br />
[[Category:Differential calculus]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Routh%E2%80%93Hurwitz_stability_criterion&diff=9137
Routh–Hurwitz stability criterion
2013-11-02T18:39:03Z
<p>99.153.64.179: Category:Stability theory</p>
<hr />
<div>'''Rado's theorem''' is a theorem from the branch of [[mathematics]] known as [[Ramsey theory]]. It is named for the German mathematician [[Richard Rado]]. It was proved in his thesis, ''Studien zur Kombinatorik''.<br />
<br />
Let ''A'''x'''''&nbsp;=&nbsp;'''0''' be a system of linear equations, where ''A'' is a matrix with integer entries. This system is said to be ''r-regular'' if, for every ''r''-coloring of the natural numbers 1,&nbsp;2,&nbsp;3,&nbsp;..., the system has a monochromatic solution. A system is ''regular'' if it is ''r-regular'' for all&nbsp;''r''&nbsp;≥&nbsp;1.<br />
<br />
Rado's theorem states that a system ''A'''x'''''='''0''' is regular if and only if the matrix ''A'' satisfies the ''columns condition''. Let ''c<sub>i</sub>'' denote the ''i''-th column of ''A''. The matrix ''A'' satisfies the columns condition provided that there exists a partition ''C''<sub>1</sub>, ''C''<sub>2</sub>, ..., ''C''<sub>''n''</sub> of the column indices such that if <math>s_i = \Sigma_{j \in C_i}c_j</math>, then<br />
<br />
# ''s''<sub>1</sub> = 0<br />
# for all ''i''&nbsp;≥&nbsp;2, ''s<sub>i</sub>'' can be written as a rational<ref>Modern graph theory by Béla Bollobás. 1st ed. 1998. ISBN 978-0-387-98488-9. Page 204</ref> linear combination of the ''c<sub>j</sub>'''s in the ''C<sub>k</sub>'' with ''k''&nbsp;<&nbsp;''i''.<br />
<br />
[[Folkman's theorem]], the statement that there exist arbitrarily large sets of integers all of whose nonempty sums are monochromatic, may be seen as a special case of Rado's theorem concerning the regularity of the system of equations<br />
:<math>x_T = \sum_{i\in T}x_{\{i\}},</math><br />
where ''T'' ranges over each nonempty subset of the set {{nowrap|{1, 2, ..., ''x''}.}}<ref name="grs">{{citation|title=Ramsey Theory|first1=Ronald L.|last1=Graham|author1-link=Ronald Graham|first2=Bruce L.|last2=Rothschild|author2-link=Bruce L. Rothschild|first3=Joel H.|last3=Spencer|author3-link=Joel Spencer|publisher=Wiley-Interscience|year=1980|contribution=3.4 Finite Sums and Finite Unions (Folkman's Theorem)|pages=65–69}}.</ref><br />
<br />
==References==<br />
<references/><br />
<br />
{{DEFAULTSORT:Rado's Theorem (Ramsey Theory)}}<br />
[[Category:Ramsey theory]]<br />
[[Category:Theorems in discrete mathematics]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Dynamical_pictures_(quantum_mechanics)&diff=30053
Dynamical pictures (quantum mechanics)
2013-11-02T17:27:15Z
<p>99.153.64.179: strong anti-merge sentiment on talk page; original proposer has bowed out.</p>
<hr />
<div>{| class="wikitable" align="right" style="margin-left:10px" width="360"<br />
!bgcolor=#e7dcc3 colspan=2|Runcitruncated tesseractic honeycomb<br />
|-<br />
|bgcolor=#ffffff align=center colspan=2|(No image)<br />
|-<br />
|bgcolor=#e7dcc3|Type||[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]<br />
|-<br />
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{4,3,3,4}<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node_1|3|node|3|node_1|4|node}}<BR>{{CDD|node_1|4|node_1|3|node|split1|nodes_11}}<br />
|-<br />
|bgcolor=#e7dcc3|4-face type||[[Runcitruncated tesseract|t<sub>0,1,3</sub>{4,3,3}]] [[File:Schlegel half-solid runcitruncated 8-cell.png|40px]]<BR>[[rectified 24-cell|t<sub>1</sub>{3,4,3}]] [[File:Schlegel half-solid cantellated 16-cell.png|40px]]<BR>[[cuboctahedral prism|t<sub>1</sub>{3,4}×{}]] [[File:Cuboctahedral prism.png|40px]]<BR>4-8 [[duoprism]] [[File:4-8 duoprism.png|40px]]<br />
|-<br />
|bgcolor=#e7dcc3|Cell type||[[Cuboctahedron]] [[File:Cuboctahedron.png|20px]]<BR><br />
[[Truncated cube]] [[File:Truncated hexahedron.png|20px]]<BR><br />
[[Cube]] [[File:Hexahedron.png|20px]]<BR><br />
[[Octagonal prism]] [[File:Octagonal prism.png|20px]]<BR><br />
[[Triangular prism]] [[File:Triangular prism.png|20px]]<br />
|-<br />
|bgcolor=#e7dcc3|Face type||{3}, {4}, {8}<br />
|-<br />
|bgcolor=#e7dcc3|[[Vertex figure]]||[[triangular prism]]-based tilted pyramid<br />
|-<br />
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_4</math> = [4,3,3,4]<BR><math>{\tilde{B}}_4</math> = [4,3,3<sup>1,1</sup>]<br />
|-<br />
|bgcolor=#e7dcc3|Dual||<br />
|-<br />
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]<br />
|}<br />
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''runcitruncated tesseractic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 4-space.<br />
<br />
== Related honeycombs==<br />
{{C4_honeycombs}}<br />
<br />
{{B4_honeycombs}}<br />
<br />
== See also ==<br />
Regular and uniform honeycombs in 4-space:<br />
*[[Tesseractic honeycomb]]<br />
*[[Demitesseractic honeycomb]]<br />
*[[24-cell honeycomb]]<br />
*[[Truncated 24-cell honeycomb]]<br />
*[[Snub 24-cell honeycomb]]<br />
* [[5-cell honeycomb]]<br />
* [[Truncated 5-cell honeycomb]]<br />
* [[Omnitruncated 5-cell honeycomb]]<br />
<br />
==Notes==<br />
{{reflist}}<br />
<br />
== References ==<br />
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]<br />
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] See p318 [http://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PA318&ots=dnT1LYgmij&dq=%22quarter%20cubic%20honeycomb%22%20q%7B4%2C3%2C4%7D&pg=PA318#v=onepage&q=%22quarter%20cubic%20honeycomb%22%20q%7B4,3,4%7D&f=false]<br />
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''<br />
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations#4D}} x3o3x *b3x4x, x4x3o3x4o - potatit - O95<br />
* {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}<br />
{{Honeycombs}}<br />
<br />
{{DEFAULTSORT:Demitesseractic Honeycomb}}<br />
[[Category:Honeycombs (geometry)]]<br />
[[Category:5-polytopes]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Track_algorithm&diff=28035
Track algorithm
2013-07-19T02:43:20Z
<p>99.153.64.179: /* Track */ typo</p>
<hr />
<div>The '''flat pseudospectral method''' is part of the family of the [[Ross–Fahroo pseudospectral method]]s introduced by [[I. Michael Ross|Ross]] and [[Fariba Fahroo|Fahroo]].<ref name ="rf-tac"> Ross, I. M. and Fahroo, F., “Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems,” IEEE Transactions on Automatic Control, Vol.49, No.8, pp. 1410–1413, August 2004.</ref> <ref name="rf-cdc-2003"> Ross, I. M. and Fahroo, F., “A Unified Framework for Real-Time Optimal Control,” Proceedings of the IEEE Conference on Decision and Control, Maui, HI, December, 2003.</ref> The method combines the concept of [[Flatness (systems theory)|differential flatness]]<ref name="fliess-95"> Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., “Flatness and defect of nonlinear systems: Introductory theory and examples,” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995. </ref> <ref name="RMM-siam98"> Rathinam, M. and Murray, R. M., “Configuration flatness of Lagrangian systems underactuated by one control” SIAM Journal on Control and Optimization, 36, 164,1998. </ref> with [[pseudospectral method]] to generate outputs in the so-called flat space.<br />
<br />
==Concept==<br />
Because the differentiation matrix, <math>D </math>, in a pseudospectral method is square, higher-order derivatives of any polynomial, <math> y </math>, can be obtained by powers of <math>D </math>,<br />
: <math><br />
\begin{align}<br />
\dot y &= D Y \\ <br />
\ddot y & = D^2 Y \\<br />
&{} \ \vdots \\<br />
y^{(\beta)} &= D^\beta Y<br />
\end{align}</math><br />
<br />
where <math> Y </math> is the pseudospectral variable and <math> \beta </math> is a finite positive integer. <br />
By differential flatness, there exists functions <math> a </math> and <math> b </math> such that the state and control variables can be written as,<br />
<br />
: <math><br />
\begin{align}<br />
x & = a(y, \dot y, \ldots, y^{(\beta)}) \\ <br />
u & = b(y, \dot y, \ldots, y^{(\beta + 1)})<br />
\end{align}<br />
</math><br />
<br />
The combination of these concepts generates the flat pseudospectral method; that is, x and u are written as,<br />
:<math> x = a(Y, D Y, \ldots, D^\beta Y) </math><br />
:<math> u = b(Y, D Y, \ldots, D^{\beta + 1}Y) </math><br />
Thus, an optimal control problem can be quickly and easiy transformed to a problem with just the Y pseudospectral variable.<ref name = "rf-tac" /><br />
<br />
==See also==<br />
*[[Ross' π lemma]]<br />
*[[Ross–Fahroo lemma]]<br />
*[[Bellman pseudospectral method]]<br />
*[http://www.cds.caltech.edu/~murray/wiki/index.php/EECI08:_Trajectory_Generation_and_Differential_Flatness Trajectory generation and differential flatness]<br />
<br />
==References==<br />
{{Reflist}}<br />
<br />
{{DEFAULTSORT:Pseudospectral Optimal Control}}<br />
[[Category:Optimal control]]<br />
[[Category:Numerical analysis]]<br />
[[Category:Control theory]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Pulse-Doppler_signal_processing&diff=26918
Pulse-Doppler signal processing
2013-07-19T02:15:52Z
