An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.Comment: 23 pages; to appear in Mediterr. J. Math., Vol. 9 (2012

Gene selection with multiple ordering criteria

BACKGROUND: A microarray study may select different differentially expressed gene sets because of different selection criteria. For example, the fold-change and p-value are two commonly known criteria to select differentially expressed genes under two experimental conditions. These two selection criteria often result in incompatible selected gene sets. Also, in a two-factor, say, treatment by time experiment, the investigator may be interested in one gene list that responds to both treatment and time effects. RESULTS: We propose three layer ranking algorithms, point-admissible, line-admissible (convex), and Pareto, to provide a preference gene list from multiple gene lists generated by different ranking criteria. Using the public colon data as an example, the layer ranking algorithms are applied to the three univariate ranking criteria, fold-change, p-value, and frequency of selections by the SVM-RFE classifier. A simulation experiment shows that for experiments with small or moderate sample sizes (less than 20 per group) and detecting a 4-fold change or less, the two-dimensional (p-value and fold-change) convex layer ranking selects differentially expressed genes with generally lower FDR and higher power than the standard p-value ranking. Three applications are presented. The first application illustrates a use of the layer rankings to potentially improve predictive accuracy. The second application illustrates an application to a two-factor experiment involving two dose levels and two time points. The layer rankings are applied to selecting differentially expressed genes relating to the dose and time effects. In the third application, the layer rankings are applied to a benchmark data set consisting of three dilution concentrations to provide a ranking system from a long list of differentially expressed genes generated from the three dilution concentrations. CONCLUSION: The layer ranking algorithms are useful to help investigators in selecting the most promising genes from multiple gene lists generated by different filter, normalization, or analysis methods for various objectives

Analysis of human meiotic recombination events with a parent-sibling tracing approach

<p>Abstract</p> <p>Background</p> <p>Meiotic recombination ensures that each child inherits distinct genetic materials from each parent, but the distribution of crossovers along meiotic chromosomes remains difficult to identify. In this study, we developed a parent-sibling tracing (PST) approach from previously reported methods to identify meiotic crossover sites of GEO GSE6754 data set. This approach requires only the single nucleotide polymorphism (SNP) data of the pedigrees of both parents and at least two of children.</p> <p>Results</p> <p>Compared to other SNP-based algorithms (identity by descent or pediSNP), fewer uninformative SNPs were derived with the use of PST. Analysis of a GEO GSE6754 data set containing 2,145 maternal and paternal meiotic events revealed that the pattern and distribution of paternal and maternal recombination sites vary along the chromosomes. Lower crossover rates near the centromeres were more prominent in males than in females. Based on analysis of repetitive sequences, we also showed that recombination hotspots are positively correlated with SINE/MIR repetitive elements and negatively correlated with LINE/L1 elements. The number of meiotic recombination events was positively correlated with the number of shorter tandem repeat sequences.</p> <p>Conclusions</p> <p>The advantages of the PST approach include the ability to use only two-generation pedigrees with two siblings and the ability to perform gender-specific analyses of repetitive elements and tandem repeat sequences while including fewer uninformative SNP regions in the results.</p

Preservation of Ranking Order in the Expression of Human Housekeeping Genes

Housekeeping (HK) genes fulfill the basic needs for a cell to survive and function properly. Their ubiquitous expression, originally thought to be constant, can vary from tissue to tissue, but this variation remains largely uncharacterized and it could not be explained by previously identified properties of HK genes such as short gene length and high GC content. By analyzing microarray expression data for human genes, we uncovered a previously unnoted characteristic of HK gene expression, namely that the ranking order of their expression levels tends to be preserved from one tissue to another. Further analysis by tensor product decomposition and pathway stratification identified three main factors of the observed ranking preservation, namely that, compared to those of non-HK (NHK) genes, the expression levels of HK genes show a greater degree of dispersion (less overlap), stableness (a smaller variation in expression between tissues), and correlation of expression. Our results shed light on regulatory mechanisms of HK gene expression that are probably different for different HK genes or pathways, but are consistent and coordinated in different tissues

SAQC: SNP Array Quality Control

<p>Abstract</p> <p>Background</p> <p>Genome-wide single-nucleotide polymorphism (SNP) arrays containing hundreds of thousands of SNPs from the human genome have proven useful for studying important human genome questions. Data quality of SNP arrays plays a key role in the accuracy and precision of downstream data analyses. However, good indices for assessing data quality of SNP arrays have not yet been developed.</p> <p>Results</p> <p>We developed new quality indices to measure the quality of SNP arrays and/or DNA samples and investigated their statistical properties. The indices quantify a departure of estimated individual-level allele frequencies (AFs) from expected frequencies via standardized distances. The proposed quality indices followed lognormal distributions in several large genomic studies that we empirically evaluated. AF reference data and quality index reference data for different SNP array platforms were established based on samples from various reference populations. Furthermore, a confidence interval method based on the underlying empirical distributions of quality indices was developed to identify poor-quality SNP arrays and/or DNA samples. Analyses of authentic biological data and simulated data show that this new method is sensitive and specific for the detection of poor-quality SNP arrays and/or DNA samples.</p> <p>Conclusions</p> <p>This study introduces new quality indices, establishes references for AFs and quality indices, and develops a detection method for poor-quality SNP arrays and/or DNA samples. We have developed a new computer program that utilizes these methods called SNP Array Quality Control (SAQC). SAQC software is written in R and R-GUI and was developed as a user-friendly tool for the visualization and evaluation of data quality of genome-wide SNP arrays. The program is available online (<url>http://www.stat.sinica.edu.tw/hsinchou/genetics/quality/SAQC.htm</url>).</p