title: Biembeddings of cycle systems using integer Heffter arrays

authors: Cavenagh, Nicholas J.

subjects: mathematics

abstract: In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2).

text: Biembeddings of cycle systems using integer Heffter arrays In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2).

year: 2020

title: Platinum(II) complexes of chelating and monodentate thiourea monoanions incorporating chiral, fluorescent or chromophoric groups

authors: Henderson, William

subjects: Crystal structures

abstract: The reaction of cis-[PtCl2(PPh3)2] with trisubstituted thioureas [R1R2NC(=S)NHR3] in refluxing methanol with triethylamine base, followed by addition of NaBPh4 gives the salts [Pt{SC(=NR1R2)NR3}(PPh3)2]BPh4 in high yield; a range of thiourea substituents, including chiral, fluorescent and chromophoric groups can be incorporated. The azo dye-derived complex [Pt{SC(=N(CH2CH2)2O)NC6H4N=NC6H4NMe2}(PPh3)2]BPh4 has been characterised by a single-crystal X-ray diffraction study. The formation of a fluorescein-derivatised platinum-thiourea complex is also described. Reaction of cis-[PtCl2(PPh3)2] with PhNHC(S)NHPh or EtNHC(S)NHEt, triethylamine and NaBPh4 gives, respectively, [Pt{SC(=NHPh)NPh}(PPh3)2]+ and the known cation [Pt{SC(=NHEt)NEt}(PPh3)2]+, isolated as tetraphenylborate salts. Reaction of cis-[PtCl2(PPh3)2] with an excess of Na[MeNHC(S)NCN] in methanol gives the bis(thiourea monoanion) complex trans-[Pt{SC(=N---CN)NHMe}2(PPh3)2], characterised by NMR spectroscopy and an X-ray crystal structure determination. When cis-[PtCl2(PPh3)2] is reacted with 1 equiv. of Na[MeNHC(S)N---CN] in methanol, with added NaBPh4, a mixture of isomers of the [Pt{SC(=NHCN)NMe}(PPh3)2]+ cation is obtained.

text: Platinum(II) complexes of chelating and monodentate thiourea monoanions incorporating chiral, fluorescent or chromophoric groups The reaction of cis-[PtCl2(PPh3)2] with trisubstituted thioureas [R1R2NC(=S)NHR3] in refluxing methanol with triethylamine base, followed by addition of NaBPh4 gives the salts [Pt{SC(=NR1R2)NR3}(PPh3)2]BPh4 in high yield; a range of thiourea substituents, including chiral, fluorescent and chromophoric groups can be incorporated. The azo dye-derived complex [Pt{SC(=N(CH2CH2)2O)NC6H4N=NC6H4NMe2}(PPh3)2]BPh4 has been characterised by a single-crystal X-ray diffraction study. The formation of a fluorescein-derivatised platinum-thiourea complex is also described. Reaction of cis-[PtCl2(PPh3)2] with PhNHC(S)NHPh or EtNHC(S)NHEt, triethylamine and NaBPh4 gives, respectively, [Pt{SC(=NHPh)NPh}(PPh3)2]+ and the known cation [Pt{SC(=NHEt)NEt}(PPh3)2]+, isolated as tetraphenylborate salts. Reaction of cis-[PtCl2(PPh3)2] with an excess of Na[MeNHC(S)NCN] in methanol gives the bis(thiourea monoanion) complex trans-[Pt{SC(=N---CN)NHMe}2(PPh3)2], characterised by NMR spectroscopy and an X-ray crystal structure determination. When cis-[PtCl2(PPh3)2] is reacted with 1 equiv. of Na[MeNHC(S)N---CN] in methanol, with added NaBPh4, a mixture of isomers of the [Pt{SC(=NHCN)NMe}(PPh3)2]+ cation is obtained.

year: 2001-08-01

title: Breaking barriers and empowering perspectives: Centering women's voices in the mongrel mob wāhine toa

authors: Arnold, Annaliese

subjects: gang research

abstract: Gang communities in Aotearoa New Zealand have traditionally been sites of masculinity where specific forms of 'maleness' - including resistance to authority, antisocial behaviour, and expressions of violence - are embraced, expressed, and enforced through various internal cultural norms and reactions to the outside world. The dominant narrative that informs public perceptions of New Zealand gangs is often centered on crime, poverty, and Indigeneity. Much less is known about the role of women in these communities, particularly as their voices have only recently emerged from one of the largest gang chapters in the country. This research, conducted in close collaboration with wāhine within the Mongrel Mob Kingdom's first female-only chapter, Mongrel Mob Wāhine Toa, uses a participatory action research approach with photovoice methodology. Drawing on the personal narratives of six wāhine, this study aims to better understand their perspectives and experiences of contributing to a fledgling movement that privileges women within a predominantly male-centered gang environment. Photovoice is used alongside semi-structured interviews to provide depth and breadth to the data, allowing the wāhine to visually and verbally express their experiences and perspectives. Themes include navigating marginality, empowerment, and resistance. The long-range goal of this work is to inform context-sensitive policies and interventions for wāhine in gangs in Aotearoa. By foregrounding the perspective of wāhine, this study contributes to breaking down barriers and reshaping discourses around women in gangs, with implications for well-being and empowerment.

text: Breaking barriers and empowering perspectives: Centering women's voices in the mongrel mob wāhine toa Gang communities in Aotearoa New Zealand have traditionally been sites of masculinity where specific forms of 'maleness' - including resistance to authority, antisocial behaviour, and expressions of violence - are embraced, expressed, and enforced through various internal cultural norms and reactions to the outside world. The dominant narrative that informs public perceptions of New Zealand gangs is often centered on crime, poverty, and Indigeneity. Much less is known about the role of women in these communities, particularly as their voices have only recently emerged from one of the largest gang chapters in the country. This research, conducted in close collaboration with wāhine within the Mongrel Mob Kingdom's first female-only chapter, Mongrel Mob Wāhine Toa, uses a participatory action research approach with photovoice methodology. Drawing on the personal narratives of six wāhine, this study aims to better understand their perspectives and experiences of contributing to a fledgling movement that privileges women within a predominantly male-centered gang environment. Photovoice is used alongside semi-structured interviews to provide depth and breadth to the data, allowing the wāhine to visually and verbally express their experiences and perspectives. Themes include navigating marginality, empowerment, and resistance. The long-range goal of this work is to inform context-sensitive policies and interventions for wāhine in gangs in Aotearoa. By foregrounding the perspective of wāhine, this study contributes to breaking down barriers and reshaping discourses around women in gangs, with implications for well-being and empowerment.

year: 2025