2,272 research outputs found
A proof of the integral identity conjecture, II
In this note, using Cluckers-Loeser's theory of motivic integration, we prove
the integral identity conjecture with framework a localized Grothendieck ring
of varieties over an arbitrary base field of characteristic zero.Comment: Changed the content in comparison with the first version. To appear
in Comptes Rendus Math\'ematiqu
Singularly Perturbed Control Systems with Noncompact Fast Variable
We deal with a singularly perturbed optimal control problem with slow and
fast variable depending on a parameter {\epsilon}. We study the asymptotic, as
{\epsilon} goes to 0, of the corresponding value functions, and show
convergence, in the sense of weak semilimits, to sub and supersolution of a
suitable limit equation containing the effective Hamiltonian. The novelty of
our contribution is that no compactness condition are assumed on the fast
variable. This generalization requires, in order to perform the asymptotic
proce- dure, an accurate qualitative analysis of some auxiliary equations posed
on the space of fast variable. The task is accomplished using some tools of
Weak KAM theory, and in particular the notion of Aubry set
Euler reflexion formulas for motivic multiple zeta functions
We introduce a new notion of \boxast-product of two integrable series with
coefficients in distinct Grothendieck rings of algebraic varieties, preserving
the integrability and commuting with the limit of rational series. In the same
context, we define a motivic multiple zeta function with respect to an ordered
family of regular functions, which is integrable and connects closely to
Denef-Loeser's motivic zeta functions. We also show that the \boxast-product
is associative in the class of motivic multiple zeta functions.
Furthermore, a version of the Euler reflexion formula for motivic zeta
functions is nicely formulated to deal with the \boxast-product and motivic
multiple zeta functions, and it is proved using the theory of arc spaces. As an
application, taking the limit for the motivic Euler reflexion formula we
recover the well known motivic Thom-Sebastiani theorem.Comment: To appear in Journal of Algebraic Geometr
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