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1.
T. Sanders 《Journal d'Analyse Mathématique》2007,101(1):123-162
The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions
ƒ ∈ L
2(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space
endowed with the norm
, and generalize this to the approximate Fourier transform on Bohr sets.
As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ
A
‖
A(ℤ/pℤ) ≫ ɛ (log p)1/3−ɛ; we improve this to ‖χ
A
‖
A(ℤ/pℤ) ≫ ɛ (log p)1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ
A
‖
A(ℤ/pℤ) ≪ log p. 相似文献
2.
If a setX ⊂E
n has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E
n has int
k
X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1. 相似文献
3.
Let {X
k
,k=1,2,…} be a sequence of independent binomial variables, with
the Fourier transform of the distribution ofY. Finally denote lim [P
k
− 1/2] byδ. We haveTheorem.
Research supported by N.S.F. Grant GP-25736.
Research supported by N.S.F. Grant GP-12365. 相似文献
4.
Camil Muscalu 《Journal of Geometric Analysis》1999,9(4):683-691
If N ∈ ℕ, 0 < p ≤ 1, and(Xk)
k=1
N
are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩
k=1
N
Xk, there exists
with
, for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ2, it is proved that the N-tuple (H(X1),…, H(Xn)) is K⊓-closed with respect to (X1,…, XN) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result. 相似文献
5.
A Borel derivative on the hyperspace 2
X
of a compactumX is a Borel monotone mapD: 2
X
→2
X
. The derivative determines a Cantor-Bendixson type rank δ:2X → ω1 ∪ {∞} . We show that ifA⊂2
X
is analytic andZ⊂A intersects stationary many layers δ−1({ξ}), then for almost all σ,F∩δ−1({ξ}) cannot be separated fromZ ∩∪
a<ξ
δ−1({a}) (and also fromZ ∩∪
a>ξ
δ−1({a}) by anyF
σ-set. Another main result involves a natural partial order on 2
X
related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper.
During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would
like to express his gratitude to the Department of Mathematics and Statistics for the hospitality. 相似文献
6.
V. V. Makeev 《Journal of Mathematical Sciences》2011,175(5):572-573
Let X be an affine cross-polytope, i.e., the convex hull of n segments A
1
B
1,…, A
n
B
n
in
\mathbbRn {\mathbb{R}^n} that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L
1 ⊂⋯ ⊂ L
n
=
\mathbbRn {\mathbb{R}^n} , where L
k
is the k-dimensional affine hull of the segments A
1
B
1,…, A
k
B
k
, k ≤ n. It is proved that each convex body K ⊂
\mathbbRn {\mathbb{R}^n} is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A
k
and B
k
to the body L
k
∩ K in the k-plane L
k
are parallel to L
k
−1.Each such X has volume at least V(K)/2
n(n−1)/2. Bibliography: 5 titles. 相似文献
7.
F. V. Petrov 《Journal of Mathematical Sciences》2007,147(6):7218-7226
Let Γ ⊂ ℝd be a bounded strictly convex surface. We prove that the number kn(Γ) of points of Γ that lie on the lattice
satisfies the following estimates: lim inf kn(Γ)/nd−2 < ∞ for d ≥ 3 and lim inf kn(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189. 相似文献
8.
Jean Bourgain Jeff Kahn Gil Kalai Yitzhak Katznelson Nathan Linial 《Israel Journal of Mathematics》1992,77(1-2):55-64
LetX be a probability space and letf: X
n
→ {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI
f
(k), as follows: Foru=(u
1,u
2,…,u
n−1) ∈X
n−1 consider the setl
k
(u)={(u
1,u
2,...,u
k−1,t,u
k
,…,u
n−1):t ∈X}.
More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI
f
(S), be the probability that assigning values to the variables not inS at random, the value off is undetermined.
Theorem 1:There is an absolute constant c
1
so that for every function f: X
n
→ {0, 1},with Pr(f
−1(1))=p≤1/2,there is a variable k so that
Theorem 2:For every f: X
n
→ {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c
2(ε)n/logn so that I
f
(S)≥1−ε.
These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.
Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science. 相似文献
9.
We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with
and
and let A = ‖a
ij
‖
n×n
, where a
ij
∈ P for i, j = 1,..., n. Let A* = ‖a
ij
′
‖
n×n
and
for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤).
Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL
n
(P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) −
, ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL
a
(P, ≤) ≅ = S
n
k
.
We give some further results concerning inversion of matrices over a pseudocomplemented lattice.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005. 相似文献
10.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D. 相似文献
11.
