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1.
In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in R3 and Rn + 1(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x1, x2, t}-space and {x1, x2,…, xn, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2].  相似文献   

2.
About spaces NR (see [2, Exercise 5I]), the following are proved: (1) dim N∪R = dim β(N∪R)?N∪R,(2)if|β(N∪R)?N∪R|<2?o, then no real-valued continuous fu ction on NR is onto (and hence, dim N∪R=0), (3) any compact metric space without isolated points is homeomorphic to some β(N∪R)?N∪R and (4)there are spaces X,X1 and X2 of the form NR such that X=X1X2,X2andX2 are zero sets of X, and dim X=n, dimX1=dimX2=0, where n=1,2,… or ∞.  相似文献   

3.
The probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in Rn (1 ≤ rn ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of Y1 = (y01,…, yr1). Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω with ‘initial’ point at the origin [ω is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and yi1 = (xi ? z)α(p2), where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in Rn invariant under the group of rotations in Rn, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x0,…, xr} for 1 ≤ rn.  相似文献   

4.
We give a characterization of manifolds modeled on R= dir lim or RnQ=dir lim Qn, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.  相似文献   

5.
In this paper we show the existence and uniqueness of a natural isomorphism øjα of Kj(A) with Kj+1(A ?αR), j ? Z/2 where (A, R, α) is a C1 dynamical R-system, K is the functor of topological K theory and A ?αR is the crossed product of A by the action of R. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace \?gt, one has \?gtø1α[u] = (12iπ) τ(δ(u)u1) for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S3) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple C1 algebra with no nontrivial idempotent.  相似文献   

6.
Let F1(Rn) denote the Fourier algebra on Rn, and D(Rn) the space of test functions on Rn. A closed subset E of Rn is said to be of spectral synthesis if the only closed ideal J in F1(Rn) which has E as its hull
h(J)={x ? Rn:f(x)=0 for all f ? J}
is the ideal
k(E)={f?F1(Rn):f(E)=0}
. We consider sufficiently regular compact subsets of smooth submanifolds of Rn with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra (k(E)∩D(Rn))?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.  相似文献   

7.
The asymptotic behaviour as t tends to +∞ of the solution of (?u?t) ? Δu + u¦u¦p ? 1 = 0 in RN × R+, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of (N + 2)N ? p and also on the rate of decay of the admissible initial data u(0, x) as ¦x¦ tends to +∞.  相似文献   

8.
In contrast to the situation in R3, where a 2-sphere with double tangent balls at each point must be tamely embedded in R3, there exist wild (n?1)-spheres in Rn for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in Rn. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.  相似文献   

9.
Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of Qp. An element xK transcendental over Qp is said to have order ≤a (a > 0) if there exists Cx > 0 such that every polynomial P(X)Qp [X] satisfies
?log;(P(x))? ?log∥P∥+cx(deg P)a
when ∥ · ∥ is the Gauss norm on Qp[X]. No xCp can have order ≤α if α < 1 but we construct some xCp with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.  相似文献   

10.
Some regularity results about the local structure of the zero set of a mapping of n + 1 real variables with values in Rn are presented. They all provide improved and generalized versions of the Morse lemma for plane curves as well as generalizations of the implicit function theorem. We describe their applications to one-parameter nonlinear problems, including an analysis of bifurcation phenomena at simple or multiple eigenvalues.  相似文献   

11.
Let (Ω, B, μ) be a measure space, X a separable Banach space, and X1 the space of all bounded conjugate linear functionals on X. Let f be a weak1 summable positive B(X, X1)-valued function defined on Ω. The existence of a separable Hilbert space K, a weakly measurable B(X, K)-valued function Q satisfying the relation Q1(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(X, X1)-valued measures, the concepts of weak1, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.  相似文献   

