Numerical Shadow

The web resource on numerical range and numerical shadow

User Tools

Site Tools


refnotes:bib

References

BibTeX

@Comment{refnotes,
  namespace = ":"
 
@article{aretaki2009investigating,
  title={Investigating the Numerical Range of Non Square Matrices},
  author={Aretaki, Aikaterini and Maroulas, John},
  journal={arXiv preprint arXiv:0904.4325},
  year={2009},
  url={https://arxiv.org/abs/0904.4325}
}
@article{marcus1985conditions,
  title={Conditions for the generalized numerical range to be real},
  author={Marcus, Marvin and Sandy, Markus},
  journal={Linear algebra and its applications},
  volume={71},
  pages={219--239},
  year={1985},
  publisher={Elsevier}
}
 
@article{chien2002davis,
  title={Davis--Wielandt shell and q-numerical range},
  author={Chien, Mao-Ting and Nakazato, Hiroshi},
  journal={Linear Algebra and its Applications},
  volume={340},
  number={1-3},
  pages={15--31},
  year={2002},
  publisher={Elsevier}
}
 
@article{chien2006q,
  title={The q-numerical range of a reducible matrix via a normal operator},
  author={Chien, Mao-Ting and Nakazato, Hiroshi},
  journal={Linear Algebra and its Applications},
  volume={419},
  number={2-3},
  pages={440--465},
  year={2006},
  publisher={North-Holland}
}
 
@article{tsing1984constrained,
  title={The constrained bilinear form and the C-numerical range},
  author={Tsing, Nam-Kiu},
  journal={Linear Algebra and its Applications},
  volume={56},
  pages={195--206},
  year={1984},
  publisher={Elsevier}
}
 
@article{li1994c,
  title={C-numerical ranges and C-numerical radii},
  author={Li, Chi-Kwong},
  journal={Linear and Multilinear Algebra},
  volume={37},
  number={1-3},
  pages={51--82},
  year={1994},
  publisher={Taylor & Francis}
}
 
@article{westwick1975theorem,
  title={A theorem on numerical range},
  author={Westwick, R},
  journal={Linear and Multilinear Algebra},
  volume={2},
  number={4},
  pages={311--315},
  year={1975},
  publisher={Taylor & Francis}
}
@article{szymanski2017classification,
  title={Classification of joint numerical ranges of three hermitian matrices of size three},
  author={Szymański, Konrad and Weis, Stephan and Życzkowski, Karol},
  journal={Linear Algebra and its Applications},
  year={2017},
  publisher={Elsevier},
  url={https://doi.org/10.1016/j.laa.2017.11.017}
}
 
@article{toeplitz1918algebraische,
  title={Das algebraische Analogon zu einem Satze von Fejer},
  author={Toeplitz, O.},
  journal={Mathematische Zeitschrift},
  volume={2},
  number={1},
  pages={187--197},
  year={1918},
  publisher={Springer},
  url={https://dx.doi.org/10.1007/BF01212904}
}
 
@article{hausdorff1919wertvorrat,
  title={Der Wertevorrat einer Bilinearform},
  author={Hausdorff, F.},
  journal={Mathematische Zeitschrift},
  volume={3},
  number={1},
  pages={314--316},
  year={1919},
  publisher={Springer},
  url={https://dx.doi.org/10.1007/BF01292610}
}
 
@article{murnaghan1932field,
  title={On the field of values of a square matrix},
  author={Murnaghan, F. D},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  volume={18},
  number={3},
  pages={246},
  year={1932},
  publisher={National Academy of Sciences},
  url={https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076200/}
}
 
@article{kippenhahn1951wertevorrat,
  title={Uber den Wertevorrat einer matrix},
  author={Kippenhahn, R.},
  journal={Mathematische Nachrichten},
  volume={6},
  number={3-4},
  pages={193--228},
  year={1951},
  publisher={Wiley Online Library},
  url={https://dx.doi.org/10.1002/mana.19510060306}
}
 
@book{horn1994topics,
  title={Topics in matrix analysis},
  author={Horn, R. and Johnson, C.},
  year={1994},
  publisher={Cambridge university press},
  url={https://www.google.pl/books?id=PlYQN0ypTwEC&lpg=PR9&ots=uArUdFZTwL&dq=Topics%20in%20matrix%20analysis&lr&hl=pl&pg=PP1#v=onepage&q&f=false}
}
 
