---
_id: '179'
abstract:
- lang: eng
  text: An asymptotic formula is established for the number of rational points of
    bounded anticanonical height which lie on a certain Zariski dense subset of the
    biprojective hypersurface x1y21+⋯+x4y24=0 in ℙ3×ℙ3. This confirms the modified
    Manin conjecture for this variety, in which the removal of a thin set of rational
    points is allowed.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Roger
  full_name: Heath Brown, Roger
  last_name: Heath Brown
citation:
  ama: Browning TD, Heath Brown R. Density of rational points on a quadric bundle
    in ℙ3×ℙ3. <i>Duke Mathematical Journal</i>. 2020;169(16):3099-3165. doi:<a href="https://doi.org/10.1215/00127094-2020-0031">10.1215/00127094-2020-0031</a>
  apa: Browning, T. D., &#38; Heath Brown, R. (2020). Density of rational points on
    a quadric bundle in ℙ3×ℙ3. <i>Duke Mathematical Journal</i>. Duke University Press.
    <a href="https://doi.org/10.1215/00127094-2020-0031">https://doi.org/10.1215/00127094-2020-0031</a>
  chicago: Browning, Timothy D, and Roger Heath Brown. “Density of Rational Points
    on a Quadric Bundle in ℙ3×ℙ3.” <i>Duke Mathematical Journal</i>. Duke University
    Press, 2020. <a href="https://doi.org/10.1215/00127094-2020-0031">https://doi.org/10.1215/00127094-2020-0031</a>.
  ieee: T. D. Browning and R. Heath Brown, “Density of rational points on a quadric
    bundle in ℙ3×ℙ3,” <i>Duke Mathematical Journal</i>, vol. 169, no. 16. Duke University
    Press, pp. 3099–3165, 2020.
  ista: Browning TD, Heath Brown R. 2020. Density of rational points on a quadric
    bundle in ℙ3×ℙ3. Duke Mathematical Journal. 169(16), 3099–3165.
  mla: Browning, Timothy D., and Roger Heath Brown. “Density of Rational Points on
    a Quadric Bundle in ℙ3×ℙ3.” <i>Duke Mathematical Journal</i>, vol. 169, no. 16,
    Duke University Press, 2020, pp. 3099–165, doi:<a href="https://doi.org/10.1215/00127094-2020-0031">10.1215/00127094-2020-0031</a>.
  short: T.D. Browning, R. Heath Brown, Duke Mathematical Journal 169 (2020) 3099–3165.
date_created: 2018-12-11T11:45:02Z
date_published: 2020-09-10T00:00:00Z
date_updated: 2023-10-17T12:51:10Z
day: '10'
department:
- _id: TiBr
doi: 10.1215/00127094-2020-0031
external_id:
  arxiv:
  - '1805.10715'
  isi:
  - '000582676300002'
intvolume: '       169'
isi: 1
issue: '16'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1805.10715
month: '09'
oa: 1
oa_version: Preprint
page: 3099-3165
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
quality_controlled: '1'
status: public
title: Density of rational points on a quadric bundle in ℙ3×ℙ3
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 169
year: '2020'
...
---
_id: '8423'
abstract:
- lang: eng
  text: In this paper we show that for a generic strictly convex domain, one can recover
    the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard
    map from the (maximal) marked length spectrum of the domain.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Guan
  full_name: Huang, Guan
  last_name: Huang
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Alfonso
  full_name: Sorrentino, Alfonso
  last_name: Sorrentino
citation:
  ama: Huang G, Kaloshin V, Sorrentino A. On the marked length spectrum of generic
    strictly convex billiard tables. <i>Duke Mathematical Journal</i>. 2017;167(1):175-209.
    doi:<a href="https://doi.org/10.1215/00127094-2017-0038">10.1215/00127094-2017-0038</a>
  apa: Huang, G., Kaloshin, V., &#38; Sorrentino, A. (2017). On the marked length
    spectrum of generic strictly convex billiard tables. <i>Duke Mathematical Journal</i>.
    Duke University Press. <a href="https://doi.org/10.1215/00127094-2017-0038">https://doi.org/10.1215/00127094-2017-0038</a>
  chicago: Huang, Guan, Vadim Kaloshin, and Alfonso Sorrentino. “On the Marked Length
    Spectrum of Generic Strictly Convex Billiard Tables.” <i>Duke Mathematical Journal</i>.
