---
_id: '2355'
abstract:
- lang: eng
text: 'The BMV conjecture for traces, which states that Tr exp(A - λB) is the Laplace
transform of a positive measure, is shown to be equivalent to two other statements:
(i) The polynomial λ → Tr(A + λB) p has only non-negative coefficients for all
A, B ≥ 0, p ∈ ℕ and (ii) λ → Tr(A + λB)-p is the Laplace transform of a positive
measure for A, B ≥ 0, p > 0.'
author:
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Lieb É, Seiringer R. Equivalent forms of the Bessis-Moussa-Villani conjecture.
*Journal of Statistical Physics*. 2004;115(1-2):185-190. doi:10.1023/B:JOSS.0000019811.15510.27
apa: Lieb, É., & Seiringer, R. (2004). Equivalent forms of the Bessis-Moussa-Villani
conjecture. *Journal of Statistical Physics*, *115*(1–2), 185–190. https://doi.org/10.1023/B:JOSS.0000019811.15510.27
chicago: 'Lieb, Élliott, and Robert Seiringer. “ Equivalent Forms of the Bessis-Moussa-Villani
Conjecture.” *Journal of Statistical Physics* 115, no. 1–2 (2004): 185–90.
https://doi.org/10.1023/B:JOSS.0000019811.15510.27.'
ieee: É. Lieb and R. Seiringer, “ Equivalent forms of the Bessis-Moussa-Villani
conjecture,” *Journal of Statistical Physics*, vol. 115, no. 1–2, pp. 185–190,
2004.
ista: Lieb É, Seiringer R. 2004. Equivalent forms of the Bessis-Moussa-Villani
conjecture. Journal of Statistical Physics. 115(1–2), 185–190.
mla: Lieb, Élliott, and Robert Seiringer. “ Equivalent Forms of the Bessis-Moussa-Villani
Conjecture.” *Journal of Statistical Physics*, vol. 115, no. 1–2, Springer,
2004, pp. 185–90, doi:10.1023/B:JOSS.0000019811.15510.27.
short: É. Lieb, R. Seiringer, Journal of Statistical Physics 115 (2004) 185–190.
date_created: 2018-12-11T11:57:11Z
date_published: 2004-04-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1023/B:JOSS.0000019811.15510.27
extern: 1
intvolume: ' 115'
issue: 1-2
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0210027
month: '04'
oa: 1
page: 185 - 190
publication: Journal of Statistical Physics
publication_status: published
publisher: Springer
publist_id: '4568'
quality_controlled: 0
status: public
title: ' Equivalent forms of the Bessis-Moussa-Villani conjecture'
type: journal_article
volume: 115
year: '2004'
...
---
_id: '2356'
abstract:
- lang: eng
text: 'Recent experimental and theoretical work has shown that there are conditions
in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function
Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical
phenomenon because it is not necessary to have a trap width that is the size of
an atom - as might have been supposed - but it suffices merely to have a trap
width such that the energy gap for motion in the transverse direction is large
compared to the energy associated with the motion along the trap. Up to now the
theoretical arguments have been based on variational - perturbative ideas or numerical
investigations. In contrast, this paper gives a rigorous proof of the one-dimensional
behavior as far as the ground state energy and particle density are concerned.
There are four parameters involved: the particle number, N, transverse and longitudinal
dimensions of the trap, r and L, and the scattering length a of the interaction
potential. Our main result is that if r/L → 0 and N → ∞ the ground state energy
and density can be obtained by minimizing a one-dimensional density functional
involving the Lieb-Liniger energy density with coupling constant ∼ a/r 2. This
density functional simplifies in various limiting cases and we identify five asymptotic
parameter regions altogether. Three of these, corresponding to the weak coupling
regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii
theory. We also show that Bose-Einstein condensation in the ground state persists
in a part of this regime. In the strong coupling regime the longitudinal motion
of the particles is strongly correlated. The Gross-Pitaevskii description is not
valid in this regime and new mathematical methods come into play.'
author:
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jakob
full_name: Yngvason, Jakob
last_name: Yngvason
citation:
ama: Lieb É, Seiringer R, Yngvason J. One-dimensional behavior of dilute, trapped
Bose gases. *Communications in Mathematical Physics*. 2004;244(2):347-393.
doi:10.1007/s00220-003-0993-3
apa: Lieb, É., Seiringer, R., & Yngvason, J. (2004). One-dimensional behavior
of dilute, trapped Bose gases. *Communications in Mathematical Physics*,
*244*(2), 347–393. https://doi.org/10.1007/s00220-003-0993-3
chicago: 'Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “One-Dimensional
Behavior of Dilute, Trapped Bose Gases.” *Communications in Mathematical Physics*
244, no. 2 (2004): 347–93. https://doi.org/10.1007/s00220-003-0993-3.'
ieee: É. Lieb, R. Seiringer, and J. Yngvason, “One-dimensional behavior of dilute,
trapped Bose gases,” *Communications in Mathematical Physics*, vol. 244,
no. 2, pp. 347–393, 2004.
ista: Lieb É, Seiringer R, Yngvason J. 2004. One-dimensional behavior of dilute,
trapped Bose gases. Communications in Mathematical Physics. 244(2), 347–393.
mla: Lieb, Élliott, et al. “One-Dimensional Behavior of Dilute, Trapped Bose Gases.”
