34,115 research outputs found
Positive Definiteness of Paired Symmetric Tensors and Elasticity Tensors
In this paper, we consider higher order paired symmetric tensors and strongly
paired symmetric tensors. Elasticity tensors and higher order elasticity
tensors in solid mechanics are strongly paired symmetric tensors. A (strongly)
paired symmetric tensor is said to be positive definite if the homogeneous
polynomial defined by it is positive definite. Positive definiteness of
elasticity and higher order elasticity tensors is strong ellipticity in solid
mechanics, which plays an important role in nonlinear elasticity theory. We
mainly investigate positive definiteness of fourth order three dimensional and
sixth order three dimensional (strongly) paired symmetric tensors. We first
show that the concerned (strongly) paired symmetric tensor is positive definite
if and only if its smallest -eigenvalue is positive. Second, we propose
several necessary and sufficient conditions under which the concerned
(strongly) paired symmetric tensor is positive definite. Third, we study the
conditions under which the homogeneous polynomial defined by a fourth order
three dimensional or sixth order three dimensional (strongly) paired symmetric
tensor can be written as a sum of squares of polynomials, and further, propose
several necessary and/or sufficient conditions to judge whether the concerned
(strongly) paired symmetric tensors are positive definite or not. Fourth, by
using semidefinite relaxation we propose a sequential semidefinite programming
method to compute the smallest -eigenvalue of a fourth order three
dimensional (strongly) paired symmetric tensor, by which we can check positive
definiteness of the concerned tensor. The preliminary numerical results
demonstrate that our method is effective
Graph Fourier Transform Based on Norm Variation Minimization
The definition of the graph Fourier transform is a fundamental issue in graph
signal processing. Conventional graph Fourier transform is defined through the
eigenvectors of the graph Laplacian matrix, which minimize the norm
signal variation. However, the computation of Laplacian eigenvectors is
expensive when the graph is large. In this paper, we propose an alternative
definition of graph Fourier transform based on the norm variation
minimization. We obtain a necessary condition satisfied by the Fourier
basis, and provide a fast greedy algorithm to approximate the Fourier
basis. Numerical experiments show the effectiveness of the greedy algorithm.
Moreover, the Fourier transform under the greedy basis demonstrates a similar
rate of decay to that of Laplacian basis for simulated or real signals
Formulating an -person noncooperative game as a tensor complementarity problem
In this paper, we consider a class of -person noncooperative games, where
the utility function of every player is given by a homogeneous polynomial
defined by the payoff tensor of that player, which is a natural extension of
the bimatrix game where the utility function of every player is given by a
quadratic form defined by the payoff matrix of that player. We will call such a
problem the multilinear game. We reformulate the multilinear game as a tensor
complementarity problem, a generalization of the linear complementarity
problem; and show that finding a Nash equilibrium point of the multilinear game
is equivalent to finding a solution of the resulted tensor complementarity
problem. Especially, we present an explicit relationship between the solutions
of the multilinear game and the tensor complementarity problem, which builds a
bridge between these two classes of problems. We also apply a smoothing-type
algorithm to solve the resulted tensor complementarity problem and give some
preliminary numerical results for solving the multilinear games
Stationary probability vectors of higher-order two-dimensional transition probability tensors
In this paper we investigate stationary probability vectors of higher-order
two-dimensional symmetric transition probability tensors. We show that there
are two special symmetric transition probability tensors of order dimension
2, which have and only have two stationary probability vectors; and any other
symmetric transition probability tensor of order dimension 2 has a unique
stationary probability vector. As a byproduct, we obtain that any symmetric
transition probability tensor of order dimension 2 has a unique positive
stationary probability vector; and that any symmetric irreducible transition
probability tensor of order dimension 2 has a unique stationary probability
vector
Cross-media Similarity Metric Learning with Unified Deep Networks
As a highlighting research topic in the multimedia area, cross-media
retrieval aims to capture the complex correlations among multiple media types.
Learning better shared representation and distance metric for multimedia data
is important to boost the cross-media retrieval. Motivated by the strong
ability of deep neural network in feature representation and comparison
functions learning, we propose the Unified Network for Cross-media Similarity
Metric (UNCSM) to associate cross-media shared representation learning with
distance metric in a unified framework. First, we design a two-pathway deep
network pretrained with contrastive loss, and employ double triplet similarity
loss for fine-tuning to learn the shared representation for each media type by
modeling the relative semantic similarity. Second, the metric network is
designed for effectively calculating the cross-media similarity of the shared
representation, by modeling the pairwise similar and dissimilar constraints.