@book{gustafson1997numerical,
  title={Numerical range: The Field of Values of Linear Operators and Matrices},
  author={Gustafson, K. E. and Rao, D. K. M.},
  year={1997},
  publisher={Springer},
  url={https://dx.doi.org/10.1007/978-1-4613-8498-4_1}
}
 
 
@article{gutkin2004toeplitz,
  title={The Toeplitz-Hausdorff theorem revisited: relating linear algebra and geometry},
  author={Gutkin, E.},
  journal={The Mathematical Intelligencer},
  volume={26},
  number={1},
  pages={8--14},
  year={2004},
  publisher={Springer},
  url={https://dx.doi.org/10.1007/BF02985393}
}
 
@article{li1996simple,
  title={A simple proof of the elliptical range theorem},
  author={Li, C. K.},
  journal={Proceedings of the American Mathematical Society},
  volume={124},
  number={7},
  pages={1985--1986},
  year={1996},
  url={https://dx.doi.org/10.1090/S0002-9939-96-03307-2}
}
 
@article{keeler1997numerical,
  title = {The numerical range of 3x3 matrices},
  journal = {Linear Algebra and its Applications},
  volume = {252},
  number = {1-3},
  pages = {115 - 139},
  year = {1997},
  url = {https://dx.doi.org/10.1016/0024-3795(95)00674-5},
  author = {Keeler, D. S.  and Rodman, L. and Spitkovsky, I. M.}
}
 
@article{crouzeix2006open,
  title={Open problems on Numerical range and functional calculus},
  author={Crouzeix, M.},
  year={2006},
  url={https://perso.univ-rennes1.fr/michel.crouzeix/publis/NopPb.pdf}
}
 
@article{johnson1981numerical,
title={Numerical ranges of principal sumbatrices},
author={Johnson, C. R.},
journal={Linear Algebra and its Applications},
volume={1},
pages={23-34},
year={1981},
url={https://www.sciencedirect.com/science/article/pii/0024379581901646?via%3Dihub}
}
 
@article{szymanski2017uncertainty,
  title={Uncertainty relations and joint numerical ranges},
  author={Szymański, Konrad},
  journal={arXiv preprint arXiv:1707.03464},
  year={2017},
  url={https://arxiv.org/abs/1707.03464}
}
 
@article{johnson1976normality,
  title={Normality and the numerical range},
  author={Johnson, C. R.},
  journal={Linear Algebra and its Applications},
  volume={37},
  number={1},
  pages={89--94},
  year={1976},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/0024-3795(76)90080-X}
}
 
@article{nylen1991numerical,
  title={Numerical range of a doubly stochastic matrix},
  author={Nylen, P. and Tam, T. Y.},
  journal={Linear Algebra and Its Applications},
  volume={153},
  pages={161--176},
  year={1991},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/0024-3795(91)90216-J}
}
 
@article{psarrakos2002numerical, 	
  title={Numerical ​range:(in) a matrix nutshell}, 	
  author={Psarrakos, Panayiotis J and Tsatsomeros, Michael J}, 
  year={2002}, 
  publisher={Department of Mathematics,​ Washington State University}, 
  url={https://​www.sci.wsu.edu/math/faculty/​tsat/​files/​numrange.pdf}
}
 
@article{chorianopoulos2009definition,
  title={A definition of numerical range of rectangular matrices},
  author={Chorianopoulos, Ch. and Karanasios, S. and Psarrakos, P.},
  journal={Linear and Multilinear Algebra},
  volume={57},
  number={5},
  pages={459--475},
  year={2009},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080802466365}
}
 
@article{shapiro2004notes,
  title={Notes on the numerical range},
  author={Shapiro, J. H.},
  journal={Lecture Notes, Michigan State University},
  year={2004},
  url={https://www.its.caltech.edu/~mneumann/publications/numrange.pdf}
}
 
@article{skoufranis2012numerical,
  title={Numerical Ranges of Operators},
  author={Skoufranis, P.},
  year={2012},
  url={https://www.math.ucla.edu/~pskoufra/OANotes-NumericalRanges.pdf}
}
 
@article{carden2009simple,
  title={A simple algorithm for the inverse field of values problem},
  author={Carden, Russell},
  journal={Inverse Problems},
  volume={25},
  number={11},
  pages={115019},
  year={2009},
  publisher={IOP Publishing},
  url={https://dx.doi.org/10.1088/0266-5611/25/11/115019}
}
 