*Communications in Mathematical Physics*, vol. 244, no. 2, Springer, 2004,
pp. 347–93, doi:10.1007/s00220-003-0993-3.
short: É. Lieb, R. Seiringer, J. Yngvason, Communications in Mathematical Physics
244 (2004) 347–393.
date_created: 2018-12-11T11:57:11Z
date_published: 2004-01-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1007/s00220-003-0993-3
extern: 1
intvolume: ' 244'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0305025
month: '01'
oa: 1
page: 347 - 393
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4569'
quality_controlled: 0
status: public
title: One-dimensional behavior of dilute, trapped Bose gases
type: journal_article
volume: 244
year: '2004'
...
---
_id: '2360'
abstract:
- lang: eng
text: An optical lattice model developed that is similar to the Bose-Hubbard model
to describe the transition between Bose-Einstein condensation (BEC) and a Mott
insulator state was analyzed. It was found that the system was a hard core lattice
gas at half of the maximum density and the optical lattice was modeled by a periodic
potential of strength λ. It was also observed that the interparticle interaction
was essential for this transition that occurred even in the ground state. The
results show that all the essential features could be proved rigorously such as
the existence of BEC for small λ and its suppression for a large λ.
author:
- first_name: Michael
full_name: Aizenman, Michael
last_name: Aizenman
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jan
full_name: Solovej, Jan P
last_name: Solovej
- first_name: Jakob
full_name: Yngvason, Jakob
last_name: Yngvason
citation:
ama: Aizenman M, Lieb É, Seiringer R, Solovej J, Yngvason J. Bose-Einstein quantum
phase transition in an optical lattice model. *Physical Review A - Atomic, Molecular,
and Optical Physics*. 2004;70(2):023612-1-0236121-2. doi:10.1103/PhysRevA.70.023612
apa: Aizenman, M., Lieb, É., Seiringer, R., Solovej, J., & Yngvason, J. (2004).
Bose-Einstein quantum phase transition in an optical lattice model. *Physical
Review A - Atomic, Molecular, and Optical Physics*, *70*(2), 023612-1-0236121-2.
https://doi.org/10.1103/PhysRevA.70.023612
chicago: 'Aizenman, Michael, Élliott Lieb, Robert Seiringer, Jan Solovej, and Jakob
Yngvason. “Bose-Einstein Quantum Phase Transition in an Optical Lattice Model.”
*Physical Review A - Atomic, Molecular, and Optical Physics* 70, no. 2 (2004):
023612-1-0236121-2. https://doi.org/10.1103/PhysRevA.70.023612.'
ieee: M. Aizenman, É. Lieb, R. Seiringer, J. Solovej, and J. Yngvason, “Bose-Einstein
quantum phase transition in an optical lattice model,” *Physical Review A -
Atomic, Molecular, and Optical Physics*, vol. 70, no. 2, pp. 023612-1-0236121-2,
2004.
ista: Aizenman M, Lieb É, Seiringer R, Solovej J, Yngvason J. 2004. Bose-Einstein
quantum phase transition in an optical lattice model. Physical Review A - Atomic,
Molecular, and Optical Physics. 70(2), 023612-1-0236121-2.
mla: Aizenman, Michael, et al. “Bose-Einstein Quantum Phase Transition in an Optical
Lattice Model.” *Physical Review A - Atomic, Molecular, and Optical Physics*,
vol. 70, no. 2, American Physical Society, 2004, pp. 023612-1-0236121-2, doi:10.1103/PhysRevA.70.023612.
short: M. Aizenman, É. Lieb, R. Seiringer, J. Solovej, J. Yngvason, Physical Review
A - Atomic, Molecular, and Optical Physics 70 (2004) 023612-1-0236121-2.
date_created: 2018-12-11T11:57:12Z
date_published: 2004-08-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1103/PhysRevA.70.023612
extern: 1
intvolume: ' 70'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/cond-mat/0403240
month: '08'
oa: 1
page: 023612 - 1-0236121-2
publication: Physical Review A - Atomic, Molecular, and Optical Physics
publication_status: published
publisher: American Physical Society
publist_id: '4567'
quality_controlled: 0
status: public
title: Bose-Einstein quantum phase transition in an optical lattice model
type: journal_article
volume: 70
year: '2004'
...
