共查询到20条相似文献,搜索用时 31 毫秒
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.
Starting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=| t| eiα, −π<α<π, t≠0, t≠1.. Let S1( x: t) be the cardinal linear spline such that S1( v: t) = tv for all v ε Z. Starting from S1( x: t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation . Here Sn( x: t) is a cardinal spline of degree n if n is odd, while
is a cardinal spline if n is even. It is shown that they have the properties Sn( v: t) = tv for v ε Z. 相似文献
3.
For every integer m ≥ 3 and every integer c, let r( m, c) be the least integer, if it exists, such that for every 2-coloring of the set {1, 2, …, r( m, c)} there exists a monochromatic solution to the equation
The values of r( m, c) were previously known for all values of m and all nonnegative values of c. In this paper, exact values of r( m, c) are found for all values of m and all values of c such that − m + 2 < c < 0 or c < − ( m − 1)( m − 2). Upper and lower bounds are given for the remaining values of c. 相似文献
4.
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
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.
Orthonormal polynomials with weight ¦τ¦
exp(−τ
4) have leading coefficients with recurrence properties which motivate the more general equations ξ
m(ξ
m − 1 + ξ
m + ξ
m + 1) = γ
m2,
M = 1, 2,…, where ξ
o is a fixed nonnegative value and γ
1, γ
2,… are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given.
相似文献
7.
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
8.
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
9.
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
10.
It is known that if
fWkp, then
ωm(
f,
t)
ptωm−1(
f′,
t)
p…. Its inverse with any constants independent of
fis not true in general. Hu and Yu proved that the inverse holds true for splines
Swith equally spaced knots, thus
ωm(
S,
t)
ptωm−1(
S′,
t)
pt2ωm−2(
S″,
t)
p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.
相似文献
11.
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
12.
The problem of approximate parameterized string searching consists of finding, for a given text
t=
t1t2…
tn and pattern
p=
p1p2…
pm over respective alphabets
Σt and
Σp, the injection
πi from
Σp to
Σt maximizing the number of matches between
πi(
p) and
titi+1…
ti+m−1 (
i=1,2,…,
n−
m+1). We examine the special case where both strings are run-length encoded, and further restrict to the case where one of the alphabets is binary. For this case, we give a construction working in time O(
n+(
rp×
rt)
α(
rt)log(
rt)), where
rp and
rt denote the number of runs in the corresponding encodings for
y and
x, respectively, and
α is the inverse of the Ackermann's function.
相似文献
13.
We prove that for
f ε
E =
C(
G) or
Lp(
G), 1
p < ∞, where
G is any compact connected Lie group, and for
n 1, there is a trigonometric polynomial
tn on
G of degree
n so that
f −
tnE Crω
r(
n−1,
f). Here ω
r(
t,
f) denotes the
rth modulus of continuity of
f. Using this and sharp estimates of the Lebesgue constants recently obtained by Giulini and Travaglini, we obtain “best possible” criteria for the norm convergence of the Fourier series of
f.
相似文献
14.
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let
X = (
X1,
X2, …,
Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ
1, Θ
2, …, Θ
n). Let θ
x* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ
x* given
X =
x to converge to a multivariate normal with mean vector 0 as |
x| tends to infinity. These same conditions imply that
E(Θ |
X =
x) − θ
x* converges to the zero vector as |
x| tends to infinity.The posterior for an observation
X = (
X1,
X2, …,
Xn is considered for a location vector Θ = (Θ
1, Θ
2, …, Θ
n) as
x gets large along a path, γ, in
Rn. Sufficient conditions are given for the distribution of γ(
t) − Θ given
X = γ(
t) to converge in law as
t → ∞. Slightly stronger conditions ensure that γ(
t) −
E(Θ |
X = γ(
t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ −
E(Θ |
X = γ(
t)) given
X = γ(
t) converges in law to a symmetric distribution as
t → ∞, it is shown that δ(γ(
t)) −
E(Θ |
X = γ(
t)) → 0 as
t → ∞.
相似文献
15.
