Properties of the linearisation of the quadratic transformation of genetic algebras

Willcox, William Denys (1982) Properties of the linearisation of the quadratic transformation of genetic algebras. MPhil thesis, Thames Polytechnic.

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Abstract

In this thesis we study the linearisation of the quadratic transformation of commutative baric algebras due to Holgate (44), elaborated and applied by Abraham (1-5). Holgate studied the quadratic transformation 4-:A—* A, x<f>= x in special train algebras and showed that they possess a plenary train. In the proof he shows that can be linearised over a higher dimensional space B in the sense that there exist a map R:A—••* B and a linear map^ on B such that = xR<4>I' (lithe projection B onto A). Abraham applies this linearisation to give explicit formulae for plenary sequences in Schafer genetic algebras for polyploidy.

Following remarks of both Abraham and Holgate our aim was to investigate the application of the linearisation to algebras corresponding to more complex modes of inheritance and to investigate the properties of algebras in which this linearisation exists with a view to obtaining a more natural characterisation of algebras arising in genetics.

Our achievements are to have extended the linearization to continuous time models, to have exhibited limitations to its further extension, to have given a method of constructing algebras possessing the linearisation and to have given an alternative technique that achieves the same ends by more standard linear algebraic methods.

We decided to include a survey of all relevant work that was scattered amongst papers ranging over some forty years when we commenced work. This year a text, WOrz-Busekros (58), has been published which does a very complete job of bringing the subject within the confines of a single volume. However she only briefly mentions linearisation and our survey is restricted to what we need to discuss this.

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