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Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation
For the tipping elements in the Earth's climate system, the most important issue to address is how stable is the desirable state against random perturbations. Extreme biotic and climatic events pose severe hazards to tropical rainforests. Their local effects are extremely stochastic and difficult to measure. Moreover, the direction and intensity of the response of forest trees to such perturbations are unknown, especially given the lack of efficient dynamical vegetation models to evaluate forest tree cover changes over time. In this study, we consider randomness in the mathematical modelling of forest trees by incorporating uncertainty through a stochastic differential equation. According to field-based evidence, the interactions between fires and droughts are a more direct mechanism that may describe sudden forest degradation in the south-eastern Amazon. In modeling the Amazonian vegetation system, we include symmetric α-stable Lévy perturbations. We report results of stability analysis of the metastable fertile forest state. We conclude that even a very slight threat to the forest state stability represents Ĺevy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year. During years of severe drought, high-intensity fires significantly accelerate the transition between a forest and savanna state
The Gauss-Green theorem in stratified groups
We lay the foundations for a theory of divergence-measure fields in
noncommutative stratified nilpotent Lie groups. Such vector fields form a new
family of function spaces, which generalize in a sense the fields. They
provide the most general setting to establish Gauss-Green formulas for vector
fields of low regularity on sets of finite perimeter. We show several
properties of divergence-measure fields in stratified groups, ultimately
achieving the related Gauss-Green theorem.Comment: 69 page
On complex-valued 2D eikonals. Part four: continuation past a caustic
Theories of monochromatic high-frequency electromagnetic fields have been
designed by Felsen, Kravtsov, Ludwig and others with a view to portraying
features that are ignored by geometrical optics. These theories have recourse
to eikonals that encode information on both phase and amplitude -- in other
words, are complex-valued. The following mathematical principle is ultimately
behind the scenes: any geometric optical eikonal, which conventional rays
engender in some light region, can be consistently continued in the shadow
region beyond the relevant caustic, provided an alternative eikonal, endowed
with a non-zero imaginary part, comes on stage. In the present paper we explore
such a principle in dimension We investigate a partial differential system
that governs the real and the imaginary parts of complex-valued two-dimensional
eikonals, and an initial value problem germane to it. In physical terms, the
problem in hand amounts to detecting waves that rise beside, but on the dark
side of, a given caustic. In mathematical terms, such a problem shows two main
peculiarities: on the one hand, degeneracy near the initial curve; on the other
hand, ill-posedness in the sense of Hadamard. We benefit from using a number of
technical devices: hodograph transforms, artificial viscosity, and a suitable
discretization. Approximate differentiation and a parody of the
quasi-reversibility method are also involved. We offer an algorithm that
restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure
Bounded Independence Fools Degree-2 Threshold Functions
Let x be a random vector coming from any k-wise independent distribution over
{-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is
determined up to an additive epsilon for k = poly(1/epsilon). This answers an
open question of Diakonikolas et al. (FOCS 2009). Using standard constructions
of k-wise independent distributions, we obtain a broad class of explicit
generators that epsilon-fool the class of degree-2 threshold functions with
seed length log(n)*poly(1/epsilon).
Our approach is quite robust: it easily extends to yield that the
intersection of any constant number of degree-2 threshold functions is
epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if
the entries of x are k-wise independent standard normals, implying for example
that bounded independence derandomizes the Goemans-Williamson hyperplane
rounding scheme.
To achieve our results, we introduce a technique we dub multivariate
FT-mollification, a generalization of the univariate form introduced by Kane et
al. (SODA 2010) in the context of streaming algorithms. Along the way we prove
a generalized hypercontractive inequality for quadratic forms which takes the
operator norm of the associated matrix into account. These techniques may be of
independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was
wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem
8.1 to m^4, not m^
Holder Continuous Solutions of Active Scalar Equations
We consider active scalar equations , where is a divergence-free velocity field, and
is a Fourier multiplier operator with symbol . We prove that when is
not an odd function of frequency, there are nontrivial, compactly supported
solutions weak solutions, with H\"older regularity . In fact,
every integral conserving scalar field can be approximated in by
such solutions, and these weak solutions may be obtained from arbitrary initial
data. We also show that when the multiplier is odd, weak limits of
solutions are solutions, so that the -principle for odd active scalars may
not be expected.Comment: 61 page
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