Brody curves in complicated sets

We prove the existence of a leaf, which is injective Brody in $\mathbb{P}^2$, in the foliation of the boundary of the set of non-escaping points for certain H\'enon mappings

Bergman and Caratheodory metrics of the Kohn-Nirenberg domains

The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.Comment: 11 page

Understanding Nanoscale Hierarchical Nature of Pore and Chemical Heterogeneity in Bone as a Function of Tissue Ages

Bone is a biocomposite material mainly composed of two major components (Type I collagen and hydroxyapatite crystals) and it is constructed in a hierarchical fashion ranging from a nano to macro scale. Bone has been known to have nanoscale mechanical heterogeneity, which is proposed to enhance bone toughening, but the underlying reasons for the nanoscale mechanical heterogeneity are unknown. Nanoscale bone structure and chemical composition are hypothesized to contribute to the nanoscale mechanical properties of bone. However, until recently, measuring nanoscale porosity and chemical composition in bone was not feasible due to a lack of appropriate characterization techniques. In this thesis, two novel characterization techniques, PALS (positron annihilation lifetime spectroscopy) and PTIR (photothermal infrared spectroscopy), were utilized and elucidated new structural and chemical information that were unknown to the field: 1) Bone has a hierarchical arrangement of nanoscale pores in bone; 2) The nanoscale mineral structure in bone is likely to be interconnected; and 3) The Amide I/mineral domain sizes range from ~50 to 500 nm, which agrees with the length scale of nanoscale mechanical maps. PALS has never been utilized to study nanoscale porosity in bone. Combining our PALS results with simulated pore size distribution (PSD) results from collagen molecule and microfibril structure, we identify pores with diameter of 0.6 nm that indicates porosity within the collagen molecule regardless of the presence of mineral and/or water. We find that water occupies three larger domain size regions with nominal mean diameters of 1.1 nm, 1.9 nm, and 4.0 nm—spaces that are hypothesized to associate with inter-collagen molecular spaces, terminal segments (d-spacing) within collagen microfibrils, and interface spacing between collagen and mineral structure, respectively. We revealed that similar to collagen and mineral structure, nanoscale porosity in bone is also constructed in a hierarchical fashion. Also, PALS data on the deproteinized bone samples showed an average spacing value between two mineral plates ranging from 5-6 nm, suggesting that the nanoscale mineral structure itself is likely to be interconnected. Combined with specific surface area (SSA) and PALS measurements, a range on the mean mineral plate thickness is deduced to 4-8 nm, which agrees with electron microscopy (EM) studies in literature. PTIR is further divided into two techniques, depending on the probe methods for signal detection; 1) atomic force microscopy-infrared spectroscopy (AFM-IR) and optical photothermal infrared spectroscopy (O-PTIR). We found that the average Amide I/mineral ratio values from AFM-IR decreased as a function of tissue ages, agreeing with the general mineralization trend measured for comparable tissue ages by Raman spectroscopy. However, in addition to the agreement of the average Amide I/mineral ratio values between AFM-IR and Raman spectroscopy, the Amide I/mineral ratio values calculated from full IR spectra and IR ratio maps collected by AFM-IR revealed reduced Amide I/mineral ratio ranges, as a function of tissue age, demonstrated by decreasing box and whisker ranges. Even though we found an overall decrease in average Amide I/mineral ratio values and ranges across the different bone samples, the Amide I/mineral ratio values from AFM-IR and O-PTIR exhibited location to location and sample to sample variations of the IR ratio values. Also, we discovered that the domain size range of the Amide I/mineral ratio maps (~50 to 500 nm) has the same length scale as nanoscale mechanical maps.PHDMacromolecular Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155230/1/tedahn_1.pd

Foliation Structure for Generalized Henon Mappings.

We consider a generalized Henon mapping f from the two dimensional complex plane to itself defined by f(z, w)=(p(z)-aw, z) where p(z) is a monic polynomial of degree greater than or equal to 2 and a is not equal to 0, and its Green function g of the two dimensional complex plane to the set of all real numbers. We show that there is no non-trivial holomorphic curve in the two dimensional projective space passing through I_+ and supported in the closure of {g=c} for c>0, where I_+ is a point of indeterminancy when f is projectivised. Next, we consider the foliation structure of a level set of the form {g=c} for c>0. It is known that each of them is foliated by conformal images of the set of all complex numbers and each leaf is dense. We prove each leaf is actually an injective Brody curve. More precisely, for any biholomorphic parametrization of any leaf, the Fubini-Study metric of its derivative is uniformly bounded.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/93830/1/taeyonga_1.pd