Seonhee Lim

Integrated multimodal network approach to PET and MRI based on multidimensional persistent homology

Finding underlying relationships among multiple imaging modalities in a coherent fashion is one of the challenging problems in multimodal analysis. In this study, we propose a novel approach based on multidimensional persistence. In the extension of the previous threshold‐free method of persistent homology, we visualize and discriminate the topological change of integrated brain networks by varying not only threshold but also mixing ratio between two different imaging modalities. The multidimensional persistence is implemented by a new bimodal integration method called 1D projection. When the mixing ratio is predefined, it constructs an integrated edge weight matrix by projecting two different connectivity information onto the one dimensional shared space. We applied the proposed methods to PET and MRI data from 23 attention deficit hyperactivity disorder (ADHD) children, 21 autism spectrum disorder (ASD), and 10 pediatric control subjects. From the results, we found that the brain networks of ASD, ADHD children and controls differ, with ASD and ADHD showing asymmetrical changes of connected structures between metabolic and morphological connectivities. The difference of connected structure between ASD and the controls was mainly observed in the metabolic connectivity. However, ADHD showed the maximum difference when two connectivity information were integrated with the ratio 0.6. These results provide a multidimensional homological understanding of disease‐related PET and MRI networks that disclose the network association with ASD and ADHD. Hum Brain Mapp 38:1387–1402, 2017.

Subword complexity and Sturmian colorings of regular trees

In this article, we discuss subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ‘infinite word’. We further give a complete characterization of the quotient graph for eventually periodic colorings.

Hyperbolic Tessellation and Colorings of Trees

We study colorings of a tree induced from isometries of the hyperbolic plane given an ideal tessellation. We show that, for a given tessellation of the hyperbolic plane by ideal polygons, a coloring can be associated with any element of Isom(), and the element is a commensurator of if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse.

Farey maps, Diophantine approximation and Bruhat-Tits tree

Based on Broise-Alamichel and Paulins work on the Gauss map corresponding to the principal convergents via the symbolic coding of the geodesic flow of the continued fraction algorithm for formal power series with coefficients in a finite field, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-one map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.

Continued fraction algorithm for Sturmian colorings of trees

Factor complexity

On the distribution of orbits of geometrically finite hyperbolic groups on the boundary

b_\phi(n)

MINIMAL VOLUME ENTROPY FOR GRAPHS

for a vertex coloring

Covering theory for complexes of groups

\phi

Counting Overlattices in Automorphism Groups of Trees

of a regular tree is the number of colored

COUNTING OVERLATTICES FOR POLYHEDRAL COMPLEXES

n