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1.
A link between Ramsey numbers for stars and matchings and the Erd
s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e ( K2m−1)→{0, 1,…, m−1} is a mapping of the edges of the complete graph on 2 m−1 vertices into {0, 1,…, m−1}, then there exists a star K 1,m in K2m−1 with edges e 1, e 2,…, e m such that c( e1)+ c( e2)++ c( em)≡0 (mod m). Theorem 8. Let m be an integer. If c : e( Kr(r+1)m−1)→{0, 1,…, m−1} is a mapping of all the r-subsets of an ( r+1) m−1 element set S into {0, 1,…, m−1}, then there are m pairwise disjoint r-subsets Z1, Z2,…, Zm of S such that c( Z1)+ c( Z2)++ c( Zm)≡0 (mod m). 相似文献
2.
In this paper we define the vertex-cover polynomial Ψ( G,τ) for a graph G. The coefficient of τ r in this polynomial is the number of vertex covers V′ of G with | V′|= r. We develop a method to calculate Ψ( G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…, m} into the vertex set V( G) of a graph G, subject to f−1( x) f−1( y)≠ for every edge xy in G. Let F( G, m) be the number of such mappings f. We show that F( G, m) can be determined from Ψ( G,τ). 相似文献
3.
Let
d−1{( x1,…, xd)
d: x21+···+ x2d=1} be the unit sphere of the d-dimensional Euclidean space
d. For r>0, we denote by Brp (1 p∞) the class of functions f on
d−1 representable in the formwhere dσ( y) denotes the usual Lebesgue measure on
d−1,
and Pλk( t) is the ultraspherical polynomial.For 1 p,q∞, the Kolmogorov N-width of Brp in Lq(
d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces XN of Lq(
d−1).The main result in this paper is that for r2( d−1) 2,where ANBN means that there exists a positive constant C, independent of N, such that C−1ANBNCAN.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions. 相似文献
4.
The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W2(1). The target is either a subset of W2(1) a point in W2(1), a subset of the Euclidean space En or a point of En. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of Em and Ei respectively with q( t) = C( t) q′( t), − p( t) = B( t) p′( t), q′( t) ε Emp′( t) ε Er and B( t) is an n × r′-matrices and C( t) an n × m-matrix. 相似文献
5.
Let Xn, n
, be i.i.d. with mean 0, variance 1, and E(¦ Xn¦ r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√ n Σ i = 1n Xi t¦ B) can be approximated by a modified Edgeworth expansion up to order o(1/ n(r − 2)/2)), if the distances of the set B from the σ-fields σ( X1, …, Xn) are of order O(1/ n(r − 2)/2)(lg n) β), where β < −( r − 2)/2 for r
and β < − r/2 for r
. An example shows that if we replace β < −( r − 2)/2 by β = −( r − 2)/2 for r
(β < − r/2 by β = − r/2 for r
) we can only obtain the approximation order O(1/ n(r − 2)/2)) for r
( O(lg lg n/ n(r − 2)/2)) for r
). 相似文献
6.
Let ( X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ( X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = ( p − m)/(2 p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate
of T(θ), based on a set of i.i.d. observations ( X1, Y1), …, ( Xn, Yn), that achieves the optimal nonparametric rate of convergence n−r in Lq-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate
of T(θ) that achieves the optimal rate ( n/log n) −r in L∞-norm restricted to compacts. 相似文献
7.
Let D be a set of positive integers. The distance graph G( Z, D) with distance set D is the graph with vertex set Z in which two vertices x, y are adjacent if and only if | x− y| D. The fractional chromatic number, the chromatic number, and the circular chromatic number of G( Z, D) for various D have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form Dm,[k,k′]={1,2,…, m}−{ k, k+1,…, k′}, where m, k, and k′ are natural numbers with m≥ k′≥ k. In particular, we completely determine the chromatic number of G( Z, Dm,[2,k′]) for arbitrary m, and k′. 相似文献
8.
For all integers m3 and all natural numbers a1, a2,…, am−1, let n= R( a1, a2,…, am−1) represent the least integer such that for every 2-coloring of the set {1,2,…, n} there exists a monochromatic solution to | Let t=min{a1,a2,…,am−1} and b=a1+a2++am−1−t. In this paper it is shown that whenever t=2, R(a1,a2,…,am−1)=2b2+9b+8.
It is also shown that for all values of t, R(a1,a2,…,am−1)tb2+(2t2+1)b+t3.
相似文献
9.
Let
f ε
Cn+1[−1, 1] and let
H[
f](
x) be the
nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials
qk associated with a distribution
dα on [−1, 1]. It is shown that if
qn+1/
qn max(
qn+1(1)/
qn(1), −
qn+1(−1)/
qn(−1)), then
f −
H[
f]
fn + 1 ·
qn+1/
qn + 1(n + 1), where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 −
t)
α (1 +
t)
β dt, α, β > −1, the condition on
qn+1/
qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.
