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 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.     

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).     

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n?k?1)\text{st}$ homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions $[B_1,\dots ,B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete k-uniform hypergraph H and where $1\in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

Extension operators and twisted sums of c0 and C(K) spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space X containing a non-complemented copy Y of $c_0$ such that the quotient space $X/Y$ is isomorphic to $C(K)$?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube $2^{{\omega }_1}$ or K is a separable scattered compact space of height 3 and weight ${\omega }_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial.We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K?L$ which do not admit an extension operator $C(K)\to C(L)$.     

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.     

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.     

Interval conjectures for level Hilbert functions

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, $(1,h_1,\dots ,h_i,\dots ,h_e)$ and $(1,h_1,\dots ,h_i+\alpha ,\dots ,h_e)$ are both level h-vectors, then $(1,h_1,\dots ,h_i+\beta ,\dots ,h_e)$ is also level for each integer $\beta =0,1,\dots ,\alpha$. In the Gorenstein case, i.e. when $h_e=1$, we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say $h_i$ and $h_{e?i}$, of a Gorenstein h-vector.These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets.We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension.     

Un théorème de Liouville pour l'opérateur de Schrödinger avec dérive

Let $(M,g)$ be a complete Riemannian manifold without boundary of dimension n and V be a $C^2$ vector field on M such that $r(x)|V(x)|$ is bounded. Suppose that ${\mathrm{Ric}}_g(x)??\min \{\lambda (r(x))?{\mu }_{\mathrm{?}V}(x),\beta (r(x))\}$ outside a compact set of M, where ${\mu }_{\mathrm{?}V}$ denotes the upper eigenvalue of ?V and $\lambda ,\beta$ are non-negative decreasing functions such that ${\lim }_{t\to +\infty }{t}^2\lambda (t)=0$. There exists positive numbers $b_n$ and $c_n$ which depend only on n and $6V{6}_{\infty }$ such that if h is a $C^2$ function defined on M with $\mathrm{\Delta }h??c_n{a}^2$ and ${\mathrm{lim?sup}}_{R\to \infty }{R}^{\mathrm{?2}}{\min }_{x\in B_p(3R)?B_p(R)}h(x)??b_n{a}^2$, where $0?a<{\mathrm{lim?inf}}_{j\to \infty }h(z_j)$, where $(z_j)$ is a sequence of M such that $r(z_j)\to \infty$, then the equation $\mathrm{\Delta }u(x)+V(x)?\mathrm{?}u(x)+h(x)u(x)=0$ has no positive $C^2$ solution on M. To cite this article: S. Asserda, C. R. Acad. Sci. Paris, Ser. I 342 (2006).