29,401 research outputs found
Quantum canonical tensor model and an exact wave function
Tensor models in various forms are being studied as models of quantum
gravity. Among them the canonical tensor model has a canonical pair of
rank-three tensors as dynamical variables, and is a pure constraint system with
first-class constraints. The Poisson algebra of the first-class constraints has
structure functions, and provides an algebraically consistent way of
discretizing the Dirac first-class constraint algebra for general relativity.
This paper successfully formulates the Wheeler-DeWitt scheme of quantization of
the canonical tensor model; the ordering of operators in the constraints is
determined without ambiguity by imposing Hermiticity and covariance on the
constraints, and the commutation algebra of constraints takes essentially the
same from as the classical Poisson algebra, i.e. is first-class. Thus one could
consistently obtain, at least locally in the configuration space, wave
functions of "universe" by solving the partial differential equations
representing the constraints, i.e. the Wheeler-DeWitt equations for the quantum
canonical tensor model. The unique wave function for the simplest non-trivial
case is exactly and globally obtained. Although this case is far from being
realistic, the wave function has a few physically interesting features; it
shows that locality is favored, and that there exists a locus of configurations
with features of beginning of universe.Comment: 17 pages. Section 2 expanded to include fuzzy-space interpretation,
and other minor change
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Fuzzy Scalar Field Theory as a Multitrace Matrix Model
We develop an analytical approach to scalar field theory on the fuzzy sphere
based on considering a perturbative expansion of the kinetic term. This
expansion allows us to integrate out the angular degrees of freedom in the
hermitian matrices encoding the scalar field. The remaining model depends only
on the eigenvalues of the matrices and corresponds to a multitrace hermitian
matrix model. Such a model can be solved by standard techniques as e.g. the
saddle-point approximation. We evaluate the perturbative expansion up to second
order and present the one-cut solution of the saddle-point approximation in the
large N limit. We apply our approach to a model which has been proposed as an
appropriate regularization of scalar field theory on the plane within the
framework of fuzzy geometry.Comment: 1+25 pages, replaced with published version, minor improvement
Physical states in the canonical tensor model from the perspective of random tensor networks
Tensor models, generalization of matrix models, are studied aiming for
quantum gravity in dimensions larger than two. Among them, the canonical tensor
model is formulated as a totally constrained system with first-class
constraints, the algebra of which resembles the Dirac algebra of general
relativity. When quantized, the physical states are defined to be vanished by
the quantized constraints. In explicit representations, the constraint
equations are a set of partial differential equations for the physical
wave-functions, which do not seem straightforward to be solved due to their
non-linear character. In this paper, after providing some explicit solutions
for , we show that certain scale-free integration of partition functions
of statistical systems on random networks (or random tensor networks more
generally) provides a series of solutions for general . Then, by
generalizing this form, we also obtain various solutions for general .
Moreover, we show that the solutions for the cases with a cosmological constant
can be obtained from those with no cosmological constant for increased .
This would imply the interesting possibility that a cosmological constant can
always be absorbed into the dynamics and is not an input parameter in the
canonical tensor model. We also observe the possibility of symmetry enhancement
in , and comment on an extension of Airy function related to the
solutions.Comment: 41 pages, 1 figure; typos correcte
Possibilistic clustering for shape recognition
Clustering methods have been used extensively in computer vision and pattern recognition. Fuzzy clustering has been shown to be advantageous over crisp (or traditional) clustering in that total commitment of a vector to a given class is not required at each iteration. Recently fuzzy clustering methods have shown spectacular ability to detect not only hypervolume clusters, but also clusters which are actually 'thin shells', i.e., curves and surfaces. Most analytic fuzzy clustering approaches are derived from Bezdek's Fuzzy C-Means (FCM) algorithm. The FCM uses the probabilistic constraint that the memberships of a data point across classes sum to one. This constraint was used to generate the membership update equations for an iterative algorithm. Unfortunately, the memberships resulting from FCM and its derivatives do not correspond to the intuitive concept of degree of belonging, and moreover, the algorithms have considerable trouble in noisy environments. Recently, we cast the clustering problem into the framework of possibility theory. Our approach was radically different from the existing clustering methods in that the resulting partition of the data can be interpreted as a possibilistic partition, and the membership values may be interpreted as degrees of possibility of the points belonging to the classes. We constructed an appropriate objective function whose minimum will characterize a good possibilistic partition of the data, and we derived the membership and prototype update equations from necessary conditions for minimization of our criterion function. In this paper, we show the ability of this approach to detect linear and quartic curves in the presence of considerable noise
Electrified Fuzzy Spheres and Funnels in Curved Backgrounds
We use the non-Abelian DBI action to study the dynamics of coincident
-branes in an arbitrary curved background, with the presence of a
homogenous world-volume electric field. The solutions are natural extensions of
those without electric fields, and imply that the spheres will collapse toward
zero size. We then go on to consider the intersection in a curved
background and find various dualities and automorphisms of the general
equations of motion. It is possible to map the dynamical equation of motion to
the static one via Wick rotation, however the additional spatial dependence of
the metric prevents this mapping from being invertible. Instead we find that a
double Wick rotation leaves the static equation invariant. This is very
different from the behaviour in Minkowski space. We go on to construct the most
general static fuzzy funnel solutions for an arbitrary metric either by solving
the static equations of motion, or by finding configurations which minimise the
energy. As a consistency check we construct the Abelian -brane world-volume
theory in the same generic background and find solutions consistent with energy
minimisation. In the 5-brane background we find time dependent solutions to
the equations of motion, representing a time dependent fuzzy funnel. These
solutions match those obtained from the -string picture to leading order
suggesting that the action in the large limit does not need corrections. We
conclude by generalising our solutions to higher dimensional fuzzy funnels.Comment: 38 pages, Latex; references adde
Fuzzy Sphere Dynamics and Non-Abelian DBI in Curved Backgrounds
We consider the non-Abelian action for the dynamics of -branes in the
background of -branes, which parameterises a fuzzy sphere using the SU(2)
algebra. We find that the curved background leads to collapsing solutions for
the fuzzy sphere except when we have branes in the background, which
is a realisation of the gravitational Myers effect. Furthermore we find the
equations of motion in the Abelian and non-Abelian theories are identical in
the large limit. By picking a specific ansatz we find that we can
incorporate angular momentum into the action, although this imposes restriction
upon the dimensionality of the background solutions. We also consider the case
of non-Abelian non-BPS branes, and examine the resultant dynamics using
world-volume symmetry transformations. We find that the fuzzy sphere always
collapses but the solutions are sensitive to the combination of the two
conserved charges and we can find expanding solutions with turning points. We
go on to consider the coincident 5-brane background, and again construct
the non-Abelian theory for both BPS and non-BPS branes. In the latter case we
must use symmetry arguments to find additional conserved charges on the
world-volumes to solve the equations of motion. We find that in the Non-BPS
case there is a turning solution for specific regions of the tachyon and radion
fields. Finally we investigate the more general dynamics of fuzzy
in the -brane background, and find collapsing solutions
in all cases.Comment: 49 pages, 3 figures, Latex; Version to appear in JHE
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