<p>99.153.64.179: /* Track */ fix bad sentence</p>
<hr />
<div>{{DISPLAYTITLE: ''q''-Hahn polynomials}}<br />
{{see also|continuous q-Hahn polynomials|dual q-Hahn polynomials|continuous dual q-Hahn polynomials}}<br />
In mathematics, the '''''q''-Hahn polynomials''' are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]]. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} give a detailed list of their properties.<br />
<br />
==Definition==<br />
<br />
The polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by <br />
:<math>\displaystyle </math><br />
<br />
==Orthogonality==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Recurrence and difference relations==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Rodrigues formula==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Generating function==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Relation to other polynomials==<br />
{{Empty section|date=September 2011}}<br />
This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.<br />
==References==<br />
<br />
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}<br />
*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}<br />
*{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}<br />
<br />
[[Category:Orthogonal polynomials]]<br />
[[Category:Q-analogs]]<br />
[[Category:Special hypergeometric functions]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Kleisli_category&diff=12730
Kleisli category
2013-06-25T19:29:11Z
<p>99.153.64.179: /* External links */ use nlab template</p>
<hr />
<div>{{Infobox manuscript<br />
<!----------Name----------><br />
| name = Moscow Mathematical Papyrus<br />
| location = [[Pushkin State Museum of Fine Arts]] in Moscow<br />
<!----------Image----------><br />
| image = File:Moskou-papyrus.jpg<br />
| width = 300px<br />
| caption = 14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)<br />
<!----------General----------><br />
| Also known as =<br />
| Type =<br />
| Date = [[Thirteenth dynasty of Egypt|13th dynasty]], [[Second Intermediate Period of Egypt]]<br />
| Place of origin = [[Thebes, Egypt|Thebes]]<br />
| Language(s) = [[Hieratic]]<br />
| Scribe(s) =<br />
| Author(s) =<br />
| Compiled by =<br />
| Illuminated by =<br />
| Patron =<br />
| Dedicated to =<br />
<!----------Form and content----------><br />
| Material =<br />
| Size = Length: {{convert|18|ft|m}}<br>Width: {{convert|1.5|in|cm}} to {{convert|3|in|cm}}<br />
| Format =<br />
| Condition =<br />
| Script =<br />
| Contents =<br />
| Illumination(s) =<br />
| Additions =<br />
| Exemplar(s) =<br />
| Previously kept =<br />
| Discovered = <br />
| Other = <br />
| below = <br />
}}<br />
<br />
The '''Moscow Mathematical Papyrus''' is an ancient [[Egyptian mathematics|Egyptian mathematical]] papyrus, also called the '''Golenishchev Mathematical Papyrus''', after its first owner, [[Egyptologist]] [[Vladimir Golenishchev]]. Golenishchev bought the papyrus in 1892 or 1893 in [[Thebes, Egypt|Thebes]]. It later entered the collection of the [[Pushkin State Museum of Fine Arts]] in Moscow, where it remains today. <br />
<br />
Based on the [[palaeography]] and orthography of the [[hieratic]] text, the text was most likely written down in the [[Thirteenth dynasty of Egypt|13th dynasty]] and based on older material probably dating to the [[Twelfth dynasty of Egypt]], roughly 1850 BC.<ref name="Clagett">Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5</ref> Approximately 18 feet long and varying between 1½ and 3 inches wide, its format was divided into 25 problems with solutions by the [[Soviet Union|Soviet]] [[oriental studies|Orientalist]] [[Vasily Vasilievich Struve]]<ref>[http://www.encspb.ru/en/article.php?kod=2804014273 Struve V.V., (1889–1965), orientalist :: ENCYCLOPAEDIA OF SAINT PETERSBURG<!-- Bot generated title -->]</ref> in 1930.<ref>Struve, Vasilij Vasil'evič, and [[Boris Turaev]]. 1930. ''Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau''. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer</ref> It is a well-known mathematical papyrus along with the ''[[Rhind Mathematical Papyrus]].'' The ''Moscow Mathematical Papyrus'' is older than the ''Rhind Mathematical Papyrus,'' while the latter is the larger of the two.<ref>''[[Great Soviet Encyclopedia]]'', 3rd edition, entry on "Папирусы математические", available online [http://slovari.yandex.ru/art.xml?art=bse/00057/24900.htm here]</ref><br />
<br />
==Exercises contained in the Moscow Papyrus==<br />
The problems in the Moscow Papyrus follow no particular order, and the solutions of the problems provide much less detail than those in the [[Rhind Mathematical Papyrus]]. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a [[frustum]] respectively. The remaining problems are more common in nature.<ref name="Clagett"/><br />
<br />
===Ship's part problems===<br />
Problems 2 and 3 are ship's part problems. One of the problems calculates the length of a ship's rudder and the other computes the length of a ship's mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long.<ref name="Clagett"/><br />
<br />
===Aha problems===<br />
{{Hiero|Aha|<hiero>P6-a:M35</hiero>|align=left|era=nk}}<br />
Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The [[Rhind Mathematical Papyrus]] also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10.<ref name="Clagett"/> In other words, in modern mathematical notation one is asked to solve <math>3/2 \times x + 4 = 10</math><br />
<br />
===Pefsu problems===<br />
Most of the problems are pefsu problems: 10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain <br />
: <math> \mbox{pefsu} = \frac{\mbox{number loaves of bread (or jugs of beer)}}{\mbox{number of heqats of grain}}</math><br />
<br />
A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example problem 8 translates as:<br />
: (1) Example of calculating 100 loaves of bread of pefsu 20<br />
: (2) If someone says to you: "You have 100 loaves of bread of pefsu 20<br />
: (3) to be exchanged for beer of pefsu 4<br />
: (4) like 1/2 1/4 malt-date beer"<br />
: (5) First calculate the grain required for the 100 loaves of the bread of pefsu 20<br />
: (6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer<br />
: (7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.<br />
: (8) Calculate 1/2 of 5 heqat, the result will be 2 1/2<br />
: (9) Take this 2 1/2 four times<br />
: (10) The result is 10. Then you say to him:<br />
: (11) "Behold! The beer quantity is found to be correct."<ref name="Clagett"/><br />
<br />
===Baku problems===<br />
Problems 11 and 23 are Baku problems. These calculate the output of workers. Problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to? Problem 23 finds the output of a shoemaker given that he has to cut and decorate sandals.<ref name="Clagett"/><br />
<br />
===Geometry problems===<br />
Seven of the twenty-five problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere (problem 10) and finding the volume of a frustum (a truncated pyramid).<ref name="Clagett"/><br />
<br />
==Two Interesting Geometry Problems==<br />
<br />
===Problem 10===<br />
The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a [[Sphere|hemisphere]] (Struve, Gillings) or possibly the area of a semi-cylinder (Peet). Below we assume that the problem refers to the area of a hemisphere.<br />
<br />
The text of problem 10 runs like this: "Example of calculating a basket. You are given a basket with a mouth of 4 1/2. What is its surface? Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1. Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18. Find the remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly."<ref name="Clagett"/><ref>Williams, Scott W. [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html#moscow10 Egyptian Mathematical Papyri]</ref><br />
<br />
The solution amounts to computing the area as <br />
: <math> \text{Area} = 2 \times \left(\frac{8}{9}\right)^2 \times (\text{diameter})^2 = 2 \times \frac{256}{81} (\text{radius})^2</math><br />
<br />
This means the scribe of the Moscow Papyrus used <math> \frac{256}{81} \approx 3.16049</math> to approximate pi.<br />
<br />
===Problem 14: Volume of frustum of square pyramid===<br />
[[Image:Pyramide-tronquée-papyrus-Moscou 14.jpg|thumb|left]]<br />
<br />
The 14th problem of the Moscow Mathematical calculates the volume of a [[frustum]]. <br />
<br />
Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units, which is correct.<ref name="Clagett"/><br />
<br />
The text of the example runs like this: "If you are told: a truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top: You are to square the 4; result 16. You are to double 4; result 8. You are to square this 2; result 4. You are to add the 16 and the 8 and the 4; result 28. You are to take 1/3 of 6; result 2. You are to take 28 twice; result 56. See, it is of 56. You will find [it] right" <ref>as given in Gunn & Peet, ''Journal of Egyptian Archaeology,'' 1929, 15: 176. See also, Van der Waerden, 1961, Plate 5</ref><br />
<br />
