Abelian fibrations on S[n]

Let $\text{S}\text{→}\text{φ}\text{P}{}^1$ be an elliptic fibration on a K3 surface S. Then the composition $\text{S}{}^{\mathrm{[n]}}\text{→}\text{π}\text{S}{}^{\mathrm{(n)}}\text{→}\text{sym}{}^{\mathrm{n}}\text{φ}\text{P}{}^{\mathrm{n}}$ gives an Abelian fibration on S^[n]. Let E be the exceptional divisor of π, then symⁿφ°π(E) is of dimension n?1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

About some problems of spectral synthesis

For a given pair $\text{(A,b)∈}\text{R}{}^{\mathrm{n\times n}}\text{×}\text{R}{}^{\mathrm{n\times 1}}$ such that A is cyclic and b is a cyclic generator (with respect to A) of $\text{R}{}^{\mathrm{n\times 1}}$, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence $\text{{f}{}_{\mathrm{j\}}}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{,f}{}_{\mathrm{j}}\text{∈}\text{R}{}^{\mathrm{1\times n}}$,so that a the zeros of the rational function det P(z), where $\text{P(z) = zI ? A ? ∑}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{z}{}^{\mathrm{-(m+j)}}\text{b?}$_f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.     

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups $\text{H>}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, U)}$ (where U means the multiplicative group) and $\text{H}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, Pic)}$ and a third sequence of groups Hⁿ(J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and Hⁿ(J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs (P, α) with P a rank one, projective Sⁿ-module and α an isomorphism from the coboundary of P (inPicS^{n + 1}) toS^{n + 1}. (Here Sⁿ denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group $\text{B(}\text{S}\text{R}\text{)}$ to H²(J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H²(J) with Ker[H²(R, U)→H²(S, U)], where H²(R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).     

On the spectral synthesis problem for hypersurfaces of RN

Let F¹(Rⁿ) denote the Fourier algebra on $\text{R}$ⁿ, and $\text{D}$($\text{R}$ⁿ) the space of test functions on $\text{R}$ⁿ. A closed subset E of $\text{R}$ⁿ is said to be of spectral synthesis if the only closed ideal J in F¹(Rⁿ) which has E as its hull

$\text{h(J)={x ? R}{}^{\mathrm{n}}\text{:f(x)=0}\text{for all}\text{f ? J}}$

is the ideal

$\text{k(E)={f?F}{}^1\text{(R}{}^{\mathrm{n}}\text{):f(E)=0}}$

. We consider sufficiently regular compact subsets of smooth submanifolds of $\text{R}$ⁿ with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra $\text{(k(E)∩D(R}{}^{\mathrm{n}}\text{))}{}^{\mathrm{?}}\text{j(E)}$, where j(E) denotes the smallest closed ideal in F¹(Rⁿ) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rⁿ) is dense in k(E). Together this solves the synthesis problem for such sets.     

Sobolev inequalities with remainder terms

The usual Sobolev inequality in $\text{R}$ⁿ, n ? 3, asserts that , with S_n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? $\text{R}$ⁿ. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with $\text{p =}\text{2}{}^1\text{2}$ and with $\text{q =}\text{n}\text{(n ? 1)}$. (ii) If ? ≠ 0 on ?Ω, then with $\text{q =}\text{2(n ? 1)}\text{(n ? 2)}$. Some further results and open problems in this area are also presented.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Delayed random walks

A delayed random walk $\text{{S}{}^1{}_{\mathrm{n}}\text{, n ≥ 0}}$ is defined here as a partial sum process of independent random variables in which the first N summands (N optional) are distributed F¹,…,F_N, respectively, while all remaining summands are distributed F₀, where {F_k, _k ≥ 0} is a sequence of proper distribution functions on the real line. Delayed random walks arise naturally in the study of certain generalized single server queues. This paper examines optional times of the process such as $\text{π = inf {n: n ≥ 1 and S}{}^1{}_{\mathrm{n}}\text{≥ 0}}$. Conditions insuring the finiteness of E {π} and E {π²} are obtained, generating functions calculated, and illustrative examples given. The bivariate functions $\text{E{r}{}^{\mathrm{\pi }}\text{explsqbitS}{}^1{}_{\mathrm{\pi }}\text{rsqb}}$ and $\text{E {}\text{∑}\text{n=0}\text{π?1}\text{explsqbitS}{}^1{}_{\mathrm{n}}\text{rsqb}}$ are studied for the case where N ≡ 1.     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

