5,284 research outputs found
Curve-counting invariants for crepant resolutions
We construct curve counting invariants for a Calabi-Yau threefold
equipped with a dominant birational morphism . Our invariants
generalize the stable pair invariants of Pandharipande and Thomas which occur
for the case when is the identity. Our main result is a PT/DT-type
formula relating the partition function of our invariants to the
Donaldson-Thomas partition function in the case when is a crepant
resolution of , the coarse space of a Calabi-Yau orbifold
satisfying the hard Lefschetz condition. In this case, our partition function
is equal to the Pandharipande-Thomas partition function of the orbifold
. Our methods include defining a new notion of stability for
sheaves which depends on the morphism . Our notion generalizes slope
stability which is recovered in the case where is the identity on .
Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence
via the Hall algebra and Joyce's integration map.Comment: In this version, Jim Bryan has been added as an author and the
required boundedness result for our stability condition has been added. arXiv
admin note: text overlap with arXiv:1002.4374 by other author
Fluid Vesicles in Flow
We review the dynamical behavior of giant fluid vesicles in various types of
external hydrodynamic flow. The interplay between stresses arising from
membrane elasticity, hydrodynamic flows, and the ever present thermal
fluctuations leads to a rich phenomenology. In linear flows with both
rotational and elongational components, the properties of the tank-treading and
tumbling motions are now well described by theoretical and numerical models. At
the transition between these two regimes, strong shape deformations and
amplification of thermal fluctuations generate a new regime called trembling.
In this regime, the vesicle orientation oscillates quasi-periodically around
the flow direction while asymmetric deformations occur. For strong enough
flows, small-wavelength deformations like wrinkles are observed, similar to
what happens in a suddenly reversed elongational flow. In steady elongational
flow, vesicles with large excess areas deform into dumbbells at large flow
rates and pearling occurs for even stronger flows. In capillary flows with
parabolic flow profile, single vesicles migrate towards the center of the
channel, where they adopt symmetric shapes, for two reasons. First, walls exert
a hydrodynamic lift force which pushes them away. Second, shear stresses are
minimal at the tip of the flow. However, symmetry is broken for vesicles with
large excess areas, which flow off-center and deform asymmetrically. In
suspensions, hydrodynamic interactions between vesicles add up to these two
effects, making it challenging to deduce rheological properties from the
dynamics of individual vesicles. Further investigations of vesicles and similar
objects and their suspensions in steady or time-dependent flow will shed light
on phenomena such as blood flow.Comment: 13 pages, 13 figures. Adv. Colloid Interface Sci., 201
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Sphaerodactylus kirbyi
Number of Pages: 2Integrative BiologyGeological Science
Wrinkling instability of vesicles in steady linear flow
We present experimental observations and numerical simulations of a wrinkling
instability that occurs at sufficiently high strain rates in the trembling
regime of vesicle dynamics in steady linear flow. Spectral and statistical
analysis of the data shows similarities and differences with the wrinkling
instability observed earlier for vesicles in transient elongation flow. The
critical relevance of thermal fluctuations for this phenomenon is revealed by a
simple model using coupled Langevin equations that reproduces the experimental
observations quite well.Comment: 6 pages, 9 figures + 2 video
The Tannakian Formalism and the Langlands Conjectures
Let H be a connected reductive group over an algebraically closed field of
characteristic zero, and let G be an abstract group. In this note we show that
every homomorphism from the Grothendieck semiring of H to that of G which maps
irreducible representations to irreducibles, comes from a group homomorphism
from G to H. We also connect this result with the Langlands conjectures.Comment: 15 page
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