The Partition Method for a Power Series Expansion: Theory and Applications - Hardcover

Über die Autorin bzw. den Autor

Dr Victor Kowalenko is a Senior Research Fellow in the Department of Mathematics and Statistics, University of Melbourne, Australia. Since 2009, he has been associated with the ARC Centre of Excellence in Mathematics and Statistics of Complex Systems. He began his research career by joining the DSTO’s railgun project in Maribyrnong in the early 1980’s before transferring to the DSTO facility at Fishermen’s Bend to work on aeronautical systems. He then returned to the Department of Physics, University of Melbourne as one of the inaugural Australian Research Fellows to work on particle-anti-particle plasmas and general relativistic magnetohydrodynamics. It was here that he introduced the partition method for a power expansion. Between 2001 and 2003, when he was a Senior Research Fellow in the School of Computer Science and Software Engineering, Monash University, he was able to develop the method further and to extend it to intractable problems in mathematics and physics.

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This book explores how the method known as the partition method for a power series expansion, which has been developed by the author, can be applied to a host of previously intractable problems in mathematics and physics.

In particular, this book describes how the method can be used to determine the Bernoulli, cosecant and reciprocal logarithm numbers, which appear as the coefficients of the resulting power series expansions and then extends the method to more complicated situations where the coefficients become polynomials or mathematical functions. From these examples a general theory for the method is presented, which enables a programming methodology to be established.

Because the coefficients in the power series expansions require the compositions of all the partitions that sum to their order, this book also presents the bivariate recursive central partition (BRCP) algorithm, which is able to process the partitions more efficiently via a tree approach as opposed to standard lexicographic methods. Another advantage of the algorithm is its ability to solve diverse problems in the theory of partitions with minor modification. Finally, the programming techniques of previous chapters are used to derive power series expansions for complex generating functions arising in the theory of partitions and in lattice models of statistical mechanics.

• Explains the partition method for a power series expansion by presenting elementary applications involving the Bernoulli, cosecant and reciprocal logarithm numbers
• Compares generating partitions via the BRCP algorithm with the standard lexicographic approaches
• Describes how to programme the partition method for a power series expansion and BRCP algorithm