title: A lower bound for the size of the smallest critical set in the back circulant latin square

authors: Cavenagh, Nicholas J.

subjects: mathematics

abstract: The back circulant latin square of order n is the latin square based on the addition table for the integers modulo n. A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. In this note we show that the size of a critical set in the back circulant latin square of order n is at least n ⁴/³/2 - n - n²/³/2 + 2n¹/³ - 1.

text: A lower bound for the size of the smallest critical set in the back circulant latin square The back circulant latin square of order n is the latin square based on the addition table for the integers modulo n. A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. In this note we show that the size of a critical set in the back circulant latin square of order n is at least n ⁴/³/2 - n - n²/³/2 + 2n¹/³ - 1.

year: 2006

title: Structural characterisation and absolute stereochemistry of some degraded carotenoids from New Zealand honeys

authors: Broom, Susan J.

abstract: The structures and absolute stereochemistries of some degraded carotenoids from New Zealand native honeys were elucidated. 

Determination of the absolute stereochemistry of 3,5,6-trihydroxy-β-ionone was attempted by an enantioselective synthesis. A racemic synthesis was achieved starting from β-ionone. The key steps involved regioselective hydroboration of 3,4-dehydro-β-ionone, followed by molybdenum-mediated stereoselective epoxidation of the resulting homoallylic alcohol. Ring opening of the epoxide afforded the required trihydroxy-ionone. Synthesis of both enantiomers of 3,5,6-trihydroxy-β-ionone via enantio- and regioselective functionalisation of 3,4-dehydro-β-ionone was not achieved. Methods of enantioselective enzyme-mediated kinetic resolution of cyclic homoallylic alcohols were investigated. A satisfactory procedure was not developed. 

The absolute stereochemistry of 3,5,6-trihydroxy-β-ionone was determined by analysis of the diastereoisomeric proton chemical shifts of (S)- and (R)-α-methoxyphenylacetate (MPA) derivatives using the extended Mosher-Trost configurational model. This analysis indicated a 3S,5R,6R absolute configuration, the same configuration as 3,5,6-trihydroxy carotenoids. The natural product was extracted from thyme (Thymus vulgaris) honey using established procedures.

Diethyl ether extracts of kamahi (Weinmannia racemosa) honey were found to contain three diastereoisomers (kamahines) of an unusual degraded carotenoid with a 14 carbon skeleton. After acetylation, one of the isomers was fully characterised by multidimensional ¹H and ¹³C NMR spectroscopy, and single-crystal X-ray crystallography, indicating the parent alcohol was 4,5-dihydro-1',5-dihydroxy-2',4,8',8'-tetramethylspiro[furan-2(3)H,7'[6']oxabicylco[3.2.1]oct[2']ene]-4'-one. Comparison of NMR-derived NOE and coupling constant data of kamahine A and B acetates with kamahine C acetate indicated the parent kamahines were C-4, C-5 epimers.
 
The absolute configurations of the kamahines were determined by NMR and a molecular modeling study (MacroModel) of the diastereoisomeric hemiacetal MPA esters. The results of the determination, coupled with a consideration of crystallographically-determined hemiacetal ester conformations obtained from a search of the Cambridge Structural Database, supported the use of the Mosher-Trost model for determination of the absolute configurations of hemiacetals. Furthermore, this analysis suggested that the Mosher-Trost model should be amended to take into account a torsion angle of +25-50° between the methine proton and the carbonyl of hemiacetal MPA esters. Kamahines A-C were shown to have the absolute configurations 1'R,2R,4R,5S,5'S, 1'R,2R,4S,5S,5'S and 1'R,2R,4R,5R,5'S, respectively. The absolute configurations of kamahines A-C were consistent with an abscisic acid precursor.

text: Structural characterisation and absolute stereochemistry of some degraded carotenoids from New Zealand honeys The structures and absolute stereochemistries of some degraded carotenoids from New Zealand native honeys were elucidated. 

