Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Modelling initial survival and growth of radiata pine in New Zealand.

A sensitive framework has been developed for modelling young radiata pine survival, its growth and its size class distribution, from time of planting to age 5 or 6 years. The data and analysis refer to the Central North Island region of New Zealand. The survival function is derived from a Weibull probability density function, to reflect diminishing mortality with the passage of time in young stands. An anamorphic family of trends was used, as very little between-tree competition can be expected in young stands. An exponential height function was found to fit best the lower portion of its sigmoid form. The most appropriate basal area/ha exponential function included an allometric adjustment which resulted in compatible mean height and basal area/ha models. Each of these equations successfully represented the effects of several establishment practices by making coefficients linear functions of site factors, management activities and their interactions. Height and diameter distribution modelling techniques that ensured compatibility with stand values were employed to represent the effects of management practices on crop variation. Model parameters for this research were estimated using data from site preparation experiments in the region and were tested with some independent data sets.

Early growth and adult health: focus on blood pressure, glucose tolerance status and body composition

Theory of developmental origins of adult health and disease proposes that experiences during critical periods of early development may have consequences on health throughout a lifespan. Thesis studies aimed to characterize associations between early growth and some components of the metabolic syndrome cluster. Participants belong to two epidemiological cohorts with data on birth measurements and, for the younger cohort, on serial recordings of weight and height during childhood. They were born as singletons between 1924-33 and 1934-44 in the Helsinki University Central Hospital, and 500 and 2003 of them, respectively, attended clinical studies at the age of 65-75 and 56-70 years, respectively. In the 65-75 year old men and women, the well-known inverse relationship between birth weight and systolic blood pressure (SBP) was confined to people who had established hypertension. Among them a 1-kg increase in birth weight was associated with a 6.4-mmHg (95% CI: 1.0 to 11.9) decrease in SBP. This relationship was further confined to people with the prevailing Pro12Pro polymorphism of the peroxisome proliferator-activated receptor-γ2 (PPARγ2) gene. People with low birth weight were more likely to receive angiotensin-converting enzyme inhibitors/angiotensin-receptor blockers (ACEI/ARB, p=0.03), and, again, this relationship was confined to the carriers of the Pro12Pro (p=0.01 for interaction). These results suggest that the inverse association between birth weight and systolic BP becomes focused in hypertensive people because pathological features of BP regulation, associated with slow fetal growth, become self-perpetuating in adult life. Insulin resistance of the Pro12Pro carriers with low birth weight may interact with the renin-angiotensin system leading to raised BP levels. Habitual physical activity protected men and women who were small at birth, and thus at increased risk for the development of type 2 diabetes, against glucose intolerance more strongly. Among subjects with birth weight ≤3000 g, the odds ratio (OR) for glucose intolerance was 5.2 (95% CI: 2.1 to 13) in those who exercised less than 3 times per week compared to regular exercisers; in those who scored their exercise light compared with moderate exercisers (defined as comparable to brisk walking) the OR was 3.5 (1.5 to 8.2). In the 56-70 year old men a 1 kg increase in birth weight corresponded to a 4.1 kg (95% CI: 3.1 to 5.1) and in women to a 2.9 kg (2.1 to 3.6) increase in adult lean mass. Rapid gain in body mass index (BMI), i.e. crossing from an original BMI percentile to a higher one, before the age of 2 years increased adult lean mass index (LMI, lean mass/height squared) without excess fat accumulation whereas rapid gain in BMI during later childhood, despite the concurrent rise in LMI, resulted in a relatively higher increase in adult body fat mass. These findings illustrate how genes, the environment and their interactions, early growth patterns, and adult lifestyle modify adult health risks which originate from early life.

Growth differentiation factor 9 signalling in the ovary

In the ovary, two new members of the large TGF-beta superfamily of growth factors were discovered in the 1990s. The oocyte was shown to express two closely related growth factors that were named growth differentiation factor 9 (GDF-9) and growth differentiation factor 9B (GDF-9B). Both of these proteins are required for normal ovarian follicle development although their individual significance varies between species. GDF-9 and GDF-9B mRNAs are expressed in the human oocytes from the primary follicle stage onwards. This thesis project was aimed to define the signalling mechanisms utilized by the oocyte secreted GDF-9. We used primary cultures of human granulosa luteal cells (hGL) as our cell model, and recombinant adenovirus-mediated gene transfer in manipulating the TGF-b family signalling cascade molecules in these cells. Overexpression of the constitutively active forms of the seven type I receptors, the activin receptor-like kinases 1-7 (ALK1-7), using recombinant adenoviruses caused a specific activation of either the Smad1 or Smad2 pathway proteins depending on the ALK used. Activation of both Smad1 and Smad2 proteins also stimulated the expression of dimeric inhibin B protein in hGL cells. Treatment with recombinant GDF-9 protein induced the specific activation of the Smad2 pathway and stimulated the expression of inhibin betaB subunit mRNA as well as inhibin B protein secretion in our cell model. Recombinant GDF-9 also activated the Smad3-responsive CAGA-luciferase reported construct, and the GDF-9 response in hGL cells was markedly potentiated upon the overexpression of Alk5 by adenoviral gene transduction. Alk5 overexpression also enhanced the GDF-9 induced inhibin B secretion by these cells. Similarly, in a mouse teratocarcinoma cell line P19, GDF-9 could activate the Smad2/3 pathway, and overexpression of ALK5 in COS7 cells rendered them responsive to GDF-9. Furthermore, transfection of rat granulosa cells with small interfering RNA for ALK5 or overexpression of the inhibitory Smad7 resulted in dose-dependent suppression of GDF-9 effects. In conclusion, this thesis shows that both Smad1 and Smad2 pathways are involved in controlling the regulation of inhibin B secretion. Therefore, in addition to endocrine control of inhibin production by the pituitary gonadotropins, also local paracrine factors within in the ovary, like the oocyte-derived growth factors, may contribute to controlling inhibin secretion. This thesis shows as well that like other TGF-beta family ligands, also GDF-9 signalling is mediated by the canonical type I and type II receptors with serine/threonine kinase activity, and the intracellular transcription factors, the Smads. Although GDF-9 binds to the BMP type II receptor, its downstream actions are specifically mediated by the type I receptor, ALK5, and the Smad2 and Smad3 proteins.