• Title, Summary, Keyword: T-N

### AN ANALOGUE OF THE HILTON-MILNER THEOREM FOR WEAK COMPOSITIONS

• Ku, Cheng Yeaw;Wong, Kok Bin
• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.1007-1025
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• 2015
• Let $\mathbb{N}_0$ be the set of non-negative integers, and let P(n, l) denote the set of all weak compositions of n with l parts, i.e., $P(n,l)=\{(x_1,x_2,{\cdots},x_l){\in}\mathbb{N}^l_0\;:\;x_1+x_2+{\cdots}+x_l=n\}$. For any element $u=(u_1,u_2,{\cdots},u_l){\in}P(n,l)$, denote its ith-coordinate by u(i), i.e., $u(i)=u_i$. A family $A{\subseteq}P(n,l)$ is said to be t-intersecting if ${\mid}\{i:u(i)=v(i)\}{\mid}{\geq}t$ for all $u,v{\epsilon}A$. A family $A{\subseteq}P(n,l)$ is said to be trivially t-intersecting if there is a t-set T of $[l]=\{1,2,{\cdots},l\}$ and elements $y_s{\in}\mathbb{N}_0(s{\in}T)$ such that $A=\{u{\in}P(n,l):u(j)=yj\;for\;all\;j{\in}T\}$. We prove that given any positive integers l, t with $l{\geq}2t+3$, there exists a constant $n_0(l,t)$ depending only on l and t, such that for all $n{\geq}n_0(l,t)$, if $A{\subseteq}P(n,l)$ is non-trivially t-intersecting, then $${\mid}A{\mid}{\leq}(^{n+l-t-l}_{l-t-1})-(^{n-1}_{l-t-1})+t$$. Moreover, equality holds if and only if there is a t-set T of [l] such that $$A=\bigcup_{s{\in}[l]{\backslash}T}\;A_s{\cup}\{q_i:i{\in}T\}$$, where $$A_s=\{u{\in}P(n,l):u(j)=0\;for\;all\;j{\in}T\;and\;u(s)=0\}$$ and $$q_i{\in}P(n,l)\;with\;q_i(j)=0\;fo\;all\;j{\in}[l]{\backslash}\{i\}\;and\;q_i(i)=n$$.

### CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS

• Kang, Jung-Im;Cho, Yeol-Je;Zhou, Hai-Yun
• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.321-328
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• 2004
• In this paper, we will prove the following: Let D be a nonempty of a normed linear space X and T : D -> X be a nonexpansive mapping. Let ${x_n}$ be a sequence in D and ${t_n}$, ${s_n}$ be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;<\;1\;and\;{\sum_{n=1}}^{\infty}\;t_n\;=\;{\infty},\;(ii)\;(a)\;0\;{\leq}\;s_n\;{\leq}\;1,\;s_n\;->\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;<\;{\infty}\;or\;(b)\;s_n\;=\;s\;for\;all\;n\;{\geq}\;1\;and\;s\;{\in}\;[0,1),\;(iii)\;x_{n+1}\;=\;(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n)\;for\;all\;n\;{\geq}\;1.$ Then, if the sequence {x_n} is bounded, then $lim_{n->\infty}\;$\mid$$\midx_n-Tx_n\mid$$\mid$\;=\;0$. This result improves and complements a result of Deng . Furthermore, we will show that certain conditions on D, X and T guarantee the weak and strong convergence of the Ishikawa iterative sequence to a fixed point of T.

