title: Hamilton-Jacobi Theory and Superintegrable Systems

authors: Armstrong, Craig Keith

subjects: Hamilton-Jacobi theory

abstract: Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found.

 We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

text: Hamilton-Jacobi Theory and Superintegrable Systems Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found.

 We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

year: 2007

title: αₙ-Designs

authors: Ruggiero, Katya

abstract: This thesis defines a broad class of resolvable incomplete block designs for multifactor experiments, called αₙ-designs. A general methodology for their construction is described and is shown to be a natural extension of the method used by Patterson and Williams (1976) in their construction of α-designs. In fact, the class of α-designs are a special case of αₙ-designs with n = 1. 

The family of αₙ-designs was primarily developed for their use in factorial experiments. While they are particularly suitable for this purpose, for some combinations of design parameters they also provide more efficient designs than the best available α-designs.

Algorithms for the generation of efficient designs require computationally expensive eigenvalue calculations. Therefore, considerable attention is devoted to the study of the structural properties of αₙ-designs and how these can be used to develop computationally fast algorithms. In this endeavour it is shown that the treatment concurrence and information matrices have an important property known as BⁿC-structure. This is used to derive a closed form mathematical expression for the average efficiency factor of any generalized interaction. It is further shown that this expression can be written recursively. These results, combined with the fact than an αₙ-design is completely specified by a small generating array, mean that computer search algorithms can be developed to quickly find highly efficient designs.

A desirable feature in a factorial design is the property of orthogonal factorial structure. The family of αₙ-designs do not, in general, admit this property. However, another class of n-dimensional cyclic designs, known as n-cyclic designs, suitable for non-resolvable designs for factorial experiments are also considered in this thesis. They possess orthogonal factorial structure and include a sizeable subset of designs which are resolvable. These resolvable n-cyclic designs are shown to be αₙ-designs.

The discovery and subsequent development of the family of αₙ-designs has stimulated a new set of interesting questions in relation to factorial experiments. These are discussed at the end of the thesis.

text: αₙ-Designs This thesis defines a broad class of resolvable incomplete block designs for multifactor experiments, called αₙ-designs. A general methodology for their construction is described and is shown to be a natural extension of the method used by Patterson and Williams (1976) in their construction of α-designs. In fact, the class of α-designs are a special case of αₙ-designs with n = 1. 

The family of αₙ-designs was primarily developed for their use in factorial experiments. While they are particularly suitable for this purpose, for some combinations of design parameters they also provide more efficient designs than the best available α-designs.

Algorithms for the generation of efficient designs require computationally expensive eigenvalue calculations. Therefore, considerable attention is devoted to the study of the structural properties of αₙ-designs and how these can be used to develop computationally fast algorithms. In this endeavour it is shown that the treatment concurrence and information matrices have an important property known as BⁿC-structure. This is used to derive a closed form mathematical expression for the average efficiency factor of any generalized interaction. It is further shown that this expression can be written recursively. These results, combined with the fact than an αₙ-design is completely specified by a small generating array, mean that computer search algorithms can be developed to quickly find highly efficient designs.

A desirable feature in a factorial design is the property of orthogonal factorial structure. The family of αₙ-designs do not, in general, admit this property. However, another class of n-dimensional cyclic designs, known as n-cyclic designs, suitable for non-resolvable designs for factorial experiments are also considered in this thesis. They possess orthogonal factorial structure and include a sizeable subset of designs which are resolvable. These resolvable n-cyclic designs are shown to be αₙ-designs.

The discovery and subsequent development of the family of αₙ-designs has stimulated a new set of interesting questions in relation to factorial experiments. These are discussed at the end of the thesis.

year: 2000

title: Comparative fish abundance in the shallow Waikato lakes Whangape and Hakanoa

authors: Hicks, Brendan J.

