共查询到20条相似文献,搜索用时 421 毫秒
1.
Alexander Koldobsky 《Israel Journal of Mathematics》2011,185(1):277-292
We say that a random vector X = (X
1, …, X
n
) in ℝ
n
is an n-dimensional version of a random variable Y if, for any a ∈ ℝ
n
, the random variables Σa
i
X
i
and γ(a)Y are identically distributed, where γ: ℝ
n
→ [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that
embeds in L
0. This result is almost optimal, as the norm of any finite-dimensional subspace of L
p
with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖
K
) is positive definite on ℝ
n
, where K is an origin symmetric star body in ℝ
n
and f: ℝ → ℝ is an even continuous function, then either the space (ℝ
n
, ‖·‖
K
) embeds in L
0 or f is a constant function. Combined with known facts about embedding in L
0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions. 相似文献
2.
For a convex body K ⊂ ℝn and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K
0 ⊂ ℝn be smooth convex bodies with boundaries of class C
2 and positive Gauss-Kronecker curvature and assume K
0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K
0 are homothetic if their i-th and j-th projection functions are proportional. When K
0 is a Euclidean ball this shows that a convex body with C
2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball.
The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953. 相似文献
3.
We prove that the Bohr radiusK
n of then-dimensional polydisc in ℂ
n
is up to an absolute constant ≥ √logn/log logn/n. This improves a result of Boas and Khavinson. 相似文献
4.
E.G. Coffman Jr. George S. Lueker Joel Spencer Peter M. Winkler 《Probability Theory and Related Fields》2001,120(4):585-599
A random rectangle is the product of two independent random intervals, each being the interval between two random points
drawn independently and uniformly from [0,1]. We prove that te number C
n
of items in a maximum cardinality disjoint subset of n random rectangles satisfies
where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constat K,
Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q
n
of items in a maximum cardinality disjoint subset of the cubes satisies
Received: 1 September 1999 / Revised version: 3 November 2000 / Published online: 14 June 2001 相似文献
5.
E. S. Dubtsov 《Journal of Mathematical Sciences》2010,166(1):23-30
Let B
n
denote the unit ball in
\mathbbC \mathbb{C}
n
, n ≥ 1. Let K \mathcal{K}
0(n) denote the class of functions defined for z ∈ B
n
as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈
\mathbbC \mathbb{C}
n
,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K}
0(n) for all f ∈ K \mathcal{K}
0(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K}
0(n), n ≥ 1. Bibliography: 15 titles. 相似文献
6.
We establish a strong regularity property for the distributions of the random sums Σ±λ
n
, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w
n+1...,w
n+ℓ) given the sum Σ
n=0
∞
w
n
λ
n
, tends to the uniform distribution on {±1}ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system
of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.
The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.). 相似文献
7.
R. Adamczak A. E. Litvak A. Pajor N. Tomczak-Jaegermann 《Constructive Approximation》2011,34(1):61-88
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by
vectors ±X
1,…,±X
N
∈ℝ
n
, (N≥n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming
probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential
tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices
are valid for the exact reconstruction process of m-sparse vectors via ℓ
1 minimization with m≤Cn/log 2(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors
with log-concave densities. We deduce that if K⊂ℝ
n
is a convex body and X
1,…,X
N
∈K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m∼n/log 2(cN/n). 相似文献
8.
On the Diaconis-Shahshahani Method in Random Matrix Theory 总被引:2,自引:0,他引:2
If Γ is a random variable with values in a compact matrix group K, then the traces Tr(Γj) (j ∊ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis
and Shahshahani computed the joint moments of any fixed number of these traces if Γ is distributed according to Haar measure
and if K is one of Un, On or Spn, where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer
algebras.
The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically,
the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to
a counting problem, which can be solved by elementary means. 相似文献
9.
LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS
n(b) of zeros of the polynomialx
n−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf
−1(S
n(b))=g
−1(S
n(b)), thenf
n=g
n. For everyn≥14, we show thatS
n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf
−1(S
n(b))=g
−1(S
n(b)), then eitherf
n=g
n, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively.
For everyn≥9, we show that the setY
n(c) of zeros of the polynomial
, (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY
n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders. 相似文献
10.
Wolfgang M. Ruppert 《manuscripta mathematica》1998,96(1):17-22
If K is a number field of degree n over Q with discriminant D
K
and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are
infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d
2 such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real
number field of prime degree n then one can find an integral generator α with .
Received: 10 January 1997 / Revised version: 13 January 1998 相似文献
11.
We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian
manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the
volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a2 and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)2/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only.
Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY
Research Award Program. 相似文献
12.
Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results
are obtained for subspaces ofl
∞
n
, while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition,
a new isometric characterization ofl
∞
n
is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern
1/2.
The research for this paper was begun while both authors were guests of the Mittag-Leffler Institute.
Supported in part by NSF-MCS 79-03042. 相似文献
13.
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
14.
We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number
of edge disjoint Hamilton cycles that the random graph G
n,p
contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G
n,p
without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.
Research supported in part by NSF grant CCR-0200945.
Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation. 相似文献
15.
Károly J. BöröczkyJr Lars Michael Hoffmann Daniel Hug 《Periodica Mathematica Hungarica》2008,57(2):143-164
Let K be a convex body in ℝ
d
, let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C
+2, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C
+3) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K.
Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953). 相似文献
16.
John E. Walsh Grace J. Kelleher 《Annals of the Institute of Statistical Mathematics》1973,25(1):87-90
The data (continuous) aren independent observations that are believed to be a random sample. The possibility exists, however, that as many asJ of the largest observations, and as many asK of the smallest observations, are outliers. That is, these observations are from populations that are different from the
population yielding the other observations (which number at leastn−J−K). The interest is in obtaining suitable estimates for the mean and variance of the population yielding the other observations.J andK are given and relatively small, with both ≦2n
A
, whereA is specified and ≦1/4. When the population yielding the other observations is continuous, has moments of all orders, and
is well-behaved in some other ways, estimates are developed that are unbiased if terms of ordern
−1+A
+2ɛ are neglected. Here, ɛ can be arbitrarily small but is positive.
Research partially supported by NASA Grant NGR 44-007-028 and by Mobil Research and Development Corporation. Also associated
with ONR Contract N00014-68-A-0515 and Dept. of Labor Grant 31-46-70-06. 相似文献
17.
We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S
2n+1 (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S
1-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4. 相似文献
18.
Greg Kuperberg 《Geometric And Functional Analysis》2008,18(3):870-892
We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on
the product v(K) = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4)
n
γ
n
, where γ
n
is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c
n
.
The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K
◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an
indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals
in the sphere S
n-1 and in hyperbolic space H
n-1.
Received: December 2006, Accepted: January 2007 相似文献
19.
Shiri Artstein-Avidan Omer Friedland Vitali Milman Sasha Sodin 《Israel Journal of Mathematics》2006,156(1):141-155
We present a quantitative form of the result of Bai and Yin from [2], and use to show that the section of ℓ
1
(1+δ)n
spanned byn random independent sign vectors is with high probability isomorphic to euclidean with isomorphism constant polynomial in
δ−1.
Partially supported by BSF grant 2002-006.
Supported by the National Science Foundation under agreement No. DMS-0111298.
Supported in part by the Israel Science Academy. 相似文献
20.
Jian Wang 《高校应用数学学报(英文版)》2008,23(3):345-350
A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that (1) m ≤ kn, (2) n ≤ km, (3) km-n = kn-m ≡ 0 (mod (k^2- 1)) and (4) λ(km-n)(kn-m) ≡ 0 (mod k(k- 1)(k^2 - 1)(m + n)). 相似文献