Abstract:
In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are "rapid") of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.

Abstract:
We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c

Abstract:
We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\partial\_t u(t, x) = \left(a(x) -- \int\_{\Omega} k(x, y)u(t, y) dy\right ) u(t, x) + \int\_{\Omega} m(x, y)[u(t, y) -- u(t, x)] dy\quad \text{ for}\quad (t, x) $\in$ \mathbb{R}\_{+} \times \Omega,$$ together with the initial condition $u(0, \cdot) = u0 \quad \text{ in }\quad \Omega$. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values $\Omega$ is a bounded subset of $\mathbb{R}^N$ , the kernels $k$ and $m$ respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on $\Omega$. Moreover, when this '' stationary '' measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in $L^{\infty} (\Omega) \cap L^1 (\Omega)$, the solution of the Cauchy problem converges to this limit measure in $L^2 (\Omega)$. We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.

Abstract:
In this article we study some spectral properties of the linear operator $\mathcal{L}\_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by :$$ \mathcal{L}\_{\Omega}[\varphi] +a\varphi:=\int\_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where $\Omega\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ defined by $$\lambda\_p(\mathcal{L}\_{\Omega}+a):= \sup\{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such that}\, \mathcal{L}\_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \, \text{in}\;\Omega\}. $$ We establish some new properties of this generalised principal eigenvalue $\lambda\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $\lambda\_{p} \left(\mathcal{L}\_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}\_{\sigma,m,\Omega}$ is defined by $\displaystyle{\mathcal{L}\_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int\_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. In particular, we prove that $$\lim\_{\sigma\to 0}\lambda\_p\left(\mathcal{L}\_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda\_1\left(\frac{D\_2(J)}{2N}\Delta +a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\varphi\_{p,\sigma}$.

Abstract:
Uncovering how natural selection and genetic drift shape the evolutionary dynamics of virus populations within their hosts can pave the way to a better understanding of virus emergence. Mathematical models already play a leading role in these studies and are intended to predict future emergences. Here, using high-throughput sequencing, we analyzed the within-host population dynamics of four Potato virus Y (PVY) variants differing at most by two substitutions involved in pathogenicity properties. Model selection procedures were used to compare experimental results to six hypotheses regarding competitiveness and intensity of genetic drift experienced by viruses during host plant colonization. Results indicated that the frequencies of variants were well described using Lotka-Volterra models where the competition coefficients βij exerted by variant j on variant i are equal to their fitness ratio, rj/ri. Statistical inference allowed the estimation of the effect of each mutation on fitness, revealing slight (s = ？0.45%) and high (s = ？13.2%) fitness costs and a negative epistasis between them. Results also indicated that only 1 to 4 infectious units initiated the population of one apical leaf. The between-host variances of the variant frequencies were described using Dirichlet-multinomial distributions whose scale parameters, closely related to the fixation index FST, were shown to vary with time. The genetic differentiation of virus populations among plants increased from 0 to 10 days post-inoculation and then decreased until 35 days. Overall, this study showed that mathematical models can accurately describe both selection and genetic drift processes shaping the evolutionary dynamics of viruses within their hosts.

Abstract:
In this research, the natural bentonite clay (from Maghnia, western Algeria) was purified (Na^{+}- montmorillonite, CEC = 91 meq/100 g), noted (puri.bent) and modified with mixed hydroxy-Fe-Al (FeAl-PILC). The purified bentonite clay and FeAl-PILC were heated at 383 K for 2 hr and characte-rized by the chemical analyses data, XRD, and N_{2} adsorption to 77 K techniques. Puri.bent and FeAl-PILC were applied to fix the organic matter (OM) present in urban wastewater from the city of Sidi Bel-Abbes (western Algeria). The adsorption of organic matter was followed by spectro-photometry at 470 nm, and the adsorption data were a good fit with Freundlich isotherm for pu-ri.bent but for FeA-lPILC, were well fit by Elovitch isotherm model. The maximum adsorption ca-pacity (q_{m}) was 571.6 mg/g for puri.bent and 1120.69 mg/g for FeAl-PLC. The degree of OM removal was 67% for puri.bent and 97% for FeAl-PILC. FeAl-PILC can be considered as a promising adsorbent for the removal of OM from wastewater.