@article{goldberg1977characterizations,
  title={On characterizations and integrals of generalized numerical ranges},
  author={Goldberg, Moshe and Straus, Ernst},
  journal={Pacific Journal of Mathematics},
  volume={69},
  number={1},
  pages={45--54},
  year={1977},
  url={https://msp.org/pjm/1977/69-1/p06.xhtml}
}
 
@article{goldberg1979certain,
  title={On certain finite dimensional numerical ranges and numerical radii†},
  author={Goldberg, Moshe},
  journal={Linear and Multilinear Algebra},
  volume={7},
  number={4},
  pages={329--342},
  year={1979},
  publisher={Taylor & Francis},
  url={https://www.tandfonline.com/doi/pdf/10.1080/03081087908817291}
}
 
@article{stampfli1970norm,
  title={The norm of a derivation},
  author={Stampfli, Joseph},
  journal={Pacific journal of mathematics},
  volume={33},
  number={3},
  pages={737--747},
  year={1970},
  publisher={Mathematical Sciences Publishers},
  url={https://msp.org/pjm/1970/33-3/pjm-v33-n3-p18-s.pdf}
}
 
@article{fiedler1981geometry,
  title={Geometry of the numerical range of matrices},
  author={Fiedler, M.},
  journal={Linear Algebra and its Applications},
  volume={37},
  pages={81--96},
  year={1981},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/0024-3795(81)90169-5}
}
 
@article{jonckheere1998differential,
  title={Differential topology of numerical range},
  author={Jonckheere, E. A. and Ahmad, F. and Gutkin, E.},
  journal={Linear algebra and its applications},
  volume={279},
  number={1},
  pages={227--254},
  year={1998},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/S0024-3795(98)00021-4}
}
 
@article{henrion2010semidefinite,
  title={Semidefinite geometry of the numerical range},
  author={Henrion, D.},
  journal={Electronic Journal of Linear Algebra},
  volume={20},
  pages={322-332},
  year={2010},
  url={https://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp322-332.pdf}
}
 
@article{helton2011possible,
  title={The possible shapes of numerical ranges},
  author={Helton, J. W. and Spitkovsky, I. M.},
  journal={arXiv preprint arXiv:1104.4587},
  year={2011},
  url={https://arxiv.org/abs/1104.4587}
}
 
@article{chien2000area,
  title={On the area of numerical range},
  author={Chien, M. T. and Lin, Y. H.},
  journal={Ssoochow Journal of Mathematics},
  volume={26},
  number={3},
  pages={255--270},
  year={2000},
  publisher={Soochow University},
  url={https://163.14.246.20/mp/pdf/S26N33.pdf}
}
 
@article{eldred2012numerical,
  title={Numerical ranges of companion matrices: flat portions on the boundary},
  author={Eldred, Jeffrey and Rodman, Leiba and Spitkovsky, Ilya},
  journal={Linear and Multilinear Algebra},
  volume={60},
  number={11-12},
  pages={1295--1311},
  year={2012},
  url={https://dx.doi.org/10.1080/03081087.2011.634415}
}
 
 
@article{maroulas1996geometrical,
  title={Geometrical properties of numerical range of matrix polynomials},
  author={Maroulas, J and Psarrakos, P},
  journal={Computers & Mathematics with Applications},
  volume={31},
  number={4},
  pages={41--47},
  year={1996},
  url={https://dx.doi.org/10.1016/0898-1221(95)00214-6}
}
 
@article{goldberg1977elementary,
  title={Elementary inclusion relations for generalized numerical ranges},
  author={Goldberg, Moshe and Straus, EG},
  journal={Linear Algebra and Its Applications},
  volume={18},
  number={1},
  pages={1--24},
  year={1977},
  publisher={Elsevier},
  url={https://www.sciencedirect.com/science/article/pii/0024379577900751}
}
@article{spitkovsky2015pre,
  title={Pre-images of extreme points of the numerical range, and applications},
  author={Spitkovsky, Ilya M and Weis, Stephan},
  journal={arXiv preprint arXiv:1509.05676},
  year={2015},
  url={https://arxiv.org/pdf/1509.05676.pdf}
}
 
