| 1. |
- Bränden, Petter, 1976-, et al.
(författare)
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Lorentzian polynomials
- 2020
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 192:3, s. 821-891
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Tidskriftsartikel (refereegranskat)abstract
- We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge-Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove that the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter q satisfies 0 < q <= 1. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave sequence.
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| 2. |
- Daelman, Bo, et al.
(författare)
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Frailty and cognitive function in middle-aged and older adults with congenital heart disease
- 2024
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Ingår i: Journal of the American College of Cardiology. - : Elsevier. - 0735-1097 .- 1558-3597. ; 83:12, s. 1149-1159
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Tidskriftsartikel (refereegranskat)abstract
- Background: Life expectancy of patients with congenital heart disease (CHD) has increased rapidly, resulting in a growing and aging population. Recent studies have shown that older people with CHD have higher morbidity, health care use, and mortality. To maintain longevity and quality of life, understanding their evolving medical and psychosocial challenges is essential.Objectives: The authors describe the frailty and cognitive profile of middle-aged and older adults with CHD to identify predictor variables and to explore the relationship with hospital admissions and outpatient visits.Methods: Using a cross-sectional, multicentric design, we included 814 patients aged ≥40 years from 11 countries. Frailty phenotype was determined using the Fried method. Cognitive function was assessed by the Montreal Cognitive Assessment.Results: In this sample, 52.3% of patients were assessed as robust, 41.9% as prefrail, and 5.8% as frail; 38.8% had cognitive dysfunction. Multinomial regression showed that frailty was associated with older age, female sex, higher physiologic class, and comorbidities. Counterintuitively, patients with mild heart defects were more likely than those with complex lesions to be prefrail. Patients from middle-income countries displayed more prefrailty than those from higher-income countries. Logistic regression demonstrated that cognitive dysfunction was related to older age, comorbidities, and lower country-level income.Conclusions: Approximately one-half of included patients were (pre-)frail, and more than one-third experienced cognitive impairment. Frailty and cognitive dysfunction were identified in patients with mild CHD, indicating that these concerns extend beyond severe CHD. Assessing frailty and cognition routinely could offer valuable insights into this aging population.
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| 3. |
- Huh, June, et al.
(författare)
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Correlation bounds for fields and matroids
- 2022
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Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society - EMS - Publishing House GmbH. - 1435-9855 .- 1435-9863. ; 24:4, s. 1335-1351
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Tidskriftsartikel (refereegranskat)abstract
- Let G be a finite connected graph, and let T be a spanning tree of G chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events e1 E T and e2 E T are negatively correlated for any distinct edges e1 and e2. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events e E B, where B is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of k-element independent sets of a matroid forms an ultra-log-concave sequence in k.
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