Théorème de Pleijel pour l'oscillateur harmonique quantique

L'objectif de ce mémoire est de démontrer certaines propriétés géométriques des fonctions propres de l'oscillateur harmonique quantique. Nous étudierons les domaines nodaux, c'est-à-dire les composantes connexes du complément de l'ensemble nodal. Supposons que les valeurs propres ont été ordonnées en ordre croissant. Selon un théorème fondamental dû à Courant, une fonction propre associée à la $n$-ième valeur propre ne peut avoir plus de $n$ domaines nodaux. Ce résultat a été prouvé initialement pour le laplacien de Dirichlet sur un domaine borné mais il est aussi vrai pour l'oscillateur harmonique quantique isotrope. Le théorème a été amélioré par Pleijel en 1956 pour le laplacien de Dirichlet. En effet, on peut donner un résultat asymptotique plus fort pour le nombre de domaines nodaux lorsque les valeurs propres tendent vers l'infini. Dans ce mémoire, nous prouvons un résultat du même type pour l'oscillateur harmonique quantique isotrope. Pour ce faire, nous utiliserons une combinaison d'outils classiques de la géométrie spectrale (dont certains ont été utilisés dans la preuve originale de Pleijel) et de plusieurs nouvelles idées, notamment l'application de certaines techniques tirées de la géométrie algébrique et l'étude des domaines nodaux non-bornés.

Crosstalk between developmental and tumour-specific signalling pathways : kallikrein-related serine peptidases and nodal in prostrate cancer

Prostate cancer is an important male health issue. The strategies used to diagnose and treat prostate cancer underscore the cell and molecular interactions that promote disease progression. Prostate cancer is histologically defined by increasingly undifferentiated tumour cells and therapeutically targeted by androgen ablation. Even as the normal glandular architecture of the adult prostate is lost, prostate cancer cells remain dependent on the androgen receptor (AR) for growth and survival. This project focused on androgen-regulated gene expression, altered cellular differentiation, and the nexus between these two concepts. The AR controls prostate development, homeostasis and cancer progression by regulating the expression of downstream genes. Kallikrein-related serine peptidases are prominent transcriptional targets of AR in the adult prostate. Kallikrein 3 (KLK3), which is commonly referred to as prostate-specific antigen, is the current serum biomarker for prostate cancer. Other kallikreins are potential adjunct biomarkers. As secreted proteases, kallikreins act through enzyme cascades that may modulate the prostate cancer microenvironment. Both as a panel of biomarkers and cascade of proteases, the roles of kallikreins are interconnected. Yet the expression and regulation of different kallikreins in prostate cancer has not been compared. In this study, a spectrum of prostate cell lines was used to evaluate the expression profile of all 15 members of the kallikrein family. A cluster of genes was co-ordinately expressed in androgenresponsive cell lines. This group of kallikreins included KLK2, 3, 4 and 15, which are located adjacent to one another at the centromeric end of the kallikrein locus. KLK14 was also of interest, because it was ubiquitously expressed among the prostate cell lines. Immunohistochemistry showed that these 5 kallikreins are co-expressed in benign and malignant prostate tissue. The androgen-regulated expression of KLK2 and KLK3 is well-characterised, but has not been compared with other kallikreins. Therefore, KLK2, 3, 4, 14 and 15 expression were all measured in time course and dose response experiments with androgens, AR-antagonist treatments, hormone deprivation experiments and cells transfected with AR siRNA. Collectively, these experiments demonstrated that prostatic kallikreins are specifically and directly regulated by the AR. The data also revealed that kallikrein genes are differentially regulated by androgens; KLK2 and KLK3 were strongly up-regulated, KLK4 and KLK15 were modestly up-regulated, and KLK14 was repressed. Notably, KLK14 is located at the telomeric end of the kallikrein locus, far away from the centromeric cluster of kallikreins that are stimulated by androgens. These results show that the expression of KLK2, 3, 4, 14 and 15 is maintained in prostate cancer, but that these genes exhibit different responses to androgens. This makes the kallikrein locus an ideal model to investigate AR signalling. The increasingly dedifferentiated phenotype of aggressive prostate cancer cells is accompanied by the re-expression of signalling molecules that are usually expressed during embryogenesis and foetal tissue development. The Wnt pathway is one developmental cascade that is reactivated in prostate cancer. The canonical Wnt cascade regulates the intracellular levels of β-catenin, a potent transcriptional co-activator of T-cell factor (TCF) transcription factors. Notably, β-catenin can also bind to the AR and synergistically stimulate androgen-mediated gene expression. This is at the expense of typical Wnt/TCF target genes, because the AR:β-catenin and TCF:β-catenin interactions are mutually exclusive. The effect of β-catenin on kallikrein expression was examined to further investigate the role of β-catenin in prostate cancer. Stable knockdown of β-catenin in LNCaP prostate cancer cells attenuated the androgen-regulated expression of KLK2, 3, 4 and 15, but not KLK14. To test whether KLK14 is instead a TCF:β-catenin target gene, the endogenous levels of β-catenin were increased by inhibiting its degradation. Although KLK14 expression was up-regulated by these treatments, siRNA knockdown of β-catenin demonstrated that this effect was independent of β-catenin. These results show that β-catenin is required for maximal expression of KLK2, 3, 4 and 15, but not KLK14. Developmental cells and tumour cells express a similar repertoire of signalling molecules, which means that these different cell types are responsive to one another. Previous reports have shown that stem cells and foetal tissues can reprogram aggressive cancer cells to less aggressive phenotypes by restoring the balance to developmental signalling pathways that are highly dysregulated in cancer. To investigate this phenomenon in prostate cancer, DU145 and PC-3 prostate cancer cells were cultured on matrices pre-conditioned with human embryonic stem cells (hESCs). Soft agar assays showed that prostate cancer cells exposed to hESC conditioned matrices had reduced clonogenicity compared with cells harvested from control matrices. A recent study demonstrated that this effect was partially due to hESC-derived Lefty, an antagonist of Nodal. A member of the transforming growth factor β (TGFβ) superfamily, Nodal regulates embryogenesis and is re-expressed in cancer. The role of Nodal in prostate cancer has not previously been reported. Therefore, the expression and function of the Nodal signalling pathway in prostate cancer was investigated. Western blots confirmed that Nodal is expressed in DU145 and PC-3 cells. Immunohistochemistry revealed greater expression of Nodal in malignant versus benign glands. Notably, the Nodal inhibitor, Lefty, was not expressed at the mRNA level in any prostate cell lines tested. The Nodal signalling pathway is functionally active in prostate cancer cells. Recombinant Nodal treatments triggered downstream phosphorylation of Smad2 in DU145 and LNCaP cells, and stably-transfected Nodal increased the clonogencity of LNCaP cells. Nodal was also found to modulate AR signalling. Nodal reduced the activity of an androgen-regulated KLK3 promoter construct in luciferase assays and attenuated the endogenous expression of AR target genes including prostatic kallikreins. These results demonstrate that Nodal is a novel example of a developmental signalling molecule that is reexpressed in prostate cancer and may have a functional role in prostate cancer progression. In summary, this project clarifies the role of androgens and changing cellular differentiation in prostate cancer by characterising the expression and function of the downstream genes encoding kallikrein-related serine proteases and Nodal. Furthermore, this study emphasises the similarities between prostate cancer and early development, and the crosstalk between developmental signalling pathways and the AR axis. The outcomes of this project also affirm the utility of the kallikrein locus as a model system to monitor tumour progression and the phenotype of prostate cancer cells.

