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1.
We consider quasi-periodic Schrödinger operators H on Z of the form H=Hλ,x,ω=λv(x+)δn,n+Δ where v is a non-constant real analytic function on the d-torus Td(d?1) and Δ denotes the discrete lattice Laplacian on Z. Denote by Lω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector ω∈Td. It is shown that for |λ|>λ0(v), there is the uniform minoration Lω(E)>12log|λ| for all E and ω. For all λ and ω, Lω(E) is a continuous function of E. Moreover, Lω(E) is jointly continuous in (ω,E), at any point 0,E0)∈Td×R such that k·ω0≠0 for all k∈Zd?{0}. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.  相似文献   

2.
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral H = ⊕L2(vt) dm(t) and the operator (L?)(t, λ) = e?iλ?(t, λ) ? 2e?iλtT ?(s, x) e(s, t) dvs(x) dm(s) on H, where e(s, t) = exp ∫stTdvλ(θ) dm(λ). Let μt be the measure defined by T?(x) dμt(x) = ∫0tT ?(x) dvs dm(s) for all continuous ?, and let ?t(z) = exp[?∫ (e + z)(e ? z)?1t(gq)]. Call {vt} regular iff for all t, ¦?t(e)¦ = ¦?(e for 1 a.e.  相似文献   

3.
Let {X(t) : t ∈ R+N} denote the N-parameter Wiener process on R+N = [0, ∞)n. For multiple sequences of certain independent random variables the authors find lower bounds for the distributions of maximum of partial sums of these random variables, and as a consequence a useful upper bound for the yet unknown function P{supt∈DnX(t) ≥ c}, c ≥ 0, is obtained where DN = Πk = 1N [0, Tk]. The latter bound is used to give three different varieties of N-parameter generalization of the classical law of iterated logarithm for the standard Brownian motion process.  相似文献   

4.
Let u∈C([0,T1[;Ln(Rn)n) be a maximal solution of the Navier–Stokes equations. We prove that u is C on ]0,T1Rn and there exists a constant ε1>0, which depends only on n, such that if T1 is finite then, for all ω∈S(Rn)n, we have limt→T16u(t)?ω6B?1,∞1.To cite this article: R. May, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

5.
A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form
XN(t)=x0+∑1NlY1N ∫t0 f1(XN(s))ds
where l∈Zt, the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies
X(t)=x0+ ∫t0 ∑ lf1(X(s))ds
Under very general conditions limN→∞XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by
ZN(t)=x0+∑1NlB1N∫t0 ft(ZN(s))ds
and
V(t)=∑ l∫t0f1(X(s))dW?1+∫t0 ?F(X(s))·V(s)ds.
Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that
XN(t)=ZN(t)+OlogNN
and
N(XN(t)?X(t))=V(t)+O log NN
  相似文献   

6.
We consider a branching diffusion {Zt}t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ2, and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Zt dies out w.p.l. if μ?μ0≡{2σ2c(m?1)}12. If μ<μ0, then itZt grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ0. Even though E{ZT}∽const.T?32 in this case, we find that P{ZT>0} is only exp{–const.T13+0(logT)2}, and conditionally on {ZT>0} there are with high probability much fewer particles alive at time T than E{ZT|ZT0}.  相似文献   

7.
This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ? 2, satisfy
n∈Nn(T)|p1pp(T).
This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π2(n), P = (p1, …, pn) and show that for TΠ2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy
i∈Ni(T)|2nn2n2(T).
More generally, we show that for TΠp(n), P = (p1, …, pn), pi ? 2, the eigenvalues are absolutely p-summable,
1p=i=1n1piandn∈Nn(T)|p1p?CpπnP(T).
We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely
n∈Nn(T)|p ? Cpn∈N αn(T)p
, 0 < p < ∞, where αn denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space.  相似文献   

8.
Let α(n1, n2) be the probability of classifying an observation from population Π1 into population Π2 using Fisher's linear discriminant function based on samples of size n1 and n2. A standard estimator of α, denoted by T1, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T1, denoted by T2, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, T11 and T21, of ET1 = τ1(n1, n2) and ET2 = τ2(n1, n2) = α(n1 ? 1, n2) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that T11 and T21 are nonincreasing functions of D2, the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of T11 and T21. It is also shown that α is a decreasing function of Δ2 = (μ1 ? μ2)′Σ?11 ? μ2). Hence, by truncating T11 and T21 (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α.  相似文献   

