共查询到20条相似文献,搜索用时 31 毫秒
1.
Zhou Songping 《分析论及其应用》1989,5(1):11-14
In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f0(x)∈C
[−1,1]
2N
, for N+1≤k≤2N, such that p
n
(k)
(f0,1)→f
0
(k)
(1), n→∞, where pn(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture
is given. 相似文献
2.
Let and suppose that f : K
n
→K
n
is nonexpansive with respect to the l
1-norm, , and satisfies f (0) = 0. Let P
3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite
problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P
3(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear
maps, in particular to periodic points of maps g : D
g→D
g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism. 相似文献
3.
S. P. Zhou 《Israel Journal of Mathematics》1992,78(1):75-83
The present paper gives a converse result by showing that there exists a functionf ∈C
[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE
n
(0)
(f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E
n(f) is the ordinary best polynomial approximation off of degreen. 相似文献
4.
W. JabŁoŃski L. Reich 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2005,75(1):179-201
We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the
group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing
homomorphisms Ф from (G, +) into the differential group Г1∞ describing the chain rules of higher order of C∞ functions with fixed point 0.
In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L1∞),Ф = (fn
n≤1,forwhich f1 = l, f2 = ... = fp+l =0,fp+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w
n
p
)n≥p+2 of universal polynomials in fp+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w
n
p
)n≥p+2 we may also use another sequence (L
n
p
)n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions. 相似文献
5.
V. A. Ustimenko 《Journal of Mathematical Sciences》2007,140(3):461-471
The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K,
i.e., a family F of nonlinear polynomial maps f
α : K
n
→ K
n
depending on “time” α ∈ {K − 0} such that f
α
−1 = f
−αM, the relation f
α1 (x) = f
α2 (x) for some x ∈ Kn implies α1 = α2, and each map f
α has no invariant points. The neighborhood {f
α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ
a = f
α1 × ⋯ × f
αs
(υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative
ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops
to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric
cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs
without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234. 相似文献
6.
The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide. 相似文献
7.
For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T
*(J
(r,s,q)(Y, R
1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T
*(J
(r,s)
(Y, R)0)*→R for any Y as above. 相似文献
8.
We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM
f(r)lnL(r),r→∞, whereM
f
(r)=max {|f(z)|: |z|=r} andL(r)=∫
0
r
l(t)dt. Ifl(r)=r
p-1
forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM
f
(r)≈r
p
.
Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996. 相似文献
9.
M. Zippin 《Israel Journal of Mathematics》2000,115(1):253-268
A projectionP on a Banach spaceX with ‖P‖≤λ0 is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα2), whereD=D(λ0) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL
p(G) is almost locally minimal. Orthogonal projections on ℓ
1
n
are investigated with respect to some weaker local minimality properties.
Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported
by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). 相似文献
10.
E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2010,62(2):241-258
It is shown that if a point x
0 ∊ ℝ
n
, n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D →
[`(\mathbb Rn)] \overline {\mathbb {R}^n} , B
f
is the set of branch points of f in D; and a point z
0 ∊
[`(\mathbb Rn)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x
0; then, for any neighborhood U containing the point x
0; the point z
0 ∊ [`(f( Bf ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x
0 or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set
[`(\mathbb Rn)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x
0. For n ≥ 3, the following relation is true:
[`(\mathbbRn )] \f( D ) ì [`(f Bf )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ¥ ? f( D ) \infty \notin f\left( D \right) , then the set f
B
f
is infinite and x0 ? [`(Bf )] x_0 \in \overline {B_f } . 相似文献
11.
Boris Rubin 《Journal d'Analyse Mathématique》1999,77(1):105-128
Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S
n :x. y ≥ 0},x ∃S
n, whereS
n is thendimensional unit sphere in ℝn+1,n ≥ 2, and Μ is a finite Borel measure onS
n. If Μ is absolutely continuous with respect to Lebesgue measuredy onS
n, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫
x.y> 0
f(y)dy and consider the following cases: (a)f ∃C
∞(Sn); (b)f ∃ Lp(S
n), 1 ≤ p < ∞; and (c)f ∃C(Sn). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining
cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL
p
γ
(Sn) is studied. For zonalf ∃ L1(S
2), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.
Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation
(Germany). 相似文献
12.
