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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.
Let G be a graph of order n with connectivity κ≥3 and let α be the independence number of G. Set σ4(G)= min{∑4
i
=1
d(x
i
):{x
1,x
2,x
3,x
4} is an independent set of G}. In this paper, we will prove that if σ4(G)≥n+2κ, then there exists a longest cycle C of G such that V(G−C) is an independent set of G. Furthermore, if the minimum degree of G is at least α, then G is hamiltonian.
Received: July 31, 1998?Final version received: October 4, 2000 相似文献
3.
We establish necessary and sufficient conditions under which a sequence x
0 = y
0 , x
n+1 = Ax
n
+ y
n+1 , n ≥ 0, is bounded for each bounded sequence
{ yn :n \geqslant 0 } ì { x ? èn = 1¥ D( An ) |supn \geqslant 0 || An x || < ¥ }\left\{ {y_n :n \geqslant 0} \right\} \subset \left\{ {\left. {x \in \bigcup\nolimits_{n = 1}^\infty {D\left( {A^n } \right)} } \right|\sup _{n \geqslant 0} \left\| {A^n x} \right\| < \infty } \right\}, where A is a closed operator in a complex Banach space with domain of definition D(A) . 相似文献
4.
V. M. Prokip 《Ukrainian Mathematical Journal》2012,63(8):1314-1320
Polynomial n × n matrices A(x) and B(x) over a field
\mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over
\mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over
\mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A
0x
- A
1, where A
0 and A
1 are n × n matrices over
\mathbbF \mathbb{F} , and A
0 is nonsingular. 相似文献
5.
W?odzimierz B?k 《Potential Analysis》2010,32(1):17-27
A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C
n
:n = 1,2,..}, allows for constructing measures nxC, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M
f(x)=?g ? Gf(g)nCx(g). f(x)=\sum_{\gamma\in\Gamma}f(\gamma)\nu^{\mathbf{C}}_x(\gamma). 相似文献
6.
Let Ω ⊂ ℝ
d
be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae
of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal
Voronoi tessellation of Ω. In this connection we come across a class of operators of the form Ln[f](x): = ?i=1n fi(x)(f(yi) + á?f(yi), x-yi?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y1,..., yn\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ
1, ..., ϕ
n
} is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L
n
with a smallest constant in an L
p
error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable
and strictly convex function f. It allows us to describe a best operator L
n
for approximating f by L
n
[f] with respect to the L
p
norm. 相似文献
7.
Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11
There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x
1, x
2, ... we have
limn ? ¥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x
1, x
2, ... of μ-distributed random elements x
i
we have
lim infn ? ¥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1]
d
) is d. 相似文献
8.
A. V. Zheleznyak 《Vestnik St. Petersburg University: Mathematics》2009,42(4):269-274
In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f
−1(x) = $
\sum\limits_{n = 0}^\infty {b_n x^n }
$
\sum\limits_{n = 0}^\infty {b_n x^n }
b
n
x
n
be such that b
0 > 0 and b
n
≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x
1, x
2, …, x
m
) with positive coefficients in order that the series f
−1(x
1, x
2, …, x
m
) = $
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
$
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
satisfies the property b
0, …, 0 > 0, $
bi_1 ,i_2 , \ldots ,i_m
$
bi_1 ,i_2 , \ldots ,i_m
≤ 0, i
12 + i
22 + … + i
m
2 > 0, which is similar to the one-dimensional case. 相似文献
9.
Consider the standard non-linear regression model y
i
= g(x
i
, θ
0)+ε
i
, i = 1, ... ,n where g(x, θ) is a continuous function on a bounded closed region X × Θ, θ
0 is the unknown parameter vector in Θ ⊂ R
p
, {x
1, x
2, ... , x
n
} is a deterministic design of experiment and {ε1, ε2, ... , ε
n
} is a
sequence of independent random variables. This paper establishes the existences of M-estimates and the asymptotic uniform linearity of M-scores in a family of non-linear regression models when the errors are independent and identically distributed. This result
is then used to obtain the asymptotic distribution of a class of M-estimators for a large class of non-linear regression models. At the same time, we point out that Theorem 2 of Wang (1995)
(J. of Multivariate Analysis, vol. 54, pp. 227–238, Corrigenda. vol. 55, p. 350) is not correct.
This research was supported by the Natural Science Foundation of China (Grant No. 19831010 and grant No. 39930160) and the
Doctoral Foundation of China 相似文献
10.
Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑
i ∈ I
e
i
with the e
i
orthogonal idempotents; (b) e
i
x = xe
i
for all i ∈ I and x ∈ R; (c) e
i
A
e
j
≠ 0 for all i, j ∈ I; (d) e
i
A
A
≇ e
j
A
A
unless i = j; (e) every e
i
Ae
i
is a local ring whenever R is; (f) e
i
A
A
≅ Hom
R
(Ae
π(i),R
R
) and
A
Ae
π(i) ≅
A
Hom
R
(e
i
A,
R
R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e
i
) = e
π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map is a tilting complex.
Dedicated to Takeshi Sumioka on the occasion of his 60th birthday. 相似文献
11.
Alexander A. Sherstov 《Combinatorica》2011,31(5):583-614
We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e.,
f(x,y)=D(|x∧y|), where D: {0,1,…,n}→{0,1} is a given predicate and x,y range over {0,1}
n
. Specifically, we show that the communication complexity of f is between Θ(k/log5
n) and Θ(k logn), where k is the number of value changes of D in {0,1,…, n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001). 相似文献
12.
We say that X=[xij]i,j=1nX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x
ij
= x
ji
and x
n − j + 1,n − i + 1, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA
T
− B|| where ||·|| is the Frobenius norm, A ∈ ℝ
m×n
, B ∈ ℝ
m×m
and X ∈ ℝ
n×n
is symmetric centrosymmetric with a specified central submatrix [x
ij
]
p ≤ i,j ≤ n − p
. Our algorithm produces a suitable X such that AXA
T
= B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical
experiments that support this claim. 相似文献
13.
We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace
K
Θ=H
2⊖ΘH
2 of the Hardy space H
2, where Θ is an inner function. First, we verify the Feichtinger conjecture for the kernels [(k)\tilde]ln=kln/||kln||\tilde{k}_{\lambda_{n}}=k_{\lambda_{n}}/\|k_{\lambda _{n}}\| under the assumption that sup
n
|Θ(λ
n
)|<1. Second, we prove the Feichtinger conjecture in the case where Θ is a one-component inner function, meaning that the
set {z:|Θ(z)|<ε} is connected for some ε∈(0,1). 相似文献
14.
We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables
ξ
1, … , ξ
n
and a vector of scalars x = (x
1, … , x
n
), and 1 ≤ k ≤ n, we provide estimates for
\mathbb E k-min1 £ i £ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and
\mathbb E k-max1 £ i £ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y
x
= (1/x
1, … , 1/x
n
). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ
1|,
G(t) = \mathbb P ({ |x1| £ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ
1 is the standard N(0, 1) Gaussian random variable, then
G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds } and
M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence. 相似文献
15.
In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed
the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine
exponent w
0(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of
\mathbbRn {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any
“inhomogeneous” vector θ ∈
\mathbbRn {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w
0(x
,
θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w
0(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈
\mathbbRn {\mathbb{R}^n} , the inhomogeneous exponent w
0(x
,
θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered
by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront
the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference
principle in all its glory. 相似文献
16.
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition
function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by
the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct
−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
RID="*"
ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
RID="**"
ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573 相似文献
17.
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
18.
Let F ì PG \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of PG {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g
n
)
n∈ω
of nonzero elements of G, there is n ∈ ω such that
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