Heaviside generalized functions and shock waves for a Burger kind equation

The purpose of the work is to study the existence and nonexistence of shock wave solutions for the Burger equations. The study is developed in the context of Colombeau's theory of generalized functions (GFs). This study uses the equality in the strict sense and the weak equality of GFs. The shock wave solutions are given in terms of GFs that have the Heaviside function, in x and ( x, t) variables, as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function, in R-n and R-n x R, in the distributional limit sense.

BASIC HYPERGEOMETRIC FUNCTIONS and ORTHOGONAL LAURENT POLYNOMIALS

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Tau-functions and dressing transformations for zero-curvature affine integrable equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda- and KdV-type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras g. Their common feature is that they have some special vacuum solutions corresponding to Lax operators lying in some Abelian (up to the central term) subalgebra of g; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of g. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the Abelian and non-Abelian affine Toda theories are discussed in detail. © 1997 American Institute of Physics.

3D-ultrastructure, functions and stress responses of rhogocytes from the freshwater gastropods Biomphalaria glabrata and Lymnaea stagnalis

Rhogocytes, also termed ‘pore cells’, exist free in the hemolymph or embedded in the connective tissue of different body parts of molluscs, notably gastropods. These unique cells can be round, elongated or irregularly shaped, and up to 30 μm in diameter. Their hallmark is the so-called slit apparatus: i.e. pocket-like invaginations of the plasma membrane creating extracellular lacunae, bridged by cytoplasmic bars. These bars form distinctive slits of ca. 20 nm width. A slit diaphragm composed of proteins establishes a molecular sieve with holes of 20 x 20 nm. Different functions have been assigned to this special molluscan cell type, notably biosynthesis of the hemolymph respiratory protein hemocyanin. It has further been proposed, but not proven, that in the case of red-blooded snail species rhogocytes might synthesize the hemoglobin. However, the secretion pathway of these hemolymph proteins, and the functional role of the enigmatic slit apparatus remained unclear. Additionally proposed functions of rhogocytes, such as heavy metal detoxification or hemolymph protein degradation, are also not well studied. This work provides more detailed electron microscopical, histological and immunobiochemical information on the structure and function of rhogocytes of the freshwater snails Biomphalaria glabrata and Lymnaea stagnalis. By in situ hybridization on mantle tissues, it proves that B. glabrata rhogocytes synthesize hemoglobin and L. stagnalis rhogocytes synthesize hemocyanin. Hemocyanin is present, in endoplasmic reticulum lacunae and in vesicles, as individual molecules or pseudo-crystalline arrays. The first 3D reconstructions of rhogocytes are provided by means of electron tomography and show unprecedented details of the slit apparatus. A highly dense material in the cytoplasmic bars close to the diaphragmatic slits was shown, by immunogold labeling, to contain actin. By immunofluorescence microscopy, the protein nephrin was localized at the periphery of rhogocytes. The presence of both proteins in the slit apparatus supports the previous hypothesis, hitherto solely based on similarities of the ultrastructure, that the molluscan rhogocytes are phylogenetically related to mammalian podocytes and insect nephrocytes. A possible secretion pathway of respiratory proteins that includes a transfer mechanism of vesicles through the diaphragmatic slits is proposed and discussed. We also studied, by electron microscopy, the reaction of rhogocytes in situ to two forms of animal stress: deprivation of food and cadmium contamination of the tank water. Significant cellular reactions to both stressors were observed and documented. Notably, the slit apparatus surface and the number of electron-dense cytoplasmic vesicles increased in response to cadmium stress. Food deprivation led to an increase in hemocyanin production. These observations are also discussed in the framework of using such animals as potential environmental biomarkers.