@incollection{militzer2017numerical,
  title={Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary},
  author={Militzer, Erin and Patton, Linda J and Spitkovsky, Ilya M and Tsai, Ming-Cheng},
  booktitle={Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics},
  pages={561--591},
  year={2017},
  publisher={Springer},
  url={https://doi.org/10.1007/978-3-319-49182-0_23}
}
@article{gutkin2004convexity,
  title={Convexity of the joint numerical range: topological and differential geometric viewpoints},
  author={Gutkin, E. and Jonckheere, E.A. and Karow, M.},
  journal={Linear algebra and its applications},
  volume={376},
  pages={143--171},
  year={2004},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/j.laa.2003.06.011}
 
}
 
@article{krupnik2006sets,
  title={Sets of matrices with given joint numerical range},
  author={Krupnik, N. and Spitkovsky, I. M.},
  journal={Linear algebra and its applications},
  volume={419},
  number={2},
  pages={569--585},
  year={2006},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/j.laa.2006.05.019}
}
 
@article{abdollahi2006the,
title={The polynomial numerical hull of a matrix and algorithms for computing the numerical range},
author={Abdollahi, A.},
journal={Applied Mathematics and Computation},
volume={180},
pages={635-640},
year={2006},
url={https://dx.doi.org/10.1016/j.amc.2005.12.042}
}
 
@article{gau2006elliptic,
title={Elliptic numerical ranges of 4x4 matrices},
author={Gau, H.},
journal={Taiwanese Journal of Mathematics},
volume={10},
pages={117-128},
year={2006},
url={https://dx.doi.org/10.11650/twjm/1500406675}
}
 
@article{puchala2011product,
  title={Product numerical range in a space with tensor product structure},
  author={Puchała, Z. and Gawron, P. and Miszczak, J.A. and Skowronek, Ł. and Choi, M.D. and Życzkowski, K.},
  journal={Linear Algebra and its Applications},
  volume={434},
  number={1},
  pages={327--342},
  year={2011},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/j.laa.2010.08.026}
}
 
@article{helton2011the,
title={The possible shapes of numerical ranges},
author={Helton, J. W. and Spitkovsky, I. M.},
journal={arXiv:1104.4587},
volume={1},
pages={1-4},
year={2011},
url={https://arxiv.org/abs/1104.4587}
}
 
@article{cheung2012elementary,
title={ Elementary proofs for some results on the circular symmetry of the numerical range},
author={Cheung, W. and Li, C.},
journal={Linear & Multilinear Algebra},
volume={ahead-of},
pages={1-7},
year={2012},
url={https://dx.doi.org/10.1080/03081087.2012.696251}
}
 
@article{jurkowski2010local,
  title={Local numerical range for a class of 2⊗ d Hermitian operators},
  author={Jurkowski, Jacek and Rutkowski, Adam and Chruściński, D},
  journal={Open Systems & Information Dynamics},
  volume={17},
  number={04},
  pages={347--359},
  year={2010},
  url={https://dx.doi.org/10.1142/S1230161210000229}
}
 
@article{choi2007higher,
  title={Higher-rank numerical ranges of unitary and normal matrices},
  author={Choi, M. D. and Holbrook, J. A. and Kribs, D. W, and Życzkowski, K.},
  journal={Operators and Matrices},
  year={2007},
  volume={1},
  pages={409--426},
  url={https://dx.doi.org/10.7153/oam-01-24}
}
 
@article{choi2008geometry,
  title={Geometry of higher-rank numerical ranges},
  author={Choi, M. D. and Giesinger, M. and Holbrook, J. A. and Kribs, D. W.},
  journal={Linear and Multilinear Algebra},
  volume={56},
  number={1-2},
  pages={53--64},
  year={2008},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080701336545}
}
 
@article{woerdeman2008higher,
  title={The higher rank numerical range is convex},
  author={Woerdeman, H. J.},
  journal={Linear and Multilinear Algebra},
  volume={56},
  number={1-2},
  pages={65--67},
  year={2008},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080701352211}
}
 
@article{li2008canonical,
  title={Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations},
  author={Li, C. K, and Sze, N. S.},
  journal={Proceedings of the American Mathematical Society},
  volume={136},
  number={9},
  pages={3013--3023},
  year={2008},
  url={https://dx.doi.org/10.1090/S0002-9939-08-09536-1 }
}
@article{li2009condition,
  title={Condition for the higher rank numerical range to be non-empty},
  author={Li, C. K. and Poon, Y. T. and Sze, N. S.},
  journal={Linear and Multilinear Algebra},
  volume={57},
  number={4},
  pages={365--368},
  year={2009},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080701786384}
}
 
