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[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

We determine all three-dimensional homogeneous and $1$ -curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.

Background Outcomes are poorer in metastatic colorectal cancer (mCRC) patients with BRAF V600E mutations than those without it, but the effect of these mutations on treatment response is unclear. This real-world study assessed the effects of antiangiogenic-based treatment and systemic inflammatory factors on outcomes in patients with BRAF V600-mutated mCRC.Methods/patients This real-world, multicenter, retrospective, observational study included patients with BRAF V600-mutated mCRC treated in eight hospitals in Spain. The primary endpoints were overall survival (OS) and progression-free survival (PFS); overall response rate (ORR) and disease control rate (DCR) were also assessed. The effect of first- and second-line treatment type on OS, PFS, ORR, and DCR were evaluated, plus the impact of systemic inflammatory markers on these outcomes. A systemic inflammation score (SIS) of 1–3 was assigned based on one point each for platelet-lymphocyte ratio (PLR) ≥200, neutrophil-lymphocyte ratio (NLR) ≥3, and serum albumin Results Of 72 patients, data from 64 were analyzed. After a median of 69.1 months, median OS was 11.9 months and median first-line PFS was 4.4 months. First-line treatment was triplet chemotherapy-antiangiogenic (12.5%), doublet chemotherapy-antiangiogenic (47.2%), doublet chemotherapy-anti-EGFR (11.1%), or doublet chemotherapy (18.1%). Although first-line treatment showed no significant effect on OS, antiangiogenic-based regimens were associated with prolonged median PFS versus non-antiangiogenic regimens. Negative predictors of survival with antiangiogenic-based treatment were NLR, serum albumin, and SIS 1–3, but not PLR. Patients with SIS 1–3 showed significantly prolonged PFS with antiangiogenic-based treatment versus non-antiangiogenic-based treatment, while those with SIS=0 showed no PFS benefit.Conclusions Antiangiogenic-based regimens, SIS, NLR, and albumin were predictors of survival in patients with mCRC, while SIS, NLR and serum albumin may predict response to antiangiogenic-based chemotherapy.Trial registration GIT-BRAF-2017-01.

We show that four-dimensional Riemannian manifolds which satisfy the Rakic duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant). Thus, since it was proved in Rakic (1999) [9] that Osserman manifolds satisfy the Rakic duality principle, both conditions are equivalent.

Natural generalizations of locally symmetric spaces are extended from the Riemannian to the Lorentzian setting and further investigated within the class of Lorentzian Walker manifolds. Explicit examples show that such conditions are meaningless in the higher signature case.