Determination of the absolute stereochemistry of 3,5,6-trihydroxy-β-ionone was attempted by an enantioselective synthesis. A racemic synthesis was achieved starting from β-ionone. The key steps involved regioselective hydroboration of 3,4-dehydro-β-ionone, followed by molybdenum-mediated stereoselective epoxidation of the resulting homoallylic alcohol. Ring opening of the epoxide afforded the required trihydroxy-ionone. Synthesis of both enantiomers of 3,5,6-trihydroxy-β-ionone via enantio- and regioselective functionalisation of 3,4-dehydro-β-ionone was not achieved. Methods of enantioselective enzyme-mediated kinetic resolution of cyclic homoallylic alcohols were investigated. A satisfactory procedure was not developed. 

The absolute stereochemistry of 3,5,6-trihydroxy-β-ionone was determined by analysis of the diastereoisomeric proton chemical shifts of (S)- and (R)-α-methoxyphenylacetate (MPA) derivatives using the extended Mosher-Trost configurational model. This analysis indicated a 3S,5R,6R absolute configuration, the same configuration as 3,5,6-trihydroxy carotenoids. The natural product was extracted from thyme (Thymus vulgaris) honey using established procedures.

Diethyl ether extracts of kamahi (Weinmannia racemosa) honey were found to contain three diastereoisomers (kamahines) of an unusual degraded carotenoid with a 14 carbon skeleton. After acetylation, one of the isomers was fully characterised by multidimensional ¹H and ¹³C NMR spectroscopy, and single-crystal X-ray crystallography, indicating the parent alcohol was 4,5-dihydro-1',5-dihydroxy-2',4,8',8'-tetramethylspiro[furan-2(3)H,7'[6']oxabicylco[3.2.1]oct[2']ene]-4'-one. Comparison of NMR-derived NOE and coupling constant data of kamahine A and B acetates with kamahine C acetate indicated the parent kamahines were C-4, C-5 epimers.
 
The absolute configurations of the kamahines were determined by NMR and a molecular modeling study (MacroModel) of the diastereoisomeric hemiacetal MPA esters. The results of the determination, coupled with a consideration of crystallographically-determined hemiacetal ester conformations obtained from a search of the Cambridge Structural Database, supported the use of the Mosher-Trost model for determination of the absolute configurations of hemiacetals. Furthermore, this analysis suggested that the Mosher-Trost model should be amended to take into account a torsion angle of +25-50° between the methine proton and the carbonyl of hemiacetal MPA esters. Kamahines A-C were shown to have the absolute configurations 1'R,2R,4R,5S,5'S, 1'R,2R,4S,5S,5'S and 1'R,2R,4R,5R,5'S, respectively. The absolute configurations of kamahines A-C were consistent with an abscisic acid precursor.

year: 1998

title: Taku Rāta Tūtahi (composed 28 Oct 2013)

authors: Maxwell, Te Kahautu

abstract: Taku Rāta Tūtahi is a lament dedicated to my daughter Kaaterina Ataria Ariha Maxwell who passed away suddenly on the 23 March 2013. This composition talks about the pain of a father for his daughter and how at times life is unbearable because of this undiscribable pain. This composition was performed by Ōpōtiki Mai Tawhiti Kapa Haka at the Mātaatua Kapa Haka Regional Competition. This composition placed 2nd in the Whakaeke discipline.

text: Taku Rāta Tūtahi (composed 28 Oct 2013) Taku Rāta Tūtahi is a lament dedicated to my daughter Kaaterina Ataria Ariha Maxwell who passed away suddenly on the 23 March 2013. This composition talks about the pain of a father for his daughter and how at times life is unbearable because of this undiscribable pain. This composition was performed by Ōpōtiki Mai Tawhiti Kapa Haka at the Mātaatua Kapa Haka Regional Competition. This composition placed 2nd in the Whakaeke discipline.

year: 2014