Matrix Roots of Eventually Positive Matrices

Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of primitive matrices. Using classical matrix function theory and Perron-Frobenius theory, we characterize, classify, and describe in terms of the real Jordan canonical form the $p$th-roots of eventually positive matrices.Comment: Accepted for publication in Linear Algebra and its Application

The peripheral spectrum of a nonnegative matrix

AbstractIn this paper we provide a necessary and sufficient condition for a collection of Jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix. We arrive at our condition by linking the Tam–Schneider condition to the level sets (with respect to the spectral radius) of a nonnegative matrix

Spectrally arbitrary ray patterns

AbstractAn n×n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a matrix in the pattern class of A such that its characteristic polynomial is f(x). In this article the authors extend the nilpotent-Jacobi method for sign patterns to ray patterns, establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary. They use this method to establish that a particular family of n×n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary. They then show that every n×n irreducible, spectrally arbitrary ray pattern has at least 3n-1 nonzeros

The minimum upper bound on the first ambiguous power of an irreducible, nonpowerful ray or sign pattern

AbstractLet A be an n×n irreducible ray or sign pattern matrix. If A is a sign pattern, it is shown that either A is powerful or else Ak has an ambiguous entry for some k⩽n2-2n+2, and further, sign patterns based on the Wielandt graph show that this bound is the best possible. If A is a ray pattern, partial results for the same bound are given

Faecal haemoglobin and faecal calprotectin as indicators of bowel disease in patients presenting to primary care with bowel symptoms

OBJECTIVE: In primary care, assessing which patients with bowel symptoms harbour significant disease (cancer, higher-risk adenoma or IBD) is difficult. We studied the diagnostic accuracies of faecal haemoglobin (FHb) and faecal calprotectin (FC) in a cohort of symptomatic patients. DESIGN: From October 2013 to March 2014, general practitioners were prompted to request FHb and FC when referring patients with bowel symptoms to secondary care. Faecal samples were analysed for haemoglobin (EIKEN OC-Sensor io) and calprotectin (BÜHLMANN Calprotectin ELISA). Patients triaged to endoscopy were investigated within 6 weeks. All clinicians and endoscopists were blind to the faecal test results. The diagnostic accuracies of FHb and FC for identification of significant bowel disease were assessed. RESULTS: 1043 patients returned samples. FHb was detectable in 57.6% (median 0.4 µg/g, 95% CI 0.4 to 0.8; range 0–200). FC at 50 µg/g or above was present in 60.0%. 755 patients (54.6% women, median age 64 years (range 16–90, IQR 52–73)) returned samples and completed colonic investigations. 103 patients had significant bowel disease; the negative predictive values of FHb for colorectal cancer, higher-risk adenoma and IBD were 100%, 97.8% and 98.4%, respectively. Using cut-offs of detectable FHb and/or 200 µg/g FC detected two further cases of IBD, one higher-risk adenoma and no additional cancers. CONCLUSIONS: In primary care, undetectable FHb is a good ‘rule-out’ test for significant bowel disease and could guide who requires investigation

Oncogenic RET Kinase domain mutations perturb the autophosphorylation trajectory by enhancing substrate presentation in trans

To decipher the molecular basis for RET kinase activation and oncogenic deregulation, we defined the temporal sequence of RET autophosphorylation by label-free quantitative mass spectrometry. Early autophosphorylation sites map to regions flanking the kinase domain core, while sites within the activation loop only form at later time points. Comparison with oncogenic RET kinase revealed that late autophosphorylation sites become phosphorylated much earlier than wild-type RET, which is due to a combination of an enhanced enzymatic activity, increased ATP affinity, and surprisingly, by providing a better intermolecular substrate. Structural analysis of oncogenic M918T and wild-type RET kinase domains reveal a cis-inhibitory mechanism involving tethering contacts between the glycine-rich loop, activation loop, and αC-helix. Tether mutations only affected substrate presentation but perturbed the autophosphorylation trajectory similar to oncogenic mutations. This study reveals an unappreciated role for oncogenic RET kinase mutations in promoting intermolecular autophosphorylation by enhancing substrate presentation