Abstract:
Plasmonic antennas offer promising opportunities to control the emission of quantum objects. As a consequence, the fluorescence enhancement factor is widely used as a figure of merit for a practical antenna realization. However, the fluorescence enhancement factor is not an intrinsic property of the antenna. It critically depends on several parameters, some of which are often disregarded. In this contribution, I explore the influence of the setup collection efficiency, emitter's quantum yield, and excitation intensity. Improperly setting these parameters may significantly alter the enhancement values, leading to potential misinterpretations. The discussion is illustrated by an antenna example of a nanoaperture surrounded by plasmonic corrugations. 1. Introduction Plasmonic antennas are receiving a large interest to interface light with nanoscale quantum emitters on dimensions much beyond the optical wavelength [1, 2]. Recent developments involve squeezing light into nanoscale volumes [3], enhancing the excitation and emission rate of individual emitters [4–8], tuning the luminescence spectrum [9, 10], polarization [11], and directivity properties [12–16]. Several plasmonic systems are being investigated to enhance the luminescence emission of fluorescent molecules or quantum dots, such as metallic nanoparticles [4, 5, 17–20], core-shell particles [21], thin films [22, 23], nanoantennas [6, 7, 15, 24], nanowires [16], nanoporous gold [25], nanopockets [26], metallic gratings [27], nanoaperture arrays [28], and single nanoapertures [29, 30]. A general review on surface-enhanced fluorescence can be found in [31]. A natural question while performing experiments on nanoantenna-enhanced luminescence deals with the quantification of the luminescence enhancement factor , which is commonly defined as the ratio of the detected radiation power per emitter with the antenna to the reference radiation power per emitter without the antenna. determines how many extra photons are detected for each emitter thanks to the use of the optical antenna. It is well known that this factor critically depends on several parameters: the antenna material and geometry, its spectral resonance, and overlap with the emitter’s absorption and luminescence spectra, as well as the emitter’s orientation and location respective to the antenna [32]. These many parameters often hide the influence of other parameters: the collection efficiency used in the experiments, the emitter’s quantum yield in the absence of the antenna, and the excitation intensity respective to the saturation process.

Abstract:
Carney complex.The complex of cardiac myxomas, endocrine overactivity and spotty pigmentation.The Carney complex (CNC) was first described in 1985 by J. Aidan Carney, as the combination of myxomas, spotty pigmentation and endocrine overactivity [1]. It is defined by the association of multiple endocrine neoplasia and cardiocutaneous manifestations. Patients previously characterized as LAMB (lentigineses, atrial myxoma, mucocutaneous myxoma, blue nevi) or NAME (nevi, atrial myxoma, myxoid neurofibroma, ephelide) could be considered as having Carney complex. Numerous organs may be involved in CNC and the manifestations vary greatly among patients. Some of them are quite specific, such as primary pigmented nodular disease (PPNAD), while others show little specificity, such as thyroid nodes or blue nevi. It is generally assumed that a patient presenting with two or more of the manifestations listed in Table 1 would be diagnosed as having Carney complex. This table lists the most frequent features of CNC and their estimated frequency [according to references [1,3-5,8] and our personal observations]. The incidence of each manifestation depends on its presentation and might not reflect true prevalence. For instance, according to autopsy studies PPNAD is a constant feature in CNC patients [8], however, reports of Cushing syndrome in the literature indicate that only 25 to 45 % of CNC patients have PPNAD. It has been established that at least two of these manifestations need to be present to confirm the diagnosis of CNC. If the patient has a germline PRKAR1A mutation and/or a first-degree relative affected by CNC, a single manifestation is sufficient for the diagnosis.CNC is a rare disease. About 500 patients have been registered by the NIH-Mayo Clinic (USA) and the Cochin center (France) [2]. Cumulative reports from these centers, plus information from the Cornell center in New York, indicate that there are about 160 index cases of CNC presently known [3-6].The manifestatio