Bao Yongguang 《分析论及其应用》1995,11(4):15-23
Let ξn −1 < ξn −2 < ξn − 2 < ... < ξ1 be the zeros of the the (n−1)-th Legendre polynomial Pn−1(x) and −1=xn<xn−1<...<x1=1, the zeros of the polynomial
. By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C
[−1,1]
1
, there exists a unique polynomial Rn(x) of degree 2n−2 (if n is even) satisfying conditions Rn(f, ξk) = f (εk) (1 ⩽ k ⩽ n −1); R1
n(f,xk)=f1(xk)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation
polynomial {Rn(f, x)} (n is even) and the main result of this paper is that if f∈C
[1,1]
r
, r≥2, n≥r+2, and n is even then |R1
n(f,x)−f1(x)|=0(1)|Wn(x)|h(x)·n3−r·E2n−r−3(f(r)) holds uniformly for all x∈[−1,1], where
. 相似文献
13.
Laurent-Padé (Chebyshev) rational approximantsP
m
(w, w
−1)/Q
n
(w, w
−1) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP
m
/Q
n
matches that of a given functionf(w, w
−1) up to terms of orderw
±(m+n)
, based only on knowledge of the Laurent series coefficients off up to terms inw
±(m+n)
. This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent
series ofP
m
matches that ofQ
n
f up to terms of orderw
±(m+n
), but based on knowledge of the series coefficients off up to terms inw
±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and
corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of
the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations
are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably
similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the
use of either. 相似文献
14.
For the mappings
f:D ? D¢, D, D¢ ì \mathbbRn f:D \to D',\,\,D,\,\,D' \subset {\mathbb{R}^n}
, n ≥ 2, satisfying certain geometric conditions in the fixed domain D, we have proved estimates of the form K
I
(x, f) ≤ Q(x) almost everywhere, where K
I
(x, f) is the inner dilatation of f at a point x, and Q(x) is a fixed real-valued function responsible for the “control” over a distortion of the families of curves in D at a mapping f. 相似文献
15.
Let r, k, s be three integers such that , or We prove the following:
Proposition.
Let Y:={y
i
}
i=1
s
be a fixed collection of distinct points y
i
∈ (-1,1) and Π (x):= (x-y
1
). ... .(x-y
s
). Let I:=[-1,1]. If f ∈ C
(r)
(I) and f'(x)Π(x) ≥ 0, x ∈ I, then for each integer n ≥ k+r-1 there is an algebraic polynomial P
n
=P
n
(x) of degree ≤ n such that P
n
'(x) Π (x) ≥ 0 and
$$ \vert f(x)-P_n(x) \vert \le B\left(\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)^r \omega_k \left(f^{(r)};\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)
\legno{(1)}$$
for all x∈ I, where ω
k
(f
(r)
;t) is the modulus of smoothness of the k -th order of the function f
(r)
and B is a constant depending only on r , k , and Y. If s=1, the constant B does not depend on Y except in the case
(r=1, k=3).
In addition it is shown that (1) does not hold for r=1, k>3.
March 20, 1995. Dates revised: March 11, 1996; December 20, 1996; and August 7, 1997. 相似文献
16.
A characteristic property of spheres 总被引:1,自引:1,他引:0
A. D. Alexandrov 《Annali di Matematica Pura ed Applicata》1962,58(1):303-315
Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that
. Then if φ(k1, ..., kn)=const on S and S is subject to some additional general conditions (those(II
0) or(II) no 1), S is a sphere.
To Enrico Bompiani on his scientific Jubilee 相似文献
17.
Extremal probabilities for Gaussian quadratic forms 总被引:1,自引:0,他引:1
Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf
Q
P(Q≤x)=P(χ2
n
/n≤x), where χ
n
2
is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z
1
,Z
2
are independent standard Gaussian rv's, then sup0≤λ≤1/2
P{λZ
1
2
+(1−λ)Z
2
2
≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic
quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.
Received: 24 June 2002 / Revised version: 26 January 2003
Published online: 15 April 2003
Research supported by NSA Grant MDA904-02-1-0091
Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10 相似文献
18.
We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r,n) such that every n-term graphic sequence π = (d1,d2,...,dn) with term sum σ(π) = d1 + d2 + ... + dn ≥ σ(Kr,r,n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. π has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined. 相似文献
19.
Hanno Lefmann 《Discrete and Computational Geometry》2008,40(3):401-413
We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1]
d
, such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ
k,d
(n), the supremum of this minimum volume over all distributions of n points in [0,1]
d
, we show that c
k,d
⋅(log n)1/(d−k+1)/n
k/(d−k+1)≤Δ
k,d
(n)≤c
k,d
′/n
k/d
for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ
k,d
(n)≤c
k,d
″/n
k/d+(k−1)/(2d(d−1)), where c
k,d
,c
k,d
′,c
k,d
″>0 are constants.
A preliminary version of this paper appeared in COCOON ’05. 相似文献
20.
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~... 相似文献