Production of trans-Neptunian binaries through chaos-assisted capture

The recent discovery of binary objects in the Kuiper-belt opens an invaluable window into past and present conditions in the trans-Neptunian part of the Solar System. For example, knowledge of how these objects formed can be used to impose constraints on planetary formation theories. We have recently proposed a binary-object formation model based on the notion of chaos-assisted capture. Here we present a more detailed analysis with calculations performed in the spatial (three-dimensional) three- and four-body Hill approximations. It is assumed that the potential binary partners are initially following heliocentric Keplerian orbits and that their relative motion becomes perturbed as these objects undergo close encounters. First, the mass, velocity, and orbital element distribu- tions which favour binary formation are identified in the circular and elliptical Hill limits. We then consider intruder scattering in the circular Hill four-body problem and find that the chaos-assisted capture mechanism is consistent with observed, apparently randomly distributed, binary mutual orbit inclinations. It also predicts asymmetric distributions of retrograde versus prograde orbits. The time-delay induced by chaos on particle transport through the Hill sphere is analogous to the formation of a resonance in a chemical reaction. Implications for binary formation rates are considered and the 'fine-tuning' problem recently identified by Noll et al. (2007) is also addressed.Comment: submitted to MNRA

Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-\beta \hbar / 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.Comment: 13 pages, 2 figure

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.Comment: submitted to JMP on April 3, 2003; still waiting for the referee's Repor

Semi-Markov Graph Dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON

Using Markov chain Monte Carlo methods for estimating parameters with gravitational radiation data

We present a Bayesian approach to the problem of determining parameters for coalescing binary systems observed with laser interferometric detectors. By applying a Markov Chain Monte Carlo (MCMC) algorithm, specifically the Gibbs sampler, we demonstrate the potential that MCMC techniques may hold for the computation of posterior distributions of parameters of the binary system that created the gravity radiation signal. We describe the use of the Gibbs sampler method, and present examples whereby signals are detected and analyzed from within noisy data.Comment: 21 pages, 10 figure

Biomarker discovery and redundancy reduction towards classification using a multi-factorial MALDI-TOF MS T2DM mouse model dataset

Diabetes like many diseases and biological processes is not mono-causal. On the one hand multifactorial studies with complex experimental design are required for its comprehensive analysis. On the other hand, the data from these studies often include a substantial amount of redundancy such as proteins that are typically represented by a multitude of peptides. Coping simultaneously with both complexities (experimental and technological) makes data analysis a challenge for Bioinformatics

Algebraic Comparison of Partial Lists in Bioinformatics

The outcome of a functional genomics pipeline is usually a partial list of genomic features, ranked by their relevance in modelling biological phenotype in terms of a classification or regression model. Due to resampling protocols or just within a meta-analysis comparison, instead of one list it is often the case that sets of alternative feature lists (possibly of different lengths) are obtained. Here we introduce a method, based on the algebraic theory of symmetric groups, for studying the variability between lists ("list stability") in the case of lists of unequal length. We provide algorithms evaluating stability for lists embedded in the full feature set or just limited to the features occurring in the partial lists. The method is demonstrated first on synthetic data in a gene filtering task and then for finding gene profiles on a recent prostate cancer dataset

Urinary MicroRNA Profiling in the Nephropathy of Type 1 Diabetes

Background: Patients with Type 1 Diabetes (T1D) are particularly vulnerable to development of Diabetic nephropathy (DN) leading to End Stage Renal Disease. Hence a better understanding of the factors affecting kidney disease progression in T1D is urgently needed. In recent years microRNAs have emerged as important post-transcriptional regulators of gene expression in many different health conditions. We hypothesized that urinary microRNA profile of patients will differ in the different stages of diabetic renal disease. Methods and Findings: We studied urine microRNA profiles with qPCR in 40 T1D with >20 year follow up 10 who never developed renal disease (N) matched against 10 patients who went on to develop overt nephropathy (DN), 10 patients with intermittent microalbuminuria (IMA) matched against 10 patients with persistent (PMA) microalbuminuria. A Bayesian procedure was used to normalize and convert raw signals to expression ratios. We applied formal statistical techniques to translate fold changes to profiles of microRNA targets which were then used to make inferences about biological pathways in the Gene Ontology and REACTOME structured vocabularies. A total of 27 microRNAs were found to be present at significantly different levels in different stages of untreated nephropathy. These microRNAs mapped to overlapping pathways pertaining to growth factor signaling and renal fibrosis known to be targeted in diabetic kidney disease. Conclusions: Urinary microRNA profiles differ across the different stages of diabetic nephropathy. Previous work using experimental, clinical chemistry or biopsy samples has demonstrated differential expression of many of these microRNAs in a variety of chronic renal conditions and diabetes. Combining expression ratios of microRNAs with formal inferences about their predicted mRNA targets and associated biological pathways may yield useful markers for early diagnosis and risk stratification of DN in T1D by inferring the alteration of renal molecular processes. © 2013 Argyropoulos et al

Quantitative model for inferring dynamic regulation of the tumour suppressor gene p53

Background: The availability of various "omics" datasets creates a prospect of performing the study of genome-wide genetic regulatory networks. However, one of the major challenges of using mathematical models to infer genetic regulation from microarray datasets is the lack of information for protein concentrations and activities. Most of the previous researches were based on an assumption that the mRNA levels of a gene are consistent with its protein activities, though it is not always the case. Therefore, a more sophisticated modelling framework together with the corresponding inference methods is needed to accurately estimate genetic regulation from "omics" datasets. Results: This work developed a novel approach, which is based on a nonlinear mathematical model, to infer genetic regulation from microarray gene expression data. By using the p53 network as a test system, we used the nonlinear model to estimate the activities of transcription factor (TF) p53 from the expression levels of its target genes, and to identify the activation/inhibition status of p53 to its target genes. The predicted top 317 putative p53 target genes were supported by DNA sequence analysis. A comparison between our prediction and the other published predictions of p53 targets suggests that most of putative p53 targets may share a common depleted or enriched sequence signal on their upstream non-coding region. Conclusions: The proposed quantitative model can not only be used to infer the regulatory relationship between TF and its down-stream genes, but also be applied to estimate the protein activities of TF from the expression levels of its target genes

A Fast Algorithm for Robust Regression with Penalised Trimmed Squares

The presence of groups containing high leverage outliers makes linear regression a difficult problem due to the masking effect. The available high breakdown estimators based on Least Trimmed Squares often do not succeed in detecting masked high leverage outliers in finite samples. An alternative to the LTS estimator, called Penalised Trimmed Squares (PTS) estimator, was introduced by the authors in \cite{ZiouAv:05,ZiAvPi:07} and it appears to be less sensitive to the masking problem. This estimator is defined by a Quadratic Mixed Integer Programming (QMIP) problem, where in the objective function a penalty cost for each observation is included which serves as an upper bound on the residual error for any feasible regression line. Since the PTS does not require presetting the number of outliers to delete from the data set, it has better efficiency with respect to other estimators. However, due to the high computational complexity of the resulting QMIP problem, exact solutions for moderately large regression problems is infeasible. In this paper we further establish the theoretical properties of the PTS estimator, such as high breakdown and efficiency, and propose an approximate algorithm called Fast-PTS to compute the PTS estimator for large data sets efficiently. Extensive computational experiments on sets of benchmark instances with varying degrees of outlier contamination, indicate that the proposed algorithm performs well in identifying groups of high leverage outliers in reasonable computational time.Comment: 27 page