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1.
A 0–1 probability space is a probability space (, 2 , P), where the sample space -{0, 1}
n
for some n. A probability space is k-wise independent if, when Y
i
is defined to be the ith coordinate or the random n-vector, then any subset of k of the Y
i
's is (mutually) independent, and it is said to be a probability space for p
1, p
2, ..., p
n
if P[ Y
i
=1]= p
i
.We study constructions of k-wise independent 0–1 probability spaces in which the p
i
's are arbitrary. It was known that for any p
1, p
2, ..., p
n
, a k-wise independent probability space of size
always exists. We prove that for some p
1, p
2, ..., p
n
[0,1], m(n,k) is a lower bound on the size of any k-wise independent 0–1 probability space. For each fixed k, we prove that every k-wise independent 0–1 probability space when each p
i
= k/n has size ( n
k
). For a very large degree of independence — k=[ n], for >1/2- and all p
i
=1/2, we prove a lower bound on the size of
. We also give explicit constructions of k-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology. 相似文献
2.
Let c( n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2 n ) edges. Let x = q/n and k = q ? n. We determine functions wk ? 1. a( x) and φ( x) such that c( n, q) ? wk( qN) enφ( x)+ a( x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4 q > (1 + ?) n log n, this formula simplifies to c( n, q) ? ( Nq) exp(– ne?2q/n). on the other hand, if k = o( n1/2), this formula simplifies to c( n, n + k) ? 1/2 wk (3/π) 1/2 ( e/12 k) k/2nn? (3k?1)/2. 相似文献
3.
Our topic is the uniform approximation of x
k
by polynomials of degree n (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann
2 and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p
k,n
, which can be identified with a certain probability. The numberp
k,n
is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn. 相似文献
4.
Let X be a Banach space with closed unit ball B. Given k
, X is said to be k-β, respectively, ( k + 1)-nearly uniformly convex (( k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence ( xn) B there are indices ( ni) ki = 1, respectively, ( ni) k + 1i = 1, such that (1/( k + 1))|| x + ∑ ki = 1 xni|| ≤ 1 − δ, respectively, (1/( k + 1))||∑ k + 1i = 1 xni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β. 相似文献
5.
Consider the hypercube [0, 1] n in Rn. This has 2 n vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let Vn be its expected volume. How large should N(n) be to pick up significant volume? Let k=2/√≈1.213, and let ? > 0. We shall show that, as n→∞, Vn→0 if N(n)?(k??) n →1 if N(n) ? ( k + ?) n. A similar result holds for sampling uniformly from within the hypercube, with constant . 相似文献
6.
It is shown that the “hit-and-run” algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O
*( n
2
R
2/ r
2), where R and r are the radii of the inscribed and circumscribed balls of K. Thus after appropriate preprocessing, hit-and-run produces an approximately uniformly distributed sample point in time O
*( n
3), which matches the best known bound for other sampling algorithms. We show that the bound is best possible in terms of R, r and n.
Received January 26, 1998 / Revised version received October 26, 1998?Published online July 19, 1999 相似文献
7.
Let f( x), x ∈ ? M, M ≥ 1, be a density function on ? M, and X1, …., Xn a sample of independent random vectors with this common density. For a rectangle B in ? M, suppose that the X's are censored outside B, that is, the value Xk is observed only if Xk ∈ B. The restriction of f( x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {( u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let Km, n( x) be the classical kernel estimator of f( x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E| Km, n( x) ? f( x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f( x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc. 相似文献
8.
Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces: Theorem. — Let x 1, x 2, …, x n be n (n≥3) independent random variables in the Hilbert space H, having their characteristic functionals f k(t) = E[e i(t,x
k)], (k=1, 2, …, n): let y 1=x 1 + x n, y 2=x 2 + x n, …, y n−1=x n−1 + x n.
If the characteristic functional f(t 1, t 2, …, t n−1) of the random variables (y 1, y 2, …, y n−1) does not vanish, then the joint distribution of (y 1, y 2, …, y n−1) determines all the distributions of x 1, x 2, …, x n up to change of location. 相似文献
9.
In this note we are interested in the graded modules M
k=I (k)/I k and
, where I is a saturated ideal in the homogeneous coordinate ring S=K[x 0,…,x n] of ℙ n, I
(k) is the symbolic power and
is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters,
we compute the Hilbert functions and we study some characteristic submodules in the special case of n+1 general points in ℙ n.
Sunto In questa nota siamo interessati ai moduli graduatiM
k=I(k)/Ik e
, doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x0,…,xn] di ℙn,I
(k) è la potenza simbolica e
è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri.
Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙn.
相似文献
10.
If μ is a positive measure, and A
2, ..., A
n
are measurable sets, the sequences S
0, ..., S
n
and P
[0], ..., P
[n] are related by the inclusion-exclusion equalities. Inequalities among the S
i
are based on the obvious P
[k]≧0. Letting
=the average average measure of the intersection of k of the sets A
i
, it is shown that (−1)
k
Δ
k
M
i
≧0 for i+ k≦ n. The case k=1 yields Fréchet’s inequalities, and k=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involving k-th order divided differences. Using convexity arguments, it is shown that for S
0=1,
when S
1≧ N−1, and
for 1≦ k< N≦ n and v=0, 1, .... Asymptotic results as n → ∞ are obtained. In particular it is shown that for fixed N,
for all sequences M
0, ..., M
n
of sufficiently large length if and only if
for 0< t<1. 相似文献
11.
