61 research outputs found
Flows driven by Banach space-valued rough paths
We show in this note how the machinery of C^1-approximate flows devised in
the work "Flows driven by rough paths", and applied there to reprove and extend
most of the results on Banach space-valued rough differential equations driven
by a finite dimensional rough path, can be used to deal with rough differential
equations driven by an infinite dimensional Banach space-valued weak geometric
Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force
in a simple way.Comment: 8 page
Robustness of the microtubule network self-organization in epithelia
Robustness of biological systems is crucial for their survival, however, for many systems its origin is an open question. Here, we analyze one subcellular level system, the microtubule cytoskeleton. Microtubules self-organize into a network, along which cellular components are delivered to their biologically relevant locations. While the dynamics of individual microtubules is sensitive to the organism’s environment and genetics, a similar sensitivity of the overall network would result in pathologies. Our large-scale stochastic simulations show that the self-organization of microtubule networks is robust in a wide parameter range in individual cells. We confirm this robustness in vivo on the tissue-scale using genetic manipulations of Drosophila epithelial cells. Finally, our minimal mathematical model shows that the origin of robustness is the separation of time-scales in microtubule dynamics rates. Altogether, we demonstrate that the tissue-scale self-organization of a microtubule network depends only on cell geometry and the distribution of the microtubule minus-ends
Numerical Schemes for Rough Parabolic Equations
This paper is devoted to the study of numerical approximation schemes for a
class of parabolic equations on (0, 1) perturbed by a non-linear rough signal.
It is the continuation of [8, 7], where the existence and uniqueness of a
solution has been established. The approach combines rough paths methods with
standard considerations on discretizing stochastic PDEs. The results apply to a
geometric 2-rough path, which covers the case of the multidimensional
fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201
Unbounded violation of tripartite Bell inequalities
We prove that there are tripartite quantum states (constructed from random
unitaries) that can lead to arbitrarily large violations of Bell inequalities
for dichotomic observables. As a consequence these states can withstand an
arbitrary amount of white noise before they admit a description within a local
hidden variable model. This is in sharp contrast with the bipartite case, where
all violations are bounded by Grothendieck's constant. We will discuss the
possibility of determining the Hilbert space dimension from the obtained
violation and comment on implications for communication complexity theory.
Moreover, we show that the violation obtained from generalized GHZ states is
always bounded so that, in contrast to many other contexts, GHZ states do in
this case not lead to extremal quantum correlations. The results are based on
tools from the theories of operator spaces and tensor norms which we exploit to
prove the existence of bounded but not completely bounded trilinear forms from
commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more
accessible for a non-specialized reade
A one-dimensional variational problem with continuous Lagrangian and singular minimizer
We construct a continuous Lagrangian, strictly convex and superlinear
in the third variable, such that the associated variational problem has a Lipschitz
minimizer which is non-differentiable on a dense set. More precisely, the upper
and lower Dini derivatives of the minimizer differ by a constant on a dense (hence
second category) set. In particular, we show that mere continuity is an insufficient
smoothness assumption for Tonelli’s partial regularity theorem
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