• Bulletin of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.83-86
• /
• 2007

We determine the largest integer i such that $0 and the coefficient of $t^{i}$ is odd in the polynomial $(1+t+t^{2}+{\cdots}+t^{n})^{n+1}$. We apply this to prove that the co-index of the tangent bundle over $FP^{n}$ is stable if $2^{r}{\leq}n<2^{r}+\frac{1}{3}(2^{r}-2)$ for some integer r.

• Honam Mathematical Journal
• /
• v.15 no.1
• /
• pp.75-85
• /
• 1993

For A and B, $-1{\leq}B<A{\leq}1$, let P[A, B] be the class of functions p analytic in the unit disk E with P(0) = 1 and subordinate to $\frac{1+Az}{1+Bz}$. We introduce the class $T_{\alpha}[A,B]$ of functions $f:f(z)=z+\sum\limits_{n=2}^{{\infty}}a_nz^n$ which are analytic in E and for $z{\in}E$, ${\alpha}{\geq}0$, $[(1-{\alpha}){\frac{f(z)}{z}}+{\alpha}f^{\prime}(z)]{\in}P[A,B]$. It is shown that, for ${\alpha}{\geq}1$, $T_{\alpha}[A,B]$ consists entirely of univalent functions and the radius of univalence for $f{\in}T_{\alpha}[A,B]$, $0<{\alpha}<1$ is obtained. Coefficient bounds and some other properties of this class are studied. Some radii problems are also solved.

• Journal of the Korean Chemical Society
• /
• v.17 no.4
• /
• pp.298-305
• /
• 1973

Following results were obtained for the mechanical degradation of polystyrene (for polystyrene itself and when blended with rubber) by roll mastication. 1) The rate of mechanical degradation for polystyrene itself can be represented by the second-order rate equation proposed by Goto. $-\frac{dP_t}{dt} = k_s(P_t-P_{\infty})^2$ Where Pt is the degree of polymerization of the degraded polymer at t minutes and $P{\infty}$ is the final degree of polymerization. 2) The mechanical degradation of polystyrene component in the polystyrene-rubber (SBR, BR) blend system occurred similarly as that of polystyrene itself. 3) Under the experimental conditions the mechanical degradation rate of the polystyrene component of the polystyrene-rubber, (SBR, BR) blend system followed approximately the same second-order equation as that for polystyrene itself.

• Bulletin of the Korean Chemical Society
• /
• v.35 no.1
• /
• pp.111-116
• /
• 2014

The kinetics and mechanism of oxidation of sulfanilic acid by N-chloro-p-toluene sulfonamide (chloramine-T) have been studied in acid medium. The species of chloramine-T were analysed on the basis of experimental observations and predominantly reactive species was taken into account for proposition of most plausible reaction mechanism. The derived rate law (1) conforms to such a mechanism. $$-\frac{d[CAT]}{dt}=\frac{kK_1[RNHCl][SA]}{K_1+[H^+]}$$ (1) All kinetic parameters were evaluated. Activation parameters such as energy and entropy of activation were calculated to be $(61.67{\pm}0.47)kJmol^{-1}$ and $(-62.71{\pm}2.48)kJmol^{-1}$ respectively employing Eyring equation.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.2
• /
• pp.547-553
• /
• 2014

The Boubaker polynomials arose from the discretization of the equations of heat transfer in pyrolysis starting from an assumed solution of the form $$\frac{1}{N}e^{\frac{A}{H/z+1}}\sum_{k=0}^{\infty}{\xi}_kJ_k(t),$$ where $J_k$ is the k-th order Bessel function of the first kind. In this paper, we investigate the distribution of zeros of the Boubaker polynomials.

• Communications of the Korean Mathematical Society
• /
• v.32 no.1
• /
• pp.193-200
• /
• 2017

Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx-uy = 1, p = x + y and q = u + v. Using this property, we show that$${\sum\limits_{1{\leq}i{\leq}x,1{\leq}j{\leq}v}}\;{t^{(i-1)q+(j-1)p}\;-\;{\sum\limits_{1{\leq}k{\leq}y,1{\leq}l{\leq}u}}\;t^{1+(k-1)q+(l-1)p}$$ is the Alexander polynomial ${\Delta}_{p,q}(t)$ of a torus knot t(p, q). Hence the number $N_{p,q}$ of non-zero terms of ${\Delta}_{p,q}(t)$ is equal to vx + uy = 2vx - 1. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8); Let q be a positive integer> 1 and let k be a positive integer. Then we have $$\begin{array}{rccl}(1)&N_{kq}+1,q&=&2k(q-1)+1\\(2)&N_{kq}+q-1,q&=&2(k+1)(q-1)-1\\(3)&N_{kq}+2,q&=&{\frac{1}{2}}k(q^2-1)+q\\(4)&N_{kq}+q-2,q&=&{\frac{1}{2}}(k+1)(q^2-1)-q\end{array}$$ where we further assume q is odd in formula (3) and (4). Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q - 2, q) in [5] agree with $N_{kq+2,q}$ and $N_{kq+q-2,q}$ respectively.

