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1.
Katalin Marton 《Probability Theory and Related Fields》1998,110(3):427-439
Summary. Let X={X
i
}
i
=−∞
∞ be a stationary random process with a countable alphabet and distribution q. Let q
∞(·|x
−
k
0) denote the conditional distribution of X
∞=(X
1,X
2,…,X
n
,…) given the k-length past:
Write d(1,x
1)=0 if 1=x
1, and d(1,x
1)=1 otherwise. We say that the process X admits a joining with finite distance u if for any two past sequences −
k
0=(−
k
+1,…,0) and x
−
k
0=(x
−
k
+1,…,x
0), there is a joining of q
∞(·|−
k
0) and q
∞(·|x
−
k
0), say dist(0
∞,X
0
∞|−
k
0,x
−
k
0), such that
The main result of this paper is the following inequality for processes that admit a joining with finite distance:
Received: 6 May 1996 / In revised form: 29 September 1997 相似文献
2.
We obtain the best approximation in L
1(ℝ), by entire functions of exponential type, for a class of even functions that includes e
−λ|x|, where λ>0, log |x| and |x|
α
, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded
degree. 相似文献
3.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
4.
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
5.
O. V. Lopotko 《Ukrainian Mathematical Journal》2011,63(6):981-992
We obtain an integral representation of even functions of two variables for which the kernel [k
1(x + y) + k
2(x − y)], x, y ∈ R
2, is positive definite. 相似文献
6.
We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ
and expansive matrix M: Λ → Λ if ρ(M
−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M
−1. We shall prove further that if the polynomial f(x) = c
0 + c
1
x + ··· + c
k
x
k
∈ Z[x], c
k
= 1 satisfies the condition |c
0| > 2 Σ
i=1
k
|c
i
| then there is a suitable digit set D for which (Z
k
, M, D) is a number system, where M is the companion matrix of f(x).
The research was supported by OTKA-T043657 and Bolyai Fellowship Committee. 相似文献
7.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
8.
It is proved that the equation (x
2−1)(y
2−1)=(z
2−1)2, |x|≠|y|, |z|≠1, is not solvable in integersx,y,z under the conditionx−y=kz, wherek is a positive integer different from 2.
Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 181–187, August, 1999. 相似文献
9.
Rinya Takahashi 《Annals of the Institute of Statistical Mathematics》1987,39(1):637-647
Summary Denote byH ak-dimensional extreme value distribution with marginal distributionH
i
(x)=Λ(x)=exp(−e
−x
),x∈R
1. Then it is proved thatH(x)=Λ(x
1)...Λ(x
k
) for anyx=(x
1, ...,x
k
) ∈R
k
, if and only if the equation holds forx=(0,...,0). Next some multivariate extensions of the results by Resnick (1971,J. Appl. Probab.,8, 136–156) on tail equivalence and asymptotic distributions of extremes are established. 相似文献
10.
Michel Talagrand 《Probability Theory and Related Fields》1998,112(4):545-563
Consider 0<α<1 and the Gaussian process Y(t) on ℝ
N
with covariance E(Y(s)Y(t))=|t|2α+|s|2α−|t−s|2α, where |t| is the Euclidean norm of t. Consider independent copies X
1,…,X
d
of Y and␣the process X(t)=(X
1(t),…,X
d
(t)) valued in ℝ
d
. When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ
k
N
/α−(
k
−1)
d
(loglog(1/ɛ))
k
. If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ
d
(log(1/ɛ) logloglog 1/ɛ)
k
. (This includes the case k=1.)
Received: 20 May 1997 / Revised version: 15 May 1998 相似文献
11.
Ignacy Kotlarski 《Annali di Matematica Pura ed Applicata》1966,74(1):129-134
Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x1, x2, …, xn be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals fk(t) = E[ei(t,x
k)], (k=1, 2, …, n): let y1=x1 + xn, y2=x2 + xn, …, yn−1=xn−1 + xn.
If the characteristic functional f(t1, t2, …, tn−1) of the random variables (y1, y2, …, yn−1) does not vanish, then the joint distribution of (y1, y2, …, yn−1) determines all the distributions of x1, x2, …, xn up to change of location. 相似文献
12.
Accuracy of several multidimensional refinable distributions 总被引:3,自引:0,他引:3
Carlos Cabrelli Chritopher Heil Ursula Molter 《Journal of Fourier Analysis and Applications》2000,6(5):483-502
Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced
from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σ
i=1
r
Σk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. 相似文献
13.
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
14.
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
15.
Wenguang Zhai 《中国科学A辑(英文版)》1999,42(11):1173-1183
Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x
1
c
1,⋯,c
k
) denote the number of primesp withp⩽x,
, where 1<c
1<⋯<c
k
are fixed constants. It is proved that π(x;c
1,⋯,c
k
) has an asymptotic formula ifc
1
−1
+⋯+c
k
−1
>k−k/(4k
2+2).
Project supported by the National Natural Science Foundation of China (Grant No. 19801021) and the Natural Science Foundation
of Shandong Province (Grant No.Q98A02110). 相似文献
16.
Z. Ditzian 《Israel Journal of Mathematics》1985,52(4):341-354
Equivalences between the condition |P
n
(k)
(x)|≦K(n
−1√1−x
2+1/n
2)
k
n
-a, whereP
n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x
2)
k
f
(2k)(x)∈C[−1,1] is established. A similar result is shown forE
n(f)=
‖f−P
n‖
C[−1,1]. Rates other thann
-a are also discussed.
Supported by NSERC grant A4816 of Canada. 相似文献
17.
Maria E. Schonbek 《Mathematische Annalen》2006,336(3):505-538
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u
t
−Δu+b(x)·∇(u|u|
q
−1)=f(x, t) in ℝ
n
×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u
0 is supposed to be in an appropriate L
p
space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant 相似文献
18.
U. Luther 《Integral Equations and Operator Theory》2006,54(4):541-554
We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N1, t ∈N2, or x=t, where N1,N2 are sets of measure zero. It is shown that such operators map weighted L∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel
k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors
of best weighted uniform approximation by algebraic polynomials. 相似文献
19.
How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity? 总被引:1,自引:0,他引:1
Anders Yeo 《Graphs and Combinatorics》1999,15(4):481-493
Let D be a semicomplete multipartite digraph, with partite sets V
1, V
2,…, V
c, such that |V
1|≤|V
2|≤…≤|V
c|. Define f(D)=|V(D)|−3|V
c|+1 and . We define the irregularity i(D) of D to be max|d
+(x)−d
−(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i
l(D) of D to be max|d
+(x)−d
−(x)| over all vertices x of D and we define the global irregularity of D to be i
g(D)=max{d
+(x),d
−(x) : x∈V(D)}−min{d
+(y),d
−(y) : y∈V(D)}. In this paper we show that if i
g(D)≤g(D) or if i
l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated
problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete
multipartite digraphs, D, with i
g(D)=i(D)=i
l(D)=g(D)+?≤f(D)+1.
Revised: September 17, 1998 相似文献
20.
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