    Duke University Press, 2017. <a href="https://doi.org/10.1215/00127094-2017-0038">https://doi.org/10.1215/00127094-2017-0038</a>.
  ieee: G. Huang, V. Kaloshin, and A. Sorrentino, “On the marked length spectrum of
    generic strictly convex billiard tables,” <i>Duke Mathematical Journal</i>, vol.
    167, no. 1. Duke University Press, pp. 175–209, 2017.
  ista: Huang G, Kaloshin V, Sorrentino A. 2017. On the marked length spectrum of
    generic strictly convex billiard tables. Duke Mathematical Journal. 167(1), 175–209.
  mla: Huang, Guan, et al. “On the Marked Length Spectrum of Generic Strictly Convex
    Billiard Tables.” <i>Duke Mathematical Journal</i>, vol. 167, no. 1, Duke University
    Press, 2017, pp. 175–209, doi:<a href="https://doi.org/10.1215/00127094-2017-0038">10.1215/00127094-2017-0038</a>.
  short: G. Huang, V. Kaloshin, A. Sorrentino, Duke Mathematical Journal 167 (2017)
    175–209.
date_created: 2020-09-17T10:42:42Z
date_published: 2017-12-08T00:00:00Z
date_updated: 2021-01-12T08:19:11Z
day: '08'
doi: 10.1215/00127094-2017-0038
extern: '1'
external_id:
  arxiv:
  - '1603.08838'
intvolume: '       167'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1603.08838
month: '12'
oa: 1
oa_version: Preprint
page: 175-209
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
quality_controlled: '1'
status: public
title: On the marked length spectrum of generic strictly convex billiard tables
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 167
year: '2017'
...
---
_id: '8505'
abstract:
- lang: eng
  text: The classical principle of least action says that orbits of mechanical systems
    extremize action; an important subclass are those orbits that minimize action.
    In this paper we utilize this principle along with Aubry-Mather theory to construct
    (Birkhoff) regions of instability for a certain three-body problem, given by a
    Hamiltonian system of 2 degrees of freedom. We believe that these methods can
    be applied to construct instability regions for a variety of Hamiltonian systems
    with 2 degrees of freedom. The Hamiltonian model we consider describes dynamics
    of a Sun-Jupiter-comet system, and under some simplifying assumptions, we show
    the existence of instabilities for the orbit of the comet. In particular, we show
    that a comet which starts close to an orbit in the shape of an ellipse of eccentricity
    e=0.66 can increase in eccentricity up to e=0.96. In the sequels to this paper,
    we extend the result to beyond e=1 and show the existence of ejection orbits.
    Such orbits are initially well within the range of our solar system. This might
    give an indication of why most objects rotating around the Sun in our solar system
    have relatively low eccentricity.
article_processing_charge: No
article_type: original
author:
- first_name: Joseph
  full_name: Galante, Joseph
  last_name: Galante
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Galante J, Kaloshin V. Destruction of invariant curves in the restricted circular
    planar three-body problem by using comparison of action. <i>Duke Mathematical
    Journal</i>. 2011;159(2):275-327. doi:<a href="https://doi.org/10.1215/00127094-1415878">10.1215/00127094-1415878</a>
  apa: Galante, J., &#38; Kaloshin, V. (2011). Destruction of invariant curves in
    the restricted circular planar three-body problem by using comparison of action.
    <i>Duke Mathematical Journal</i>. Duke University Press. <a href="https://doi.org/10.1215/00127094-1415878">https://doi.org/10.1215/00127094-1415878</a>
  chicago: Galante, Joseph, and Vadim Kaloshin. “Destruction of Invariant Curves in
    the Restricted Circular Planar Three-Body Problem by Using Comparison of Action.”
    <i>Duke Mathematical Journal</i>. Duke University Press, 2011. <a href="https://doi.org/10.1215/00127094-1415878">https://doi.org/10.1215/00127094-1415878</a>.
  ieee: J. Galante and V. Kaloshin, “Destruction of invariant curves in the restricted
    circular planar three-body problem by using comparison of action,” <i>Duke Mathematical
    Journal</i>, vol. 159, no. 2. Duke University Press, pp. 275–327, 2011.
  ista: Galante J, Kaloshin V. 2011. Destruction of invariant curves in the restricted
    circular planar three-body problem by using comparison of action. Duke Mathematical
    Journal. 159(2), 275–327.
  mla: Galante, Joseph, and Vadim Kaloshin. “Destruction of Invariant Curves in the
    Restricted Circular Planar Three-Body Problem by Using Comparison of Action.”
