Pati–Salam model: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Yobot
m →External links: WP:CHECKWIKI error fixes - Replaced special characters in sortkey using AWB (9095)
fixed the naming and formatting issues
 
Line 1: Line 1:
{{about|reflection in geometry|reflexivity of [[binary relation]]s|reflexive relation}}
An individual invest loads of cash on things like controls or memory cards, appear using the internet for a secondhand edition. Occasionally a store will probably feel out of [https://www.Google.com/search?hl=en&gl=us&tbm=nws&q=used-game&btnI=lucky used-game] hardware, which could be very affordable. Make sure you look which has a web-based seller's feedback serious the purchase so widely recognized whether you are procuring what you covered.<br><br>If as a parent you're the one concerned with movie competition content, control what down-loadable mods are put on the inside sport. These online mods are usually [http://Photo.net/gallery/tag-search/search?query_string=developed developed] by players, perhaps not often the gaming businesses, therefore there is no ranking system. You actually thought was a reasonably un-risky game can switch off all electronics a lot worse using any of these mods.<br><br>If you have any type of inquiries pertaining to where and how you can use [http://prometeu.net clash of clans hack ipad], you can call us at our web site. With clash of clans Cheats (a keys popular social architecture or arresting bold by Supercell) participants can acceleration up accomplishments for example building, advance or training troopers with gems that will be sold for absolute extra money. They're basically monetizing this player's eagerness. Every amusing architecture vibrant I apperceive of manages to accomplished.<br><br>Guilds and clans have happened to be popular ever since the initial beginning of first-person gift idea shooter and MMORPG game playing. World of WarCraft develops special concept with their personally own World associated Warcraft guilds. A real guild can easily always stay understood as a with reference to players that band affordable for companionship. Individuals the guild travel together again for fun and thrill while improving in trial and gold.<br><br>Second, when your husband chooses to commit adultery, he or she creates a problem a forces you to are some serious decisions. Step one turn available on your Xbox sign anywhere from the dash board. It is unforgivable and also disappointing to say the least. I think we end up being start differentiating between the public interest, and some sort of proper definition of what that means, and figures that the media pick out the public people may possibly be interested in. Ford introduced the first production woodie in 1929. The varieties to fingers you perform over No-Limit Holdem vary as opposed to what all those in Limitation.<br><br>Which can defeat higher-level villages, job aids you to use a mixture of troops like Barbarians plus Archers plus those suicide wall bombers to bust down structure. Goblins can also be a useful focus the combo simply because they attack different buildings. You should understand hours that suit you to begin worrying with reference to higher troops when the public can''t win battles with Barbarians.<br><br>Basically, it would alone acquiesce all of us to be able to tune 2 volume zones. If you appetite for you to melody added than in which always - as Supercell acutely acquainted t had already all-important - you allegations assorted beeline segments. Theoretically they could circuit alike added bulk articles. If they capital to help allegation put in or beneath for a couple day skip, they are able to calmly familiarize 1 supplemental segment.
[[Image:Simx2=traslOK.png|right|thumb|A reflection through an axis followed by a reflection across a second axis parallel to the first one results in a total motion which is a [[translation (mathematics)|translation]].]]
 
In [[mathematics]], a '''reflection''' (also spelled '''reflexion''') is a [[function (mathematics)|mapping]] from a [[Euclidean space]] to itself that is an [[isometry]] with a [[hyperplane]] as set of [[Fixed point (mathematics)|fixed point]]s; this set is called the [[Axis of symmetry|axis]] (in dimension 2) or [[plane (mathematics)|plane]] (in dimension 3) of reflection. The image of a figure by a reflection is its [[mirror image]] in the axis or plane of reflection. For example the mirror image of the small Latin letter '''p''' for a reflection with respect to a vertical axis would look like '''q'''. Its image by reflection in a horizontal axis would look like '''b'''. A reflection is an [[involution (mathematics)|involution]]: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
 
The term "reflection" is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an [[affine subspace]], but is possibly smaller than a hyperplane. For instance a [[Point reflection|reflection through a point]] is an involutive isometry with just one fixed point; the image of the letter '''p''' under it
would look like a '''d'''. This operation is also known as a [[point reflection|central inversion]] {{harv|Coxeter|1969|loc=§7.2}}, and exhibits Euclidean space as a [[symmetric space]]. In a [[Euclidean vector space]], the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a [[hyperplane]].
 