Compared to the existing methods which mostly ignore the dissimilar constraints
and only use sample distance metric as Euclidean distance separately, our UNCSM
approach unifies the representation learning and distance metric to preserve
the relative similarity as well as embrace more complex similarity functions
for further improving the cross-media retrieval accuracy. The experimental
results show that our UNCSM approach outperforms 8 state-of-the-art methods on
4 widely-used cross-media datasets.Comment: 19 pages, submitted to Multimedia Tools and Application
Copositivity Detection of Tensors: Theory and Algorithm
A symmetric tensor is called copositive if it generates a multivariate form
taking nonnegative values over the nonnegative orthant. Copositive tensors have
found important applications in polynomial optimization and tensor
complementarity problems. In this paper, we consider copositivity detection of
tensors both from theoretical and computational points of view. After giving
several necessary conditions for copositive tensors, we propose several new
criteria for copositive tensors based on the representation of the multivariate
form in barycentric coordinates with respect to the standard simplex and
simplicial partitions. It is verified that, as the partition gets finer and
finer, the concerned conditions eventually capture all strictly copositive
tensors. Based on the obtained theoretical results with the help of simplicial
partitions, we propose a numerical method to judge whether a tensor is
copositive or not. The preliminary numerical results confirm our theoretical
findings
Quantum Reflections of Nonlocal Optical Solitons in a Cold Rydberg Atomic Gas
Quantum reflection refers to a non-vanishing reflection probability in the
absence of a classically turning point. Much attention has been paid to such
reflections due to their fundamental, intriguing physics and potential
practical applications. Here we propose a scheme to realize a quantum
reflection of nonlocal nonlinear optical beams in a cold Rydberg atomic gas via
electromagnetically induced transparency working in a dispersion regime. Based
on the long-range interaction between Rydberg atoms, we found that the system
supports low-power nonlocal optical solitons. Such nonlocal solitons can
display a sharp transition between reflection, trapping, and transmission when
scattered by a linear attractive potential, created by gate photons stored in
another Rydberg state. Different from conventional physical systems explored up
to now, the quantum reflection of the nonlocal optical solitons in the Rydberg
atomic gas exhibits interesting anomalous behaviors, which can be actively
manipulated by tuning the incident velocity and intensity of the probe field,
as well as the nonlocality of the Kerr nonlinearity inherent in the Rydberg
atomic gas. The results reported here are not only useful for developing
Rydberg nonlinear optics but also helpful for characterizing the physical
property of the Rydberg gas and for designing novel nonlinear optical devices
Estimates for eigenvalues of Lr operator on self-shrinkers
Let be an -dimensional compact
self-shrinker in with smooth boundary . In this
paper, we study eigenvalues of the operator on , where
is defined by with denoting a positive
definite (0,2)-tensor field on . We obtain "universal" inequalities for
eigenvalues of the operator . These inequalities generalize the
result of Cheng and Peng in \cite{ChengPeng2013}.
Furthermore, we also consider the case that equalities occur.Comment: 19 page
The Contributions of Neutral Higgs Bosons to Charmless Nonleptonic B Decays in MSSM
We investigate the contributions of neutral Higgs bosons to nonleptonic
transition under the supersymmetric context. Their
effects to decay width and CP violation in corresponding exclusive decays are
explored. The anomalous dimension matrices of the operators which have to be
incorporated to include the contributions of neutral Higgs bosons are given. We
find that when tan is large (say, 50) and neutral Higgs bosons are not
too heavy (say, 100 GeV), contributions of neutral Higgs penguin can dominate
electroweak penguin contributions, and for some processes, they can greatly
modify both decay width and CP asymmetry.Comment: 11 pages, typo corrected, references added, minor revisions mad
Lossless Airy Surface Polaritons in a Metamaterial via Active Raman Gain
We propose a scheme to realize a lossless propagation of linear and nonlinear
Airy surface polaritons (SPs) via active Raman gain (ARG). The system we
suggest is a planar interface superposed by a negative index metamaterial
(NIMM) and a dielectric, where three-level quantum emitters are doped. By using
the ARG from the quantum emitters and the destructive interference effect
between the electric and magnetic responses from the NIMM, we show that not
only the Ohmic loss of the NIMM but also the light absorption of the quantum
emitters can be completely eliminated. As a result, non-diffractive Airy SPs
may propagate for very long distance without attenuation. We also show that the
Kerr nonlinearity of the system can be largely enhanced due to the introduction
of the quantum emitters and hence lossless Airy surface polaritonic solitons
with very low power can be generated in the system
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