Distance-preserving approximations of polygonal paths

Given a polygonal path P with vertices $p_1,p_2,\dots ,p_n\in {\mathbb{R}}^d$ and a real number $t⩾1$, a path $Q=(p_{i_1},p_{i_2},\dots ,p_{i_k})$ is a t-distance-preserving approximation of P if $1=i_1<i_2<\cdots <i_k=n$ and each straight-line edge $(p_{i_j},p_{i_{j+1}})$ of Q approximates the distance between $p_{i_j}$ and $p_{i_{j+1}}$ along the path P within a factor of t. We present exact and approximation algorithms that compute such a path Q that minimizes k (when given t) or t (when given k). We also present some experimental results.     

On the longest common increasing binary subsequence

Let $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ be two independent sequences of iid Bernoulli random variables with parameter 1/2. Let ${\mathit{LCI}}_n$ be the length of the longest increasing sequence which is a subsequence of both finite sequences $X_1,\dots ,X_n$ and $Y_1,\dots ,Y_n$. We prove that, as n goes to infinity, $n^{?1/2}({\mathit{LCI}}_n?n/2)$ converges in law to a Brownian functional that we identify. To cite this article: C. Houdré et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Le régulateur p-adique

For all $n\in {\mathbf{N}}^1$, we define a p-adic regulator $R_p:K_{2n+1}({\mathbf{Q}}_p)\to {\mathbf{Q}}_p$ given by an explicit formula and we show that $R_p$ is non-trivial for $n=1$. The main ideas come from a Note published in 2000 for the transcendental case. To cite this article: N. Hamida, C. R. Acad. Sci. Paris, Ser. I 342 (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 the proportion of rank 0 twists of elliptic curves

Let E be an elliptic curve defined over Q, let $E_d$ denote its dth quadratic twist, and $r_{E_d}:=\mathrm{rank}{E}_d(\mathbf{Q})$. We prove, that, for any positive integer k there are pairwise non-isogenous elliptic curves $E^1,\dots ,E^k$ such that $r_{{E^1}_p}=\cdots =r_{{E^k}_p}=0$ for a positive proportion of primes p. To cite this article: A. Dąbrowski, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the triplex substitution – combinatorial properties

If a substitution τ over a three-letter alphabet has a positively linear complexity, that is, $P_{\tau }(n)=C_{1}n+C_2$ ($n⩾1$) with $C_1,C_2⩾0$, there are only 4 possibilities: $P_{\tau }(n)=3$, $n+2$, $2n+1$ or 3n. The first three cases have been studied by many authors, but the case 3n remained unclear. This leads us to consider the triplex substitution $\sigma :a\mapsto ab$, $b\mapsto acb$, $c\mapsto acc$. Studying the factor structure of its fixed point, which is quite different from the other cases, we show that it is of complexity 3n. We remark that the triplex substitution is also a typical example of invertible substitution over a three-letter alphabet. To cite this article: B. Tan et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number $\mathrm{adrn}(G)$ is the least k such that every set of k dacards determines G. We determine $\mathrm{adrn}(D_{m,n,p})$, where the double-broom $D_{m,n,p}$ with $p\ge 2$ is the tree with $m+n+p$ vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine $\mathrm{adrn}(D_{m,n,p})$ for all $m,n,p$. For $2\le m\le n$, usually $\mathrm{adrn}(D_{m,n,p})=m+2$, except $\mathrm{adrn}(D_{m,m+1,p})=m+1$ and $\mathrm{adrn}(D_{m,m+2,p})=m+3$. There are exceptions when $(m,n)=(2,3)$ or $p=4$. For $m=1$ the usual value is 4, with exceptions when $p\in \{2,3\}$ or $n=2$.     

