55 research outputs found
Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence
Consider the generalized iterated wreath product of symmetric groups. We give a complete description of the traversal
for the generalized iterated wreath product. We also prove an existence of a
bijection between the equivalence classes of ordinary irreducible
representations of the generalized iterated wreath product and orbits of labels
on certain rooted trees. We find a recursion for the number of these labels and
the degrees of irreducible representations of the generalized iterated wreath
product. Finally, we give rough upper bound estimates for fast Fourier
transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv
admin note: text overlap with arXiv:1409.060
Generalized iterated wreath products of cyclic groups and rooted trees correspondence
Consider the generalized iterated wreath product where . We
prove that the irreducible representations for this class of groups are indexed
by a certain type of rooted trees. This provides a Bratteli diagram for the
generalized iterated wreath product, a simple recursion formula for the number
of irreducible representations, and a strategy to calculate the dimension of
each irreducible representation. We calculate explicitly fast Fourier
transforms (FFT) for this class of groups, giving literature's fastest FFT
upper bound estimate.Comment: 15 pages, to appear in Advances in the Mathematical Science
Representation Theoretical Methods in Image Processing
Image processing refers to the various operations performed on pictures that are digitally stored as an aggregate of pixels. One can enhance or degrade the quality of an image, artistically transform the image, or even find or recognize objects within the image. This paper is concerned with image processing, but in a very mathematical perspective, involving representation theory. The approach traces back to Cooley and Tukey’s seminal paper on the Fast Fourier Transform (FFT) algorithm (1965). Recently, there has been a resurgence in investigating algebraic generalizations of this original algorithm with respect to different symmetry groups. My approach in the following chapters is as follows. First, I will give necessary tools from representation theory to explain how to generalize the Discrete Fourier Transform (DFT). Second, I will introduce wreath products and their application to images. Third, I will show some results from applying some elementary filters and compression methods to spectrums of images. Fourth, I will attempt to generalize my method to noncyclic wreath product transforms and apply it to images and three-dimensional geometries
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing
measurements of some quantity of interest. Usually, as a first step, features
of the time series need to be extracted. These are numerical quantities that
aim to succinctly describe the data and to dampen the influence of noise. In
some applications, these features are also required to satisfy some invariance
properties. In this paper, we concentrate on time-warping invariants. We show
that these correspond to a certain family of iterated sums of the increments of
the time series, known as quasisymmetric functions in the mathematics
literature. We present these invariant features in an algebraic framework, and
we develop some of their basic properties.Comment: 18 pages, 1 figur
- …