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We present a Lanczos algorithm utilizing multiple grids that reduces the memory requirements both on disk and in working memory by one order of magnitude for RBC/UKQCD’s 48I and 64I ensembles at the physical pion mass. The precision of the resulting eigenvectors is on par with exact deflation.

The width difference ΔΓ s that can be extracted from lifetime measurements of the two mass eigenstates of the B s 0 − B ¯ s 0 system is one of the key flavor precision observables and has been experimentally measured at per cent level accuracy. The current theory prediction is much less accurate and a sizable reduction of scale uncertainties can only be achieved by means of evaluating the uncalculated 2- and 3-loop QCD corrections. This is precisely the issue addressed in this work where we report on the results that have been obtained so far and explain some of the technical and conceptual challenges that we encountered in the course of our calculations.

Eigenvector mapping techniques are widely used by ecologists and evolutionary biologists to describe and control for spatial and/or phylogenetic patterns in their data. The selection of an appropriate subset of eigenvectors is a critical step (misspecification can lead to highly biased results and interpretations), and there is no consensus yet on how to proceed. We conducted a ten-year review of the practices of eigenvector selection and highlighted three main procedures: selecting the subset of descriptors minimising the Akaike information criterion (AIC), using a forward selection with double stopping criterion after testing the global model significance (FWD), and selecting the subset minimising the autocorrelation in the model residuals (MIR). We compared the type I error rates, statistical power, and R ² estimation accuracy of these methods using simulated data. Finally, a real dataset was analysed using variation partitioning analysis to illustrate to what extent the different selection approaches affected the ecological interpretation of the results. We show that, while the FWD and MIR approaches presented a correct type I error rate and were accurate, the AIC approach displayed extreme type I error rates (100%), and strongly overestimated the R ². Moreover, the AIC approach resulted in wrong ecological interpretations, as it overestimated the pure spatial fraction (and the joint spatial-environmental fraction to a lesser extent) of the variation partitioning. Both the FWD and MIR methods performed well at broad and medium scales but had a very low power to detect fine-scale patterns. The FWD approach selected more eigenvectors than the MIR approach but also returned more accurate R ² estimates. Hence, we discourage any future use of the AIC approach, and advocate choosing between the MIR and FWD approaches depending on the objective of the study: controlling for spatial or phylogenetic autocorrelation (MIR) or describing the patterns as accurately as possible (FWD).

We present a method for calculating eigenvectors of the staggered Dirac operator based on the Golub-Kahan-Lanczos bidiagonalization algorithm. Instead of using orthogonalization during the bidiagonalization procedure to increase stability, we choose to stabilize the method by combining it with an outer iteration that refines the approximate eigenvectors obtained from the inner bidiagonalization procedure. We discuss the performance of the current implementation using QEX and compare with other methods.

Eigenvector-mapping methods such as Moran’s eigenvector maps (MEM) are derived from a spatial weighting matrix (SWM) that describes the relations among a set of sampled sites. The specification of the SWM is a crucial step, but the SWM is generally chosen arbitrarily, regardless of the sampling design characteristics. Here, we compare the statistical performances of different types of SWMs (distance-based or graph-based) in contrasted realistic simulation scenarios. Then, we present an optimization method and evaluate its performances compared to the arbitrary choice of the most-widely used distance-based SWM. Results showed that the distance-based SWMs generally had lower power and accuracy than other specifications, and strongly underestimated spatial signals. The optimization method, using a correction procedure for multiple tests, had a correct type I error rate, and had higher power and accuracy than an arbitrary choice of the SWM. Nevertheless, the power decreased when too many SWMs were compared, resulting in a trade-off between the gain of accuracy and the loss of power. We advocate that future studies should optimize the choice of the SWM using a small set of appropriate candidates. R functions to implement the optimization are available in the adespatial package and are detailed in a tutorial.

