17 research outputs found
Three Dimensional Strongly Symmetric Circulant Tensors
In this paper, we give a necessary and sufficient condition for an even order
three dimensional strongly symmetric circulant tensor to be positive
semi-definite. In some cases, we show that this condition is also sufficient
for this tensor to be sum-of-squares. Numerical tests indicate that this is
also true in the other cases
Regularly Decomposable Tensors and Classical Spin States
A spin- state can be represented by a symmetric tensor of order and
dimension . Here, can be a positive integer, which corresponds to a
boson; can also be a positive half-integer, which corresponds to a fermion.
In this paper, we introduce regularly decomposable tensors and show that a
spin- state is classical if and only if its representing tensor is a
regularly decomposable tensor. In the even-order case, a regularly decomposable
tensor is a completely decomposable tensor but not vice versa; a completely
decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an
SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the
odd-order case, the first row tensor of a regularly decomposable tensor is
regularly decomposable and its other row tensors are induced by the regular
decomposition of its first row tensor. We also show that complete
decomposability and regular decomposability are invariant under orthogonal
transformations, and that the completely decomposable tensor cone and the
regularly decomposable tensor cone are closed convex cones. Furthermore, in the
even-order case, the completely decomposable tensor cone and the PSD tensor
cone are dual to each other. The Hadamard product of two completely
decomposable tensors is still a completely decomposable tensor. Since one may
apply the positive semi-definite programming algorithm to detect whether a
symmetric tensor is an SOS tensor or not, this gives a checkable necessary
condition for classicality of a spin- state. Further research issues on
regularly decomposable tensors are also raised.Comment: published versio
SOS-Hankel Tensors: Theory and Application
Hankel tensors arise from signal processing and some other applications. SOS
(sum-of-squares) tensors are positive semi-definite symmetric tensors, but not
vice versa. The problem for determining an even order symmetric tensor is an
SOS tensor or not is equivalent to solving a semi-infinite linear programming
problem, which can be done in polynomial time. On the other hand, the problem
for determining an even order symmetric tensor is positive semi-definite or not
is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are
two known positive semi-definite Hankel tensor classes: even order complete
Hankel tensors and even order strong Hankel tensors. We show complete Hankel
tensors are strong Hankel tensors, and even order strong Hankel tensors are
SOS-Hankel tensors. We give several examples of positive semi-definite Hankel
tensors, which are not strong Hankel tensors. However, all of them are still
SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel
tensor? The answer to this question remains open. If the answer to this
question is no, then the problem for determining an even order Hankel tensor is
positive semi-definite or not is solvable in polynomial-time. An application of
SOS-Hankel tensors to the positive semi-definite tensor completion problem is
discussed. We present an ADMM algorithm for solving this problem. Some
preliminary numerical results on this algorithm are reported
Positive Semi-Definiteness of Generalized Anti-Circulant Tensors
Anti-circulant tensors have applications in exponential data fitting. They
are special Hankel tensors. In this paper, we extend the definition of
anti-circulant tensors to generalized anti-circulant tensors by introducing a
circulant index such that the entries of the generating vector of a Hankel
tensor are circulant with module . In the special case when , where
is the dimension of the Hankel tensor, the generalized anticirculant tensor
reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors
are still special Hankel tensors. For the cases that , and some other cases, including the matrix case that , we give
necessary and sufficient conditions for positive semi-definiteness of even
order generalized anti-circulant tensors, and show that in these cases, they
are SOS tensors. This shows that, in these cases, there are no PNS (positive
semidefinite tensors which are not sum of squares) Hankel tensors
Supersymmetric deformations of 3D SCFTs from tri-sasakian truncation
We holographically study supersymmetric deformations of and
superconformal field theories (SCFTs) in three dimensions using
four-dimensional gauged supergravity coupled to three-vector multiplets
with non-semisimple gauge
group. This gauged supergravity can be obtained from a truncation of
eleven-dimensional supergravity on a tri-sasakian manifold and admits both
supersymmetric and stable non-supersymmetric critical points.
We analyze the BPS equations for singlet scalars in details and study
possible supersymmetric solutions. A number of RG flows to non-conformal field
theories and half-supersymmetric domain walls are found, and many of them can
be given analytically. Apart from these "flat" domain walls, we also consider
-sliced domain wall solutions describing two-dimensional conformal
defects with supersymmetry within the dual field theory while
this type of solutions does not exist in the case.Comment: 33 pages, 4 figures, references added, typos and parts of the
analysis corrected with numerical solutions include