Unfoldings of holomorphic foliations

The objective of this paper is to give a criterium for an unfolding of a holomorphic foliation with singularities to be holomorphically trivial

Differential clinical characteristics and prognosis of intraventricular conduction defects in patients with chronic heart failure

Intraventricular conduction defects (IVCDs) can impair prognosis of heart failure (HF), but their specific impact is not well established. This study aimed to analyse the clinical profile and outcomes of HF patients with LBBB, right bundle branch block (RBBB), left anterior fascicular block (LAFB), and no IVCDs. Clinical variables and outcomes after a median follow-up of 21 months were analysed in 1762 patients with chronic HF and LBBB (n = 532), RBBB (n = 134), LAFB (n = 154), and no IVCDs (n = 942). LBBB was associated with more marked LV dilation, depressed LVEF, and mitral valve regurgitation. Patients with RBBB presented overt signs of congestive HF and depressed right ventricular motion. The LAFB group presented intermediate clinical characteristics, and patients with no IVCDs were more often women with less enlarged left ventricles and less depressed LVEF. Death occurred in 332 patients (interannual mortality = 10.8%): cardiovascular in 257, extravascular in 61, and of unknown origin in 14 patients. Cardiac death occurred in 230 (pump failure in 171 and sudden death in 59). An adjusted Cox model showed higher risk of cardiac death and pump failure death in the LBBB and RBBB than in the LAFB and the no IVCD groups. LBBB and RBBB are associated with different clinical profiles and both are independent predictors of increased risk of cardiac death in patients with HF. A more favourable prognosis was observed in patients with LAFB and in those free of IVCDs. Further research in HF patients with RBBB is warranted

A law of conservation of number for local Euler characteristics

Let T and V be complex analytic spaces, with T reduced and locally irreducible and π:T×V→T the projection map to the first factor. Let K ∗ be a complex 0→K n → X n K n−1 → X n−1 …→ X 2 K 1 → X 1 K 0 →0 of O T×V coherent sheaves, where all X j are O T×V -linear, all sheaves K j are O T -flat and such that the support of the homology sheaves H j (K ∗ ) is π -finite. For every t∈T , V t denotes {t}×V≅V ; K ∗ t denotes the complex obtained by tensoring K ∗ with O V t ; K ∗ t;p denotes the complex formed by the germs at (t,p) of K ∗ t and H j (K ∗ t;p ) denotes the j th homology group of the complex K ∗ t;p . The Euler characteristic of the complex of sheaves K ∗ t at a point (t,p)∈V t is defined as χ(K ∗ t;p )=∑ j=0 n (−1) j dim C H j (K ∗ t;p ). The authors show that, for every (t 0 ,p 0 )∈T×V there are neighbourhoods T ′ and V ′ of t 0 and p 0 , respectively, such that for every t∈T ′ , χ(K ∗ t 0 ;p 0 )=∑ q∈V ′ χ(K ∗ t;p )

Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces.

In this paper the authors review some work relating the topology to algebraic invariants. Three cases are considered: hypersurfaces, vector fields and vector fields tangent to hypersurfaces. An example is the case of "real Milnor fibres". Let f R :B R →R be a real analytic function which extends to a function f on the closed unit ball B in C n. Assume that 0 is the only critical point of f and denote the real hypersurface f −1 R (δ) by V R + for small positive real δ . Let A denote A/(f 0 ,⋯,f n ) , the quotient of the ring of germs of real analytic functions by the partial derivatives of f . Choose any R-linear map L:A→R which sends the Hessian of f to a positive number; then one has a non-degenerate bilinear form ⟨ , ⟩:A×A→ . A→ L R , for which the signature σ gives χ(V R + )=1−σ . The authors use a result of D. Eisenbud and H. I. Levine [Ann. Math. (2) 106 (1977), no. 1, 19–44, to calculate the signature of the bilinear form; this is notoriously difficult to compute in practice

On the index of vector fields tangent to hypersurfaces with non-isolated singularities

Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu