title: Variable separation for Heat and Schrödinger equations

authors: Reid, Gregory John

abstract: All R-separable coordinate systems are classified for the equation
Δₘ Ψ + 2ε ∂/∂t Ψ = EΨ (*)
where Δₘ = ᵐΣ ᵤ₌₁ ∂²/(∂yᵘ)² and the variables t and yᵘ are real. An R-separable coordinate system for (*) is a coordinate system yᵘ = yᵘ (xᵏ), t = t (xᵏ) such that (*) admits a solution of form Ψ = eᴿ⁽ˣⁱ⁾ ᵐ⁺¹Πᵢ₌₁ Ψᵢ(xⁱ;c₁, ···, cₘ₊₁). For ε = ⁱ/₂ and ε = - ¹/₂, E = 0, (*) yields standardised versions of the Schrӧdinger and Heat equations respectively. Recognition of (*) as a symmetry-reduced version of the Helmholtz equation on m + 2 dimensional Minkowski space, for which there is a well developed theory of separation, is central to the solution of the classification problem. The solution is built from the case ε = 0 which has been solved by Kalnins and Miller in an article submitted to the Transactions of the American Mathematical Society.
We find the operators and separation equations for (*) and give a detailed treatment of the physically interesting case m =3. The operators are always in the enveloping algebra of the Schrӧdinger algebra. This means that much of the special function theory relating to the separated solutions can be reduced to problems in the representation theory of this algebra.
Various complex extensions of (*) are treated. In particular a class of systems is uncovered leading to nontrivial R-separation for the Hamilton-Jacobi equation corresponding to (*). A formula is derived for expressing the second order Killing tensors describing separation in flat spaces in terms of the enveloping algebra. This formula forms the basis of a program written in the symbolic language MACSYMA capable of producing all the time consuming details of separation.

text: Variable separation for Heat and Schrödinger equations All R-separable coordinate systems are classified for the equation
Δₘ Ψ + 2ε ∂/∂t Ψ = EΨ (*)
where Δₘ = ᵐΣ ᵤ₌₁ ∂²/(∂yᵘ)² and the variables t and yᵘ are real. An R-separable coordinate system for (*) is a coordinate system yᵘ = yᵘ (xᵏ), t = t (xᵏ) such that (*) admits a solution of form Ψ = eᴿ⁽ˣⁱ⁾ ᵐ⁺¹Πᵢ₌₁ Ψᵢ(xⁱ;c₁, ···, cₘ₊₁). For ε = ⁱ/₂ and ε = - ¹/₂, E = 0, (*) yields standardised versions of the Schrӧdinger and Heat equations respectively. Recognition of (*) as a symmetry-reduced version of the Helmholtz equation on m + 2 dimensional Minkowski space, for which there is a well developed theory of separation, is central to the solution of the classification problem. The solution is built from the case ε = 0 which has been solved by Kalnins and Miller in an article submitted to the Transactions of the American Mathematical Society.
We find the operators and separation equations for (*) and give a detailed treatment of the physically interesting case m =3. The operators are always in the enveloping algebra of the Schrӧdinger algebra. This means that much of the special function theory relating to the separated solutions can be reduced to problems in the representation theory of this algebra.
Various complex extensions of (*) are treated. In particular a class of systems is uncovered leading to nontrivial R-separation for the Hamilton-Jacobi equation corresponding to (*). A formula is derived for expressing the second order Killing tensors describing separation in flat spaces in terms of the enveloping algebra. This formula forms the basis of a program written in the symbolic language MACSYMA capable of producing all the time consuming details of separation.

year: 1984

title: Cultivating microbial dark matter - A combination of traditional and molecular techniques

