Quaternionic matrix - Revision history

en>Monkbot: /* Determinants */Task 6k: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

2014-12-12T15:01:17Z

Determinants: Task 6k: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

← Older revision		Revision as of 17:01, 12 December 2014
Line 1:		Line 1:
	~~[[Image:Fermi Surface.svg\|thumb\|200 px\|right\|Luttinger's theorem relates a Fermi liquid's particle density~~ to the ~~volume~~ of ~~its Fermi surface~~.]]		There are also many businesses that are now offering the fixed gear bike, kromica fixie and takara fixie bikes, much to the delight of many biking aficionados. Above the very last ten many years the trails of this place have become smaller and scaled-down. If you have any sort of concerns concerning where and how you can utilize [http://platformarcade.com/profile/bestout Choosing the right ride for you mountain bike sizing.], you could contact us at the web site. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. noted, Valhalla was designed "by the people from Whistler" (the company Gravity Logic) so "it's special. That kept me looking forward to the book launch and beyond. <br><br> Similarly mountain bike wheels are also an important feature. in summary let me give some procuations you bshould take when ridding it , ridding a bike at high speed can lead to accidents just maintain a minimum and also follow the traffic lights , if it is for sport just make enough practice and enjoy the game. Fitting is logical, the instructions are clear few tools are required. Just to give you an idea, mine has three settings, those being; Completely off, so that I am riding it as a normal bike; Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac. <br><br>you know, right before making myself all nice and pretty before climbing on my bike and heading to work. Mulberry Gap Mountain Bike Get-A-Way is located 12 miles from downtown Ellijay in the Chattahoochee National Forest with some of the best single track mountain bike trails including the Georgia Pinhoti trail system. But when it comes to mountain biking it is not enough that you own a bike. Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. The most common sized crankset is the 170 millimeter. <br><br>June 2011 saw the successful launch of my very first book, WELCOME TO YOUR LIFE – simple insights for your inspiration & empowerment (. It is important for the rider to note that not all frames are compatible with every type of forks. Mountain biking is all about stretching your limits, enjoying the ambience, feel the mood swings of mountains weather and then reaching to the final destination. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. One of the problems is that they just haven't caught on and become commercially available enough to make them an everyday sight. <br><br>When the helmet is in the correct position on the head, the straps can be adjusted so that they are snug across the side of the head and under the chin. Try finding the best bike for yourself at local shops, where you can take a test ride before you finally decide. The best way of getting mountain bike components is shopping online. Mountain bike wheels provide a good mixture of traction, stability, and sturdiness ; and different riding conditions need specific types of mountain bicycle wheels. If they connect with the wrong part, it could cause you to stop on a dime, which will most likely result in you taking a tumble over the handlebars.

	~~In condensed matter physics, '''Luttinger's theorem'''<ref>~~
	~~{{Cite journal~~
	~~\| author = Luttinger, J. M.; Ward, J. C.~~
	~~\| year = 1960~~
	~~\| title = Ground-State Energy~~ of ~~a Many~~-~~Fermion System~~. II
	~~\| journal =~~ [~~[Physical Review]]~~
	~~\| volume = 118 \| issue = 5 \| pages = 1417–1427~~
	~~\| doi = 10~~.~~1103~~/~~PhysRev~~.~~118~~.~~1417~~
	~~\|bibcode = 1960PhRv~~..~~118~~.~~1417L }}~~<~~/ref~~><~~ref~~>
	~~{{Cite journal~~
	~~\| last = Luttinger \| first = J~~. M.
	~~\| year = 1960~~
	~~\| title = Fermi Surface~~ and ~~Some Smple Equilibrium Properties of a System of Interacting Fermions~~
	~~\| journal = [[Physical Review]]~~
	~~\| volume = 119 \| issue = 4 \| pages = 1153–1163~~
	~~\| doi =~~ ~~10.1103/PhysRev.119.1153~~
	~~\|bibcode = 1960PhRv..119.1153L }}</ref>~~ is ~~a result derived by [[Joaquin Mazdak Luttinger\|J. M. Luttinger]]~~ and ~~[[John Clive Ward\|J~~. C. ~~Ward]] in 1960~~ that ~~has broad implications~~ in ~~the field~~ of ~~electron transport. It arises frequently in theoretical models~~ of ~~correlated electrons, such~~ as the high-~~temperature superconductors,~~ and ~~in photoemission~~, ~~where~~ a ~~metal's Fermi surface can be directly observed~~.

	~~==Definition==~~

	~~Luttinger's theorem states that '''the volume enclosed by a material's [[Fermi surface]] is directly proportional~~ to ~~the particle density'''~~.

