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: Interfacial microstructure and shear strength of Ti-6Al-4V/TiAl laminate composite sheet fabricated by hot packed rolling

authors: Kong, Fantao

subjects: Intermetallics

abstract: A two layer Ti-6Al-4V(wt%)/Ti-43Al-9V-Y(at%) laminate composite sheet with a uniform interfacial microstructure and no discernible defects at the interfaces has been prepared by hot-pack rolling, and its interfacial microstructure and shear strength were characterized. Characterization of the interfacial microstructure shows that there was an interfacial region of uniform thickness of about 250 μm which consisted of two layers: Layer I on the TiAl side which was 80 μm thick and Layer II on the Ti-6Al-4V side which was 170 μm thick. The microstructure of Layer I consisted of massive γ phases, needlelike γ phases and B2 phase matrix, while the microstructure of Layer II consisted of α₂ phase. The microstructure of the interfacial region is the result of the interdiffusion of Ti element from Ti-6Al-4V alloy layer into the TiAl alloy layer and Al element from the TiAl alloy layer into the Ti-6Al-4V alloy layer. The shear strength measurement demonstrated that the bonding strength between the TiAl alloy and Ti-6Al-4V alloy layers in the laminate composite sheet was very high. This means that the quality of the interfacial bonding between the two layers achieved by the multi-path rolling is high, and the interface between the layers is very effective in transferring loading, causing significantly improved toughness and plasticity of the TiAl/Ti-6Al-4V laminate composite sheet.

text: Interfacial microstructure and shear strength of Ti-6Al-4V/TiAl laminate composite sheet fabricated by hot packed rolling A two layer Ti-6Al-4V(wt%)/Ti-43Al-9V-Y(at%) laminate composite sheet with a uniform interfacial microstructure and no discernible defects at the interfaces has been prepared by hot-pack rolling, and its interfacial microstructure and shear strength were characterized. Characterization of the interfacial microstructure shows that there was an interfacial region of uniform thickness of about 250 μm which consisted of two layers: Layer I on the TiAl side which was 80 μm thick and Layer II on the Ti-6Al-4V side which was 170 μm thick. The microstructure of Layer I consisted of massive γ phases, needlelike γ phases and B2 phase matrix, while the microstructure of Layer II consisted of α₂ phase. The microstructure of the interfacial region is the result of the interdiffusion of Ti element from Ti-6Al-4V alloy layer into the TiAl alloy layer and Al element from the TiAl alloy layer into the Ti-6Al-4V alloy layer. The shear strength measurement demonstrated that the bonding strength between the TiAl alloy and Ti-6Al-4V alloy layers in the laminate composite sheet was very high. This means that the quality of the interfacial bonding between the two layers achieved by the multi-path rolling is high, and the interface between the layers is very effective in transferring loading, causing significantly improved toughness and plasticity of the TiAl/Ti-6Al-4V laminate composite sheet.

year: 2011

title: Ngā mahi: Kaupapa Māori outcomes and indicators kete

authors: Jefferies, Richard

subjects: New Zealand

abstract: As part of the Planning Under Cooperative Mandates research (PUCM) we developed a kaupapa Māori outcomes and indicators framework. This framework reflects tikanga Māori and Māori values. It also considers issues (including environmental issues) according to those tikanga that particular issues invoke. For example, tikanga brought into play in relation to council treatment and disposal of sewerage include tapu (sacred), and might also impact on the mauri (life-force) of water if treated effluent were to enter waterways.

The intention of our research has been to clarify and define key Māori environmental concepts so that stakeholders (including council staff) will have a terms of reference against which they can compare desired environmental outcomes from different perspectives and be better placed to integrate Māori environmental outcomes into planning processes. The end-point is this Nga Mahi kete report containing tikanga-based worksheets and the guidelines for using them.

text: Ngā mahi: Kaupapa Māori outcomes and indicators kete As part of the Planning Under Cooperative Mandates research (PUCM) we developed a kaupapa Māori outcomes and indicators framework. This framework reflects tikanga Māori and Māori values. It also considers issues (including environmental issues) according to those tikanga that particular issues invoke. For example, tikanga brought into play in relation to council treatment and disposal of sewerage include tapu (sacred), and might also impact on the mauri (life-force) of water if treated effluent were to enter waterways.

The intention of our research has been to clarify and define key Māori environmental concepts so that stakeholders (including council staff) will have a terms of reference against which they can compare desired environmental outcomes from different perspectives and be better placed to integrate Māori environmental outcomes into planning processes. The end-point is this Nga Mahi kete report containing tikanga-based worksheets and the guidelines for using them.

year: 2009-06-30