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 radial distribution of Julia sets of entire solutions of f(n)+A(z)f=0

This paper is devoted to studying the dynamical properties of solutions of $f^{(n)}+A(z)f=0$, where $n(?2)$ is an integer, and $A(z)$ is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of $E(z)$ provided that $E=f_1{f}_2?f_n$ and $\{f_1,f_2,\dots ,f_n\}$ is a solution base of such equations.     

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 $t\to \infty$.     

The Banach algebra generated by a C0-semigroup

Let $\mathbf{T}={\{T(t)\}}_{t?0}$ be a bounded $C_0$-semigroup on a Banach space with generator A. We define $A_{\mathbf{T}}$ as the closure with respect to the operator-norm topology of the set $\{\stackrel{?}{f}(\mathbf{T}):f\in L^1({\mathbb{R}}_{+})\}$, where $\stackrel{?}{f}(\mathbf{T})={\int }_0^{\infty }f(t)T(t)\mathrm{d}t$ is the Laplace transform of $f\in L^1({\mathbb{R}}_{+})$ with respect to the semigroup T. Then $A_{\mathbf{T}}$ is a commutative Banach algebra. It is shown that if the unitary spectrum $\sigma (A)\cap \mathrm{i}\mathbb{R}$ of A is at most countable, then the Gelfand transform of $S\in A_{\mathbf{T}}$ vanishes on $\sigma (A)\cap \mathrm{i}\mathbb{R}$ if and only if, ${\lim }_{t\to \infty }6T(t)S6=0$. Some applications to the semisimplicity problem are given. To cite this article: H. Mustafayev, C. R. Acad. Sci. Paris, Ser. I 342 (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).     

Regular expression constrained sequence alignment

We introduce regular expression constrained sequence alignment as the problem of finding the maximum alignment score between given strings $S_1$ and $S_2$ over all alignments such that in these alignments there exists a segment where some substring $s_1$ of $S_1$ is aligned to some substring $s_2$ of $S_2$, and both $s_1$ and $s_2$ match a given regular expression R, i.e. $s_1,s_2\in L(R)$ where $L(R)$ is the regular language described by R. For complexity results we assume, without loss of generality, that $n=|S_1|⩾|m|=|S_2|$. A motivation for the problem is that protein sequences can be aligned in a way that known motifs guide the alignments. We present an $\mathrm{O}(nmr)$ time algorithm for the regular expression constrained sequence alignment problem where $r=\mathrm{O}(t^4)$, and t is the number of states of a nondeterministic finite automaton N that accepts $L(R)$. We use in our algorithm a nondeterministic weighted finite automaton M that we construct from N. M has $\mathrm{O}(t^2)$ states where the transition-weights are obtained from the given costs of edit operations, and state-weights correspond to optimum alignment scores we compute using the underlying dynamic programming solution for sequence alignment. If we are given a deterministic finite automaton D accepting $L(R)$ with $t_d$ states then our construction creates a deterministic finite automaton $M_d$ with $t_d^2$ states. In this case, our algorithm takes $\mathrm{O}(t_d^{2}nm)$ time. Using $M_d$ results in faster computation than using M when $t_d<t^2$. If we only want to compute the optimum score, the space required by our algorithm is $\mathrm{O}(t^{2}n)$ ($\mathrm{O}(t_d^{2}m)$ if we use a given $M_d$). If we also want to compute an optimal alignment then our algorithm uses $\mathrm{O}(t^{2}m+t^2|s_1||s_2|)$ space ($\mathrm{O}(t_d^{2}m+t_d^2|s_1||s_2|)$ space if we use a given $M_d$) where $s_1$ and $s_2$ are substrings of $S_1$ and $S_2$, respectively, $s_1,s_2\in L(R)$, and $s_1$ and $s_2$ are aligned together in the optimal alignment that we construct. We also show that our method generalizes for the case of the problem with affine gap penalties, and for finding optimal regular expression constrained local sequence alignments.     

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.     

Étude en temps petit des solutions d'EDS conduites par des mouvements browniens fractionnaires

We study, in small times, the properties of the operator ${\mathbf{P}}_t(f)(x)=\mathbb{E}(f(X_t^x))$, where ${(X_t^x)}_{t?0}$ is the solution of a stochastic differential equation driven by fractional Brownian motions with the same Hurst parameter $H>\frac{1}{4}$. To cite this article: F. Baudoin, L. Coutin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).