12.
The theory of inner-outer factorization in the Hardy spaces Hp in the unit disc D is well known and has many applications. It does not carry over to the spaces Hp on the polydisc Dn or the ball Bn when n > 1. However, for Lumer's Hardy spaces (LH)p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)p has a factorization f = Iε × Eε, where Iε is internal and Eε is external, and Eε? (LH)p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when p = 2m, where m is an integer.  相似文献   

13.
In this paper, minimization problems in L1(R3) are considered. These problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids (rotating stars). Under nearly optimal assumptions a minimizer is proved to exist by a direct variational method, which heavily uses the symmetry of the problem in order to get some compactness. Finally, by looking directly at the Euler equation, we give some existence results (of solutions of the Euler equation) even if the infimum is not finite.  相似文献   

14.
We consider functionals of the form: If(u) = ∝Tf[t, u(t)]μ(dt), which are defined on spaces Lp(T, Rk), and we study for these functionals the properties of a convergence for which the conjugacy If → If1 is a continuous operator.  相似文献   

15.
Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D1, D2, h) of two countable dense subsets D1 and D2 and a homeomorphism h: X?D1X?D2, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for X = R, such atriple is shown to exist.  相似文献   

16.
The main theorem of this paper is that if χ is a character of a connected closed normal subgroup of a connected Lie group, then every matrix element of the induced representation Uχ vanishes at infinity modulo the kernel of that representation. As a consequence, it is shown that every faithful irreducible unitary representation of a connected motion group vanishes at infinity. In the course of the development a generalization of the classical Riemann-Lebesgue lemma is proved. Suppose M is an analytic submanifold of Rn which is not contained in any proper hyperplane. Then the Fourier transform of any measure, which is concentrated on M and which is absolutely continuous with respect to the “Lebesgue” measure on M, vanishes at infinity.  相似文献   

17.
Let H′ be either the space K1 of distributions of exponential growth or the space S′ of tempered distributions, and let OC(H′:H′) be the space of convolution operators in H′. In each case H′ is the dual of a space H of C-functions which are in OC(H′:H′). We establish necessary and sufficient conditions on the Fourier transform S? of ? of S ? OC(H′:H′) in order that every distribution u? OC(H′:H′) with S1u?H be in H. If H′ = K1, the condition is equivalent to S×H1=H1.  相似文献   

18.
Three main results are obtained: (1) If D is an atomic maximal Abelian subalgebra of B(H), P is the projection of B(H) onto D and h is a complex homomorphism on D, then h ° P is a pure state on B(H). (2) If {Pn} is a sequence of mutually orthogonal projections with rank(Pn) = n and ∑ Pn = I, P is the projection of B(H) onto {Pn}″ given by P(T)=∑tracen(T)Pn and h is a homomorphism on {Pn}″ such that h(Pn) = 0 for all n then h ° P induces a type II factor representation of the Calkin algebra. (3) If M is a nonatomic maximal Abelian subalgebra of B(H) then there is an atomic maximal Abelian subalgebra D of B(H) and a large family {Φα} of 1-homomorphisms from D onto M such that for each α, Φα ° P is an extreme point in the set of projections from B(H) onto M. (Here P denotes the projection of B(H) onto D.)  相似文献   

19.
A theorem on regularly varying functions in R2 is proved and applied to domains of attraction of stable laws with index 1 ? α ? 2. We also present a theory of Π-variation in R2. Unlike the situation in R1 the latter is not connected with domain of attraction theory. The situation in Rd (d > 1) is more complicated but not essentially different; for simplicity we limit ourselves to R2. This article complements de Haan and Resnick (1979) where the situation for 0 < α < 1 was considered.  相似文献   

20.
In their paper on pseudo-boundaries and pseudo-interiors R. Geoghegan and R.R. Summerhill construct k-dimensional pseudo-boundaries in Rn, where they used West's notion of a pseudo-boundary, rather than Toruńczyk's. In this paper we construct pseudo-boundaries in the sense of Toruńczyk (skeletoids) in Rn and use this result to find k-dimensional skeletoids in the Hilbert cube.  相似文献   

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