Nonlinear Leray–Schauder Alternatives for Decomposable Operators in Dunford–Pettis Spaces and Application to Nonlinear Eigenvalue Problems

We present some new variants of Leray–Schauder type fixed point theorems and eigenvalue results for decomposable single-valued nonlinear weakly compact operators in Dunford-Pettis spaces.     

Essential Spectra of a 3 × 3 Operator Matrix and an Application to Three-Group Transport Equations

In this paper we study spectral properties of a 3 × 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. It is shown that, under certain conditions, this block operator matrix defines a closed operator, and the essential spectra of this operator are determined. These results are applied to a three-group transport equation.     

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 $n\ge 1$.     

Dynamical study of fractional order differential equations of predator‐pest models

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.     

Preparation and characterization of nanocrystalline CoFe<Subscript>2</Subscript>O<Subscript>4</Subscript> deposited on SiO<Subscript>2</Subscript>: in situ sol–gel process

In situ base catalyst assisted sol–gel process is used for the synthesis of nanocrystalline CoFe₂O₄ deposition on SiO₂ particles. The SiO₂ particles were prepared using base catalyst assisted sol–gel process and the consecutive formation and deposition of nanocrystalline CoFe₂O₄ on SiO₂ particles was monitored using Powder X ray Diffraction (XRD), Fourier Transform Infrared Spectroscopy (FTIR), Thermo Gravimetric And Differential Thermal Analysis (TG/DTA), Scanning Electron Microscopy and Energy Dispersive X ray Spectroscopy (SEM–EDS) and High Resolution Transmission Electron Microscopy (HRTEM). The crystallite size of CoFe₂O₄ is calculated using Scherrer’s formula and it is found to be 8 nm. The HRTEM images and selective area electron diffraction (SAED) results confirmed the formation of nanocrystalline CoFe₂O₄ particles deposited over SiO₂ spheres.     

Toward Electrochemically Controllable Tristable Three‐Station [2]Catenanes

Encouraged by the prospect of producing an electrochemical, color‐switchable red–green–blue (RGB) dye compound, we have designed, synthesized, and characterized two three‐station [2]catenanes. Both are composed of macrocyclic polyethers containing three π‐electron‐rich stations, which act as recognition sites for a π‐electron‐deficient tetracationic cyclophane. The molecular structures of the two three‐station [2]catenanes were characterized fully by mass spectrometry and ¹H NMR spectroscopy. To anticipate the relative occupancies of the three stations in each [2]catenane by the cyclophane, model compounds with the same constitutions in the vicinity of the stations were synthesized. The relative ground‐state populations of the three stations occupied in both [2]catenanes were estimated from the thermodynamic parameters for 1:1 complexes between all these model compounds and the cyclophane, obtained from isothermal titration calorimetry (ITC). The electrochemical and electromechanical properties of the three‐station [2]catenanes were analyzed by cyclic voltammetry (CV), differential pulse voltammetry (DPV), and spectroelectrochemistry (SEC). The first three‐station [2]catenane was found to behave like a bistable system, whereas the second can be described as a quasi‐tristable system.     

One step regioselective synthesis of 5-aminoisoxazoles from nitrile oxides and alpha-cyanoenamines

The 1,3-dipolar cycloaddition of nitrile oxides to 1-cyanoenamines gives 5-aminoisoxazoles regioselectively. Moderate to good yields could be obtained depending on the method used to generate the nitrile oxides. The intermediate isoxazolines could not be isolated.     

One-step,stereoselective synthesis of octahydrochromanes via the Prins reaction and their cannabinoid activities

Novel, functionalized octahydrochromane derivatives were synthesized in a single step via the Prins reaction. Enantiomerically pure (+)-isopulegol was reacted with benzaldehyde to stereoselectively yield the corresponding octahydro-2H-chromen-4-ol derivative containing five stereocenters. A total of 10 compounds were synthesized by altering the enantiomer of isopulegol and the substituted benzaldehyde, and the resulting enantiopure octahydrochromanes were screened in vitro against the cannabinoid receptor isoforms CB1 and CB2. Compounds containing an olefin at the C4 position [(+)-3c, (?)-3c, (?)-7c, (?)-9c and (?)-11c] of the octahydrochromane scaffold were found to exhibit reasonable displacement of [³H] CP55,940 from the CB receptors, whereas the corresponding hydroxy analogs [(+)-3a, (+)-3b, (?)-3a, (?)-3b and (+)-5a] had very little or no effect.