Cyclic subgroup separability of certain graph products of subgroup separable groups

Kok Bin Wong,Peng Choon Wong 대한수학회 2013 대한수학회보 Vol.50 No.5

In this paper, we show that tree products ofcertain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

Polygonal products of residually finite groups

Kok Bin Wong,Peng Choon Wong 대한수학회 2007 대한수학회보 Vol.44 No.1

CYCLIC SUBGROUP SEPARABILITY OF CERTAIN GRAPH PRODUCTS OF SUBGROUP SEPARABLE GROUPS

Wong, Kok Bin,Wong, Peng Choon Korean Mathematical Society 2013 대한수학회보 Vol.50 No.5

In this paper, we show that tree products of certain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

Wong, Kok-Bin,Wong, Peng-Choon Korean Mathematical Society 2007 대한수학회보 Vol.44 No.1

A group G is called cyclic subgroup separable for the cyclic subgroup H if for each $x\;{\in}\;G{\backslash}H$, there exists a normal subgroup N of finite index in G such that $x\;{\not\in}\;HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-by-finite groups.

Investigation on the effect of polymer and starch on the tablet properties of lyophilized orally disintegrating tablet

Kai Bin Liew,Kok Khiang Peh 대한약학회 2021 Archives of Pharmacal Research Vol.44 No.8

Orally disintegrating tablet (ODT) is a userfriendly and convenient dosage form. The study aimed toinvestigate the effect of polymers and wheat starch on thetablet properties of lyophilized ODT, with dapoxetine asmodel drug. Three polymers (hydroxypropylmethyl cellulose,carbopol 934P and Eudragit EPO) and wheat starchwere used as matrix forming materials in preparation oflyophilized ODT. The polymeric dispersion was casted intoa mould and kept in a freezer at -20 C for 4 h beforefreeze dried for 12 h. It was found that increasing in HPMCand Carbopol 934P concentrations produced tablets withhigher hardness and longer disintegration time. In contrast,Eudragit EPO was unable to form tablet with sufficienthardness at various concentrations. Moreover, HPMCseems to have a stronger effect on tablet hardness comparedto Carbopol 934P at the same concentration level. ODT of less friable was obtained. Wheat starch acted asbinder which strengthen the hardness of ODTs and prolongedthe disintegration time. ODT comprising of HPMCand wheat starch at ratio of 2:1 was found to be optimumbased upon the tablet properties. The optimum formulationwas palatable and 80 % of the drug was released within30 min in the dissolution study.

An Erdos-Ko-Rado theorem for minimal covers

Cheng Yeaw Ku,Kok Bin Wong 대한수학회 2017 대한수학회보 Vol.54 No.3

Let $[n]=\{1,2,\dots, n\}$. A set $\mathbf A=\{A_1,A_2,\dots ,A_l\}$ is a minimal cover of $[n]$ if $\bigcup_{1\leq i\leq l} A_i =[n]$ and \[ \bigcup_{\substack{1\leq i\leq l,\\ i\neq j_0}} A_i \neq [n] \quad\textnormal{for all $j_0\in [l]$}. \] Let $\mathcal{C}(n)$ denote the collection of all minimal covers of $[n]$, and write $C_{n} = \vert \mathcal{C}(n)\vert$. Let $\mathbf A \in \mathcal{C}(n)$. An element $u \in [n]$ is critical in $\mathbf A$ if it appears exactly once in $\mathbf A$. Two minimal covers $\mathbf A$, $\mathbf B \in \mathcal{C}(n)$ are said to be restricted $t$-intersecting if they share at least $t$ sets each containing an element which is critical in both $\mathbf A$ and $\mathbf B$. A family $\A \subseteq \mathcal{C}(n)$ is said to be restricted $t$-intersecting if every pair of distinct elements in $\A$ are restricted $t$-intersecting. In this paper, we prove that there exists a constant $n_{0}=n_{0}(t)$ depending on $t$, such that for all $n \ge n_{0}$, if $\A \subseteq \mathcal{C}(n)$ is restricted $t$-intersecting, then $|\A| \le C_{n-t}$. Moreover, the bound is attained if and only if $\A$ is isomorphic to the family $\mathcal{D}_{0}(t)$ consisting of all minimal covers which contain the singleton parts $\{1\}$, $\ldots$, $\{t\}$. A similar result also holds for restricted $r$-cross intersecting families of minimal covers.

