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On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction
On a formula for sets of constant width in 2d
Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany |
A formula for smooth orbiforms originating from Euler can be adjusted to describe all sets of constant width in 2d. Moreover, the formula allows short proofs of some laborious approximation results for sets of constant width.
References:
[1] |
A. D. Alexandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Ann. Uchenye Zapiski Math., Ser. 6, (1939), 3–35., |
[2] |
E. Barbier, Note sur le problème de l'aiguille et le jeu du joint couvert, J. Math. Pures Appl., 2e série, tome 5 (1860), 273–286. |
[3] |
W. Blaschke,
Einige Bemerkungen über Kurven und Flächen von konstanter Breite, Ber. Verh. Sächs. Akad. Leipzig, 67 (1915), 290-297.
|
[4] |
W. Blaschke,
Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathem. Annalen, 176 (1915), 504-513.
doi: 10.1007/BF01458221. |
[5] |
J. Böhm, Convex bodies of constant width, in Mathematical Models from the Collections of Universities and Museums (ed. G. Fischer), Vieweg, Braunschweig, (1986), 49–56., |
[6] |
J. Böhm and E. Quaisser, Schönheit und Harmonie Geometrischer Formen–Sphäroformen und Symmetrische Körper, Akademie Verlag, Berlin, 1991., |
[7] |
G. D. Chakerian,
Sets of Constant Width, Pacific J. Math., 19 (1966), 13-21.
|
[8] |
G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in Convexity and its Applications (eds. P. M. Gruber and J. M. Wills), Birkhauser, (1983), 49–96., |
[9] |
L. Euler, De curvis triangularibus, Acta Academiae Scientarum Imperialis Petropolitinae, 1778 (1781), 3-30 (Opera Omnia: Series 1, Volume 28, pp. 298-321). |
[10] |
P. C. Hammer and A. Sobczyk,
Planar line families. Ⅰ, Proc. Amer. Math. Soc., 4 (1953), 226-233.
doi: 10.2307/2031796. |
[11] |
P. C. Hammer and A. Sobczyk,
Planar line families. Ⅱ, Proc. Amer. Math. Soc., 4 (1953), 341-349.
doi: 10.2307/2032127. |
[12] |
P. C. Hammer,
Constant breadth curves in the plane, Proc. Amer.Math. Soc., 6 (1955), 333-334.
doi: 10.2307/2032370. |
[13] |
D. Hilbert and St. Cohn-Vossen, Geometry and The Imagination, AMS Chelsea, Providence, R.I., 1952 (transl. from the German: Anschauliche Geometrie, Springer, Berlin, 1932)., |
[14] |
I. M. Jaglom and W. G. Boltjanski, Konvexe Figuren, VEB Deutscher Verlag der Wissenschaften, 1956., |
[15] |
B. Kawohl and Ch. Weber,
Meissner's mysterious bodies, The Mathematical Intelligencer, 33 (2011), 94-101.
doi: 10.1007/s00283-011-9239-y. |
[16] |
T. Lachand-Robert and É. Oudet,
Bodies of constant width in arbitrary dimension, Mathe-matische Nachrichten, 280 (2007), 740-750.
doi: 10.1002/mana.200510512. |
[17] |
F. Malagoli,
An optimal control theory approach to the Blaschke-Lebesgue theorem, J. Convex Anal., 16 (2009), 391-407.
|
[18] |
E. Meissner,
Über die Anwendung der Fourier-Reihen auf einige Aufgaben der geometrie und kinematik, Vierteljahrsschr. Nat.forsch. Ges. Zür, 54 (1909), 309-329.
|
[19] |
E. Meissner,
Über Punktmengen konstanter Breite, Vierteljahrsschr. Nat.forsch. Ges. Zür, 56 (1911), 42-50.
|
[20] |
H. Rademacher and O. Toeplitz, Von Zahlen und Figuren. Proben mathematischen Denkens für Liebhaber der Mathematik, 2. Aufl. Julius Springer, Berlin, 1933., |
[21] |
Sh. Tanno,
$C^{\infty }$-approximation of continuous ovals of constant width, J. Math. Soc. Japan, 28 (1976), 384-395.
doi: 10.2969/jmsj/02820384. |
[22] |
B. Wegner,
Analytic approximation of continuous ovals of constant width, J. Math. Soc. Japan, 29 (1977), 537-540.
