-
Previous Article
Spectral approximation of the curl operator in multiply connected domains
- DCDS-S Home
- This Issue
-
Next Article
Stabilized Galerkin for transient advection of differential forms
On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists
1. | Boston University, ECE Dept, 8 Saint Mary's Street, Boston, MA 02215, United States |
References:
[1] |
F. J. Almgren and W. P. Thurston, Examples of unknotted curves which bound only surfaces of high genus within their convex hulls, Annals of Math. $2^{nd}$ Ser., 105 (1977), 527-538.
doi: 10.2307/1970922. |
[2] |
M. F. Atiyah, Mathematics in the $20^{th}$ century, Bull. LMS, 34 (2002), 1-15.
doi: 10.1112/S0024609301008566. |
[3] |
G. Burde and H. Zeischang, Knots, $2^{nd}$ ed., De Gruyter Studies in Math No.5, Walter de Gruyter, 2003. |
[4] |
J. C. Crager and P. R. Kotiuga, Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields, IEEE Trans. Mag., 38 (2002), 1309-1312. |
[5] |
D. A. Ellwood, P. S. Ozsvath, A. I. Stipicz and Z. Szabo. Eds., Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc. Vol.5, AMS, AMS Providence, RI, 2006. |
[6] |
J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 103-141. |
[7] |
G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere, Math. Intelligencer, 2 (1980), 200-203.
doi: 10.1007/BF03028603. |
[8] |
P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Annalen, 102 (1930), 785-789.
doi: 10.1007/BF01782377. |
[9] |
E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs, Proc. Int'l Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405-414. |
[10] |
H. Goda, Circle valued Morse theory for knots and links, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 71-99. |
[11] |
P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph No. 48, Camb. U. Press, 2004.
doi: 10.1017/CBO9780511756337. |
[12] |
R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Annali di Matematica Pura ed Applicata, 191 (2012), 431-457.
doi: 10.1007/s10231-011-0189-y. |
[13] |
P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions, Jour. Appl. Phys., 61 (1987), 3916-3918.
doi: 10.1063/1.338583. |
[14] |
P. R. Kotiuga, An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method, IEEE Trans. Mag., 25 (1989), 4129-4131.
doi: 10.1109/INTMAG.1989.690333. |
[15] |
P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions, IEEE Trans. Mag., 25 (1989), 2813-2815.
doi: 10.1109/20.34293. |
[16] |
P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields, IEEE Trans. Mag., 40 (2004), 1108-1111.
doi: 10.1109/TMAG.2004.824590. |
[17] |
J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves, Journ. Diff. Geom., 68 (2004), 1-29. |
[18] |
S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90 (1958), 281-290. |
[19] |
W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347.
doi: 10.1090/S0002-9939-1975-0375366-7. |
[20] |
H. Whitney, Moscow 1935: Topology moving toward America, reprinted in Hassler Whitney Collected Papers, Vol I (eds. J. Eells and D. Toledo), Birkhauser, Boston, 1992, 1-21. |
show all references
References:
[1] |
F. J. Almgren and W. P. Thurston, Examples of unknotted curves which bound only surfaces of high genus within their convex hulls, Annals of Math. $2^{nd}$ Ser., 105 (1977), 527-538.
doi: 10.2307/1970922. |
[2] |
M. F. Atiyah, Mathematics in the $20^{th}$ century, Bull. LMS, 34 (2002), 1-15.
doi: 10.1112/S0024609301008566. |
[3] |
G. Burde and H. Zeischang, Knots, $2^{nd}$ ed., De Gruyter Studies in Math No.5, Walter de Gruyter, 2003. |
[4] |
J. C. Crager and P. R. Kotiuga, Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields, IEEE Trans. Mag., 38 (2002), 1309-1312. |
[5] |
D. A. Ellwood, P. S. Ozsvath, A. I. Stipicz and Z. Szabo. Eds., Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc. Vol.5, AMS, AMS Providence, RI, 2006. |
[6] |
J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 103-141. |
[7] |
G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere, Math. Intelligencer, 2 (1980), 200-203.
doi: 10.1007/BF03028603. |
[8] |
P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Annalen, 102 (1930), 785-789.
doi: 10.1007/BF01782377. |
[9] |
E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs, Proc. Int'l Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405-414. |
[10] |
H. Goda, Circle valued Morse theory for knots and links, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 71-99. |
[11] |
P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph No. 48, Camb. U. Press, 2004.
doi: 10.1017/CBO9780511756337. |
[12] |
R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Annali di Matematica Pura ed Applicata, 191 (2012), 431-457.
doi: 10.1007/s10231-011-0189-y. |
[13] |
P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions, Jour. Appl. Phys., 61 (1987), 3916-3918.
doi: 10.1063/1.338583. |
[14] |
P. R. Kotiuga, An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method, IEEE Trans. Mag., 25 (1989), 4129-4131.
doi: 10.1109/INTMAG.1989.690333. |
[15] |
P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions, IEEE Trans. Mag., 25 (1989), 2813-2815.
doi: 10.1109/20.34293. |
[16] |
P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields, IEEE Trans. Mag., 40 (2004), 1108-1111.
doi: 10.1109/TMAG.2004.824590. |
[17] |
J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves, Journ. Diff. Geom., 68 (2004), 1-29. |
[18] |
S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90 (1958), 281-290. |
[19] |
W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347.
doi: 10.1090/S0002-9939-1975-0375366-7. |
[20] |
H. Whitney, Moscow 1935: Topology moving toward America, reprinted in Hassler Whitney Collected Papers, Vol I (eds. J. Eells and D. Toledo), Birkhauser, Boston, 1992, 1-21. |
[1] |
Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983 |
[2] |
Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579 |
[3] |
Diego Rapoport. Random representations of viscous fluids and the passive magnetic fields transported on them. Conference Publications, 2001, 2001 (Special) : 327-336. doi: 10.3934/proc.2001.2001.327 |
[4] |
Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic and Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331 |
[5] |
Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339 |
[6] |
Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086 |
[7] |
Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems and Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 |
[8] |
Vincenzo Ambrosio. Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 781-815. doi: 10.3934/dcds.2020062 |
[9] |
Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems and Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033 |
[10] |
Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038 |
[11] |
Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 |
[12] |
Frank Jochmann. A singular limit in a nonlinear problem arising in electromagnetism. Communications on Pure and Applied Analysis, 2011, 10 (2) : 541-559. doi: 10.3934/cpaa.2011.10.541 |
[13] |
Denis Serre. Non-linear electromagnetism and special relativity. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435 |
[14] |
Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 605-629. doi: 10.3934/dcdss.2012.5.605 |
[15] |
Bruce Hughes. Geometric topology of stratified spaces. Electronic Research Announcements, 1996, 2: 73-81. |
[16] |
Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929 |
[17] |
Fengbo Hang, Fanghua Lin. Topology of Sobolev mappings IV. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1097-1124. doi: 10.3934/dcds.2005.13.1097 |
[18] |
Robert Cardona. The topology of Bott integrable fluids. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022054 |
[19] |
D. Alderson, H. Chang, M. Roughan, S. Uhlig, W. Willinger. The many facets of internet topology and traffic. Networks and Heterogeneous Media, 2006, 1 (4) : 569-600. doi: 10.3934/nhm.2006.1.569 |
[20] |
M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]