On the fractional susceptibility function of piecewise expanding maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ &gt;1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0&lt;\Re \eta &lt;1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>

Barley RIC157 is involved in RACB-mediated susceptibility to powdery mildew

Successful pathogens often benefit from certain cellular host processes. For the biotrophic ascomycete fungus Blumeria graminis f.sp. hordei (Bgh) it has been shown that barley RACB, a small monomeric G-protein (ROP, RHO of plants), is required for full susceptibility to fungal penetration. The susceptibility function of RACB probably lies in its role in cell polarisation, which may be co-opted by the pathogen for invasive ingrowth of its haustorium. However, the actual mechnism of how RACB supports the fungal penetration success is little understood. RIC proteins are considered scaffold proteins which can interact directly with ROPs via a conserved CRIB motif. Here we describe a yet uncharacterised RIC protein, RIC157, which can interact directly with RACB. We could show that RIC157 undergoes a recruitment from the cytoplasm to the cell periphery in the presence of activated RACB. During fungal infection, RIC157 and activated RACB colocalise at the penetration site, particularly at the haustorial neck. In a RACB-dependent manner, transiently overexpressed RIC157 renders barley epidermal cells more susceptible to fungal penetration. We conclude that RIC157 promotes fungal penetration into barley epidermal cells via its function as downstream executor in RACB-signaling.

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

For a piecewise expanding unimodal interval map$f$with unique absolutely continuous invariant probability measure$\mu $, a perturbation$X$, and an observable$\varphi $, the susceptibility function is$\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps.Comm. Math. Phys.275(2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps.Nonlinearity21(2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory.Adv. Math.226(2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet.Ann. Sci. Éc. Norm. Supér.(3)66(1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that$\Psi _\varphi (z)$has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of$\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation$X$is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of$\Psi _\varphi (z)$as$z\to 1$exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of$\Psi _\varphi (z)$as$z\to e^{i\omega }$for real$\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

Sensitivity to deliberate sea salt seeding of marine clouds – observations and model simulations

Sea salt seeding of marine clouds to increase their albedo is a proposed technique to counteract or slow global warming. In this study, we first investigate the susceptibility of marine clouds to sea salt injections, using observational data of cloud droplet number concentration, cloud optical depth, and liquid cloud fraction from the MODIS (Moderate Resolution Imaging Spectroradiometer) instruments on board the Aqua and Terra satellites. We then compare the derived susceptibility function to a corresponding estimate from the Norwegian Earth System Model (NorESM). Results compare well between simulations and observations, showing that stratocumulus regions off the west coast of the major continents along with large regions over the Pacific and the Indian Oceans are susceptible. At low and mid latitudes the signal is dominated by the cloud fraction. We then carry out geo-engineering experiments with a uniform increase over ocean of 10−9 kg m−2 s−1 in emissions of sea salt particles with a dry modal radius of 0.13 μm, an emission strength and areal coverage much greater than proposed in earlier studies. The increased sea salt concentrations and the resulting change in marine cloud properties lead to a globally averaged forcing of −4.8 W m−2 at the top of the atmosphere, more than cancelling the forcing associated with a doubling of CO2 concentrations. The forcing is large in areas found to be sensitive by using the susceptibility function, confirming its usefulness as an indicator of where to inject sea salt for maximum effect. Results also show that the effectiveness of sea salt seeding is reduced because the injected sea salt provides a large surface area for water vapor and gaseous sulphuric acid to condense on, thereby lowering the maximum supersaturation and suppressing the formation and lifetime of sulphate particles. In some areas, our simulations show an overall reduction in the CCN concentration and the number of activated cloud droplets decreases, resulting in a positive forcing.

Sensitivity to deliberate sea salt seeding of marine clouds – observations and model simulations

Sea salt seeding of marine clouds to increase their albedo is a proposed technique to counteract or slow global warming. In this study, we first investigate the susceptibility of marine clouds to sea salt injections, using observational data of cloud droplet number concentration, cloud optical depth, and liquid cloud fraction from the MODIS (Moderate Resolution Imaging Spectroradiometer) instruments on board the Aqua and Terra satellites. We then compare the derived susceptibility function to a corresponding estimate from the Norwegian Earth System Model (NorESM). Results compare well between simulations and observations, showing that stratocumulus regions off the west coast of the major continents along with large regions in the Pacific and the Indian Oceans are susceptible. We then carry out geo-engineering experiments with a uniform increase of 10−9 kg m−2 s−1 in emissions of sea salt particles with a modal radius of 0.13 μm. The increased sea salt concentrations and the resulting change in marine cloud properties lead to a globally averaged forcing of −4.8 W m−2 at the top of the atmosphere, more than cancelling a doubling of CO2 concentrations. The forcing is large in areas found to be sensitive by using the susceptibility function, confirming its usefulness as an indicator of where to inject sea salt for maximum effect. Results also show that the effectiveness of sea salt seeding is reduced because the injected sea salt provide a large surface area for water vapor and gaseous sulphuric acid to condense on, thereby lowering the maximum supersaturation and suppressing the formation and lifetime of sulphate particles. In some areas, our simulations show an overall reduction in the CCN concentration and the number of activated cloud droplets decreases, resulting in a positive globally averaged forcing.