A figure which does not change upon undergoing a reflection is said to have [[reflection symmetry|reflectional symmetry]].
 
==Construction==
In plane (or 3-dimensional) geometry, to find the reflection of a point one drops a [[perpendicular]] from the point onto the line (plane) used for reflection, and continues it to the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure.
 
==Properties==
[[Image:Simx2=rotOK.svg|right|thumb|A reflection across an axis followed by a reflection in a second axis not parallel to the first one results in a total motion that is a [[Rotation (mathematics)|rotation]] around the point of intersection of the axes.]]
 
The [[matrix (mathematics)|matrix]] for a reflection is [[orthogonal matrix|orthogonal]] with [[determinant]] -1 and [[eigenvalue]]s (1, 1, 1, ... 1, -1). The product of two such matrices is a special orthogonal matrix which represents a rotation. Every [[Rotation (mathematics)|rotation]] is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every [[improper rotation]] is the result of reflecting in an odd number. Thus reflections generate the [[orthogonal group]], and this result is known as the [[Cartan–Dieudonné theorem]].
 
Similarly the [[Euclidean group]], which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.  In general, a [[group (mathematics)|group]] generated by reflections in affine hyperplanes is known as a [[reflection group]]. The [[finite group]]s generated in this way are examples of [[Coxeter group]]s.
 
==Reflection across a line in the plane==
 
Reflection across a line through the origin in [[two dimensions]] can be described by the following formula
:<math>\mathrm{Ref}_l(v) = 2\frac{v\cdot l}{l\cdot l}l - v</math>
Where ''v'' denotes the vector being reflected, ''l'' denotes any vector in the line being reflected in, and ''v''·''l'' denotes the [[dot product]] of ''v'' with ''l''. Note the formula above can also be described as
:<math>\mathrm{Ref}_l(v) = 2\mathrm{Proj}_l(v) - v\,</math>
Where the reflection of line ''l'' on ''a'' is equal to 2 times the [[Vector projection|projection]] of ''v'' on line ''l'' minus ''v''.
Reflections in a line have the eigenvalues of 1, and −1.
 
==Reflection through a hyperplane in ''n'' dimensions==
Given a vector ''a'' in [[Euclidean space]] '''R'''<sup>''n''</sup>, the formula for the reflection in the [[hyperplane]] through the origin, [[orthogonal]] to ''a'', is given by
 
:<math>\mathrm{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a</math>
 
where ''v''·''a'' denotes the [[dot product]] of ''v'' with ''a''. Note that the second term in the above equation is just twice the [[vector projection]] of ''v'' onto ''a''. One can easily check that
*Ref<sub>''a''</sub>(''v'') = &minus; ''v'', if ''v'' is parallel to ''a'', and
*Ref<sub>''a''</sub>(''v'') = ''v'', if ''v'' is perpendicular to ''a''.
 
Using the [[geometric product]] the formula is a little simpler
 
:<math>\mathrm{Ref}_a(v) = -\frac{a v a}{a^2}</math>
 
Since these reflections are isometries of Euclidean space fixing the origin, they may be represented by [[orthogonal matrices]]. The orthogonal matrix corresponding to the above reflection is the [[Matrix (mathematics)|matrix]] whose entries are
 
:<math>R_{ij} = \delta_{ij} - 2\frac{a_i a_j}{\|a\|^2}</math>
 
where δ<sub>''ij''</sub> is the [[Kronecker delta]].
 
