Moments of the characteristic polynomial in the three ensembles of random matrices

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n\times n$ determinant and an $m\times m$ determinant and another between the pfaffian of a $2n\times 2n$ anti-symmetric matrix and a sum of $m\times m$ determinants.Comment: tex, 1 file, 15 pages [SPhT-T01/016], published J. Phys. A: Math. Gen. 34 (2001) 1-1

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Block orthogonal polynomials: I. Definition and properties

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean scalar product, to a given $i$-dimensional subspace ${\cal E}_i$ of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct scalar products. An explicit two step Gram-Schmidt orthogonalization (G-SO) procedure to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of ${\cal E}_i$ equal to the subspace of polynomials of degree less than $i$. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.Comment: This is a reduced version of the initial manuscript, the number of pages being reduced from 34 to 2

Genetic basis for variation in plasma IL-18 levels in persons with chronic hepatitis C virus and human immunodeficiency virus-1 infections

Inflammasomes are multi-protein complexes integrating pathogen-triggered signaling leading to the generation of pro-inflammatory cytokines including interleukin-18 (IL-18). Hepatitis C virus (HCV) and human immunodeficiency virus (HIV) infections are associated with elevated IL-18, suggesting inflammasome activation. However, there is marked person-to-person variation in the inflammasome response to HCV and HIV. We hypothesized that host genetics may explain this variation. To test this, we analyzed the associations of plasma IL-18 levels and polymorphisms in 10 genes in the inflammasome cascade. About 1538 participants with active HIV and/or HCV infection in three ancestry groups are included. Samples were genotyped using the Illumina Omni 1-quad and Omni 2.5 arrays. Linear regression analyses were performed to test the association of variants with log IL-18 including HCV and HIV infection status, and HIV RNA in each ancestry group and then meta-analyzed. Eleven highly correlated single-nucleotide polymorphisms (r²=0.98–1) in the IL-18-BCO2 region were significantly associated with log IL-18; each T allele of rs80011693 confers a decrease of 0.06 log pg ml⁻¹ of IL-18 after adjusting for covariates (rs80011693; rs111311302 β=−0.06, P-value=2.7 × 10⁻⁴). In conclusion, genetic variation in IL-18 is associated with IL-18 production in response to HIV and HCV infection, and may explain variability in the inflammatory outcomes of chronic viral infections

Effect of alirocumab on mortality after acute coronary syndromes. An analysis of the ODYSSEY OUTCOMES randomized clinical trial

Background: Previous trials of PCSK9 (proprotein convertase subtilisin-kexin type 9) inhibitors demonstrated reductions in major adverse cardiovascular events, but not death. We assessed the effects of alirocumab on death after index acute coronary syndrome. Methods: ODYSSEY OUTCOMES (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab) was a double-blind, randomized comparison of alirocumab or placebo in 18 924 patients who had an ACS 1 to 12 months previously and elevated atherogenic lipoproteins despite intensive statin therapy. Alirocumab dose was blindly titrated to target achieved low-density lipoprotein cholesterol (LDL-C) between 25 and 50 mg/dL. We examined the effects of treatment on all-cause death and its components, cardiovascular and noncardiovascular death, with log-rank testing. Joint semiparametric models tested associations between nonfatal cardiovascular events and cardiovascular or noncardiovascular death. Results: Median follow-up was 2.8 years. Death occurred in 334 (3.5%) and 392 (4.1%) patients, respectively, in the alirocumab and placebo groups (hazard ratio [HR], 0.85; 95% CI, 0.73 to 0.98; P=0.03, nominal P value). This resulted from nonsignificantly fewer cardiovascular (240 [2.5%] vs 271 [2.9%]; HR, 0.88; 95% CI, 0.74 to 1.05; P=0.15) and noncardiovascular (94 [1.0%] vs 121 [1.3%]; HR, 0.77; 95% CI, 0.59 to 1.01; P=0.06) deaths with alirocumab. In a prespecified analysis of 8242 patients eligible for ≥3 years follow-up, alirocumab reduced death (HR, 0.78; 95% CI, 0.65 to 0.94; P=0.01). Patients with nonfatal cardiovascular events were at increased risk for cardiovascular and noncardiovascular deaths (P<0.0001 for the associations). Alirocumab reduced total nonfatal cardiovascular events (P<0.001) and thereby may have attenuated the number of cardiovascular and noncardiovascular deaths. A post hoc analysis found that, compared to patients with lower LDL-C, patients with baseline LDL-C ≥100 mg/dL (2.59 mmol/L) had a greater absolute risk of death and a larger mortality benefit from alirocumab (HR, 0.71; 95% CI, 0.56 to 0.90; Pinteraction=0.007). In the alirocumab group, all-cause death declined wit h achieved LDL-C at 4 months of treatment, to a level of approximately 30 mg/dL (adjusted P=0.017 for linear trend). Conclusions: Alirocumab added to intensive statin therapy has the potential to reduce death after acute coronary syndrome, particularly if treatment is maintained for ≥3 years, if baseline LDL-C is ≥100 mg/dL, or if achieved LDL-C is low. Clinical Trial Registration: URL: https://www.clinicaltrials.gov. Unique identifier: NCT01663402

Characterisation of Elastic and Acoustic Properties of an Agar-Based Tissue Mimicking Material

As a first step towards an acoustic localisation device for coronary stenosis to provide a non-invasive means of diagnosing arterial disease, measurements are reported for an agar-based tissue mimicking material (TMM) of the shear wave propagation velocity, attenuation and viscoelastic constants, together with one dimensional quasi-static elastic moduli and Poisson’s ratio. Phase velocity and attenuation coefficients, determined by generating and detecting shear waves piezo-electrically in the range 300 Hz–2 kHz, were 3.2–7.5 ms−1 and 320 dBm−1. Quasi-static Young’s modulus, shear modulus and Poisson’s ratio, obtained by compressive or shear loading of cylindrical specimens were 150–160 kPa; 54–56 kPa and 0.37–0.44. The dynamic Young’s and shear moduli, derived from fitting viscoelastic internal variables by an iterative statistical inverse solver to freely oscillating specimens were 230 and 33 kPa and the corresponding relaxation times, 0.046 and 0.036 s. The results were self-consistent, repeatable and provide baseline data required for the computational modelling of wave propagation in a phantom