335 research outputs found
Asymptotics for optimal design problems for the Schr\"odinger equation with a potential
We study the problem of optimal observability and prove time asymptotic
observability estimates for the Schr\"odinger equation with a potential in
, with , using spectral theory.
An elegant way to model the problem using a time asymptotic observability
constant is presented. For certain small potentials, we demonstrate the
existence of a nonzero asymptotic observability constant under given conditions
and describe its explicit properties and optimal values. Moreover, we give a
precise description of numerical models to analyze the properties of important
examples of potentials wells, including that of the modified harmonic
oscillator
Trivial Witt groups of flag varieties
Let G be a split semi-simple linear algebraic group over a field, let P be a
parabolic subgroup and let L be a line bundle on the projective homogeneous
variety G/P. We give a simple condition on the class of L in Pic(G/P)/2 in
terms of Dynkin diagrams implying that the Witt groups W^i(G/P,L) are zero for
all integers i. In particular, if B is a Borel subgroup, then W^i(G/B,L) is
zero unless L is trivial in Pic(G/B)/2.Comment: 3 pages, 1 figur
Structurable algebras and groups of type E_6 and E_7
It is well-known that every algebraic group of type F_4 is the automorphism
group of an exceptional Jordan algebra, and that up to isogeny all groups of
type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm
forms of such Jordan algebras. We describe a similar relationship between
groups of type E_6 and groups of type E_7 and use it to give explicit
descriptions of the homogeneous projective varieties associated to groups of
type E_7 with trivial Tits algebras. The underlying algebraic structure for the
relationship considered here are a sort of 56-dimensional structurable algebra
which are forms of an algebra constructed from an exceptional Jordan algebra.Comment: 35 pages, AMSLaTeX -- error in final section correcte
On standard norm varieties
Let be a prime integer and a field of characteristic 0. Let be
the {\em norm variety} of a symbol in the Galois cohomology group
(for some ), constructed in the proof of
the Bloch-Kato conjecture. The main result of the paper affirms that the
function field has the following property: for any equidimensional
variety , the change of field homomorphism \CH(Y)\to\CH(Y_{F(X)}) of Chow
groups with coefficients in integers localized at is surjective in
codimensions . One of the main ingredients of the proof is a
computation of Chow groups of a (generalized) Rost motive (a variant of the
main result not relying on this is given in Appendix). Another important
ingredient is {\em -triviality} of , the property saying that the degree
homomorphism on \CH_0(X_L) is injective for any field extension with
. The proof involves the theory of rational correspondences
reviewed in Appendix.Comment: 38 pages; final version, to appear in Ann. Sci. \'Ec. Norm. Sup\'er.
(4
Invariants of degree 3 and torsion in the Chow group of a versal flag
We prove that the group of normalized cohomological invariants of degree 3
modulo the subgroup of semidecomposable invariants of a semisimple split linear
algebraic group G is isomorphic to the torsion part of the Chow group of
codimension 2 cycles of the respective versal G-flag. In particular, if G is
simple, we show that this factor group is isomorphic to the group of
indecomposable invariants of G. As an application, we construct nontrivial
cohomological classes for indecomposable central simple algebras.Comment: Appendix with computations for the PGO8-case is adde
Excellence of function fields of conics
For every generalized quadratic form or hermitian form over a division
algebra, the anisotropic kernel of the form obtained by scalar extension to the
function field of a smooth projective conic is defined over the field of
constants. The proof does not require any hypothesis on the characteristic
- …