1,310 research outputs found
On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups
We show by explicit estimates that the SubRiemannian distance in a Carnot
group of step two is locally semiconcave away from the diagonal if and only if
the group does not contain abnormal minimizing curves. Moreover, we prove that
local semiconcavity fails to hold in the step-3 Engel group, even in the weaker
"horizontal" sense.Comment: Revised version. To appear on J. Math. Anal- App
On the subRiemannian cut locus in a model of free two-step Carnot group
We characterize the subRiemannian cut locus of the origin in the free Carnot
group of step two with three generators. We also calculate explicitly the cut
time of any extremal path and the distance from the origin of all points of the
cut locus. Finally, by using the Hamiltonian approach, we show that the cut
time of strictly normal extremal paths is a smooth explicit function of the
initial velocity covector. Finally, using our previous results, we show that at
any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va
A Hadamard-type open map theorem for submersions and applications to completeness results in Control Theory
We prove a quantitative openness theorem for submersions under suitable
assumptions on the differential. We then apply our result to a class of
exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a
classical completeness result by Palais.Comment: 12 pages. Revised version. Minor changes. To appear on Annali di
Matematic
Nonsmooth Hormander vector fields and their control balls
We prove a ball-box theorem for nonsmooth Hormander vector fields of step s.Comment: Final version. Trans. Amer. Math. Soc. (2012 to appear
Multiexponential maps in Carnot groups with applications to convexity and differentiability
We analyze some properties of a class of multiexponential maps appearing
naturally in the geometric analysis of Carnot groups. We will see that such
maps can be useful in at least two interesting problems. First, in relation to
the analysis of some regularity properties of horizontally convex sets. Then,
we will show that our multiexponential maps can be used to prove the Pansu
differentiability of the subRiemannian distance from a fixed point.Comment: Revised version. Published on Annali di Matematica Pura ed Applicat
Anisotropic estimates of subelliptic type
We discuss some estimates of subelliptic type related with vector fields satisfying the Hormander condition. Our approach makes use of a class of approximate exponentials studied in our previous papers. Such kind of estimates arises naturally in the study of regularity theory of weak solutions of degenerate elliptic equations
Cosmological Measurements with General Relativistic Galaxy Correlations
We investigate the cosmological dependence and the constraining power of
large-scale galaxy correlations, including all redshift-distortions,
wide-angle, lensing and gravitational potential effects on linear scales. We
analyze the cosmological information present in the lensing convergence and in
the gravitational potential terms describing the so-called "relativistic
effects," and we find that, while smaller than the information contained in
intrinsic galaxy clustering, it is not negligible. We investigate how
neglecting them does bias cosmological measurements performed by future
spectroscopic and photometric large-scale surveys such as SKA and Euclid. We
perform a Fisher analysis using the CLASS code, modified to include
scale-dependent galaxy bias and redshift-dependent magnification and evolution
bias. Our results show that neglecting relativistic terms introduces an error
in the forecasted precision in measuring cosmological parameters of the order
of a few tens of percent, in particular when measuring the matter content of
the Universe and primordial non-Gaussianity parameters. Therefore, we argue
that radial correlations and integrated relativistic terms need to be taken
into account when forecasting the constraining power of future large-scale
number counts of galaxy surveys.Comment: 18 pages, 10 figure
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