Isomorphisme de Dolbeault dans les variétés CR

Let M be a q-concave CR generic C^∞-smooth submanifold of real codimension k in an n-dimensional complex manifold X, then the truncated tangential Cauchy-Riemann complexes τ^≤q−1[ɛ^p,*](M), τ^{≥n−k−q+}2[ɛ^{p, *}](M) of C^∞-smooth forms and τ^≤q−1 [D′^p.*](M), τ^{≥n−k−q+}2[D′^p,*(M)] of currents on M are quasi-isomorphic for 0 ≤ p ≤ n.     

涉及复代数超曲面的全纯映射正规族

本文研究了涉及固定超曲面的全纯映照的正规性问题。利用Aladro 和Krantz对全纯映射族正规性的刻画和Shirosahi建立的一系列涉及一些特殊复代数超曲面的Picard 型定理,得到了全纯映射族的一些正规定则。     

关于可三角剖分的紧致流形的模2示嵌类

<正> 引言 关于复合形或更一般的空間在欧氏空間中的实現問題,Whitney和Thom分別有下面的結果: 定理.(Whitney)n維紧致微分流形M~n可微分实現于R~N中的必要条件为 W~k(M~n)=0,k≥N-n.(1) 定理.(Thom)一个有可数基而局部可縮的紧致Hausdorff空間X可以拓扑实現     

Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae

Let X be a semialgebraic set in Rⁿ defined by a Boolean combination of atomic formulae of the kind h * 0 where * \in { >, \ge, = }, deg(h) < d, and the number of distinct polynomials h is k. We prove that the sum of Betti numbers of X is less than O(k²d)ⁿ.     

The spectral remainder estimates for non-isotropic polyharmonic operators on bounded and quasibounded domains

Summary Approximation numbers of the embedding operator of the nonisotropic Sobolev space W^(m), 2(Q) into L²(Q), Q being an n-dimensional open box, are applied to the study of spectral asymptotics. We deal with the nonisotropic polyharmonic operators on bounded and quasibounded open sets in Rⁿ. The quasibounded open sets are required to satisfy theG _k,a ^(m),2 -condition. Not only the leading terms of the asymptotic formulae but also remainder terms are obtained. An application is made to a quasibounded domain whose width has a negative power-type decay at infinity.     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

Estimates on conjectures of Minkowski and woods II

Let ? ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ? ⁿ of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.     

Variations on Tilings in the Manhattan Metric

We investigate tilings of the integer lattice in the Euclidean n-dimensional space. The tiles considered here are the union of spheres defined by the Manhattan metric. We give a necessary condition for the existence of such a tiling for Z ⁿ when n 2. We prove that this condition is sufficient when n=2. Finally, we give some tilings of Z ⁿ when n 3.     

关于Extremal Ray的除子型收缩

设X是定义在复数域上的n维光滑射影簇且它的典范除子Kx不是数字有效的,A是X上的一个ample除子。本文详细研究了由Kx (n-k)A确定的关于X的除子型收缩映射（1≤k≤n-1),对其例外休的结构作了比较完整的分类,特别地,当1≤k≤3时,即得到Fujita,Sommese等人的相关结果。     

由随机过程序列产生的滑动平均过程部分和的弱收敛

本文讨论了由ρ-混合随机过程序列产生的形如Xk(t)=∑j=0∞ajεk-j(t),0≤t≤1,其中{aj;j≥0)为一实数序列,满足∑j=0∞|aj|<∞的滑动平均过程部分和的弱收敛性;同时也讨论了由此滑动平均过程产生的形如Yn(s,t)=1/n~(1/2)∑k=1[n,s]Xk(t),0≤s,t ≤ 1的随机过程的弱收敛性,以及随机足标和SNn(t)=∑k=1NnXk(t)的弱收敛性.     

The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n ⁴ log n) + log w , the condition number k(M) satisfies k(M) ≤ w n ^5/2 with probability at least 1-2w ^-1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length. March 7, 2001. Final version received: June 7, 2001.     

