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Power Series Solutions and Special Functions
Published in Steven G. Krantz, Differential Equations, 2022
Abel's genius was recognized when he was still young. In spite of grinding poverty, he managed to attend the University of Oslo. When only 21 years old, Abel produced a proof that the fifth degree polynomial cannot be solved by an elementary formula (involving only arithmetic operations and radicals). Recall that the quadratic equation can be solved by the quadratic formula, and cubic and quartic equations can be solved by similar but more complicated formulas. This was an age-old problem, and Abel's solution was a personal triumph. He published the proof in a small pamphlet at his own expense. This was typical of the poor luck and lack of recognition that plagued Abel's short life.
Power Series Solutions and Special Functions
Published in Steven G. Krantz, Differential Equations, 2015
Abel’s genius was recognized when he was still young. In spite of grinding poverty, he managed to attend the University of Oslo. When only 21 years old, Abel produced a proof that the fifth degree polynomial cannot be solved by an elementary formula (involving only arithmetic operations and radicals). Recall that the quadratic equation can be solved by the quadratic formula, and cubic and quartic equations can be solved by similar but more complicated formulas. This was an age-old problem, and Abel’s solution was a personal triumph. He published the proof in a small pamphlet at his own expense. This was typical of the poor luck and lack of recognition that plagued Abel’s short life.
A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages
Published in Mechanics Based Design of Structures and Machines, 2020
Juan Ignacio Valderrama-Rodríguez, José M. Rico, J. Jesús Cervantes-Sánchez
About 15 years ago Foster and Pennock (2003) presented a graphical technique that was able to find all the instantaneous centers of indeterminate linkages and improved substantially the approach proposed by Klein. A few years later, Di Gregorio (2008a) introduced an analytical technique that generated a system of equations involving both, closure equations of the planar linkages and locations of the indeterminate instantaneous rotation centers, so that He was able to find the location of the indeterminate rotation centers. In 2009, Kung and Wang (2009) employed graph theory to obtain a graph associated with the instantaneous centers of a linkage. This graph allowed Kung and Wang to formulate a system of iterative equations, whose unknowns are the coordinates of the indeterminate instantaneous rotation centers. The solution of the problem is reduced to solving a quadratic equation and then a quartic equation.