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1.
Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer
n and any x
1,…,x
n
∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x
i
−x
j
‖≤‖L(x
i
)−L(x
j
)‖≤O(1)⋅‖x
i
−x
j
‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion
22O(log*n)2^{2^{O(\log^{*}n)}}
. On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E
n
⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function. 相似文献
2.
Guizhen LIU 《Frontiers of Mathematics in China》2009,4(2):311-323
Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g
−, g
+) and ƒ = (ƒ
−, ƒ
+) be pairs of positive integer-valued functions defined on V(G) such that g
−(x) ⩽ ƒ
−(x) and g
+(x) ⩽ ƒ
+(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g
−(x) ⩽ id
H
(x) ⩽ ƒ
−(x) and g
+(x) ⩽ od
H
(x) ⩽ ƒ
+(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let
= {F
1, F
2,…, F
m} and H be a factorization and a subdigraph of G, respectively.
is called k-orthogonal to H if each F
i
, 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g
−(x), g
+(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.
相似文献
3.
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
4.
A polynomial Q = Q(X
1, …, X
n
) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α
1(F), …, α
m
(F)), where q(u
1, …, u
m
) is a polynomial in u
1, …, u
m
and α
j
(F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X
1 + … + X
n) = 0 holds if and only if q(α
1(θ), …, α
m
(θ)) ≡ 0, where α
j
(θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X
1 + … + X
n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F
θ(x) = ∫−∞
x
e
θu−k(θ)
dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for
natural exponential families though, at the first glance, the latter seem unrelated to the former. 相似文献
5.
Yossi Moshe 《Journal d'Analyse Mathématique》2006,99(1):267-294
Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X
i)
i
0/∞
over ℂ. Assume that theX
i's are chosen from a finite set {D
0,D
1...,D
t-1(ℂ), withP(X
i=Dj)>0, and that the monoid generated byD
0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case
where theX
i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22].
Our results on λ enable us to provide an approximation for the numberN
≠0(F(x)n,r) of nonzero coefficients inF(x)
n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN
≠0(F(x)n,r) ≈n
α for “almost” everyn.
Supported in part by FWF Project P16004-N05 相似文献
6.
V. V. Makeev 《Journal of Mathematical Sciences》2011,175(5):572-573
Let X be an affine cross-polytope, i.e., the convex hull of n segments A
1
B
1,…, A
n
B
n
in
\mathbbRn {\mathbb{R}^n} that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L
1 ⊂⋯ ⊂ L
n
=
\mathbbRn {\mathbb{R}^n} , where L
k
is the k-dimensional affine hull of the segments A
1
B
1,…, A
k
B
k
, k ≤ n. It is proved that each convex body K ⊂
\mathbbRn {\mathbb{R}^n} is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A
k
and B
k
to the body L
k
∩ K in the k-plane L
k
are parallel to L
k
−1.Each such X has volume at least V(K)/2
n(n−1)/2. Bibliography: 5 titles. 相似文献
7.
Wei Cao 《Czechoslovak Mathematical Journal》2007,57(1):253-268
A set S={x
1,...,x
n
} of n distinct positive integers is said to be gcd-closed if (x
i
, x
j
) ∈ S for all 1 ⩽ i, j ⩽ n. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n ⩽ k(t), then the power LCM matrix ([x
i
, x
j
]
t
) defined on any gcd-closed set S={x
1,...,x
n
} is nonsingular, but for n ⩾ k(t) + 1, there exists a gcd-closed set S={x
1,...,x
n
} such that the power LCM matrix ([x
i
, x
j
]
t
) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed
set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2. 相似文献
8.
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(
n
n+x
). Here we prove that the order x Veronese embedding ofP
n
is not weakly (k−1)-defective, i.e. for a general S⊃P
n
such that #(S) = k+1 the projective space | I
2S
(x)| of all degree t hypersurfaces ofP
n
singular at each point of S has dimension (
n
/n+x
)−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I
2S
(x)| has an ordinary double point at each P∈ S and Sing (F)=S.
The author was partially supported by MIUR and GNSAGA of INdAM (Italy). 相似文献
9.
N. G. Aharonyan H. S. Harutyunyan V. K. Ohanyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(6):348-356
Under the assumption that S is a segment of the length l and D is a bounded, convex domain in the Euclidean plane ℝ2, the paper considers the randomly moving copy L of S, under the condition that it hits D. Denote by L| the length of L ∩ D. In the paper an elementary expression for the distribution function F
L
(x) of the random variable L| is obtained. Note that F
L
(x) can have a jump at the point l or can be a continuous function depending on l and the domainD. In particular, a relation between chord length distribution functions of D and F
L
(x) is given. Moreover, we derive explicit forms of F
L
(x) for the disk and regular n-gons with n = 3÷7. 相似文献
10.