---
_id: '2417'
alternative_title:
- 'Contemporary Mathematics '
author:
- first_name: László
full_name: Lovász, László
last_name: Lovász
- first_name: Katalin
full_name: Vesztergombi, Katalin
last_name: Vesztergombi
- first_name: Uli
full_name: Uli Wagner
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Emo
full_name: Welzl, Emo
last_name: Welzl
citation:
ama: 'Lovász L, Vesztergombi K, Wagner U, Welzl E. Convex quadrilaterals and k-sets
. In: Pach J, ed. *Towards a Theory of Geometric Graphs*. Vol 342. American
Mathematical Society; 2004:139-148. doi:10.1090/conm/342'
apa: Lovász, L., Vesztergombi, K., Wagner, U., & Welzl, E. (2004). Convex quadrilaterals
and k-sets . In J. Pach (Ed.), *Towards a Theory of Geometric Graphs* (Vol.
342, pp. 139–148). American Mathematical Society. https://doi.org/10.1090/conm/342
chicago: Lovász, László, Katalin Vesztergombi, Uli Wagner, and Emo Welzl. “Convex
Quadrilaterals and K-Sets .” In *Towards a Theory of Geometric Graphs*, edited
by János Pach, 342:139–48. American Mathematical Society, 2004. https://doi.org/10.1090/conm/342.
ieee: L. Lovász, K. Vesztergombi, U. Wagner, and E. Welzl, “Convex quadrilaterals
and k-sets ,” in *Towards a Theory of Geometric Graphs*, vol. 342, J. Pach,
Ed. American Mathematical Society, 2004, pp. 139–148.
ista: Lovász L, Vesztergombi K, Wagner U, Welzl E. 2004. Convex quadrilaterals and
k-sets . Towards a Theory of Geometric Graphs. , Contemporary Mathematics , vol.
342. 139–148.
mla: Lovász, László, et al. “Convex Quadrilaterals and K-Sets .” *Towards a Theory
of Geometric Graphs*, edited by János Pach, vol. 342, American Mathematical
Society, 2004, pp. 139–48, doi:10.1090/conm/342.
short: L. Lovász, K. Vesztergombi, U. Wagner, E. Welzl, in:, J. Pach (Ed.), Towards
a Theory of Geometric Graphs, American Mathematical Society, 2004, pp. 139–148.
date_created: 2018-12-11T11:57:32Z
date_published: 2004-01-01T00:00:00Z
date_updated: 2019-04-26T07:22:12Z
day: '01'
doi: 10.1090/conm/342
editor:
- first_name: János
full_name: Pach, János
last_name: Pach
extern: 1
intvolume: ' 342'
month: '01'
page: 139 - 148
publication: Towards a Theory of Geometric Graphs
publication_status: published
publisher: American Mathematical Society
publist_id: '4508'
quality_controlled: 0
status: public
title: 'Convex quadrilaterals and k-sets '
type: book_chapter
volume: 342
year: '2004'
...
---
_id: '2425'
abstract:
- lang: eng
text: A finite set N ⊂ Rd is a weak ε-net for an n-point set X ⊂ Rd (with respect
to convex sets) if N intersects every convex set K with |K ∩ X| ≥ εn. We give
an alternative, and arguably simpler, proof of the fact, first shown by Chazelle
et al., that every point set X in Rd admits a weak ε-net of cardinality O(ε-dpolylog(1/ε)).
Moreover, for a number of special point sets (e.g., for points on the moment curve),
our method gives substantially better bounds. The construction yields an algorithm
to construct such weak ε-nets in time O(n ln(1/ε)).
author:
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Uli
full_name: Uli Wagner
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Matoušek J, Wagner U. New constructions of weak ε-nets. *Discrete & Computational
Geometry*. 2004;32(2):195-206. doi:10.1007/s00454-004-1116-4
apa: Matoušek, J., & Wagner, U. (2004). New constructions of weak ε-nets. *Discrete
& Computational Geometry*, *32*(2), 195–206. https://doi.org/10.1007/s00454-004-1116-4
chicago: 'Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak ε-Nets.” *Discrete
& Computational Geometry* 32, no. 2 (2004): 195–206. https://doi.org/10.1007/s00454-004-1116-4.'
ieee: J. Matoušek and U. Wagner, “New constructions of weak ε-nets,” *Discrete
& Computational Geometry*, vol. 32, no. 2, pp. 195–206, 2004.
ista: Matoušek J, Wagner U. 2004. New constructions of weak ε-nets. Discrete &
Computational Geometry. 32(2), 195–206.
mla: Matoušek, Jiří, and Uli Wagner. “New Constructions of Weak ε-Nets.” *Discrete
& Computational Geometry*, vol. 32, no. 2, Springer, 2004, pp. 195–206,
doi:10.1007/s00454-004-1116-4.
short: J. Matoušek, U. Wagner, Discrete & Computational Geometry 32 (2004) 195–206.
date_created: 2018-12-11T11:57:35Z
date_published: 2004-07-01T00:00:00Z
date_updated: 2019-04-26T07:22:12Z
day: '01'
doi: 10.1007/s00454-004-1116-4
extern: 1
intvolume: ' 32'
issue: '2'
month: '07'
page: 195 - 206
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '4500'
quality_controlled: 0
status: public
title: New constructions of weak ε-nets
type: journal_article
volume: 32
year: '2004'
...