2,401 research outputs found
Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules
Given a family of Laurent polynomials, we will construct a morphism between
its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The
kernel and cokernel of this morphism are very simple and consist of free
O-modules. The result above enables us to put a mixed Hodge module structure on
certain classes of GKZ systems and shows that they have quasi-unipotent
monodromy.Comment: 34 page
Logarithmic degenerations of Landau-Ginzburg models for toric orbifolds and global tt^* geometry
We discuss the behavior of Landau-Ginzburg models for toric orbifolds near
the large volume limit. This enables us to express mirror symmetry as an
isomorphism of Frobenius manifolds which aquire logarithmic poles along a
boundary divisor. If the toric orbifold admits a crepant resolution we
construct a global moduli space on the B-side and show that the associated
tt^*-geometry exists globally.Comment: 40 page
Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules
We describe mirror symmetry for weak toric Fano manifolds as an equivalence
of D-modules equipped with certain filtrations. We discuss in particular the
logarithmic degeneration behavior at the large radius limit point, and express
the mirror correspondence as an isomorphism of Frobenius manifolds with
logarithmic poles. The main tool is an identification of the Gauss-Manin system
of the mirror Landau-Ginzburg model with a hypergeometric D-module, and a
detailed study of a natural filtration defined on this differential system. We
obtain a solution of the Birkhoff problem for lattices defined by this
filtration and show the existence of a primitive form, which yields the
construction of Frobenius structures with logarithmic poles associated to the
mirror Laurent polynomial. As a final application, we show the existence of a
pure polarized non-commutative Hodge structure on a Zariski open subset of the
complexified Kaehler moduli space of the variety
Beyond Bayesian model averaging over paths in probabilistic programs with stochastic support
The posterior in probabilistic programs with stochastic support decomposes as a weighted sum of the local posterior distributions associated with each possible program path. We show that making predictions with this full posterior implicitly performs a Bayesian model averaging (BMA) over paths. This is potentially problematic, as BMA weights can be unstable due to model misspecification or inference approximations, leading to sub-optimal predictions in turn. To remedy this issue, we propose alternative mechanisms for path weighting: one based on stacking and one based on ideas from PAC-Bayes. We show how both can be implemented as a cheap post-processing step on top of existing inference engines. In our experiments, we find them to be more robust and lead to better predictions compared to the default BMA weights
Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion
We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit
- …