The hot spots conjecture - Revision history

Teorth at 00:22, 22 March 2018

2018-03-22T00:22:04Z

Teorth at 02:11, 9 February 2018

2018-02-09T02:11:56Z

← Older revision		Revision as of 19:11, 8 February 2018
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	(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)		(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)

			An affirmative solution to the conjecture was claimed in [MJ2018].

	== Threads ==		== Threads ==

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2018-02-09T02:11:19Z

Bibliography

← Older revision		Revision as of 19:11, 8 February 2018
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	* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009) [http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf Article]		* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009) [http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf Article]
	* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164		* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164
			* [MJ2018] S. Mondal, C. Judge, [https://arxiv.org/abs/1802.01800 Euclidean Triangles Have No Hot Spots], preprint.
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]
	* [S2013] B. Siudeja, [http://pages.uoregon.edu/siudeja/hotspots.pdf On the hot spots conjecture for triangles], preprint. [http://arxiv.org/abs/1308.3005 arXiv]		* [S2013] B. Siudeja, [http://pages.uoregon.edu/siudeja/hotspots.pdf On the hot spots conjecture for triangles], preprint. [http://arxiv.org/abs/1308.3005 arXiv]

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2013-08-15T07:43:14Z

Bibliography

← Older revision		Revision as of 00:43, 15 August 2013
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	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]
	* [S2013] B. Siudeja, [http://pages.uoregon.edu/siudeja/hotspots.pdf On the hot spots conjecture for triangles], preprint.		* [S2013] B. Siudeja, [http://pages.uoregon.edu/siudeja/hotspots.pdf On the hot spots conjecture for triangles], preprint. [http://arxiv.org/abs/1308.3005 arXiv]

Teorth: /* Threads */

2013-08-09T19:52:58Z

Threads

← Older revision		Revision as of 12:52, 9 August 2013
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	* [http://www.math.missouri.edu/~evanslc/Polymath/ Initial web page for proposal]		* [http://www.math.missouri.edu/~evanslc/Polymath/ Initial web page for proposal]
	* [http://www.math.missouri.edu/~evanslc/Polymath/Polymath.pdf A more mathematical description of the problem]		* [http://www.math.missouri.edu/~evanslc/Polymath/Polymath.pdf A more mathematical description of the problem]
	* [http://polymathprojects.org/2012/06/03/polymath-proposal-the-hot-spots-conjecture-for-acute-triangles First research thread] (June 3, 2012) '''Inactive'''		* [http://polymathprojects.org/2012/06/03/polymath-proposal-the-hot-spots-conjecture-for-acute-triangles First research thread] (June 3, 2012) ''Inactive''
	* [http://polymathprojects.org/2012/06/09/polymath7-discussion-thread/ Current discussion thread] (June 9, 2012) '''Active'''		* [http://polymathprojects.org/2012/06/09/polymath7-discussion-thread/ Current discussion thread] (June 9, 2012) '''Active'''
	* [http://polymathprojects.org/2012/06/12/polymath7-research-thread-1-the-hot-spots-conjecture/ Research thread 1] (June 12, 2012) '''Inactive'''		* [http://polymathprojects.org/2012/06/12/polymath7-research-thread-1-the-hot-spots-conjecture/ Research thread 1] (June 12, 2012) ''Inactive''
	* [http://polymathprojects.org/2012/06/15/polymath7-research-threads-2-the-hot-spots-conjecture/ Research thread 2] (June 15, 2012) '''Inactive'''		* [http://polymathprojects.org/2012/06/15/polymath7-research-threads-2-the-hot-spots-conjecture/ Research thread 2] (June 15, 2012) ''Inactive''
	* [http://polymathprojects.org/2012/06/24/polymath7-research-threads-3-the-hot-spots-conjecture/ Research thread 3] (June 24, 2012) '''Inactive'''		* [http://polymathprojects.org/2012/06/24/polymath7-research-threads-3-the-hot-spots-conjecture/ Research thread 3] (June 24, 2012) ''Inactive''
	* [http://polymathprojects.org/2012/09/10/polymath7-research-threads-4-the-hot-spots-conjecture Research thread 4] (September 10, 2012) ''~~'Active'~~''		* [http://polymathprojects.org/2012/09/10/polymath7-research-threads-4-the-hot-spots-conjecture Research thread 4] (September 10, 2012) ''Inactive''
			* [http://polymathprojects.org/2013/08/09/polymath7-research-thread-5-the-hot-spots-conjecture/ Research thread 5] (August 9, 2013)