The solution to the problem indicates that the Egyptians knew the correct formula for obtaining the [[volume]] of a [[frustum|truncated pyramid]]:<br />
:<math>V = \frac{1}{3} h(a^2 + a b +b^2).</math><br />
<br />
It remains unknown how the Egyptians arrived at the formula for the volume of a [[frustum]].{{Citation needed|date=May 2012}}<br />
<br />
==Other papyri==<br />
Other mathematical texts from Ancient Egypt include:<br />
*[[Berlin Papyrus 6619]]<br />
*[[Egyptian Mathematical Leather Roll]]<br />
*[[Lahun Mathematical Papyri]]<br />
*[[Rhind Mathematical Papyrus]]<br />
<br />
General papyri:<br />
*[[Papyrus Harris I]]<br />
*[[Rollin Papyrus]]<br />
<br />
For the 2/n tables see:<br />
*[[RMP 2/n table]]<br />
<br />
==References==<br />
<references/><br />
<br />
===Full Text of the Moscow Mathematical Papyrus===<br />
*Struve, Vasilij Vasil'evič, and [[Boris Turaev]]. 1930. ''Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau''. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer<br />
<br />
===Other references===<br />
*Allen, Don. April 2001. [http://www.math.tamu.edu/~don.allen/history/egypt/node4.html ''The Moscow Papyrus''] and [http://www.math.tamu.edu/~don.allen/history/egypt/node5.html ''Summary of Egyptian Mathematics''].<br />
*Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.<br />
*Mathpages.com. [http://www.mathpages.com/home/kmath189/kmath189.htm ''The Prismoidal Formula''].<br />
*O'Connor and Robertson, 2000. [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html ''Mathematics in Egyptian Papyri''].<br />
*Truman State University, Math and Computer Science Division. '''Mathematics and the Liberal Arts:''' [http://math.truman.edu/~thammond/history/AncientEgypt.html ''Ancient Egypt''] and [http://math.truman.edu/~thammond/history/MoscowPapyrus.html ''The Moscow Mathematical Papyrus''].<br />
*Williams, Scott W. [http://www.math.buffalo.edu/mad/index.html ''Mathematicians of the African Diaspora''], containing a page on [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html ''Egyptian Mathematics Papyri''].<br />
*Zahrt, Kim R. W. [http://www.iusb.edu/~journal/static/volumes/2000/zahrt.html ''Thoughts on Ancient Egyptian Mathematics''].<br />
<br />
[[Category:Egyptian mathematics]]<br />
[[Category:Egyptian fractions]]<br />
[[Category:Ancient Egyptian literature]]<br />
[[Category:Egyptian papyri]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Talk:Calculus_of_constructions&diff=321953
Talk:Calculus of constructions
2013-06-25T18:18:04Z
<p>99.153.64.179: {{Maths rating|class=Start|priority=Low|field=foundations}}</p>
<hr />
<div>Market Analysis Analyst Harry Grippo from Barwick, likes to spend time caravaning, [http://churchofrng.com/activity/p/186568/ property developers in singapore list] developers in singapore and dancing. Loves to discover new cities and locales like Historic Town of Goslar.</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Exponential_object&diff=10267
Exponential object
2013-06-16T05:48:18Z
<p>99.153.64.179: power set provides a different definition of power objects</p>
<hr />
<div>In [[quantum field theory]], a '''fermionic field''' is a [[quantum field]] whose quanta are [[fermion]]s; that is, they obey [[Fermi–Dirac statistics]]. Fermionic fields obey [[canonical anticommutation relation]]s rather than the [[canonical commutation relation]]s of [[bosonic field]]s.<br />
<br />
The most prominent example of a fermionic field is the Dirac field, which describes fermions with [[spin (physics)|spin]]-1/2: [[electron]]s, [[proton]]s, [[quarks]], etc. The Dirac field can be described as either a 4-component [[spinor]] or as a pair of 2-component Weyl spinors. Spin-1/2 [[Majorana fermion]]s, such as the hypothetical [[neutralino]], can be described as either a dependent 4-component [[Majorana spinor]] or a single 2-component Weyl spinor. It is not known whether the [[neutrino]] is a Majorana fermion or a [[Dirac fermion]] (see also [[Double beta decay#Neutrinoless double beta decay|Neutrinoless double-beta decay]] for experimental efforts to determine this). <br />
<br />
==Basic properties==<br />
Free (non-interacting) fermionic fields obey [[canonical anticommutation relation]]s, i.e., involve the [[anticommutator]]s {''a'',''b''} = ''ab'' + ''ba'' rather than the commutators [''a'',''b''] = ''ab'' − ''ba'' of bosonic or standard quantum mechanics. Those relations also hold for interacting fermionic fields in the [[interaction picture]], where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states.<br />
<br />
It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the [[Pauli exclusion principle]]: two fermionic particles cannot occupy the same state at the same time.<br />
<br />
==Dirac fields==<br />
The prominent example of a spin-1/2 fermion field is the '''Dirac field''' (named after [[Paul Dirac]]), and denoted by ψ(''x''). The equation of motion for a free field is the [[Dirac equation]],<br />
<br />
:<math>(i\gamma^{\mu} \partial_{\mu} - m) \psi(x) = 0.\,</math><br />
<br />
where γ<sup>μ</sup> are [[gamma matrices]] and ''m'' is the mass. The simplest possible solutions to this equation are plane wave solutions, <math>\psi_{1}(x) = u(p)e^{-ip.x}\,</math> and <math>\psi_{2}(x) = v(p)e^{ip.x}\,</math>. These [[plane wave]] solutions form a basis for the Fourier components of ψ(''x''), allowing for the general expansion of the Dirac field as follows,<br />
<br />
<math>\psi(x) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left(<br />
a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\,</math><br />
<br />
The ''a'' and ''b'' labels are spinor indices and the ''s'' indices represent spin labels and so for the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. Since ψ(''x'') can be thought of as an operator, the coefficients of its Fourier modes must be operators too. Hence, <math>a^{s}_{\textbf{p}}</math> and <math>b^{s \dagger}_{\textbf{p}}</math> are operators. The properties of these operators can be discerned from the properties of the field. ψ(''x'') and <math>\psi(y)^{\dagger}</math> obey the anticommutation relations<br />
<br />
:<math>\{\psi_a(\textbf{x}),\psi_b^{\dagger}(\textbf{y})\} = \delta^{(3)}(\textbf{x}-\textbf{y})\delta_{ab},</math><br />
<br />
By putting in the expansions for ψ(''x'') and ψ(''y''), the anticommutation relations for the coefficients can be computed.<br />
<br />
:<math>\{a^{r}_{\textbf{p}},a^{s \dagger}_{\textbf{q}}\} = \{b^{r}_{\textbf{p}},b^{s \dagger}_{\textbf{q}}\}=(2 \pi)^{3} \delta^{3} (\textbf{p}-\textbf{q}) \delta^{rs},\,</math><br />
<br />
In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that <math>a^{s \dagger}_{\textbf{p}}</math> creates a fermion of momentum '''p''' and spin s, and <math>b^{r \dagger}_{\textbf{q}}</math> creates an antifermion of momentum '''q''' and spin ''r''. The general field ψ(''x'') is now seen to be a weighed (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, <math>\bar{\psi} \ \stackrel{\mathrm{def}}{=}\ \psi^{\dagger} \gamma^{0}</math>, is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions. <br />
<br />
With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity <math>\overline{\psi}\psi\,</math>. This makes the reason for the choice of <math>\bar{\psi} = \psi^{\dagger} \gamma^{0}</math>clear. This is because the general Lorentz transform on ψ is not [[Unitary transformation|unitary]] so the quantity <math>\psi^{\dagger}\psi</math> would not be invariant under such transforms, so the inclusion of <math>\gamma^{0}\,</math> is to correct for this. The other possible non-zero [[Lorentz covariance|Lorentz invariant]] quantity, up to an overall conjugation, constructible from the fermionic fields is <math>\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi</math>. <br />
<br />
Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the [[Lagrangian#Lagrangians and Lagrangian densities in field theory|Lagrangian density]] for the Dirac field by the requirement that the [[Euler–Lagrange equation]] of the system recover the Dirac equation.<br />
<br />
:<math>\mathcal{L}_{D} = \bar{\psi}(i\gamma^{\mu} \partial_{\mu} - m)\psi\,</math><br />
<br />
Such an expression has its indices suppressed. When reintroduced the full expression is<br />
<br />
:<math>\mathcal{L}_{D} = \bar{\psi}_{a}(i\gamma^{\mu}_{ab} \partial_{\mu} - m\mathbb{I}_{ab})\psi_{b}\,</math><br />
<br />
Given the expression for ψ(''x'') we can construct the Feynman [[propagator]] for the fermion field:<br />
<br />
:<math> D_{F}(x-y) = \langle 0| T(\psi(x) \bar{\psi}(y))| 0 \rangle </math><br />
<br />
we define the [[time-ordered]] product for fermions with a minus sign due to their anticommuting nature<br />
<br />
:<math> T(\psi(x) \bar{\psi}(y)) \ \stackrel{\mathrm{def}}{=}\ \theta(x^{0}-y^{0}) \psi(x) \bar{\psi}(y) - \theta(y^{0}-x^{0})\bar\psi(y) \psi(x) .</math><br />
<br />
Plugging our plane wave expansion for the fermion field into the above equation yields: <br />
<br />
:<math> D_{F}(x-y) = \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}+i \epsilon}e^{-ip \cdot (x-y)}</math><br />