doi: 10.2969/jmsj/02930537. |
[23] |
Mathematical Etudes Foundation, Russia. Available from http://www.etudes.ru/en/etudes/reuleaux-triangle/ |
show all references
References:
[1] |
A. D. Alexandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Ann. Uchenye Zapiski Math., Ser. 6, (1939), 3–35., |
[2] |
E. Barbier, Note sur le problème de l'aiguille et le jeu du joint couvert, J. Math. Pures Appl., 2e série, tome 5 (1860), 273–286. |
[3] |
W. Blaschke,
Einige Bemerkungen über Kurven und Flächen von konstanter Breite, Ber. Verh. Sächs. Akad. Leipzig, 67 (1915), 290-297.
|
[4] |
W. Blaschke,
Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathem. Annalen, 176 (1915), 504-513.
doi: 10.1007/BF01458221. |
[5] |
J. Böhm, Convex bodies of constant width, in Mathematical Models from the Collections of Universities and Museums (ed. G. Fischer), Vieweg, Braunschweig, (1986), 49–56., |
[6] |
J. Böhm and E. Quaisser, Schönheit und Harmonie Geometrischer Formen–Sphäroformen und Symmetrische Körper, Akademie Verlag, Berlin, 1991., |
[7] |
G. D. Chakerian,
Sets of Constant Width, Pacific J. Math., 19 (1966), 13-21.
|
[8] |
G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in Convexity and its Applications (eds. P. M. Gruber and J. M. Wills), Birkhauser, (1983), 49–96., |
[9] |
L. Euler, De curvis triangularibus, Acta Academiae Scientarum Imperialis Petropolitinae, 1778 (1781), 3-30 (Opera Omnia: Series 1, Volume 28, pp. 298-321). |
[10] |
P. C. Hammer and A. Sobczyk,
Planar line families. Ⅰ, Proc. Amer. Math. Soc., 4 (1953), 226-233.
doi: 10.2307/2031796. |
[11] |
P. C. Hammer and A. Sobczyk,
Planar line families. Ⅱ, Proc. Amer. Math. Soc., 4 (1953), 341-349.
doi: 10.2307/2032127. |
[12] |
P. C. Hammer,
Constant breadth curves in the plane, Proc. Amer.Math. Soc., 6 (1955), 333-334.
doi: 10.2307/2032370. |
[13] |
D. Hilbert and St. Cohn-Vossen, Geometry and The Imagination, AMS Chelsea, Providence, R.I., 1952 (transl. from the German: Anschauliche Geometrie, Springer, Berlin, 1932)., |
[14] |
I. M. Jaglom and W. G. Boltjanski, Konvexe Figuren, VEB Deutscher Verlag der Wissenschaften, 1956., |
[15] |
B. Kawohl and Ch. Weber,
Meissner's mysterious bodies, The Mathematical Intelligencer, 33 (2011), 94-101.
doi: 10.1007/s00283-011-9239-y. |
[16] |
T. Lachand-Robert and É. Oudet,
Bodies of constant width in arbitrary dimension, Mathe-matische Nachrichten, 280 (2007), 740-750.
doi: 10.1002/mana.200510512. |
[17] |
F. Malagoli,
An optimal control theory approach to the Blaschke-Lebesgue theorem, J. Convex Anal., 16 (2009), 391-407.
|
[18] |
E. Meissner,
Über die Anwendung der Fourier-Reihen auf einige Aufgaben der geometrie und kinematik, Vierteljahrsschr. Nat.forsch. Ges. Zür, 54 (1909), 309-329.
|
[19] |
E. Meissner,
Über Punktmengen konstanter Breite, Vierteljahrsschr. Nat.forsch. Ges. Zür, 56 (1911), 42-50.
|
[20] |
H. Rademacher and O. Toeplitz, Von Zahlen und Figuren. Proben mathematischen Denkens für Liebhaber der Mathematik, 2. Aufl. Julius Springer, Berlin, 1933., |
[21] |
Sh. Tanno,
$C^{\infty }$-approximation of continuous ovals of constant width, J. Math. Soc. Japan, 28 (1976), 384-395.
doi: 10.2969/jmsj/02820384. |
[22] |
B. Wegner,
Analytic approximation of continuous ovals of constant width, J. Math. Soc. Japan, 29 (1977), 537-540.
doi: 10.2969/jmsj/02930537. |
[23] |
Mathematical Etudes Foundation, Russia. Available from http://www.etudes.ru/en/etudes/reuleaux-triangle/ |




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