3 research outputs found
A Euclid style algorithm for MacMahon's partition analysis
Solutions to a linear Diophantine system, or lattice points in a rational
convex polytope, are important concepts in algebraic combinatorics and
computational geometry. The enumeration problem is fundamental and has been
well studied, because it has many applications in various fields of
mathematics. In algebraic combinatorics, MacMahon's partition analysis has
become a general approach for linear Diophantine system related problems. Many
algorithms have been developed, but "bottlenecks" always arise when dealing
with complex problems. While in computational geometry, Barvinok's important
result asserts the existence of a polynomial time algorithm when the dimension
is fixed. However, the implementation by the LattE package of De Loera et. al.
does not perform well in many situations. By combining excellent ideas in the
two fields, we generalize Barvinok's result by giving a polynomial time
algorithm for MacMahon's partition analysis in a suitable condition. We also
present an elementary Euclid style algorithm, which might not be polynomial but
is easy to implement and performs well. As applications, we contribute the
generating series for magic squares of order 6.Comment: 28 pages, some modification, add the link to the Maple package
CTEucli
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The life and work of Major Percy Alexander MacMahon
This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context