The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is {1+\sqrt{2}}

In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is ${\mu=\sqrt{2+\sqrt{2}}}$ . A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with ${n\in [-2,2]}$ (the case n = 0 corresponding to self-avoiding walks). We modify this model by restricting to a half-plane and introducing a surface fugacity y associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov’s identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be ${y_{\rm c}=1+2/\sqrt{2-n}}$ . This value plays a crucial role in our generalized identity, just as the value of the growth constant did in Smirnov’s identity. For the case n = 0, corresponding to self-avoiding walks interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height T, taken at its critical point 1/μ, tends to 0 as T increases, as predicted from SLE theory.     

MSE-MRI sequence optimisation for measurement of bi- and tri-exponential T2 relaxation in a phantom and fruit

The transverse relaxation signal from vegetal cells can be described by multi-exponential behaviour, reflecting different water compartments. This multi-exponential relaxation is rarely measured by conventional MRI imaging protocols; mono-exponential relaxation times are measured instead, thus limiting information about of the microstructure and water status in vegetal cells. In this study, an optimised multiple spin echo (MSE) MRI sequence was evaluated for assessment of multi-exponential transverse relaxation in fruit tissues. The sequence was designed for the acquisition of a maximum of 512 echoes. Non-selective refocusing RF pulses were used in combination with balanced crusher gradients for elimination of spurious echoes. The study was performed on a bi-compartmental phantom with known T₂ values and on apple and tomato fruit. T₂ decays measured in the phantom and fruit were analysed using bi- and tri-exponential fits, respectively. The MRI results were compared with low field non-spatially resolved NMR measurements performed on the same samples.     

NMR Study of Water Distribution inside Tomato Cells: Effects of Water Stress

Tomato pericarp tissue was studied by low-field nuclear magnetic resonance (NMR) relaxometry. Two kinds of experiments were performed to investigate the correlation between multi-exponential NMR relaxation and the subcellular compartments. The longitudinal (T ₁) versus transverse (T ₂) relaxation times were first measured on fresh samples and then the transverse relaxation time was measured on samples exposed to water stress. Four signal components were found in all experiments. The results showed that all signal components corresponded to the water in different cell compartments, and that no signal from non-exchangeable protons was present. Moreover, we demonstrated that NMR relaxation is suitable for the continuous monitoring of water rebalancing between subcellular compartments of plant tissues.     

On the normal form of a system of differential equations with nilpotent linear part

We consider prenormal forms associated to generic perturbations of the system $\text{x}\text{?}\text{=2y,}\text{y}\text{?}\text{=3x}{}^2$. It is known that they have a formal normal form $\text{x}\text{?}\text{=2y+2x}\text{Δ}{}^1\text{,}\text{y}\text{?}\text{=3x}{}^2\text{+3y}\text{Δ}{}^1$, where $\text{Δ}{}^1\text{=x+A}{}_0\text{(y}{}^2\text{?x}{}^3\text{)}$ [Differential Equations 158 (1) (1999) 152–173]. We show that the series A₀ and the normalizing transformations are divergent, but 1-summable. To cite this article: M. Canalis-Durand, R. Schäfke, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Quantification of microporosity in fruit by MRI at various magnetic fields: comparison with X-ray microtomography

Microstructure determines the mechanical and transport properties of fruit tissues. One important characteristic of the microstructure is the relative volume fraction of gas-filled intercellular spaces, i.e., the tissue microporosity. Quantification of this microporosity is fundamental for investigating the relationship between gas transfer and various disorders in fruit.     

Stealth Luminescent Organic Nanoparticles Made from Quadrupolar Dyes for Two-Photon Bioimaging: Effect of End-Groups and Core

Molecular-based Fluorescent Organic Nanoparticles (FONs) are versatile light-emitting nano-tools whose properties can be rationally addressed by bottom-up molecular engineering. A challenging property to gain control over is the interaction of the FONs’ surface with biological systems. Indeed, most types of nanoparticles tend to interact with biological membranes. To address this limitation, we recently reported on two-photon (2P) absorbing, red to near infrared (NIR) emitting quadrupolar extended dyes built from a benzothiadiazole core and diphenylamino endgroups that yield spontaneously stealth FONs. In this paper, we expand our understanding of the structure-property relationship between the dye structure and the FONs 2P absorption response, fluorescence and stealthiness by characterizing a dye-related series of FONs. We observe that increasing the strength of the donor end-groups or of the core acceptor in the quadrupolar (D-π-A-π-D) dye structure allows for the tuning of optical properties, notably red-shifting both the emission (from red to NIR) and 2P absorption spectra while inducing a decrease in their fluorescence quantum yield. Thanks to their strong 1P and 2P absorption, all FONs whose median size varies between 11 and 28 nm exhibit giant 1P (10⁶ M⁻¹.cm⁻¹) and 2P (10⁴ GM) brightness values. Interestingly, all FONs were found to be non-toxic, exhibit stealth behaviour, and show vanishing non-specific interactions with cell membranes. We postulate that the strong hydrophobic character and the rigidity of the FONs building blocks are crucial to controlling the stealth nano-bio interface.