27,033 research outputs found
On the size of the fibers of spectral maps induced by semialgebraic embeddings
Let be the ring of (continuous) semialgebraic functions on
a semialgebraic set and its subring
of bounded semialgebraic functions. In this work we compute the size of the
fibers of the spectral maps and induced by the
inclusion of a semialgebraic subset of .
The ring can be understood as the localization of at the multiplicative subset of those bounded
semialgebraic functions on with empty zero set. This provides a natural
inclusion that reduces both problems above to an analysis of
the fibers of the spectral map . If we denote , it holds that the restriction map
is a homeomorphism.
Our problem concentrates on the computation of the size of the fibers of at the points of . The size of the fibers of prime ideals
`close' to the complement provides valuable information
concerning how is immersed inside . If is dense in , the map
is surjective and the generic fiber of a prime ideal
contains infinitely many elements. However, finite fibers
may also appear and we provide a criterium to decide when the fiber is a finite set for .Comment: 33 pages, 3 figure
On the one dimensional polynomial and regular images of
In this work we present a full geometric characterization of the
1-dimensional polynomial and regular images of and we compute for
all of them the invariants and , already introduced in
\cite{fg2}
Tuning High-Harmonic Generation by Controlled Deposition of Ultrathin Ionic Layers on Metal Surfaces
High harmonic generation (HHG) from semiconductors and insulators has become
a very active area of research due to its great potential for developing
compact HHG devices. Here we show that by growing monolayers (ML) of insulators
on single-crystal metal surfaces, one can tune the harmonic spectrum by just
varying the thickness of the ultrathin layer, not the laser properties. This is
shown from numerical solutions of the time-dependent Schr\"odinger equation for
ML NaCl/Cu(111) systems () based on realistic potentials available
in the literature. Remarkably, the harmonic cutoff increases linearly with
and as much as an order of magnitude when going from 1 to 30, while
keeping the laser intensity low and the wavelength in the near-infrared range.
Furthermore, the degree of control that can be achieved in this way is much
higher than by varying the laser intensity. The origin of this behavior is the
reduction of electronic "friction" when moving from the essentially discrete
energy spectrum associated with a few-ML system to the continuous energy
spectrum (bands) inherent to an extended periodic system.Comment: 6 pages, 4 figure
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