<br />
where we have employed the [[Feynman slash]] notation. This result makes sense since the factor <br />
<br />
:<math>\frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}}</math> <br />
<br />
is just the inverse of the operator acting on ψ(''x'') in the Dirac equation. Note that the Feynman propagator for the Klein–Gordon field has this same property. Since all reasonable observables (such as energy, charge, particle number, etc.) are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented [[Lorentz invariance]] for the Dirac field, and preserved [[causality]].<br />
<br />
More complicated field theories involving interactions (such as [[Yukawa theory]], or [[quantum electrodynamics]]) can be analyzed too, by various perturbative and non-perturbative methods.<br />
<br />
Dirac fields are an important ingredient of the [[Standard Model]].<br />
<br />
==See also==<br />
*[[Dirac equation]]<br />
*[[Einstein-Maxwell-Dirac equations|Einstein–Maxwell–Dirac equations]]<br />
*[[Spin-statistics theorem]]<br />
*[[Spinor]]<br />
<br />
==References==<br />
*{{cite journal |last=Edwards |first=D. |year=1981 |title=The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories |journal=International J. of Theor. Phys. |volume=20 |issue=7 |pages=503–517 |doi=10.1007/BF00669437 |bibcode = 1981IJTP...20..503E }}<br />
* Peskin, M and Schroeder, D. (1995). ''An Introduction to Quantum Field Theory,'' Westview Press. (See pages 35–63.)<br />
* Srednicki, Mark (2007). ''[http://www.physics.ucsb.edu/~mark/qft.html Quantum Field Theory]'', Cambridge University Press, ISBN 978-0-521-86449-7.<br />
* Weinberg, Steven (1995). ''The Quantum Theory of Fields,'' (3 volumes) Cambridge University Press.<br />
<br />
[[Category:Quantum field theory]]<br />
[[Category:Spinors]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Opposite_category&diff=4366
Opposite category
2013-06-16T03:16:30Z
<p>99.153.64.179: /* Properties */ opposite functor</p>
<hr />
<div>{{multiple issues|<br />
{{technical|date=October 2010}}<br />
{{Refimprove|date=October 2010}}<br />
}}<br />
{{Merge from|Magic tesseract|discuss=Talk:Magic hypercube#Merge from 4|date=November 2010}}<br />
{{Merge from|Nasik magic hypercube|discuss=Talk:Magic hypercube#Nasik|date=November 2010}}<br />
<br />
In [[mathematics]], a '''magic hypercube''' is the [[dimension|''k''-dimensional]] generalization of [[magic square]]s, [[magic cube]]s and [[magic tesseract]]s; that is, a number of [[integers]] arranged in an ''n'' × ''n'' × ''n'' × ... × ''n'' pattern such that the sum of the numbers on each pillar (along any axis) as well as the main [[space diagonal]]s is equal to a single number, the so-called [[magic constant]] of the [[hypercube]], denoted ''M''<sub>''k''</sub>(''n''). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., ''n''<sup>''k''</sup>, then it has magic number<br />
<br />
:<math>M_k(n) = \frac{n(n^k+1)}{2}</math><br />
<br />
If, in addition, the numbers on every [[cross section (geometry)|cross section]] diagonal also sum up to the hypercube's magic number, the hypercube is called a [[perfect magic hypercube]]; otherwise, it is called a [[semiperfect magic hypercube]]. The number ''n'' is called the order of the magic hypercube.<br />
<br />
Five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by [[J. R. Hendricks]].<br />
<br />
Marian Trenkler proved the following theorem:<br />
A ''p''-dimensional magic hypercube of order ''n'' exists if and only if<br />
''p'' > 1 and ''n'' is different from 2 or ''p'' = 1. A construction of a magic hypercube follows from the proof.<br />
<br />
The [[R (programming language)|R programming language]] includes a module, <tt> library(magic)</tt>, that will create magic hypercubes of any dimension (with ''n'' a multiple of 4).<br />
<br />
Change to more modern conventions here-after (basically k ==> n and n ==> m)<br />
<br />
==Conventions==<br />
<br />
It is customary to denote the [[dimension]] with the letter 'n' and the [[cardinality|order]] of a hypercube with the letter 'm'.<br />
<br />
*'''(''n'') Dimension''' : the number of directions within a hypercube.<br />
*'''(''m'') Order''' : the number of numbers along a direction.<br />
<br />
Further: In this article the analytical number range [0..m<sup>n</sup>-1] is being used. For the regular number range [1..m<sup>n</sup>] you can add 1 to each number. This has absolutely no effect on the properties of the hypercube.<br />
<br />
==Notations==<br />
<br />
in order to keep things in hand a special notation was developed:<br />
*'''[ <sub>k</sub>i; k=[0..n-1]; i=[0..m-1] ]''': positions within the hypercube<br />
*'''&lt; <sub>k</sub>i; k=[0..n-1]; i=[0..m-1] &gt;''': vector through the hypercube<br />
<br />
Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and order can be added to it thus forming: <sup>n</sup>[<sub>k</sub>i]<sub>m</sub><br />
<br />
As is indicated 'k' runs through the dimensions, while the coordinate 'i' runs through all possible values, when values 'i' are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of m, as the magic hypercube resides in n-dimensional modular space.<br />
<br />
There can be multiple 'k' between bracket, these can't have the same value, though in undetermined order, which explains the equality of:<br />
[ <sub>1</sub>i, <sub>k</sub>j ] = [ <sub>k</sub>j, <sub>1</sub>i ]<br />
Of course given 'k' also one value 'i' is referred to.<br /><br />
When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #k=1 as in:<br />
[<sub>k</sub>1 ; #k=1] = [<sub>k</sub>1 <sub>j</sub>0 ; #k=1; #j=n-1] ("axial"-neighbor of [<sub>k</sub>0])<br />
(#j=n-1 can be left unspecified) j now runs through all the values in [0..k-1,k+1..n-1].<br />
<br />
Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)<br />
<br />
Note: as far as I now this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.<br />
<br />
Further: "'''perm(0..n-1)'''" specifies a [[permutation]] of the n numbers 0..n-1.<br />
<br />
==Construction==<br />
<br />
Besides more specific constructions two more general construction method are noticeable:<br />
<br />
===KnightJump construction===<br />
This construction generalizes the movement of the chessboard horses (vectors &lt;1,2&gt;, &lt;1,-2&gt;, &lt;-1,2&gt;, &lt;-1,-2&gt;) to more general movements (vectors &lt;<sub>k</sub>i&gt;). The method starts at the position P<sub>0</sub> and further numbers are sequentially placed at positions V<sub>0</sub> further until (after m steps) a position is reached that is already occupied, a further vector is needed to find the next free position. Thus the method is specified by the n by n+1 matrix:<br />
[P<sub>0</sub>, V<sub>0</sub> .. V<sub>n-1</sub>]<br />
This positions the number 'k' at position:<br />
P<sub>k</sub> = P<sub>0</sub> + <sub>l=0</sub>∑<sup>n-1</sup>((k\m<sup>l</sup>)%m) V<sub>l</sub>; k = 0 .. m<sup>n</sup>-1.<br />
'''C. Planck''' gives in his 1905 article [http://www.magichypercubes.com/Encyclopedia/k/PathNasiks.zip "'''The theory of Path Nasiks'''"] conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.<br />
<br />
===Latin prescription construction===<br />
(modular equations).<br />
This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector [x<sub>0</sub>,..,x<sub>n-1</sub>,1], After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:<br />
LP<sub>k</sub> = ( <sub>l=0</sub>∑<sup>n-1</sup> LP<sub>k,l</sub> x<sub>l</sub> + LP<sub>k,n</sub> ) % m<br />
of radix m numbers (also called "'''digits'''"). On these LP<sub>k</sub>'s "'''digit changing'''" (?i.e. Basic manipulation) are generally applied before these LP<sub>k</sub>'s are combined into the hypercube:<br />
<sup>n</sup>H<sub>m</sub> = <sub>k=0</sub>∑<sup>n-1</sup> LP<sub>k</sub> m<sup>k</sup><br />
<br />
'''J.R.Hendricks''' often uses modular equation, conditions to make hypercubes of various quality can be found on [http://www.magichypercubes.com/Encyclopedia http://www.magichypercubes.com/Encyclopedia] at several places (especially p-section)<br />
<br />
Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)<br />
<br />
===Multiplication===<br />
Amongst the various ways of compounding, the multiplication<ref>this is a n-dimensional version of (pe.): [http://mathforum.org/alejandre/magic.square/adler/product.html Alan Adler magic square multiplication]</ref> can be considered as the most basic of these methods. The '''basic multiplication''' is given by:<br />
<sup>n</sup>H<sub>m<sub>1</sub></sub> * <sup>n</sup>H<sub>m<sub>2</sub></sub> : <sup>n</sup>[<sub>k</sub>i]<sub>m<sub>1</sub>m<sub>2</sub></sub> = <sup>n</sup>[ [<nowiki>[</nowiki><sub>k</sub>i \ m<sub>2</sub>]<sub>m<sub>1</sub></sub></sub>m<sub>1</sub><sup>n</sup>]<sub>m<sub>2</sub></sub> + [<sub>k</sub>i % m<sub>2</sub>]<sub>m<sub>2</sub></sub>]<sub>m<sub>1</sub></sub><sub>m<sub>2</sub></sub><br />
<br />
Most compounding methods can be viewed as variations of the above, As most qualfiers are invariant under multiplication one can for example place any aspectial variant of <sup>n</sup>H<sub>m<sub>2</sub></sub> in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.<br />
