Invertibility of circulant matrices of arbitrary size

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in $\mathbb{Z}_p$. Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant $0, 1$-matrices.Comment: 18 page

Adjunctive biomarkers for improving diagnosis of tuberculosis and monitoring therapeutic effects

SummaryObjectivesTo identify host biomarkers associated with latent tuberculosis infection (LTBI), active tuberculosis (TB), and nontuberculous mycobacteria (NTM) diseases to improve diagnosis and effective anti-TB treatment.MethodsActive TB and NTM patients at diagnosis, recent TB contacts, and normal healthy subjects were recruited. Tuberculin skin tests, QuantiFERON-TB Gold In-Tube tests, and multiplex bead arrays with 17 analytes were performed. TB patients were re-evaluated after 2 and 6 months of treatment.ResultsMycobacterium tuberculosis (M. tb) antigen-specific IFN-γ, IL-2, and CXCL10 responses were significantly higher in active TB and LTBI compared with controls (P < 0.01). Only serum VEGF levels varied between the active TB and LTBI groups (AUC = 0.7576, P < 0.001). Active TB and NTM diseases were differentiated by serum IL-2, IL-9, IL-13, IL-17, TNF-α and sCD40L levels (P < 0.05). Increased sCD40L and decreased M. tb antigen-specific IFN-γ levels correlated with sputum clearance of M. tb after 2 months of treatment (P < 0.001).ConclusionsSerum IL-2, IL-9, IL-13, IL-17, TNF-α, sCD40L and VEGF-A levels may be adjunctive biomarkers for differential diagnosis of active TB, LTBI, and NTM disease. Assessment of serum sCD40L and M. tb antigen-specific IFN-γ, TNF-α, and IL-2 levels could help predict successful anti-TB treatment in conjunction with M. tb clearance

Wavelet-based identification of DNA focal genomic aberrations from single nucleotide polymorphism arrays

<p>Abstract</p> <p>Background</p> <p>Copy number aberrations (CNAs) are an important molecular signature in cancer initiation, development, and progression. However, these aberrations span a wide range of chromosomes, making it hard to distinguish cancer related genes from other genes that are not closely related to cancer but are located in broadly aberrant regions. With the current availability of high-resolution data sets such as single nucleotide polymorphism (SNP) microarrays, it has become an important issue to develop a computational method to detect driving genes related to cancer development located in the focal regions of CNAs.</p> <p>Results</p> <p>In this study, we introduce a novel method referred to as the wavelet-based identification of focal genomic aberrations (WIFA). The use of the wavelet analysis, because it is a multi-resolution approach, makes it possible to effectively identify focal genomic aberrations in broadly aberrant regions. The proposed method integrates multiple cancer samples so that it enables the detection of the consistent aberrations across multiple samples. We then apply this method to glioblastoma multiforme and lung cancer data sets from the SNP microarray platform. Through this process, we confirm the ability to detect previously known cancer related genes from both cancer types with high accuracy. Also, the application of this approach to a lung cancer data set identifies focal amplification regions that contain known oncogenes, though these regions are not reported using a recent CNAs detecting algorithm GISTIC: SMAD7 (chr18q21.1) and FGF10 (chr5p12).</p> <p>Conclusions</p> <p>Our results suggest that WIFA can be used to reveal cancer related genes in various cancer data sets.</p

Caplets: wavelet representations without wavelets

MultiResolution (MR) is among the most effective and the most popular approaches for data representation. In that approach, the given data are organized into a sequence of resolution layers, and then the “difference ” between each two consecutive layers is recorded in terms of detail coefficients. Wavelet decomposition is the best known representation methodology in the MR category. The major reason for the popularity of wavelet decompositions is their implementation and inversion by a fast algorithm, the so-called fast wavelet transform (FWT). Another central reason for the success of wavelets is that the wavelet coefficients capture very accurately the smoothness class of the function hidden behind the data. This is essential for the understanding of the performance of key wavelet-based algorithms in compression, in denoising, and in other applications. On the downside, constructing wavelets with good space-frequency localization properties becomes involved as the spatial dimension grows. An alternative to the sometime-hard-to-construct wavelet representations is the always-easy-to-construct (and slightly older) non-orthogonal pyramidal algorithms. Similar to wavelets, the (linear, regular, isotropic) pyramidal representations are based on some method for linear coarsening (by a decomposition filter) of their data, and a complementary method for linear prediction (by a prediction filter) of the original data from the coarsened one. The first step creates the resolution layers and the second allows for trivial extractions of suitable detail coefficients. Th