The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

LIMIT CYCLES OF A GENERALIZED LINARD DIFFERENTIAL EQUATION VIA AVERAGING THEORY

We apply the averaging theory of first and second orders to a generalized Liénard differential equation to study the maximum number of limit cycles of the system.     

High surface area thermally stabilized porous iron oxide/silica nanocomposites via a formamide modified sol-gel process

Iron oxide/silica (Fe:Si as 1:10 atomic ratio) composite materials have been prepared by calcination for 3 h at different temperatures (400-900 °C) of xerogel precursor obtained via a formamide modified sol-gel process. The process involved TEOS and iron(III) nitrate, nitric acid and formamide. Genesis of the composite materials from the xerogel precursor has been investigated by TGA, DSC, FTIR, XRD, SEM and EDX. Results indicated that all the calcined composites are mainly composed of amorphous iron oxide dispersed as finely divided particles in amorphous silica matrixes. Nitrogen adsorption/desorption isotherms revealed a reversible type I of isotherms indicative of microporosity. However, high S_BET surface area and microsporosity were observed for the calcined composite materials (e.g. S_BET = 625 m² g⁻¹, and S_αs = 556 m² g⁻¹ for the composite calcined at 400 °C). Formation of the porous texture was discussed in terms of the action of formamide, which enhanced strengthening of the silica gel network during evaporation of the more volatile components within the composite body during the drying process.     

Non-isothermal crystallization kinetics and nucleation behavior of isotactic polypropylene composites with micro-talc

Journal of Thermal Analysis and Calorimetry - The current investigation has presented a new synthesis technique to prepare pentaethylene glycol-treated graphene nanoplatelets (PEG-GnP) and...     

Sur le principe de superposition et l'équation de Riccati

In this Note we study the conditions under which a system of ordinary differential equations admits a nonlinear superposition of the solutions, and also the linearization of such systems. A particular study is established for the Riccati equation. To cite this article: S. Rezzag et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Smart Therapeutic Strategy of pH-Responsive Gold Nanoparticles for Combating Multidrug Resistance

As several multi-target drug delivery approaches are successfully identified through preclinical screening, their clinical success is often hampered by challenges such as poor circulation stability, dissimilarities in the pharmacokinetics of different drugs, as well as targeting inefficiency. Gold nanoparticles (AuNPs) are adopted as promising nanocarriers in the co-delivery of multiple therapeutic drugs for combination therapy. The pH-responsive AuNPs are synthesized and incorporated with multiple chemotherapeutic drugs, such as doxorubicin and bleomycin. Such structures can work as drug carriers to treat cervical carcinoma by adopting a quality by design approach. The designed nanocarrier is characterized by adopting a range of physicochemical and morphological techniques. In vitro drug release and cytotoxicity of optimized nanocarriers are assessed to cervical tumor epithelial cells. The results highlight the notable advantages of colloidal AuNPs, including sustained drug release, therapeutic agent delivery with high stability, and biocompatibility for more effective treatment of cervical carcinoma. Furthermore, by improving the biodistribution and/or bioavailability profiles, it is believed that the two-in-one approach may therefore give evidence on the fate of co-loaded nanocarrier as a promising trajectory for successful clinical translation against ovarian carcinoma to achieve maximum therapeutic synergy for an individual patient.     

Graft copolymerization of acrylic acid onto polyamide fibers

The grafting of acrylic acid (AA) monomer (CH₂CHCOOH) on polyamide 6.6 monofilaments (PA 6.6) using benzoyl peroxide (BPO) as initiator was carried out in order to enhance the hydrophilic nature of fibers. The grafting rate depends on the AA concentration, the BPO concentration, the time and the temperature of reaction.The best conditions for optimum rate of grafting were obtained with a AA concentration of 0.5 M, a BPO concentration of 0.03 M, a reaction temperature of T = 85 °C and a reaction time of 120 mn.The fiber surface has been investigated by many experimental techniques of characterization such as Fourier transform infrared spectroscopy (FTIR), calorimetric analysis (DSC), scanning electron microscopy (SEM), and contact angle measurements.The effect of grafting of acrylic acid onto PA 6.6 fibers on their moisture and mechanical resistances was analyzed from water sorption and elongation at break measurements.The analysis of the experimental data shows clearly the efficiency of the grafting reaction used, leading to a significant increase of the hydrophilic character of the PA 6.6 surface.     

Synthesis of a novel highly efficient flame-retardant coating for cotton fabrics with low combustion toxicity and antibacterial properties

Cellulose - The synthesis of multi-function flame retardants is widely increasing to fulfill industrial and economic goals. In this work, a novel flame retardant, melamine salt of tannic phosphate...     

LIMIT CYCLES OF THIRD-ORDER DIFFERENTIAL EQUATION

In this paper, we investigate a third-order differential equation. Based on the averaging theory, we obtain sufficient conditions for the existence of periodic solutions to the equation.