共查询到20条相似文献,搜索用时 31 毫秒
1.
Let f:?R??R be integrable in a neighbourhood of x??R. If there are real numbers ?? 0,?? 2,??,?? 2n?2 such that $$\lim_{s\to\infty}s^{2n+1} \int_0^\delta e^{-st}\left[\frac{f(x+t)+f(x-t)}{2}-\sum_{i=0}^{n-1}\frac{t^{2i}}{(2i)!}\alpha_{2i}\right]\, dt$$ exists for some ??>0 then the limit is called the 2n-th symmetric Laplace derivative at x. There is a corresponding definition of (2n+1)-th symmetric Laplace derivative. It is shown that this derivative is a generalization of the symmetric d.l.V.P. derivative. Some properties of this derivative are studied. 相似文献
2.
R. A. Kerman 《Analysis Mathematica》1988,14(1):3-9
Пустьk-мерное евклид ово пространствоR k рассматривается как подмножествоR n . Зафиксируемр, 1<р<∞ иα >(n?k)/p, α≠п. Как обычно, бесселев потенциалJαf обобщенной функции Шварцаf наR n определяется с помощ ью ее преобразования Фурь е \((\widehat{G_\alpha f})(\xi ) = (2\pi )^{ - n/2} [1 + |\xi |^2 ]^{\alpha /2} f(\xi ), \xi \in R^n .B\) , ξ∈R n . В работе характ еризуются положител ьные весовые функцииw(x 1,...,x k ), которые при продолжении наR n с помощью равенстваw(x 1,...,x k ,...,x n )=w(x 1, ...,x k ) обладают с ледующим свойством: существует числос>0, не зависящее отf, такое, что $$\begin{gathered} \int\limits_{R^k } {|(G_\alpha f)(x_1 ,...,x_k ,0,...,0)w(x_1 ,...,x_k )|^p dx_1 ...dx_k \leqq } \hfill \\ \leqq C\int\limits_{R^n } {|f(x_1 ,...,x_n )w(x_1 ,...,x_n )|^p dx_1 ...dx_n } \hfill \\ \end{gathered} $$ 相似文献
3.
LiuHuizhao JiangDaqing 《高校应用数学学报(英文版)》1999,14(1):1-6
By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation 相似文献
4.
Paul Feit 《Israel Journal of Mathematics》1990,69(3):289-320
Letf(X; T 1, ...,T n) be an irreducible polynomial overQ. LetB be the set ofb teZ n such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F j}{F j } j?1 ∞ be a Følner sequence inZ n — that is, a sequence of finite nonempty subsetsF j ?Z n such that for eachvteZ n , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP n ?A n and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ n is identified withA n (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ . 相似文献
5.
Björn G. Walther 《Arkiv f?r Matematik》1999,37(2):381-393
For Ξ∈R n ,t∈R andf∈S(R n ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys 0 such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R n ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL 2(R n ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity. 相似文献
6.
Let r ∈ N, α, t ∈ R, x ∈ R 2, f: R 2 → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R 2, R + 2 }, and ψ n ∈ L 1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R + such that A ? E, f ∈ L p (G), α ∈ [0, 2π] when G =R 2 and α ∈ [0, π/2] when G = R + 2 Denote Δ n, k = ∫ A t k ψ n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ m, r > 0, Δ m, r + 1 < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R
2, f: R
2 → C, and denote
In this paper, we investigate the relation between the behavior of the quantity
as n → ∞ (here, E ⊂ R, G ∈ {R
2, R
+2}, and ψ
n
∈ L
1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity:
Here is one of the results obtained.
Theorem 1. Let E and A be intervals in
R
+
such that A ⊂ E, f ∈ L
p
(G), α ∈ [0, 2π] when G =R
2
and α ∈ [0, π/2] when G = R
+2
Denote Δ
n, k
= ∫
A
t
k
ψ
n
(t)dt. If there exists an r ∈ N
such that, for any m ∈ N, we have Δ
m, r
> 0, Δ
m, r + 1 < ∞, and
then the relations
are equivalent. Particular methods of approximation are considered. We establish
Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and
Then the relations and are equivalent.
Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1.
Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33. 相似文献
7.
Shi Hongjian 《分析论及其应用》1989,5(1):87-96
Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 w p (lr) to L W P (lr) if and only if w ∈ AP(R); for 1≤p<∞, N is bounded from L W P (lr) to weak L W P (lr) if and only if W ∈ AP(R). Here we say W∈Ap (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ , 相似文献
8.
LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2 (x) are finite. Let {p n (W 2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW 2, and let {α n } 1 ∞ and {β n } 1 ∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec n 1 ∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR. 相似文献
9.
Jun Tateoka 《Analysis Mathematica》1994,20(2):133-145
Штрихарц [3] дал характе ристику пространствL p (R n ) бесселевых потенциа лов порядка α функций из п ространстваL p (R n ) с пом ощьюL p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF α p (P 0) (0<p<1, α>1/р?1) в терминах функц ионаловS r α f(x) (1τ≤2), которые в случаеR n } соответствуютD α f (x), гдеР 0 - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов. 相似文献
10.
