Generalized inverses of Hankel and Toeplitz mosaic matrices

AbstractHankel and Toeplitz mosaic matrices are block matrices with Hankel or Toeplitz blocks, respectively. It is shown that Hankel and Toeplitz mosaic matrices possess reflexive generalized inverses which are Bezoutians. Furthermore the Bezoutian structure of the Moore-Penrose and group inverses is investigated

Matrix representations of Toeplitz-plus-Hankel matrix inverses

AbstractInverses of Toeplitz-plus-Hankel matrices and, more generally, T+H-Bezoutians are represented as sums of products of triangular Toeplitz and Hankel matrices. The parameters occurring in these representations can be determined with the help of (1) solutions of “fundamental equations,” (2) solutions of a certain homogeneous equation, and (3) columns and rows of the inverse matrix

Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication

AbstractRepresentations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n × n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O(n) operations

Vandermonde factorization and canonical representations of block hankel matrices

AbstractWe study to which extent well-known facts concerning Vandermonde factorization or canonical representation of scalar Hankel matrices transfer to block Hankel matrices with p × q blocks. It is shown that nonsingular block Hankel matrices can be factored, like in the scalar case, into nonconfluent Vandermonde matrices and that the theorem on full-rank factorization of arbitrary Hankel matrices transfers (in a weak version) to the 2 × 2 block case but not to larger block sizes. In general, the minimal rank of a Vandermonde factorization (both with finite nodes and affine) is described in terms of the Hankel matrix. The main tools are realization, partial realization, and Moebius transformations

Partial realization for singular systems in standard form

AbstractThe partial realization problem under consideration consists in finding, for a given sequence s=(sk)0N−1 of blocks, matrices (A,E,B,C) of appropriate size such that si=CEN−1−iAiB and the identity matrix is a linear combination of A and E. We discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying Hankel matrices. We show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general. The paper strengthens results obtained by Manthey et al. [cf. W. Manthey, U. Helmke, D. Hinrichsen, in: U. Helmke et al. (Eds.), Operators, Systems, and Linear Algebra, Teubner, Stuttgart, 1997, pp. 138–156]. The main tools are the results of the authors obtained in connection with Vandermonde factorization of block Hankel matrices. Finally, an interpretation of the problem in periodic discrete-time systems is given

Current Challenges in Plant Eco-Metabolomics

The relatively new research discipline of Eco-Metabolomics is the application of metabolomics techniques to ecology with the aim to characterise biochemical interactions of organisms across different spatial and temporal scales. Metabolomics is an untargeted biochemical approach to measure many thousands of metabolites in different species, including plants and animals. Changes in metabolite concentrations can provide mechanistic evidence for biochemical processes that are relevant at ecological scales. These include physiological, phenotypic and morphological responses of plants and communities to environmental changes and also interactions with other organisms. Traditionally, research in biochemistry and ecology comes from two different directions and is performed at distinct spatiotemporal scales. Biochemical studies most often focus on intrinsic processes in individuals at physiological and cellular scales. Generally, they take a bottom-up approach scaling up cellular processes from spatiotemporally fine to coarser scales. Ecological studies usually focus on extrinsic processes acting upon organisms at population and community scales and typically study top-down and bottom-up processes in combination. Eco-Metabolomics is a transdisciplinary research discipline that links biochemistry and ecology and connects the distinct spatiotemporal scales. In this review, we focus on approaches to study chemical and biochemical interactions of plants at various ecological levels, mainly plant–organismal interactions, and discuss related examples from other domains. We present recent developments and highlight advancements in Eco-Metabolomics over the last decade from various angles. We further address the five key challenges: (1) complex experimental designs and large variation of metabolite profiles; (2) feature extraction; (3) metabolite identification; (4) statistical analyses; and (5) bioinformatics software tools and workflows. The presented solutions to these challenges will advance connecting the distinct spatiotemporal scales and bridging biochemistry and ecology

Meta-analysis identifies seven susceptibility loci involved in the atopic March

Eczema often precedes the development of asthma in a disease course called the a 'atopic march'. To unravel the genes underlying this characteristic pattern of allergic disease, we conduct a multi-stage genome-wide association study on infantile eczema followed by childhood asthma in 12 populations including 2,428 cases and 17,034 controls. Here we report two novel loci specific for the combined eczema plus asthma phenotype, which are associated with allergic disease for the first time; rs9357733 located in EFHC1 on chromosome 6p12.3 (OR 1.27; P=2.1 × 10 a'8) and rs993226 between TMTC2 and SLC6A15 on chromosome 12q21.3 (OR 1.58; P=5.3 × 10 a'9). Additional susceptibility loci identified

52 Genetic Loci Influencing Myocardial Mass.

BACKGROUND: Myocardial mass is a key determinant of cardiac muscle function and hypertrophy. Myocardial depolarization leading to cardiac muscle contraction is reflected by the amplitude and duration of the QRS complex on the electrocardiogram (ECG). Abnormal QRS amplitude or duration reflect changes in myocardial mass and conduction, and are associated with increased risk of heart failure and death. OBJECTIVES: This meta-analysis sought to gain insights into the genetic determinants of myocardial mass. METHODS: We carried out a genome-wide association meta-analysis of 4 QRS traits in up to 73,518 individuals of European ancestry, followed by extensive biological and functional assessment. RESULTS: We identified 52 genomic loci, of which 32 are novel, that are reliably associated with 1 or more QRS phenotypes at p < 1 × 10(-8). These loci are enriched in regions of open chromatin, histone modifications, and transcription factor binding, suggesting that they represent regions of the genome that are actively transcribed in the human heart. Pathway analyses provided evidence that these loci play a role in cardiac hypertrophy. We further highlighted 67 candidate genes at the identified loci that are preferentially expressed in cardiac tissue and associated with cardiac abnormalities in Drosophila melanogaster and Mus musculus. We validated the regulatory function of a novel variant in the SCN5A/SCN10A locus in vitro and in vivo. CONCLUSIONS: Taken together, our findings provide new insights into genes and biological pathways controlling myocardial mass and may help identify novel therapeutic targets

Chebyshev–Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices

AbstractCentrosymmetric Toeplitz-plus-Hankel matrices are investigated on the basis of their “splitting property”, which is their similarity to the direct sum of two special Toeplitz-plus-Hankel matrices. These matrices can be considered as Hankel matrices (moment matrices) in bases of Chebyshev polynomials and are called Chebyshev–Hankel matrices. Chebyshev–Hankel matrices have similar properties like Hankel matrices. This concerns inversion formulas and fast algorithms. A superfast algorithm for solving Chebyshev–Hankel and centrosymmetric Toeplitz-plus-Hankel systems is presented that is based on real trigonometric transforms. The main tool of investigation is the interpretation of Chebyshev–Hankel matrices as matrices of restricted multiplication operators with respect to Chebyshev bases