Sums,differences, products,and ratios of hypergeometric beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, differences, products and ratios of independent beta random variables. In this Note we extend Professor Pham-Gia's results when $X_1$ and $X_2$ are independent random variables distributed according to the confluent and Gauss hypergeometric distributions (which are generalizations of the beta distribution). For each of these distributions, we derive exact expressions for the densities of $S=X_1+X_2$, $D=X_1?X_2$, $P=X_1{X}_2$, and $R=X_2/X_1$. The expressions turn out to involve the hypergeometric functions of one and two variables. To cite this article: S. Nadarajah, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Version explicite du théorème de Beilinson pour la courbe modulaire X1(N)

We state an explicit version of Beilinson's theorem for the modular curve $X_1(N)$. We deduce from it, for any elliptic curve E of prime conductor N, a formula giving $L(E,2)$ in terms of the twisted values $L(E,\chi ,1)$, where χ is a character modulo N. We illustrate this result and its consequences in the case of the elliptic curve $E=X_1(11)$. To cite this article: F. Brunault, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Estimation of certain exponential sums arising in complexity theory

It is shown that the correlation on ${\{0,1\}}^n$ between parity and a polynomial $p(x_1,\dots ,x_n)\in \mathbb{Z}[X_1,\dots ,X_n](\mathrm{mod}q)$, q a fixed odd number and $p(X)$ of degree d arbitrary but fixed, is exponentially small in n as $n\to \infty$. An application to circuit complexity, from where the problem originates, is given. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

La forme seconde trace d'une algèbre simple centrale de degré 4 de caractéristique 2

Let A be a central simple algebra of degree 4 over a field k of characteristic 2 and let $q_A$ be the quadratic form on A given by the second coefficient of the reduced characteristic polynomial. We show that A uniquely determines a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ such that $q_A=[1,1]+q_2+q_4$ in the Witt group of k, where $[1,1]$ is the form $x^2+xy+y^2$. The form $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A{?}_{k}A$, and $q_4$ is hyperbolic if and only if A is cyclic. To cite this article: J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Solvable PST-groups,strong Sylow bases and mutually permutable products

A subgroup H of a group G is said to permute with the subgroup K of G if $HK=KH$. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and $H\cap K=1$, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let $\{p_1,\dots ,p_n,p_{n+1},\dots ,p_k\}$ be the set of prime divisors of the order of a finite group G with $\{p_1,\dots ,p_n\}$ the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups $\{P_1,\dots ,P_n,P_{n+1},\dots ,P_k\}$, $P_i\in {\mathrm{Syl}}_{p_i}(G)$, form a strong Sylow system with respect to N if $P_i{P}_j$ is a mutually permutable product for all $i\in \{1,2,\dots ,n\}$ and $j\in \{1,2,\dots ,k\}$. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that $G/N$ is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and $G/N^{″}$ is a solvable PST-group.     

Deformation of central charges,vertex operator algebras whose Griess algebras are Jordan algebras

If a vertex operator algebra $V={\oplus }_{n=0}^{\infty }{V}_n$ satisfies $\dim V_0=1$, $V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set ${\mathrm{Sym}}_d(\mathbb{C})$ of symmetric matrices of degree d becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge c and its Griess algebra is isomorphic to ${\mathrm{Sym}}_d(\mathbb{C})$ for any complex number c and a positive integer d.     

Sequences of dilations and translations in function spaces

Let $f\in L^1[0,1]$ be a mean zero function and let $f_n$, $n=1,2,\dots$, be the dyadic dilations and translations of f. We investigate conditions on f, under which the linear operator $T_f$ defined by $T_f{h}_n=f_n$, $n=1,2,\dots$, where $h_n$, $n=1,2,\dots$, are mean zero Haar functions, can be continuously extended to the closed linear span $[h_n]$ in a certain function space X. Among other results we prove that $T_f$ is bounded in every symmetric space with nontrivial Boyd indices whenever $f\in BM{O}_d$ and f has “good” Haar spectral properties. In the special case of so-called Haar chaoses the above results can be essentially refined and sharpened. In particular, we find necessary and sufficient conditions, under which the operator $T_f$, generated by a Haar chaos f of order 1, is continuously invertible in $L^p$ for all $1<p<\infty$.     

Normal,unipotent subgroup schemes of reductive groups

Let G be a reductive group over a field k of characteristic p. Let $k^{\mathrm{sep}}$ be a separable closure of k. If $p\ne 2$, there exists a linear representation of G that is faithful and semisimple; moreover, any unipotent, normal subgroup scheme of G is trivial. For $p=2$, these two properties hold if and only if $G_{k^{\mathrm{sep}}}$ has no direct factor that is isomorphic to ${\mathbf{SO}}_{2n+1}$ for some $n?1$. To cite this article: A. Vasiu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).