231 research outputs found
Polyhedral Gauss Sums, and polytopes with symmetry
We define certain natural finite sums of 'th roots of unity, called
, that are associated to each convex integer polytope , and which
generalize the classical -dimensional Gauss sum defined over , to higher dimensional abelian groups and integer polytopes.
We consider the finite Weyl group , generated by the reflections
with respect to the coordinate hyperplanes, as well as all permutations of the
coordinates; further, we let be the group generated by
as well as all integer translations in . We prove
that if multi-tiles under the action of , then we
have the closed form . Conversely, we also prove
that if is a lattice tetrahedron in , of volume , such
that , for , then there is
an element in such that is the fundamental tetrahedron
with vertices , , , .Comment: 18 pages, 2 figure
Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective
The theory of intrinsic volumes of convex cones has recently found striking
applications in areas such as convex optimization and compressive sensing. This
article provides a self-contained account of the combinatorial theory of
intrinsic volumes for polyhedral cones. Direct derivations of the General
Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the
Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In
addition, a connection between the characteristic polynomial of a hyperplane
arrangement and the intrinsic volumes of the regions of the arrangement, due to
Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page
New Strings for Old Veneziano Amplitudes III. Symplectic Treatment
A d-dimensional rational polytope P is a polytope whose vertices are located
at the nodes of d-dimensional Z-lattice. Consider a number of points inside the
inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart
polynomial of P counts the number of such lattice points (nodes) inside the
inflated P and (may be) at its faces (including vertices). In Part I
(hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude
is just the Laplace transform of the generating function (considered as a
partition function in the sence of statistical mechanics) for the Ehrhart
polynomial for the regular inflated simplex obtained as a deformation retract
of the Fermat (hyper) surface living in complex projective space. This
observation is sufficient for development of new symplectic (this work) and
supersymmetric (hep-th/0411241)physical models reproducing the Veneziano (and
Veneziano-like) amplitudes. General ideas (e.g.those related to the properties
of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use
of mirror symmetry for explanation of available experimental data on pion-pion
scattering) worked out in some detail. Obtained final results are in formal
accord with those earlier obtained by Vergne [PNAS 93 (1996) 14238].Comment: 48 pages J.Geom.Phys.(in press, available on line
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Discrete Differential Geometry
This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry
On stringy cohomology spaces
We modify the definition of the families of and stringy cohomology
spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces
have the same dimension as the ones defined in the joint paper with Mavlyutov
\cite{BM}, but they admit natural flat connections with respect to the
appropriate parameters. This solves a longstanding question of relating GKZ
hypergeometric system to stringy cohomology. We construct products on these
spaces by vertex algebra techniques. In the process, we fix a minor gap in
\cite{BM} and prove a statement on intersection cohomology of dual cones that
may be of independent interest.Comment: 21 pages, LaTe
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