26,401 research outputs found
Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical
points of discrete total curvature functional on triangulated 3-manifolds. We
study DQE-metrics by introducing some combinatorial curvature flows. We prove
that these flows produce solutions which converge to discrete quasi-Einstein
metrics when the initial energy is small enough. The proof relies on a careful
analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix
of the curvature map. As a consequence, combinatorial curvature flow provides
an algorithm to compute discrete sphere packing metrics with prescribed
curvatures.Comment: 20 pages, 1 figure
A combinatorial Yamabe problem on two and three dimensional manifolds
In this paper, we introduce a new combinatorial curvature on two and three
dimensional triangulated manifolds, which transforms in the same way as that of
the smooth scalar curvature under scaling of the metric and could be used to
approximate the Gauss curvature on two dimensional manifolds. Then we use the
flow method to study the corresponding constant curvature problem, which is
called combinatorial Yamabe problem.Comment: We add a proof of the discrete maximal principle in this versio
2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry
For triangulated surfaces locally embedded in the standard hyperbolic space,
we introduce combinatorial Calabi flow as the negative gradient flow of
combinatorial Calabi energy. We prove that the flow produces solutions which
converge to ZCCP-metric (zero curvature circle packing metric) if the initial
energy is small enough. Assuming the curvature has a uniform upper bound less
than , we prove that combinatorial Calabi flow exists for all time.
Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.Comment: 15 page
3-Dimensional Discrete curvature flows and discrete Einstein metric
We introduce the discrete Einstein metrics as critical points of discrete
energy on triangulated 3-manifolds, and study them by discrete curvature flow
of second (fourth) order. We also study the convergence of the discrete
curvature flow. Discrete curvature flow of second order is an analogue of
smooth Ricci flow.Comment: 15 pages, 1 figure
Energy Optimal Interpolation in Quantum Evolution
We introduce the concept of interpolation in quantum evolution and present a
general framework to find the energy optimal Hamiltonian for a quantum system
evolving among a given set of middle states using variational and geometric
methods. A few special cases are carefully studied. The quantum brachistochrone
problem is proved as a special case.Comment: 8 pages, 0 figure
Some extremal results on hypergraph Tur\'{a}n problems
For two -graphs and , let
be the maximum number of copies of
in an -vertex -free -graph. In this paper,
using the random algebraic method, we prove that if is sufficiently larger
than , then
where
is an -graph with vertices and edges. In particular,
when is an edge or a complete bipartite -graph, we can
determine their asymptotics. We show that if is sufficiently larger than
, then
and
where and .
Meanwhile, we provide an explicit construction giving
This improves the previous best known lower bound
obtained by probabilistic method
Parallel Data Augmentation for Formality Style Transfer
The main barrier to progress in the task of Formality Style Transfer is the
inadequacy of training data. In this paper, we study how to augment parallel
data and propose novel and simple data augmentation methods for this task to
obtain useful sentence pairs with easily accessible models and systems.
Experiments demonstrate that our augmented parallel data largely helps improve
formality style transfer when it is used to pre-train the model, leading to the
state-of-the-art results in the GYAFC benchmark dataset.Comment: Accepted by ACL 2020. arXiv admin note: text overlap with
arXiv:1909.0600
New Theoretical Bounds and Constructions of Permutation Codes under Block Permutation Metric
Permutation codes under different metrics have been extensively studied due
to their potentials in various applications. Generalized Cayley metric is
introduced to correct generalized transposition errors, including previously
studied metrics such as Kendall's -metric, Ulam metric and Cayley metric
as special cases. Since the generalized Cayley distance between two
permutations is not easily computable, Yang et al. introduced a related metric
of the same order, named the block permutation metric. Given positive integers
and , let denote the maximum size of a
permutation code in with minimum block permutation distance . In this
paper, we focus on the theoretical bounds of and the
constructions of permutation codes under block permutation metric. Using a
graph theoretic approach, we improve the Gilbert-Varshamov type bound by a
factor of , when is fixed and goes into infinity. We
also propose a new encoding scheme based on binary constant weight codes.
Moreover, an upper bound beating the sphere-packing type bound is given when
is relatively close to
Wyner's Common Information: Generalizations and A New Lossy Source Coding Interpretation
Wyner's common information was originally defined for a pair of dependent
discrete random variables. Its significance is largely reflected in, hence also
confined to, several existing interpretations in various source coding
problems. This paper attempts to both generalize its definition and to expand
its practical significance by providing a new operational interpretation. The
generalization is two-folded: the number of dependent variables can be
arbitrary, so are the alphabet of those random variables. New properties are
determined for the generalized Wyner's common information of N dependent
variables. More importantly, a lossy source coding interpretation of Wyner's
common information is developed using the Gray-Wyner network. In particular, it
is established that the common information equals to the smallest common
message rate when the total rate is arbitrarily close to the rate distortion
function with joint decoding. A surprising observation is that such equality
holds independent of the values of distortion constraints as long as the
distortions are within some distortion region. Examples about the computation
of common information are given, including that of a pair of dependent Gaussian
random variables.Comment: 31 pages, 5 figures. Submitted to IEEE Transactions on Information
Theor
Formality Style Transfer with Hybrid Textual Annotations
Formality style transformation is the task of modifying the formality of a
given sentence without changing its content. Its challenge is the lack of
large-scale sentence-aligned parallel data. In this paper, we propose an
omnivorous model that takes parallel data and formality-classified data jointly
to alleviate the data sparsity issue. We empirically demonstrate the
effectiveness of our approach by achieving the state-of-art performance on a
recently proposed benchmark dataset of formality transfer. Furthermore, our
model can be readily adapted to other unsupervised text style transfer tasks
like unsupervised sentiment transfer and achieve competitive results on three
widely recognized benchmarks
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