	== Basic concepts ==		== Basic concepts ==

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2013-08-09T18:36:25Z

Bibliography

← Older revision		Revision as of 11:36, 9 August 2013
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	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]
			* [S2013] B. Siudeja, [http://pages.uoregon.edu/siudeja/hotspots.pdf On the hot spots conjecture for triangles], preprint.

Teorth: /* Bibliography */

2013-08-04T14:28:52Z

Bibliography

← Older revision		Revision as of 07:28, 4 August 2013
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	* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.		* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.
	* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.		* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.
	* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009)		* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009) [http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf Article]
	* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164		* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]

Teorth: /* Bibliography */

2013-08-03T01:20:00Z

Bibliography

← Older revision		Revision as of 18:20, 2 August 2013
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	* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.		* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.
	* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.		* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.
			* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009)
	* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164		* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]

Teorth: /* Bibliography */

2013-08-03T01:18:24Z

Bibliography

← Older revision		Revision as of 18:18, 2 August 2013
Line 507:		Line 507:
	* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.		* [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011.
	* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.		* [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968.
			* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
	* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]		* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]

Teorth: /* Bibliography */

2013-07-19T16:02:21Z

Bibliography

← Older revision		Revision as of 09:02, 19 July 2013
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	* [HR2002] E. Hernandez, R. Rodriguez, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01467-9/S0025-5718-02-01467-9.pdf Finite element approximation of spectral problems with Neumann boundary conditions on curved domains], MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1099--1115.		* [HR2002] E. Hernandez, R. Rodriguez, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01467-9/S0025-5718-02-01467-9.pdf Finite element approximation of spectral problems with Neumann boundary conditions on curved domains], MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1099--1115.
	* [JN2000] D. Jerison, N. Nadirashvili, [http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00346-5/S0894-0347-00-00346-5.pdf The "Hot Spots" Conjecture for Domains with Two Axes of Symmetry], Journal of the American Mathematical Society Vol. 13, No. 4 (Oct., 2000), pp. 741-772		* [JN2000] D. Jerison, N. Nadirashvili, [http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00346-5/S0894-0347-00-00346-5.pdf The "Hot Spots" Conjecture for Domains with Two Axes of Symmetry], Journal of the American Mathematical Society Vol. 13, No. 4 (Oct., 2000), pp. 741-772
			* [KKLS] T. Kulczycki, M. Kwasnicki, J. Matecki, A. Stos, Spectral properties of the Cauchy process on half-line and interval, Proc. London Math. Soc. (2010) 101 (2): 589-622. [http://plms.oxfordjournals.org/content/101/2/589 Article]
	* [LS2009] R. Laugesen, B. Siudeja, [http://arxiv.org/abs/0907.1552 Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality], Journal of Differential Equations, Volume 249, Issue 1, 1 July 2010, Pages 118–135.		* [LS2009] R. Laugesen, B. Siudeja, [http://arxiv.org/abs/0907.1552 Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality], Journal of Differential Equations, Volume 249, Issue 1, 1 July 2010, Pages 118–135.
	* [LS2009b] R. Laugesen, B. Siudeja, [http://dx.doi.org/10.1016/j.jde.2010.02.020 Maximizing Neumann fundamental tones of triangles], 22 pages. Journal of Mathematical Physics, 50:112903, 2009.		* [LS2009b] R. Laugesen, B. Siudeja, [http://dx.doi.org/10.1016/j.jde.2010.02.020 Maximizing Neumann fundamental tones of triangles], 22 pages. Journal of Mathematical Physics, 50:112903, 2009.
Line 508:		Line 509:
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
	* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.		* [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
			* [SWL2009] J. Shen, L. Wang, H. Li, A triangular spectral element method using fully tensorial rational basis functions, Siam J. Numer. Anal. 47 (2009), 1619--1650. [http://www.math.purdue.edu/~shen/pub/SWL_SINUM09.pdf Article]

