title: Jacobi elliptic coordinates, functions of Heun and Lame type and the Niven transform

authors: Kalnins, Ernie G.

subjects: Lame and Heun function

abstract: Lame and Heun functions arise via separation of the Laplace equation in general Jacobi ellipsoidal or conical coordinates. In contrast to hypergeometric functions that also arise via variable separation in the Laplace equation, Lame and Heun functions have received relatively little attention, since they are rather intractable. Nonetheless functions of Heun type do have remarkable properties, as was pointed out in the classical book \Modern Analysis" by Whittaker and Watson who devoted an entire chapter to the subject. Unfortunately the beautiful identities appearing in this chapter have received little notice, probably because the methods of proof seemed obscure. In this paper we apply the modern operator characterization of variable separation and exploit the conformal symmetry of the Laplace equation to obtain product identities for Heun type functions. We interpret the Niven transform as an intertwining operator under the action of the conformal group. We give simple operator derivations of some of the basic formulas presented by Whittaker and Watson and then show how to generalize their results to more complicated situations and to higher dimensions.

text: Jacobi elliptic coordinates, functions of Heun and Lame type and the Niven transform Lame and Heun functions arise via separation of the Laplace equation in general Jacobi ellipsoidal or conical coordinates. In contrast to hypergeometric functions that also arise via variable separation in the Laplace equation, Lame and Heun functions have received relatively little attention, since they are rather intractable. Nonetheless functions of Heun type do have remarkable properties, as was pointed out in the classical book \Modern Analysis" by Whittaker and Watson who devoted an entire chapter to the subject. Unfortunately the beautiful identities appearing in this chapter have received little notice, probably because the methods of proof seemed obscure. In this paper we apply the modern operator characterization of variable separation and exploit the conformal symmetry of the Laplace equation to obtain product identities for Heun type functions. We interpret the Niven transform as an intertwining operator under the action of the conformal group. We give simple operator derivations of some of the basic formulas presented by Whittaker and Watson and then show how to generalize their results to more complicated situations and to higher dimensions.

year: 2005

title: NIR technologies for high speed fruit grading

authors: Fraser, Daniel Grant

abstract: Today’s consumers seek information about food products so they can purchase with confidence, even if it translates to higher priced items. It is the aim of this thesis to improve the instruments available for the prediction of food quality, in particular, to advance the technology used for the rapid, low cost, non-destructive analysis of the internal quality of fruit. This thesis contains both experimental and theoretical work covering two main areas of research: the first deals with issues related to the practical grading of fruit, the second with the NIR light penetration and distribution within a fruit. 

A ‘rotary grader’ has been built to enable the rapid, non-destructive assessment of fruit. Up to twenty fruit can be placed on a rotating disk, which can be spun at a controlled speed or moved to fixed positions for stationary studies. A variety of optical arrangements can be mounted to enable the acquisition of spectra from individual fruit in different ‘modes’ of operation. A comparison between the different optical arrangements is presented. It was found that transmission mode (where the light passes right through the fruit) performs better than reflectance mode (where the detected light comes primarily from near the surface of the fruit). This result has been previously predicted and has now been demonstrated with the equipment used in this work. Different angles of transmission are compared as well as different fruit orientations for their influence on fruit quality predictions. It was found that 90° transmission (where the light is detected at right angles to the illumination of the fruit) might work best for randomly orientated kiwifruit; if they are hand placed then 180° or 150° may be more effective. A study of mandarins suggested that the acid content cannot be assessed directly, but it can be inferred from surface properties such as colour and chlorophyll content. This is contrary to several published reports and questions the ability of NIR spectroscopy to measure low concentrations of acid in fruit. A third experiment showed that soluble solids content (SSC) and dry matter (DM) can be assessed from moving apples with a prediction error of 0.54 ± 0.07 °Brix (R²ₚ  = 0.65) and 0.66 ± 0.08 % (R²ₚ  = 0.80) respectively. A unique perspective is presented showing where the useful information is attained from the continuously moving fruit. Transmission and reflectance spectra from apples moving at speeds of up to 3 fruit/s have been recorded, and the optimal spectra from the sequence were combined for improved predictions. 

To investigate the NIR light distribution and penetration in fruit a fibre optic probe was built to take direct measurements of the light levels inside fruit. This enabled, for the first time, the degree of light penetration to be assessed without altering the optical boundaries of the fruit (for example cutting away sections of the fruit). Corresponding simulations using Monte Carlo photon tracing, match the experimental measurements. It was found that fruit such as mandarin exhibit strong internal reflection from the skin, a result that has not previously been reported. The influence of the core and other features in the fruit on the light penetration and distribution is also studied showing perturbations in the internal light levels mapping out these features.

text: NIR technologies for high speed fruit grading Today’s consumers seek information about food products so they can purchase with confidence, even if it translates to higher priced items. It is the aim of this thesis to improve the instruments available for the prediction of food quality, in particular, to advance the technology used for the rapid, low cost, non-destructive analysis of the internal quality of fruit. This thesis contains both experimental and theoretical work covering two main areas of research: the first deals with issues related to the practical grading of fruit, the second with the NIR light penetration and distribution within a fruit. 

