• Journal for History of Mathematics
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• v.27 no.3
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• pp.165-182
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• 2014

After algebraic expressions for the roots of 3rd and 4th degree polynomial equations were given in the mid 16th century, seeking such a formula for the 5th and greater degree equations had been one main problem for algebraists for almost 200 years. Lagrange made careful and thorough investigation of various solving methods for equations with the purpose of finding a principle which could be applicable to general equations. In the process of doing this, he found a relation between the roots of the original equation and its auxiliary equation using permutations of the roots. Lagrange's ingenious idea of using permutations of roots of the original equation is regarded as the key factor of the Abel's proof of unsolvability by radicals of general 5th degree equations and of Galois' theory as well. This paper intends to examine Lagrange's contribution in the theory of polynomial equations, providing a detailed analysis of various solving methods of Lagrange and others before him.

• Journal for History of Mathematics
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• v.34 no.2
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• pp.39-54
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• 2021

One of the topics in school mathematics is the relation between the roots and the coefficients of equations. It deals with the way to find the roots out of the coefficients of equations. One of the concepts derived from the theory of equations is symmetric functions. Symmetry is a kind of functionality of human cognition. It is, in mathematics, geometrically related to the congruence and the similarity of figures, and algebraically a kind of invariants. We look at stories on the appearance of symmetric functions through the development of the theory of equations.

• Proceedings of the KSME Conference
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• 2007.05b
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• pp.3300-3305
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• 2007

Ignition delay of second injection of HSDI diesel engine was usually much shorter than that of first injection. It is due to the interaction between radicals generated during the combustion process, and mixed gas of second injection. In this paper, To analyze combustion phenomena of multiple injection mode in HSDI diesel engine effectively, two-dimensional flamelet combustion model was modified. To reduce calculation time, two-dimensional flamelet equations were only applied near stoichiometric region. If this region was ignited, species and temperature of other region were changed to the steady-state solutions of one dimensional flamelet equations. By this method calculation time for solving flamelet equations was reduced to 20 percents, thought the results were almost same. Modified flamelet combustion model was coupled to commercial CFD code interactively using user subroutine.

• Proceedings of the Korean Vacuum Society Conference
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• 2014.02a
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• pp.221-221
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• 2014

The present work proposes an improved numerical simulator for design and modification of large area capacitively coupled plasma (CCP) processing chamber. CCP, as notoriously well-known, demands the tremendously huge computational cost for carrying out transient analyses in realistic multi-dimensional models, because electron dissociations take place in a much smaller time scale (${\Delta}t{\approx}10-8{\sim}10-10$) than time scale of those happened between neutrals (${\Delta}t{\approx}10-1{\sim}10-3$), due to the rf drive frequencies of external electric field. And also, for spatial discretization of electron flux (Je), exponential scheme such as Scharfetter-Gummel method needs to be used in order to alleviate the numerical stiffness and resolve exponential change of spatial distribution of electron temperature (Te) and electron number density (Ne) in the vicinity of electrodes. Due to such computational intractability, it is prohibited to simulate CCP deposition in a three-dimension within acceptable calculation runtimes (<24 h). Under the situation where process conditions require thickness non-uniformity below 5%, however, detailed flow features of reactive gases induced from three-dimensional geometric effects such as gas distribution through the perforated plates (showerhead) should be considered. Without considering plasma chemistry, we therefore simulated flow, temperature and species fields in three-dimensional geometry first, and then, based on that data, boundary conditions of two-dimensional plasma discharge model are set. In the particular case of SiH4-NH3-N2-He CCP discharge to produce deposition of SiNxHy thin film, a cylindrical showerhead electrode reactor was studied by numerical modeling of mass, momentum and energy transports for charged particles in an axi-symmetric geometry. By solving transport equations of electron and radicals simultaneously, we observed that the way how source gases are consumed in the non-isothermal flow field and such consequences on active species production were outlined as playing the leading parts in the processes. As an example of application of the model for the prediction of the deposited thickness uniformity in a 300 mm wafer plasma processing chamber, the results were compared with the experimentally measured deposition profiles along the radius of the wafer varying inter-electrode gap. The simulation results were in good agreement with experimental data.