Functions and effects of creatine in the central nervous system

Creatine kinase catalyses the reversible transphosphorylation of creatine by ATP. In the cell, creatine kinase isoenzymes are specifically localized at strategic sites of ATP consumption to efficiently regenerate ATP in situ via phosphocreatine or at sites of ATP generation to build-up a phosphocreatine pool. Accordingly, the creatine kinase/phosphocreatine system plays a key role in cellular energy buffering and energy transport, particularly in cells with high and fluctuating energy requirements like neurons. Creatine kinases are expressed in the adult and developing human brain and spinal cord, suggesting that the creatine kinase/phosphocreatine system plays a significant role in the central nervous system. Functional impairment of this system leads to a deterioration in energy metabolism, which is phenotypic for many neurodegenerative and age-related diseases. Exogenous creatine supplementation has been shown to reduce neuronal cell loss in experimental paradigms of acute and chronic neurological diseases. In line with these findings, first clinical trials have shown beneficial effects of therapeutic creatine supplementation. Furthermore, creatine was reported to promote differentiation of neuronal precursor cells that might be of importance for improving neuronal cell replacement strategies. Based on these observations there is growing interest on the effects and functions of this compound in the central nervous system. This review gives a short excursion into the basics of the creatine kinase/phosphocreatine system and aims at summarizing findings and concepts on the role of creatine kinase and creatine in the central nervous system with special emphasis on pathological conditions and the positive effects of creatine supplementation.

IDENTIFICATION OF GENES CONTROLLING BIOLOGICAL PROCESSES AND PATHWAYS THROUGH STATISTICAL ANALYSIS AND NETWORK RECONSTRUCTION

The developmental processes and functions of an organism are controlled by the genes and the proteins that are derived from these genes. The identification of key genes and the reconstruction of gene networks can provide a model to help us understand the regulatory mechanisms for the initiation and progression of biological processes or functional abnormalities (e.g. diseases) in living organisms. In this dissertation, I have developed statistical methods to identify the genes and transcription factors (TFs) involved in biological processes, constructed their regulatory networks, and also evaluated some existing association methods to find robust methods for coexpression analyses. Two kinds of data sets were used for this work: genotype data and gene expression microarray data. On the basis of these data sets, this dissertation has two major parts, together forming six chapters. The first part deals with developing association methods for rare variants using genotype data (chapter 4 and 5). The second part deals with developing and/or evaluating statistical methods to identify genes and TFs involved in biological processes, and construction of their regulatory networks using gene expression data (chapter 2, 3, and 6). For the first part, I have developed two methods to find the groupwise association of rare variants with given diseases or traits. The first method is based on kernel machine learning and can be applied to both quantitative as well as qualitative traits. Simulation results showed that the proposed method has improved power over the existing weighted sum method (WS) in most settings. The second method uses multiple phenotypes to select a few top significant genes. It then finds the association of each gene with each phenotype while controlling the population stratification by adjusting the data for ancestry using principal components. This method was applied to GAW 17 data and was able to find several disease risk genes. For the second part, I have worked on three problems. First problem involved evaluation of eight gene association methods. A very comprehensive comparison of these methods with further analysis clearly demonstrates the distinct and common performance of these eight gene association methods. For the second problem, an algorithm named the bottom-up graphical Gaussian model was developed to identify the TFs that regulate pathway genes and reconstruct their hierarchical regulatory networks. This algorithm has produced very significant results and it is the first report to produce such hierarchical networks for these pathways. The third problem dealt with developing another algorithm called the top-down graphical Gaussian model that identifies the network governed by a specific TF. The network produced by the algorithm is proven to be of very high accuracy.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Partial Quasi-Bilateral Generating Function Involving some Special Functions

A group-theoretic method of obtaining more general class of generating functions from a given class of partial quasi-bilateral generating functions involving Hermite, Laguerre and Gegenbaur polynomials are discussed.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

EGCL: an extended G-Code Language with flow control, functions and mnemonic variables

In the context of computer numerical control (CNC) and computer aided manufacturing (CAM), the capabilities of programming languages such as symbolic and intuitive programming, program portability and geometrical portfolio have special importance -- They allow to save time and to avoid errors during part programming and permit code re-usage -- Our updated literature review indicates that the current state of art presents voids in parametric programming, program portability and programming flexibility -- In response to this situation, this article presents a compiler implementation for EGCL (Extended G-code Language), a new, enriched CNC programming language which allows the use of descriptive variable names, geometrical functions and flow-control statements (if-then-else, while) -- Our compiler produces low-level generic, elementary ISO-compliant Gcode, thus allowing for flexibility in the choice of the executing CNC machine and in portability -- Our results show that readable variable names and flow control statements allow a simplified and intuitive part programming and permit re-usage of the programs -- Future work includes allowing the programmer to define own functions in terms of EGCL, in contrast to the current status of having them as library built-in functions

Australia : The Challenges of Poverty, Pedagogy, and Pathways

A discussion of 2008/2009 developments in Australian educational policy, with specific reference to the adoption of US and UK trends in accountability, testing and school reform.

