title: A representation of the natural numbers by means of cycle-numbers, with consequences in number theory

authors: Turner, John C.

subjects: Cycle-number

abstract: In this paper we give rules for creating a number triangle T in a manner analogous to that for producing Pascal's arithmetic triangle; but all of its elements belong to {0, 1}, and cycling of its rows is involved in the creation. The method of construction of any one row of T from its preceding rows will be defined, and that, together with starting and boundary conditions, will suffice to define the whole triangle, by sequential continuation. We shall use this triangle in order to define the so-called cycle-numbers, which can be mapped to the natural numbers. T will be called the 'cyclenumber triangle'. First we shall give some theorems about relationships between the cyclenumbers and the natural numbers, and discuss the cycling of patterns within the triangle's rows and diagonals. We then begin a study of figures (i.e. (0,1)- patterns, found on lines, triangles and squares, etc.) within T. In particular, we shall seek relationships which tell us something about the prime numbers. For our later studies, we turn the triangle onto its side and work with a doubly-infinite matrix C. We shall find that a great deal of cycling of figures occurs within T and C, and we exploit this fact whenever we can. The phenomenon of cycling patterns leads us to muse upon a 'music of the integers', indeed a 'symphony of the integers', being played out on the cycle-number triangle or on C. Like Pythagoras and his 'music of the spheres', we may well be the only persons capable of hearing it!.

text: A representation of the natural numbers by means of cycle-numbers, with consequences in number theory In this paper we give rules for creating a number triangle T in a manner analogous to that for producing Pascal's arithmetic triangle; but all of its elements belong to {0, 1}, and cycling of its rows is involved in the creation. The method of construction of any one row of T from its preceding rows will be defined, and that, together with starting and boundary conditions, will suffice to define the whole triangle, by sequential continuation. We shall use this triangle in order to define the so-called cycle-numbers, which can be mapped to the natural numbers. T will be called the 'cyclenumber triangle'. First we shall give some theorems about relationships between the cyclenumbers and the natural numbers, and discuss the cycling of patterns within the triangle's rows and diagonals. We then begin a study of figures (i.e. (0,1)- patterns, found on lines, triangles and squares, etc.) within T. In particular, we shall seek relationships which tell us something about the prime numbers. For our later studies, we turn the triangle onto its side and work with a doubly-infinite matrix C. We shall find that a great deal of cycling of figures occurs within T and C, and we exploit this fact whenever we can. The phenomenon of cycling patterns leads us to muse upon a 'music of the integers', indeed a 'symphony of the integers', being played out on the cycle-number triangle or on C. Like Pythagoras and his 'music of the spheres', we may well be the only persons capable of hearing it!.

year: 2013

title: Optimal Task Scheduling in a Flexible Manufacturing System using Model Checking

authors: Malik, Robi

subjects: Applications

abstract: This paper demonstrates the use of model checking to solve the problem of optimal task scheduling in a flexible manufacturing system. The system is modelled as a discrete event system, for which the least restrictive safe behaviour is synthesised according to supervisory control theory. Then timing constraints are added to the model in the form of extended finite-state machines, and time-optimal schedules are computed using the discrete event systems and model checking tool Supremica. In the case study considered in this paper, which previously was only solved heuristically, the method successfully produces optimal schedules to manufacture up to 30 products of two different types. The method is furthermore used to find an optimal cycle, solving the scheduling problem of the case study for an arbitrary number of products in optimal or asymptotically close to optimal time.

text: Optimal Task Scheduling in a Flexible Manufacturing System using Model Checking This paper demonstrates the use of model checking to solve the problem of optimal task scheduling in a flexible manufacturing system. The system is modelled as a discrete event system, for which the least restrictive safe behaviour is synthesised according to supervisory control theory. Then timing constraints are added to the model in the form of extended finite-state machines, and time-optimal schedules are computed using the discrete event systems and model checking tool Supremica. In the case study considered in this paper, which previously was only solved heuristically, the method successfully produces optimal schedules to manufacture up to 30 products of two different types. The method is furthermore used to find an optimal cycle, solving the scheduling problem of the case study for an arbitrary number of products in optimal or asymptotically close to optimal time.

