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1.
Melvin Hausner 《Combinatorica》1985,5(3):215-225
Ifμ is a positive measure, andA
2, ...,A
n
are measurable sets, the sequencesS
0, ...,S
n
andP
[0], ...,P
[n] are related by the inclusion-exclusion equalities. Inequalities among theS
i
are based on the obviousP
[k]≧0. Letting
=the average average measure of the intersection ofk of the setsA
i
, it is shown that (−1)
k
Δ
k
M
i
≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS
0=1,
whenS
1≧N−1, and
for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN,
for all sequencesM
0, ...,M
n
of sufficiently large length if and only if
for 0<t<1. 相似文献
2.
Summary In analogy to the well-known tilings of the euclidean plane
by Penrose rhombs (or, to be more precise, to the equivalent tilings by Robinson triangles) we give a construction of an
inflation rule based on then-fold symmetryD
nfor everyn greater than 3 and not divisible by 3. For givenn the inflation factor η can be any quotient
as well as any product
where
. The construction is based on the system ofn tangents of the well-known deltoidD, which form angles with the ζ-axis of typevπ/n. None of these tilings permits two linearly independent translations. We conjecture that they have no period at all. For
some of them the Fourier transform contains a ℤ-module of Dirac deltas.
Editors' note: This paper was accepted for the special issue ofDiscrete & Computational Geometry (Volume 13, Numbers 3–4) devoted to the László Fejes Tóth, Festschrift, but was not received in final form in time to appear
in that issue.
Research supported by the DFG and the Fritz Thyssen Stiftung. 相似文献
3.
Kate Okikiolu 《Geometric And Functional Analysis》2008,17(5):1655-1684
Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g
Γ, denote the area element by dV and the Laplace–Beltrami operator by Δ
g
. We define the Robin mass m(x) at the point x
M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δ
g
−1 is then defined by trace Δ−1 = ∫
M
m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g
Γ with trace Δ
g
−1 < trace then the minimum of trace Δ−1 over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ−1 over Γ is equal to trace . In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator
A of degree n which is conformally covariant in the sense that for the metric g we have . In this case the role of is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem.
Received: February 2006, Accepted: March 2006 相似文献
4.
We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several
arguments. Let
be a measurable function defined on the product of finitely many standard probability spaces (Xi,
, μi), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g1, ..., gn) of measure preserving automorphisms of the respective spaces (Xi,
, μi) such that f(g1
x
1, ..., gnxn) = f(x1, ..., xn) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence,
we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable
functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 57–67. 相似文献
5.
V. V. Makeev 《Journal of Mathematical Sciences》2007,140(4):558-563
Let ℝn be the n-dimensional Euclidean space, and let { · } be a norm in Rn. Two lines ℓ1 and ℓ2 in ℝn are said to be { · }-orthogonal if their { · }-unit direction vectors e
1 and e
2 satisfy {e
1 + e
2} = {e
1 − e
2}. It is proved that for any two norms { · } and { · }′ in ℝn there are n lines ℓ1, ..., ℓn that are { · }-and { · }′-orthogonal simultaneously. Let
be a continuous function on the unit sphere
with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in
, and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even,
then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e
1, ..., e
n such that for 1 ≤ i ≤ j ≤ n we have
. Bibliography: 8 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117. 相似文献
6.
N. I. Osetinskii 《Computational Mathematics and Modeling》2008,19(1):45-56
We consider the classification of generalized linear controllable systems over the field
= ℂ or
= ℝ under transformations defined by the action of the group GL
n
(
) × GL
n
(
). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O’Halloran, and others on the geometric structure
of the set of orbits C
n,m
(
) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of C
n,m
(
) and evaluate its dimension. We show that C
n,m
(
) is a natural compactification of the set of orbits of ordinary linear controllable systems ∑
n,m
(
) and the boundary C
n,m
(
) − ∑
n,m
(
) consists of the orbits of singular generalized systems.
__________
Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 153–166, 2004. 相似文献
7.
Mohammad Sal Moslehian 《Bulletin of the Brazilian Mathematical Society》2007,38(4):611-622
In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation
相似文献
8.
Paweł Hitczenko 《Israel Journal of Mathematics》1993,84(1-2):161-178
Letf
n
= Σ
k=1
n
v
k
r
k
,n=1,…, be a martingale transform of a Rademacher sequence (r
n)and let (r
n
′
) be an independent copy of (r
n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true:
In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that
for every sequence of scalars (a
n)one has
where
is theK-interpolation norm between ℓ1 and ℓ2. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence.
This research was supported in part by an NSF grant and an FRPD grant at NCSU. 相似文献
9.
Let
be a set of finite groups. A group G is saturated with groups from
if every finite subgroup of G is contained in a subgroup isomorphic to some member of
. It is proved that a periodic group G saturated with groups from the set {L3(2m)|m = 1, 2, …} is isomorphic to L3(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite.
__________
Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007. 相似文献
10.
Xiao Hong CAO Mao Zheng GUO Bin MENG 《数学学报(英文版)》2006,22(1):169-178
When A ∈ B(H) and B ∈ B(K) are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = ( A0 CB). In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm (lower semi-Fredholm, or Fredholm) operator for some C ∈B(K,H). In addition, let σSF+(A) = {λ ∈ C : A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B(H) and let σrsF- (A) = {λ∈ C : A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±(A) U σSF±(B) to σSF±(Mc) is accomplished by removing certain open subsets of σSF-(A) ∩σSF+ (B) from the former, that is, there is an equality σSF±(A) ∪σSF± (B) = σSF± (Mc) ∪& where L is the union of certain of the holes in σSF±(Mc) which ilappen to be subsets of σSF- (A) A σSF+ (B). Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space. 相似文献
11.