On a property of the set of radiation patterns

Consider the exterior boundary value problem (▽² + K²) u = 0, in Ω, k >0. $\text{u¦}{}_{\mathrm{\Gamma }}\text{= h}$, where Γ is a smooth closed connected surface in $\text{R}$³, $\text{u ~}\text{exp}\text{(ik ¦x¦)¦x¦}{}^{\mathrm{?1}}\text{∝(k, n)}\text{as}\text{¦X¦→ ∞, n = x¦x¦}{}^{\mathrm{?1}}$, ∝ is called the radiation pattern. We prove that when h runs through any dense set in L²(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L²(S²) for any k >0, where S² is the unit sphere in $\text{R}$³.     

Positive operators on the n-dimensional ice cream cone

Let $\text{K}{}_{\mathrm{n}}\text{= {x ? R}{}^{\mathrm{n}}\text{: (x}{}_1{}^2\text{+ · +x}{}^2{}_{\mathrm{n?1}}\text{)}\text{1}\text{2}\text{? x}{}_{\mathrm{n}}\text{}}$ be the n-dimensional ice cream cone, and let Γ(K_n) be the cone of all matrices in $\text{R}$ⁿⁿ mapping K_n into itself. We determine the structure of Γ(K_n), and in particular characterize the extreme matrices in Γ(K_n).     

Some properties of Q-matrices

This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ $\text{R}$^n×1, the linear complementarity problem

$\text{Iw ? Mz = q}$

,

$\text{w ? 0, z ? 0, and w}{}^{\mathrm{T}}\text{z = 0}$

, has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q.     

W1,p-quasiconvexity and variational problems for multiple integrals

Variational problems for the multiple integral $\text{I}{}_{\mathrm{\Omega }}\text{(u) = ∝}{}_{\mathrm{\Omega }}\text{g(▽u(x))dx}$, where $\text{Ω?R}{}^{\mathrm{m}}$ and $\text{u:Ω→R}{}^{\mathrm{n}}$ are studied. A new condition on g, called W^1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of $\text{I}{}_{\mathrm{\Omega }}$ in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$ and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$, p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

A Note on pinching sphere theorem

Let M²ⁿ be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S²ⁿ be the unit sphere of 2n+1 dimension Euclidean space $\text{R}{}^{\mathrm{2n+1}}$. We prove in this note that if the sectional curvature K_M varies in (0,1] and the volume V(M) is not larger than $\text{(}\text{3}\text{2}\text{+η)V(S}{}^{\mathrm{2n}}\text{)}$ for some positive number η depending only on n, then M²ⁿ is homeomorphic to S²ⁿ. To cite this article: Y. Wen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Monotone Splitting of matrices

Given the iterative scheme xⁱ⁺¹ = BTxⁱ + r where B, T are fixed n×:n real matrices, r a fixed real n-vector and xⁱ a real n-vector we investigate the convergence and monotonicity of schemes of the type

$\text{v}{}^{\mathrm{i+1}}\text{w}{}^{\mathrm{i+1}}\text{=}\text{B}\text{O}\text{O}\text{B}\text{S}{}_{11}\text{?S}{}_{12}\text{?S}{}_{21}\text{S}{}_{22}\text{v}{}^{\mathrm{i}}\text{w}{}^{\mathrm{i}}\text{+}\text{r}\text{r}$

, where S_ij are n×:n real matrices related to T. The n-vector iterates vⁱ,wⁱ bracket in a certain sense solutions x of x = BTx + r. We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself xⁱ⁺¹ = BTxⁱ + r. This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Random Euclidean embeddings in spaces of bounded volume ratio

Let $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$ be the space $\text{R}{}^{\mathrm{N}}$ equipped with a norm 6·6 whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space $\text{(}\text{R}{}^{\mathrm{n}}\text{,|·|)}$ onto its image in $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$: there exist α,β>0 such that for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, $\text{α}\text{N}\text{|x|?6Γx6?β}\text{N}\text{|x|}$. This solves a conjecture of Schechtman on random embeddings of ?₂ⁿ into ?₁^N. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).