An improved local well-posedness result for the one-dimensional Zakharov system

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the parameters k,l and 1<p\le 2, where \|u_0\|_{\hat{H^{k,p}}} := \| ^k \hat{u_0}\|_{L^{p'}}, generalizing the results for p=2 by Ginibre, Tsutsumi, and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<-1/2, i.e. data u_0 \not\in L^2 and (n_0,n_1)\not\in H^{-1/2}\times H^{-3/2}, which was excluded in the case p=2.Comment: 17 pages. Final version to appear in Journal of Mathematical Analysis and Application

An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

We give an algorithm to compute the following cohomology groups on U = \C^n \setminus V(f) for any non-zero polynomial f \in \Q[x_1, ..., x_n]; 1. H^k(U, \C_U), \C_U is the constant sheaf on $U$ with stalk \C. 2. H^k(U, \Vsc), \Vsc is a locally constant sheaf of rank 1 on $U$. We also give partial results on computation of cohomology groups on $U$ for a locally constant sheaf of general rank and on computation of H^k(\C^n \setminus Z, \C) where $Z$ is a general algebraic set. Our algorithm is based on computations of Gr\"obner bases in the ring of differential operators with polynomial coefficients.Comment: 38 page

The Erd\H{o}s--Moser equation 1k+2k+...+(m−1)k=mk1^k+2^k+...+(m-1)^k=m^k revisited using continued fractions

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.Comment: 17 page

Extremal Problems in Bergman Spaces and an Extension of Ryabykh's Theorem

We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$. These results extend and provide a partial converse to a theorem of Ryabykh.Comment: 16 pages. To appear in the Illinois Journal of Mathematic

Infrared and microwaves at 5.8 GHz in a catalytic reactor

An improved micro-reactor cell for IR spectroscopic studies of heterogeneous catalysis was built around a 5.8 GHz microwave cavity. The reactor can operate at 20 bars and with conventional heating up to 720 K, with reactant gas flows velocities (GHSV) from 25 000 to 50 000 h−1. The temperature of the sample under microwave irradiation was measured by time resolved IR emission spectroscopy. The first experiment performed was the IR monitoring of the desorption of carbonates induced by irradiating an alumina sample by microwaves at 5.8 GHz