2,580 research outputs found
On Quantum de Rham Cohomology Theory
We define quantum exterior product wedge_h and quantum exterior differential
d_h on Poisson manifolds (of which symplectic manifolds are an important class
of examples). Quantum de Rham cohomology, which is a deformation quantization
of de Rham cohomology, is defined as the cohomology of d_h. We also define
quantum Dolbeault cohomology. A version of quantum integral on symplectic
manifolds is considered and the correspoding quantum Stokes theorem is proved.
We also derive quantum hard Lefschetz theorem. By replacing d by d_h and wedge
by wedge_h in the usual definitions, we define many quantum analogues of
important objects in differential geometry, e.g. quantum curvature. The quantum
characteristic classes are then studied along the lines of classical Chern-Weil
theory. Quantum equivariant de Rham cohomology is defined in the similar
fashion.Comment: 8 pages, AMSLaTe
Degenerate Chern-Weil Theory and Equivariant Cohomology
We develop a Chern-Weil theory for compact Lie group action whose generic
stabilizers are finite in the framework of equivariant cohomology. This
provides a method of changing an equivariant closed form within its
cohomological class to a form more suitable to yield localization results. This
work is motivated by our work on reproving wall crossing formulas in
Seiberg-Witten theory, where the Lie group is the circle. As applications, we
derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions
on compact manifolds with boundary. One of the formulas is then used to yield a
very simple proof of a localization formula due to Jeffrey-Kirwan in the case
of G = SU(2) or SO(3).Comment: 23 pages, AMSLaTe
Frobenius Manifold Structure on Dolbeault Cohomology and Mirror Symmetry
We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from
Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold
structure on Dolbeault cohomology.Comment: 10 pages, AMS LaTe
Identification of Two Frobenius Manifolds In Mirror Symmetry
We identify two Frobenius manifolds obtained from two different differential
Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is
constructed on the Dolbeault cohomology, and the other on the de Rham
cohomology. Our result can be considered as a generalization of the
identification of the Dolbeault cohomology ring with the complexified de Rham
cohomology ring on a Kaehler manifold.Comment: 12 pages, AMS LaTe
On Quantum de Rham Cohomology
We define quantum exterior product wedge_h and quantum exterior differential
d_h on Poisson manifolds, of which symplectic manifolds are an important class
of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We
also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is
proved. We also define a version of quantum integral, and prove the quantum
Stokes theorem. By the trick of replacing d by d_h and wedge by wedge_h in the
usual definitions, we define many quantum analogues of important objects in
differential geometry, e.g. quantum curvature. The quantum characteristic
classes are then studied along the lines of classical Chern-Weil theory, i.e.,
they can be represented by expressions of quantum curvature. Quantum
equivariant de Rham cohomology is defined in a similar fashion. Calculations
are done for some examples, which show that quantum de Rham cohomology is
different from the quantum cohomology defined using pseudo-holomorphic curves.Comment: 36 pages, AMS LaTe
DGBV Algebras and Mirror Symmetry
We describe some recent development on the theory of formal Frobenius
manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk
(DGBV) algebras and formulate a version of mirror symmetry conjecture: the
extended deformation problems of the complex structure and the Poisson
structure are described by two DGBV algebras; mirror symmetry is interpreted in
term of the invariance of the formal Frobenius manifold structures under
quasi-isomorphism.Comment: 11 pages, to appear in Proceedings of ICCM9
Formal Frobenius manifold structure on equivariant cohomology
For a closed K\"{a}hler manifold with a Hamiltonian action of a connected
compact Lie group by holomorphic isometries, we construct a formal Frobenius
manifold structure on the equivariant cohomology by exploiting a natural DGBV
algebra structure on the Cartan model.Comment: AMS-LaTex, 14 page
On quasi-isomorphic DGBV algebras
One of the methods to obtain Frobenius manifold structures is via DGBV
(differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An
important problem is how to identify Frobenius manifold structures constructed
from two different DGBV algebras. For DGBV algebras with suitable conditions,
we show the functorial property of a construction of deformations of the
multiplicative structures of their cohomology. In particular, we show that
quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold
structures.Comment: 16 pages, AMS-LaTe
BRITS: Bidirectional Recurrent Imputation for Time Series
Time series are widely used as signals in many classification/regression
tasks. It is ubiquitous that time series contains many missing values. Given
multiple correlated time series data, how to fill in missing values and to
predict their class labels? Existing imputation methods often impose strong
assumptions of the underlying data generating process, such as linear dynamics
in the state space. In this paper, we propose BRITS, a novel method based on
recurrent neural networks for missing value imputation in time series data. Our
proposed method directly learns the missing values in a bidirectional recurrent
dynamical system, without any specific assumption. The imputed values are
treated as variables of RNN graph and can be effectively updated during the
backpropagation.BRITS has three advantages: (a) it can handle multiple
correlated missing values in time series; (b) it generalizes to time series
with nonlinear dynamics underlying; (c) it provides a data-driven imputation
procedure and applies to general settings with missing data.We evaluate our
model on three real-world datasets, including an air quality dataset, a
health-care data, and a localization data for human activity. Experiments show
that our model outperforms the state-of-the-art methods in both imputation and
classification/regression accuracies
An axion-like scalar field environment effect on binary black hole merger
Environment, such as the accretion disk, could modify the signal of the
gravitational wave from the astrophysical black hole binaries. In this article,
we model the matter field around the intermediate-mass binary black holes by
means of an axion-like scalar field and investigate their joint evolution. In
details, we consider the equal mass binary black holes surrounded by a shell of
axion-like scalar field both in spherical symmetric and non-spherical symmetric
cases, and with different strength of the scalar field. Our result shows that
the environmental scalar field could essentially modify the dynamics. Firstly,
in the spherical symmetric case, with increasing of the scalar field strength,
the number of circular orbit of the binary black hole is reduced. It means that
the scalar field could significantly accelerate the merger process. Secondly,
once the scalar field strength exceeds certain critical value, the scalar field
could collapse into a third black hole with its mass being larger than the
binary. Consequently, the new black hole collapsed from the environmental
scalar field could accrete the binary promptly and the binary collides head-on
between each other. In this process, there is almost no any quadrupole signal
produced, namely the gravitational wave is greatly suppressed. Thirdly, when
the scalar field strength is relatively smaller than the critical value, the
black hole orbit could develop eccentricity through the accretion of the scalar
field. Fourthly, during the initial stage of the inspire, the gravitational
attractive force from the axion-like scalar field could induce a sudden turn in
the binary orbits, hence result in a transient wiggle in the gravitational
waveform. Finally, in the non-spherical case, the scalar field could
gravitationally attract the binary moving toward the mass center of the scalar
field and slow down the merger process.Comment: 14 pages, 12 figure
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