    <i>Duke Mathematical Journal</i>, vol. 159, no. 2, Duke University Press, 2011,
    pp. 275–327, doi:<a href="https://doi.org/10.1215/00127094-1415878">10.1215/00127094-1415878</a>.
  short: J. Galante, V. Kaloshin, Duke Mathematical Journal 159 (2011) 275–327.
date_created: 2020-09-18T10:47:41Z
date_published: 2011-08-04T00:00:00Z
date_updated: 2021-01-12T08:19:45Z
day: '04'
doi: 10.1215/00127094-1415878
extern: '1'
intvolume: '       159'
issue: '2'
keyword:
- General Mathematics
language:
- iso: eng
month: '08'
oa_version: None
page: 275-327
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
quality_controlled: '1'
status: public
title: Destruction of invariant curves in the restricted circular planar three-body
  problem by using comparison of action
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 159
year: '2011'
...
---
_id: '2730'
abstract:
- lang: eng
  text: We give the leading order semiclassical asymptotics for the sum of the negative
    eigenvalues of the Pauli operator (in dimension two and three) with a strong non-homogeneous
    magnetic field. This result can be used to prove that the magnetic Thomas-Fermi
    theory gives the leading order ground state energy of large atoms. We develop
    a new localization scheme well suited to the anisotropic character of the strong
    magnetic field. We also use the basic Lieb-Thirring estimate obtained earlier
    (1996). (orig.) 19 refs.
acknowledgement: The first author gratefully acknowledges financial support from the
  Eidgen6ssiche Technische Hochschule, Forschungsinstitut für Mathematik, Zürich,
  where this work was started. He is also grateful for the hospitality and support
  of Aarhus University during his visits there.
article_processing_charge: No
article_type: original
author:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Jan
  full_name: Solovej, Jan
  last_name: Solovej
citation:
  ama: 'Erdös L, Solovej J. Semiclassical eigenvalue estimates for the Pauli operator
    with strong nonhomogeneous magnetic fields, I: Nonasymptotic Lieb-Thirring-type
    estimate. <i>Duke Mathematical Journal</i>. 1999;96(1):127-173. doi:<a href="https://doi.org/10.1215/S0012-7094-99-09604-7">10.1215/S0012-7094-99-09604-7</a>'
  apa: 'Erdös, L., &#38; Solovej, J. (1999). Semiclassical eigenvalue estimates for
    the Pauli operator with strong nonhomogeneous magnetic fields, I: Nonasymptotic
    Lieb-Thirring-type estimate. <i>Duke Mathematical Journal</i>. Duke University
    Press. <a href="https://doi.org/10.1215/S0012-7094-99-09604-7">https://doi.org/10.1215/S0012-7094-99-09604-7</a>'
  chicago: 'Erdös, László, and Jan Solovej. “Semiclassical Eigenvalue Estimates for
    the Pauli Operator with Strong Nonhomogeneous Magnetic Fields, I: Nonasymptotic
    Lieb-Thirring-Type Estimate.” <i>Duke Mathematical Journal</i>. Duke University
    Press, 1999. <a href="https://doi.org/10.1215/S0012-7094-99-09604-7">https://doi.org/10.1215/S0012-7094-99-09604-7</a>.'
  ieee: 'L. Erdös and J. Solovej, “Semiclassical eigenvalue estimates for the Pauli
    operator with strong nonhomogeneous magnetic fields, I: Nonasymptotic Lieb-Thirring-type
    estimate,” <i>Duke Mathematical Journal</i>, vol. 96, no. 1. Duke University Press,
    pp. 127–173, 1999.'
  ista: 'Erdös L, Solovej J. 1999. Semiclassical eigenvalue estimates for the Pauli
    operator with strong nonhomogeneous magnetic fields, I: Nonasymptotic Lieb-Thirring-type
    estimate. Duke Mathematical Journal. 96(1), 127–173.'