year: 2018

title: Evaluation of Rongo Atea: Alcohol and Other Drug Treatment Centre for Adolescents

authors: Paki, Helen Mary

subjects: Alcohol

abstract: Abstract Rongo Ātea is a residential abstinence-based Kaupapa Māori alcohol and other drug programme accommodating youth between the ages of 13 and 17. This evaluation investigates the role that staff play at Rongo Ātea along with a number of other factors including the physical environment; stages of change; programme implementation and aftercare. As a Kaupapa Māori programme, the role of how culture and identity can influence positive change is also explored along with the development of a youth-focused approach. I utilised a three phase framework borrowed from the work of Mason Durie (2008) to organise themes. These three phases, which Durie (2008) based on marae encounters, include Whakapiri (Engagement); Whakamarama (Enlightenment); and Whakamana (Empowerment) as they aptly reflect a three phase intervention approach: detoxification and early programme engagement; learning and development through ongoing programme commitment; and post treatment outcomes and aftercare. The data collection phase of this evaluation took place in 2006 and was initiated by the manager of Rongo Ātea who requested an evaluation to identify programme strengths and limitations from the perspectives of staff and students, and to make recommendations to Rongo Ātea that would assist with further programme developments. I utilised a collaborative and participatory approach (Bishop, 1996; Patton, 1990). Kaupapa Māori research principles were reflected in the use of ‘kanohi kitea’— face to face contact (Smith, 1999). With appropriate training and management support, staff could have a greater influence on programme outcomes. Evaluation findings suggest that drug and alcohol intervention and treatment for young people in New Zealand is significantly under-resourced, particularly in the areas of detoxification and aftercare. To be effective, residential treatment programmes should consist of a three stage programme covering detoxification; treatment; and aftercare incorporating an integrated approach. A greater emphasis on working with whānau alongside the young person is recommended.

text: Evaluation of Rongo Atea: Alcohol and Other Drug Treatment Centre for Adolescents Abstract Rongo Ātea is a residential abstinence-based Kaupapa Māori alcohol and other drug programme accommodating youth between the ages of 13 and 17. This evaluation investigates the role that staff play at Rongo Ātea along with a number of other factors including the physical environment; stages of change; programme implementation and aftercare. As a Kaupapa Māori programme, the role of how culture and identity can influence positive change is also explored along with the development of a youth-focused approach. I utilised a three phase framework borrowed from the work of Mason Durie (2008) to organise themes. These three phases, which Durie (2008) based on marae encounters, include Whakapiri (Engagement); Whakamarama (Enlightenment); and Whakamana (Empowerment) as they aptly reflect a three phase intervention approach: detoxification and early programme engagement; learning and development through ongoing programme commitment; and post treatment outcomes and aftercare. The data collection phase of this evaluation took place in 2006 and was initiated by the manager of Rongo Ātea who requested an evaluation to identify programme strengths and limitations from the perspectives of staff and students, and to make recommendations to Rongo Ātea that would assist with further programme developments. I utilised a collaborative and participatory approach (Bishop, 1996; Patton, 1990). Kaupapa Māori research principles were reflected in the use of ‘kanohi kitea’— face to face contact (Smith, 1999). With appropriate training and management support, staff could have a greater influence on programme outcomes. Evaluation findings suggest that drug and alcohol intervention and treatment for young people in New Zealand is significantly under-resourced, particularly in the areas of detoxification and aftercare. To be effective, residential treatment programmes should consist of a three stage programme covering detoxification; treatment; and aftercare incorporating an integrated approach. A greater emphasis on working with whānau alongside the young person is recommended.

year: 2010