Dynamical Anomalous Subvarieties: Structure and Bounded Height Theorems

According to Medvedev and Scanlon, a polynomial $f(x)\in \bar{\mathbb Q}[x]$ of degree $d\geq 2$ is called disintegrated if it is not linearly conjugate to $x^d$ or $\pm C_d(x)$ (where $C_d(x)$ is the Chebyshev polynomial of degree $d$). Let $n\in\mathbb{N}$, let $f_1,\ldots,f_n\in \bar{\mathbb Q}[x]$ be disintegrated polynomials of degrees at least 2, and let $\varphi=f_1\times\ldots\times f_n$ be the corresponding coordinate-wise self-map of $({\mathbb P}^1)^n$. Let $X$ be an irreducible subvariety of $({\mathbb P}^1)^n$ of dimension $r$ defined over $\bar{\mathbb Q}$. We define the \emph{$\varphi$-anomalous} locus of $X$ which is related to the \emph{$\varphi$-periodic} subvarieties of $({\mathbb P}^1)^n$. We prove that the $\varphi$-anomalous locus of $X$ is Zariski closed; this is a dynamical analogue of a theorem of Bombieri, Masser, and Zannier \cite{BMZ07}. We also prove that the points in the intersection of $X$ with the union of all irreducible $\varphi$-periodic subvarieties of $({\mathbb P}^1)^n$ of codimension $r$ have bounded height outside the $\varphi$-anomalous locus of $X$; this is a dynamical analogue of Habegger's theorem \cite{Habegger09} which was previously conjectured in \cite{BMZ07}. The slightly more general self-maps $\varphi=f_1\times\ldots\times f_n$ where each $f_i\in \bar{\mathbb Q}(x)$ is a disintegrated rational map are also treated at the end of the paper.Comment: Minor mistakes corrected, slight reorganizatio

ManiNetCluster: a novel manifold learning approach to reveal the functional links between gene networks.

BACKGROUND:The coordination of genomic functions is a critical and complex process across biological systems such as phenotypes or states (e.g., time, disease, organism, environmental perturbation). Understanding how the complexity of genomic function relates to these states remains a challenge. To address this, we have developed a novel computational method, ManiNetCluster, which simultaneously aligns and clusters gene networks (e.g., co-expression) to systematically reveal the links of genomic function between different conditions. Specifically, ManiNetCluster employs manifold learning to uncover and match local and non-linear structures among networks, and identifies cross-network functional links. RESULTS:We demonstrated that ManiNetCluster better aligns the orthologous genes from their developmental expression profiles across model organisms than state-of-the-art methods (p-value <2.2×10-16). This indicates the potential non-linear interactions of evolutionarily conserved genes across species in development. Furthermore, we applied ManiNetCluster to time series transcriptome data measured in the green alga Chlamydomonas reinhardtii to discover the genomic functions linking various metabolic processes between the light and dark periods of a diurnally cycling culture. We identified a number of genes putatively regulating processes across each lighting regime. CONCLUSIONS:ManiNetCluster provides a novel computational tool to uncover the genes linking various functions from different networks, providing new insight on how gene functions coordinate across different conditions. ManiNetCluster is publicly available as an R package at https://github.com/daifengwanglab/ManiNetCluster

A Tight Lower Bound to the Outage Probability of Discrete-Input Block-Fading Channels

In this correspondence, we propose a tight lower bound to the outage probability of discrete-input Nakagami-m block-fading channels. The approach permits an efficient method for numerical evaluation of the bound, providing an additional tool for system design. The optimal rate-diversity trade-off for the Nakagami-m block-fading channel is also derived and a tight upper bound is obtained for the optimal coding gain constant.Comment: 22 pages, 4 figures. This work has been accepted for IEEE Transactions on Information Theory and has been presented in part at the 2007 IEEE International Symposium on Information Theory, Nice, France, June 200

Wearable Sensor Data Based Human Activity Recognition using Machine Learning: A new approach

Recent years have witnessed the rapid development of human activity recognition (HAR) based on wearable sensor data. One can find many practical applications in this area, especially in the field of health care. Many machine learning algorithms such as Decision Trees, Support Vector Machine, Naive Bayes, K-Nearest Neighbor, and Multilayer Perceptron are successfully used in HAR. Although these methods are fast and easy for implementation, they still have some limitations due to poor performance in a number of situations. In this paper, we propose a novel method based on the ensemble learning to boost the performance of these machine learning methods for HAR