<br />
==Aspects==<br />
A hypercube knows '''n! 2<sup>n</sup>''' Aspectial variants, which are obtained by coordinate reflection ([<sub>k</sub>i] --&gt; [<sub>k</sub>(-i)]) and coordinate permutations ([<sub>k</sub>i] --&gt; [<sub>perm[k]</sub>i]) effectively giving the Aspectial variant:<br />
<sup>n</sup>H<sub>m</sub><sup>~R perm(0..n-1)</sup>; R = <sub>k=0</sub>∑<sup>n-1</sup> ((reflect(k)) ? 2<sup>k</sup> : 0) ; perm(0..n-1) a permutation of 0..n-1<br />
Where reflect(k) true iff coordinate k is being reflected, only then 2<sup>k</sup> is added to R.<br />
As is easy to see, only n coordinates can be reflected explaining 2<sup>n</sup>, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!<br />
<br />
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in '''"normal position"''' by:<br />
[<sub>k</sub>0] = min([<sub>k</sub>θ ; θ ε {-1,0}]) (by reflection)<br />
[<sub>k</sub>1 ; #k=1] &lt; [<sub>k+1</sub>1 ; #k=1] ; k = 0..n-2 (by coordinate permutation)<br />
(explicitly stated here: [<sub>k</sub>0] the minimum of all corner points. The axial neighbour sequentially based on axial number)<br />
<br />
==Basic manipulations==<br />
<br />
Besides more specific manipulations, the following are of more general nature<br />
<br />
*'''#[perm(0..n-1)]''' : component permutation<br />
*'''^[perm(0..n-1)]''' : coordinate permutation (n == 2: transpose)<br />
*'''_2<sup>axis</sup>[perm(0..m-1)]''' : monagonal permutation (axis ε [0..n-1])<br />
*'''=[perm(0..m-1)]''' : digit change<br />
<br />
Note: <nowiki>'#'</nowiki>, '^', '_' and '=' are essential part of the notation and used as manipulation selectors.<br />
<br />
===Component permutation===<br />
Defined as the exchange of components, thus varying the factor m<sup>'''k'''</sup> in m<sup>'''perm(k)'''</sup>, because there are n component hypercubes the permutation is over these n components<br />
<br />
===Coordinate permutation===<br />
The exchange of coordinate [<sub>'''k'''</sub>i] into [<sub>'''perm(k)'''</sub>i], because of n coordinates a permutation over these n directions is required.<br /><br />
The term '''transpose''' (usually denoted by <sup>t</sup>) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.<br />
<br />
===Monagonal permutation===<br />
Defined as the change of [<sub>k</sub>'''i'''] into [<sub>k</sub>'''perm(i)'''] alongside the given "axial"-direction. Equal permutation along various axes can be combined by adding the factors 2<sup>axis</sup>. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.<br />
<br />
Noted be that '''reflection''' is the special case:<br />
~R = _R[n-1,..,0]<br />
Further when all the axes undergo the same ;permutation (R = 2<sup>n</sup>-1) an '''n-agonal permutation''' is achieved, In this special case the 'R' is usually omitted so:<br />
_[perm(0..n-1)] = _(2<sup>n</sup>-1)[perm(0..n-1)]<br />
<br />
===Digitchanging===<br />
Usually being applied at component level and can be seen as given by '''[<sub>k</sub>i]''' in '''perm([<sub>k</sub>i]''') since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.<br />
<br />
==Pathfinders==<br />
J. R. Hendricks called the directions within a hypercubes "'''pathfinders'''", these directions are simplest denoted in a ternary number system as:<br />
Pf<sub>p</sub> where: p = <sub>k=0</sub>∑<sup>n-1</sup> (<sub>k</sub>i + 1) 3<sup>k</sup> &lt;==&gt; &lt;<sub>k</sub>i&gt; ; i ε {-1,0,1}<br />
<br />
This gives 3<sup>n</sup> directions. since every direction is traversed both ways one can limit to the upper half [(3<sup>n</sup>-1)/2,..,3<sup>n</sup>-1)] of the full range.<br />
<br />
With these pathfinders any line to be summed over (or r-agonal) can be specified:<br />
[ <sub>j</sub>0 <sub>k</sub>p <sub>l</sub>q ; #j=1 #k=r-1 ; k > j ] &lt; <sub>j</sub>1 <sub>k</sub>θ <sub>l</sub>0 ; θ ε {-1,1} &gt; ; p,q ε [0,..,m-1]<br />
<br />
which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:<br />
[ <sub>j</sub>0 <sub>k</sub>0 <sub>l</sub>-1 <sub>s</sub>p ; #j=1 #k+#l=r-1 ; k,l > j ] &lt; <sub>j</sub>1 <sub>k</sub>1 <sub>l</sub>-1 <sub>s</sub>0 &gt;<br />
<br />
==Qualifications==<br />
<br />
A hypercube <sup>n</sup>H<sub>m</sub> with numbers in the analytical numberrange [0..m<sup>n</sup>-1] has the magic sum:<br />
<sup>n</sup>S<sub>m</sub> = m (m<sup>n</sup> - 1) / 2.<br />
<br />
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"<br />
<br />
*{'''r-agonal'''} : all main (unbroken) r-agonals are summing.<br />
*{'''pan r-agonal'''} : all (unbroken and broken) r-agonals are summing.<br />
*{'''magic'''} : {1-agonal n-agonal}<br />
*{'''perfect'''} : {pan r-agonal; r = 1..n}<br />
<br />
Note: This series doesn't start with 0 since a nill-agonal doesn't exist, the numbers correspond with the usual name-calling: 1-agonal = monagonal, 2-agonal = diagonal, 3-agonal = triagonal etc.. Aside from this the number correspond to the amount of "-1" and "1" in the corresponding pathfinder.<br />
<br />
In case the hypercube also sum when all the numbers are raised to the power p one gets p-multimagic hypercubes. The above qualifiers are simply prepended onto the p-multimagic qualifier. This defines qualifications as {r-agonal 2-magic}. Here also "2-" is usually replaced by "bi", "3-" by "tri" etc. ("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using [[Faulhaber's formula]] and divide it by m<sup>n-1</sup>.<br />
<br />
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the [[perfect magic cube|Trump/Boyer {diagonal} cube]] is technically seen {1-agonal 2-agonal 3-agonal}.<br />
<br />
[[Nasik magic hypercube]] gives arguments for using {'''nasik'''} as synonymous to {'''perfect'''}. The strange generalization of square 'perfect' to using it synonymous to {diagonal} in cubes is however also resolve by putting curly brackets around qualifiers, so {'''perfect'''} means {pan r-agonal; r = 1..n} (as mentioned above).<br />
<br />
some minor qualifications are:<br />
*{'''<sup>n</sup>compact'''} : {all order 2 subhyper cubes sum to 2<sup>n</sup> <sup>n</sup>S<sub>m</sub> / m}<br />
*{'''<sup>n</sup>complete'''} : {all pairs halve an n-agonal apart sum equal (to (m<sup>n</sup> - 1)}<br />
<br />
{'''<sup>n</sup>compact'''} might be put in notation as : '''<sub>(k)</sub>∑ [<sub>j</sub>i + <sub>k</sub>1] = 2<sup>n</sup> <sup>n</sup>S<sub>m</sub> / m'''.<br /><br />
{'''<sup>n</sup>complete'''} can simply written as: '''[<sub>j</sup>i] + [<sub>j</sup>i + <sub>k</sub>(m/2) ; #k=n ] = m<sup>n</sup> - 1'''.<br /><br />
Where:<br /><br />
<sub>(k)</sub>∑ is symbolic for summing all possible k's, there are 2<sup>n</sup> possibilities for <sub>k</sub>1.<br /><br />
[<sub>j</sub>i + <sub>k</sub>1] expresses [<sub>j</sub>i] and all its r-agonal neighbors.<br /><br />
for {complete} the complement of [<sub>j</sup>i] is at position [<sub>j</sup>i + <sub>k</sub>(m/2) ; #k=n ].<br />
<br />
for squares: {'''<sup>2</sup>compact <sup>2</sup>complete'''} is the "modern/alternative qualification" of what Dame [[Kathleen Ollerenshaw]] called [[most-perfect magic square]], {<sup>n</sup>compact <sup>n</sup>complete} is the qualifier for the feature in more than 2 dimensions<br /><br />
Caution: some people seems to equate {compact} with {<sup>2</sup>compact} instead of {<sup>n</sup>compact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript <sup>n</sup> to both these qualifiers (which are defined as shown) <br /><br />
consequences of {<sup>n</sup>compact} is that several figures also sum since they can be formed by adding/subtracting order 2 sub-hyper cubes. Issues like these go beyond this articles scope.<br />
<br />
==Special hypercubes==<br />
The following hypercubes serve special purposes;<br />
<br />
===The "normal hypercube"===<br />
<sup>n</sup>N<sub>m</sub> : [<sub>k</sub>i] = <sub>k=0</sub>∑<sup>n-1</sup> <sub>k</sub>i m<sup>k</sup><br />
This hypercube can be seen as the source of all numbers. A procedure called [http://www.magichypercubes.com/Encyclopedia/d/DynamicNumbering.html "Dynamic numbering"] makes use of the [[isomorphism]] of every hypercube with this normal, changing the source, changes the hypercube. Usually these sources are limited to direct products of normal hypercubes or normal [[Magic hyperbeam|hyperbeams]] (defined as having possibly other orders along the various directions).<br />
<br />
===The "constant 1"===<br />
<sup>n</sup>1<sub>m</sub> : [<sub>k</sub>i] = 1<br />
The hypercube that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hypercubes are of course multiples of this one.<br />
<br />
== See also ==<br />
<br />
*[[Magic hyperbeam]]<br />
*[[Nasik magic hypercube]]<br />
*[[Space diagonal]]<br />
*[[John R. Hendricks]]<br />
<br />
==File format==<br />
Based on [[XML]], the file format [http://www.magichypercubes.com/Encyclopedia/x/XmlHypercubes.html Xml-Hypercubes] is developed to describe various hypercubes to ensure human readability as well as programmatical usability. Besides full listings the format offers the ability to invoke mentioned constructions (amongst others)<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
