A characterization of systems of first order differential equations with (classical) complete solutions is given. Systems with (classical) complete solutions that consist of hyperplanes are also characterized.

We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.

Given a real analytic vector field tangent to a hypersurface $V$ with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra $\frac{\mathbf{B}}{{\mathrm{Ann}}_{\mathbf{B}}\left(h\right)}$ associated with the singularity of the vector field on $V$. We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...

We prove that if f:(ℝⁿ,0) → (ℝⁿ,0) is an analytic map germ such that $f^{-1}\left(0\right)=0$ and f satisfies a certain non-degeneracy condition with respect to a Newton polyhedron Γ₊ ⊆ ℝⁿ, then the index of f only depends on the principal parts of f with respect to the compact faces of Γ₊. In particular, we obtain a known result on the index of semi-weighted-homogeneous map germs. We also discuss non-degenerate vector fields in the sense of Khovanskiĭand special applications of our results to planar analytic vector fields....

Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.

In this paper we study equivalence classes of generic $1$-parameter germs of real analytic families ${𝒬}_{\epsilon }$ unfolding codimension $1$ germs of diffeomorphisms ${𝒬}_0:(ℝ,0)\to (ℝ,0)$ with a fixed point at the origin and multiplier $-1,$ under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, ${𝒫}_{\epsilon }={𝒬}_{\epsilon }^{\circ 2},$ are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle...

In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection $𝒦$ and if $\xi$ is a smooth Lipschitz-Fredholm vector field on $M$ with respect to $𝒦$ which satisfies condition (WCV), then, for any smooth functional $l$ on $M$ which is associated to $\xi$, the set of the critical values of $l$ is of first category in $ℝ$. Therefore,...

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\hslash )$, and gives improvements and new proofs for many of the results in the field. In the completely resonant...

In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space $H_2^p(M,T^{*}M\otimes T^{*}M)$ with $p\>\frac{n}{2}$ and there exist a point $P\in M$ and $\delta \>0$ such that $g$ is smooth in the ball $B_p\left(\delta \right)$. We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to $g$ and of volume $1$. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...