  mla: 'Erdös, László, and Jan Solovej. “Semiclassical Eigenvalue Estimates for the
    Pauli Operator with Strong Nonhomogeneous Magnetic Fields, I: Nonasymptotic Lieb-Thirring-Type
    Estimate.” <i>Duke Mathematical Journal</i>, vol. 96, no. 1, Duke University Press,
    1999, pp. 127–73, doi:<a href="https://doi.org/10.1215/S0012-7094-99-09604-7">10.1215/S0012-7094-99-09604-7</a>.'
  short: L. Erdös, J. Solovej, Duke Mathematical Journal 96 (1999) 127–173.
date_created: 2018-12-11T11:59:18Z
date_published: 1999-01-15T00:00:00Z
date_updated: 2023-02-20T07:34:48Z
day: '15'
doi: 10.1215/S0012-7094-99-09604-7
extern: '1'
intvolume: '        96'
issue: '1'
language:
- iso: eng
month: '01'
oa_version: None
page: 127 - 173
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
publist_id: '4162'
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous
  magnetic fields, I: Nonasymptotic Lieb-Thirring-type estimate'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 96
year: '1999'
...
---
_id: '2713'
acknowledgement: Work supported by the NSF grant PHY90-19433 A02 and by the Alfred
  Sloan Foundation dissertation fellowship.
article_processing_charge: No
article_type: original
author:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Erdös L. Estimates on stochastic oscillatory integrals and on the heat kernel
    of the magnetic Schrödinger operator. <i>Duke Mathematical Journal</i>. 1994;76(2):541-566.
    doi:<a href="https://doi.org/10.1215/S0012-7094-94-07619-9">10.1215/S0012-7094-94-07619-9</a>
  apa: Erdös, L. (1994). Estimates on stochastic oscillatory integrals and on the
    heat kernel of the magnetic Schrödinger operator. <i>Duke Mathematical Journal</i>.
    Duke University Press. <a href="https://doi.org/10.1215/S0012-7094-94-07619-9">https://doi.org/10.1215/S0012-7094-94-07619-9</a>
  chicago: Erdös, László. “Estimates on Stochastic Oscillatory Integrals and on the
    Heat Kernel of the Magnetic Schrödinger Operator.” <i>Duke Mathematical Journal</i>.
    Duke University Press, 1994. <a href="https://doi.org/10.1215/S0012-7094-94-07619-9">https://doi.org/10.1215/S0012-7094-94-07619-9</a>.
  ieee: L. Erdös, “Estimates on stochastic oscillatory integrals and on the heat kernel
    of the magnetic Schrödinger operator,” <i>Duke Mathematical Journal</i>, vol.
    76, no. 2. Duke University Press, pp. 541–566, 1994.
  ista: Erdös L. 1994. Estimates on stochastic oscillatory integrals and on the heat
    kernel of the magnetic Schrödinger operator. Duke Mathematical Journal. 76(2),
    541–566.
  mla: Erdös, László. “Estimates on Stochastic Oscillatory Integrals and on the Heat
    Kernel of the Magnetic Schrödinger Operator.” <i>Duke Mathematical Journal</i>,
    vol. 76, no. 2, Duke University Press, 1994, pp. 541–66, doi:<a href="https://doi.org/10.1215/S0012-7094-94-07619-9">10.1215/S0012-7094-94-07619-9</a>.
  short: L. Erdös, Duke Mathematical Journal 76 (1994) 541–566.
date_created: 2018-12-11T11:59:13Z
date_published: 1994-11-01T00:00:00Z
date_updated: 2022-06-03T11:59:06Z
day: '01'
doi: 10.1215/S0012-7094-94-07619-9
extern: '1'
intvolume: '        76'
issue: '2'
language:
- iso: eng
main_file_link:
- url: https://projecteuclid.org/journals/duke-mathematical-journal/volume-76/issue-2/Estimates-on-stochastic-oscillatory-integrals-and-on-the-heat-kernel/10.1215/S0012-7094-94-07619-9.short
month: '11'
oa_version: None
page: 541 - 566
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
publist_id: '4183'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Estimates on stochastic oscillatory integrals and on the heat kernel of the
  magnetic Schrödinger operator
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 76
year: '1994'
...