### STRONG CONVERGENCE OF COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS

• Gu, Feng
• East Asian mathematical journal
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• v.24 no.1
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• pp.35-43
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• 2008
• Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let ${\{T_i\}}^N_{i=1}$ be N nonexpansive self-mappings of K with $F\;=\;{\cap}^N_{i=1}F(T_i)\;{\neq}\;{\theta}$ (here $F(T_i)$ denotes the set of fixed points of $T_i$). Suppose that one of the mappings in ${\{T_i\}}^N_{i=1}$ is semi-compact. Let $\{{\alpha}_n\}\;{\subset}\;[{\delta},\;1-{\delta}]$ for some ${\delta}\;{\in}\;(0,\;1)$ and $\{{\beta}_n\}\;{\subset}\;[\tau,\;1]$ for some ${\tau}\;{\in}\;(0,\;1]$. For arbitrary $x_0\;{\in}\;K$, let the sequence {$x_n$} be defined iteratively by $\{{x_n\;=\;{\alpha}_nx_{n-1}\;+\;(1-{\alpha}_n)T_ny_n,\;\;\;\;\;\;\;\;\; \atop {y_n\;=\;{\beta}nx_{n-1}\;+\;(1-{\beta}_n)T_nx_n},\;{\forall}_n{\geq}1,}$, where $T_n\;=\;T_{n(modN)}$. Then {$x_n$} convergence strongly to a common fixed point of the mappings family ${\{T_i\}}^N_{i=1}$. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of ${\beta}_n\;{\equiv}\;1$ or N=1 are also new.

### On (H, μn) Summability of Fourier Series

• CHANDRA, SATISH
• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.513-518
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• 2003
• In this paper, we have proved a theorem on Hausdorff summability of Fourier series which generalizes various known results. We prove that if $${\int}_{o}^{t}\;{\mid}{\phi}(u){\mid}\;du=o(t)\;as\;t{\rightarrow}0\; and\;\lim_{n{\rightarrow}{\infty}}{\int}^{\eta}_{{\pi}/n}{\frac{{\mid}{\phi}(t)-{\phi}(t+{\pi}/n){\mid}}{t}}dt=o(n)$$ where 0 < ${\eta}$ < 1, then the Fourier series is (H, ${\mu}_n$) summable to s at t = x where the sequence ${\mu}_n$ is given by ${\mu}_n={\int}^1_0x^n{\chi}(x)\;dx\;n=0,1,2\;and\;K_n(t)=\limits\sum_{{\nu}=0}^n(\array {n\\{\nu}})({\Delta}^{{n}-{\nu}}{\mu}_{\nu}){\frac{sin{\nu}t}{t}}$.

### Photodecomposition of N-t-Butyl-N-chloro-$\omega$-phenylalkanesulfonamides in the Presence of Oxygen

• Lee, Jong Hun;Kim Kyongtae
• Bulletin of the Korean Chemical Society
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• v.13 no.6
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• pp.676-680
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• 1992
• Irradiation of N-t-butyl-N-chloro-3-phenylpropanesulfonamide (1a) in benzene at $20^{\circ}C$ using 450 W high pressure mercury arc lamp in the presence of oxygen affored N-t-butyl-3-phenylpropanesulfonamide (2), N-t-butyl-3-chloro-3-phenylpropanesulfonamide (3), and N-t-butyl-3-oxo-3-phenylpropanesulfonamide (4). Similarly, N-t-butyl-4- (5), N-t-butyl-4-chloro-4- (6), and N-t-butyl-4-phenylbutanesulfonamides (7) were obtained from N-t-butyl-N-chloro-4-phenylbutanesulfonamide (1b). However, irradiation of N-t-butyl-N-chloro-5-phenylpentanesulfonamide (1c) under the same conditions gave complex mixtures. These results indicate that sulfonamidyl radical generated from each of 1a and 1b can abstract intramolecularly a hydrogen atom from the benzylic position only by forming six and seven-membered transition states, respectively.

### THE COMPUTATION METHOD OF THE MILNOR NUMBER OF HYPERSURFACE SINGULARITIES DEFINED BY AN IRREDUCIBLE WEIERSTRASS POLYNOMIAL $z^n$+a(x,y)z+b(x,y)=0 in $C^3$ AND ITS APPLICATION

• Kang, Chung-Hyuk
• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.169-173
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• 1989