== External links ==<br />
* [http://www.magichypercubes.com/Encyclopedia/index.html The Magic Encyclopedia] Articles by Aale de Winkel<br />
* [http://members.shaw.ca/hdhcubes/ Magic Cubes - Introduction] by Harvey D. Heinz<br />
* [http://math.ku.sk/~trenkler/Cube-Ref.html Magic Cubes and Hypercubes - References] Collected by Marian Trenkler<br />
** [http://math.ku.sk/~trenkler/05-MagicCube.pdf An algorithm for making magic cubes] by Marian Trenkler<br />
* [http://www.multimagie.com/ multimagie.com ] Articles by Christian Boyer<br />
<br />
==Further reading==<br />
<br />
* J.R.Hendricks: Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9<br />
* Planck, C., M.A.,M.R.C.S., The Theory of Paths Nasik, 1905, printed for private circulation. Introductory letter to the paper<br />
<br />
{{DEFAULTSORT:Magic Hypercube}}<br />
[[Category:Recreational mathematics]]<br />
[[Category:Magic squares|*]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=End_(category_theory)&diff=12646
End (category theory)
2013-06-13T15:24:34Z
<p>99.153.64.179: /* Examples */ make layout prettier</p>
<hr />
<div>In [[fluid mechanics]] the term '''static pressure''' has several uses:<br />
* In the design and operation of [[aircraft]], ''static pressure'' is the air pressure in the aircraft’s [[Pitot-static system#Static pressure|static pressure system]].<br />
* In [[fluid dynamics]], ''static pressure'' is the [[pressure]] at a nominated point in a fluid. Many authors use the term ''static pressure'' in place of ''pressure'' to avoid ambiguity.<br />
* The term ''static pressure'' is also used by some authors in [[fluid statics]].<br />
<br />
== Static pressure in design and operation of aircraft ==<br />
An aircraft’s [[altimeter]] is operated by the [[Pitot-static system#Static pressure|static pressure system]]. An aircraft’s [[airspeed indicator]] is operated by the static pressure system and the [[Pitot-static system#Pitot pressure|pitot pressure system]]<br />
.<ref>Lombardo, D.A., ''Aircraft Systems'', 2nd edition – chapter 2</ref><br />
<br />
The static pressure system is open to the exterior of the aircraft to sense the pressure of the atmosphere at the [[Altitude#Altitude in aviation|altitude]] at which the aircraft is flying. This small opening is called the [[Pitot-static system#Pitot-static pressure|static port]]. In flight the air pressure is slightly different at different positions around the exterior of the aircraft. The aircraft designer must select the position of the [[Pitot-static system#Pitot-static pressure|static port]] carefully. There is no position on the exterior of an aircraft at which the air pressure, for all [[angle of attack|angles of attack]], is identical to the atmospheric pressure at the altitude at which the aircraft is flying.<ref>"It is virtually impossible to find a position where the static pressure is always exactly the same as the pressure in the free airstream away from the aircraft". Kermode, A.C., ''Mechanics of Flight'', 10th edition – page 65</ref> The difference in pressure causes a small error in the altitude indicated on the altimeter, and the [[airspeed]] indicated on the airspeed indicator. This error in indicated altitude and airspeed is called [[position error]].<ref>Kermode, A.C., ''Mechanics of Flight'', 10th Edition – page 65</ref><br />
<ref>"Of these errors the error in detection of static pressure is generally the most serious and has the special name, ''position error''." Dommasch, D.O., Sherby, S.S., and Connolly, T.F. (1967) ''Airplane Aerodynamics'', 4th edition – page 51, Pitman Publishing Corp., New York</ref><br />
<br />
When selecting the position for the static port, the aircraft designer’s objective is to ensure the pressure in the aircraft’s static pressure system is as close as possible to the atmospheric pressure at the altitude at which the aircraft is flying, across the operating range of weight and airspeed. Many authors describe the atmospheric pressure at the altitude at which the aircraft is flying as the ''[[freestream]] static pressure''. At least one author takes a different approach in order to avoid a need for the expression ''freestream static pressure''. Gracey has written "The static pressure is the atmospheric pressure at the flight level of the aircraft".<br />
<ref>Gracey, William, [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800015804_1980015804.pdf ''Measurement of aircraft speed and altitude''] NASA, RP-1046, page 1</ref><ref>Gracey, William, ''Measurement of Aircraft Speed and Altitude'', page 1</ref> Gracey then refers to the air pressure at any point close to the aircraft as the ''local static pressure''.<br />
<br />
== Static pressure in fluid dynamics ==<br />
The concept of [[pressure]] is central to the study of fluids. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be [[Pressure measurement|measured]] using an [[Barometer#Aneroid barometers|aneroid]], [[Pressure measurement#Bourdon|Bourdon tube]], mercury column, or various other methods.<br />
<br />
The concepts of ''total pressure'' and ''[[dynamic pressure]]'' arise from [[Bernoulli's equation]] and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense - they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to [[pressure]] in fluid dynamics, many authors use the term ''static pressure'' to distinguish it from ''total pressure'' and ''dynamic pressure''. ''Static pressure'' is identical to pressure and can be identified for every point in a fluid flow field.<br />
<br />
In ''Aerodynamics'', L.J. Clancy<ref>Clancy, L.J., ''Aerodynamics'', page 21</ref> writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."<br />
<br />
[[Bernoulli's equation]] is fundamental to the dynamics of incompressible fluids. In many fluid flow situations of interest, changes in elevation are insignificant and can be ignored. With this simplification, Bernoulli’s equation for incompressible flows can be expressed as<ref>Clancy, L.J., ''Aerodynamics'', equation 3.13</ref><ref>Hurt, H.H. Jr, (1960), ''Aerodynamics for Naval Aviators'', page 9, A National Flightshop Reprint, Florida</ref><ref>Anderson, J.D. Jr, ''Fundamentals of Aerodynamics'', 4th edition – page 212, McGraw-Hill, New York. ISBN 978-0-07-295046-5</ref><br />
<br />
:<math>P + \tfrac12 \rho v^2 = P_0,</math><br />
<br />
where:<br />
*<math>P\;</math> is static pressure,<br />
*<math>\tfrac12 \rho v^2</math> is [[dynamic pressure]], usually denoted by <math>q\;</math>,<br />
*<math>\rho\,</math> is the [[density]] of the fluid,<br />
*<math>v\,</math> is the [[flow velocity]], and<br />
*<math>P_0\;</math> is total pressure which is constant along any [[Streamlines, streaklines, and pathlines|streamline]].<br />
<br />
Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own static pressure <math>P</math>, dynamic pressure <math>q</math>, and total pressure <math>P_0</math>. Static pressure and dynamic pressure are likely to vary significantly throughout the fluid but total pressure is constant along each streamline. In [[Conservative vector field#Irrotational flows|irrotational flow]], total pressure is the same on all streamlines and is therefore constant throughout the flow.<ref>A.M. Kuethe and J.D. Schetzer (1959), ''Foundations of Aerodynamics'', Section 3.5 (2nd edition), John Wiley & Sons, Inc. New York ISBN 0-471-50952-3</ref><br />
<br />
The simplified form of Bernoulli's equation can be summarised in the following memorable word equation:<br />
<ref>Clancy, L.J., ''Aerodynamics'', Section 3.5</ref><br />
<ref>”The total pressure is composed of two parts, the static pressure and the dynamic pressure”. Streeter, V.L., ''Fluid Mechanics'' 4th edition – page 404</ref><br />
<ref>[http://www.grc.nasa.gov/WWW/K-12/airplane/bern.html NASA's guide to Bernoulli's Equation]</ref><br />
:''static pressure + dynamic pressure = total pressure''.<br />
<br />
This simplified form of Bernoulli’s equation is fundamental to an understanding of the design and operation of ships, low speed aircraft, and airspeed indicators for low speed aircraft – that is aircraft whose maximum speed will be less than about 30% of the [[speed of sound]].<br />
<br />
As a consequence of the widespread understanding of the term ''static pressure'' in relation to Bernoulli’s equation, many authors <br />
<ref>For example: Abbott, I.H. and Von Doenhoff, A.E. (1949) ''Theory of Wing Sections'', Navier-Stokes equations - section 5.4. Dover Publications, Inc., New York. Standard Book Number 486-60586-8</ref><br />
in the field of fluid dynamics also use ''static pressure'' rather than ''pressure'' in applications not directly related to Bernoulli’s equation.<br />
<br />
The [[British Standards Institution]], in its Standard<ref>British Standard BS 185: Part 1: 1950 ''Glossary of Aeronautical Terms''</ref> ''Glossary of Aeronautical Terms'', gives the following definition:<br />
