On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].     

On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

Two remarks on liftings of maps with values into S1

Given a map $u\in L_{\mathrm{loc}}^1(\Omega ,S^1)$ with some regularity: ${|u|}_X=R<\infty$, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u={\mathrm{e}}^{\mathrm{i}\phi }$) with the same regularity and with an optimal control ${|\phi |}_X?g(R)$. Two cases are treated here:(i) ${|?|}_X$ is a $W^{s,p}(0,1)$-seminorm, with $0<s<1<p$ and $sp>1$. We find a lifting φ such that ${|\phi |}_{W^{s,p}(I)}?C(R+R^{1/s})$ and we show that the exponent $1/s$ cannot be improved.(ii) ${|?|}_X$ is the $\mathit{BV}(\Omega )$-seminorm where $\Omega ?{\mathbf{R}}^d$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^1$, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $\phi \in \mathit{BV}(\Omega )$ satisfying ${|\phi |}_{\mathit{BV}}?2R$. To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Estimation du nombre de dérivées d'un processus Gaussien

We consider a real Gaussian process X with unknown smoothness $r_0\in \mathbb{N}$ where the mean-square derivative $X^{(r_0)}$ is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations $X(t_1),\dots ,X(t_n)$, we study reconstruction of $X(t)$, $t\in [0,1]$, with ${\stackrel{?}{X}}_r(t)$, a piecewise polynomial interpolation of degree $r?1$. We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as $r?r_0$. Next, from an interpolation-based empirical criterion, we derive an estimator $\stackrel{?}{r}$ of $r_0$ and prove its strong consistency by giving an exponential inequality for $P(\stackrel{?}{r}\ne r_0)$. Finally, we prove the strong convergence of ${\stackrel{?}{X}}_{\stackrel{?}{r}}(t)$ toward $X(t)$ with a similar rate as in the case ‘$r_0$ known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On sub-harmonic bifurcations

Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family $h_u$ of transformations near $(u_0,x_0)$ when $h_{u_0}(x_0)=x_0$ and $\mathrm{d}{h}_{u_0}(x_0)$ has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On a tree-partition problem

If $T=(V,E)$ is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees $T_1$, $T_2$, so that $V(T_1)\cap V(T_2)=\{v\}$, $V(T_1)\cup (T_2)=V(T)$, $E(T_1)\cap E(T_2)=\varnothing$, $E(T_1)\cup E(T_2)=E(T)$, $T[V(T_1)\backslash \{v\}]\subseteq E(T_1)$ and $T[V(T_2)\backslash \{v\}]\subseteq E(T_2)$. We call such a partition a $T_v-partition$ of T. We study the following em: Given a graph G belonging to –T. Is it true that for any $T_v$-partition $T_1$, $T_2$ of T there exists a partition $\{V_1,V_2\}$ of the vertices of G such that $G[V_1]\in -T_1$ and $G[V_2]\in -T_2$? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.     

L'inégalité de Bohr pour les séries de Dirichlet

We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if $f(s)={\sum }_{n=1}^{\infty }{a}_n{n}^{?s}$, with $6f{6}_{\infty }:={\sup }_{\mathfrak{R}s>0}|f(s)|<\infty$, then ${\sum }_{n=1}^{\infty }|a_n|n^{\mathrm{?2}}?6f{6}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }|a_n|n^{?1/2}?C6f{6}_{\infty }$, C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On mixed codes with covering radius 1 and minimum distance 2: (extended abstract)

Let R, S and T be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when r divides s, when $r=s-1$, when s is large, relative to r, when r is large, relative to s, as well as $K(3r,2r,2r;2)$. Finally, a table with bounds on $K(r,s,s;2)$ is given.