← Older revision		Revision as of 17:22, 21 March 2018
Line 18:		Line 18:
	(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)		(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)

	An affirmative solution to the conjecture was claimed in [~~MJ2018~~].		An affirmative solution to the conjecture was claimed in [JM2018].

	== Threads ==		== Threads ==
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	* [HR2002] E. Hernandez, R. Rodriguez, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01467-9/S0025-5718-02-01467-9.pdf Finite element approximation of spectral problems with Neumann boundary conditions on curved domains], MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1099--1115.		* [HR2002] E. Hernandez, R. Rodriguez, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01467-9/S0025-5718-02-01467-9.pdf Finite element approximation of spectral problems with Neumann boundary conditions on curved domains], MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1099--1115.
	* [JN2000] D. Jerison, N. Nadirashvili, [http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00346-5/S0894-0347-00-00346-5.pdf The "Hot Spots" Conjecture for Domains with Two Axes of Symmetry], Journal of the American Mathematical Society Vol. 13, No. 4 (Oct., 2000), pp. 741-772		* [JN2000] D. Jerison, N. Nadirashvili, [http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00346-5/S0894-0347-00-00346-5.pdf The "Hot Spots" Conjecture for Domains with Two Axes of Symmetry], Journal of the American Mathematical Society Vol. 13, No. 4 (Oct., 2000), pp. 741-772
			* [JM2018] C. Judge, S. Mondal, [https://arxiv.org/abs/1802.01800 Euclidean Triangles Have No Hot Spots], preprint.
	* [KKLS] T. Kulczycki, M. Kwasnicki, J. Matecki, A. Stos, Spectral properties of the Cauchy process on half-line and interval, Proc. London Math. Soc. (2010) 101 (2): 589-622. [http://plms.oxfordjournals.org/content/101/2/589 Article]		* [KKLS] T. Kulczycki, M. Kwasnicki, J. Matecki, A. Stos, Spectral properties of the Cauchy process on half-line and interval, Proc. London Math. Soc. (2010) 101 (2): 589-622. [http://plms.oxfordjournals.org/content/101/2/589 Article]
	* [LS2009] R. Laugesen, B. Siudeja, [http://arxiv.org/abs/0907.1552 Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality], Journal of Differential Equations, Volume 249, Issue 1, 1 July 2010, Pages 118–135.		* [LS2009] R. Laugesen, B. Siudeja, [http://arxiv.org/abs/0907.1552 Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality], Journal of Differential Equations, Volume 249, Issue 1, 1 July 2010, Pages 118–135.
Line 512:		Line 513:
	* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009) [http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf Article]		* [M2009] Y. Miyamoto, [http://jmp.aip.org/resource/1/jmapaq/v50/i10/p103530_s1?isAuthorized=no The “hot spots” conjecture for a certain class of planar convex domains], J. Math. Phys. 50, 103530 (2009) [http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf Article]
	* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164		* [M2013] Y. Miyamoto, [http://link.springer.com/article/10.1007/s13160-012-0091-z A planar convex domain with many isolated “ hot spots” on the boundary], Japan Journal of Industrial and Applied Mathematics February 2013, Volume 30, Issue 1, pp 145--164
	* [MJ2018] S. Mondal, C. Judge, [https://arxiv.org/abs/1802.01800 Euclidean Triangles Have No Hot Spots], preprint.
	* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).		* [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986).
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