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| In [[mathematics]], the '''indefinite orthogonal group''', O(''p'',''q'') is the [[Lie group]] of all [[linear transformation]]s of a ''n''-[[dimension of a vector space|dimensional]] real [[vector space]] which leave invariant a [[nondegenerate form|nondegenerate]], [[symmetric bilinear form]] of [[signature of a quadratic form|signature]] (''p'',''q''), where {{nowrap|1=''n'' = ''p'' + ''q''}}. The dimension of the group is {{nowrap|''n''(''n'' − 1)/2}}.
| | BSOD or the Blue Screen of Death, (additionally known because blue screen physical memory dump), is an error that occurs on a Windows system - whenever the computer simply shuts down or automatically reboots. This error may occur merely because your computer is booting up or certain Windows application is running. Whenever the Windows OS discovers an unrecoverable error it hangs the program or leads to memory dumps.<br><br>Windows Defender - this does come standard with numerous Windows OS Machines, but otherwise is download from Microsoft for free. It will aid protect against spyware.<br><br>Windows is actually fairly dumb. It only knows how to follow commands plus instructions, meaning which when we install a system, that program has to tell Windows what to do. This really is done by storing an "training file" inside the registry of your system. All your computer programs place these "manuals" into the registry, allowing the computer to run a broad array of programs. Whenever we load up one of those programs, Windows just looks up the program file in the registry, plus carries out its instructions.<br><br>In order to remove the programs on a computer, Windows Installer should be inside a healthy state. If its installation is corrupted we may get error 1721 inside Windows 7, Vista and XP during the system removal process. Just re-registering its component files would resolve a problem.<br><br>After that, I additionally bought the Regtool [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014] Software, plus it further secure my computer having program crashes. All my registry difficulties are fixed, plus I may function peacefully.<br><br>If you think that there are issues with all the d3d9.dll file, then you must substitute it with a brand-new functioning file. This is performed by performing a series of steps and you can commence by downloading "d3d9.zip" from the server. Next you need to unzip the "d3d9.dll" file found on the difficult drive of your computer. Proceed by locating "C:\Windows\System32" and then acquiring the existing "d3d9.dll" on your PC. When found, rename the file "d3d9.dll to d3d9BACKUP.dll" and then copy-paste this modern file to "C:\Windows\System32". After which, hit "Start" followed by "Run" or search "Run" on Windows Vista & 7. As shortly as a box shows up, sort "cmd". A black screen will then appear and we have to type "regsvr32d3d9.dll" plus then click "Enter". This process usually help we to replace the old file with the fresh copy.<br><br>As the hub center of the computer, the important settings are stored the registry. Registry is structured because keys plus each key relates to a system. The system reads the keys and uses the info to launch and run programs. However, the big issue is the fact that there are too several unwelcome settings, useless information occuping the useful room. It makes the system run slowly plus huge amounts of settings become unreadable.<br><br>Another important system you'll wish To receive is a registry cleaner. The registry is a big list of everything installed on the computer, and Windows references it when it opens a system or uses a device connected to the computer. If you delete a program, its registry entry could moreover be deleted, yet occasionally it's not. A registry cleaner could do away with these old entries so Windows could look the registry faster. It also deletes or corrects any entries that viruses have corrupted. |
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| The '''indefinite special orthogonal group''', SO(''p'',''q'') is the [[subgroup]] of O(''p'',''q'') consisting of all elements with [[determinant]] 1. Unlike in the definite case, SO(''p'',''q'') is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO<sup>+</sup>(''p'',''q'') and O<sup>+</sup>(''p'',''q''), which has 2 components – see the [[#Topology|topology section]] for definition and discussion.
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| The signature of the form determines the group up to [[isomorphism]]; interchanging ''p'' with ''q'' amounts to replacing the metric by its negative, and so gives the same group. If either ''p'' or ''q'' equals zero, then the group is isomorphic to the ordinary [[orthogonal group]] O(''n''). We assume in what follows that both ''p'' and ''q'' are positive.
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| The group O(''p'',''q'') is defined for vector spaces over the [[real number|real]]s. For [[Complex number|complex]] spaces, all groups {{nowrap|O(''p'',''q''; '''C''')}} are isomorphic to the usual [[orthogonal group]] {{nowrap|O(''p'' + ''q''; '''C''')}}, since the transform <math>z_j \mapsto iz_j</math> changes the signature of a form.
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| In even dimension, the middle group O(''n'',''n'') is known as the [[#Split orthogonal group|split orthogonal group]], and is of particular interest. In odd dimension, split form is the almost-middle group {{nowrap|O(''n'',''n'' + 1)}}.
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| == Examples ==
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| [[File:Squeeze r=1.5.svg|thumb|[[Squeeze mapping]]s, here ''r'' = 3/2, are the basic hyperbolic symmetries.]]
