• Journal of the military operations research society of Korea
• /
• v.10 no.2
• /
• pp.75-83
• /
• 1984

Consider an m machine flow shop with blocking. The processing time of job j,j=1,..., n on each one of the m machines is equal to the same random variable $X_j$ and is distributed according to $F_i$. We assume that the processing times are stochastically ordered, i.e., $F_{1_{-st}}{<}F_{2_{st}}{<}cdots_{-st}{<}F_n$. We show that the sequence 1,3,5,...,n-1,n,n-2,...,6,4,2 when n is even and sequence 1,3,5,...,n-2,n,n-1 ... 6,4,2 when n is odd minimizes the expected makespan and that the sequence 1,...,n minimizes the expected flow time.

• Journal of applied mathematics & informatics
• /
• v.24 no.1_2
• /
• pp.179-190
• /
• 2007

In this paper, we consider the general variational inequality GVI(F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type method $u_{n+l}=(1-{\alpha}+{\theta}_{n+1})Tu_n+{\alpha}u_n-{\theta}_{n+1g}(Tu_n)-{\lambda}_{n+1}{\mu}F(Tu_n),\;n{\geq}0$. for solving the general variational inequalities. The sequence $\{x_n}\$ is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.2
• /
• pp.273-286
• /
• 2011

Using the fixed point method, we prove the Hyers-Ulam stability of the following quadratic functional equations $${cf\left({\displaystyle\sum_{i=1}^n\;xi}\right)+{\displaystyle\sum_{i=2}^nf}{\left(\displaystyle\sum_{i=1}^n\;x_i-(n+c-1)x_j\right)}\\ {=(n+c-1)\;\left(f(x_1)+c{\displaystyle\sum_{i=2}^n\;f(x_i)}+{\displaystyle\sum_{i in fuzzy Banach spaces.

• East Asian mathematical journal
• /
• v.19 no.2
• /
• pp.165-171
• /
• 2003

We show that any F-harmonic map from a compact manifold M to N is necessarily constant if N possesses a strictly-convex function, and prove 'Liouville type theorems' for F-harmonic maps. Finally, when the target manifold is the real line, we get a result for F-subharmonic functions.

• Proceedings of the Korean Vacuum Society Conference
• /
• 2010.02a
• /
• pp.458-458
• /
• 2010

The process window for the etch selectivity of silicon nitride ($Si_3N_4$) layers to extreme ultra-violet (EUV) resist and variation of line edge roughness (LER) of EUV resist were investigated durin getching of $Si_3N_4$/EUV resist structure in a dual-frequency superimposed capacitive coupled plasma (DFS-CCP) etcher by varying the process parameters, such as the $CH_2F_2$ and $N_2$ gas flow rate in $CH_2F_2/N_2$/Ar plasma. The $CH_2F_2$ and $N_2$ flow rate was found to play a critical role in determining the process window for infinite etch selectivity of $Si_3N_4$/EUV resist, due to disproportionate changes in the degree of polymerization on $Si_3N_4$ and EUV resist surfaces. The preferential chemical reaction between hydrogen and carbon in the hydrofluorocarbon ($CH_xF_y$) polymer layer and the nitrogen and oxygen on the $Si_3N_4$, presumably leading to the formation of HCN, CO, and $CO_2$ etch by-products, results in a smaller steady-state hydrofluorocarbon thickness on $Si_3N_4$ and, in turn, in continuous $Si_3N_4$ etching due to enhanced $SiF_4$ formation, while the $CH_xF_y$ layer is deposited on the EUV resist surface. Also critical dimension (and line edge roughness) tend to decrease with increasing $N_2$ flow rate due to decreased degree of polymerization.

• Journal of applied mathematics & informatics
• /
• v.28 no.1_2
• /
• pp.13-30
• /
• 2010

Let C be a nonempty closed convex subset of a real Hilbert space H. Consider the following iterative algorithm given by $x_0\;{\in}\;C$ arbitrarily chosen, $x_{n+1}\;=\;{\alpha}_n{\gamma}f(W_nx_n)+{\beta}_nx_n+((1-{\beta}_n)I-{\alpha}_nA)W_nP_C(I-s_nB)x_n$, ${\forall}_n\;{\geq}\;0$, where $\gamma$ > 0, B : C $\rightarrow$ H is a $\beta$-inverse-strongly monotone mapping, f is a contraction of H into itself with a coefficient $\alpha$ (0 < $\alpha$ < 1), $P_C$ is a projection of H onto C, A is a strongly positive linear bounded operator on H and $W_n$ is the W-mapping generated by a finite family of nonexpansive mappings $T_1$, $T_2$, ${\ldots}$, $T_N$ and {$\lambda_{n,1}$}, {$\lambda_{n,2}$}, ${\ldots}$, {$\lambda_{n,N}$}. Nonexpansivity of each $T_i$ ensures the nonexpansivity of $W_n$. We prove that the sequence {$x_n$} generated by the above iterative algorithm converges strongly to a common fixed point $q\;{\in}\;F$ := $\bigcap^N_{i=1}F(T_i)\;\bigcap\;VI(C,\;B)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)q,\;p\;-\;q{\rangle}\;{\leq}\;0$ for all $p\;{\in}\;F$. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.4
• /
• pp.831-841
• /
• 2009

