Sökning: WFRF:(Thim Oliver 1997)
> (2021) >
Crystallographic Gr...
Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions
-
- Rowlett, Julie, 1978 (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
-
- Blom, Max (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
-
- Nordell, Henrik (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
-
visa fler...
-
- Thim, Oliver, 1997 (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
-
- Vahnberg, Jack (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
-
visa färre...
-
(creator_code:org_t)
- 2021-04-27
- 2021
- Engelska.
-
Ingår i: American Mathematical Monthly. - : Informa UK Limited. - 1930-0972 .- 0002-9890. ; 128:5, s. 387-406
- Relaterad länk:
-
https://research.cha... (primary) (free)
-
visa fler...
-
https://www.tandfonl...
-
https://research.cha...
-
https://doi.org/10.1...
-
https://gup.ub.gu.se...
-
visa färre...
Abstract
Ämnesord
Stäng
- The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Berard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Berard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. To conclude, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.
Ämnesord
- NATURVETENSKAP -- Matematik -- Algebra och logik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Algebra and Logic (hsv//eng)
- NATURVETENSKAP -- Matematik -- Geometri (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Geometry (hsv//eng)
- NATURVETENSKAP -- Matematik -- Matematisk analys (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Mathematical Analysis (hsv//eng)
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Nyckelord
- Primary 20H15
- 47A75
- Secondary 20F55
- 51M20
- 47A75
- 51M20
- Primary 20H15
- Secondary 20F55
Publikations- och innehållstyp
- art (ämneskategori)
- ref (ämneskategori)
Hitta via bibliotek
Till lärosätets databas