3,937 research outputs found
Moduli of surfaces with an anti-canonical cycle
We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth
projective rational surface and D is an effective anti-canonical divisor which
is a cycle of rational curves. This Torelli theorem was conjectured by Friedman
in 1984. In addition, we construct natural universal families for such pairs.Comment: Final version. Much simplified proofs. To appear in Compositi
Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups
of toric varieties. This enables us to provide, among other results, an
elementary geometric proof of the Laurent phenomenon for cluster algebras (of
geometric type), extend Speyer's example of an upper cluster algebra which is
not finitely generated, and show that the Fock-Goncharov dual basis conjecture
is usually false.Comment: 50 pages, to appear in Algebraic Geometr
Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
We formulate hydrodynamic equations and spectrally accurate numerical methods
for investigating the role of geometry in flows within two-dimensional fluid
interfaces. To achieve numerical approximations having high precision and level
of symmetry for radial manifold shapes, we develop spectral Galerkin methods
based on hyperinterpolation with Lebedev quadratures for -projection to
spherical harmonics. We demonstrate our methods by investigating hydrodynamic
responses as the surface geometry is varied. Relative to the case of a sphere,
we find significant changes can occur in the observed hydrodynamic flow
responses as exhibited by quantitative and topological transitions in the
structure of the flow. We present numerical results based on the
Rayleigh-Dissipation principle to gain further insights into these flow
responses. We investigate the roles played by the geometry especially
concerning the positive and negative Gaussian curvature of the interface. We
provide general approaches for taking geometric effects into account for
investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure
Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to a pair
(Y,D) of a smooth projective surface with an anti-canonical cycle of rational
curves, as the spectrum of an explicit algebra defined in terms of counts of
rational curves on Y meeting D in a single point. In the case D is
contractible, the family gives a smoothing of the dual cusp, and thus a proof
of Looijenga's 1981 cusp conjecture.Comment: 144 pages, 3 figures, Second version significantly shorter, 109
pages. The first version has a lot of material (particularly in the
introduction and material on cyclic quotient singularities) which does not
appear in the new version. Download version 1 if this material is desired.
Third and final version, small changes from Version 2, to appear in Publ.
IHE
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
Yesterday, today and tomorrow: A perspective of CFD at NASA's Ames Research Center
The opportunity to reflect on the computational fluid dynamics (CFD) progam at the NASA Ames Research Center (its beginning, its present state, and its direction for the future) is afforded. Essential elements of the research program during each period are reviewed, including people, facilities, and research problems. The burgeoning role that CFD is playing in the aerospace business is discussed, as is the necessity for validated CFD tools. The current aeronautical position of this country is assessed, as are revolutionary goals to help maintain its aeronautical supremacy in the world
Progress and future directions in computational fluid dynamics
Computational fluid dynamics (CFD) has made great strides in the detailed simulation of complex fluid flows, including the fluid physics of flows heretofore not understood. It is now being routinely applied to some rather complicated problems, and starting to impact the design cycle of aerospace vehicles and their components. In addition, it is being used to complement and is being complemented by experimental studies. In this paper some major elements of contemporary CFD research, such as code validation, turbulence physics, and hypersonic flows are discussed, along with a review of the principal pacing items that currently govern CFD. Several examples are presented to illustrate the current state of the art. Finally, prospects for the future of the development and application of CFD are suggested
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