Vitamin D in health and disease : an insight into traditional functions and new roles for the 'sunshine vitamin'

Vitamin D is unique among the vitamins in that humans can synthesize it via the action of UV radiation upon the skin. This combined with its ability to act on specific target tissues via Vitamin D Receptor’s (VDR) make its classification as a steroid hormone more appropriate. While Vitamin D deficiency is a recognized problem in some northern latitude countries, recent studies have shown even in sunny countries such as Australia, vitamin D deficiency may be more prevalent than first thought. Vitamin D is most well known for its role in bone health, however, the discovery of VDR’s on a wide variety of tissue types has also opened up roles for vitamin D far beyond traditional bone health. These include possible associations with autoimmune diseases such as multiple sclerosis and inflammatory bowel diseases, cancer, cardiovascular diseases and muscle strength. Firstly, this paper presents an overview of the two sources of vitamin D: exposure to ultraviolet-B radiation and food sources of vitamin D, with particular focus on both Australian and international studies on dietary vitamin D intake and national fortification strategies. Secondly, the paper reviews recent epidemiological and experimental evidence linking vitamin D and its role in health and disease for the major conditions linked to suboptimal vitamin D, while identifying significant gaps in the research and possible future directions for research.

The effect of prolonged reading on visual functions and reading performance in students with low vision

