74,418 research outputs found
Ray casting implicit fractal surfaces with reduced affine arithmetic
A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store
An Abundance of Heterotic Vacua
We explicitly construct the largest dataset to date of heterotic vacua
arising from stable vector bundles on Calabi-Yau threefolds. Focusing on
elliptically fibered Calabi-Yau manifolds with spectral cover bundles, we show
that the number of heterotic models with non-zero number of generations is
finite. We classify these models according to the complex base of their
Calabi-Yau threefold and to the unification gauge group that they preserve in
four dimensions. This database of the order of models, which includes
potential Standard Model candidates, is subjected to some preliminary
statistical analyses. The additional constraint that there should be three net
generations of particles gives a dramatic reduction of the number of vacua.Comment: 27 pages, 12 figures, added reference
Zero entropy subgroups of mapping class groups
Let be a compact surface with boundary. We are interested in the question
of how a group action on permutes a finite invariant set . More precisely, how the algebraic properties of the induced group of
permutations of a finite invariant set affects the dynamical properties of the
group. Our main result shows that in many circumstances if the induced
permutation group is not solvable then among the homeomorphisms in the group
there must be one with a pseudo-Anosov component. We formulate this in terms of
the mapping class group relative to the finite set and show the stronger result
that in many circumstances (e.g. if ) this mapping
class group is itself solvable if it has no elements with pseudo-Anosov
components
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