In this paper, the Hierarchical Model is studied near a non-trivial fixed point φɛ of its renormalization group. Our analysis is an extension of work of Bleher and Sinai. We prove the validity of the ɛ-expansion for φɛ. We then show that the renormalization transformations around φɛ have an unstable manifold which is completely characterized by the tangent map and can be brought to normal form. We then establish relations between this result and the critical behaviour of the model in the thermodynamic limit.

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A note on cyclic difference sets

The epsilon -expansion for the hierarchical model.

Following the Backlund transformation and using the theorems proved by the same authors, previously we obtained sets of new solutions to the KdV equation and the nonlinear equation \({y_t} + {y_{xxx}} - 6{y^2}{y_x} + 6\lambda {y_x} = 0\) which transforms into the modified KdV equation when λ = 0 In this paper we present another new series of solutions to the above nonlinear equation

A Series of New Exact Solutions to the Nonlinear Equation \({y_t} + {y_{xxx}} - 6{y^2}{y_x} + 6\lambda {y_x} = 0\)

Peter Neumann was born in Oxford on 28 December 1940, the first son of mathematicians Bernhard and Hanna Neumann, who came to the UK from Germany in the 1930s. Peter obtained his DPhil in 1963 under the supervision of Graham Higman, after which he was appointed to a fellowship at Queen's and a university lectureship. He spent his entire career at Oxford, retiring in 2008. Peter's lifelong contribution to mathematics in the UK and worldwide was monumental and wide-ranging: through his research in algebra and the history of algebra; his supervision of over 40 doctoral students, many of whom went on to have distinguished academic careers; his extensive service to the London Mathematical Society; and his enormous contribution to mathematics education. Peter was a leading figure in algebra for over 50 years, publishing around 100 papers and books on varieties of groups, soluble groups, group enumeration, permutation groups, computational algebra, and the history of algebra. He was a great collaborator, publishing with 38 different co-authors, and holding visiting positions at many places around the world. His contributions to research and scholarship were recognised by the London Mathematical Society with the award of the Senior Whitehead Prize in 2003 and with the joint LMS–IMA David Crighton Medal in 2012. Peter also made an enormous contribution to mathematics education in the UK. He was the founding Chairman of UK Mathematics Trust, serving from 1996 to 2005. During his period as chairman, Peter led UKMT in taking on the staging of the 2002 International Mathematical Olympiad. For his services to mathematics education, Peter was awarded an OBE in 2008. Peter was a superb lecturer, writer and expositor, and delightful company with a sparkling sense of humour. Above all, Peter is remembered for his kindness and generosity, his infectious and inspirational enthusiasm, and his boundless energy. 

Peter Michael Neumann, 1940–2020

/pdf/decomposition-fields-of-difference-sets-3ju7sdsz79.pdf

A note on cyclic difference sets

Citations

The epsilon -expansion for the hierarchical model.

A Series of New Exact Solutions to the Nonlinear Equation \({y_t} + {y_{xxx}} - 6{y^2}{y_x} + 6\lambda {y_x} = 0\)

Peter Michael Neumann, 1940–2020

References

Decomposition fields of difference sets

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