For a map f : A $\rightarrow$ X, we define and study a concept of $G^f$-space for a map, which is a generalized one of a G-space. Any G-space is a $G^f$-space, but the converse does not hold. In fact, $S^2$ is a $G^{\eta}$-space, but not G-space. We show that X is a $G^f$-space if and only if $G_n$(A, f,X) = $\pi_n(X)$ for all n. It is clear that any $H^f$-space is a $G^f$-space and any $G^f$-space is a $W^f$-space. We can also obtain some results about $G^f$-spaces in Postnikov systems for spaces, which are generalization of Haslam's results about G-spaces.

• Journal of Korean Ophthalmic Optics Society
• /
• v.9 no.2
• /
• pp.439-446
• /
• 2004

Many studies have reported that retinal defocus cause and increase refractive error specially myopia. Uncorrected astigmatism may be one factor of retinal defocus factors. To understand the relationship between myopia and astigmatism 62 college students were participated in this study. Spherical refractive error and astigmatism were measured using N-vision 5001 autorefractor (Shinnippon). Co-relations between spherical refractive error and astigmatism were high both in the with-the-rule astigmatism group(r=0.53; ANOVA F=32.40, N=87, P<0.05) and oblique astigmatism group (r=0.53ANOVA F=5.14, N=15, P<0.001). However it was very low (r=0.09; ANOVA F=0.18, N=22, P<0.001)in the against-the-rule stigmatism group. In the total group co-relation was also high (r=0.56: ANOVA F=77.80, N=173, P<0.001). Uncorrected astigmatism may cause and increase spherical refractive error.

• Journal of the Korea Institute of Information and Communication Engineering
• /
• v.9 no.2
• /
• pp.380-389
• /
• 2005

The Goldschmidt iterative algorithm for a floating point divide calculates it by performing a fixed number of multiplications. In this paper, a variable latency Goldschmidt's divide algorithm is proposed, that performs multiplications a variable number of times until the error becomes smaller than a given value. To calculate a floating point divide '$\frac{N}{F}$', multifly '$T=\frac{1}{F}+e_t$' to the denominator and the nominator, then it becomes ’$\frac{TN}{TF}=\frac{N_0}{F_0}$'. And the algorithm repeats the following operations: ’$R_i=(2-e_r-F_i),\;N_{i+1}=N_i{\ast}R_i,\;F_{i+1}=F_i{\ast}R_i$, i$\in${0,1,...n-1}'. The bits to the right of p fractional bits in intermediate multiplication results are truncated, and this truncation error is less than ‘$e_r=2^{-p}$'. The value of p is 29 for the single precision floating point, and 59 for the double precision floating point. Let ’$F_i=1+e_i$', there is $F_{i+1}=1-e_{i+1},\;e_{i+1}',\;where\;e_{i+1}, If '$[F_i-1]<2^{\frac{-p+3}{2}}$ is true, ’$e_{i+1}<16e_r$' is less than the smallest number which is representable by floating point number. So, ‘$N_{i+1}$ is approximate to ‘$\frac{N}{F}$'. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation is derived from many reciprocal tables ($T=\frac{1}{F}+e_t$) with varying sizes. 1'he superiority of this algorithm is proved by comparing this average number with the fixed number of multiplications of the conventional algorithm. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a divider. Also, it can be used to construct optimized approximate reciprocal tables. The results of this paper can be applied to many areas that utilize floating point numbers, such as digital signal processing, computer graphics, multimedia, scientific computing, etc

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.15 no.4
• /
• pp.319-328
• /
• 2011

Let $\mathbf{c}=\{c_n\}_{n{\in}\mathbb{Z}}\in{\ell}^1(\mathbb{Z})$ and $\{f_n\}_{n{\in}\mathbb{Z}}$ be a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$. We obtain necessary and sufficient conditions of $\mathbf{c}$ under which $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$, where ${\lambda}$ > 0 and $(\mathbf{c}{\ast}_{\lambda}f)(t)\;:=\;{\sum}_{n{\in}\mathbb{Z}}c_nf(t-n{\lambda})$. When $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame of $L^2(\mathbb{R})$, we present its frame operator and the canonical dual frame in a simple form. Some interesting examples are included.