:''4412 '''Static pressure''' The pressure at a point on a body moving with the fluid.''<br />
<br />
== Static pressure in fluid statics ==<br />
<br />
The term ''static pressure'' is sometimes used in [[fluid statics]] to refer to the [[pressure]] of a fluid at a nominated depth in the fluid. In fluid statics the fluid is stationary everywhere and the concepts of dynamic pressure and total pressure are not applicable. Consequently there is little risk of ambiguity in using the term ''pressure'', but some authors<ref>For example: "The pressure in cases where no motion is occurring is referred to as static pressure." Curtis D. Johnson, [http://zone.ni.com/devzone/cda/ph/p/id/190 Process Control Instrumentation Technology], Prentice Hall (1997)</ref> choose to use ''static pressure'' in some situations.<br />
<br />
== See also ==<br />
* [[Hydrostatic pressure]]<br />
* [[Pascal's law]]<br />
* [[Stagnation pressure]]<br />
* [[Standard conditions for temperature and pressure]]<br />
<br />
==Notes==<br />
{{Reflist|2}}<br />
<br />
== References ==<br />
'''Aircraft design and operation'''<br />
* {{Citation<br />
| first = William<br />
| last = Gracey<br />
| title = Measurement of static pressure on aircraft<br />
| year = 1958<br />
| place = Langley Research Center<br />
| publisher = NACA<br />
| url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930092348_1993092348.pdf<br />
| accessdate = 2008-04-26<br />
| id = TR-1364 }}.<br />
* {{Citation<br />
| first = William<br />
| last = Gracey<br />
| title = Measurement of aircraft speed and altitude<br />
| year = 1980<br />
| place = Langley Research Center<br />
| publisher = NASA<br />
| url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800015804_1980015804.pdf<br />
| accessdate = 2008-04-26<br />
| id = RP-1046 }}.<br />
* {{Citation<br />
| last = Gracey<br />
| first = William<br />
| title = Measurement of Aircraft Speed and Altitude<br />
| publisher = John Wiley & Sons<br />
| year = 1981<br />
| location = New York<br />
| isbn = 0-471-08511-1 }}<br />
* Kermode, A.C. (1972) ''Mechanics of Flight'', Longman Group Limited, London ISBN 0-582-23740-8<br />
* Lombardo, D.A., ''Aircraft Systems'', 2nd edition, McGraw-Hill (1999), New York ISBN 0-07-038605-6<br />
<br />
'''Fluid dynamics'''<br />
* Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London ISBN 0-273-01120-0<br />
* Streeter, V.L. (1966), ''Fluid Mechanics'', McGraw-Hill, New York<br />
<br />
[[Category:Aerodynamics]]<br />
[[Category:Aircraft instruments]]<br />
[[Category:Fluid dynamics]]</div>
99.153.64.179
https://en.formulasearchengine.com/index.php?title=Representable_functor&diff=6701
Representable functor
2013-06-13T14:37:37Z
<p>99.153.64.179: /* Examples */ heap (mathematics) in example</p>
<hr />
<div>[[File:Magnetic Tunnel Junction.png|thumb|right|Magnetic tunnel junction (schematic)]]<br />
<br />
'''Tunnel magnetoresistance''' (TMR) is a [[magnetoresistance|magnetoresistive effect]] that occurs in a '''magnetic tunnel junction''' (MTJ), which is a component consisting of two [[ferromagnet]]s separated by a thin [[Insulator (electrical)|insulator]]. If the insulating layer is thin enough (typically a few [[nanometer]]s), [[electron]]s can [[Quantum tunneling|tunnel]] from one ferromagnet into the other. Since this process is forbidden in classical physics, the tunnel magnetoresistance is a strictly [[Quantum mechanics|quantum mechanical]] phenomenon.<br />
<br />
Magnetic tunnel junctions are manufactured in [[thin film]] technology. On an industrial scale the film deposition is done by magnetron [[sputter deposition]]; on a laboratory scale [[molecular beam epitaxy]], [[pulsed laser deposition]] and [[electron beam physical vapor deposition]] are also utilized. The junctions are prepared by [[photolithography]].<br />
<br />
== Phenomenological description ==<br />
<br />
The direction of the two [[magnetization]]s of the ferromagnetic films can be switched individually by an external [[magnetic field]]. If the magnetizations are in a parallel orientation it is more likely that [[electron]]s will tunnel through the insulating film than if they are in the oppositional (antiparallel) orientation. Consequently, such a junction can be switched between two states of [[electrical resistance]], one with low and one with very high resistance.<br />
<br />
== History ==<br />
<br />
The effect was originally discovered in 1975 by M. Jullière (University of Rennes, France) in [[iron|Fe]]/[[germanium|Ge]]-[[oxygen|O]]/[[cobalt|Co]]-junctions at 4.2 K. The relative change of resistance was around 14%, and did not attract much attention.<ref>{{cite journal | author=M. Julliere | title=Tunneling between ferromagnetic films | journal=Phys. Lett.| year=1975 | volume=54A | pages=225–226|doi=10.1016/0375-9601(75)90174-7 |bibcode = 1975PhLA...54..225J }}</ref> In 1991 Terunobu Miyazaki ([[Tohoku University]], Japan) found an effect of 2.7% at room temperature. Later, in 1994, Miyazaki found 18% in junctions of iron separated by an [[amorphous]] [[aluminum oxide]] insulator <ref>{{cite journal | author=T. Miyazaki and N. Tezuka | title =Giant magnetic tunneling effect in Fe/Al<sub>2</sub>O<sub>3</sub>/Fe junction|title=Giant magnetic tunneling effect in Fe/Al<sub>2</sub>O<sub>3</sub>/Fe junction | journal=J. Magn. Magn. Mater. | year=1995 | volume=139 | pages=L231–L234|bibcode = 1995JMMM..139L.231M |doi = 10.1016/0304-8853(95)90001-2 }}</ref> and [[Jagadeesh Moodera]] found 11.8% in junctions with electrodes of CoFe and Co.<ref>{{cite journal | author=J. S. Moodera ''et al.'' | title=Large Magnetoresistance at Room Temperature in Ferromagnetic Thin Film Tunnel Junctions | journal=Phys. Rev. Lett. | year=1995 | volume=74 | pages=3273–3276 | doi=10.1103/PhysRevLett.74.3273 | pmid=10058155 | issue=16 | bibcode=1995PhRvL..74.3273M}}</ref> The highest effects observed to date with aluminum oxide insulators are around 70% at room temperature.<br />
<br />
Since the year 2000, tunnel barriers of [[crystalline]] [[magnesium oxide]] (MgO) have been under development. In 2001 Butler and Mathon independently made the theoretical prediction that using [[iron]] as the ferromagnet and [[MgO]] as the insulator, the tunnel magnetoresistance can reach several thousand percent.<ref>{{cite journal | author=W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren | title=Spin-dependent tunneling conductance of Fe/MgO/Fe sandwiches | journal=Phys. Rev. B | year=2001 | volume=63 | page=054416 | doi=10.1103/PhysRevB.63.054416|bibcode = 2001PhRvB..63e4416B | issue=5 }}</ref><ref>{{cite journal | author=J. Mathon and A. Umerski | title=Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe (001) junction | journal=Phys. Rev. B | year=2001 | volume=63 | page=220403 | doi=10.1103/PhysRevB.63.220403|bibcode = 2001PhRvB..63v0403M | issue=22 }}</ref> The same year, Bowen et al. were the first to report experiments showing a significant TMR in a MgO based magnetic tunnel junction [Fe/MgO/FeCo(001)].<ref>{{cite journal | author=M. Bowen ''et al''. | title=Large magnetoresistance in Fe/MgO/FeCo(001)
epitaxial tunnel junctions on GaAs(001
) | journal=Appl. Phys. Lett. | year=2001 | volume=79 | page=1655 | doi=10.1063/1.1404125|bibcode = 2001ApPhL..79.1655B | issue=11 }}</ref> <br />
In 2004, Parkin and Yuasa were able to make Fe/MgO/Fe junctions that reach over 200% TMR at room temperature.<ref>{{cite journal | author=S Yuasa, T Nagahama, A Fukushima, Y Suzuki, and K Ando | title=Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions | journal=Nat. Mat. | year=2004 | volume=3 | pages=868–871 | doi=10.1038/nmat1257 | pmid=15516927 | issue=12|bibcode = 2004NatMa...3..868Y }}</ref><ref>{{cite journal | author=S. S. P. Parkin et al. | title=Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers | journal=Nat. Mat. | year=2004 | volume=3 | pages=862–867 | doi=10.1038/nmat1256 | pmid=15516928 | issue=12|bibcode = 2004NatMa...3..862P }}</ref> In 2009, effects of up to 600% at room temperature and more than 1100% at 4.2 K were observed in junctions of CoFeB/MgO/CoFeB.<ref>{{cite journal | author=S. Ikeda, J. Hayakawa, Y. Ashizawa, Y.M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. Matsukura and H. Ohno | title=Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudo-spin-valves annealed at high temperature | journal=Appl. Phys. Lett. | year=2008 | volume=93 | page=082508 | doi=10.1063/1.2976435|bibcode = 2008ApPhL..93h2508I | issue=8 }}</ref><br />
<br />
== Applications ==<br />
<br />
The [[Disk read-and-write head|read-heads]] of modern [[hard disk drive]]s work on the basis of magnetic tunnel junctions. TMR, or more specifically the magnetic tunnel junction, is also the basis of [[MRAM]], a new type of [[non-volatile memory]]. The 1st generation technologies relied on creating cross-point magnetic fields on each bit to write the data on it, although this approach has a scaling limit at around 90–130&nbsp;nm.<ref name="white paper">Barry Hoberman [http://www.crocus-technology.com/pdf/BH%20GSA%20Article.pdf The Emergence of Practical MRAM]. Crocus Technologies</ref> There are two 2nd generation techniques currently being developed: [[Thermal Assisted Switching]] (TAS)<ref name="white paper"/> and [[Spin Torque Transfer]] (STT). Magnetic tunnel junctions are also used for sensing applications.<br />
