Periodic Solutions of the Duffing Differential Equation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.     

Periodic Solutions of a Class of Duffing Differential Equations

In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form $x^{\prime\prime}+ c x^{\prime}+ a(t) x +b(t) x^3 = h(t)$, where $c$ is a real parameter, $a(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions. Our results are proved using three different results on the averaging theory of first order.     

Effects of modified starch and fat on the rheological characteristics of newly formulated processed cheese: Use of experimental design method

This study aims to investigate the impact of the incorporation of starch (modified and non-modified) on the texture of the processed cheese in order to appreciate a new recipe on one hand and to try to reduce fat content (FC) on the other. The modeling of the experimental results has been performed. An experimental design was used (the variable factors are the amount of starch and FC). The dry extract (DE) of cheese and parameters derived from the modeling of rheological results are the selected responses. The cheese with a concentration of modified starch that is greater than 2% has a high viscosity at rest, while the cheese with non-modified starch has a lower viscosity. When the FC exceeds 15%, the viscosity increases as the concentration of the starch exceeds 2%. Below this concentration, the FC effect is negligible except when the starch concentration is 1%, in this case the viscosity decreases.     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

Investigations of the Effects of Xanthan and Sodium Caseinate on the Formation and Stability of an Oil-in-Water Emulsion Stabilized by a Nonionic Surfactant Using a Response Surface Method

Effects of xanthan and sodium caseinate concentrations on the rheological properties and stability of an oil-in water emulsion stabilized by a nonionic surfactant (Tween 20) were investigated. In order to evaluate the influence of component concentrations on the emulsion properties, a response surface method was applied. It was shown by polynomial simulation and modelization of responses of storage modulus and creaming index that in absence of Tween 20, the oily droplets were flocculated by bridging of sodium caseinate macromolecules, when those are insufficient to cover the entire interface. Additionally, it was noted that xanthan reinforces the polymeric network in the aqueous phase when the caseinate is not moved toward the phase. On the other hand, in presence of nonadsorbed caseinate, the polysaccharide develops segregative interactions by thermodynamic incompatibility. These nonadsorbed macromolecules can also flocculate the fatty globules by depletion.     

A new efficient short-step projective interior point method for linear programming

In this paper, we are interested in the performance of Karmarkar’s projective algorithm for linear programming. We propose a new displacement step to accelerate and improve the convergence of this algorithm. This purpose is confirmed by numerical experimentations showing the efficiency and the robustness of the obtained algorithm over Schrijver’s one for small problem dimensions.