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1.
A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d
H
(x, y) ≤ d
G
(x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q
n
has at least clog2
n · 2
n
edges.
József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board
#06139 and #07048, and OTKA 049398.
Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian
Foundation for Basic Research. 相似文献
2.
Y. Lacroix 《Israel Journal of Mathematics》2002,132(1):253-263
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ
0
∞
G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ
U
(x)=inf{k⩾1:T
k
xεU}, and defineG
U
(t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U
n
)
n≥1
of neighbourhoods ofx such that {x}=∩
n
U
n
, and for anyG ∈G, there exists a subsequence (n
k
)
k≥1
withG
U
n
k
↑U weakly.
We also construct a uniquely ergodic Toeplitz flowO(x
∞,S, μ), the orbit closure of a Toeplitz sequencex
∞, such that the above conclusion still holds, with moreover the requirement that eachU
n
be a cylinder set.
In memory of Anzelm Iwanik 相似文献
3.
A classic theorem of Pólya shows that 2
z
is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of
Gel’fond concerns entire functions that are integer valued on the setX
a={a
n:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X
a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X
a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed. 相似文献
4.
LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for
certain sequences of subsetsA
n
⊃ X and μ-almost every pointx∈X the inclusionT
n
x∈A
n
holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs
measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical
Borel-Cantelli lemma.
Partially supported by NSF grant DMS-9732728.
Partially supported by NSF grant DMS-9704489. 相似文献
5.
Summary Consider the stationary sequenceX
1=G(Z
1),X
2=G(Z
2),..., whereG(·) is an arbitrary Borel function andZ
1,Z
2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z
1
Z
k+1) satisfyingr(0)=1 and
k=1
|r(k)|
m
< , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X
n
}, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF
n
(·) be the empirical distribution function ofX
1,...,X
n
. We prove that, asn, the empirical processn
1/2{F
n
(·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067 相似文献
6.
We study equidistribution properties of nil-orbits (b
n
x)
n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ ℝ \ ℤ is positive, then the sequence $
(b^{[n^c ]} x)_{n \in \mathbb{N}}
$
(b^{[n^c ]} x)_{n \in \mathbb{N}}
is equidistributed in X for every x ∈ X. This is also the case when n
c
is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials,
and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established
by Boshernitzan when X is the circle. 相似文献
7.
We investigate when the set of finite products of distinct terms of a sequence 〈x
n
〉
n=1∞ in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x
n
〉
n=1∞) has any or all of the first three properties if and only if {x
n+1−∑
t=1
n
x
t
:n∈ℕ} is bounded from above.
N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803. 相似文献
8.
Consider the retarded difference equationx
n
−x
n−1
=F(−f(x
n
)+g(x
n−k
)), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞.
Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan 相似文献
9.
On weak positive supercyclicity 总被引:1,自引:0,他引:1
A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT
n
x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ
p
(T
*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT
n
x: n ∈ ℂ, r ∈ ℝ+} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators
defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further
directions.
Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225.
Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225 相似文献
10.
Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ... 相似文献
11.
General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF
∞ (resp., noncommutative disc algebraA
n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric
representations of the quotient algebraF
∞/J on Hilbert spaces whereJ is anyw
*-closed, 2-sided ideal ofF
∞, are obtained and used to construct aw
*-continuous,F
∞/J-functional calculus associated to row contractionsT=[T
1,…,T
n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF
∞/J are considered.
The second author was partially supported by NSF DMS-9531954. 相似文献
12.
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
13.
Daniel Wulbert 《Israel Journal of Mathematics》2001,126(1):363-380
LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L
1 (X, Σ,μ) bem-dimensional.
Theorem:There is an l ∈ E* and real numbers −∞=x
0<x
1<x
2<…<x
n<x
n+1=∞with n≤m, such that for all g ∈ G,
相似文献
14.
Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open
disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature
as that of [8], where d(x) is the degree of x, F(x) is the multiset of all open faces σ in the embedding such that the closure contains x (the multiplicity of σ is the number of times that x is visited along ∂σ), and |σ| is the number of sides of edges bounding the face σ. In this paper, we first show that if the
absolute total curvature ∑
x∈V(G) |Φ
G
(x)| is finite, then G has only finite number of vertices of non-vanishing curvature. Next we present a Gauss-Bonnet formula for embedded infinite
graphs with finite number of accumulation points. At last, for a finite simple graph G with 3 ≤ d
G
(x) < ∞ and Φ
G
(x) > 0 for every x ∈ V(G), we have (i) if G is embedded in a projective plane and #(V(G)) = n ≥ 1722, then G is isomorphic to the projective wheel P
n
; (ii) if G is embedded in a sphere and #(V(G)) = n ≥ 3444, then G is isomorphic to the sphere annulus either A
n
or B
n
; and (iii) if d
G
(x) = 5 for all x ∈ V(G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs.
Guantao Chen: The second author was partially supported by NSF DMS-0070514 and NSA-H98230-04-1-0300. 相似文献
15.
We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG
1(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y).
Research Associate at the National Fund of Scientific Research of Belgium.
Research partially supported by NSF Grant DMS-8901904. 相似文献
16.
A. K. Aleskeviciene 《Lithuanian Mathematical Journal》2005,45(4):359-367
Let X
1, X
2,... be independent identically distributed random variables with distribution function F, S
0 = 0, S
n
= X
1 + ⋯ + X
n
, and Sˉ
n
= max1⩽k⩽n
S
k
. We obtain large-deviation theorems for S
n
and Sˉ
n
under the condition 1 − F(x) = P{X
1 ⩾ x} = e−l(x), l(x) = x
α
L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005. 相似文献
17.
K. V. Storozhuk 《Siberian Mathematical Journal》2011,52(6):1104-1107
Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X
0 = {x ∈ X ∣ T
n
x → 0}. Assume given a compact set K ⊂ X such that lim inf
n→∞
ρ{T
n
x, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If $\eta < \tfrac{1}
{2}
$\eta < \tfrac{1}
{2}
, then codim X
0 < ∞. This is true in X reflexive for $\eta \in [\tfrac{1}
{2},1)
$\eta \in [\tfrac{1}
{2},1)
, but fails in the general case. 相似文献
18.
Extremes of independent Gaussian processes 总被引:1,自引:0,他引:1
Zakhar Kabluchko 《Extremes》2011,14(3):285-310
For every n ∈ ℕ, let X
1n
,..., X
nn
be independent copies of a zero-mean Gaussian process X
n
= {X
n
(t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a
n
(M
n
− b
n
), where M
n
(t) = max
i = 1,...,n
X
in
(t), and a
n
, b
n
are normalizing constants. We also provide an analogous characterization for the limits of the process a
n
L
n
, where L
n
(t) = min
i = 1,...,n
|X
in
(t)|. 相似文献
19.
Walter Feit 《Israel Journal of Mathematics》1983,46(4):274-300
Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm
p (χ) denote the Schur index of χ overQ
p
, the completion ofQ atp. It is shown that ifx is ap′-element ofG such that
for all irreducible charactersX
u ofG thenm
p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E
n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all
local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still
an open question concerning one character of the double cover of Suz).
This work was partly supported by NSF Grant MCS-8201333. 相似文献
20.
Given a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:V→W is a polynomial mapping andT is a measure preserving action ofW onX. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish
a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a
polynomial Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following
fact:Let
be a finite family of polynomials with integer coefficients and zero constant term. For any α>0, there exists N ∈ ℕ such
that whenever F is a field with |F|≥N and E ⊆F with |E|/|F|≥α, there exist u∈F, u≠0, and w∈E such that w+ϕ(u)∈E for all ϕ∈
.
The first two authros are supported by NSF, grant DMS-0070566 and DMS-0245350. The second author was supported by the A. Sloan
Foundation. The third author is supported by NSF, grant DMS-0200700. 相似文献