If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of ...

We prove that every graph of girth at least five which admits an embedding in the Klein bottle is 3-colorable. This solves a problem raised by Woodburn, and complements a result of Thomassen who proved the same for projective planar and toroidal graphs. 15 July 1998, Revised 26 April 2004. To appear in J. Combin. Theory Ser. B. 1 Partially supported by NSF under Grant No. DMS-9623031 and by NSA...

This paper studies deterministic movement over toroidal grids, integrating local information, bounded memory and choice at individual nodes. The research is motivated by recent work on deterministic random walks, and applications in multi-agent systems. Several results regarding passing tokens through toroidal grids are discussed, as well as some open questions.

4-branes that wrap the long and short fundamental 1-cycles of a rectangular toroidal orbifold T 2 /(Z2) 3 , which has Planck-scale compactification radii, intersect after 43 and 109 wrappings, respectively, in a 3-brane upon which the scale factor, equal to a power of the inverse of compactification length, has precisely the same value when measured parallel to either of the fundamental 1-cycle...

We discuss the resolution of toroidal orbifolds. For the resulting smooth Calabi–Yau manifolds, we calculate the intersection ring and determine the divisor topologies. In a next step, the orientifold quotients are constructed.

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We study toroidal compactification of Matrix theory, using ideas and results of noncommutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT form...

für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution 4.0 International License. Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namen...

We consider compactifications of theN = 1, d = 6, E8 theory on tori to five, four, and three dimensions and learn about some properties of this theory. As a by-product we derive the SL(2,Z) duality of the N = 2, d = 4, SU(2) theory with Nf = 4. Using this theory on a D-brane probe we shed new light on the singularities of F-theory compactifications to eight dimensions. As another application we...

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