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| The basic example is the [[squeeze mapping]]s, which is the group SO<sup>+</sup>(1,1) of (the identity component of) linear transforms preserving the [[unit hyperbola]]. Concretely, these are the matrices <math>\left[\begin{smallmatrix} \lambda & 0 \\ 0 & 1/\lambda\end{smallmatrix}\right],</math> and can be interpreted as ''hyperbolic rotations,'' just as the group SO(2) can be interpreted as ''circular rotations.''
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| In physics, the [[Lorentz group]] O(1,3) is of central importance, being the setting for [[electromagnetism]] and [[special relativity]].
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| ==Matrix definition==
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| One can define O(''p'',''q'') as a group of [[matrix (mathematics)|matrices]], just as for the classical orthogonal group O(''n''). The standard inner product on '''R'''<sup>''p'',''q''</sup> is given in coordinates by the [[diagonal matrix]]:
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| :<math>\eta = \mathrm{diag}(\underbrace{1,\cdots,1}_{p},\underbrace{-1,\cdots,-1}_{q}).\,</math>
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| As a quadratic form,
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| <math>Q(x_1,\dots,x_n) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_{p+q}^2.</math> | |
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| The group O(''p'',''q'') is then the group of a ''n''×''n'' matrices ''M'' (where ''n'' = ''p''+''q'') such that <math>Q(Mv)=Q(v)</math>; as a bilinear form,
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| :<math>M^T\eta M = \eta.\,</math>
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| Here ''M''<sup>''T''</sup> denotes the [[transpose]] of the matrix ''M''. One can easily verify that the set of all such matrices forms a group. The inverse of ''M'' is given by
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| :<math>M^{-1} = \eta^{-1}M^T\eta.\,</math>
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| One obtains an isomorphic group (indeed, a conjugate subgroup of GL(V)) by replacing η with any [[symmetric matrix]] with ''p'' positive eigenvalues and ''q'' negative ones (such a matrix is necessarily [[nonsingular matrix|nonsingular]]); equivalently, any quadratic form with signature (''p'',''q''). Diagonalizing this matrix gives a conjugation of this group with the standard group O(''p'',''q'').
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| ==Topology==
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| Assuming both ''p'' and ''q'' are nonzero, neither of the groups O(''p'',''q'') or SO(''p'',''q'') are [[connected space|connected]], having four and two components respectively.
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| {{nowrap|1=''π''<sub>0</sub>(O(''p'',''q'')) ≅ ''C''<sub>2</sub> × ''C''<sub>2</sub>}} is the [[Klein four-group]], with each factor being whether an element preserves or reverses the respective orientations on the ''p'' and ''q'' dimensional subspaces on which the form is definite; note that reserving orientation on only one of these subspaces reverses orientation on the whole space. The special orthogonal group has components {{nowrap|1=''π''<sub>0</sub>(SO(''p'',''q'')) = {(1,1),(−1,−1)}}} which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.
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| The [[identity component]] of O(''p'',''q'') is often denoted SO<sup>+</sup>(''p'',''q'') and can be identified with the set of elements in SO(''p'',''q'') which preserves both orientations. This notation is related to the notation O<sup>+</sup>(1,3) for the [[orthochronous Lorentz group]], where the + refers to preserving the orientation on the first (temporal) dimension.
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| The group O(''p'',''q'') is also not [[compact space|compact]], but contains the compact subgroups O(''p'') and O(''q'') acting on the subspaces on which the form is definite. In fact, {{nowrap|1=O(''p'') × O(''q'')}} is a [[maximal compact subgroup]] of O(''p'',''q''), while {{nowrap|1=S(O(''p'') × O(''q''))}} is a maximal compact subgroup of SO(''p'',''q'').
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| Likewise, {{nowrap|1=SO(''p'') × SO(''q'')}} is a maximal compact subgroup of SO<sup>+</sup>(''p'', ''q'').
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| Thus up to homotopy, the spaces are products of (special) orthogonal groups, from which algebro-topological invariants can be computed.