K R PARTHASARATHY 《Proceedings Mathematical Sciences》2013,123(1):75-84
Let T be a bijective map on ? n such that both T and T ???1 are Borel measurable. For any θ?∈?? n and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? n with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ j ???θ 0, 1?≤?j?≤?n } is a basis for ? n . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε j ?>???1 for every j and {u j , 1?≤?j?≤?n } is a basis of unit vectors in ? n with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u j . Suppose N(0, I)T ???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 n diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E s } is a collection of 2 n Borel subsets of ? n such that { E s } and {V s U (E s )} are partitions of ? n modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404). 相似文献
11.
12.
K. Kopotun 《Constructive Approximation》1996,12(1):67-94
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ωυ(f(k),δ) denotes the usual vth modulus of smoothness off (k), and Moreover, for no 0≤k≤r can (1?x 2)( r?k+1)/(r?k+m)(1/n2)(m?1)/(r?k+m) be replaced by (1-x2)αkn2αk-2, with αk>(r-k+a)/(r-k+m). 相似文献
13.
Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ1 and Δ2 from ${R^n \to R}$ such that Δ1(δ 2(x), x, . . . ,x) = 0 for all ${x \in R,}$ where δ 2 is the trace of Δ2. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ 1, δ 2, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ 1 δ 2(x) = f(x) holds for all ${x \in R}$ , then R is commutative. 相似文献
14.
Biagio Ricceri 《Positivity》2012,16(3):455-470
In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y ?? R a C 1 function with locally Lipschitzian derivative, and ${\varphi : Y \to [0, + \infty[}$ a C 1 convex function with locally Lipschitzian derivative at 0 and ${\varphi^{-1}(0) = \{0\}}$ . Then, for each ${x_0 \in Y}$ for which J??(x 0)??? 0, there exists ???> 0 such that, for each ${r \in ]0, \delta[}$ , the restriction of J to B(x 0, r) has a unique global minimum u r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all ${x \in B(x_0, r)}$ , where ${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$ 相似文献
15.
Shu-Yu Hsu 《manuscripta mathematica》2013,140(3-4):441-460
Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u t = Δu m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u 0(x) ≈ A|x|?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g ij /?t = ?Rg ij with u(x, 0) = u 0(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation. 相似文献
16.
LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa n =a n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP n(x) of degree at mostn, the sup norm ofP n(x)W(a n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p n (x)W(a n x)} 1 ∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p n (x)W(a n x)} 1 ∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL p analogue for 0W(x)=exp(?|x| α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| p exp(?|x| α ),α > 0,p > ?1. 相似文献
17.
Iskander Aliev 《Discrete and Computational Geometry》2008,39(1-3):59-66
Let L(x)=a 1 x 1+a 2 x 2+???+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that there exist linearly independent vectors x 1,…,x n?1∈? n such that L(x i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a 1,a 2,…,a n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry. 相似文献
18.
W. Banaszczyk 《Discrete and Computational Geometry》1996,16(3):305-311
The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR n . The Minkowski functional? n ofU, the polar bodyU 0, the dual latticeL *, the covering radius μ(L, U), and the successive minima λ i ,i=1, …,n, are defined in the usual way. Let $\mathcal{L}_n $ be the family of all lattices inR n . Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andx∈R n /(L+U), somey∈L * with ∥y∥ U 0?sd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC 1,C 2,C 3 are some numerical constants andK(R U n ) is theK-convexity constant of the normed space (R n , ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU. 相似文献
19.
Yu. V. Netrusov 《Journal of Mathematical Sciences》1996,78(2):199-217
Definition
Let A??n, 0<β≤∞. Define $$h_{\varphi ,\beta } (A) = \inf \left( {\sum\limits_{i = 0}^{ + \infty } {\left( {m_j \varphi (2^{ - i} } \right)^\beta } } \right)^{1/\beta } $$ where the infinum is taken over all coverings of A by a countable number of balls, whose radii rj do not exceed 1, while mi is the number of balls from this covering whose radii rj belong to the set (2?i?1, 2?i], i∈N0.Theorem 1
Let p≤1, θ=∞, and let the function ?(t)tlp?n increase. Then the following conditions are 2quivalent;- for any compact set K, K??n, if $\overline {cap} (K, X) = 0$ , then h?,∞(K)=0;
Theorem 2
Let θ<1. Then for any set A the inequalities $c_1 \overline {cap} (A,X) \leqslant h_{t^{n - lp} ,\theta /p} (A) \leqslant c_2 \overline {cap} (A,X)$ hold. Bibliography:6 titles. 相似文献20.
Konrad Engel 《Combinatorica》1984,4(2-3):133-140
LetP be that partially ordered set whose elements are vectors x=(x 1, ...,x n ) withx i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx i =y i orx i =0 for alli. LetN i (P)={x εP : |{j:x j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN 1(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f 0, ...,f n ) for which there is an Erdös-Ko-Rado familyF inP such that |N i (P) ∩F|=f i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉. 相似文献