• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.593-604
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• 2018

Recently, a new version of expansiveness which is closely attached to some certain weak version of hyperbolicity was given for $C^1$ vector fields as following: a $C^1$ vector field X will be called rescaling expansive on a compact invariant set ${\Lambda}$ of X if for any ${\epsilon}$ > 0 there is ${\delta}$ > 0 such that, for any $x,\;y{\in}{\Lambda}$ and any time reparametrization ${\theta}:{\mathbb{R}}{\rightarrow}{\mathbb{R}}$, if $d({\varphi}_t(x),\,{\varphi}_{{\theta}(t)}(y)){\leq}{\delta}{\parallel}X({\varphi}_t(x)){\parallel}$ for all $t{\in}{\mathbb{R}}$, then ${\varphi}_{{\theta}(t)}(y){\in}{\varphi}_{(-{\epsilon},{\epsilon})}({\varphi}_t(x))$ for all $t{\in}{\mathbb{R}}$. In this paper, some equivalent definitions for rescaled expansiveness are given.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.987-1028
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• 2014

In a recent systematic study, C. Sandon and F. Zanello offered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for multipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ramanujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.761-766
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• 2013

We derive several modular equations and present their proofs based on concise algebraic computations. In addition, we establish explicit relations and formulas for some parameterizations for the theta functions ${\varphi}$ and ${\psi}$ and show some applications of the modular equations to evaluations of the cubic continued fraction and the theta function ${\psi}$.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.983-996
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• 2015

In the paper A new parameter for Ramanujan's theta-functions and explicit values, Arab J. Math. Sc., 18 (2012), 105-119, Saikia studied the parameter $A_{k,n}$ involving Ramanujan's theta-functions ${\phi}(q)$ and ${\psi}(q)$ for any positive real numbers k and n and applied it to find explicit values of ${\psi}(q)$. As more application to the parameter $A_{k,n}$, in this paper we prove a new general theorem for explicit evaluation of Ramanujan-$G{\ddot{o}}llnitz$-Gordon continued fraction K(q) in terms of the parameter $A_{k,n}$ and give examples. We also find some new explicit values of the parameter $A_{k,n}$ and offer reciprocity theorems for the continued fraction K(q).

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.389-394
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• 1997

Consider a tetrhedron is composed of six dihedral angles .phi.(i=1,2..., 6), and a vertex of a tetrahedron is also three dihedral angles. It will assume that a vertex A, for an example, is composed of there angles definded such as .alpha..betha. and .gamma. !. then there is a corresponding angle can be given as .phi1.,.phi2.,.phi3.. Here, in order to differentiate between a conventional triangle and dihedral angle, if a dihedral angle degined in this paper is symbolized as .phi..LAMBDA.,the value of cos.theta.of .phi./sab a/, in a trigonometric function rule,can be defined to tecos.phi..LAMBD/sab A/., and it is defined as a tetradedral cosine .phi. or simply called a tecos.phi.. Moreover, in a simillar method, the dihedral angle of tetrahedron .phi..LAMBDA. is given as : value of sin .theta. can defind a tetra-sin.phi..LAMBDA., and value of tan .theta. of .phi..LAMBDA. is a tetra-tan .phi..LAMBDA. By induction it can derive that a tetrahedral geometry on the basis of suggesting a geometric tetrahedron

• Proceedings of the KIEE Conference
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• 1989.07a
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• pp.633-636
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• 1989

The simulation results of the third harmonic conversion with 1.064 um Nd :YAG laser using polarization mismatch method are reported. The third harmonic conversion efficiency was over 65% for the Gaussian incident pulse of $300{\sim}400MW/cm^2$ peak intensity, and over 80% for $3{\sim}4GW/cm^2$ peak intensity pulse. The dependence of the third harmonic conversion efficiency' on the variation of incident polarization angle ${\theta}_p$ and angular discrepancy ${\Delta}{\theta}$ of phase matched angle ${\theta}_m$ is discussed.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 2000.05b
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• pp.237-241
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• 2000

The effects of Si addition on the precipitation processes of in Al-Cu-Si alloy films were studied by the transmission electron microscopy. Deposition of an Al-1.5Cu-1.5Si (wt. %) film at $305^{\circ}C$ resulted in formation of fine, uniformly distributed spherical $\theta$-phase particles due to the precipitation of the $\theta$ and Si phase particles during deposition. For deposition at $435^{\circ}C$, fine $\theta$-phase particles precipitated during wafer cooldown, while coarse Si nodules formed at the sublayer interface during deposition. The film susceptibility to corrosion is discussed in relation to the film microstructure and deposition temperature.

• Publications of The Korean Astronomical Society
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• v.7 no.1
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• pp.25-30
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• 1992

Recent redshift surveys suggest that most galaxies may be distributed on the surfaces of bubbles surrounding large voids. To investigate the quantitative consistency of this qualitative picture of large-scale structure, we study analytically the clustering properties of galaxies in a universe filled with spherical shells. In this paper, we report the results of the calculations for the spatial and angular two-point correlation functions of galaxies. With ${\sim}20%$ of galaxies in clusters and a power law distribution of shell sizes, $n_{sh}(R){\sim}R^{-{\alpha}}$, ${\alpha}\;{\simeq}\;4$, the observed slope and amplitude of the spatial two-point correlation function ${\xi}_{gg}(r)$ can be reproduced. (It has been shown that the same model parameters reproduce the enhanced cluster two-point correlation function, ${\xi}_{cc}(r)$). The corresponding angular two-point correlation function $w({\theta})$ is calculated using the relativistic form of Limber's equation and the Schecter-type luminosity function. The calculated w(${\theta}$) agrees with the observed one quite well on small separations (${\theta}{\lesssim}2deg$).

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.10a
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• pp.539-542
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• 1997

We present a micro-positioning stage that has minimized geometrical error and can drive in the 4-axis. This stage divided into two parts: $Z\theta_x$ $\theta_y$, motion stage and$\theta_z$ motion stage. These stages are constructed in flexure hinges, piezoelectric actuators and displacement scnsors. The dynamic model for each stage is obtained and their FE (finite element) models are made. Using the Lagrange's equation, the motion of equation is found. Through the parametric analysis and FE analysis, sensitiv~ty of the design parameters is executed. Finally, fundamental frequencies, maximum stress, and displacement sensitivity for each stage are obtained. We expect that this micro-positioning stage be a useful micro-alignment device for various applications.

• East Asian mathematical journal
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• v.33 no.1
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• pp.83-102
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• 2017

The main purpose of this paper is to introduce a pair new class of primal and dual problem associated with an operator. We prove the sufficient optimality theorem, weak duality theorem and strong duality theorem for these problems. The equivalence between the generalized optimization problems and the generalized variational inequality problems is studied in ordered topological vector space modeled in Hilbert spaces. We introduce the concept of partial differential associated (PDA)-operator, PDA-vector function and PDA-antisymmetric function to show the existence of a new class of function called, ($T_{\eta};{\xi}_{\theta}$)-invex functions. We discuss first and second kind of ($T_{\eta};{\xi}_{\theta}$)-invex functions and establish their existence theorems in ordered topological vector spaces.