相似文献
10.
We introduce a new multidimensional pattern matching problem that is a natural generalization of string matching, a well studied problem[1]. The motivation for its algorithmic study is mainly theoretical. Let
A[1:
n1,…,1:
nd] be a text matrix with
N =
n1…
ndentries and
B[1:
m1,…,1:
mr] be a pattern matrix with
M =
m1…
mrentries, where
d ≥
r ≥ 1 (the matrix entries are taken from an ordered alphabet Σ). We study the problem of checking whether some
r-dimensional submatrix of
Ais equal to
B(i.e., a
decisionquery).
Acan be preprocessed and
Bis given on-line. We define a new data structure for preprocessing
Aand propose CRCW-PRAM algorithms that build it in
O(log
N) time with
N2/
nmaxprocessors, where
nmax = max(
n1,…,
nd), such that the decision query for
Btakes
O(
M) work and
O(log
M) time. By using known techniques, we would get the same preprocessing bounds but an
O((
dr)
M) work bound for the decision query. The latter bound is undesirable since it can depend exponentially on
d; our bound, in contrast, is independent of
dand optimal. We can also answer, in optimal work, two further types of queries: (a) an
enumerationquery retrieving all the
r-dimensional submatrices of
Athat are equal to
Band (b) an
occurrencequery retrieving only the distinct positions in
Athat correspond to all of these submatrices. As a byproduct, we also derive the first efficient sequential algorithms for the new problem.
相似文献
11.
The basic result of the paper states: Let
F1, …,
Fn,
F1′,…,
Fn′ have proportional hazard functions with λ
1 ,…, λ
n , λ
1′ ,…, λ
n′ as the constants of proportionality. Let
X(1) ≤ … ≤
X(n) (
X(1)′ ≤ … ≤
X(n)′) be the order statistics in a sample of size
n from the heterogeneous populations {
F1 ,…,
Fn}({
F1′ ,…,
Fn′}). Then (λ
1 ,…, λ
n) majorizes (λ
1′ ,…, λ
n′) implies that (
X(1) ,…,
X(n)) is stochastically larger than (
X(1)′ ,…,
X(n)′). Earlier results stochastically comparing
individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables.
相似文献
12.
Let
A = (
aij) be an
n × n Toeplitz matrix with bandwidth
k + 1,
K =
r +
s, that is,
aij =
aj−i,
i,
J = 1,… ,
n,
ai = 0 if
i >
s and if
i < -
r. We compute
p(λ)= det(
A - λ
I), as well as
p(λ)/
p′(λ), where
p′(λ) is the first derivative of
p(λ), by using
O(
k log
k log
n) arithmetic operations. Moreover, if
ai are
m × m matrices, so that
A is a banded Toeplitz block matrix, then we compute
p(λ), as well as
p(λ)/
p′(λ), by using
O(
m3k(
log2 k + log
n) +
m2k log
k log
n) arithmetic operations. The algorithms can be extended to the computation of det(
A − λB) and of its first derivative, where both
A and
B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix
A and of the generalized eigenvalue problem for
A and
B.
相似文献
13.
In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is
yt =
Bzt +
vt. The unobserved error sequence {
vt} is a sequence of martingale differences with conditional covariance matrices {Σ
t} and satisfying sup
t=1,…, n
{
v′tvtI(
v′tvt>
a) |
zt,
vt−1,
zt−1, …}
0 as
a → ∞. The sample covariance of the independent variables
z1, …,
zn, is assumed to have a probability limit
M, constant and nonsingular; max
t=1,…,nz′tzt/
n
0. If (1/
n)Σ
t=1
nΣ
t
Σ, constant, then √
nvec(
n−
B)
N(
0,M−1Σ) and
n
Σ. The autoregression model is
xt =
Bxt − 1 +
vt with the maximum absolute value of the characteristic roots of
B less than one, the above conditions on {
vt}, and (1/
n)Σ
t=max(r,s)+1(Σ
tvt−1−rv′t−1−s)
δ
rs(ΣΣ), where δ
rs is the Kronecker delta. Then √
nvec(
n−
B)
N(0,Γ
−1Σ), where Γ = Σ
s = 0∞BsΣ(
B′)
s.
相似文献
14.
Let
Lq (1
q<∞) be the space of functions
f measurable on
I=[−1,1] and integrable to the power
q, with norm
L∞ is the space of functions measurable on
I with normWe denote by
AC the set of all functions absolutely continuous on
I. For
nN,
q[1,∞] we set
Wn,q={
f:
f(n−1)AC,
f(n)Lq}.In this paper, we consider the problem of accuracy of constants
A,
B in the inequalities
(1)||
f(m)||
qA||
f||
p+
B||
f(m+k+1)||
r,
mN,
kW;
p,
q,
r[1,∞],
fWm+k+1,r.
相似文献
15.
