25,523 research outputs found
The E-Eigenvectors of Tensors
We first show that the eigenvector of a tensor is well-defined. The
differences between the eigenvectors of a tensor and its E-eigenvectors are the
eigenvectors on the nonsingular projective variety . We show that a generic
tensor has no eigenvectors on . Actually, we show that a generic
tensor has no eigenvectors on a proper nonsingular projective variety in
. By these facts, we show that the coefficients of the
E-characteristic polynomial are algebraically dependent. Actually, a certain
power of the determinant of the tensor can be expressed through the
coefficients besides the constant term. Hence, a nonsingular tensor always has
an E-eigenvector. When a tensor is nonsingular and symmetric, its
E-eigenvectors are exactly the singular points of a class of hypersurfaces
defined by and a parameter. We give explicit factorization of the
discriminant of this class of hypersurfaces, which completes Cartwright and
Strumfels' formula. We show that the factorization contains the determinant and
the E-characteristic polynomial of the tensor as irreducible
factors.Comment: 17 page
Convergence of a Second Order Markov Chain
In this paper, we consider convergence properties of a second order Markov
chain. Similar to a column stochastic matrix is associated to a Markov chain, a
so called {\em transition probability tensor} of order 3 and dimension
is associated to a second order Markov chain with states. For this ,
define as on the dimensional standard simplex
. If 1 is not an eigenvalue of on and is
irreducible, then there exists a unique fixed point of on . In
particular, if every entry of is greater than , then 1 is not
an eigenvalue of on . Under the latter condition, we
further show that the second order power method for finding the unique fixed
point of on is globally linearly convergent and the
corresponding second order Markov process is globally -linearly convergent.Comment: 16 pages, 3 figure
Excitation function of initial temperature of heavy flavor quarkonium emission source in high energy collisions
The transverse momentum spectra of , , and produced in proton-proton (+), proton-antiproton
(+), proton-lead (+Pb), gold-gold (Au+Au), and lead-lead (Pb+Pb)
collisions over a wide energy range are analyzed by the (two-component) Erlang
distribution, the Hagedorn function (the inverse power-law), and the
Tsallis-Levy function. The initial temperature is obtained from the color
string percolation model due to the fit by the (two-component) Erlang
distribution in the framework of multisource thermal model. The excitation
functions of some parameters such as the mean transverse momentum and initial
temperature increase from dozens of GeV to above 10 TeV. The mean transverse
momentum and initial temperature decrease (increase slightly or do not change
obviously) with the increase of rapidity (centrality). Meanwhile, the mean
transverse momentum of is larger than that of
and , and the initial temperature for
emission is higher than that for and emission, which shows
a mass-dependent behavior.Comment: 26 pages, 12 figures. Advances in High Energy Physics, accepte
Non-local Geometry inside Lifshitz Horizon
Based on the quantum renormalization group, we derive the bulk geometry that
emerges in the holographic dual of the fermionic U(N) vector model at a nonzero
charge density. The obstruction that prohibits the metallic state from being
smoothly deformable to the direct product state under the renormalization group
flow gives rise to a horizon at a finite radial coordinate in the bulk. The
region outside the horizon is described by the Lifshitz geometry with a
higher-spin hair determined by microscopic details of the boundary theory. On
the other hand, the interior of the horizon is not described by any Riemannian
manifold, as it exhibits an algebraic non-locality. The non-local structure
inside the horizon carries the information on the shape of the filled Fermi
sea.Comment: 20 page
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
- …