Let k and n be positive integers such that k n. Let Sn ( F) denote the space of all n× n symmetric matrices over the field F with char F≠2. A subspace L of Sn ( F) is said to be a k-subspace if rank A k for every A? L. Now suppose that k is even, and write k=2 r. We say a k∥-subspace of Sn ( F) is decomposable if there exists in Fn a subspace W of dimension n? r such that xtAx=0 for every x? W A? L. We show here, under some mild assumptions on k n and F, that every k∥-subspace of Sn ( F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of Fm,n . 相似文献
12.
We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1} n, and k parties, who have noisy version of the source bits yi ∈ {0, 1} n, when for all i and j, it holds that P [ y = xj] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions fi: {0, 1} n → {0, 1} such that P [ f1( y1) = … = fk( yk)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function fi may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by fi( yi) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 相似文献
13.
本文给出Hermite-Fejér插值的若干收敛准则.其中之一是:Hermite-Fejer插值算子对每一个连续函数一致收敛当且仅当该算子范数一致有界且该算子对两个单项式x及x 2一致收敛. 相似文献
14.
We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter wi, where wi denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n(τ‐2)/(τ‐1). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013 相似文献
15.
Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC 0, that is, a pseudorandom generator that maps n‐bit strings to m‐bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4 n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC 0 generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC 0 generator can sample an ε‐biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of k, we construct generators that map n bits to bits such that every XOR of outputs has bias . We also present a polynomial‐time distinguisher for k = 4, m ≥ 24 n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2 kn?k/2?). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2 ?Ω(n) and which maps n bits to Ω( n2) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 相似文献
16.
Let { S
k
, k ≥ 0} be a symmetric random walk on , and an independent random field of centered i.i.d. random variables with tail decay . We consider a random walk in random scenery, that is . We present asymptotics for the probability, over both randomness, that { X
n
> n
β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local
times process , where l
n
( x) is the number of visits of site x up to time n.
相似文献
17.
Let σ be a nontrivial permutation of order n. A semigroup S is said to be σ- permutable if x
1
x
2
...x
n
=x
σ(1)
x
σ(2)
...x
σ(n)
, for every ( x
1
,x
2, ...,x
n
)∈ S
n
. A semigroup S is called (r,t)-commutative, where r,t are in ℕ *, if x
1
...x
r
x
r+1
...x
r+t
=x
r+1
...x
r+t
x
1
...x
r
, for every ( x
1
,x
2,..., x
r+t
∈ S
r+t
. According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation and S is a σ-permutable semigroup then there exists k ∈ ℕ * such that S is (1, k)-commutative. In [8], S. Lajos raises up the problem to determine the least k=k(n) ∈ ℕ * such that, for every fixed-point-free permutation σ of order n, every σ-permutable semigroup is also (1, k)-commutative. In this paper this problem is solved for any n less than or equal to eight and also when n is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of order n then inevitably it also satisfies some permutation identities of order n+1. 相似文献
18.
For a family $\boldmath{F}(k) = \{{\mathcal F}_1^{(k)}, {\mathcal F}_2^{(k)},\ldots,{\mathcal F}_t^{(k)}\}$ of k‐uniform hypergraphs let ex ( n, F ( k)) denote the maximum number of k‐tuples which a k‐uniform hypergraph on n vertices may have, while not containing any member of F ( k). Let rk( n) denote the maximum cardinality of a set of integers Z?[ n], where Z contains no arithmetic progression of length k. For any k≥3 we introduce families $\boldmath{F}(k) = \{{\mathcal F}_1^{(k)},{\mathcal F}_2^{(k)}\}$ and prove that nk?2rk( n)≤ex ( nk2, F ( k))≤ cknk?1 holds. We conjecture that ex( n, F ( k))= o( nk?1) holds. If true, this would imply a celebrated result of Szemerédi stating that rk( n)= o( n). By an earlier result o Ruzsa and Szemerédi, our conjecture is known to be true for k=3. The main objective of this article is to verify the conjecture for k=4. We also consider some related problems. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 131–164, 2002. 相似文献
19.
We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction ( O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 相似文献
20.
Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of the Erdős matching conjecture: For positive integers n, k and m with n ⩾ km, if each of the families \(F_{1},\ldots,F_{m}\subseteq\left(\begin{array}{c}[n]\\ k\end{array}\right)\) has size more than \(\max\{\left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-m+1\\ k\end{array}\right),\left(\begin{array}{c}km-1\\ k\end{array}\right)\}\), then there exist pairwise disjoint subsets e1,…,em such that ei ∈ Fi for all i ∈ [m]. We prove that there exists an absolute constant n0 such that this rainbow version holds for k = 3 and n ⩾ n0. We convert this rainbow matching problem to a matching problem on a special hypergraph H. We then combine several existing techniques on matchings in uniform hypergraphs: Find an absorbing matching M in H; use a randomization process of Alon et al. (2012) to find an almost regular subgraph of H − V(M); find an almost perfect matching in H − V(M). To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of Łuczak and Mieczkowska (2014) and might be of independent interest. 相似文献
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