A. I. Perov 《Functional Analysis and Its Applications》2010,44(1):69-72
Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R
n
, where K is the cone of nonnegative vectors in R
n
. A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x
k
= Fx
k-1, k = 1, 2,..., starting from an arbitrary point x
0 in M, and the following error estimates hold: ρ (x*, x
k
) ⩽ Q
k
(I - Q)-1ρ(x
1, x
0) ⩽ (I - Q)-1
Q
k
ρ(x
1, x
0), k = 1, 2,.... Generally the mappings (I - Q)-1 and Q
k
do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle. 相似文献
11.
Jan-Ove Larsson 《Israel Journal of Mathematics》1986,55(2):153-161
Isomorphic embeddings ofl
l
m
intol
∞
n
are studied, and ford(n, k)=inf{‖T ‖ ‖T
−1 ‖;T varies over all isomorphic embeddings ofl
1
[klog2n]
intol
∞
n
we have that lim
n→∞
d(n, k)=γ(k)−1,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k
−1ln4.
Here [x] denotes the integer part of the real numberx. 相似文献
12.
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
13.
For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT
α=Σ
j=1
∞
a
j
T
j
, where {a
j
} are the coefficients in the power series expansion (1-t)α=1-Σ
j=1
∞
a
j
t
j
in the open unit disk, which satisfya
j
>0 anda
j
>0 and Σ
j=1
∞
a
j
=1. The operator calculus justifies the notation(I−T)
α
:=I−T
α
(e.g., (I−T
1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for
T if there is anx∈X such that(I−T)
α
x=y, i.e.,y is a coboundary forT
α
. The fractional Poisson equation forT is the Poisson equation forT
α
. We show that if(I−T)X is not closed, then(I−T)
α
X strictly contains(I−T)X (but has the same closure).
ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ
k=1
∞
T
k
y/k
1-α converges in norm, and conclude that lim
n
‖(1/n
1-α)Σ
k=1
n
T
k
y‖=0 for suchy.
For a Dunford-Schwartz operatorT onL
1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T)
α
L
1 for some 0<α<1, then the one-sided Hilbert transform Σ
k=1
∞
T
k
f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T)
α
L
p
with α>1−1/p=1/q, then Σ
k=1
∞
T
k
f/k
1/p
converges a.e., and thus (1/n
1/p
) Σ
k=1
n
T
k
f converges a.e. to zero. Whenf∈(I−T)
1/q
L
p
(the case α=1/q), we prove that (1/n
1/p
(logn)1/q
)Σ
k=1
n
T
k
f converges a.e. to zero. 相似文献
14.
Y. Lacroix 《Israel Journal of Mathematics》2002,132(1):253-263
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ
0
∞
G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ
U
(x)=inf{k⩾1:T
k
xεU}, and defineG
U
(t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U
n
)
n≥1
of neighbourhoods ofx such that {x}=∩
n
U
n
, and for anyG ∈G, there exists a subsequence (n
k
)
k≥1
withG
U
n
k
↑U weakly.
We also construct a uniquely ergodic Toeplitz flowO(x
∞,S, μ), the orbit closure of a Toeplitz sequencex
∞, such that the above conclusion still holds, with moreover the requirement that eachU
n
be a cylinder set.
In memory of Anzelm Iwanik 相似文献
15.
J. Sunklodas 《Lithuanian Mathematical Journal》2005,45(4):475-486
We derive a lower bound of L
p
norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X
1,..., X
n
with
under the boundedness condition ℙ{|X
i
| ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑
r⩾1
r
2α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005. 相似文献
16.
17.
18.
D. R. Heath-Brown 《Proceedings Mathematical Sciences》1994,104(1):13-29
LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ
n
;F(x)=0, |x|⩽X}, where
. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X
n−2+2/n
for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields.
However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.
Dedicated to the memory of Professor K G Ramanathan 相似文献
19.
Andrés del Junco 《Israel Journal of Mathematics》1983,44(2):160-188
LetT
α be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=T
α(x−1) forx∈[1,3/2) andTx = T
α
x forx∈[1/2, 1), i.e.T isT
α with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {p
n
/q
n
} with |α−p
n
/q
n
|=o(q
n
−2)) and λ is a measure onX
k
all of whose one-dimensional marginals are Lebesgue and which is ⊗
i − 1
k
T
1 invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction.
A connection is established with the POD (proximal orbit dense) condition in topological dynamics.
Research supported in part by NSF contract MCS-8003038. 相似文献
20.
Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F
1,F
2,⃛,F
1| ofG, if |E(H)∩E(F
1)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G).
Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences. 相似文献