This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.

Nous prouvons un encadrement optimal pour la quantité $H\left( x,\,s \right)\,=\,\sum{_{n\ge 1}\,\frac{1}{{{\left( x+n \right)}^{s}}}}$ pour $x\,\ge \,0$ et $s\,>\,1$, qui améliore l'encadrement standard par des intégrales. Cet encadrement entraîne des inégalités sur la fonction $\zeta $ de Riemann, et amène à conjecturer la monotonie de la fonction $s\,\mapsto \,{{[(s\,-\,1)\text{ }\!\!\zeta\!\!\text{ (}s\text{)}]}^{\frac{1}{S-1}}}$. On donne des applications à l'étude de la convexité de fonctions liées à la fonction $\Gamma $ d'Euler et à la majoration optimale des fonctions élémentaires intervenant dans les opérateurs de Baskakov. Puis, nous étendons aux fonctions complètement monotones sur ]$0,\,+\infty $[ les résultats établis pour la fonction $x\,\mapsto \,{{x}^{-s}}$, et nous en déduisons des preuves élémentaires du comportement, quand $z$ tend vers 1, des séries génératrices de certaines fonctions arithmétiques. Enfin, nous prouvons qu'une partie du résultat se généralise à une classe de fonctions convexes positives décroissantes.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in ${{\mathbf{R}}^{n}}$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by “analytic” and “meromorphic” solutions of the equation $Lu\,=\,0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

We study the resonances of the operator $P(h)\,=\,-{{\Delta }_{x}}\,+\,V(x)\,+\,\varphi (hx)$. Here $V$ is a periodic potential, $\varphi $ a decreasing perturbation and $h$ a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of ${{P}_{0\,}}=\,-{{\Delta }_{x}}\,+\,V(x)$, and we give its asymptotic expansions in powers of ${{h}^{\frac{1}{2}}}$.

We investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then expanded in a Puiseux series with respect to the small parameter and its coefficients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous center ${{\mathcal{S}}_{3}}$. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension

1

$$\left\{ \begin{align} & i{{u}_{t}}\,+\,\frac{1}{2}{{u}_{xx}}\,+\,{{{\bar{u}}}^{3}}\,=\,0,\,\,\,\,\,t\,\in \,\mathbf{R},\,x\,\in \,\mathbf{R}, \\ & u(0,\,x)\,=\,{{u}_{0}}(x),\,\,\,\,\,\,\,\,\,\,\,x\,\in \,\mathbf{R}. \\ \end{align} \right.$$

Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data ${{u}_{0}}\,\in \,{{\mathbf{H}}^{1,\,0}}\,\cap \,{{\mathbf{H}}^{0,\,1}}$ are small and such that ${{\sup }_{|\xi |\le 1}}\left| \arg \mathcal{F}{{u}_{0}}(\text{ }\xi \text{ )}\text{0}\frac{\pi n}{2} \right|<\frac{\pi }{8}$ for some n ∈ Z, and ${{\inf }_{\left| \xi \right|\le 1}}\,\left| \mathcal{F}{{u}_{0}}(\xi ) \right|\,>\,0$, then the solution has an additional logarithmic timedecay in the short range region $\left| x \right|\,\le \,\sqrt{t}$. In the far region $\left| x \right|\,>\,\sqrt{t}$ the asymptotics have a quasilinear character.

We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in $\left[ \text{P} \right]$.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.