@article{gau2010higher,
  title={Higher-rank numerical ranges and dilations},
  author={Gau, H. L. and Li, C. K. and Wu, P. Y.},
  journal={Journal of Operator Theory},
  volume={63},
  number={1},
  pages={181},
  year={2010},
  url={https://people.wm.edu/cklixx/glwJOT.pdf}
}
 
@article{choi2006higher,
  title={Higher-rank numerical ranges and compression problems},
  author={Choi, M. D. and Kribs, D. W. and Życzkowski, K.},
  journal={Linear algebra and its applications},
  volume={418},
  number={2},
  pages={828--839},
  year={2006},
  url={https://dx.doi.org/10.1016/j.laa.2006.03.019}
}
 
@article{chien2011boundary,
  title={The boundary of higher rank numerical ranges},
  author={Chien, Mao-Ting and Nakazato, Hiroshi},
  journal={Linear Algebra and its Applications},
  volume={435},
  number={11},
  pages={2971--2985},
  year={2011},
  url={https://dx.doi.org/10.1016/j.laa.2011.05.020}
}
 
 
 
@article{gau2013higher,
  title={Higher-rank numerical ranges and Kippenhahn polynomials},
  author={Gau, Hwa-Long and Wu, Pei Yuan},
  journal={Linear Algebra and its Applications},
  volume={438},
  number={7},
  pages={3054--3061},
  year={2013},
  url={https://dx.doi.org/10.1016/j.laa.2012.11.017}
}
 
@article{gau2011higher,
  title={Higher rank numerical ranges of normal matrices},
  author={Gau, Hwa-Long and Li, Chi-Kwong and Poon, Yiu-Tung and Sze, Nung-Sing},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume={32},
  number={1},
  pages={23--43},
  year={2011},
  publisher={SIAM},
  url={https://dx.doi.org/10.1137/09076430X}
}
 
@article{holbrook2015bounds,
  title={Bounds on polygons of higher rank numerical ranges},
  author={Holbrook, John and Mudalige, Nishan and Newman, Mike and Pereira, Rajesh},
  journal={Linear Algebra and its Applications},
  volume={474},
  pages={230--242},
  year={2015},
  publisher={Elsevier},
  url={https://dx.doi.org/10.1016/j.laa.2014.12.025}
}
 
@article{najarbashi2007geometry,
  title={Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations},
  author={Najarbashi, G and Ahadpour, S and Fasihi, MA and Tavakoli, Y},
  journal={Journal of Physics A: Mathematical and Theoretical},
  volume={40},
  number={24},
  pages={6481},
  year={2007},
  url={https://dx.doi.org/10.1088/1751-8113/40/24/014}
}
 
@article{kribs2009research,
  title={Research problems on numerical ranges in quantum computing},
  author={Kribs, D. W. and Pasieka, A. and Laforest, M. and Ryan, C. and da Silva, M. P.},
  journal={Linear and Multilinear Algebra},
  volume={57},
  number={5},
  pages={491--502},
  year={2009},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080802677441}
}
 
@article{schulte2008significance,
  title={The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information},
  author={Schulte-Herbruggen, T. and Dirr, G. and Helmke, U. and Glaser, S. J.},
  journal={Linear and Multilinear Algebra},
  volume={56},
  number={1-2},
  pages={3--26},
  year={2008},
  publisher={Taylor & Francis},
  url={https://dx.doi.org/10.1080/03081080701544114}
}
 
@article{gawron2010restricted,
  title={Restricted numerical range: A versatile tool in the theory of quantum information},
  author={Gawron, P. and Puchała, Z. and Miszczak, J.A. and Skowronek, Ł. and Życzkowski, K.},
  journal={Journal of Mathematical Physics},
  volume={51},
  pages={102204},
  year={2010},
  url={https://dx.doi.org/10.1088/1751-8113/45/41/415309}
}
 
@article{zyczkowski2009generalized,
  title={Generalized numerical range as a versatile tool to study quantum entanglement},
  author={Życzkowski, K. and Choi, M.-D. and Dunkl, C. and Holbrook, J. and Gawron, P. and Miszczak, J. A. and  Puchala, Z. and Skowronek, Ł.},
  journal={Oberwolfach Report},
  volume={59},
  number={1-2},
  pages={34-37},
  year={2009},
  url={https://chaos.if.uj.edu.pl/~karol/pdf2/zyczkowski2dec09.pdf}
}
 
