4,523 research outputs found
Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors
We define a simplicial differential calculus by generalizing divided
differences from the case of curves to the case of general maps, defined on
general topological vector spaces, or even on modules over a topological ring
K. This calculus has the advantage that the number of evaluation points growths
linearly with the degree, and not exponentially as in the classical, "cubic"
approach. In particular, it is better adapted to the case of positive
characteristic, where it permits to define Weil functors corresponding to
scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).Comment: V2: minor changes, and chapter 3: new results included; to appear in
Forum Mathematicu
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
Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm
In this paper we study the focusing cubic wave equation in 1+5 dimensions
with radial initial data as well as the one-equivariant wave maps equation in
1+3 dimensions with the model target manifolds and
. In both cases the scaling for the equation leaves the
-norm of the solution
invariant, which means that the equation is super-critical with respect to the
conserved energy. Here we prove a conditional scattering result: If the
critical norm of the solution stays bounded on its maximal time of existence,
then the solution is global in time and scatters to free waves both forwards
and backwards in infinite time. The methods in this paper also apply to all
supercritical power-type nonlinearities for both the focusing and defocusing
radial semi-linear equation in 1+5 dimensions, yielding analogous results.Comment: 59 pages, minor typos have been correcte
Non-commutative Geometry and Kinetic Theory of Open Systems
The basic mathematical assumptions for autonomous linear kinetic equations
for a classical system are formulated, leading to the conclusion that if they
are differential equations on its phase space , they are at most of the 2nd
order. For open systems interacting with a bath at canonical equilibrium they
have a particular form of an equation of a generalized Fokker-Planck type. We
show that it is possible to obtain them as Liouville equations of Hamiltonian
dynamics on with a particular non-commutative differential structure,
provided certain geometric in character, conditions are fulfilled. To this end,
symplectic geometry on is developped in this context, and an outline of the
required tensor analysis and differential geometry is given. Certain questions
for the possible mathematical interpretation of this structure are also
discussed.Comment: 22 pages, LaTe
Non-linear Realizations of Conformal Symmetry and Effective Field Theory for the Pseudo-Conformal Universe
The pseudo-conformal scenario is an alternative to inflation in which the
early universe is described by an approximate conformal field theory on flat,
Minkowski space. Some fields acquire a time-dependent expectation value, which
breaks the flat space so(4,2) conformal algebra to its so(4,1) de Sitter
subalgebra. As a result, weight-0 fields acquire a scale invariant spectrum of
perturbations. The scenario is very general, and its essential features are
determined by the symmetry breaking pattern, irrespective of the details of the
underlying microphysics. In this paper, we apply the well-known coset technique
to derive the most general effective lagrangian describing the Goldstone field
and matter fields, consistent with the assumed symmetries. The resulting action
captures the low energy dynamics of any pseudo-conformal realization, including
the U(1)-invariant quartic model and the Galilean Genesis scenario. We also
derive this lagrangian using an alternative method of curvature invariants,
consisting of writing down geometric scalars in terms of the conformal mode.
Using this general effective action, we compute the two-point function for the
Goldstone and a fiducial weight-0 field, as well as some sample three-point
functions involving these fields.Comment: 49 pages. v2: minor corrections, added references. v3: minor edits,
version appearing in JCA
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