We consider the Tikhonov regularizer f
λ of a smooth function
f ε
H2m[0, 1], defined as the solution (see [1]) to We prove that if
f(j)(0) =
f(j)(1) = 0,
J =
m, …,
k < 2
m − 1, then ¦
f −
fλ¦
j2 Rλ
(2k − 2j + 3)/2m,
J = 0, …,
m. A detailed analysis is given of the effect of the boundary on convergence rates.
相似文献
16.
In this paper new lower bounds for the cardinality of minimal
m-blocking sets are determined. Let
r2(
q) be the number such that
q+
r2(
q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order
q. If
B is a minimal
m-blocking set in PG(
n,
q) that contains at most
qm+
qm−1+…+
q+1+
r2(
q)·(∑
i=2m−n′m−1qi) points for an integer
n′ satisfying
mn′2
m, then the dimension of
B is at most
n′. If the dimension of
B is
n′, then the following holds. The cardinality of
B equals
qm+
qm−1+…+
q+1+
r2(
q)(∑
i=2m−n′m−1qi). For
n′=
m the set
B is an
m-dimensional subspace and for
n′=
m+1 the set
B is a cone with an (
m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality
q+
r2(
q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For
n′>
m+1 and
q not a prime the number
q is a square and for
q16 the set
B is a Baer cone. If
q is odd and |
B|<
qm+
qm−1+…+
q+1+
r2(
q)(
qm−1+
qm−2), it follows from this result that the subspace generated by
B has dimension at most
m+1. Furthermore we prove that in this case, if
, then
B is an
m-dimensional subspace or a cone with an (
m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality
q+
r2(
q)+1 in a plane skew to the vertex. For
q=p3h, p7 and
q not a square we show this assertion for |
B|
qm+
qm−1+…+
q+1+
q2/3·(
qm−1+…+1).
相似文献
17.
The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the
mth order cardinal
B-spline
Nm. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry (or antisymmetry), minimum support, “shift-invariance,” and inter-orthogonality. For
Nm, it turns out that symmetry can be achieved for even
m and antisymmetry for odd
m, that minimum support and shift-invariance can be attained by considering the frame generators with two-scale symbols 2
−m(1−
z)
m and 2
−mz(1−
z)
m, and that inter-orthogonality is always achievable, but sometimes at the sacrifice of symmetry. The results in this paper are valid for all compactly supported refinable functions that are reasonably smooth, such as piecewise Lipα for some α>0, as long as the corresponding two-scale Laurent polynomial symbols vanish at
z=−1. Furthermore, the methods developed here can be extended to the more general setting, such as arbitrary integer scaling factors, multi-wavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function).
相似文献
18.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium
N*=1/(
a+∑
i=0mbi) of the following differential equation with piecewise constant arguments:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, ∑
i=0mbi>0,
bi0,
i=0,1,2,…,
m, and
a+∑
i=0mbi>0. These new conditions depend on
a,
b0 and ∑
i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case
m=0 and
r(
t)≡
r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, 1−
ax−
g(
x,
x,…,
x)=0 has a unique solution
x*>0 and
g(
x0,
x1,…,
xm)
C1[(0,+∞)×(0,+∞)××(0,+∞)].
相似文献
19.
Consider the neutral delay differential equation (*) (
d/
dt)[
y(
t) +
py(
t − τ)] +
qy(
t − σ) = 0,
t t0, where τ,
q, and σ are positive constants, while
p ε (−∞, −1) (0, + ∞). (For the case
p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume
p < − 1. Then every nonoscillatory solution
y(
t) of Eq. (*) tends to ± ∞ as
t → ∞. Theorem 2. Assume
p < − 1, τ > σ, and
q(σ − τ)/(1 +
p) > (1/
e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume
p > 0. Then every nonoscillatory solution
y(
t) of Eq. (*) tends to zero as
t → ∞. Theorem 4. Assume
p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume
p > 0, σ > τ, and
q(σ − τ)/(1 +
p) > (1/
e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.
相似文献
20.
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