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1.
In the paper, the equation
is considered in the scale of the weighted spaces H β s (ℝ n ) (q > 1, a ∈ ℂ). We prove that if the expression
does not vanish on the set {ξ ∈ ℝ n ∖ 0, |z| ≤ q βs+n /2−2m}, then this equation has a unique solution uH β s+2m (ℝ n ) for every function fH β s (ℝ n ) provided that β, s ≠ ∈ ℝ, βsn/2 + p, and βs − 2m ≠ − n/2 − p (p = 0, 1, ...). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 37–55, 2007.  相似文献   

2.
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.  相似文献   

3.
The aim of the paper is to prove that every fL 1([0,1]) is of the form f = , where j n,k is the characteristic function of the interval [k- 1 / 2 n , k / 2 n ) and Σ n=0Σ k=12n |a n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b n,k ) n≧0 k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).   相似文献   

4.
This paper deals with conditions for the existence of solutions of the equations
considered in the whole space ℝn, n ≥ 2. The functions A i (x, u, ξ), i = 1,…, n, A 0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution uW loc 1,p (ℝn) under the condition p > n. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 133–147, 2006.  相似文献   

5.
Two Inequalities for Convex Functions   总被引:1,自引:0,他引:1  
Let a 0 < a 1 < ··· < a n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a 0 < a 1 < ··· < a n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α 0, α 1, ... , α n , β 0, β 1, ... , β n being real numbers in [A, B] with α 0α 1 ≤ ··· ≤ α n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α 0, α 1, ... , α n , β 0, β 1, ... , β n , α' 0, α' 1, ... , α' n , β' 0 , β' 1 , ... , β' n be real numbers in [A, B] with α 0α 1 ≤ ··· ≤ α n , α' 0α' 1 ≤ ··· ≤ α' n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k 0, k 1, ... , k n being nonnegative real numbers and α 0, α 1, ... , α n , β 0, β 1, ... , β n being real numbers in [A, B] with α 0α 1 ≤ ··· ≤ α n , Then   相似文献   

6.
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα n ≡0, then Bn (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q n (k) (α n ,f(t))=d k /dx k B n+k (α n ,F k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα n ≡0, k=1, then Qn (1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα n =0, k=2, then Qn (2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]: . As usual, for the space Lp [a,b](1≤p<∞), we have and L[a, b]=l1[a, b]. Letα n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials [3] [4]. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.  相似文献   

7.
Let U n be the unit polydisk in C n and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω 1, ..., ω n ), ω j S(1 ≤ jn) and fH(U n ). The function f is said to be in holomorphic Besov space B p (ω) if
$ \left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}} {{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty } $ \left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}} {{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty }   相似文献   

8.
The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial is known. The identical random coefficients aj(ω) are normally distributed defined on a probability space , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Qn(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Qn(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of .  相似文献   

9.
We introduce new potential type operators Jab = (E+(-D)b/2)-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with Jαβ. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).  相似文献   

10.
Let f∈C [−1,1] (r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn (f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x k } are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.  相似文献   

11.
In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \mathbbFn \mathbb{F}^n be the n dimensional linear space over the field \mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from \mathbbFn \mathbb{F}^n to itself {A i } iI such that for every linear subspace V ⊂ \mathbbFn \mathbb{F}^n of dimension dim(V)<n/2 we have
dim( ?i ? I Ai (V) ) \geqslant (1 + a) ·dim(V),\dim \left( {\sum\limits_{i \in I} {A_i (V)} } \right) \geqslant (1 + \alpha ) \cdot \dim (V),  相似文献   

12.
The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x| α u p−1, u > 0, xB R (0) ⊂ ℝ n (n ⩾ 3), u = 0, x ∈ ∂B R (0), where $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} from left side, α > 0.  相似文献   

13.
According to S. Bochner [6, 7]: IfD =B +i n is a tube domain in ℂ n , where B is a domain in ℝ n , and if [(B)\tilde]\tilde B is the convex envelope of B, then any holomorphic function on D extends to the tube domain [(D)\tilde] = [(B)\tilde] + i\mathbbRn \tilde D = \tilde B + i\mathbb{R}^n , which is a univalent envelope of holomorphy of D. We give a generalization of this result to (nonunivalent) tube domains over a complex Lie group which admit a closed sub-group as a real form. Application: If (V, φ) is a tube domain over ℂ n and if B is the convex envelope of ϕ(V)∩ℝ n in ℝ n , then [(V)\tilde] = B + i\mathbbRn \tilde V = B + i\mathbb{R}^n is an envelope of holomorphy of (V, φ).  相似文献   

14.
We consider a new Sobolev type function space called the space with multiweighted derivatives $ W_{p,\bar \alpha }^n $ W_{p,\bar \alpha }^n , where $ \bar \alpha $ \bar \alpha = (α 0, α 1,…, α n ), α i ∈ ℝ, i = 0, 1,…, n, and $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} ,
$ D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d} {{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n $ D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d} {{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n   相似文献   

15.
16.
Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.  相似文献   

17.
We investigate the correlation between the constants K(ℝn) and , where
is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line, , L l p, p (G n) is the set of functions ƒL p (G n ) such that the partial derivative belongs to L p (G n ), , 1 ≤ p ≤ ∞, l ∈ ℕn, α ∈ ℕ 0 n = (ℕ ∪ 〈0〉)n, D α f is the mixed derivative of a function ƒ, 0 < μi < 1, , and ∑ i=0 n . If G n = ℝ, then μ0=1−∑ i=0 n i /l i ), μi = αi/l i , if , then μ0=1−∑ i=0 n i /l i ) − ∑ i=0 n (λ/l i ), μi = αi/ l i + λ/l i , , λ ≥ 0. We prove that, for λ = 0, the equality is true. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 597–606, May, 2006.  相似文献   

18.
Let K2 be the Milnor functor and let Фn (x)∈ Q[X] be the n-th cyclotomic polynomial. Let Gn(Q) denote a subset consisting of elements of the form {a, Фn(a)}, where a ∈ Q^* and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn(Q) is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn(Q) is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h(Q(ζp)) is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.  相似文献   

19.
Carleman estimates for one-dimensional degenerate heat equations   总被引:1,自引:0,他引:1  
In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=xα (1 − x)β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=xα with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday  相似文献   

20.
IfE andF aren-dimensional Banach spaces, ifE has cotype 2, and if the ball ofF* has a small number of extreme points, then the Banach-Mazur distanced(E, F)Cnlogn. The techniques lead to the formally stronger result: IfE andF* have type 2 constantsa andb, respectively, thend(E, F)≦√n(a+b). IfE isn-dimensional, the identity map onE, when restricted to a large subspace ofE, factors through with normCn. The authors’ work was supported in part, respectively, by NSF grant numbers MCS 78-02194, MCS 79-02489, and MCS 77-04174.  相似文献   

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