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1.
T. O. Banach 《Mathematical Notes》1998,64(3):295-302
For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ℤ
n
, the cardinalχ(ℤ
n
) is equal to the minimal cardinality of an essential subset of ℤ
n
, i.e., a of a subsetA ⊂ ℤ
n
such that, for any coloring of the group ℤ
n
inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten(
n + l)/2 ≤χ(ℤ
n
) < 2n is proved for alln and the relationχ(ℤ
n
) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(ℤ
n
) in ℤ
n
is completely described forn ≤ 3.
Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 341–350, September, 1998. 相似文献
2.
J. J. Macys 《Lithuanian Mathematical Journal》2005,45(3):284-291
We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n
−1(2n)!!2/(2n − 1)!!2 → π. We improve the Wallis inequalities (n + 1/2)−1(2n)!!2/(2n − 1)!!2 < π < n
−1(2n)!!2/(2n − 1)!!2 for π and obtain new estimates of factorial quotients with error order not worse than 1/n
2.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005. 相似文献
3.
Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()α≤≤ ()
n
−ω. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour
number μ(G) of G: n− (n−ω)()
n
−ω≤μ(G)≤n−α() α.
Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002 相似文献
4.
J. J. Grobler 《Israel Journal of Mathematics》1988,64(1):32-38
LetA be a unital Banach lattice algebra and leta εA
+ satisfy ‖a ‖≦1. Then either ‖a
n+1 −a
n ‖=2 for alln≧0 or else ‖a
n+1 −a
n ‖ → 0 asn → ∞. Cyclicity of the peripheral spectrum ofa is also established. 相似文献
5.
Stefan Eilertsen 《Arkiv f?r Matematik》2000,38(1):53-75
A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)μ inR
n
,n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev.
As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there
is an interval [A
n
, 1/2n−A
n
], whereA
n
→1, asn→∞, with μ-values for which the property does not hold. This interval is non-empty forn≥8. 相似文献
6.
Gabor Toth 《Geometriae Dedicata》1987,22(2):251-254
The space of range-equivalence classes of full orthogonal multiplications F: ℝ
n
×ℝ
n
→ℝ
p
, n≤p ≤n
2, is shown to be a compact convex body lying in so(n)⊗so(n). Furthermore, the dimension of the space of equivalence classes is determined to be (n
2(n−1)2)/4−n(n−1). 相似文献
7.
We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain
the rate of convergence n
−1/2
β
n
log 3/2
n in which β
n
can be particularly taken as (log log n)1/σ
with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n
−1/2(log log n)1/2log 2
n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n
−1/2
β
n
log 1/2
n for the above β
n
under the given covariance function, which improves the relevant one n
−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results,
which extend and improve some known results. 相似文献
8.
Takashi Shioya 《manuscripta mathematica》1999,99(4):509-516
Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥n−1. We prove that for 1 ≤k≤n, the k
th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S
n
−
k
}Γ of the unit (n−k)-sphere by a finite group Γ⊂O(n−k+1) acting isometrically on S
n
−
k
⊂ℝ
n
−
k
+.
Received: 21 September 1998 / Revised version: 23 February 1999 相似文献
9.
Given a sequence (x
n
)
n=1∞ of real numbers in the interval [0, 1) and a sequence (δ
n
)
n=1∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’
by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x
n
| < δ
n
holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x
n
)
n=1∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ
n
)
n=1∞ converges to zero. On the other hand, for any sequence of positive numbers (δ
n
)
n=1∞ tending to zero, there is a well distributed sequence (x
n
)
n=1∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’. 相似文献
10.
In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑
n=1∞
F
2n−1−1, ∑
n=1∞
F
2n−1−2, ∑
n=1∞
F
2n−1−3 and write each ∑
n=1∞
F
2n−1−s
(s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including
the reciprocal sums of odd terms in Lucas numbers.
相似文献
11.