This study is the first to investigate the effect of prolonged reading on reading performance and visual functions in students with low vision. The study focuses on one of the most common modes of achieving adequate magnification for reading by students with low vision, their close reading distance (proximal or relative distance magnification). Close reading distances impose high demands on near visual functions, such as accommodation and convergence. Previous research on accommodation in children with low vision shows that their accommodative responses are reduced compared to normal vision. In addition, there is an increased lag of accommodation for higher stimulus levels as may occur at close reading distance. Reduced accommodative responses in low vision and higher lag of accommodation at close reading distances together could impact on reading performance of students with low vision especially during prolonged reading tasks. The presence of convergence anomalies could further affect reading performance. Therefore, the aims of the present study were 1) To investigate the effect of prolonged reading on reading performance in students with low vision 2) To investigate the effect of prolonged reading on visual functions in students with low vision. This study was conducted as cross-sectional research on 42 students with low vision and a comparison group of 20 students with normal vision, aged 7 to 20 years. The students with low vision had vision impairments arising from a range of causes and represented a typical group of students with low vision, with no significant developmental delays, attending school in Brisbane, Australia. All participants underwent a battery of clinical tests before and after a prolonged reading task. An initial reading-specific history and pre-task measurements that included Bailey-Lovie distance and near visual acuities, Pelli-Robson contrast sensitivity, ocular deviations, sensory fusion, ocular motility, near point of accommodation (pull-away method), accuracy of accommodation (Monocular Estimation Method (MEM)) retinoscopy and Near Point of Convergence (NPC) (push-up method) were recorded for all participants. Reading performance measures were Maximum Oral Reading Rates (MORR), Near Text Visual Acuity (NTVA) and acuity reserves using Bailey-Lovie text charts. Symptoms of visual fatigue were assessed using the Convergence Insufficiency Symptom Survey (CISS) for all participants. Pre-task measurements of reading performance and accuracy of accommodation and NPC were compared with post-task measurements, to test for any effects of prolonged reading. The prolonged reading task involved reading a storybook silently for at least 30 minutes. The task was controlled for print size, contrast, difficulty level and content of the reading material. Silent Reading Rate (SRR) was recorded every 2 minutes during prolonged reading. Symptom scores and visual fatigue scores were also obtained for all participants. A visual fatigue analogue scale (VAS) was used to assess visual fatigue during the task, once at the beginning, once at the middle and once at the end of the task. In addition to the subjective assessments of visual fatigue, tonic accommodation was monitored using a photorefractor (PlusoptiX CR03™) every 6 minutes during the task, as an objective assessment of visual fatigue. Reading measures were done at the habitual reading distance of students with low vision and at 25 cms for students with normal vision. The initial history showed that the students with low vision read for significantly shorter periods at home compared to the students with normal vision. The working distances of participants with low vision ranged from 3-25 cms and half of them were not using any optical devices for magnification. Nearly half of the participants with low vision were able to resolve 8-point print (1M) at 25 cms. Half of the participants in the low vision group had ocular deviations and suppression at near. Reading rates were significantly reduced in students with low vision compared to those of students with normal vision. In addition, there were a significantly larger number of participants in the low vision group who could not sustain the 30-minute task compared to the normal vision group. However, there were no significant changes in reading rates during or following prolonged reading in either the low vision or normal vision groups. Individual changes in reading rates were independent of their baseline reading rates, indicating that the changes in reading rates during prolonged reading cannot be predicted from a typical clinical assessment of reading using brief reading tasks. Contrary to previous reports the silent reading rates of the students with low vision were significantly lower than their oral reading rates, although oral and silent reading was assessed using different methods. Although the visual acuity, contrast sensitivity, near point of convergence and accuracy of accommodation were significantly poorer for the low vision group compared to those of the normal vision group, there were no significant changes in any of these visual functions following prolonged reading in either group. Interestingly, a few students with low vision (n =10) were found to be reading at a distance closer than their near point of accommodation. This suggests a decreased sensitivity to blur. Further evaluation revealed that the equivalent intrinsic refractive errors (an estimate of the spherical dioptirc defocus which would be expected to yield a patient’s visual acuity in normal subjects) were significantly larger for the low vision group compared to those of the normal vision group. As expected, accommodative responses were significantly reduced for the low vision group compared to the expected norms, which is consistent with their close reading distances, reduced visual acuity and contrast sensitivity. For those in the low vision group who had an accommodative error exceeding their equivalent intrinsic refractive errors, a significant decrease in MORR was found following prolonged reading. The silent reading rates however were not significantly affected by accommodative errors in the present study. Suppression also had a significant impact on the changes in reading rates during prolonged reading. The participants who did not have suppression at near showed significant decreases in silent reading rates during and following prolonged reading. This impact of binocular vision at near on prolonged reading was possibly due to the high demands on convergence. The significant predictors of MORR in the low vision group were age, NTVA, reading interest and reading comprehension, accounting for 61.7% of the variances in MORR. SRR was not significantly influenced by any factors, except for the duration of the reading task sustained; participants with higher reading rates were able to sustain a longer reading duration. In students with normal vision, age was the only predictor of MORR. Participants with low vision also reported significantly greater visual fatigue compared to the normal vision group. Measures of tonic accommodation however were little influenced by visual fatigue in the present study. Visual fatigue analogue scores were found to be significantly associated with reading rates in students with low vision and normal vision. However, the patterns of association between visual fatigue and reading rates were different for SRR and MORR. The participants with low vision with higher symptom scores had lower SRRs and participants with higher visual fatigue had lower MORRs. As hypothesized, visual functions such as accuracy of accommodation and convergence did have an impact on prolonged reading in students with low vision, for students whose accommodative errors were greater than their equivalent intrinsic refractive errors, and for those who did not suppress one eye. Those students with low vision who have accommodative errors higher than their equivalent intrinsic refractive errors might significantly benefit from reading glasses. Similarly, considering prisms or occlusion for those without suppression might reduce the convergence demands in these students while using their close reading distances. The impact of these prescriptions on reading rates, reading interest and visual fatigue is an area of promising future research. Most importantly, it is evident from the present study that a combination of factors such as accommodative errors, near point of convergence and suppression should be considered when prescribing reading devices for students with low vision. Considering these factors would also assist rehabilitation specialists in identifying those students who are likely to experience difficulty in prolonged reading, which is otherwise not reflected during typical clinical reading assessments.