165,972 research outputs found
Gravitational instantons with faster than quadratic curvature decay (I)
In this paper, we study gravitational instantons (i.e., complete hyperk\"aler
4-manifolds with faster than quadratic curvature decay).
We prove three main theorems:
1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH.
2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact
complex elliptic surface minus a divisor. Thus, we confirm a long-standing
question of Yau in ALG and ALH cases.
3.In ALF-D_k case, it must have an O(4)-multiplet
Gravitational instantons with faster than quadratic curvature decay (II)
This is our second paper in a series to study gravitational instantons, i.e.
complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay.
We prove two main theorems:
1.The asymptotic rate of gravitational instantons to the standard models can
be improved automatically.
2.Any ALF-D_k gravitational instanton must be the
Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric.Comment: We add a corollary and the applications, correct the asymptotic rate
of the multi-Taub-NUT metri
G manifolds with nodal singularities along circles
The goal of this paper is the construction of a compact manifold with G
holonomy and nodal singularities along circles using twisted connected sum
method. This paper finds matching building blocks by solving the Calabi
conjecture on certain asymptotically cylindrical manifolds with nodal
singularities. However, by comparison to the untwisted connected sum case, it
turns out that the obstruction space for the singular twisted connected sum
construction is infinite dimensional. By analyzing the obstruction term, there
are strong evidences that the obstruction may be resolved if a further gluing
is performed in order to get a compact manifold with G holonomy and
isolated conical singularities with link .Comment: There is a simplification about homogenous harmonic forms on the
nodal cone in this versio
On J-equation
In this paper, we prove that for any K\"ahler metrics and
on , there exists
satisfying the
J-equation if and only if
is uniformly J-stable. As a corollary, we can find many
constant scalar curvature K\"ahler metrics with . Using the same method,
we also prove a similar result for the deformed Hermitian-Yang-Mills equation
when the angle is in
Shi-type estimates and finite time singularities of flows of G structures
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to
more general flows of G structures including the modified Laplacian
co-flow. Then we prove a version of -non-collapsing theorem. We will
use both of them to study finite time singularities of general flows of G
structures.Comment: The previous version is too short. So it is merged into a new pape
Rate of asymptotic convergence near isolated singularity of G manifold
In this paper, a metric with G holonomy and slow rate of convergence to
the cone metric is constructed on a ball inside the cone over the flag
manifold
On - covering sequences
Recently, motivated by Stanley sequences, Kiss, S\' andor and Yang introduced
a new type sequence: a sequence of nonnegative integers is called an
- covering sequence if there exists an integer such that if ,
then there exist ,
such that form a -term arithmetic progression.
They prove that there exists an - covering sequence such that
. In this note, we prove
that there exists an - covering sequence such that
.Comment: 5 page
Representations of bicircular lift matroids
Bicircular lift matroids are a class of matroids defined on the edge set of a
graph. For a given graph , the circuits of its bicircular lift matroid are
the edge sets of those subgraphs of that contain at least two cycles, and
are minimal with respect to this property. The main result of this paper is a
characterization of when two graphs give rise to the same bicircular lift
matoid, which answers a question proposed by Irene Pivotto. In particular,
aside from some appropriately defined "small" graphs, two graphs have the same
bicircular lift matroid if and only if they are -isomorphic in the sense of
Whitney
Asymptotic performance of regularized multi-task learning
This paper analyzes asymptotic performance of a regularized multi-task
learning model where task parameters are optimized jointly. If tasks are
closely related, empirical work suggests multi-task learning models to
outperform single-task ones in finite sample cases. As data size grows
indefinitely, we show the learned multi-classifier to optimize an average
misclassification error function which depicts the risk of applying multi-task
learning algorithm to making decisions. This technique conclusion demonstrates
the regularized multi-task learning model to be able to produce reliable
decision rule for each task in the sense that it will asymptotically converge
to the corresponding Bayes rule. Also, we find the interaction effect between
tasks vanishes as data size growing indefinitely, which is quite different from
the behavior in finite sample cases
DenseImage Network: Video Spatial-Temporal Evolution Encoding and Understanding
Many of the leading approaches for video understanding are data-hungry and
time-consuming, failing to capture the gist of spatial-temporal evolution in an
efficient manner. The latest research shows that CNN network can reason about
static relation of entities in images. To further exploit its capacity in
dynamic evolution reasoning, we introduce a novel network module called
DenseImage Network(DIN) with two main contributions. 1) A novel compact
representation of video which distills its significant spatial-temporal
evolution into a matrix called DenseImage, primed for efficient video encoding.
2) A simple yet powerful learning strategy based on DenseImage and a
temporal-order-preserving CNN network is proposed for video understanding,
which contains a local temporal correlation constraint capturing temporal
evolution at multiple time scales with different filter widths. Extensive
experiments on two recent challenging benchmarks demonstrate that our
DenseImage Network can accurately capture the common spatial-temporal evolution
between similar actions, even with enormous visual variations or different time
scales. Moreover, we obtain the state-of-the-art results in action and gesture
recognition with much less time-and-memory cost, indicating its immense
potential in video representing and understanding.Comment: 7 page
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