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1.
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation yp ? y = β generates a tower of extensions through Ki = Ki?1(yi) where y = [y1, y2,…, yn]. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form (Bi) = qΠpjλj. In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants.  相似文献   

2.
Let {Xt, t ≥ 0} be Brownian motion in Rd (d ≥ 1). Let D be a bounded domain in Rd with C2 boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” Ex{exp(∝0τDq(Xt)dt)1A(XτD)}, where τD is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any ? ? C0(?D), φ(x) = Ex{exp(∝0τDq(Xt)dt)?(XτD)} is the unique solution in C2(D) ∩ C0(D?) of the Schrödinger boundary value problem (12Δ + q)φ = 0 in D, φ = ? on ?D.  相似文献   

3.
Let L = ∑j = 1mXj2 be sum of squares of vector fields in Rn satisfying a Hörmander condition of order 2: span{Xj, [Xi, Xj]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X1,…, Xm are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator ?2?x2 + x2?2?t2 in R2. (b) The real part of the Kohn Laplacian on the Heisenberg group j ? 1n (??xj + 2yj??t)2 + (??yj ? 2xj??t)2 in R2n + 1. In contrast to non-characteristic points, C regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.  相似文献   

4.
Let (Vn, g) be a C compact Riemannian manifold without boundary. Given the following changes of metric: g′?± = g + Hess ? ± lα2(▽ ? ? ▽?), g?± = ±?g + α2Hess ?, where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)±Log(¦g′?±¦ ¦g¦?1) = F(P,▽?;?), (2)±Logg??±¦ ¦g¦?1) = F(P, ▽?; ?). We first solve Eq. (2)?, under some simple assumptions on F?C. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)?, we extend to Eq.(1)? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)+ is not locally invertible, and does not even admit a solution for all F = λ? + ?, λ > 0, f ? C(Vn), a similar change of functions leads to partial results about Eq. (1)+, via C2 and C3 estimates for Eq. (2)+. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353).  相似文献   

5.
Let H1 = ?∑i = 1Ni + V(xi)) + ∑1 ? i <j ? N¦xi ? xj¦?1, V(xi) = N ∝ ¦x ? y¦?1 ?(y)dy, with ? a normalized Gaussian. Suppose E ≠ 0 and that H = H1 + E · (∑i = 1Nxi) has no eigenfunctions in L2(R3N. If H1ψ = μψ with μ < infσess(H1), then (ψ, e?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.  相似文献   

6.
The K-theory of the C1-algebra C1(V, F) associated to C-foliations (V, F) of a manifold V in the simplest non-trivial case, i.e., dim V = 2, is studied. Since the case of the Kronecker foliation was settled by Pimsner and Voiculescu (J. Operator Theory4 (1980), 93–118), the remaining problem deals with foliations by Reeb components. The K-theory of C1(V, F) for the Reeb foliation of S3 is also computed. In these cases the C1-algebra C1(V, F) is obtained from simpler C1-algebras by means of pullback diagrams and short exact sequences. The K-groups K1(C1(V, F)) are computed using the associated Mayer-Vietoris and six-term exact sequences. The results characterize the C1-algebra of the Reeb foliation of T2 uniquely as an extension of C(S1) by C(S1). For the foliations of T2 it is found that the K-groups count the number of Reeb components separated by stable compact leaves. A C-foliation of T2 such that K1(C1(T2, F)) has infinite rank is also constructed. Finally it is proved, by explicit calculation using (M. Penington, “K-Theory and C1-Algebras of Lie Groups and Foliations,” D. Phil. thesis, Oxford, 1983), that the natural map μ: K1,τ(BG) → K1(C1(V, F)) is an isomorphism for foliations by Reeb components of T2 and S3. In particular this proves the Baum-Connes conjecture (P. Baum and A. Connes, Geometric K-theory for Lie groups, preprint, 1982; A. Connes, Proc. Symp. Pure Math.38 (1982), 521–628) when V = T2.  相似文献   

7.
Let H and K be symmetric linear operators on a C1-algebra U with domains D(H) and D(K). H is defined to be strongly K-local if ω(K(A)1K(A)) = 0 implies ω(H(A)1 H(A)) = 0 for A?D(H) ∩ D(K) and ω in the state space of U, and H is completely strongly K-local if Ω(K(A)1K(A))=0 implies Ω(H(A)1H(A))=0 for AD(H) ∩ D(K) and Ω in the state of U, and H is cpmpletely strongly K-local if H??n is K??n-local on U?Mn for all n ? 1, where 1n is the identity on the n × n matrices Mn. If U is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of 1-automorphisms of U with generator δ0 and δ is a derivation which commutes with τ and is completely strongly δ0-local then δ generates a group α of 1-automorphisms of U. Various characterizations of α are given and the particular case of periodic τ is discussed.  相似文献   

8.
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   

9.
Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, 1N)21λN?(x,Hx)(x,H?1x)?1. Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products M(y)1H and M(y)1H?1, where M(y) is a matrix defined by M(y)=(δij+(1?δij)yiyj. Subsequently, these results are extended to improve the spectral bounds of M(y)1H.  相似文献   