• Journal of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1385-1405
• /
• 2021

Let A be a positive bounded linear operator acting on a complex Hilbert space (𝓗, ⟨·,·⟩). Let ω_A(T) and ║T║_A denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space (𝓗, ⟨·,·⟩_A), respectively, where ⟨x, y⟩_A := ⟨Ax, y⟩ for all x, y ∈ 𝓗. In this paper, we show with different techniques from that used by Kittaneh in [24] that $$\frac{1}{4}{\parallel}T^{{\sharp}_A}T+TT^{{\sharp}_A}{\parallel}_A{\leq}{\omega}^2_A(T){\leq}\frac{1}{2}{\parallel}T^{{\sharp}_A}T+TT^{{\sharp}_A}{\parallel}_A.$$ Here T^#_A denotes a distinguished A-adjoint operator of T. Moreover, a considerable improvement of the above inequalities is proved. This allows us to compute the 𝔸-numerical radius of the operator matrix $\(\array{I&T\\0&-I}\)$ where 𝔸 = diag(A, A). In addition, several A-numerical radius inequalities for semi-Hilbert space operators are also established.

• Kyungpook Mathematical Journal
• /
• v.47 no.3
• /
• pp.335-340
• /
• 2007

Let M be a hyperbolic 3-manifold such that ${\partial}M$ has at least two boundary tori ${\partial}_oM$ and ${\partial}_1M$. Suppose that M contains an essential orientable surface P of genus $g$ with one outer boundary component ${\partial}_oP$, lying in ${\partial}_oM$ and having slope ${\lambda}$ in ${\partial}_oM$, and $p$ inner boundary components ${\partial}_iP$, $i=1$, ${\cdots}$, $p$, each having slope ${\alpha}$ in ${\partial}_1M$. Let ${\beta}$ be a slope in ${\partial}_1M$ and suppose that $M({\beta})$ is toroidal. Let $\hat{T}$ be a minimal essential torus in $M({\beta})$, which means that $\hat{T}$ is pierced a minimal number of times by the core of the ${\beta}$-Dehn filling, among all essential tori in $M({\beta})$. Let $T=\hat{T}{\cap}M$ and denote by $t$ the number of components of ${\partial}T$. In this paper we prove: (i) if $t{\geq}3$, then ${\Delta}({\alpha},{\beta}){\leq}6+\frac{10g-5}{p}$, (ii) If $t=2$, then ${\Delta}({\alpha},{\beta}){\leq}13+\frac{24g-12}{p}$, (iii) If $t=1$, then ${\Delta}({\alpha},{\beta}){\leq}1$.

• Journal of the Chungcheong Mathematical Society
• /
• v.31 no.2
• /
• pp.231-237
• /
• 2018

Hardy's inequality is refined non-trivially as the form $${\int_{0}^{{\infty}}}\{{\frac{1}{x}}{\int_{0}^{x}}f(t)dt\}^pdx{\leq}Q_f{\times}({\frac{p}{p-1}})^p{\int_{0}^{x}}f^p(x)dx$$ for some $Q_f:0{\leq}Q_f{\leq}1$. $P{\acute{o}}lya$-Knopp's inequality is also refined by the similar form.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.9 no.2
• /
• pp.91-97
• /
• 2005

If ${\beta}\;>\;1$, then every non-negative number x has a ${\beta}-expansion$, i.e., $$x\;=\;{\epsilon}_0(x)\;+\;{\frac{\epsilon_1(x)}{\beta}}\;+\;{\frac{\epsilon_2(x)}{\beta}}\;+\;{\cdots}$$ where ${\epsilon}_0(x)\;=\;[x],\;{\epsilon}_1(x)\;=\;[\beta(x)],\;{\epsilon}_2(x)\;=\;[\beta(({\beta}x))]$, and so on ([x] denotes the integral part and (x) the fractional part of the real number x). Let T be a transformation on [0,1) defined by $x\;{\rightarrow}\;({\beta}x)$. It is well known that the relative frequency of $k\;{\in}\;\{0,\;1,\;{\cdots},\;[\beta]\}$ in ${\beta}-expansion$ of x is described by the T-invariant absolutely continuous measure ${\mu}_{\beta}$. In this paper, we show the mod M normality of the sequence $\{{\in}_n(x)\}$.