	~~While~~ the ~~theorem is an immediate result~~ of the ~~Pauli exclusion principle in~~ the ~~case of noninteracting particles,~~ it ~~remains true even as interactions between particles are taken into consideration provided~~ that the ~~appropriate definitions of Fermi surface~~ and ~~particle density~~ are ~~adopted. Specifically, in the interacting case the Fermi surface must be defined according~~ to the ~~criteria~~ that
	~~:<math>G(\omega=0,\,p) \to 0</math> or <math>\infty,</math>~~
	~~where <math>G</math>~~ is ~~the single-particle Green function in terms of frequency and momentum~~. ~~Then Luttinger's theorem can be recast into~~ the ~~form<ref>~~
	~~{{cite book~~
	~~\| author = Alexei M~~. ~~Tsvelik~~
	~~\| year = 2003~~
	~~\| title = Quantum Field Theory in Condensed Matter Physics~~
	~~\| edition=2nd \| page = 327~~
	~~\| publisher = [[Cambridge University Press]]~~
	~~\| isbn = 0-521-82284-X~~
	}}<~~/ref~~>
	:<~~math~~>~~n = 2 \int_{G(\omega=0~~,~~p)>0}\frac{d^D k}{~~(~~2\pi)^D}</math>,~~
	~~where <math>G</math> is as above and <math>d^D k</math>~~ is the ~~differential volume~~ of ~~<math>k</math>-space in <math>D</math> dimensions~~.

	~~==See also==~~

	* [[Fermi liquid]]
	* [[Fermi surface]]

	~~==References==~~
	~~===Inline===~~
	~~{{reflist}}~~

	~~===General===~~
	* {{cite arxiv \|eprint=0711.0952 \|author1=Behnam Farid \|title=On the ~~Luttinger theorem concerning number~~ of ~~particles in~~ the ~~ground states~~ of ~~systems of interacting fermions \|class=cond-mat.str-el \|~~year~~=2007}}~~
	* {{cite arxiv \|eprint=0909.~~2886 \|author1=Behnam Farid \|author2=Tsvelik \|title=Comment on "Breakdown~~ of the ~~Luttinger sum rule within the Mott-Hubbard insulator", by J. Kokalj~~ and P. ~~Prelovšek, ''Phys. Rev.'' B 78, 153103 (2008) \|class=cond-mat.str-el \|year=2009}}~~
	* {{cite arxiv \|eprint=0909.2887 \|author1=Behnam Farid \|title=Comment on "Violation of the ~~Luttinger sum rule within~~ the ~~Hubbard model~~ on ~~a triangular lattice"~~, ~~by J. Kokalj~~ and P. ~~Prelovšek~~, ~~''Eur. Phys. J.'' B 63, 431 (2008) \|class=cond-mat.str-el \|year=2009}}~~
	* {{cite arxiv \|eprint=1207.4201 \|author1=Kiaran B. ~~Dave\|author2=Philip W. Phillips\|author3=Charles L. Kane \|title=Absence~~ of ~~Luttinger's theorem \|class=cond-mat~~.~~str-el \|year=2012}}~~
	* {{cite arxiv \|eprint=1211.5612 \|author1=Behnam Farid \|title=Comment on "Absence of ~~Luttinger's theorem"~~, ~~by Kiaran B. Dave~~, ~~Philip W. Phillips~~ and ~~Charles L~~. ~~Kane~~, ~~arXiv:1207.4201 \|class=cond-mat.str-el \|year=2012}}~~
	* {{cite journal \|author=M. Oshikawa \|year=2000 \|title=Topological Approach to ~~Luttinger's Theorem and the Fermi Surface of~~ a ~~Kondo Lattice \|journal=[[Physical Review Letters]] \|volume=84 \|issue=15 \|pages=3370–3373 \|doi=10.1103/PhysRevLett.84.3370\|arxiv = cond-mat/0002392 \|bibcode = 2000PhRvL..84.3370O }}~~
	*{{cite conference \|author=F. D. M. Haldane \|year=2005 \|title=Luttinger's Theorem and Bosonization of the Fermi Surface \|booktitle=Proceedings of the International School of Physics "Enrico Fermi", ~~Course CXXI "Perspectives~~ in ~~Many-Particle Physics" \|editors=R~~. ~~A. Broglia and J. R. Schrieffer \|publisher=[[North-Holland Publishing Company\|North-Holland]] \|pages=5-29 \|arxiv=cond-mat/0505529}}~~

	~~[[Category:Condensed matter physics]]~~
	~~[[Category:Fermions]]~~

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q3029670

2013-03-07T19:39:37Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q3029670

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en>Stephen J. Brooks: /* Matrix operations */ Must define scalar multiplication both sides! (noncommutative)

