Local Isometric immersions of pseudo-spherical surfaces and evolution equations

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in ${\bf E}^{3}$ from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in ${\bf E}^{3}$. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper [11], we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for $u(x,t)$ of order $k\geq 3$ describing pseudo-spherical surfaces. We show that whenever an isometric immersion in ${\bf E}^3$ exists, depending on a jet of finite order of $u$, then the coefficients of the second fundamental forms are functions of the independent variables $x$ and $t$ only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and Applications, pp.N

BIODIESEL PRODUCTION FROM MICROWAVE IRRADIATED REACTOR USING HOMOGENEOUS AND HETEROGENEOUS CATALYSIS

Biodiesel was successful produced in a microwave irradiation reactor using homogeneous and heterogeneous catalysis. The biodiesel was production by the trasesterification reaction of soybean oil using metanol. Sodium methylate (30% solution in metanol) was used for the homogeneous catalyst and the heterogeneous catalyst was developed using wasted eggshells. The eggshells were calcined and tested pure and doped with potassium hydroxide in 10, 30 and 50% of weight. The power and temperature of the microwave were kept constant in every reaction being 800W and 200º Celsius, respectively. The reaction time was significantly reduced using microwave compared to the conventional process. In only one minute of reaction, the methyl ester (FAME) conversion obtained was 98.9% with the homogeneous catalyst and within 15 minutes, the heterogeneous catalysis accomplished 100%. For heterogeneous catalyst, the best results were acquired when the doped catalyst contained 50% of KOH. The results indicated that the eggshells treated with KOH has a great potential to be used for microwave-assisted transesterification reactions of oils with mild operations conditions: molar ratio oil/alcochol 1:6 and just 5% of catalyst. In addition, the heterogenous catalyst was recovered and reused in other reactions with a relatively satisfying results. The physico-chemical properties of the catalysts were characterized by X-ray diffraction and thermogravimectric analysis

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Efect of bovine milk addition to buffalo's milk on different characteristics of artisanal Marajó "cream cheese" type.

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