Consider the difference equation xn+1=α+∑i=0kaixn−i+∑i=0k∑j=ikaijxn−ixn−jβ+∑i=0kbixn−i+∑i=0k∑j=ikbijxn−ixn−j,n=0,1,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1} = \frac{\alpha+ \sum_{i=0}^{k} a_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} a_{i j} x_{n-i} x_{n-j} }{\beta+ \sum_{i=0}^{k} b_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} b_{ij} x_{n-i} x_{n-j}}, \quad n=0,1, \ldots, $$\end{document} where all parameters α, β, ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}$\end{document}, bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{i}$\end{document}, aij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{ij}$\end{document}, bij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{ij}$\end{document}, i,j=0,1,…,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i,j=0,1,\ldots, k$\end{document}, and the initial conditions xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{i}$\end{document}, i∈{−k,…,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i \in\{-k, \ldots, 0 \}$\end{document}, are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

We investigate the global behavior of a cubic second order difference equation xn+1=Axn3+Bxn2xn−1+Cxnxn−12+Dxn−13+Exn2+Fxnxn−1+Gxn−12+Hxn+Ixn−1+J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{n+1}=Ax_{n}^{3}+ Bx_{n}^{2}x_{n-1}+Cx_{n}x_{n-1}^{2}+Dx_{n-1}^{3}+Ex_{n}^{2} +Fx_{n}x_{n-1}+Gx_{n-1}^{2}+Hx_{n}+Ix_{n-1}+J$\end{document}, n=0,1,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n=0,1,\ldots$\end{document} , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.

Irritable bowel syndrome is a disorder diagnosed on symptom-based criteria without inclusion of any objective parameter measurable by known diagnostic methods. Heterogeneity of the disorder and overlapping with more serious organic diseases increase uncertainty for the physician's work and increase the cost of confirming the diagnosis. This paper is an attempt to summarize the efforts to find adequate biomarkers for irritable bowel syndrome, which should shorten the time to diagnosis and reduce the cost. Most of the reviewed papers were observational studies from secondary care institutions. Since publication of the Rome III criteria in 2006, most recent studies use these for the recruitment of IBS patients. This is a positive step forward as future studies should use the same criteria, facilitating comparison of their results. Among the studied biomarkers, most evidence is provided for fecal calprotectin. Cutoff values for fecal calprotectin have still to be investigated prior to inclusion in the irritable bowel syndrome diagnostic algorithm.

Motivation: CYP2D6 is highly polymorphic gene which encodes the (CYP2D6) enzyme, involved in the metabolism of 20–25% of all clinically prescribed drugs and other xenobiotics in the human body. CYP2D6 genotyping is recommended prior to treatment decisions involving one or more of the numerous drugs sensitive to CYP2D6 allelic composition. In this context, high-throughput sequencing (HTS) technologies provide a promising time-efficient and cost-effective alternative to currently used genotyping techniques. To achieve accurate interpretation of HTS data, however, one needs to overcome several obstacles such as high sequence similarity and genetic recombinations between CYP2D6 and evolutionarily related pseudogenes CYP2D7 and CYP2D8, high copy number variation among individuals and short read lengths generated by HTS technologies. Results: In this work, we present the first algorithm to computationally infer CYP2D6 genotype at basepair resolution from HTS data. Our algorithm is able to resolve complex genotypes, including alleles that are the products of duplication, deletion and fusion events involving CYP2D6 and its evolutionarily related cousin CYP2D7. Through extensive experiments using simulated and real datasets, we show that our algorithm accurately solves this important problem with potential clinical implications. Availability and implementation: Cypiripi is available at http://sfu-compbio.github.io/cypiripi. Contact: cenk@sfu.ca.