@article{dunkl2011numerical1,
  title={Numerical shadow and geometry of quantum states},
  author={Dunkl, C.F. and Gawron, P. and Holbrook, J.A. and Miszczak, J.A. and Puchała, Z. and Życzkowski, K.},
  journal={Journal of Physics A: Mathematical and Theoretical},
  volume={44},
  pages={335301},
  year={2011},
  publisher={IOP Publishing},
  url={https://dx.doi.org/10.1088/1751-8113/44/33/335301}
}
 
@article{dunkl2011numerical2,
  title={Numerical shadows: measures and densities on the numerical range},
  author={Dunkl, C.F. and Gawron, P. and Holbrook, J.A. and Puchała, Z. and Zyczkowski, K.},
  journal={Linear Algebra and its Applications},
  volume={434},
  number={9},
  pages={2042--2080},
  year={2011},
  publisher={North-Holland},
  url={https://dx.doi.org/10.1016/j.laa.2010.12.003}
}
 
@article{bengtsson2011geometry,
title={Geometry of the set of mixed quantum states: An apophatic approach},
author={Bengtsson, I. and Weis, S. and Życzkowski, K.},
journal={arxiv},
volume={1},
pages={1-13},
year={2011},
url={https://arxiv.org/abs/1112.2347}
}
 
@article{gutkin2013joint,
title={Joint numerical ranges, quantum maps, and joint numerical shadows},
author={Gutkin, E. and Zyczkowski, K. },
journal={Linear Algebra Appl.},
volume={438},
pages={2394-2404},
year={2013},
url={https://dx.doi.org/10.1016/10.1016/j.laa.2012.10.043}
}
 
@article{gallay2010the,
title={The numerical measure of a complex matrix},
author={Gallay, T. and Serre, D.},
journal={arXiv:1009.1522},
volume={1},
pages={1-41},
year={2010},
url={https://arxiv.org/abs/1009.1522}
}
 
@article{puchala2012restricted,a
  title={Restricted numerical shadow and geometry of quantum entanglement},
  author={Puchała, Z. and Miszczak, J.A. and Gawron, P. and Dunkl, C.F. and Holbrook, J.A. and Życzkowski, K.},
  journal={Journal of Physics A: Mathematical and Theoretical},
  volume={45},
  number={41},
  pages={415309},
  year={2012},
  url={https://dx.doi.org/10.1088/1751-8113/45/41/415309}
}
 
@article{dunkl2014real,
  title={Real numerical shadow and generalized B-splines},
  author={Dunkl, Charles F and Gawron, Piotr and Pawela, Łukasz and Puchała, Zbigniew and Życzkowski, Karol},
  journal={Linear Algebra and its Applications},
  year={2015},
  volume={479},
  pages={12--51},
  url={https://dx.doi.org/10.1016/j.laa.2015.03.029}
}
 
@article{brody2007on,
title={On quantum microcanonical equilibrium},
author={Brody, D. C. and Hook, D. W. and Hughston, L. P.},
journal={Journal of Physics: Conference Series},
volume={67},
pages={012025},
year={2007},
url={https://dx.doi.org/10.1088/1742-6596/67/1/012025}
}
 
@article{brody2007quantum,
title={Quantum phase transitions without thermodynamic limits},
author={Brody, D. C. and Hook, D. W. and Hughston, L. P.},
journal={ Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science},
volume={463},
pages={2021-2030},
year={2007},
url={https://dx.doi.org/10.1098/rspa.2007.1865}
}
 
@article{brody2005microcanonical,
title={Microcanonical distributions for quantum systems},
author={Brody, D. C. and Hook, D. W. and Hughston, L. P.},
journal={arxiv},
volume={1},
pages={1-8},
year={2005},
url={https://arxiv.org/abs/quant-ph/0506163}
}
 
@article{brody1998the,
title={The quantum canonical ensemble},
author={Brody, D. C. and Hughston, L. P.},
journal={Journal of Mathematical Physics},
volume={39},
pages={6502-6508},
year={1998},
url={https://dx.doi.org/10.1063/1.532661}
}
 
@article{zanardi2012probability,
title={Probability density of quantum expectation values},
author={Zanardi, L. C. V. P.},
journal={arXiv preprint},
volume={1},
pages={1-8},
year={2012},
url={https://arxiv.org/abs/1202.4810}
}
refnotes/bib.txt · Last modified: 2019/06/12 11:35 by plewandowska