The development of a simple multi-nodal tool to identify performance issues in existing commercial buildings

Australia’s building stock includes many older commercial buildings with numerous factors that impact energy performance and indoor environment quality. The built environment industry has generally focused heavily on improving physical building design elements for greater energy efficiency (such as retrofits and environmental upgrades), however there are noticeable ‘upper limits’ to performance improvements in these areas. To achieve a stepchange improvement in building performance, the authors propose that additional components need to be addressed in a whole of building approach, including the way building design elements are managed and the level of stakeholder engagement between owners, tenants and building managers. This paper focuses on the opportunities provided by this whole-of-building approach, presenting the findings of a research project undertaken through the Sustainable Built Environment National Research Centre (SBEnrc) in Australia. Researchers worked with a number of industry partners over two years to investigate issues facing stakeholders at base building and tenancy levels, and the barriers to improving building performance. Through a mixed-method, industry-led research approach, five ‘nodes’ were identified in whole-of-building performance evaluation, each with interlinking and overlapping complexities that can influence performance. The nodes cover building management, occupant experience, indoor environment quality, agreements and culture, and design elements. This paper outlines the development and testing of these nodes and their interactions, and the resultant multi-nodal tool, called the ‘Performance Nexus’ tool. The tool is intended to be of most benefit in evaluating opportunities for performance improvement in the vast number of existing low-performing building stock.

QTL for nodal root angle in sorghum (Sorghum bicolor L. Moench) co-locate with QTL for traits associated with drought adaptation.

