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Why quintic polynomial equations are not solvable in radicals

Křížek, MichalSomer, Lawrence — 2015

Application of Mathematics 2015

We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations + , - , · , : , and · n . Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

On simplicial red refinement in three and higher dimensions

Korotov, SergeyKřížek, Michal — 2013

Applications of Mathematics 2013

We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original...

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