We show that, for every l, the family
of circuits of length at least l satisfies the Erdős–Pósa property, with f(k)=13l(k−1)(k−2)+(2l+3)(k−1), thereby sharpening a result of C. Thomassen. We obtain as a corollary that graphs without k disjoint circuits of length l or more have tree-width O(lk2). 相似文献
12.
P. C. Allaart 《Acta Mathematica Hungarica》2008,121(3):243-275
This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable
function. Let F(x):=∑
n=1∞
ε
n
ϕ(2
n−1
x), x ∈ R, where ɛ
1, ɛ
2, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ
1 = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ
n
}, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa
n − 1
ɛ
n
ϕ(2
n − 1
x), where 0 < a < 1.
相似文献
13.
Let X and Y be two complex manifolds, let D⊂X and G⊂Y be two nonempty open sets, let A (resp. B) be an open subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Under a geometric condition on the boundary sets A and B, we show that every function locally bounded, separately continuous on W, continuous on A×B, and separately holomorphic on (A×G)∪(D×B) “extends” to a function continuous on a “domain of holomorphy” and holomorphic on the interior of . 相似文献
14.
Wolfgang Lusky 《Israel Journal of Mathematics》2004,143(1):239-251
LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR
n:X→X such thatR
nRm=Rmin(n,m) ifn≠m and lim
n→∞
R
n
x=x for allx∈X. We prove that, ifR
n−Rn
−1 factors uniformly through somel
p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL
Λ=closed span
, where
, has an unconditional basis. Examples include the Hardy space
. 相似文献
15.
Summary Given two subspaces A0 ⊂ A1 ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those
adjoint pairs A, A* satisfying A0 ⊂ A ⊂ A1, A
1
*
⊂ A∗ ⊂ A
0
*
⊂ W+=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace
extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary
differential subspaces in W=Lq(ι) ⊕ Lr(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C(
) ⊕ C(
), where
is a compact interval.
Entrata in Redazione il 21 febbraio 1977.
The work of EarlA. Coddington was supported in part by the National Science Foundation under NSF Grant No. MCS-76-05855. 相似文献
16.
Letx
1,x
2, ...,x
n
ben unit vectors in a normed spaceX and defineM
n
=Ave{‖Σ
i=1
n
ε1
x
i
‖:ε1=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality
such that {x
i
}
i∈A
is 16M
n
-isomorphic to the natural basis ofl
∞
k
. This result implies a significant improvement of the known results concerning embedding ofl
∞
k
in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX
n
of dimensionn either contains a (1+∈)-isomorphic copy ofl
2
m
for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl
∞
k
for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.
The contribution of the first author to this paper forms part of his Ph.D. Thesis written under the supervision of Prof. M.
A. Perles from the Hebrew University. 相似文献
17.
In this paper the structure of subspaces and quotients ofl
p
N
of dimension very close toN is studied, for 1≤p≤∞. In particular, the maximal dimensionk=k(p, m, N) so that an arbitrarym-dimensional subspaceX ofl
p
N
contains a good copy ofl
p
k
, is investigated form=N−o(N). In several cases the obtained results are sharp. 相似文献
18.
Under consideration is the problem of constructing a square Booleanmatrix A of order n without “rectangles” (it is a matrix whose every submatrix of the elements that are in any two rows and two columns does
not consist of 1s). A linear transformation modulo two defined by A has complexity o(ν(A) − n) in the base {⊕}, where ν(A) is the weight of A, i.e., the number of 1s (the matrices without rectangles are called thin). Two constructions for solving this problem are given. In the first construction, n = p
2 where p is an odd prime. The corresponding matrix H
p
has weight p
3 and generates the linear transformation of complexity O(p
2 log p log log p). In the second construction, the matrix has weight nk where k is the cardinality of a Sidon set in ℤ
n
. We may assume that
$
k = \Theta \left( {\sqrt n } \right)
$
k = \Theta \left( {\sqrt n } \right)
相似文献
19.
Suppose thatX andY are real Banach spaces,U ⊂X is an open bounded set star-shaped with respect to some point,n, k ∈ ℕ,k <n, andMn, k (U,Y) is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for anyM ≥M
n,k
(U, Y) there exists a constantB > 0 such that for any functionf ∈C
n
(U, Y) the following inequality holds:
20.
If A⊂ L0(X, μ) is a convex solid subset of L0(X, μ), then there exist disjoint X0 and X1 with X = X0∪ X1 such that A| X_0 is dense in L0(X0, μ) and A|X_1 is bounded in measure in L0(X1, μ). 相似文献
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