共查询到20条相似文献,搜索用时 31 毫秒
1.
Ignacy Kotlarski 《Annali di Matematica Pura ed Applicata》1966,74(1):129-134
Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x1, x2, …, xn be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals fk(t) = E[ei(t,x
k)], (k=1, 2, …, n): let y1=x1 + xn, y2=x2 + xn, …, yn−1=xn−1 + xn.
If the characteristic functional f(t1, t2, …, tn−1) of the random variables (y1, y2, …, yn−1) does not vanish, then the joint distribution of (y1, y2, …, yn−1) determines all the distributions of x1, x2, …, xn up to change of location. 相似文献
2.
A. Dubickas 《Journal of Mathematical Sciences》2006,137(2):4654-4657
Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence
x0 = g, xn = x
n−1
n
+ F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number,
is considered. We show, for instance, that the sequence x0 = 2, xn = x
n−1
n
+ 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except
for three prime numbers p = 23, 47, 167 that do not divide xn. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences [ζn!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 76–82. 相似文献
3.
Eugene Wesley 《Israel Journal of Mathematics》1973,14(1):104-114
Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions:
LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S
t
def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf
n: T→T such thatS
t={fn(t)|nεω} for alltε[0,1],W
x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW
x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable. 相似文献
4.
M. Ivette Gomes 《Annals of the Institute of Statistical Mathematics》1984,36(1):71-85
Summary Let {X
n}n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM
n=max (X
1,…,X
n), suitably normalized with attraction coefficients {αn}n≧1(αn>0) and {b
n}n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s
which better approximate the d.f. of(M
n−bn)/an thanG(x), for moderaten. We thus consider differences of the formF
n(anx+bn)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф
α(x)=exp (−x−α), x>0] or a type III [Ψ
α(x)= exp (−(−x)α), x<0] d.f. of largest values, much closer toF
n(anx+bn) than the ultimate itself. 相似文献
5.
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
6.
Let G=(I
n
,E) be the graph of the n-dimensional cube. Namely, I
n
={0,1}
n
and [x,y]∈E whenever ||x−y||1=1. For A⊆I
n
and x∈A define h
A
(x) =#{y∈I
n
A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I
n
of size 2
n−1
we have for a universal constant K independent of n. We prove a related lower bound for graphs: Let G=(V,E) be a graph with . Then , where d(x) is the degree of x. Equality occurs for the clique on k vertices.
Received: January 7, 2000
RID="*"
ID="*" Supported in part by BSF and by the Israeli academy of sciences 相似文献
7.
V. V. Kapustin 《Journal of Mathematical Sciences》2007,141(5):1538-1542
Let θ be an inner function, let K
θ
= H
2 ⊖ θH
2, and let Sθ : Kθ → Sθ be defined by the formula Sθf = Pθzf, where f ∈ Kθ is the orthogonal projection of H2 onto Kθ. Consider the set A of all trace class operators L : Kθ → Kθ, L = ∑(·,un)vn, ∑∥un∥∥vn∥ < ∞ (un, vn ∈ Kθ), such that ∑ūn vn ∈ H
0
1
. It is shown that trace class commutators of the form XSθ − SθX (where X is a bounded linear operator on Kθ) are dense in A in the trace class norm. Bibliography: 2 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61. 相似文献
8.
Z. Xu 《Acta Mathematica Hungarica》2010,127(4):301-319
Let f be a primitive positive integral binary quadratic form of discriminant −D, and r
f
(n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $
\sum\limits_{n \leqq x} {r_f (n)^\beta }
$
\sum\limits_{n \leqq x} {r_f (n)^\beta }
for any positive integer β. Next, we give an estimate of ∫1
T
|E(x)|2
x
−3/2
dx when β = 1. 相似文献
9.
R. J. Cook 《Proceedings Mathematical Sciences》1989,99(2):147-153
Letf(x)=θ1
x
1
k
+...+θ
s
x
s
k
be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ
1,…,θ
s
, are algebraic ands = 4k then there are integersx
1,…,x
s
, satisfying l ≤x
1,≤ N and ∥f(x)∥ ≤ N
E
, withE = − 1 + 2/e.
Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ
1,…,θ
s
, be algebraic then the result holds for almost all values of θεℝ
s
. Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate. 相似文献
10.
Li Banghe 《Commentarii Mathematici Helvetici》1982,57(1):135-144
Under the assumption of (f, M
n
,N
2n−1) being trivial, the classification of immersions homotopic tof: M
n
→N
2n−1 is obtained in many cases. The triviality of (f, M
n
,P
2n−1) is proved for anyM
n
andf.
LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N]
f
the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N]
f
is nonempty for anyf. In this paper we will determine the setI[M, N]
f
in some cases.
For example, ifN=P
2n−1 or more generally, the lens spacesS
m
2n−1
=Z
m
/S
2n−1,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N]
f is determined completely.