2012-08-01T14:53:07Z

Matrix operations: Must define scalar multiplication both sides! (noncommutative)

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← Older revision		Revision as of 17:01, 12 December 2014
Line 1:		Line 1:
	~~[[Image:Fermi Surface.svg\|thumb\|200 px\|right\|Luttinger's theorem relates a Fermi liquid's particle density~~ to the ~~volume~~ of ~~its Fermi surface~~.]]		There are also many businesses that are now offering the fixed gear bike, kromica fixie and takara fixie bikes, much to the delight of many biking aficionados. Above the very last ten many years the trails of this place have become smaller and scaled-down. If you have any sort of concerns concerning where and how you can utilize [http://platformarcade.com/profile/bestout Choosing the right ride for you mountain bike sizing.], you could contact us at the web site. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. noted, Valhalla was designed "by the people from Whistler" (the company Gravity Logic) so "it's special. That kept me looking forward to the book launch and beyond. <br><br> Similarly mountain bike wheels are also an important feature. in summary let me give some procuations you bshould take when ridding it , ridding a bike at high speed can lead to accidents just maintain a minimum and also follow the traffic lights , if it is for sport just make enough practice and enjoy the game. Fitting is logical, the instructions are clear few tools are required. Just to give you an idea, mine has three settings, those being; Completely off, so that I am riding it as a normal bike; Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac. <br><br>you know, right before making myself all nice and pretty before climbing on my bike and heading to work. Mulberry Gap Mountain Bike Get-A-Way is located 12 miles from downtown Ellijay in the Chattahoochee National Forest with some of the best single track mountain bike trails including the Georgia Pinhoti trail system. But when it comes to mountain biking it is not enough that you own a bike. Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. The most common sized crankset is the 170 millimeter. <br><br>June 2011 saw the successful launch of my very first book, WELCOME TO YOUR LIFE – simple insights for your inspiration & empowerment (. It is important for the rider to note that not all frames are compatible with every type of forks. Mountain biking is all about stretching your limits, enjoying the ambience, feel the mood swings of mountains weather and then reaching to the final destination. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. One of the problems is that they just haven't caught on and become commercially available enough to make them an everyday sight. <br><br>When the helmet is in the correct position on the head, the straps can be adjusted so that they are snug across the side of the head and under the chin. Try finding the best bike for yourself at local shops, where you can take a test ride before you finally decide. The best way of getting mountain bike components is shopping online. Mountain bike wheels provide a good mixture of traction, stability, and sturdiness ; and different riding conditions need specific types of mountain bicycle wheels. If they connect with the wrong part, it could cause you to stop on a dime, which will most likely result in you taking a tumble over the handlebars.

	~~In condensed matter physics, '''Luttinger's theorem'''<ref>~~
	~~{{Cite journal~~
	~~\| author = Luttinger, J. M.; Ward, J. C.~~
	~~\| year = 1960~~
	~~\| title = Ground-State Energy~~ of ~~a Many~~-~~Fermion System~~. II
	~~\| journal =~~ [~~[Physical Review]]~~
	~~\| volume = 118 \| issue = 5 \| pages = 1417–1427~~
	~~\| doi = 10~~.~~1103~~/~~PhysRev~~.~~118~~.~~1417~~
	~~\|bibcode = 1960PhRv~~..~~118~~.~~1417L }}~~<~~/ref~~><~~ref~~>
	~~{{Cite journal~~
	~~\| last = Luttinger \| first = J~~. M.
	~~\| year = 1960~~
	~~\| title = Fermi Surface~~ and ~~Some Smple Equilibrium Properties of a System of Interacting Fermions~~
	~~\| journal = [[Physical Review]]~~
	~~\| volume = 119 \| issue = 4 \| pages = 1153–1163~~
	~~\| doi =~~ ~~10.1103/PhysRev.119.1153~~
	~~\|bibcode = 1960PhRv..119.1153L }}</ref>~~ is ~~a result derived by [[Joaquin Mazdak Luttinger\|J. M. Luttinger]]~~ and ~~[[John Clive Ward\|J~~. C. ~~Ward]] in 1960~~ that ~~has broad implications~~ in ~~the field~~ of ~~electron transport. It arises frequently in theoretical models~~ of ~~correlated electrons, such~~ as the high-~~temperature superconductors,~~ and ~~in photoemission~~, ~~where~~ a ~~metal's Fermi surface can be directly observed~~.

	~~==Definition==~~

	~~Luttinger's theorem states that '''the volume enclosed by a material's [[Fermi surface]] is directly proportional~~ to ~~the particle density'''~~.