J. Sunklodas 《Lithuanian Mathematical Journal》2007,47(3):327-335
We estimate the difference
for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z
v
= B
v
−1
∑
i=0
∞
v
i
X
i
and
with discount factor ν such that 0 < ν < 1. Here {X
n
, n ≥ 0} is a sequence of strongly mixing random variables with
, and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)1/2).
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007.
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07. 相似文献
12.
Keiji Izuchi 《Journal d'Analyse Mathématique》1998,75(1):135-154
Letb be a Blaschke product with zeros {z
n
} in the open unit disk Δ. Let
be the set of sequences of non-negative integersp=(p
1,p
2,…) such that ∑
n=1
∞
p
n
(1 − |z
n
|) < ∞ andp
n
→∞ asn→∞. We study the class of weak infinite powers ofb,
Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z
n
}. It is proved thatS(b)=∂Δ if and only if
, the Douglas algebra generated by
. Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that
. 相似文献
13.
The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity 总被引:1,自引:0,他引:1
Lun Chuan ZHANG 《数学学报(英文版)》2007,23(8):1413-1418
In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx : X→L(E) be a continuous map, and each R(Tx) be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^+: X→L(E) is continuous if and only if ||Tx^+|| is locally bounded, where Tx^+ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement: For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that (0, λ^2)lohtatn in ∩x∈∪0C/σ(Tx^+Tx), where a(T) denotes the spectrum of operator T. 相似文献
14.
Jeremy Berman 《Israel Journal of Mathematics》1978,31(3-4):383-393
Forn≧1, letS
n=ΣX
n,i (1≦i≦r
n <∞), where the summands ofS
n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some
for allt≧1 and all values ofx.
Theorem.For centering constants c
n,let S
n
− c
n
converge in distribution to a random variable S. (A)In order that Ef(Sn − cn) converge to a limit L, it is necessary and sufficient that there exist a common limit
(B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R.
Applications are given to infinite series of independent random variables, and to normed sums of independent, identically
distributed random variables. 相似文献
15.
M. A. Vronskii 《Mathematical Notes》2000,68(4):444-451
In this paper, for the partial sumsS
n
of a stationary associated random process it is proved that the logarithmic averages
converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their
limiting value is established.
Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000. 相似文献
16.
Eberhard MALKOWSKY M. MURSALEEN Suthep SUANTAI 《数学学报(英文版)》2007,23(3):521-532
Let p = (pk)k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = (xk)k=0^∞ is any sequence of complex numbers we write Δ(m)x for the sequence of the m th order differences of x and Δu^(m)X = {x=(x)k=0^∞ uΔ(m)x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^(m)X for X=co(p),c(p),l∞(p) and characterize some matrix transformations between these spaces Δ^(m)X. 相似文献
17.
We investigate the correlation between the constants K(ℝn) and
, where
is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line,
, L
l
p, p
(G
n) is the set of functions ƒ ∈ L
p
(G
n
) such that the partial derivative
belongs to L
p
(G
n
),
, 1 ≤ p ≤ ∞, l ∈ ℕn, α ∈ ℕ
0
n
= (ℕ ∪ 〈0〉)n, D
α
f is the mixed derivative of a function ƒ, 0 < μi < 1,
, and ∑
i=0
n
. If G
n
= ℝ, then μ0=1−∑
i=0
n
(α
i
/l
i
), μi = αi/l
i
,
if
, then μ0=1−∑
i=0
n
(α
i
/l
i
) − ∑
i=0
n
(λ/l
i
), μi = αi/ l
i
+ λ/l
i
,
, λ ≥ 0. We prove that, for λ = 0, the equality
is true.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 597–606, May, 2006. 相似文献
18.
Chen Hua 《数学学报(英文版)》1997,13(1):81-89
In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle
, where Ω is a bounded open subset of ℝ
n
(n≥2) with non-smooth boundary ∂Ω and connected complement Ω
e
=ℝ
n
. We can prove that if Ω satisfies a certain geometrical condition, then
where
,d
n>0 depending only onn, and |·|
j
(j = n - l, n) is aj- dimensional Lebesgue measure.
Research partially supported by the Natural Science Foundation of China and the Grant of Chinese State Education Committee 相似文献
19.
V. V. Andrievskii 《Journal d'Analyse Mathématique》2005,96(1):283-295
LetG⊂C be a quasidisk,K ⊂ G be a compact set, andp
n be a non-constant complex polynomial of degree at mostn. We establish the inequality
whereα
n < 0 depends onn, K,
and the geometrical structure of ϖG. 相似文献
20.
We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ
n
withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T
NM in a neighborhood of a pointz
o.We identify in natural manner
and assume that for a holomorphic vector fieldL tangent toM and verifying
we have that the Levi form
takes a value
. Then we prove thatCR functions onV extend ∀ω to a wedgeV
1 “attached” toV in direction of a vector fieldiV such that |pr(iV(z
0))−iv
0| < ε (where pr is the projection pr:T
NX →T
MX |
N
).We then prove that when the Levi cone “relative to Σ”iZ
Σ = convex hull
is open inT
MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT
NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ
Σ =T
MX, and is continuous up toN, thenf is in fact holomorphic atz
o. 相似文献