7,403 research outputs found
About Projections of Solutions for Fuzzy Differential Equations
In this paper we propose the concept of fuzzy projections on subspaces of , obtained from Zadeh's extension of canonical projections in , and we study some of the main properties of such projections. Furthermore, we will review some properties of fuzzy projection solution of fuzzy differential equations. As we will see, the concept of fuzzy projection can be interesting for the graphical representation of fuzzy solutions
A Gauge Theory on Fuzzy Extra Dimensions
In this article, we explore the low energy structure of a gauge theory
over spaces with fuzzy sphere(s) as extra dimensions. In particular, we
determine the equivariant parametrization of the gauge fields, which transform
either invariantly or as vectors under the combined action of rotations
of the fuzzy spheres and those gauge transformations generated by carrying the spin irreducible representation of . The
cases of a single fuzzy sphere and a particular direct sum of
concentric fuzzy spheres, , covering the monopole bundle
sectors with windings are treated in full and the low energy degrees of
freedom for the gauge fields are obtained. Employing the parametrizations of
the fields in the former case, we determine a low energy action by tracing over
the fuzzy sphere and show that the emerging model is abelian Higgs type with
gauge symmetry and possess vortex solutions on , which we discuss in some detail. Generalization of our formulation to
the equivariant parametrization of gauge fields in theories is also
briefly addressed.Comment: 27+1 page
Multiple M0-brane system in an arbitrary eleven dimensional supergravity background
The equations of motion of multiple M0{brane (multiple M-wave or mM0) system
in an arbitrary D = 11 supergravity superspace, which generalize the Matrix
model equations for the case of inter- action with a generic 11D supergravity
background, are obtained in the frame of superembedding approach. We also
derive the BPS equations for supersymmetric bosonic solutions of these mM0
equations and show that the set of 1/2 BPS solutions contain a fuzzy sphere
modeling M2 brane as well as that the Nahm equation appears as a particular
case of the 1/4 BPS equations.Comment: RevTeX4, 20 pages, no figures. V2: misprints corrected, minor
changes, published in Phys. Rev. D82, 105030 (2010)). V3. Dec. 2011 :
misprints in coeffs of Eqs.(5.10) correcte
Matrix Quantum Mechanics and Soliton Regularization of Noncommutative Field Theory
We construct an approximation to field theories on the noncommutative torus
based on soliton projections and partial isometries which together form a
matrix algebra of functions on the sum of two circles. The matrix quantum
mechanics is applied to the perturbative dynamics of scalar field theory, to
tachyon dynamics in string field theory, and to the Hamiltonian dynamics of
noncommutative gauge theory in two dimensions. We also describe the adiabatic
dynamics of solitons on the noncommutative torus and compare various classes of
noncommutative solitons on the torus and the plane.Comment: 70 pages, 4 figures; v2: References added and update
Twisted submanifolds of R^n
We propose a general procedure to construct noncommutative deformations of an
embedded submanifold of determined by a set of smooth
equations . We use the framework of Drinfel'd twist deformation of
differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006),
1883]; the commutative pointwise product is replaced by a (generally
noncommutative) -product determined by a Drinfel'd twist. The twists we
employ are based on the Lie algebra of vector fields that are tangent
to all the submanifolds that are level sets of the ; the twisted Cartan
calculus is automatically equivariant under twisted tangent infinitesimal
diffeomorphisms. We can consistently project a connection from the twisted
to the twisted if the twist is based on a suitable Lie
subalgebra . If we endow with a metric
then twisting and projecting to the normal and tangent vector fields commute,
and we can project the Levi-Civita connection consistently to the twisted ,
provided the twist is based on the Lie subalgebra
of the Killing vector fields of the metric; a
twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can
be characterized in terms of generators and polynomial relations. We present in
some detail twisted cylinders embedded in twisted Euclidean and
twisted hyperboloids embedded in twisted Minkowski [these are
twisted (anti-)de Sitter spaces ].Comment: Latex file, 48 pages, 1 figure. Slightly adapted version to the new
preprint arXiv:2005.03509, where the present framework is specialized to
quadrics and other algebraic submanifolds of R^n. Several typos correcte
Trefftz Difference Schemes on Irregular Stencils
The recently developed Flexible Local Approximation MEthod (FLAME) produces
accurate difference schemes by replacing the usual Taylor expansion with
Trefftz functions -- local solutions of the underlying differential equation.
This paper advances and casts in a general form a significant modification of
FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the
exact match of the approximate solution at the stencil nodes. As a consequence
of that, FLAME schemes can now be generated on irregular stencils with the
number of nodes substantially greater than the number of approximating
functions. The accuracy of the method is preserved but its robustness is
improved. For demonstration, the paper presents a number of numerical examples
in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering
of electromagnetic (acoustic) waves, and wave propagation in a photonic
crystal. The examples explore the role of the grid and stencil size, of the
number of approximating functions, and of the irregularity of the stencils.Comment: 28 pages, 12 figures; to be published in J Comp Phy
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