Soon-Mo Jung Byungbae Kim 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》1999,69(1):293-308
A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]n →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]n withx + y, x - y ∈ [-t, t]n, then there exists a quadratic functionq: ℝn →E such that ∥f(x) -q(x)∥ < (2912n2 + 1872n + 334)δ for anyx ∈ [-t, t]
n
. 相似文献
12.
V. F. Gaposhkin 《Mathematical Notes》1998,64(3):316-321
The asymptotic behavior asn → ∞ of the normed sumsσn =n
−1 Σ
k
=0n−1
Xk for a stationary processX = (X
n
,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup
k≥n
|σ
k
| ≥ε) asn → ∞ are obtained.
Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998. 相似文献
13.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
14.
Janusz Konieczny 《Semigroup Forum》1992,44(1):393-402
We prove that there are exactlyn numbers greater than 2
n−1 that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2
n−1+1,2
n−1+2,2
n−1+4, ..., 2
n−1+2
n−2, 2
n
. Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2
n−1+n−1. The second is that the numbers listed above are precisely the numbers greater than 2
n−1 that can serve as the cardinalities of topologies on a finite setX withn elements. 相似文献
15.
Joseph H.G. Fu 《manuscripta mathematica》1998,97(2):175-187
A rectifiable current of dimension n−1 in the sphere bundle Sℝn≃ℝn×S
n
−1 for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the
curvature measures of such a set. We prove that the projection to ℝn of a carrier of a general such T is C
2-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its
singular skeleta, are C
2-rectifiable.
In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid
for the curvature measures of subsets of space forms.
Received: 3 December 1997/ Revised version: 25 May 1998 相似文献
16.
A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ
n
, considered as a subgroup of the affine group on ℝ
n
, admits wavelets ψ ∈ L2(ℝ
n
) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization
of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup Dx for the transpose action of D on ℝ
n
must be compact for a. e. x. ∈ ℝ
n
; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ
n
there exists an ε > 0 for which the ε-stabilizer D
x
ε
is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups. 相似文献
17.
N. J. A. Sloane Vinay A. Vaishampayan Sueli I. R. Costa 《Discrete and Computational Geometry》2011,46(3):472-478
It is shown that, given any (n−1)-dimensional lattice Λ, there is a vector v∈ℤ
n
such that the orthogonal projection of ℤ
n
onto v
⊥ is, up to a similarity, arbitrarily close to Λ. The problem arises in attempting to find the largest cylinder anchored at
two points of ℤ
n
and containing no other points of ℤ
n
. 相似文献
18.
We explicitly solve the equation Ax
n
− By
n
= ±1 and, along the way, we obtain new results for a collection of equations Ax
n
− By
n
= z
m
with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p
α
q
β
with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for
linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations
of low degree. 相似文献
19.
Gerald Kuba 《Mathematica Slovaca》2009,59(3):349-356
Let ℛ
n
(t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ
n
(t)| of the set ℛ
n
(t) by showing that, as t → ∞, t
2 log t ≪ |ℛ2(t)| ≪ t
2 log t and t
n
≪ |ℛ
n
(t)| ≪ t
n
for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ
k,n
(t) ⊂ ℛ
n
(t) such that p(X) ∈ ℛ
k,n
(t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t
k+1 ≪ |ℛ
k,n
(t)| ≪ t
k+1 and hence |ℛ
n−1,n
(t)| ≫ |ℛ
n
(t)| so that ℛ
n−1,n
(t) is the dominating subclass of ℛ
n
(t) since we can show that |ℛ
n
(t)∖ℛ
n−1,n
(t)| ≪ t
n−1(log t)2.On the contrary, if R
n
s
(t) is the total number of all polynomials in ℛ
n
(t) which split completely into linear factors over ℤ, then t
2(log t)
n−1 ≪ R
n
s
(t) ≪ t
2 (log t)
n−1 (t → ∞) for every fixed n ≥ 2.
相似文献
20.
We establish a strong regularity property for the distributions of the random sums Σ±λ
n
, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w
n+1...,w
n+ℓ) given the sum Σ
n=0
∞
w
n
λ
n
, tends to the uniform distribution on {±1}ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system
of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.
The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.). 相似文献