Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dP_t^β = V_t^β dt with P₀^β = p₀, dV_t^β = βV_t^βdt − βω²P_t^β + βdW_t with V₀^β = v₀, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that P^β converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P^β at level u during the time interval [0, t] · (N_t^Pβ (u)) to the local time of P(L_t^P (u)).     

部分和乘积的几乎处处中心极限定理

设X_n, n≥1是独立同分布正的随机变量序列, E(X₁)=u >0, Var(X₁)=σ², E|X₁|³<∞, 记S_n==∑^N_k=1X_k, 变异系数γ=σ/u.g是满足一定条件的无界可测函数, 证明了 lim_N→∞1/logN∑^N_n=11/n g((∏ⁿ_k=1S_k/n!uⁿ )^1/γ√n )=∫^∞₀g(x)dF(x),a.s., 其中 F(•) 是随机变量e^√2ξ 的分布函数, ξ 是服从标准正态分布的随机变量.     

4-Dimensional topological bordism

We show that two oriented, topological, closed 4-manifolds M⁴₁, M⁴₂ are topologically cobordant iff σ(M⁴₁) = σ(M⁴₂) and either both of M⁴₁, M⁴₂ are stably smoothable or neither of M⁴₁, M⁴₂ is stably smoothable.     

幂次为3 和4 的整数变量非线性型的整数部分

本文利用Davenport-Heilbronn 方法证明了对于自然数x_j, 表达式
λ₁x₁³+λ₂x₂³+λ₃x₃³+λ₄x₄⁴+λ₅x₅⁴+λ₆x₆⁴
的整数部分在给定条件下可表示无穷多素数, 深化了Brüdern 等人的广义Riemann 假设下等幂次的结果.     

On a partition identity

Two proofs are given, one combinatorial, the other by character theory, for the identity, ∏_λa₁! a₂! … a_n! = ∏_λ1^a₁2^a₂ … n^a_n, where λ ranges over all partitions λ = (1^a₁2^a₂ … n^a_n) of n. The two demonstrations yield a simple proof of the known formula, det²T(n) = [∏_λ1^a₁2^a₂ … n^a_n]², where T(n) is the matrix formed by the character table of S_n. Finally a sufficient condition is given so that the permanent of T(n) is zero.     

Determinacy of Wadge classes and subsystems of second order arithmetic

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA₀*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ₁⁰ comprehension and Π₀⁰ induction, and which is known as a weaker system than the popularbase theory RCA₀: 1. Bisep(Δ₁⁰, Σ₁⁰)‐Det* ? WKL₀, (1) 2. Bisep(Δ₁⁰, Σ₂⁰)‐Det* ? ATR₀ + Σ₁¹ induction, (2) 3. Bisep(Σ₁⁰, Σ₂⁰)‐Det* ? Sep(Σ₁⁰, Σ₂⁰)‐Det* ? Π₁¹‐CA₀, (3) 4. Bisep(Δ₂⁰, Σ₂⁰)‐Det* ? Π₁¹‐TR₀, (4) where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On universal graphs without instances of CH

We first prove the consistency of: there is a universal graph of power ?₁<2^?₀ = 2^?₁=?₂. The consistency of the non-existence of a universal graph of power ?₁ is trivial. Add ?₂ Cohen generic reals. We then show that we can have 2^?₀=?₂<2^?₁, and get similar results for other cardinals.     

一类非线性分数阶多点边值问题的可解性

研究下面一类非线性分数阶微分方程多点边值问题■通过应用Mawhin重合度理论得到解的存在性结果.此结论拓展了在分数阶多点边值问题这个领域的以前的结果.     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Rotational analysis of the C2Σ+ — X2Σ+ system of AlO

The 2-0, 1-0, 0-0 and 0-1 bands of the C²Σ⁺-X²Σ⁺ system of AlO, lying at 2391.88 Å, 2438.35 Å, 2487.32 Å and 2548.51 Å respectively were excited in a low pressure arc and photographed on a 6.6-metre concave grating spectrograph in the third order at a dispersion of 0.37 Å/mm. From the rotational analysis of these bands, the following constants have been determined: B_e′ = 0.565₃ cm.^?1 a_e′ = 0.004₈ cm.^?1 T_e =40267.0 cm.^?1 B_e″ = 0.641₃ cm.^?1 a_{″ = 0.005 7} cm.^?1     

Minimal free resolutions of monomial curves in P3k

Minimal free resolutions for prime ideals with generic zero (tn3,tn3?n10tn11,tn3?n20tn2, tn31), n1<n2<n3 positive integers, (n1,n2,n3)=1, are determined.     

Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.     

Some descriptive set theory and core models

We analyse the trees given by sharps for Π¹₂ sets via inner core models to give a canonical decomposition of such sets when a core model is Σ¹₃ absolute. This is by way of analogy with Solovay's analysis of Π¹₁ sets into ω₁ Borel sets — Borel in codes for wellorders. We find that Π¹₂ sets are also unions of ω₁ Borel sets — but in codes for mice and wellorders. We give an application of this technique in showing that if a core model, K, is Σ¹₃ absolute thenTheorem. Every real is in K iff every Π¹₃ set of reals contains a Π¹₃ singleton.     

Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

In a partially ordered space, the method x_n+1 = L⁺x _n ⁺ – N⁺x _n ^- – L^–y⁺ + N^– y _n ^- + r, y_n+1 = N⁺y⁺ – L⁺y _n ^- – N^–x _n ⁺ + L^–x^– + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x₀ y₀, x₀ x₁, y₁ y₀ related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

     

The magid-ryan conjecture for 4-dimensional affine spheres

In this paper we prove the Magid-Ryan conjecture for 4-dimensional affine hyperspheres in R⁵. This conjecture states that every affine hypersphere with non-zero Pick invariant and constant sectional curvature is affinely equivalent with either (x ₁ ² ±x ₂ ² )(x ₃ ² ±x ₄ ² ...(x _2m−1 ² ±x _2m ² ) = 1 or (x ₁ ² ±x ₂ ² (x ₃ ² ±x ₄ ² )...(x _2m−1 ² ±x _2m ² )x _2m+1 = 1 where the dimensionn satisfiesn = 2m orn =2m + 1. This conjecture was proved in [11] in case the metric is positive definite and in [2] in case the metric is Lorentzian.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ɛ-approximations in classes of smooth functions

Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W _∞ ^r (ℝ⁺;W ₂ ^2+k (Ω)), W ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) and S ₂ ^r (ℝ⁺;W ₂ ^2+k (Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W _∞ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), W ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)) and S ₂ ^r-1 (ℝ⁺;W ₂ ^k (Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations in the smoothness classes W _∞ ^r−1 (ℝ; W ₂ ^2+k (Ω)) and W ₂ ^r−1 (0, T; W ₂ ^2+k (Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W _∞ ^r−1 (ℝ⁺; W ₂ ^k (Ω)) and W ₂ ^r−1 (0,T; W ₂ ^k (Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {v^ɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems for the equations of Kelvin-Voight fluids. Bibliography: 27 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242. Translated by T. N. Surkova.     

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.