arXiv subjects

Xiang-Yang Wang

Publications and source records attributed to Xiang-Yang Wang.

4 recordsLinked to original sources

arXiv2020

Gromov Hyperbolic Graphs Arising From Iterations

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor $K$. This setup was taken up to study the probabilistic potential theory on $K$, and the bi-Lipschitz equivalence on $K$. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of "weighted" IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

math.MG↗

arXiv2020

Immunological determinants of clinical outcomes in COVID-19: A quantitative perspective

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has a variable clinical presentation that ranges from asymptomatic, to severe disease with cytokine storm. The mortality rates also differ across the globe, ranging from 0.5-13%. This variation is likely due to both pathogen and host factors. Host factors may include genetic differences in the immune response genes as well as variation in HLA and KIR allotypes. To better understand what impact these genetic variants in immune response genes may have in the differences observed in the immune response to SARS-CoV-2, a quantitative analysis of a dynamical systems model that considers both, the magnitude of viral growth, and the subsequent innate and adaptive response required to achieve control of infection is considered. Based on this broad quantitative framework it may be posited that the spectrum of symptomatic to severely symptomatic presentations of COVID19 represents the balance between innate and adaptive immune responses. In asymptomatic patients, prompt and adequate adaptive immune response quells infection, whereas in those with severe symptoms a slower inadequate adaptive response leads to a runaway cytokine cascade fueled by ongoing viral replication. Polymorphisms in the various components of the innate and adaptive immune response may cause altered immune response kinetics that would result in variable severity of illness. Understanding how this genetic variation may alter the response to SARS-CoV-2 infection is critical to develop successful treatment strategies.

q-bio.CB↗

arXiv2016

On Hyperbolic graphs induced by iterated function systems

For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the Hölder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies (\cite {[Ka]}, \cite{[LW1]}, \cite{[W]}) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed.

math.MG↗

arXiv2012

Graphs induced by iterated function systems

For an iterated function system (IFS) of simillitidues, we define two graphs on the representing symbolic space. We show that if the self-similar set $K$ has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic, moreover the hyperbolic boundaries are homeomorphic to the self-similar sets.

math.FA↗