For
X one observation on a
p-dimensional (
p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of
X (the best invariant procedure) are found with respect to general quadratic loss,
L(δ, θ) = (δ − θ)′
D(δ − θ) where
D is a known
p ×
p positive definite matrix. For
C a
p ×
p known positive definite matrix, conditions are given under which estimators of the form δ
a,r,C,D(
X) = (
I − (
ar(|
X|
2))
D−1/2CD1/2 |
X|
−2)
X are minimax with smaller risk than
X. For the problem of estimating the mean when
n observations
X1,
X2, …,
Xn are taken on a
p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(
X1,
X2, …,
Xn), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.
相似文献
16.
It is known that a geometry with rank
rand no minor isomorphic to the (
q+2)-point line has at most (
qr−1)/(
q−1) points, with strictly fewer points if
r>3 and
qis not a prime power. For
qnot a prime power and
r>3, we show that
qr−1−1 is an upper bound. For
qa prime power and
r>3, we show that any rank-
rgeometry with at least
qr−1points and no (
q+2)-point-line minor is representable over
GF(
q). We strengthen these bounds to
qr−1−(
qr−2−1)/(
q−1)−1 and
qr−1−(
qr−2−1)/(
q−1) respectively when
qis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor.
相似文献
17.
Let
W(2
n+1,
q),
n1, be the symplectic polar space of finite order
q and (projective) rank
n. We investigate the smallest cardinality of a set of points that meets every generator of
W(2
n+1,
q). For
q even, we show that this cardinality is
q
n+1+
q
{n–1, and we characterize all sets of this cardinality. For
q odd, better bounds are known.
相似文献
18.
We give a direct formulation of the invariant polynomials
μGq(n)(, Δ
i,;,
xi,i + 1,) characterizing
U(
n) tensor operators
p,
q, …,
q, 0, …, 0 in terms of the symmetric functions
Sλ known as Schur functions. To this end, we show after the change of variables Δ
i = γ
i − δ
i and
xi, i + 1 = δ
i − δ
i + 1 that
μGq(n)(,Δ
i;,
xi, i + 1,) becomes an integral linear combination of products of Schur functions
Sα(, γ
i,) ·
Sβ(, δ
i,) in the variables {γ
1,…, γ
n} and {δ
1,…, δ
n}, respectively. That is, we give a direct proof that
μGq(n)(,Δ
i,;,
xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ
1,…, γ
n} and {δ
1,…, δ
n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials
μmGq(n)(γ
1,…, γ
n; δ
1,…, δ
m). These new symmetries enable us to give an explicit formula for both
μmG1(n)(γ; δ) and
1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for
μmGq(n)(γ; δ).
相似文献
19.
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix
S:
p ×
p Wp(
n, Σ) and an independent normal random matrix
X:
p ×
k N(ξ, Σ
Ik) with ξ(
p ×
k) unknown. Denote the columns of
X by
X(1) ,…,
X(k) and set ψ
(0)(
S,
X) = {(
n −
p + 2)!/(
n + 2)!} |
S |, ψ
(i)(
X,
X) = min[ψ
(i−1)(
S,
X), {(
n −
p +
i + 2)!/(
n +
i + 2)!} |
S +
X(1) X′
(1) + +
X(i) X′
(i) |] and Ψ
(i)(
S,
X) = min[ψ
(0)(
S,
X), {(
n −
p +
i + 2)!/(
n +
i + 2)!}|
S +
X(1) X′
(1) + +
X(i) X′
(i) |],
i = 1,…,
k. Our result is that the minimax, best affine equivariant estimator ψ
(0)(
S,
X) is dominated by each of Ψ
(i)(
S,
X),
i = 1,…,
k and for every
i, ψ
(i)(
S,
X) is better than ψ
(i−1)(
S,
X). In particular, ψ
(k)(
S,
X) = min[{(
n −
p + 2)!/(
n + 2)!} |
S |, {(
n −
p + 2)!/(
n + 2)!} |
S +
X(1)X′
(1)|,…,| {(
n −
p +
k + 2)!/(
n +
k + 2)!} |
S +
X(1)X′
(1) + +
X(k)X′
(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (
Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.
相似文献
20.
Let
B1:
n×
N1→
m1,
B2:
n×
N2→
m2and
Q:
m2→
m1be bilinear forms which are related as follows: if
μand
νsatisfy
B1(
ξ,
μ)=0 and
B2(
ξ,
ν)=0 for some
ξ≠0, then
μτQν=0. Suppose
p−1+
q−1=1. Coifman, Lions, Meyer and Semmes proved that, if
uLp(
n) and
vLq(
n), and the first order systems
B1(
D,
u)=0,
B2(
D,
v)=0 hold, then
uτQvbelongs to the Hardy space
H1(
n), provided that both (i)
p=
q=2, and (ii) the ranks of the linear maps
Bj(
ξ, ·) :
Nj→
m1are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition when
p=
q=2 involves the use of a deep result of Lojasiewicz from singularity theory.
相似文献