4,521 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)
The alternating sum of volumes (ASV) decomposition is a widely used
technique for converting a B-Rep into a CSG model. The obtained CSG
tree has convex primitives at its leaf nodes, while the contents of
its internal nodes alternate between the set union and difference
operators.
This work first shows that the obtained CSG tree T can also be
expressed as the regularized Exclusive-OR operation among all the
convex primitives at the leaf nodes of T, regardless the structure and
internal nodes of T. This is an important result in the case in which
EVM represented orthogonal polyhedra are used because in this model
the Exclusive-OR operation runs much faster than set union and
difference operations. Therefore this work applies this result to EVM
represented orthogonal polyhedra. It also presents experimental
results that corroborate the theoretical results and includes some
practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version
Scaling behaviour of three-dimensional group field theory
Group field theory is a generalization of matrix models, with triangulated
pseudomanifolds as Feynman diagrams and state sum invariants as Feynman
amplitudes. In this paper, we consider Boulatov's three-dimensional model and
its Freidel-Louapre positive regularization (hereafter the BFL model) with a
?ultraviolet' cutoff, and study rigorously their scaling behavior in the large
cutoff limit. We prove an optimal bound on large order Feynman amplitudes,
which shows that the BFL model is perturbatively more divergent than the
former. We then upgrade this result to the constructive level, using, in a
self-contained way, the modern tools of constructive field theory: we construct
the Borel sum of the BFL perturbative series via a convergent ?cactus'
expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our
method shows how the ?sum over trian- gulations' in quantum gravity can be
tamed rigorously, and paves the way for the renormalization program in group
field theory
Renormalization of the commutative scalar theory with harmonic term to all orders
The noncommutative scalar theory with harmonic term (on the Moyal space) has
a vanishing beta function. In this paper, we prove the renormalizability of the
commutative scalar field theory with harmonic term to all orders by using
multiscale analysis in the momentum space. Then, we consider and compute its
one-loop beta function, as well as the one on the degenerate Moyal space. We
can finally compare both to the vanishing beta function of the theory with
harmonic term on the Moyal space.Comment: 16 page
Dynamical Symmetry Breaking in SYM Theories as a Non-Semiclassical Effect
We study supersymmetry breaking effects in N=1 SYM from the point of view of
quantum effective actions. Restrictions on the geometry of the effective
potential from superspace are known to be problematic in quantum effective
actions, where explicit supersymmetry breaking can and must be studied. On the
other hand the true ground state can be determined from this effective action,
only. We study whether some parts of superspace geometry are still relevant for
the effective potential and discuss whether the ground states found this way
justify a low energy approximation based on this geometry. The answer to both
questions is negative: Essentially non-semiclassical effects change the
behavior of the auxiliary fields completely and demand for a new interpretation
of superspace geometry. These non-semiclassical effects can break
supersymmetry.Comment: 37 pages, LaTex. Version 3: many important changes, extended
discussion of the topi
Combinatorial Quantum Field Theory and Gluing Formula for Determinants
We define the combinatorial Dirichlet-to-Neumann operator and establish a
gluing formula for determinants of discrete Laplacians using a combinatorial
Gaussian quantum field theory. In case of a diagonal inner product on cochains
we provide an explicit local expression for the discrete Dirichlet-to-Neumann
operator. We relate the gluing formula to the corresponding Mayer-Vietoris
formula by Burghelea, Friedlander and Kappeler for zeta-determinants of
analytic Laplacians, using the approximation theory of Dodziuk. Our argument
motivates existence of gluing formulas as a consequence of a gluing principle
on the discrete level.Comment: 26 pages, accepted for publication at Letters in Math. Physic
Second-order mixed-moment model with differentiable ansatz function in slab geometry
We study differentiable mixed-moment models (full zeroth and first moment,
half higher moments) for a Fokker-Planck equation in one space dimension.
Mixed-moment minimum-entropy models are known to overcome the zero net-flux
problem of full-moment minimum entropy models. Realizability theory for
these modification of mixed moments is derived for second order. Numerical
tests are performed with a kinetic first-order finite volume scheme and
compared with , classical and a reference scheme.Comment: arXiv admin note: text overlap with arXiv:1611.01314,
arXiv:1511.0271
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