IntroductionHypertrophic cardiomyopathy (HCM) is a common heritable cardiac muscle disease and a major cause of sudden cardiac death in young adults.1 It is characterised by unexplained left ventricular hypertrophy (LVH), typically caused by rare gene variants affecting cardiac sarcomere.2 Genetic screening identifies asymptomatic sarcomere variant carriers before LVH develops, i.e. genotype-positive phenotype-negative G+P-(HCM) subjects who require regular surveillance. However, current phenotyping is limited to the macroscopic scale. Diffusion tensor cardiovascular magnetic resonance (DT-CMR) non-invasively probes the myocardial microstructure in vivo, 3 characterising cardiomyocytes and their functional units (sheetlets) and can provide new insights into the pathophysiology of HCM. We aimed to investigate if DT-CMR can detect abnormal myocardial microstructure in G+P-(HCM) subjects before phenotypic conversion.Materials and Methods In-vivo DT-CMR of mid-LV short axis slice was performed using stimulated echo acquisition mode sequence on a 3T Vida scanner3 in G+P-(HCM) patients and healthy volunteers (HVOL). G+P-(HCM) patients carried a HCM-causing sarcomere variant confirmed by Sanger sequencing and had maximal LV wall thickness (LVWT) <13 mm on CMR. This study was ethically approved (13/LO/1830).Results25 G+P-(HCM) were matched with 20 HVOL for age and gender. There was no difference between the cohorts in the LVWT at the imaged by DT-CMR slice, LV volumes, mass, ejection fraction, systolic strain, native T1 and T2 values, ECV% and LGE%. G+P-(HCM) subjects had significantly elevated second eigenvector angle (E2A), i.e. sheetlet angle when compared to HVOL: diastolic E2A 18°(14–21) vs 15°(12–17) and systolic E2A 70°(±5) vs 64°(±5). There was no difference in sheetlet mobility, magnitude of diffusivity (mean diffusivity) or its anisotropy (fractional anisotropy). DiscussionThis is the first in human study demonstrating that in G+P-(HCM) patients myocardial sheetlet abnormality affects both diastole and systole and precedes irreversible changes (LVH and myocardial tissue alteration). This likely reflects primary sarcomere defect caused by rare gene variants, occurring early in the pathophysiological cascade of HCM development. E2A abnormality should be investigated further as a potential novel pre-phenotypic imaging biomarker of sarcomere dysfunction in G+P-(HCM) and a target for developing therapeutics aimed at mitigating phenotypic conversion before irreversible myocardial tissue changes occur. Abstract 14 Figure 1Examples of absolute E2A in G+P-(HCM) and HVOL. Examples of DT-CMR absolute secondary eigenvector angle (E2A) maps representing myocardial sheetlet orientation relative to the LV epicardial wall. G+P-(HCM) patient has elevated E2A, especially noticeable in diastole, with more red pixels (coding for high E2A, i.e. myocardial sheetlets form a larger angle with the epicardial wall) compared to HVOL whose diastolic E2A map is predominately in blue colour (coding for low E2A, i.e. myocardial sheetlets are almost parallel to the epicardial wall). SAX (short axis slice); LV (left ventricle); LGE PSIR (late gadolinium enhancement phase sensitive inversion recovery) *Mid LV refers to the short axis slice at mid LV level at which DT-CMR imaging was acquiredAbstract 14 Figure 2Comparison of myocardial sheetlet configuration (absolute E2A) in G+P-(HCM) vs HVOL. Compared to HVOL, in G+P-(HCM) myocardial sheetlets orientate with a larger angle to the LV epicardial wall, denoted by elevated secondary eigenvector value (E2A) during both diastole and systoleConclusionDT-CMR detects myocardial sheetlet abnormality which precedes the development of irreversible phenotypic changes in G+P-(HCM) and may be an early and potentially modifiable marker of disease.AcknowledgementsThis study is supported by the British Heart Foundation (BHF programme grant RG 19/1/34160). We are also grateful to our collaborators Professor Sanjay Prasad, Dr Antonis Pantazis, Dr Tessa Homfray and Dr Deborah Morris-Rosendahl.ReferencesO’Mahony C, Jichi F, Pavlou M, Monserrat L, Anastasakis A, Rapezzi C, Biagini E, Gimeno JR, Limongelli G, McKenna WJ, Omar RZ, Elliott PM, Ortiz-Genga M, Fernandez X, Vlagouli V, Stefanadis C, Coccolo F, Sandoval MJO, Pacileo G, et al. A novel clinical risk prediction model for sudden cardiac death in hypertrophic cardiomyopathy (HCM Risk-SCD). Eur Heart J. 2014;35(30):2010–2020.Walsh R, Buchan R, Wilk A, John S, Felkin LE, Thomson KL, Chiaw TH, Loong CCW, Pua CJ, Raphael C, Prasad S, Barton PJ, Funke B, Watkins H, Ware JS, Cook SA. Defining the genetic architecture of hypertrophic cardiomyopathy: re-evaluating the role of non-sarcomeric genes. Eur Heart J. 2017;38(46):3461–3468.Nielles-Vallespin S, Khalique Z, Ferreira PF, de Silva R, Scott AD, Kilner P, McGill LA, Giannakidis A, Gatehouse PD, Ennis D, Aliotta E, Al-Khalil M, Kellman P, Mazilu D, Balaban RS, Firmin DN, Arai AE, Pennell DJ. Assessment of myocardial microstructural dynamics by in vivo diffusion tensor cardiac magnetic resonance. J Am Coll Cardiol. 2017;69(6):661–676.

Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors, few results are tight for entrywise analyses, which are critical for a number of problems such as community detection. This paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low rank, which helps settle the conjecture in Abbe, Bandeira and Hall (2014) that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps. The key is a first-order approximation of eigenvectors under the ℓ ∞ norm: u k ≈ A u k * λ k * , where {uk } and { u k * } are eigenvectors of a random matrix A and its expectation 𝔼A, respectively. The fact that the approximation is both tight and linear in A facilitates sharp comparisons between uk and u k * . In particular, it allows for comparing the signs of uk and u k * even if ‖ u k − u k * ‖ ∞ is large. The results are further extended to perturbations of eigenspaces, yielding new ℓ ∞- type bounds for synchronization (ℤ₂-spikedWigner model) and noisy matrix completion.

This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis. The classical forward selection method presents two problems: a highly inflated Type I error and an overestimation of the amount of explained variance. Correcting these problems will greatly improve the performance of this very useful method in ecological modeling. To prevent the first problem, we propose a two-step procedure. First, a global test using all explanatory variables is carried out. If, and only if, the global test is significant, one can proceed with forward selection. To prevent overestimation of the explained variance, the forward selection has to be carried out with two stopping criteria: (1) the usual alpha significance level and (2) the adjusted coefficient of multiple determination ($R_{a}^{2}$) calculated using all explanatory variables. When forward selection identifies a variable that brings one or the other criterion over the fixed threshold, that variable is rejected, and the procedure is stopped. This improved method is validated by simulations involving univariate and multivariate response data. An ecological example is presented using data from the Bryce Canyon National Park, Utah, USA.

IntroductionMicrovascular disease (MVD) and disarray are myocardial substrates that associate with adverse events in HCM. How these relate to sarcomeric mutation and LVH are poorly understood. Advances in CMR allow disarray and MVD to be detected and quantified. We aimed to investigate their prevalence and severity as phenotype evolves.Materials and Methods89 patients with overt HCM, (44 G+LVH+, 45 G-LVH+), 75 individuals with sarcomeric mutations and no LVH (G+LVH-) and 28 healthy volunteers (HV) underwent quantitative adenosine stress perfusion CMR and cardiac diffusion tensor imaging (cDTI) (measuring fractional anisotropy (FA), mean diffusivity (MD) and Second Eigenvector Angle (E2A)).ResultsDemographics were as follows: HCM: Age 56(47–61) years, female 21%, G+LVH-: Age 34(24–41), female 55%, HV: Age 34(32–40) years and female 50%.Patients with overt HCM had more disarray compared to both G+LVH- and HV (lower FA:0.28±0.03 vs FAG+LVH-0.32±0.02 p<0.001, higher MD:1.57±0.08 × 10-3 mm2/s vs MDG+LVH-1.50±0.05 × 10-3 mm2/s, p<0.001 and higher E2A60±9° vs E2AG+LVH- 49±9° p<0.001). When considering G+LVH+ vs G-LVH+, there was no significant differences in FA, or MD, however G-LVH+ adopted steeper systolic sheetlet angles compared to G+LVH+ (62±6° vs 58±11° p=0.047). Compared to G+LVH-, overt HCM had lower stress MBF (1.7±0.5 vs 2.5±0.5ml/g/min p<0.001) and a higher prevalence of perfusion defects (91%(84)vs 27%(19) p<0.001). Perfusion defects were ubiquitous in G+LVH+, occurring in all 44 patients but only 84%(35) in G-LVH+ p=0.005.Compared to HV, G+LVH- had lower FA (0.32±0.02 vs 0.34±0.02 p<0.001), higher MD (1.50±0.05 × 10-3 mm2/s vs 1.48±0.04 × 10-3 mm2/s p<0.001 and higher E2A (49±9° vs 40±9° p<0.001). Compared to HV, G+LVH- had no significant difference in global stress perfusion values, and a higher prevalence of perfusion defects (25%(19) vs 0 HV p=0.005). G+LVH- with perfusion defects had lower FA than G+LVH- without perfusion defects (0.31±0.3 vs 0.32±0.02 p=0.018).Abstract 1 Figure 1Abnormalities in perfusion and disarray (low FA, high MD, steeper E2A angulation) occurring pre-hypertrophy (G+LVH-) and more severely in overt disease (G+LVH+ & G-LVH+) FA – Fractional Anisotropy, MD – Mean Diffusivity, E2A – Second Eigenvector angle, MYH7 – myosin heavy chain 7, TPM1 – tropomyosin 1, TNNI3, Troponin I3Abstract 1 Figure 2FA- Fractional Anistropy, MD, mean diffusivity, E2A, Second Eigenvector Angle, MBF, myocardial blood flow. *p<0.05, **p<0.01, ***p<0.001ConclusionMVD and disarray are present, detectable and associate in G+LVH-. Overt HCM is characterised by more myocyte disarray and MVD than both G+LVH- and health. Subtle differences in MVD and disarray occur between G+LVH+ and G-LVH+ with perfusion defects being ubiquitous in genotype positive HCM.

In this paper, a Goodwin model of business cycle with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the linear stability of the equilibrium is investigated. Numerical simulations are carried out to study the excitation conditions of the long-periodic Goodwin oscillations.