abstract: The Waikato-Tainui College for Research and Development contracted the University of Waikato to conduct boat electrofishing estimates of the fish abundance in lakes Whangape and Hakanoa; both are shallow, riverine lakes in the lower Waikato River floodplain and are about 2,000 years old. Lake Whangape (latitude 37.46853°S, longitude 175.05120°W) is a large (1,450 ha), shallow (maximum depth 2.7 m) lake to the west of the Waikato River; Lake Hakanoa (latitude 37.55258°S, longitude 175.16859°W) is a 52-ha lake in suburban Huntly with a maximum depth of 2.5 m.
We fished 10 sites for 10 mins in both lakes Whangape and Lake Hakanoa; sites fished were 253-609 m long (1,012-2,436 m² in area) in Lake Whangape and 125-276 m long (500-1,104 m² in area) in Lake Hakanoa.
No submerged aquatic macrophytes were seen in either lake. Fishing was conducted close to the water's edge in Lake Hakanoa, but the shallow margins in Lake Whangape restricted fishing at most sites to 30-70 m from the shoreline. The shoreline was much more accessible in Lake Hakanoa, and was dominated by raupo (Typha orientalis) with willows (Salix spp.) in places. Submerged tree trunks and logs in Lake Hakanoa made navigation difficult in parts of the lake margins.
We caught a total of 118 fish in Lake Whangape, where shortfin eels (Anguilla australis) were the most abundant fish, and 594 fish in Lake Hakanoa, where gambusia (Gambusia affinis) the most abundant species. In Lake Whangape, shortfin eels comprised the greatest total biomass (12.4 kg), with koi carp (Cyprinus carpio) almost equally abundant (11.4 kg). In Lake Hakanoa, koi carp were most abundant (74.4 kg), followed by shortfin eels (22.1 kg).
Fish were unevenly spread among sites in Lake Whangape and koi carp were relatively sparsely among sites. In Lake Hakanoa, in contrast, fish were more evenly spread among sites, but with higher concentrations of catfish (Ameiurus nebulosus) and goldfish (Carassius auratus) at a few sites. Biomass of fish caught in each of the ten 10-min fishing shots reflected the numerical distributions, with some wide variations between sites. Fish densities were generally lower in Lake Whangape than in Lake Hakanoa; however, common smelt (Retropinna retropinna) and grey mullet (Mugil cephalus) occurred in Whangape but not in Hakanoa. Areal fish biomass was dominated by shortfin eels and koi carp in Lake Whangape, but koi carp dominated the fish biomass in Lake Hakanoa, despite the smaller number of carp than eels. This was because of the large mean weight of koi carp (about 1,000 g) compared to the smaller shortfin eels (mean weight about 200 g in both lakes).

text: Comparative fish abundance in the shallow Waikato lakes Whangape and Hakanoa The Waikato-Tainui College for Research and Development contracted the University of Waikato to conduct boat electrofishing estimates of the fish abundance in lakes Whangape and Hakanoa; both are shallow, riverine lakes in the lower Waikato River floodplain and are about 2,000 years old. Lake Whangape (latitude 37.46853°S, longitude 175.05120°W) is a large (1,450 ha), shallow (maximum depth 2.7 m) lake to the west of the Waikato River; Lake Hakanoa (latitude 37.55258°S, longitude 175.16859°W) is a 52-ha lake in suburban Huntly with a maximum depth of 2.5 m.
We fished 10 sites for 10 mins in both lakes Whangape and Lake Hakanoa; sites fished were 253-609 m long (1,012-2,436 m² in area) in Lake Whangape and 125-276 m long (500-1,104 m² in area) in Lake Hakanoa.
No submerged aquatic macrophytes were seen in either lake. Fishing was conducted close to the water's edge in Lake Hakanoa, but the shallow margins in Lake Whangape restricted fishing at most sites to 30-70 m from the shoreline. The shoreline was much more accessible in Lake Hakanoa, and was dominated by raupo (Typha orientalis) with willows (Salix spp.) in places. Submerged tree trunks and logs in Lake Hakanoa made navigation difficult in parts of the lake margins.
We caught a total of 118 fish in Lake Whangape, where shortfin eels (Anguilla australis) were the most abundant fish, and 594 fish in Lake Hakanoa, where gambusia (Gambusia affinis) the most abundant species. In Lake Whangape, shortfin eels comprised the greatest total biomass (12.4 kg), with koi carp (Cyprinus carpio) almost equally abundant (11.4 kg). In Lake Hakanoa, koi carp were most abundant (74.4 kg), followed by shortfin eels (22.1 kg).
Fish were unevenly spread among sites in Lake Whangape and koi carp were relatively sparsely among sites. In Lake Hakanoa, in contrast, fish were more evenly spread among sites, but with higher concentrations of catfish (Ameiurus nebulosus) and goldfish (Carassius auratus) at a few sites. Biomass of fish caught in each of the ten 10-min fishing shots reflected the numerical distributions, with some wide variations between sites. Fish densities were generally lower in Lake Whangape than in Lake Hakanoa; however, common smelt (Retropinna retropinna) and grey mullet (Mugil cephalus) occurred in Whangape but not in Hakanoa. Areal fish biomass was dominated by shortfin eels and koi carp in Lake Whangape, but koi carp dominated the fish biomass in Lake Hakanoa, despite the smaller number of carp than eels. This was because of the large mean weight of koi carp (about 1,000 g) compared to the smaller shortfin eels (mean weight about 200 g in both lakes).

year: 2016