740 research outputs found
Z_3-graded exterior differential calculus and gauge theories of higher order
We present a possible generalization of the exterior differential calculus,
based on the operator d such that d^3=0, but d^2\not=0. The first and second
order differentials generate an associative algebra; we shall suppose that
there are no binary relations between first order differentials, while the
ternary products will satisfy the cyclic relations based on the representation
of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the
first order differentials and grade 2 to the second order differentials; under
the associative multiplication law the grades add up modulo 3. We show how the
notion of covariant derivation can be generalized with a 1-form A, and we give
the expression in local coordinates of the curvature 3-form. Finally, the
introduction of notions of a scalar product and integration of the Z_3-graded
exterior forms enables us to define variational principle and to derive the
differential equations satisfied by the curvature 3-form. The Lagrangian
obtained in this way contains the invariants of the ordinary gauge field tensor
F_{ik} and its covariant derivatives D_i F_{km}.Comment: 13 pages, no figure
The cubic chessboard
We present a survey of recent results, scattered in a series of papers that
appeared during past five years, whose common denominator is the use of cubic
relations in various algebraic structures. Cubic (or ternary) relations can
represent different symmetries with respect to the permutation group S_3, or
its cyclic subgroup Z_3. Also ordinary or ternary algebras can be divided in
different classes with respect to their symmetry properties. We pay special
attention to the non-associative ternary algebra of 3-forms (or ``cubic
matrices''), and Z_3-graded matrix algebras. We also discuss the Z_3-graded
generalization of Grassmann algebras and their realization in generalized
exterior differential forms. A new type of gauge theory based on this
differential calculus is presented. Finally, a ternary generalization of
Clifford algebras is introduced, and an analog of Dirac's equation is
discussed, which can be diagonalized only after taking the cube of the
Z_3-graded generalization of Dirac's operator. A possibility of using these
ideas for the description of quark fields is suggested and discussed in the
last Section.Comment: 23 pages, dedicated to A. Trautman on the occasion of his 64th
birthda
General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems
An asymptotic method for finding instabilities of arbitrary -dimensional
large-amplitude patterns in a wide class of reaction-diffusion systems is
presented. The complete stability analysis of 2- and 3-dimensional localized
patterns is carried out. It is shown that in the considered class of systems
the criteria for different types of instabilities are universal. The specific
nonlinearities enter the criteria only via three numerical constants of order
one. The performed analysis explains the self-organization scenarios observed
in the recent experiments and numerical simulations of some concrete
reaction-diffusion systems.Comment: 21 pages (RevTeX), 8 figures (Postscript). To appear in Phys. Rev. E
(April 1st, 1996
On a graded q-differential algebra
Given a unital associatve graded algebra we construct the graded
q-differential algebra by means of a graded q-commutator, where q is a
primitive N-th root of unity. The N-th power (N>1) of the differential of this
graded q-differential algebra is equal to zero. We use our approach to
construct the graded q-differential algebra in the case of a reduced quantum
plane which can be endowed with a structure of a graded algebra. We consider
the differential d satisfying d to power N equals zero as an analog of an
exterior differential and study the first order differential calculus induced
by this differential.Comment: 6 pages, submitted to the Proceedings of the "International
Conference on High Energy and Mathematical Physics", Morocco, Marrakech,
April 200
Examples of derivation-based differential calculi related to noncommutative gauge theories
Some derivation-based differential calculi which have been used to construct
models of noncommutative gauge theories are presented and commented. Some
comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour
of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry,
Homology and Fundamental Interactions". To appear in a special issue of
International Journal of Geometric Methods in Modern Physic
Z-graded differential geometry of quantum plane
In this work, the Z-graded differential geometry of the quantum plane is
constructed. The corresponding quantum Lie algebra and its Hopf algebra
structure are obtained. The dual algebra, i.e. universal enveloping algebra of
the quantum plane is explicitly constructed and an isomorphism between the
quantum Lie algebra and the dual algebra is given.Comment: 17 page
Scenarios of domain pattern formation in a reaction-diffusion system
We performed an extensive numerical study of a two-dimensional
reaction-diffusion system of the activator-inhibitor type in which domain
patterns can form. We showed that both multidomain and labyrinthine patterns
may form spontaneously as a result of Turing instability. In the stable
homogeneous system with the fast inhibitor one can excite both localized and
extended patterns by applying a localized stimulus. Depending on the parameters
and the excitation level of the system stripes, spots, wriggled stripes, or
labyrinthine patterns form. The labyrinthine patterns may be both connected and
disconnected. In the the stable homogeneous system with the slow inhibitor one
can excite self-replicating spots, breathing patterns, autowaves and
turbulence. The parameter regions in which different types of patterns are
realized are explained on the basis of the asymptotic theory of instabilities
for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101
(1996). The dynamics of the patterns observed in our simulations is very
similar to that of the patterns forming in the ferrocyanide-iodate-sulfite
reaction.Comment: 15 pages (REVTeX), 15 figures (postscript and gif), submitted to
Phys. Rev.
Self-replication and splitting of domain patterns in reaction-diffusion systems with fast inhibitor
An asymptotic equation of motion for the pattern interface in the
domain-forming reaction-diffusion systems is derived. The free boundary problem
is reduced to the universal equation of non-local contour dynamics in two
dimensions in the parameter region where a pattern is not far from the points
of the transverse instabilities of its walls. The contour dynamics is studied
numerically for the reaction-diffusion system of the FitzHugh-Nagumo type. It
is shown that in the asymptotic limit the transverse instability of the
localized domains leads to their splitting and formation of the multidomain
pattern rather than fingering and formation of the labyrinthine pattern.Comment: 9 pages (ReVTeX), 5 figures (postscript). To be published in Phys.
Rev.
Theory of Spike Spiral Waves in a Reaction-Diffusion System
We discovered a new type of spiral wave solutions in reaction-diffusion
systems --- spike spiral wave, which significantly differs from spiral waves
observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of
these waves in Gray-Scott model. We derive the kinematic relations describing
the shape of this spiral and find the dependence of its main parameters on the
control parameters. The theory does not rely on the specific features of
Gray-Scott model and thus is expected to be applicable to a broad range of
reaction-diffusion systems.Comment: 4 pages (REVTeX), 2 figures (postscript), submitted to Phys. Rev.
Let
On the construction of generalized Grassmann representatives of state vectors
Generalized -graded Grassmann variables are used to label coherent
states related to the nilpotent representation of the q-oscillator of
Biedenharn and Macfarlane when the deformation parameter is a root of unity.
These states are then used to construct generalized Grassmann representatives
of state vectors.Comment: 8 page
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