A ‘rotary grader’ has been built to enable the rapid, non-destructive assessment of fruit. Up to twenty fruit can be placed on a rotating disk, which can be spun at a controlled speed or moved to fixed positions for stationary studies. A variety of optical arrangements can be mounted to enable the acquisition of spectra from individual fruit in different ‘modes’ of operation. A comparison between the different optical arrangements is presented. It was found that transmission mode (where the light passes right through the fruit) performs better than reflectance mode (where the detected light comes primarily from near the surface of the fruit). This result has been previously predicted and has now been demonstrated with the equipment used in this work. Different angles of transmission are compared as well as different fruit orientations for their influence on fruit quality predictions. It was found that 90° transmission (where the light is detected at right angles to the illumination of the fruit) might work best for randomly orientated kiwifruit; if they are hand placed then 180° or 150° may be more effective. A study of mandarins suggested that the acid content cannot be assessed directly, but it can be inferred from surface properties such as colour and chlorophyll content. This is contrary to several published reports and questions the ability of NIR spectroscopy to measure low concentrations of acid in fruit. A third experiment showed that soluble solids content (SSC) and dry matter (DM) can be assessed from moving apples with a prediction error of 0.54 ± 0.07 °Brix (R²ₚ  = 0.65) and 0.66 ± 0.08 % (R²ₚ  = 0.80) respectively. A unique perspective is presented showing where the useful information is attained from the continuously moving fruit. Transmission and reflectance spectra from apples moving at speeds of up to 3 fruit/s have been recorded, and the optimal spectra from the sequence were combined for improved predictions. 

To investigate the NIR light distribution and penetration in fruit a fibre optic probe was built to take direct measurements of the light levels inside fruit. This enabled, for the first time, the degree of light penetration to be assessed without altering the optical boundaries of the fruit (for example cutting away sections of the fruit). Corresponding simulations using Monte Carlo photon tracing, match the experimental measurements. It was found that fruit such as mandarin exhibit strong internal reflection from the skin, a result that has not previously been reported. The influence of the core and other features in the fruit on the light penetration and distribution is also studied showing perturbations in the internal light levels mapping out these features.

year: 2001

title: Mā mahi, ka ora: by work, we prosper--traditional healers and workforce development.

authors: Ahuriri-Driscoll, Annabel

subjects: Adult

abstract: AIM: Rongoā Māori practitioners make a valuable contribution towards Māori health outcomes, albeit with limited resourcing or formal training. This paper reports on a survey of healers/healing practices-specifically healers' aspirations for professional development and training-and considers the implications for healing practice and future training undertakings. 
METHODS: Healers in seven districts around the country were surveyed about rongoā practice and service delivery during 2013. Consenting healers completed surveys either in person, via phone, or returned them via post, according to their preference and convenience. Resulting data were analysed and reported according to frequency of responses. 
RESULTS: Thirty-eight healers/rongoā clinics completed the survey--a 79% response rate. Respondents were primarily Māori (88%), female (69%), aged 50 years or older (60%), and worked as volunteers. Informal training modes focused on te reo, mātauranga and tikanga were the most common means of skill/ knowledge acquisition, and preferred modes for further training. 
CONCLUSIONS: The survey highlights the pressing need for expansion of the rongoā Māori workforce and training/service funding, to sustain rongoā practice. The findings add to what little is known about the training pathways and aspirations of practising healers, identified targets of the Māori Health Workforce Development Plan 2006.

text: Mā mahi, ka ora: by work, we prosper--traditional healers and workforce development. AIM: Rongoā Māori practitioners make a valuable contribution towards Māori health outcomes, albeit with limited resourcing or formal training. This paper reports on a survey of healers/healing practices-specifically healers' aspirations for professional development and training-and considers the implications for healing practice and future training undertakings. 
METHODS: Healers in seven districts around the country were surveyed about rongoā practice and service delivery during 2013. Consenting healers completed surveys either in person, via phone, or returned them via post, according to their preference and convenience. Resulting data were analysed and reported according to frequency of responses. 
RESULTS: Thirty-eight healers/rongoā clinics completed the survey--a 79% response rate. Respondents were primarily Māori (88%), female (69%), aged 50 years or older (60%), and worked as volunteers. Informal training modes focused on te reo, mātauranga and tikanga were the most common means of skill/ knowledge acquisition, and preferred modes for further training. 
CONCLUSIONS: The survey highlights the pressing need for expansion of the rongoā Māori workforce and training/service funding, to sustain rongoā practice. The findings add to what little is known about the training pathways and aspirations of practising healers, identified targets of the Māori Health Workforce Development Plan 2006.

year: 2015