## Journals

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### Open Access Journals

NACO

In this paper, we study the $l^k$-eigenvalues/vectors of a real
symmetric square tensor. Specially, we investigate some properties on the related $l^k$-spectral radius of a real
nonnegative symmetric square tensor.

JIMO

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors
in this paper. Hilbert tensors are symmetric Cauchy tensors. An
even order symmetric Cauchy tensor is positive semi-definite if and
only if its generating vector is positive. An even order symmetric
Cauchy tensor is positive definite if and only if its generating
vector has positive and mutually distinct entries. This extends
Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy
tensors. Then, it is proven that the positive semi-definiteness
character of an even order symmetric Cauchy tensor can be
equivalently checked by the monotone increasing property of a
homogeneous polynomial related to the Cauchy tensor. The homogeneous
polynomial is strictly monotone increasing in the nonnegative
orthant of the Euclidean space when the even order symmetric Cauchy
tensor is positive definite. At last, bounds of the largest
H-eigenvalue of a positive semi-definite
symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.

keywords:
generating vector.
,
Cauchy tensor
,
eigenvalue
,
positive definiteness
,
Positive semi-definiteness

JIMO

Circulant tensors naturally arise from stochastic process and
spectral hypergraph theory. The joint moments of stochastic
processes are symmetric circulant tensors. The adjacency, Laplacian
and signless Laplacian tensors of circulant hypergraphs are also
symmetric circulant tensors. The adjacency, Laplacian and
signless Laplacian tensors of directed circulant hypergraphs are
circulant tensors, but they are not symmetric in general. In this
paper, we study spectral properties of circulant tensors and their
applications in spectral hypergraph theory and stochastic process.
We show that in certain cases, the largest H-eigenvalue of a
circulant tensor can be explicitly identified. In particular, the
largest H-eigenvalue of a nonnegative circulant tensor can be
explicitly identified. This confirms the results in circulant
hypergraphs and directed circulant hypergraphs. We prove that an
even order circulant B$_0$ tensor is always positive semi-definite.
This shows that the Laplacian tensor and the signless Laplacian
tensor of a directed circulant even-uniform hypergraph are positive
semi-definite. If a stochastic process is $m$th order stationary,
where $m$ is even, then its $m$th order moment, which is a circulant
tensor, must be positive semi-definite. In this paper, we give
various conditions for an even order circulant tensor to be positive
semi-definite.

JIMO

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

JIMO

In this paper, we investigate the reformulation
of the steady state security region problem for electrical power
systems. Firstly, a simple security region problem with one
changeable parameter
is reformulated into a system of semismooth equations, which is composed by
the normal power flow equations and an additional piecewise smooth
equation. Then the semismooth Newton method and the smoothing Newton
method can be applied to solve the problem. Preliminary numerical
results show that the method is promising. Finally, by using the
smoothing technique, a more complicated security region problem, the
Euclidean security region problem, is reformulated as an equality
constrained optimization problem. These works provide a possibility
to implement on-line calculation of the security region of
electrical power systems.

JIMO

In this paper, we propose a network model called simple manufacturing
network. Our model is a combined version of the ordinary multicommodity network and the
manufacturing network flow model. It can be used to characterize the complicated manufacturing
scenarios. By formulating the model as a minimum cost flow problem plus several bounded variables,
we present a modified network simplex method, which exploits the special structure of the model and
can perform the computation on the network. A numerical example is provided for illustrating our
method.

JIMO

In this paper, we present a new approach for finding a stable
solution of a system of nonlinear equations arising from dynamical
systems. We introduce the concept of
stability functions and use this idea to construct
stability solution models of several
typical small signal stability problems in dynamical systems.
Each model consists of a system of constrained
semismooth equations. The advantage of the new models is twofold.
Firstly, the stability requirement of dynamical systems is
controlled by nonlinear inequalities. Secondly, the semismoothness
property of the stability functions makes the models solvable by
efficient numerical methods. We introduce smoothing functions for
the stability functions and present a smoothing Newton method
for solving the problems. Global and local quadratic convergence of
the algorithm is established. Numerical examples from dynamical
systems are also given to illustrate the efficiency of the new
approach.

IPI

Diffusion Kurtosis Imaging (DKI) is a new Magnetic
Resonance Imaging (MRI) model to characterize the non-Gaussian
diffusion behavior in tissues. In reality, the term
$bD_{app}-\frac{1}{6}b^2D_{app}^2K_{app}$ in the extended Stejskal
and Tanner equation of DKI should be positive for an appropriate range of $b$-values
to make sense
physically. The positive definiteness of the above term reflects the
signal attenuation in tissues during imaging. Hence, it is essential
for the validation of DKI.

In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.

In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.

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