## Welcome from Jim Hamilton Adams

## This is the mathematics archive.

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The mathematics archive contains work which led up to the production of the three eBooks, *Superexponential algebra*, *Innovation in mathematics* and *Elementary methods in number theory*. The text that accompanies the files is the last such text before transfer to this archive.

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__Historical background__

1. Archived Quintic Solution Comparison Method Attempts Attempts at solving the quintic with adjoined roots using comparison methods against a solvable sextic and octic.

2. Algebra, Space and Logic Chapter headings of an intended third eBook, now incorporated in Volume III of Superexponential algebra.

3. Algorithms and Consistency was a sketch of future work which will now become chapter XVI of * Superexponential algebra*.

4A. Foundations of Hyperintricate Matrices This was the 2013 version of Superexponential Algebra, saying: material will be added to chapters VII, VIII and XI, but otherwise this work is ready. The article is an overview of the subject in eleven chapters.

4B. Applications of Hyperintricate Matrices This 2013 version contains chapter headings.

5. Polynomial Equations for Non Commutative Algebras This developmental work demonstrated that Galois theory carries over to 2 X 2 matrix variables when abelian coefficients operate with them, to 'J-abelian' 2^{n} X 2^{n} matrices and also purported to show Galois theory carries over for all matrices. For non-commutative algebras new solutions may be found when they are available in the classical cases (degree less than 5). The genesis of this idea is based on many fruitless hours investigating Galois theory. We show unique factorisation breaks down for matrix polynomials. Solution by hyperintricate parts is an additional constraint which can be removed.

6. Hyperintricate Rings This developmental work was to be be extended to deal with modules and that of Lie and other groups, which provides an alternative methodology. Completion is not now scheduled.

7. E-mail to Doly Garcia gave the reason at the time for research into Galois theory. This note has been superseded by development of the theory.

8. Superexponentiation Since some of these ideas had leaked out, this uncompleted work gave a description of work then in progress on higher order exponentiation, involving the study of 'intricate numbers' - a particular representation of GL(n). These were around in July 2008.

__The topology work __will now become part of Volume III of * Superexponential algebra* - it describes branched spaces and their Euler characteristics. I would like to thank James Hirschfeld for getting a research student to look at the homology and cohomology.

1B. The Semantics of Branched Spaces Chapter 1 is ready - a topology popularisation that describes branched spaces and their Euler characteristics.

1C. Ladder Numbers for Branched Spaces Chapter 2 is ready and introduces ladder numbers for branched spaces, which will be later applied to explosions. The theory is a slight modification of that given in the Mathematics Introduction part of this website.

1D. Branched Boundaries Chapter 3, just begun, uses the tree definition of a group to describe branched spaces and their boundaries.

2. Chromotopic Algebra This topology research will describe homotopy for branched spaces, called chromotopy. We investigate the Euler characteristic derived from article 1A above, in terms of chromotopy. There is a third part now indicated briefly in this paper, and is the connection with lattice gauge theory and quark confinement. It is clear that a quark branched space is incompatible with an exterior "familiar space" topology, which humans inhabit. Scheduled completion is by May 2015.

3A. Explosions This is an announcement to develop explosion analysis, combining the work on branched spaces above with that of ladder numbers in section 5 of the Mathematics - Introduction.

3B. A Model of Explosions In this preliminary sketch, a 'Garcia model' of explosions uses the branched Euler characteristic combined with the results on ladder numbers in 5 of the Mathematics - Introduction and item 2 of this part.

3C. Branched Space de Rham Theory This second preliminary sketch discusses differential and integral structures for branched spaces.

3D. Chromotopic Functors This describes extraction and assembly of chromotopic objects, the basic building blocks of chromotopic algebra.

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1A. Rescaling Polynomial Probabilities This work is ready and is part of research which I undertook 35 years ago on probability logics. An extension on probability toposes has been inserted. I generalise to a non-linear polynomial for the NOT operation. There are corresponding polynomials fitting the "certain" (true) and "impossible" (false) edges of a truth table in more than one variable and polynomials fitting edges with values in probabilities. The logic of these operations in the interior is generally non-commutative. The reason I investigated these was that I was interested in dependent probabilities of events which might happen in nuclear power stations, where I thought the mathematical basis of calculations was invalid, being based on independent probabilities.

1B. Classical Polynomial Rings, U Theory and Probability A pot pourri combining zeros of polynomial rings, multiplicative inverses of polynomials and relating it to probability for polynomials.

1C. Varieties and Ideals Not ready at all. I found that I was referring to the above two articles when I began to clarify my ideas on this subject by writing about it.

2A. Beal's Conjecture investigates Beal's conjecture by elementary means.

2B. Fermat's Last Theorem proves Fermat's last theorem by elementary methods in the 'e = 1' case.

2C. Elementary Methods and Fermat's Last Theorem discusses whether it is feasible to prove Fermat's last theorem by elementary methods.

3. Hyperintricate Number Theory is an assortment of number theoretic results.

4A. Exponentiation. Over 100 theorems covering generalised Fermat and Mersenne numbers, prime number theorems also related to differences and sums of powers and reciprocity theorems.

4B. Quadronacci describes a formula for a sequence of complex numbers, each of which is a sum of factors of the previous four.

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