• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.937-956
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• 2018

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm [26] to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

• Korean Journal of Mathematics
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• v.20 no.4
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• pp.441-449
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• 2012

The quasi-(${\theta}$, s)-continuity is a weakened form of the weak (${\theta}$, s)-continuity and equivalent to the weak quasi-continuity. The basic properties of those functions are investigated in concern with the other weakened continuous functions. It turns out that the open property of a function and the extremall disconnectedness of the spaces are crucial tools for the survey of these functions.

• Korean Journal of Mathematics
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• v.24 no.2
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• pp.297-305
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• 2016

In this paper, by using fixed point theorem, we obtain the stability of the following functional equations $$f(pr,qs)+g(ps,qr)={\theta}(p,q,r,s)f(p,q)h(r,s)\\f(pr,qs)+g(ps,qr)={\theta}(p,q,r,s)g(p,q)h(r,s)$$, where G is a commutative semigroup, ${\theta}:G^4{\rightarrow}{\mathbb{R}}_k$ a function and f, g, h are functionals on $G^2$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.269-278
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• 2008

Let ${\partial}D$ be the boundary of the open unit disk D in the complex plane and $L^p({\partial}D)$ the class of all complex, Lebesgue measurable function f for which $\{\frac{1}{2\pi}{\int}_{-\pi}^{\pi}{\mid}f(\theta){\mid}^pd\theta\}^{1/p}<{\infty}$. Let P be the orthogonal projection from $L^p({\partial}D)$ onto ${\cap}_{n<0}$ ker $a_n$. For $f{\in}L^1({\partial}D)$, ${\hat{f}}(z)=\frac{1}{2\pi}{\int}_{-\pi}^{\pi}P_r(t-\theta)f(\theta)d{\theta}$ is the harmonic extension of f. Let ${\hat{P}}$ be the composition of P with the harmonic extension. In this paper, we will show that if $1, then ${\hat{P}}:L^p({\partial}D){\rightarrow}H^p(D)$ is bounded. In particular, we will show that ${\hat{P}}$ is unbounded on $L^{\infty}({\partial}D)$.

• Journal of the Korean Magnetics Society
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• v.25 no.4
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• pp.129-137
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• 2015

The Cu(radius ra = $95{\mu}m$)/$Ni_{80}Fe_{20}$(outer radius $r_b$ = $120{\mu}m$) core/shell composite wire is fabricated by electrodeposition. The two diagonal components of impedance tensor for the Cu/$Ni_{80}Fe_{20}$ core/shell composite wire in cylindrical coordinates, $Z_{zz}$ and $Z_{{\theta}{\theta}}$, are measured as a function of frequency in 10 kHz~10 MHz and external static magnetic field in 0 Oe~200 Oe. The equations expressing the diagonal $Z_{zz}$ and $Z_{{\theta}{\theta}}$ in terms of diagonal components of complex permeability tensor, ${\mu}^*_{zz}$ and ${\mu}^*_{{\theta}{\theta}}$, are derived from Maxwell's equations. The real and imaginary parts of ${\mu}^*_{zz}$(f) and ${\mu}^*_{{\theta}{\theta}}$(f) spectra are extracted from the measured $Z_{zz}$(f) and $Z_{{\theta}{\theta}}$(f) spectra, respectively. It is presened that the extraction of ${\mu}^*_{zz}$(f) and ${\mu}^*_{{\theta}{\theta}}$(f) spectra from the diagonal impedance spectra can be a versatile tool to investigate dymanic magnetization process in the core/shell composite wire.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.67-77
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• 2011

We find some Eisenstein series related to modulus 4 using a theta function identity of McCullough and Shen and residue theorem for elliptic functions.

• 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.

• 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

• Korean Journal of Materials Research
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• v.9 no.2
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• pp.111-116
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• 1999

The microstructure of vitreous bonded abrasives, which are used as the essential materials in the precise grinding, was investigated theoretically using two particle model. In this paper, a general equation applicable for a case in which there is a gap between abrasive grits is suggested. As a result, it was known that both the volume ratio of grit to glassy bond(V\ulcorner/V\ulcorner) and porosity(V\ulcorner) are the function of $\alpha$(the ratio of distance between grit to diameter of grit) and $\theta$(the angle from the center of pore to that of grit). Because the value $\alpha$ and $\theta$ can be get easily by using these suggested equations, the microstructure could be explained quantitatively. Also the raised error with the increasing amount of bond was modified by the simple assumption. As a result, in that case, both V\ulcorner/V\ulcorner and V\ulcorner were known to be the function of $\alpha$ and $\theta$(the ratio diameter of pore to that of grit).

• Kyungpook Mathematical Journal
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• v.51 no.4
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• pp.385-393
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• 2011

The aim of this paper is to introduce and study the sequence spaces [w, ${\theta}$, F, p, q]$_{\infty}({\Delta}_{\upsilon}^m)$, [w, ${\theta}$, F, p, q]$_1({\Delta}_{\upsilon}^m)$ and [w, ${\theta}$, F, p, q]$_0({\Delta}_{\upsilon}^m)$, which arise from the notions of generalized difference sequence space, lacunary convergence, invariant mean and a sequence of Moduli $F=(f_k)$. We establish some inclusion relations between these spaces under some conditions.