In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ $ {\mathbb {R}}^{\alpha },$ study Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ one its hyperbolic sectors. We prove (theorem 1.1) that derivative $\partial _...

Abstract In this paper we characterise the minimisers of a one-parameter family nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n ...

Summary In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: deal with quadratic polynomials and extensions [5], [4]. First introduce polynomials, their discriminants prove the midnight formula. Then show that - case discriminant of p being non square adjoining a root ’s results splitting . Finally these are only degree 2, e.g. an extension E F is if there Elem...