• The Pure and Applied Mathematics
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• v.19 no.1
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• pp.37-42
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• 2012

We give an elementary proof of Descartes' theorem for polyhedra. Since Descartes' theorem is equivalent to Euler's theorem for polyhedra, this also gives an elementary proof of Euler's theorem.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.495-498
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• 2011

We give a converse of the well-known Euler's theorem for convex polyhedra.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.151-156
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• 2011

In this research paper, motivated by the extension of the Euler type I transformation obtained very recently by Rathie and Paris, the authors aim at presenting the extensions of Euler type II transformation. In addition to this, a natural extension of the classical Saalsch$\ddot{u}$tz's summation theorem for the series $_3F_2$ has been investigated. Two interesting applications of the newly obtained extension of classical Saalsch$\ddot{u}$tz's summation theorem are given.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.17 no.1
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• pp.59-68
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• 2017

The flower and woman are beautiful but Euler's theorem and the symmetry are the best. Shannon applied his theorem to information and communication based on Euler's theorem. His theorem is the root of wireless communication and information theory and the principle of today smart phone. Their meeting point is $e^{-SNR}$ of MIMO(multiple input and multiple output) multiple antenna diversity. In this paper, Euler, who discovered the most beautiful formula($e^{{\pi}i}+1=0$) in the world, briefly guided Shannon's formula ($C=Blog_2(1+{\frac{S}{N}})$) to discover the origin of wireless communication and information communication, and these two masters prove a meeting at the Shannon limit, It reveals something what this secret. And we find that it is symmetry and element-wise inverse are the hidden secret in algebraic coding theory and triangular function.

• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Journal of Institute of Control, Robotics and Systems
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• v.20 no.4
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• pp.408-413
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• 2014

In this paper, we propose a fault diagnosis method based on Park's Vector Approach using the Euler's theorem. If we interpreted it as Euler's theorem, it is possible to easily find the phase angle difference between the healthy condition and the fault condition. And, we analyzed the variation of the phase angle and performed the diagnostic method of the induction motor using feature vectors that were obtained by using a Fourier transform. The analysis of time and speed variation of the motor was performed and, as a result, we could find more soft variations than rough variations. In particular, the analysis of the distortion through each phase shows that two-turn and four-turn shorted motors are linearly separable. In this experiment, we know that the maximum breakdown threshold value for determining steady-state fault detection is 49.0788. Simulation and experimental results show the more detectable than conventional method.

• Advances in nano research
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• v.12 no.2
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• pp.139-150
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• 2022

This paper presents sets of explicit analytical equations that compute the static displacements of nanobeams by adopting the nonlocal elasticity theory of Eringen within the framework of Euler Bernoulli and Timoshenko beam theories. Castigliano's theorem is applied to an equivalent Virtual Local Beam (VLB) made up of linear elastic material to compute the displacements. The first derivative of the complementary energy of the VLB with respect to a virtual point load provides displacements. The displacements of the VLB are assumed equal to those of the nonlocal beam if nonlocal effects are superposed as additional stress resultants on the VLB. The illustrative equations of displacements are relevant to a few types of loadings combined with a few common boundary conditions. Several equations of displacements, thus derived, matched precisely in similar cases with the equations obtained by other analytical methods found in the literature. Furthermore, magnitudes of maximum displacements are also in excellent agreement with those computed by other numerical methods. These validated the superposition of nonlocal effects on the VLB and the accuracy of the derived equations.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.509-512
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• 2007

We aim at deriving Gauss's second summation theorem for the series $_2F_1(1/2)$ by using Euler's integral representation for $_2F_1$. It seems that this method of proof has not been tried.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.701-706
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• 2013

Summation theorems for hypergeometric series $_2F_1$ and generalized hypergeometric series $_pF_q$ play important roles in themselves and their diverse applications. Some summation theorems for $_2F_1$ and $_pF_q$ have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series $_3F_2$(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.551-560
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• 2009

A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated $Poincar{\acute{e}}$ Recurrence Theorem is pointed out.