Nodal root angle in sorghum influences vertical and horizontal root distribution in the soil profile and is thus relevant to drought adaptation. In this study, we report for the first time on the mapping of four QTL for nodal root angle (qRA) in sorghum, in addition to three QTL for root dry weight, two for shoot dry weight, and three for plant leaf area. Phenotyping was done at the six leaf stage for a mapping population (n = 141) developed by crossing two inbred sorghum lines with contrasting root angle. Nodal root angle QTL explained 58.2% of the phenotypic variance and were validated across a range of diverse inbred lines. Three of the four nodal root angle QTL showed homology to previously identified root angle QTL in rice and maize, whereas all four QTL co-located with previously identified QTL for stay-green in sorghum. A putative association between nodal root angle QTL and grain yield was identified through single marker analysis on field testing data from a subset of the mapping population grown in hybrid combination with three different tester lines. Furthermore, a putative association between nodal root angle QTL and stay-green was identified using data sets from selected sorghum nested association mapping populations segregating for root angle. The identification of nodal root angle QTL presents new opportunities for improving drought adaptation mechanisms via molecular breeding to manipulate a trait for which selection has previously been very difficult.

Genetic control of nodal root angle in sorghum and its implications on water extraction

Genotypic variability in root system architecture has been associated with root angle of seedlings and water extraction patterns of mature plants in a range of crops. The potential inclusion of root angle as a selection criterion in a sorghum breeding program requires (1) availability of an efficient screening method, (2) presence of genotypic variation with high heritability, and (3) an association with water extraction pattern. The aim of this study was to determine the feasibility for inclusion of nodal root angle as a selection criterion in sorghum breeding programs. A high-throughput phenotypic screen for nodal root angle in young sorghum plants has recently been developed and has been used successfully to identify significant variation in nodal root angle across a diverse range of inbred lines and a mapping population. In both cases, heritabilities for nodal root angle were high. No association between nodal root angle and plant size was detected. This implies that parental inbred lines could potentially be used to asses nodal root angle of their hybrids, although such predictability is compromised by significant interactions. To study effects of nodal root angle on water extraction patterns of mature plants, four inbred lines with contrasting nodal root angle at seedling stage were grown until at least anthesis in large rhizotrons. A consistent trend was observed that nodal root angle may affect the spatial distribution of root mass of mature plants and hence their ability to extract soil water, although genotypic differences were not significant. The potential implications of this for specific adaptation to drought stress are discussed. Results suggest that nodal root angle of young plants can be a useful selection criterion for specific drought adaptation, and could potentially be used in molecular breeding programs if QTLs for root angle can be identified. (C) 2012 Elsevier B.V. All rights reserved.

Alginate encapsulation of shoot tips and nodal segments for short-term storage and distribution of the eucalypt Corymbia torelliana × C. citriodora

A protocol was developed for short-term preservation and distribution of the plantation eucalypt, Corymbia torelliana × C. citriodora, using alginate-encapsulated shoot tips and nodes as synthetic seeds. Effects of sowing medium, auxin concentration, storage temperature and planting substrate on shoot regrowth or conversion into plantlets were assessed for four different clones. High frequencies of shoot regrowth (76–100%) from encapsulated explants were consistently obtained in hormone-free half- and full-strength Murashige and Skoog (MS) sowing media. Conversion into plantlets from synthetic seeds was achieved on half-strength MS medium by treating shoot tips or nodes with 4.9–78.4 μM IBA prior to encapsulation. Pre-treatment with 19.6 μM IBA provided 62–100% conversion, and 95–100% of plantlets survived after acclimatisation under nursery conditions. Synthetic seeds containing explants pre-treated with IBA were stored for 8 weeks much more effectively at 25°C than at 4°C, with regrowth frequencies of 50–84% at 25°C compared with 0–4% at 4°C. To eliminate the in vitro culture step after encapsulation, synthetic seeds were allowed to pre-convert before sowing directly onto a range of ex vitro non-sterile planting substrates. Highest frequencies (46–90%) of plantlet formation from pre-converted synthetic seeds were obtained by transferring shoot tip-derived synthetic seeds onto an organic compost substrate. These plantlets exhibited almost 100% survival in the nursery without mist irrigation. Pre-conversion of non-embryonic synthetic seeds is a novel technique that provides a convenient alternative to somatic embryo-derived artificial seeds.

Nodal length fluctuations for arithmetic random waves

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (''arithmetic random waves''). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

Nodal domains of the equilateral triangle billiard

We characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, nu of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrodinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and nonseparable integrable billiards, nu satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of m mod kn, given a particular k, for a set of quantum numbers, m, n. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. (C) 2014 Elsevier Inc. All rights reserved.