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| In particular, the [[fundamental group]] of SO<sup>+</sup>(''p'',''q'') is the product of the fundamental groups of the components, {{nowrap|1=''π''<sub>1</sub>(SO<sup>+</sup>(''p'',''q'')) = ''π''<sub>1</sub>(SO(''p'')) × ''π''<sub>1</sub>(SO(''q''))}}, and is given by:
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| :{| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"
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| !style="background:#efefef;"| ''π''<sub>1</sub>(SO<sup>+</sup>(''p'',''q''))
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| !style="background:#efefef;"| ''p'' = 1
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| !style="background:#efefef;"| ''p'' = 2
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| !style="background:#efefef;"| ''p'' ≥ 3
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| |-
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| !style="background:#efefef;"| ''q'' = 1
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| | {1} || '''Z''' || '''Z'''<sub>2</sub>
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| |-
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| !style="background:#efefef;"| ''q'' = 2
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| | '''Z''' || '''Z''' × '''Z'''
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| || '''Z''' × '''Z'''<sub>2</sub>
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| |-
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| !style="background:#efefef;"| ''q'' ≥ 3
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| | '''Z'''<sub>2</sub> || '''Z'''<sub>2</sub> × '''Z'''
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| || '''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>
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| |}
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| ==Split orthogonal group==
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| In even dimension, the middle group O(''n'',''n'') is known as the '''split orthogonal group''', and is of particular interest. It is the [[split Lie group]] corresponding to the complex [[Lie algebra]] so<sub>2''n''</sub> (the Lie group of the [[split real form]] of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. In this sense it is opposite to the definite orthogonal group O(''n'') := O(''n'',0) = O(0,''n''), which is the [[compact real form|''compact'' real form]] of the complex Lie algebra.
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| The case (1,1) corresponds to the [[split-complex number]]s.
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| In terms of being a [[group of Lie type]] – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are [[Chevalley group]]s, while the non-split orthogonal groups require a slightly more complicated construction, and are [[Steinberg group (Lie theory)|Steinberg groups]].
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| Split orthogonal groups are used to construct the [[generalized flag variety]] over non-algebraically closed fields.
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| {{Expand section|date=March 2011}}
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| In odd dimension, the split form is the almost-middle group O(''n'',''n''+1).
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| ==See also==
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| *[[Pin group]]
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| ==References==
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| {{refbegin}}
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| *{{springer|id=O/o070300|title=Orthogonal group|author=[[Vladimir L. Popov|V. L. Popov]]}}
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| *[[Anthony Knapp]], ''Lie Groups Beyond an Introduction'', Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. ISBN 0-8176-4259-5 – see page 372 for a description of the indefinite orthogonal group
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| *[[Joseph A. Wolf]], ''Spaces of constant curvature'', (1967) page. 335.
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| {{refend}}
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| [[Category:Lie groups]]
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BSOD or the Blue Screen of Death, (additionally known because blue screen physical memory dump), is an error that occurs on a Windows system - whenever the computer simply shuts down or automatically reboots. This error may occur merely because your computer is booting up or certain Windows application is running. Whenever the Windows OS discovers an unrecoverable error it hangs the program or leads to memory dumps.
Windows Defender - this does come standard with numerous Windows OS Machines, but otherwise is download from Microsoft for free. It will aid protect against spyware.
Windows is actually fairly dumb. It only knows how to follow commands plus instructions, meaning which when we install a system, that program has to tell Windows what to do. This really is done by storing an "training file" inside the registry of your system. All your computer programs place these "manuals" into the registry, allowing the computer to run a broad array of programs. Whenever we load up one of those programs, Windows just looks up the program file in the registry, plus carries out its instructions.
In order to remove the programs on a computer, Windows Installer should be inside a healthy state. If its installation is corrupted we may get error 1721 inside Windows 7, Vista and XP during the system removal process. Just re-registering its component files would resolve a problem.
After that, I additionally bought the Regtool tuneup utilities 2014 Software, plus it further secure my computer having program crashes. All my registry difficulties are fixed, plus I may function peacefully.
If you think that there are issues with all the d3d9.dll file, then you must substitute it with a brand-new functioning file. This is performed by performing a series of steps and you can commence by downloading "d3d9.zip" from the server. Next you need to unzip the "d3d9.dll" file found on the difficult drive of your computer. Proceed by locating "C:\Windows\System32" and then acquiring the existing "d3d9.dll" on your PC. When found, rename the file "d3d9.dll to d3d9BACKUP.dll" and then copy-paste this modern file to "C:\Windows\System32". After which, hit "Start" followed by "Run" or search "Run" on Windows Vista & 7. As shortly as a box shows up, sort "cmd". A black screen will then appear and we have to type "regsvr32d3d9.dll" plus then click "Enter". This process usually help we to replace the old file with the fresh copy.
As the hub center of the computer, the important settings are stored the registry. Registry is structured because keys plus each key relates to a system. The system reads the keys and uses the info to launch and run programs. However, the big issue is the fact that there are too several unwelcome settings, useless information occuping the useful room. It makes the system run slowly plus huge amounts of settings become unreadable.
Another important system you'll wish To receive is a registry cleaner. The registry is a big list of everything installed on the computer, and Windows references it when it opens a system or uses a device connected to the computer. If you delete a program, its registry entry could moreover be deleted, yet occasionally it's not. A registry cleaner could do away with these old entries so Windows could look the registry faster. It also deletes or corrects any entries that viruses have corrupted.