10.
Let Sp(H) be the symplectic group for a complex Hibert space H. Its Lie algebra sp(H) contains an open invariant convex cone C0; each element of C0 commutes with a unique sympletic complex structure. The Cayley transform C: X∈ sp(H)→(I + X)1∈ Sp(H) is analyzed and compared with the exponential mapping. As an application we consider equations of the form (ddt) S = A(t)S, where t → A(t) ? C?0 is strongly continuous, and show that if ∝?∞A(t)∥ dt < 2 and ∝? t8A(t) dt?C0, the (scattering) operator
S=s?limt→∞t′→?∞ St(t)
, where St(t) is the solution such that St(t′) = I, is in the range of B restricted to C0. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp(H) to a unitary operator.  相似文献   

11.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If Ln denotes the symmetric group on {1,2,…,n} then we define the projection PL : Tn(V) → Sn(V) by the formula (n!)?1Σσ ? Ln Pσ, where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If xi ? V1, 1 ? i ? n, then x1?x2? … ?xn denotes the member of Tn(V) such that (x1?x2· ? ? ?xn)(y1,y2,…,yn) = Пni=1xi(yi) for each y1 ,2,…,yn in V, and x1·x2xn denotes PL(x1?x2? … ?xn). If B? Sn(V) and there exists x i ? V1, 1 ? i ? n, such that B = x1·x2xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.  相似文献   

12.
Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space C1(G). We show that the corresponding property is no longer true for the space of double cosets K\GK. If G is of real-rank 1, we construct liner functionals Tp ? (Cc(K\GK))′ for each p, 0 < p ? 2, such that Tp(f 1 f1) ? 0, ?f ? Cc(K\GK) but Tp does not extend to a continuous functional on Cp(K\GK). In particular, if p ? 1, Tv does not extend to a continuous functional on C1(K\GK). We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that T(f 1 f1) ? 0, ?f ? Cc(K\GK). The main tool used is a theorem of Trombi-Varadarajan.  相似文献   

13.
Let Kn= {x ? Rn: (x12 + · +x2n?1)12 ? xn} be the n-dimensional ice cream cone, and let Γ(Kn) be the cone of all matrices in Rnn mapping Kn into itself. We determine the structure of Γ(Kn), and in particular characterize the extreme matrices in Γ(Kn).  相似文献   

14.
Let (Wt) = (W1t,W2t,…,Wdt), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from R+ into the space of d × d skew-symmetric matrices and x(t) such a function into Rd. A class of stochastic processes (LtA,x), a particular example of which is Levy's “stochastic area” Lt = 120?t (W1s,dW2s ? W2s,dW1s), is dealt with.The joint characteristic function of Wt and L1A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (Wt,LtA,x) is given.  相似文献   

15.
16.
Let p, q be arbitrary parameter sets, and let H be a Hilbert space. We say that x = (xi)i?q, xi ? H, is a bounded operator-forming vector (?HFq) if the Gram matrixx, x〉 = [(xi, xj)]i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on lq2, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from lq2 to lp2. Then exists a linear operator ǎ from (the Banach space) HFq to HFp on D(A) = {x:x ? HFq, A〈x, x〉12 is p × q bounded on lq2} such that y = ǎx satisfies yj?σ(x) = {space spanned by the xi}, 〈y, x〉 = Ax, x〉 and 〈y, y〉 = A〈x, x〉12(A〈x, x〉12)1. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.  相似文献   

17.
18.
Ck estimates for convex domains of finite type in Cn are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of Cn, of finite type m and m′ such that D?D′, for all q=1,…,n?2, there exists a linear operator T1q from C0,q(D′?D) to C0,q?1(D′?D) such that for all k∈N and all (0,q)-form f, ??-closed of regularity Ck up to the boundary, T1qf is of regularity Ck+1/max(m,m′) up to the boundary and ??Tq1f=f. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the ?? problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

19.
On a compact Kähler manifold of complex dimension m ? 2, let us consider the change of Kähler metric g′λ\?gm = gλ\?gm + ?λ\?gmφ. Let F?C(V × R) be a function everywhere > 0 and v a real number ≠ 0. When 0 < C?1 ? F(x, t) ? C(¦t¦a + 1) for all (x, t) ?V × ] ?∞, t0], where C and t0 are constants and 1 ? a < m(m ? 1), one exhibits a function φ?C (V) such that ¦g′∥g¦?1 = eν\?gfF(x, φ ? \?gf) (¦g¦ and ¦g′¦ the determinants of the metrics g and g′, \?gf = (mes V)?1 ∝ φ dV).  相似文献   

20.
We study a continuous time growth process on Zd (d?1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call Pd the law of such a process and S0d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set Cd?Rd, such that for every ε>0, Pd-a.s. eventually in t, the set Sd0(t) is within an ε neighborhood of the set [Cdt], where for A?Rd we define [A]:=A∩Zd. Moreover, for d large enough, the set Cd is not a ball under the Euclidean norm. We also show that the empirical density of particles within Sd0(t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ram??rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826.  相似文献   

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