9.
Let x(t), t = 1,…, T, be generated by a zero mean stationary process and let I(ω) = |Σx(t)expitω|2T be the periodogram. Under general conditions, and in particular assuming only a finite 2nd moment, it is shown that maxωI(ω){2πf(ω)logT} ≤ 1, a.s., and under stricter conditions it is shown that equality holds.  相似文献   

10.
Let π = (a1, a2, …, an), ? = (b1, b2, …, bn) be two permutations of Zn = {1, 2, …, n}. A rise of π is pair ai, ai+1 with ai < ai+1; a fall is a pair ai, ai+1 with ai > ai+1. Thus, for i = 1, 2, …, n ? 1, the two pairs ai, ai+1; bi, bi+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ωn denotes the number of pairs with RR forbidden, it is proved that 0ωnznn!n! = 1?(z), ?(z) = ∑0(-1) nznn!n!. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then 1 + ∑n=1znn!n!n?1k=0 ω(n, k)xk = (1 ? x)(?(z(1 ? x)) ? x).  相似文献   

11.
Given a cocycle a(t) of a unitary group {U1}, ?∞ < t < ∞, on a Hilbert space H, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;t0Usxds + β(t) for a unique x ? H, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as σ2({rmUt}, a(t)) = limT→∞(1T)∥∝t0 Us x ds∥2 if existing. For a stationary diffusion process on R1, with Ω1, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on Ω1, based on the paths restricted to the time interval [0, 1]. It is shown that I(Ω) is continuous and bounded on Ω1. The shift τt, defines a unitary representation {Ut}. Assuming Ω1 I dm = 0, dm being the stationary measure defined by the transition probabilities and the invariant measure on J, I(Ω) has a C spectral density function f;. It is then shown that σ2({Ut}, I) = f;(O).  相似文献   

12.
Let Ω be a simply connected domain in the complex plane, and A(Ωn), the space of functions which are defined and analytic on Ωn, if K is the operator on elements u(t, a1, …, an) of A(Ωn + 1) defined in terms of the kernels ki(t, s, a1, …, an) in A(Ωn + 2) by Ku = ∑i = 1naitk i(t, s, a1, …, an) u(s, a1, …, an) ds ? A(Ωn + 1) and I is the identity operator on A(Ωn + 1), then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on A(Ωn + 1) defined in terms of a kernel w(t, s, a1, …, an) in A(Ωn + 2) by Wu = ∝antw(t, s, a1, …, an) u(s, a1, …, an) ds. ΠW is the operator; ΠWu = ∝an ? 1w(t, s, a1, …, an) u(s, a1, …, an) ds. ΠK is the operator; ΠKu = ∑i = 1n ? 1aitki(t, s, a1, …, an) ds + ∝an ? 1tkn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where m ? A(Ωn + 1) and maps elements of A(Ωn + 1) into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator K1 on functions u in A(Ωn + 2), by K1u = ∑i = 1n ? 1ait ki(t, s, a1, …, an) u(s, a, …, an + 1) ds + ∝an + 1t kn(t, s, a1, …, an) u((s, a1, …, an + 1) ds. A determinant δ(I ? K1) of the operator I ? K1 is defined as an element m1(t, a1, …, an + 1) of A(Ωn + 2). This is mapped into A(Ωn + 1) by setting an + 1 = t to give m(t, a1, …, an). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in A(Ωn + 1), explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.  相似文献   

13.
For certain types of stochastic processes {Xn | n ∈ N}, which are integrable and adapted to a nondecreasing sequence of σ-algebras Fn on a probability space (Ω, F, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {Fn | n ∈ N}, when issupsΩ | Xσ | dPfinite, and when is there a stopping time σ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.  相似文献   

14.
If r, k are positive integers, then Tkr(n) denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r = 1. An explicit formula for Tkr(n) is derived and it is shown that limn→∞Tkr(n)nk = 1ζ(rk).If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2psas; piS and ai ≥ 0 for all i} and Tkr(S, n) denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for Tkr(S, n) are derived and it is shown that limn→∞Tkr(S, n)nk = (p1 … pa)rkζ(rk)(p1rk ? 1) … (psrk ? 1).  相似文献   

