335 research outputs found

    Asymptotics for optimal design problems for the Schr\"odinger equation with a potential

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    We study the problem of optimal observability and prove time asymptotic observability estimates for the Schr\"odinger equation with a potential in L(Ω)L^{\infty}(\Omega), with ΩRd\Omega\subset \mathbb{R}^d, 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

    On standard norm varieties

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    Trivial Witt groups of flag varieties

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    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

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    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

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    Let pp be a prime integer and FF a field of characteristic 0. Let XX be the {\em norm variety} of a symbol in the Galois cohomology group Hn+1(F,μpn)H^{n+1}(F,\mu_p^{\otimes n}) (for some n1n\geq1), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F(X)F(X) has the following property: for any equidimensional variety YY, the change of field homomorphism \CH(Y)\to\CH(Y_{F(X)}) of Chow groups with coefficients in integers localized at pp is surjective in codimensions <(dimX)/(p1)< (\dim X)/(p-1). 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 AA-triviality} of XX, the property saying that the degree homomorphism on \CH_0(X_L) is injective for any field extension L/FL/F with X(L)X(L)\ne\emptyset. 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

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    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

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    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
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