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1.
It is shown that ifA andB are non-empty subsets of {0, 1}
n
(for somenεN) then |A+B|≧(|A||B|)α where α=(1/2) log2 3 here and in what follows. In particular if |A|=2
n-1 then |A+A|≧3
n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2
n-1 then |A+A|=3
n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system. 相似文献
2.
Ákos Kisvölcsey 《Combinatorica》2006,26(1):65-82
A flat antichain is a collection of incomparable subsets of a finite ground set, such that |B|−|C|≤1 for every two members B, C. Using Lieby’s results, we prove the Flat Antichain Conjecture, which says that for any antichain there exists a flat antichain
having the same cardinality and average set size. 相似文献
3.
Intersection theorems with geometric consequences 总被引:3,自引:0,他引:3
In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦(
s
n
).
As a consequence we show that ifR
n
is covered bym sets withm<(1+o(1)) (1.2)
n
then there is one set within which all the distances are realised.
It is left open whether the same conclusion holds for compositep. 相似文献
4.
We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4.
The problem originates from Razborov and Vereshchagin’s work on decision tree complexity.
Research supported in part by the Hungarian Research Fund under grant OTKA T-032969. 相似文献
5.
B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R
n
, n ≥ 3:
- Du = V(x)u, u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right. 相似文献
6.
Arrangements and cohomology 总被引:11,自引:0,他引:11
Michael Falk 《Annals of Combinatorics》1997,1(1):135-157
7.
We show the relative consistency of ℵ1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless
inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in
ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin
are if families of finite subsets of ω1 satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω1, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ1 many infinite subsets of ω1, one can find a family ? of finite subsets of ω1 which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?.
We then pose some open problems related to the question.
Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001 相似文献
8.
Claude Berge 《Combinatorica》1982,2(3):213-222
Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ1, μ2, ..., μk such tht |μi ∩S|=1 for alli.
Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic
number; here again, we do not know if there exists an optimal coloring (S
1,S
2, ...,S
k) and a path μ such that |μ ∩S
i|=1 for alli.
In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable
setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal
coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the
perfect graph conjecture.
Dedicated to Tibor Gallai on his seventieth birthday 相似文献
9.
It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a
1
, . . . , a
n
) = c for some c ∈ A and all the a
1
, . . . , a
n
∈ A) or a projection (i.e., f(a
1
, . . . , a
n
) = a
i
for some i and all the a
1
, . . . , a
n
∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x
a
= y
a
for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a
2 for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all
the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction
on the number of elements. 相似文献
10.
Mario V. Wüthrich 《Probability Theory and Related Fields》1998,112(3):299-319
We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|ξ, whereas the distance fluctuation is of order |y|χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for
this conjecture.
Received: 31 December 1997 / Revised version: 14 April 1998 相似文献
11.
Jean-Pierre Levinski 《Israel Journal of Mathematics》1984,48(2-3):225-243
We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that
if ρ is a cardinal such that ρ ≧ ω1, and (ρ++,ρ+↠(ρ+, ρ), then 0+ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries,
we get:
Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧
K
+ ≧ω2 and (λ+, λ)↠(K
+,K. Then 0+ exists.
Theorem B.Assume that there exists a singularcardinal κ such that(K
+,K↠(ω1, ω0. Then 0+ exists.
Theorem C.Assume that (λ
++, λ). Then 0+ exists (also ifK=ω
0.
Remark. Here, as in the paper of Donder and Koepke, “O+ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model.
Theorem D.Assume that (λ
++, λ)↠(K
+, K) and thatK≧ω
1. Then 0+ exists.
Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω
2, or that κ is inaccessible. We shall recall this in due place.
Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K
+, K↠(ωn +1, ωn), for κ singular andn≧0. 0266 0152 V 3
The consistency of several statements of this sort—a prototype of which is (N
ω+1,N
ω)↠(ω1, ω0) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker
than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the
author of the present paper. 相似文献
12.
Let b denote the unboundedness number of ωω. That is, b is the smallest cardinality of a subset
such that for everyg∈ωω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that
, and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ0, ℵ1, ℵ1, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ0, ℵ0, λ, 1〉, andB is not well-generated, then λ≥b. 相似文献
13.
A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}.
We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2
ω
, ω
2}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω
1-resolvable.
Any MN space of cardinality < ℵ
ω
is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ
ω
such that no subspace of X is ω
2-resolvable.
The preparation of this paper was supported by OTKA grant no. 61600 相似文献
14.
The Besov spacesB
p
α,μ
(Γ) and Triebel-Lizorkin spacesF
p
α,μ
(Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference
characterization of such spaces in the case |α|<1 is given also.
The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China. 相似文献
15.
Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2
x
if for everyA εA we have
α(B)=0. The 0-configurations form a vector space of dimension 2
n
− |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension
, we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6). 相似文献
16.
Rajendra Bhatia 《印度理论与应用数学杂志》2010,41(1):99-111
Lipschitz continuity of the matrix absolute value |A| = (A*A)1/2 is studied. Let A and B be invertible, and let M
1 = max(‖A‖, ‖B‖), M
2 = max(‖A
−1‖, ‖B
−1‖). Then it is shown that
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