5 research outputs found
Stochastic matrices and the Boyle and Handelman conjecture
No full textLet A be an stochastic matrix with rank(). A reformulation of the Boyle and Handelman Conjecture is det( for all real numbers t satisfying .In this thesis, we prove that this conjecture is true for stochastic matrices whose rank exceeds
A reverse Hadamard inequality
No full textAbstractLet U be an n × n unitary matrix with determinant equal to 1. Let A be an n × n real matrix with rank(A) ⩽ 2 and entries satisfying aij ⩾ 1 for 1 ⩽ i, j ⩽ n. Then it follows that det(A ∘ U) ⩾ 1. This reversal of the Hadamard inequality can be obtained easily from an old result of Fiedler. In this article we present a different proof of this fact and discuss its ramifications
Some remarks on a conjecture of Boyle and Handelman
No full textAbstractBoyle and Handelman have conjectured that whenever A is an n × n nonnegative matrix with rank A ⩽ r and Perron root λ1, the inequality det(λI − tA) ⩽ λn−r(λr−λ1r) holds for all real numbers λ satisfying λ ⩾ λ1. We introduce an analogous conjecture involving nonnegative central (class) functions on the permutation group Sn. The analogue of the rank condition in this context is a condition on the support of the nonabelian Fourier transform of the central function. We are able to establish that both conjectures are true in case 2r ⩾ n
Association of FADS1/2 Locus Variants and Polyunsaturated Fatty Acids With Aortic Stenosis.
No full textIMPORTANCE: Aortic stenosis (AS) has no approved medical treatment. Identifying etiological pathways for AS could identify pharmacological targets. OBJECTIVE: To identify novel genetic loci and pathways associated with AS. DESIGN, SETTING, AND PARTICIPANTS: This genome-wide association study used a case-control design to evaluate 44 703 participants (3469 cases of AS) of self-reported European ancestry from the Genetic Epidemiology Research on Adult Health and Aging (GERA) cohort (from January 1, 1996, to December 31, 2015). Replication was performed in 7 other cohorts totaling 256 926 participants (5926 cases of AS), with additional analyses performed in 6942 participants from the Cohorts for Heart and Aging Research in Genomic Epidemiology (CHARGE) Consortium. Follow-up biomarker analyses with aortic valve calcium (AVC) were also performed. Data were analyzed from May 1, 2017, to December 5, 2019. EXPOSURES: Genetic variants (615 643 variants) and polyunsaturated fatty acids (ω-6 and ω-3) measured in blood samples. MAIN OUTCOMES AND MEASURES: Aortic stenosis and aortic valve replacement defined by electronic health records, surgical records, or echocardiography and the presence of AVC measured by computed tomography. RESULTS: The mean (SD) age of the 44 703 GERA participants was 69.7 (8.4) years, and 22 019 (49.3%) were men. The rs174547 variant at the FADS1/2 locus was associated with AS (odds ratio [OR] per C allele, 0.88; 95% CI, 0.83-0.93; P = 3.0 × 10-6), with genome-wide significance after meta-analysis with 7 replication cohorts totaling 312 118 individuals (9395 cases of AS) (OR, 0.91; 95% CI, 0.88-0.94; P = 2.5 × 10-8). A consistent association with AVC was also observed (OR, 0.91; 95% CI, 0.83-0.99; P = .03). A higher ratio of arachidonic acid to linoleic acid was associated with AVC (OR per SD of the natural logarithm, 1.19; 95% CI, 1.09-1.30; P = 6.6 × 10-5). In mendelian randomization, increased FADS1 liver expression and arachidonic acid were associated with AS (OR per unit of normalized expression, 1.31 [95% CI, 1.17-1.48; P = 7.4 × 10-6]; OR per 5-percentage point increase in arachidonic acid for AVC, 1.23 [95% CI, 1.01-1.49; P = .04]; OR per 5-percentage point increase in arachidonic acid for AS, 1.08 [95% CI, 1.04-1.13; P = 4.1 × 10-4]). CONCLUSIONS AND RELEVANCE: Variation at the FADS1/2 locus was associated with AS and AVC. Findings from biomarker measurements and mendelian randomization appear to link ω-6 fatty acid biosynthesis to AS, which may represent a therapeutic target
An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices
No full textsummary:Suppose that is an nonnegative matrix whose eigenvalues are . Fiedler and others have shown that , for all , with equality for any such if and only if is the simple cycle matrix. Let be the signed sum of the determinants of the principal submatrices of of order , . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: , for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of : If are (all) the nonzero eigenvalues of , then . We prove this conjecture for the case when the spectrum of is real