同伦群和上同调群的上乘积

<正> 在[1]中,作者研究了一个(N—1)维连通空间的同伦群和同调群的密切关系,那里所考虑的同伦群的维数是在(2N—2)以内,本文可以看成[1]的继续,我偿将考虑维数在(2N—1)以上而又在(3N—3)以下的同伦群,Betti 数;和上乘积的密切关系.我们     

Random systems of equations in free abelian groups

We study the solvability of random systems of equations on the free abelian group ? ^m of rank m. Denote by SAT(? ^m , k, n) and \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) the sets of all systems of n equations of k unknowns in ? ^m satisfiable in ? ^m and ? ^m respectively. We prove that the asymptotic density \(\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)\) of the set \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) equals 1 for n ≤ k and 0 for n > k. As regards, SAT(? ^m , k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π _j=k?n+1 ^k ζ(j))^?1 and \(\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n\) , where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(? ^m , k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(? ^m , n, n). We prove that ρ(SAT(? ^m , k, n)) = 0 for n > k.     

Some properties of the distance function and a conjecture of De Giorgi

In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ ^n+m ,moving by the gradient of the functional

Inscribed simplexes of a convex body

This paper advances the theorem that for any smooth point M of the boundary of a convex body K Rⁿ there exists a nondegenerate simplex inscribed in K with a vertex at M that is similar to a given n-dimensional simplex. Similar problems are considered and unanswered questions are posed.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 47–49, 1992.     

The brauer group and ramified double covers of surfaces

Let Z be a nonsingular plane curve of even degree and U = P ² — Z. Let : X —> P ² be tbe double cover ramified over Z and V = ^—1(U). It is shown that the kernel of the restriction map on Brauer groups^" : B(U) —> B(V), is isomorphic to Z/2(²) where ρ— 2 < r <ρ— 1, p being the Picard number of X and if ^" : Pic P² —> Pic X is an isomorphism, exactly half of the algebra classes of order 2 in B(X) are restrictions of algebra classes of order 2 in B(U) whose ramification has been split by V. The kernel of the corestriction map 2 B(V) — ₂B(U) is shown to be the subgroup consisting of elements fixed by the Galois group.     

一类子空间格

令Ｖ_n（ｑ）是具ｑ个元素的有限域上的ｎ维向量空间．Ｃ［ｎ,ｋ］是Ｖ_n（ｑ）中与某ｋ维子空间相交不为零空间之子空间全体按包含关系所成偏序集,Ｗ_m为其Ｗｈｉｔｎｅｙ数（０≤ｍ≤ｎ）．本文证明了Ｃ［ｎ,ｋ］具Ｓｐｅｒｎｅｒ性质和单峰性质．进一步地,W_m²-qW_m-1W_m+1作为ｑ的多项式具有非负系数,并且W₀≤W_n≤W₁≤W_n-1≤W₂≤…．     

Simultaneous approximation by algebraic polynomials

Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C ^r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP _n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ω^υ(f^(k),δ) denotes the usual vth modulus of smoothness off ^(k), and Moreover, for no 0≤k≤r can (1?x ²)( ^{r?k+1)/(r?k+m)}(1/n²)^{(m?1)/(r?k+m)} be replaced by (1-x²)^αkn^2αk-2, with α_k>(r-k+a)/(r-k+m).     

Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

Let Ω be a bounded, smooth domain in ?²ⁿ, n ≥ 2. The well‐known Moser‐Trudinger inequality ensures the nonlinear functional J_ρ(u) is bounded from below if and only if ρ ≤ ρ_2n := 2²ⁿn!(n ? 1)!ω_2n, where in , and ω_2n is the area of the unit sphere ??^{2n ? 1} in ?²ⁿ. In this paper, we prove the inf_u∈X J_ρ(u) is always attained for ρ ≤ ρ_2n. The existence of minimizers of J_ρ at the critical value ρ = ρ_2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for J_ρ at the critical value ρ = ρ_2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.     

Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

LetP _n,k ^c denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[?1, 1], let $B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$ , andM _n,k ^* ?max{n(k+1),nlogn},M _n,k ?n(k+1). It is shown that $M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$ for everyx∈[?1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[?1,1] denotes the supremum norm on [?1,1]. This result should be compared with the inequalities $c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$ , for everyx∈[?1,1], wherec ₃>0 andc ₄>0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.