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1.
In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H 01×L 2(Ω) and D(AH 01(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.  相似文献   

2.
Exact controllability for the wave equation with variable coefficients   总被引:2,自引:0,他引:2  
We consider in this paper the evolution systemy″−Ay=0, whereA = i(aijj) anda ijC 1 (ℝ+;W 1,∞ (Ω)) ∩W 1,∞ (Ω × ℝ+), with initial data given by (y 0,y 1) ∈L 2(Ω) ×H −1 (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.  相似文献   

3.
In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk(Ω,R2) for any k ≥ 1, which attracts any bounded set of Hk(Ω,R2) in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong's conclusion.  相似文献   

4.
 Let X 1 ,X 2 ,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that |X i+1 +...+X j |≥a for some integers 0≤i<j<∞. For each k≥2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms of P(A), the probability that A occurs at least once. More generally, let X=(X 1 ,X 2 ,...)Ω=Π j ≥1Ω j be a random element in a product probability space (Ω,ℬ,P=⊗ j ≥1 P j ). We are interested in events AB that are (at most contable) unions of finite-dimensional cylinders. We term such sets sequentially searchable. Let L(A) denote the (random) number of disjoint intervals (i,j] such that the value of X (i,j] =(X i+1 ,...,X j ) ensures that XA. By definition, for sequentially searchable A, P(A)≡P(L(A)≥1)=P(𝒩−ln (P(Ac)) ≥1), where 𝒩γ denotes a Poisson random variable with some parameter γ>0. Without further assumptions we prove that, if 0<P(A)<1, then P(L(A)≥k)<P(𝒩−ln (P(Ac)) k) for all integers k≥2. An application to sums of independent Banach space random elements in l is given showing how to extend our theorem to situations having dependent components. Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003 RID="*" ID="*" Supported by NSF Grant DMS-99-72417. RID="†" ID="†" Supported by the Swedish Research Council. Mathematics Subject Classification (2000): Primary 60E15, 60G50 Key words or phrases: Tail probability inequalities – Hoffmann-Jo rgensen inequality – Poisson bounds – Number of event recurrences – Number of entrance times – Product spaces  相似文献   

5.
Consider a setA of symmetricn×n matricesa=(a i,j) i,jn . Consider an independent sequence (g i) in of standard normal random variables, and letM=Esupa∈Ai,j⪯nai,jgigj|. Denote byN 2(A, α) (resp.N t(A, α)) the smallest number of balls of radiusα for thel 2 norm ofR n 2 (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN 2(A, α))1/4KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN tK(δ)M. Work partially supported by an N.S.F. grant.  相似文献   

6.
Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 −n where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1μ) in the metric d is
almost surely on the event of nonextinction, where h(μ) is the entropy of the measure μ and q(i, j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of HAWKES [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log2 α, where α is the Malthusian parameter. Received: 30 June 1998 / Revised: 4 February 1999  相似文献   

7.
A family {A i | iI} of sets in ℝ d is antipodal if for any distinct i, jI and any pA i , qA j , there is a linear functional ϕ:ℝ d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ iI A i . We study the existence of antipodal families of large finite or infinite sets in ℝ3. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.  相似文献   

8.
The paper considers a class of regular, hypoelliptic in x 1, two-dimensional operators P(D) = P(D 1,D 2) in rather wide strip Ω H = {x = (x 1; x 2) ∈ $ \mathbb{E} $ \mathbb{E} 2, |x 1| < H, x 2 ∈ $ \mathbb{E} $ \mathbb{E} 1}. It is proved the infinite differentiability in Ω H of those generalized solutions of the equation P(D) u = 0, for which D 2 j uL 2 H ), j = 0, …, ord x2 P.  相似文献   

9.
We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.  相似文献   

10.
A hypersurface x : MS n+1 without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ?2 i (e i (H) + ∑ j (h ij ? ij )e j (log ρ))θ i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S n +1 without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ−2 i (e i (H) + ∑ j (h ij Hδ ij )e j (log ρ))θ i vanishes and its M?bius shape operator ? = ρ−1(SHid) has constant eigenvalues. Here {e i } is a local orthonormal basis for I = dx·dx with dual basis {θ i }, II = ∑ ij h ij θ i ⊗θ i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S n +1 is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S n +1 with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S n +1 can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002  相似文献   

11.
The paper gives a necessary and sufficient condition for the embedding of the Orlicz-Sobolev spaceW kLA (Ω) inC(Ω). The same condition is also found to be necessary and sufficient so that a continuous function inW kLA (Ω) be differentiable of orderk almost everywhere in Ω.  相似文献   

12.
Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x 1,…,x n X, there exists a linear mapping L:XF, where FX is a linear subspace of dimension O(log n), such that ‖x i x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i x j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 22O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E n Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.  相似文献   

13.
Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg iBigi −1 andA+B i, whereg i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB i possess the following property: ‖B iA−1gBjA−1‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.  相似文献   

14.
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ n . Let L(P m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ n ×(ℝ\{ 0}) for any QL(P m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P m must be locally hyperbolic. Received: 24 January 2000  相似文献   

15.
两类惯量惟一的对称符号模式   总被引:4,自引:0,他引:4  
§ 1  IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .   For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…  相似文献   

16.
Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, jn, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM n (Ω[ξ]) generated by the matricesY k=({ie65-2}), 1 ≦i, jn, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X 1, …,X m) is a polynomial linear in all theX i but one (similar to Formanek’s central polynomials for matrix rings) andf 2 is central forM n (Ω), thenf is central forM n (Ω). (The existence of a polynomial not central forM n (Ω), but whose square is central forM n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.  相似文献   

17.
LetK be an algebraically closed field of characteristic zero. ForAK[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ i=1 n(λ) A iλ k μ whereA iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.  相似文献   

18.
Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2N, and real edge-weights w(e) for eE. Given a convex n-gon P in the plane with vertices x0,x1,…,xn-1 which are arranged in this order clockwisely, let each node iN correspond to the vertex xi and define the area AP(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size cH(i,j,k) in H is defined as the sum of weights of edges eE such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:
AP(H)?AP(H) for all convex n-gons P.
cH(i,j,k)?cH(i,j,k) for all convex three-cuts C(i,j,k).
From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H satisfy “AP(H)?AP(H) for all convex n-gons P” is immediately obtained.  相似文献   

19.
We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ 0, A 0) ∈ L 2(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L 3(Ω)) using the Lorentz gauge.   相似文献   

20.
Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),ABI(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT -1 for all AB(H), or Φ(A) = TA*T -1 for all AB(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.  相似文献   

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