authors: King, Charlotte

abstract: The services bacteria offer the world are immense and impact people's lives daily. They were first described by Antoni van Leeuwenhoek in 1676, but bacteria have been utilised in the medical, agricultural, and food and beverage industries even before they were formally identified. Microbes’ roles in the environment for cycling of nutrients, climate regulation and pollution are becoming more well understood, and they are also utilised to answer a wide variety of research questions in disparate fields. Traditional cultivation techniques were initially developed to study bacteria and understand their functions as isolates. Such cultivation techniques have become rarer as new molecular and bioinformatic tools have been developed. These cultivation- independent techniques have allowed the study of microbial life genetic data which led to the finding that 99% of all microbial species were yet to be cultured and studied in a laboratory. The applications of these unstudied organisms could be unprecedented, though their cultivation remains challenging due to the requirement of specific nutrients that are hard to identify, very slow growth rates, environmental conditions that are difficult to replicate in a laboratory setting, live in symbiosis with other organisms, or live in extreme climates. Therefore, much of the information may be overlooked and hidden in this microbial dark matter. Methods such as metagenomics, metagenome-assembled genomes (MAGs), and single amplified genomes (SAGs) have provided us with genomic data for previously uncharacterised microbes from diverse environments. However, such data is still limited. Omics tools can provide insights into what is living there and what it can potentially do. However, this data is regularly incomplete due to the processes involved in sequencing, and conclusions are often drawn from associations relating to cultured organisms. This can lead to potentially spurious conclusions as a gene may behave differently in different species, and the absence of genes may depend on how the data was analysed. Therefore, cultured isolates of the organism are vital. To combat this issue of an uncultivated majority, it is important to have a targeted approach to culture, what is wanted, not just what is easy.

This thesis aimed to provide an easy-to-follow, straightforward workflow to cultivate target environmental organisms using standard molecular tools and methods readily accessible in any standard microbiological laboratory. The idea is to isolate and culture what is specifically wanted or needed, not just what is easy. Specifically, it combined classical cultivation and molecular techniques to target a sulphur-reducing, obligate anaerobe from an environmental sample— namely a Desulfurella sp. from a geothermal pool in Rotorua, New Zealand.
In order to provide the desired targeted approach, specific primers needed to be designed and assessed to first find the right sampling site for the initial inoculum and second to ensure the target is not lost throughout the culturing process. I designed and evaluated primers for targeting specific taxa at different stringency levels. Initially, I tested published primers (EPS_F/EPS_R, Gittel et al., 2012) aimed at Epsilonproteobacteria, which were reclassified into the Campylobacterota phylum (including Desulfurella sp.). These primers did not target Desulfurella sp., so I modified them (EPS_FM/EPS_RL) to produce a longer amplicon for broader Campylobacterota detection. However, these also amplified non-target species (e.g., Desulfobacterota). Finally, I designed a third set of primers (DS_F/DS_R) specific to Desulfurella sp., confirmed by in silico testing. These primers allow assessment of Desulfurella sp. and broader Campylobacterota before sequencing.
The second experimental chapter focuses on enriching the target using the previously described primers, classical cultivation techniques, and molecular tools. Site selection was based on the 1000 Springs Project (Power et al., 2018). Culturing was performed originally with DSMZ Desulfurella medium and later modified, investigating pH (4 and 6) and temperature (50°C and 30°C) variations and adapting it for solid media by replacing elemental sulphur with sodium polysulfide. I used a vacuum food sealer to cultivate anaerobes, an accessible and portable alternative to expensive anaerobic chambers. Automated Ribosomal Intergenic Spacer Analysis (ARISA) was used to assess axenicity before sending samples for 16S rRNA gene and whole genome sequencing. Desulfurella was detected in each culture using species-specific primers (DS_F/DS_R), enabling its phylogenetic placement. However, despite Desulfurella presence in the 16S rRNA gene amplicon sequencing, a metagenome-assembled genome (MAG) could not be produced. This thesis discusses lessons learned and suggests improvements for future studies.

text: Cultivating microbial dark matter - A combination of traditional and molecular techniques The services bacteria offer the world are immense and impact people's lives daily. They were first described by Antoni van Leeuwenhoek in 1676, but bacteria have been utilised in the medical, agricultural, and food and beverage industries even before they were formally identified. Microbes’ roles in the environment for cycling of nutrients, climate regulation and pollution are becoming more well understood, and they are also utilised to answer a wide variety of research questions in disparate fields. Traditional cultivation techniques were initially developed to study bacteria and understand their functions as isolates. Such cultivation techniques have become rarer as new molecular and bioinformatic tools have been developed. These cultivation- independent techniques have allowed the study of microbial life genetic data which led to the finding that 99% of all microbial species were yet to be cultured and studied in a laboratory. The applications of these unstudied organisms could be unprecedented, though their cultivation remains challenging due to the requirement of specific nutrients that are hard to identify, very slow growth rates, environmental conditions that are difficult to replicate in a laboratory setting, live in symbiosis with other organisms, or live in extreme climates. Therefore, much of the information may be overlooked and hidden in this microbial dark matter. Methods such as metagenomics, metagenome-assembled genomes (MAGs), and single amplified genomes (SAGs) have provided us with genomic data for previously uncharacterised microbes from diverse environments. However, such data is still limited. Omics tools can provide insights into what is living there and what it can potentially do. However, this data is regularly incomplete due to the processes involved in sequencing, and conclusions are often drawn from associations relating to cultured organisms. This can lead to potentially spurious conclusions as a gene may behave differently in different species, and the absence of genes may depend on how the data was analysed. Therefore, cultured isolates of the organism are vital. To combat this issue of an uncultivated majority, it is important to have a targeted approach to culture, what is wanted, not just what is easy.