<br />
== Physical explanation ==<br />
<br />
[[File:TunnelSchema TMR.png|thumb|right|Two-current model for parallel and anti-parallel alignment of the magnetizations]]<br />
<br />
The relative resistance change—or effect amplitude—is defined as<br />
<br />
:<math>\mathrm{TMR} := \frac{R_{\mathrm{ap}}-R_{\mathrm{p}}}{R_{\mathrm{p}}}</math><br />
<br />
where <math>R_\mathrm{ap}</math> is the electrical resistance in the anti-parallel state, whereas <math>R_\mathrm{p}</math> is the resistance in the parallel state.<br />
<br />
The TMR effect was explained by Jullière with the [[spin polarization]]s of the ferromagnetic electrodes. The spin polarization ''P'' is calculated from the [[spin (physics)|spin]] dependent [[density of states]] (DOS) <math>\mathcal{D}</math> at the [[Fermi energy]]:<br />
<br />
<math>P = \frac{\mathcal{D}_\uparrow(E_\mathrm{F}) - \mathcal{D}_\downarrow(E_\mathrm{F})}{\mathcal{D}_\uparrow(E_\mathrm{F}) + \mathcal{D}_\downarrow(E_\mathrm{F})}</math><br />
<br />
The spin-up electrons are those with spin orientation parallel to the external magnetic field, whereas the spin-down electrons have anti-parallel alignment with the external field. The relative resistance change is now given by the spin polarizations of the two ferromagnets, ''P<sub>1</sub>'' and ''P<sub>2</sub>'':<br />
<br />
<math>\mathrm{TMR} = \frac{2 P_1 P_2}{1 - P_1 P_2}</math><br />
<br />
If no [[voltage]] is applied to the junction, electrons tunnel in both directions with equal rates. With a bias voltage ''U'', electrons tunnel preferentially to the positive electrode. With the assumption that spin is [[conservation law|conserved]] during tunneling, the current can be described in a two-current model. The total current is split in two partial currents, one for the spin-up electrons and another for the spin-down electrons. These vary depending on the magnetic state of the junctions.<br />
<br />
There are two possibilities to obtain a defined anti-parallel state. First, one can use ferromagnets with different [[coercivity|coercivities]] (by using different materials or different film thicknesses). And second, one of the ferromagnets can be coupled with an [[antiferromagnet]] ([[exchange bias]]). In this case the magnetization of the uncoupled electrode remains "free".<br />
<br />
The TMR decreases with both increasing temperature and increasing bias voltage. Both can be understood in principle by [[magnon]] excitations and interactions with magnons.<br />
<br />
It is obvious that the TMR becomes infinite if ''P<sub>1</sub>'' and ''P<sub>2</sub>'' equal 1, i.e. if both electrodes have 100% spin polarization. In this case the magnetic tunnel junction becomes a switch, that switches magnetically between low resistance and infinite resistance. Materials that come into consideration for this are called ''ferromagnetic half-metals''. Their conduction electrons are fully spin polarized. This property is theoretically predicted for a number of materials (e.g. CrO<sub>2</sub>, various [[Heusler alloy]]s) but has not been experimentally confirmed to date.<br />
<br />
==Spin-filtering in Tunnel Barriers==<br />
Prior to the introduction of epitaxial [[magnesium oxide]] (MgO), amorphous aluminum oxide was used as the tunnel barrier of the MTJ and typical room temperature TMR was in the range of tens of percent. MgO barriers increased TMR to hundreds of percent due to the ability to filter spin, which is complementary to the electrode spin polarization effect described above. The physical origin of this spin filtering is actually symmetry filtering because the electron wavefunctions of opposite spin originate from different bands at the Fermi level. These bands correspond to different orbitals for majority and minority spin and thus have different symmetries. The MgO conduction and valence bands have the same symmetry as majority spin electrons, so they experience a lower barrier height than minority spin electrons. This exponentially increases the tunneling probability so parallel configuration current exceeds anti-parallel current by a much larger amount.<br />
<br />
==Spin-transfer torque in Magnetic Tunnel Junctions (MTJs)==<br />
The effect of [[spin-transfer torque]] (STT) has been studied in MTJs where there is an tunnelling barrier sandwiched between a set of 2 ferromagnetic electrodes such that there is (free) magnetization of the right electrode, while assuming that the left electrode (with fixed magnetization) acts as spin-polarizer. This would then be pinned to some selecting transistor in an MRAM device. <br />
<br />
The STT vector, driven by the linear response voltage, can be computed from the expectation value of the torque operator:<br />
<br />
<math> \mathbf{T} = \mathrm{Tr}[\hat{\mathbf{T}} \hat{\rho}_\mathrm{neq}] </math><br />
<br />
where <math> \hat{\rho}_\mathrm{neq} </math> is the [[Gauge theory|gauge-invariant]] nonequilibrium [[density matrix]] for the steady-state transport, in the zero-temperature limit, in the linear-response regime,<ref>[F. Mahfouzi, N. Nagaosa, and B. K. Nikolić, ''Spin-orbit coupling induced spin-transfer torque and current polarization in topological-insulator/ferromagnet vertical heterostructures'', Phys. Rev. Lett. '''109''', 166602 (2012). Eq. (13)]</ref> and the torque operator <math> \hat{\mathbf{T}} </math> is obtained from the time derivative of the spin operator:<br />
<br />
<math><br />
\hat{\mathbf{T}} = \frac{d\hat{\mathbf{S}}}{dt}= -\frac{i}{\hbar}\left[\frac{\hbar}{2}\boldsymbol{\sigma},\hat{H}\right]<br />
</math><br />
<br />
Using the general form of a 1D tight-binding Hamiltonian:<br />
<br />
<math> \hat{H}=\hat{H}_0 - \Delta (\boldsymbol{\sigma} \cdot \mathbf{m})/2 </math><br />
<br />
where total magnetization (as macrospin) is along the unit vector <math> \mathbf{m}</math> and the Pauli matrices properties involving arbitrary classical vectors <math> \mathbf{p},\mathbf{q} </math>, given by<br />
<br />
<math> (\boldsymbol{\sigma} \cdot \mathbf{p})(\boldsymbol{\sigma} \cdot \mathbf{q}) = \mathbf{p} \cdot \mathbf{q} + i(\mathbf{p}\times\mathbf{q})\cdot \boldsymbol{\sigma} </math><br />
<br />
<math> (\boldsymbol{\sigma} \cdot \mathbf{p}) \boldsymbol{\sigma} = \mathbf{p} + i \boldsymbol{\sigma} \times \mathbf{p} </math><br />
<br />
<math> \boldsymbol{\sigma} (\boldsymbol{\sigma} \cdot \mathbf{q}) = \mathbf{q} + i \mathbf{q} \times \boldsymbol{\sigma} </math><br />
<br />
it is then possible to first obtain an analytical expression for <math> \hat{\mathbf{T}} </math> (which can be expressed in compact form using <math> \Delta, \mathbf{m} </math>, and the vector of Pauli spin matrices <math> \boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z) </math>). <br />
<br />
The STT vector in general MTJs has two components: a parallel and perpendicular component:<br />
<br />
A parallel component:<br />
<math> T_{\parallel}=\sqrt{T_x^2+T_z^2} </math><br />
<br />
And a perpendicular component:<br />
<math> T_{\perp}=T_y </math><br />
<br />
While in symmetric MTJs (made of electrodes with the same geometry and exchange splitting), the STT vector has only one active component, as the perpendicular component disappears: <br />
<br />
<math> T_{\perp} \equiv 0 </math>.<ref>[S.-C. Oh ''et. al.'', ''Bias-voltage dependence of perpendicular spin-transfer torque in a symmetric MgO-based magnetic tunnel junctions'', Nature Phys. '''5''', 898 (2009). [http://www.nature.com/nphys/journal/v5/n12/abs/nphys1427.html &#91;PDF&#93;]</ref> <br />
<br />
Therefore, only <math> T_{\parallel} </math> vs. <math> \theta </math> needs to be plotted at the site of the right electrode to characterise tunnelling in symmetric MTJs, making them appealing for production and characterisation at an industrial scale.<br />
<br />
Note: <br />
In these calculations the active region (for which it is necessary to calculate the retarded [[Green's function (many-body theory)|Green's function]]) should consist of the tunnel barrier + the right ferromagnetic layer of finite thickness (as in realistic devices). The active region is attached to the left ferromagnetic electrode (modeled as semi-infinite tight-binding chain with non-zero [[Zeeman effect|Zeeman splitting]]) and the right N electrode (semi-infinite tight-binding chain without any Zeeman splitting), as encoded by the corresponding self-energy terms.<br />
<br />
==References==<br />
{{reflist|35em}}<br />
<br />
==See also==<br />
<br />
* [[Quantum tunnelling]]<br />
* [[Magnetoresistance]]<br />
* [[Giant Magnetoresistance]]<br />
* [[Spin-transfer torque]]<br />
<br />
{{DEFAULTSORT:Tunnel Magnetoresistance}}<br />
[[Category:Electric and magnetic fields in matter]]<br />
[[Category:Spintronics]]</div>
99.153.64.179
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