AN ERDŐS-KO-RADO THEOREM FOR MINIMAL COVERS

Ku, Cheng Yeaw,Wong, Kok Bin Korean Mathematical Society 2017 대한수학회보 Vol.54 No.3

Let $[n]=\{1,2,{\ldots},n\}$. A set ${\mathbf{A}}=\{A_1,A_2,{\ldots},A_l\}$ is a minimal cover of [n] if ${\cup}_{1{\leq}i{\leq}l}A_i=[n]$ and $$\bigcup_{{1{\leq}i{\leq}l,}\\{i{\neq}j_0}}A_i{\neq}[n]\text{ for all }j_0{\in}[l]$$. Let ${\mathcal{C}}(n)$ denote the collection of all minimal covers of [n], and write $C_n={\mid}{\mathcal{C}}(n){\mid}$. Let ${\mathbf{A}}{\in}{\mathcal{C}}(n)$. An element $u{\in}[n]$ is critical in ${\mathbf{A}}$ if it appears exactly once in ${\mathbf{A}}$. Two minimal covers ${\mathbf{A}},{\mathbf{B}}{\in}{\mathcal{C}}(n)$ are said to be restricted t-intersecting if they share at least t sets each containing an element which is critical in both ${\mathbf{A}}$ and ${\mathbf{B}}$. A family ${\mathcal{A}}{\subseteq}{\mathcal{C}}(n)$ is said to be restricted t-intersecting if every pair of distinct elements in ${\mathcal{A}}$ are restricted t-intersecting. In this paper, we prove that there exists a constant $n_0=n_0(t)$ depending on t, such that for all $n{\geq}n_0$, if ${\mathcal{A}}{\subseteq}{\mathcal{C}}(n)$ is restricted t-intersecting, then ${\mid}{\mathcal{A}}{\mid}{\leq}{\mathcal{C}}_{n-t}$. Moreover, the bound is attained if and only if ${\mathcal{A}}$ is isomorphic to the family ${\mathcal{D}}_0(t)$ consisting of all minimal covers which contain the singleton parts $\{1\},{\ldots},\{t\}$. A similar result also holds for restricted r-cross intersecting families of minimal covers.

The residual finiteness of certain HNN extensions

Wong Peng Choon,Wong Kok Bin 대한수학회 2005 대한수학회보 Vol.42 No.3

In this note we give characterizations for certain HNNextensions with central associated subgroups to be residually nite.We then apply our results to HNN extensions of polycyclic-by-finite groups.

THE RESIDUAL FINITENESS OF CERTAIN HNN EXTENSIONS

Choon, Wong-Peng,Bin, Wong-Kok Korean Mathematical Society 2005 대한수학회보 Vol.42 No.3

In this note we give characterizations for certain HNN extensions with central associated subgroups to be residually finite. We then apply our results to HNN extensions of polycyclic-by-finite groups.

AN ANALOGUE OF THE HILTON-MILNER THEOREM FOR WEAK COMPOSITIONS

Cheng Yeaw Ku,Kok Bin Wong 대한수학회 2015 대한수학회보 Vol.52 No.3

Let N0 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) = {(x1, x2, . . . , xl) ∈ Nl 0 : x1 + x2 + · · · + xl = n}. For any element u = (u1, u2, . . . , ul) ∈ P(n, l), denote its ith-coordinate by u(i), i.e., u(i) = ui. A family A ⊆ P(n, l) is said to be t-intersecting if |{i : u(i) = v(i)}| ≥ t for all u, v ∈ A. A family A ⊆ P(n, l) is said to be trivially t-intersecting if there is a t-set T of [l] = {1, 2, . . . , l} and elements ys ∈ N0 (s ∈ T) such that A = {u ∈ P(n, l) : u(j) = yj for all j ∈ T}. We prove that given any positive integers l, t with l ≥ 2t + 3, there exists a constant n0(l, t) depending only on l and t, such that for all n ≥ n0(l, t), if A ⊆ P(n, l) is non-trivially t-intersecting, then |A| ≤ (n + l − t − 1 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 = [∪ s∈[l]\T As ∪ {qi : i ∈ T} , where As = {u ∈ P(n, l) : u(j) = 0 for all j ∈ T and u(s) = 0} and qi ∈ P(n, l) with qi(j) = 0 for all j ∈ [l] \ {i} and qi(i) = n.