This thesis aimed to provide an easy-to-follow, straightforward workflow to cultivate target environmental organisms using standard molecular tools and methods readily accessible in any standard microbiological laboratory. The idea is to isolate and culture what is specifically wanted or needed, not just what is easy. Specifically, it combined classical cultivation and molecular techniques to target a sulphur-reducing, obligate anaerobe from an environmental sample— namely a Desulfurella sp. from a geothermal pool in Rotorua, New Zealand.
In order to provide the desired targeted approach, specific primers needed to be designed and assessed to first find the right sampling site for the initial inoculum and second to ensure the target is not lost throughout the culturing process. I designed and evaluated primers for targeting specific taxa at different stringency levels. Initially, I tested published primers (EPS_F/EPS_R, Gittel et al., 2012) aimed at Epsilonproteobacteria, which were reclassified into the Campylobacterota phylum (including Desulfurella sp.). These primers did not target Desulfurella sp., so I modified them (EPS_FM/EPS_RL) to produce a longer amplicon for broader Campylobacterota detection. However, these also amplified non-target species (e.g., Desulfobacterota). Finally, I designed a third set of primers (DS_F/DS_R) specific to Desulfurella sp., confirmed by in silico testing. These primers allow assessment of Desulfurella sp. and broader Campylobacterota before sequencing.
The second experimental chapter focuses on enriching the target using the previously described primers, classical cultivation techniques, and molecular tools. Site selection was based on the 1000 Springs Project (Power et al., 2018). Culturing was performed originally with DSMZ Desulfurella medium and later modified, investigating pH (4 and 6) and temperature (50°C and 30°C) variations and adapting it for solid media by replacing elemental sulphur with sodium polysulfide. I used a vacuum food sealer to cultivate anaerobes, an accessible and portable alternative to expensive anaerobic chambers. Automated Ribosomal Intergenic Spacer Analysis (ARISA) was used to assess axenicity before sending samples for 16S rRNA gene and whole genome sequencing. Desulfurella was detected in each culture using species-specific primers (DS_F/DS_R), enabling its phylogenetic placement. However, despite Desulfurella presence in the 16S rRNA gene amplicon sequencing, a metagenome-assembled genome (MAG) could not be produced. This thesis discusses lessons learned and suggests improvements for future studies.

year: 2024

title: Significance of an early date from an adzed tree stump in Northland

authors: Higham, Thomas F.G.

abstract: Ogle, Jones, Sutton and Wallace (1998) have recently investigated the radiocarbon dating of a tree stump ('Ogle's stump’) which had evidence of adze marks, near the site of Taumatawhana pa, Houhora, Northland. The conventional radiocarbon age of 938 ± 31 BP was obtained at the Institute of Geological and Nuclear Sciences Ltd (formerly INS) laboratory in 1971. Ogle et al.(1998) considered the 'inbuilt age' of the wood (see McFadgen, 1982) and assessed its contribution to the radiocarbon determination. In this brief paper we re-evaluate that methodology and question the conclusion of Ogle et al. (1998) that the date is reliable. We are particularly concerned with the comparison of the date with an associated radiocarbon result obtained by Elliot et al. (1995) from a nearby site at Lake Taumatawhana, which we think is misleading.

text: Significance of an early date from an adzed tree stump in Northland Ogle, Jones, Sutton and Wallace (1998) have recently investigated the radiocarbon dating of a tree stump ('Ogle's stump’) which had evidence of adze marks, near the site of Taumatawhana pa, Houhora, Northland. The conventional radiocarbon age of 938 ± 31 BP was obtained at the Institute of Geological and Nuclear Sciences Ltd (formerly INS) laboratory in 1971. Ogle et al.(1998) considered the 'inbuilt age' of the wood (see McFadgen, 1982) and assessed its contribution to the radiocarbon determination. In this brief paper we re-evaluate that methodology and question the conclusion of Ogle et al. (1998) that the date is reliable. We are particularly concerned with the comparison of the date with an associated radiocarbon result obtained by Elliot et al. (1995) from a nearby site at Lake Taumatawhana, which we think is misleading.

year: 1998