	~~While~~ the ~~theorem is an immediate result~~ of the ~~Pauli exclusion principle in~~ the ~~case of noninteracting particles,~~ it ~~remains true even as interactions between particles are taken into consideration provided~~ that the ~~appropriate definitions of Fermi surface~~ and ~~particle density~~ are ~~adopted. Specifically, in the interacting case the Fermi surface must be defined according~~ to the ~~criteria~~ that
	~~:<math>G(\omega=0,\,p) \to 0</math> or <math>\infty,</math>~~
	~~where <math>G</math>~~ is ~~the single-particle Green function in terms of frequency and momentum~~. ~~Then Luttinger's theorem can be recast into~~ the ~~form<ref>~~
	~~{{cite book~~
	~~\| author = Alexei M~~. ~~Tsvelik~~
	~~\| year = 2003~~
	~~\| title = Quantum Field Theory in Condensed Matter Physics~~
	~~\| edition=2nd \| page = 327~~
	~~\| publisher = [[Cambridge University Press]]~~
	~~\| isbn = 0-521-82284-X~~
	}}<~~/ref~~>
	:<~~math~~>~~n = 2 \int_{G(\omega=0~~,~~p)>0}\frac{d^D k}{~~(~~2\pi)^D}</math>,~~
	~~where <math>G</math> is as above and <math>d^D k</math>~~ is the ~~differential volume~~ of ~~<math>k</math>-space in <math>D</math> dimensions~~.

	~~==See also==~~

	* [[Fermi liquid]]
	* [[Fermi surface]]

	~~==References==~~
	~~===Inline===~~
	~~{{reflist}}~~

	~~===General===~~
	* {{cite arxiv \|eprint=0711.0952 \|author1=Behnam Farid \|title=On the ~~Luttinger theorem concerning number~~ of ~~particles in~~ the ~~ground states~~ of ~~systems of interacting fermions \|class=cond-mat.str-el \|~~year~~=2007}}~~
	* {{cite arxiv \|eprint=0909.~~2886 \|author1=Behnam Farid \|author2=Tsvelik \|title=Comment on "Breakdown~~ of the ~~Luttinger sum rule within the Mott-Hubbard insulator", by J. Kokalj~~ and P. ~~Prelovšek, ''Phys. Rev.'' B 78, 153103 (2008) \|class=cond-mat.str-el \|year=2009}}~~
	* {{cite arxiv \|eprint=0909.2887 \|author1=Behnam Farid \|title=Comment on "Violation of the ~~Luttinger sum rule within~~ the ~~Hubbard model~~ on ~~a triangular lattice"~~, ~~by J. Kokalj~~ and P. ~~Prelovšek~~, ~~''Eur. Phys. J.'' B 63, 431 (2008) \|class=cond-mat.str-el \|year=2009}}~~
	* {{cite arxiv \|eprint=1207.4201 \|author1=Kiaran B. ~~Dave\|author2=Philip W. Phillips\|author3=Charles L. Kane \|title=Absence~~ of ~~Luttinger's theorem \|class=cond-mat~~.~~str-el \|year=2012}}~~
	* {{cite arxiv \|eprint=1211.5612 \|author1=Behnam Farid \|title=Comment on "Absence of ~~Luttinger's theorem"~~, ~~by Kiaran B. Dave~~, ~~Philip W. Phillips~~ and ~~Charles L~~. ~~Kane~~, ~~arXiv:1207.4201 \|class=cond-mat.str-el \|year=2012}}~~
	* {{cite journal \|author=M. Oshikawa \|year=2000 \|title=Topological Approach to ~~Luttinger's Theorem and the Fermi Surface of~~ a ~~Kondo Lattice \|journal=[[Physical Review Letters]] \|volume=84 \|issue=15 \|pages=3370–3373 \|doi=10.1103/PhysRevLett.84.3370\|arxiv = cond-mat/0002392 \|bibcode = 2000PhRvL..84.3370O }}~~
	*{{cite conference \|author=F. D. M. Haldane \|year=2005 \|title=Luttinger's Theorem and Bosonization of the Fermi Surface \|booktitle=Proceedings of the International School of Physics "Enrico Fermi", ~~Course CXXI "Perspectives~~ in ~~Many-Particle Physics" \|editors=R~~. ~~A. Broglia and J. R. Schrieffer \|publisher=[[North-Holland Publishing Company\|North-Holland]] \|pages=5-29 \|arxiv=cond-mat/0505529}}~~

	~~[[Category:Condensed matter physics]]~~
	~~[[Category:Fermions]]~~