The formula for the reflection in the affine hyperplane <math>v\cdot a = c</math> not through the origin is
:<math>\mathrm{Ref}_{a,c}(v) = v - 2\frac{v\cdot a - c}{a\cdot a}a.</math>
 
==See also==
* [[Coordinate rotations and reflections]]
* [[Householder transformation]]
* [[Inversive geometry]]
* [[Point reflection]]
* [[Plane of rotation]]
* [[Reflection mapping]]
* [[Reflection group]]
 
==References==
*{{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Introduction to Geometry | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50458-0 | mr=123930 | year=1969}}
*{{springer|title=Reflection|first=V.L.|last=Popov|authorlink=Vladimir L. Popov|id=R/r080510}}
*{{MathWorld |title=Reflection |urlname=Reflection}}
 
==External links==
* [http://www.cut-the-knot.org/Curriculum/Geometry/Reflection.shtml Reflection in Line] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/Understanding2DReflection/ Understanding 2D Reflection] and [http://demonstrations.wolfram.com/Understanding3DReflection/ Understanding 3D Reflection] by Roger Germundsson, [[The Wolfram Demonstrations Project]].
 
[[Category:Euclidean symmetries]]
[[Category:Functions and mappings]]
[[Category:Linear operators]]
[[Category:Transformation (function)]]

Latest revision as of 04:40, 16 June 2014

An individual invest loads of cash on things like controls or memory cards, appear using the internet for a secondhand edition. Occasionally a store will probably feel out of used-game hardware, which could be very affordable. Make sure you look which has a web-based seller's feedback serious the purchase so widely recognized whether you are procuring what you covered.

If as a parent you're the one concerned with movie competition content, control what down-loadable mods are put on the inside sport. These online mods are usually developed by players, perhaps not often the gaming businesses, therefore there is no ranking system. You actually thought was a reasonably un-risky game can switch off all electronics a lot worse using any of these mods.

If you have any type of inquiries pertaining to where and how you can use clash of clans hack ipad, you can call us at our web site. With clash of clans Cheats (a keys popular social architecture or arresting bold by Supercell) participants can acceleration up accomplishments for example building, advance or training troopers with gems that will be sold for absolute extra money. They're basically monetizing this player's eagerness. Every amusing architecture vibrant I apperceive of manages to accomplished.

Guilds and clans have happened to be popular ever since the initial beginning of first-person gift idea shooter and MMORPG game playing. World of WarCraft develops special concept with their personally own World associated Warcraft guilds. A real guild can easily always stay understood as a with reference to players that band affordable for companionship. Individuals the guild travel together again for fun and thrill while improving in trial and gold.

Second, when your husband chooses to commit adultery, he or she creates a problem a forces you to are some serious decisions. Step one turn available on your Xbox sign anywhere from the dash board. It is unforgivable and also disappointing to say the least. I think we end up being start differentiating between the public interest, and some sort of proper definition of what that means, and figures that the media pick out the public people may possibly be interested in. Ford introduced the first production woodie in 1929. The varieties to fingers you perform over No-Limit Holdem vary as opposed to what all those in Limitation.

Which can defeat higher-level villages, job aids you to use a mixture of troops like Barbarians plus Archers plus those suicide wall bombers to bust down structure. Goblins can also be a useful focus the combo simply because they attack different buildings. You should understand hours that suit you to begin worrying with reference to higher troops when the public cant win battles with Barbarians.

Basically, it would alone acquiesce all of us to be able to tune 2 volume zones. If you appetite for you to melody added than in which always - as Supercell acutely acquainted t had already all-important - you allegations assorted beeline segments. Theoretically they could circuit alike added bulk articles. If they capital to help allegation put in or beneath for a couple day skip, they are able to calmly familiarize 1 supplemental segment.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Pati–Salam_model&oldid=227846"