WhenN=R
2n−1, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R
2n−1 we obtain their results again, except for the casen≡2 mod 4 andM nonorientable.
Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R5] is known by Wu [4]). 相似文献
11.
A. K. Aleškevičienė 《Lithuanian Mathematical Journal》2006,46(2):129-145
Let X,X
1,X
2, … be independent identically distributed random variables, F(x) = P{X < x}, S
0 = 0, and S
n
=Σ
i=1
n
X
i
. We consider the random variables, ladder heights Z
+ and Z
− that are respectively the first positive sum and the first negative sum in the random walk {S
n
}, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z
+ and Z
− in the qualitatively different cases EX > 0, EX < 0, and EX = 0.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006. 相似文献
12.
J. Feldman 《Israel Journal of Mathematics》1993,81(3):281-287
Letμ be a probability measure on [0, 1), invariant underS:x ↦px mod 1, and for which almost every ergodic component has positive entropy. Ifq is a real number greater than 1 for which logq/ logp is irrational, andT
n sendsx toq
nx mod 1, then for any ε>0 the measureμT
n
−1
will — for a set ofn of positive lower density — be within ε of Lebesgue measure. 相似文献
13.
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
14.
StrongwayShi 《高校应用数学学报(英文版)》2000,15(1):45-54
Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u, 相似文献
15.
If dμ is the Fourier transform of a smooth measure dμ on the hypersphere Sn−1 (n≥2) then there exists a constant C dependent only on n such that ⋎dμ(y)⋎≤C(1+⋎y⋎)−(n−1)/2 for all y∈Rn. In this paper, we show that the above statement is false for non-smooth measures. And we present the corresponding estimations
for the Fourier transforms of certain non-smooth measures on Sn−1.
This research is supported by a grant of NSF of P. R. China. 相似文献
16.
We determine all square-free odd positive integers n such that the 2-Selmer groups S
n
and Ŝ
n
of the elliptic curve E
n
: y
2 = x(x − n)(x − 2n) and its dual curve ê
n
: y
2 = x
3 + 6nx
2 + n
2
x have the smallest size: S
n
= {1}, Ŝ
n
= {1, 2, n, 2n}. It is well known that for such integer n, the rank of group E
n
(ℚ) of the rational points on E
n
is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves E
n
with rank zero and such series of integers n are non-congruent numbers.
Dedicated to Professor Sheng GONG on the occasion of his 75th birthday 相似文献
17.
I. J. Schoenberg 《Israel Journal of Mathematics》1971,10(3):261-274
Letx
v
=cos (πν/n) (v=0, 1, …,n). It is shown that theB-splineM(x)=M(x; x
0
,x
1
,…, x
n
) is such thatM
n
(n)
(x) has a constant absolute value (=2
n−2 (n−1)!) in [−1, 1]. Its integralf
0(x)=∫
−1
x
M(t)dt is shown to have an optimal property that allows to solveexplicitly a certain time-optimal control problem. 相似文献
18.
Kazem Khashyarmanesh 《Proceedings Mathematical Sciences》2010,120(1):35-43
Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom
R
(−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x
1, …, x
n
is a regular sequence on M contained in α, then H
(x1, …,xnR
n
D(H
a
n
(M))) is a homomorphic image of D(M), where H
b
i
(−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H
(x1, …,xn)R
n
(D(H
a
n
(M)))) ⋟ D(D(M)). 相似文献
19.
Let {S
n
} be a random walk on ℤ
d
and let R
n
be the number of different points among 0, S
1,…, S
n
−1. We prove here that if d≥ 2, then ψ(x) := lim
n
→∞(−:1/n) logP{R
n
≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper.
We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ
d
let Λ
t
= Λ
t
(A) be the d-dimensional Lebesgue measure of the `sausage' ∪0≤
s
≤
t
(B(s) + A). Then φ(x) := lim
t→∞:
(−1/t) log P{Λ
t
≥tx exists for x≥ 0 and has similar properties as ψ.
Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001 相似文献
20.
Roger L. Jones 《Journal d'Analyse Mathématique》1993,61(1):29-45
LetU
1,U
2, …,U
n denoten commuting ergodic invertible measure preserving flows on a probability space (X,Σ,m). LetS
r denote the sphere of radiusr inR
n
, and αr the rotationally invariant unit measure onS
r. WriteU
tx to denote
x wheret=(t
1 …,tn). Define the ergodic averaging operator
. This paper shows that these averages converge for eachf ∈L
p(X), p>n/(n−1), n≥3. This is closely related to the work on differentiation by E. M. Stein, S. Wainger, and others. Because of their work,
the necessary maximal inequality transfers quite easily. The difficulty is to show that we have convergence on a dense subspace.
This is done with the aid of a maximal variational inequality.
Partially supported by NSF grant DMS-8910947. 相似文献