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We study the bifurcation of Riemann solutions due to parameter change that alters the type of an umbilic point existing in state space. Solutions with data near generic umbilic points are primarily determined by the local quadratic expansion of flux functions. We observe that near an umbilic point, the bifurcation of the solution is essentially local and its behavior depends solely on the cubic expansion of the flux functions. These phenomena are illustrated for immiscible three-phase flow in porous media, which looses strict hyperbolicity at an isolated point in the interior of the oil-water-gas saturation triangle.Mathematics Subject Classification. 35B32 · 35L65 · 35L67 · 35L81 · 35Q86.

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Having the property to modify only the geometry of a polygonal structure, preserving its physical magnitudes, the Conformal Mapping is an exceptional tool to solve electromagnetism problems with known boundary conditions. This work aims to introduce a new developed mathematical operator, based on polynomial extrapolation. This operator has the capacity to accelerate an optimization method applied in conformal mappings, to determinate the equipotential lines, the field lines, the capacitance, and the permeance of some polygonal geometry electrical devices with an inner dielectric of permittivityε. The results obtained in this work are compared with other simulations performed by the software of finite elements method, Flux2D.

This paper presents a method for calculating the parameters of the Schwarz-Christoffel inverse transformation using the Genetic Algorithm. It is shown that for problems involving a polygonal geometry with axial symmetry, the number of estimated parameters is reduced by half. Simulations are performed with polygonal figures with several different geometries and with up to 27 sides. A four-sided polygon is inversely mapped in order to verify the accuracy of the method. The method is also applied for calculating the induction lines and the flux lines in the region of a doubly slotted airgap of an electrical machine. Finally, based on the length of the midline of induction in the airgap, a new and very simple formula to calculate the Carter factor of an electrical machine is presented, which takes into account the actual geometry of the airgap.