38 research outputs found
Homomesies on permutations -- an analysis of maps and statistics in the FindStat database
In this paper, we perform a systematic study of permutation statistics and
bijective maps on permutations in which we identify and prove 122 instances of
the homomesy phenomenon. Homomesy occurs when the average value of a statistic
is the same on each orbit of a given map. The maps we investigate include the
Lehmer code rotation, the reverse, the complement, the Foata bijection, and the
Kreweras complement. The statistics studied relate to familiar notions such as
inversions, descents, and permutation patterns, and also more obscure
constructs. Beside the many new homomesy results, we discuss our research
method, in which we used SageMath to search the FindStat combinatorial
statistics database to identify potential homomesies
Loops, Knots, Gauge Theories
This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication