Wodzicki Residue for Operators on Manifolds with Cylindrical Ends

We define the Wodzicki Residue TR(A) for A in a space of operators with double order (m_1,m_2). Such operators are globally defined initially on R^n and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function. Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator belonging the mentioned class in the case m_1=m_2.Comment: 24 pages, picture changed, added references, corrected typo

Laboratory studies of the ARUM process : progress report

The following is a progress report on the developmental programme of the ARUM process conducted at the Microbiology Services Laboratory of Dearborn Chemical Company Limited. It covers the period from the progress review meeting held at the Biohydrometallurgy Conference in August, 1989 to April, 1990

Laboratory studies of the ARUM process : progress report

The following is a progress report on the developmental programme of the ARUM process conducted at the Microbiology Services Laboratory of Dearborn Chemical Company Limited. It covers the period from April 1990 to December 199

A Note on the Einstein-Hilbert action and the Dirac operator on R^n

We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised Wodzicki Residue for a class of operators globally defined on R^n. The result is then obtained by using the properties of heat kernels and generalised Laplacians.Comment: 10 pages, 1 figure. Corrected typos; expanded references and section 1; slightly modified figure, abstract, introduction and some notation; material of section 3 reorganised, according to changes in previous section

Early Predictors of Objectively Measured Physical Activity and Sedentary Behaviour in 8–10 Year Old Children: The Gateshead Millennium Study

With a number of studies suggesting associations between early life influences and later chronic disease risk, it is suggested that associations between early growth and later physical activity (PA) may be a mediator. However, conflicting evidence exists for association between birth weight and childhood PA. In addition, it is important to know what other, potentially modifiable, factors may influence PA in children given its' association with childhood and later adiposity. We used the Gateshead Millennium Study (GMS) to identify predictors of childhood PA levels. The GMS is a cohort of 1029 infants born in 1999–2000 in Gateshead in northern England. Throughout infancy and early childhood, detailed information was collected. Assessments at age 9 years included body composition, objective measures of habitual PA and a range of lifestyle factors. Mean total volumes of PA (accelerometer count per minute, cpm) and moderate-vigorous intensity PA (MVPA), and the percentage of time spent in sedentary behaviour (%SB) were quantified and related to potential predictors using linear regression and path analysis. Children aged 8–10 years were included. Significant differences were seen in all three outcome variables between sexes and season of measurement (p<0.001). Restricting children’s access to television was associated with decreased MVPA. Increased paternal age was associated with significant increases in %SB (p = 0.02), but not MVPA or total PA. Increased time spent in out of school sports clubs was significantly associated with decreased %SB (p = 0.02). No significant associations were seen with birth weight. A range of factors, directly or indirectly, influenced PA and sedentary behaviour. However, associations differed between the different constructs of PA and %SB. Exploring further the sex differences in PA would appear to be useful, as would encouraging children to join out of school sports clubs

Cancer Biomarker Discovery: The Entropic Hallmark

Background: It is a commonly accepted belief that cancer cells modify their transcriptional state during the progression of the disease. We propose that the progression of cancer cells towards malignant phenotypes can be efficiently tracked using high-throughput technologies that follow the gradual changes observed in the gene expression profiles by employing Shannon's mathematical theory of communication. Methods based on Information Theory can then quantify the divergence of cancer cells' transcriptional profiles from those of normally appearing cells of the originating tissues. The relevance of the proposed methods can be evaluated using microarray datasets available in the public domain but the method is in principle applicable to other high-throughput methods. Methodology/Principal Findings: Using melanoma and prostate cancer datasets we illustrate how it is possible to employ Shannon Entropy and the Jensen-Shannon divergence to trace the transcriptional changes progression of the disease. We establish how the variations of these two measures correlate with established biomarkers of cancer progression. The Information Theory measures allow us to identify novel biomarkers for both progressive and relatively more sudden transcriptional changes leading to malignant phenotypes. At the same time, the methodology was able to validate a large number of genes and processes that seem to be implicated in the progression of melanoma and prostate cancer. Conclusions/Significance: We thus present a quantitative guiding rule, a new unifying hallmark of cancer: the cancer cell's transcriptome changes lead to measurable observed transitions of Normalized Shannon Entropy values (as measured by high-throughput technologies). At the same time, tumor cells increment their divergence from the normal tissue profile increasing their disorder via creation of states that we might not directly measure. This unifying hallmark allows, via the the Jensen-Shannon divergence, to identify the arrow of time of the processes from the gene expression profiles, and helps to map the phenotypical and molecular hallmarks of specific cancer subtypes. The deep mathematical basis of the approach allows us to suggest that this principle is, hopefully, of general applicability for other diseases