15.
We consider the mixed boundary value problem Au = f in Ω, B0u = g0in Γ?, B1u = g1in Γ+, where Ω is a bounded open subset of Rn whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on Ω and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on Γ+. The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset T of the reals such that, if Ds = {u ? Hs(Ω): Au = 0} then for s ? = 12(mod 1), (B0,B1): Ds → Hs ? 12?) × Hs ? 32+) is a Fredholm operator if and only if s ∈T . Moreover, T = ?xewTx, where the sets Tx are determined algebraically by the coefficients of the operators at x. If n = 2, Tx is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, Tx is either an open interval of length 1 or is empty; and finally, if n ? 4, Tx is an open interval of length 1.  相似文献   

16.
Let X={X(t)}t∈R be a continuous-time strictly stationary and strongly mixing process. In this paper, we prove in the setting of spectral density estimation, at first, under some hard conditions on the spectral density φX (because of aliasing phenomenon), the uniformly complete convergence of the spectral density estimate from periodic sampling. Afterwards, to overcome aliasing, we consider the sampled process {X(tn)}n∈Z, where {tn} is a stationary point process independent from X. The uniform complete convergence of the spectral estimate based on the discrete time observations {X(tk),tk} is also obtained. The convergence rates are also established. To cite this article: M. Rachdi, C. R. Acad. Sci. Paris, Ser. I 337 (2003).  相似文献   

17.
We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family C=?m?NCm of closed sets, interior-preserving over each member C of which is a countable family {Dn(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈C and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: DDv(Γ))?U.  相似文献   

18.
Let q=1,…,n?1 and D be a bounded convex domain in Cn of finite type m. We construct two integral operators Tq and Tq such that for all p∈N,Tq,Tq:Cp0,q(bD)→Cp+1/m0,q?1(bD) are continuous, and for all (0,q)-forms h continuous on bD with ?bh continuous on bD too, with the additional hypothesis when q=n?1 that ∫bDhφ=0 for all φCn,0(bD) ??b-fermée, we show h=??b(Tq?Tq)h+(Tq+1?Tq+1)??bh. For this construction, we use the Diederich–Fornæss support function of Alexandre (Publ. IRMA Lille 54 (III) (2001)). To prove the continuity of Tq, we integrate by parts and take care of the tangential derivatives. The normal component in z of the kernel of Tq will have a bad behaviour, so, in order to find a good representative of its equivalence class, we isolate the tangential component of the kernel and then integrate by parts again. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  相似文献   

19.
Given a set S of positive integers let ZkS(t) denote the number of k-tuples 〈m1, …, mk〉 for which mi ∈ S ? [1, t] and (m1, …, mk) = 1. Also let PkS(n) denote the probability that k integers, chosen at random from S ? [1, n], are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1pr) = 1}, then ZkS(t) = (td(S))k Πν?P(1 ? 1pk) + O(tk?1) if k ≥ 3 and Z2S(t) = (td(S))2 Πp?P(1 ? 1p2) + O(t log t) where d(S) denotes the natural density of S. From this result it follows immediately that PkS(n) → Πp?P(1 ? 1pk) = (ζ(k))?1 Πp∈P(1 ? 1pk)?1 as n → ∞. This result generalizes an earlier result of the author's where P = ? and S is then the whole set of positive integers. It is also shown that if S = {p1x1prxr : xi = 0, 1, 2,…}, then PkS(n) → 0 as n → ∞.  相似文献   

20.
A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by
DF(s,p,α)= α∈Zn?{0}F(α)?s e(ρF(α)+〈α, α〉)
where ? ∈ Q, α ∈ Qn, 〈x, y〉 = x1y1 + ? + xnyn, e(a) = exp (2πia) for aR, and s = σ + ti is a complex number. The author proves that: (1) DF(s, ?, α) has analytic continuation into the whole s-plane, (2) DF(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at s = nδ. The residue at s = nδ is given explicitly. (3) ? = 0, α ? Zn, DF(s, 0, α) is analytic for α>, n(δ ? 1).  相似文献   

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