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1.
M. J. Pelling 《Aequationes Mathematicae》1989,37(1):15-37
Thepositive half A
+ of an ordered abelian groupA is the set {x Ax 0} andM
A
+ is amodule if for allx, y M alsox + y, |x – y| M. If A
+
\M thenM() is the module generated byM and. S
M isunbounded inM if(x M)(y S)(x y) and isdense inM if (x1, x2 M)(y S) (x1 <>2 x1 y x2). IfM is a module, or a subgroup of any abelian group, a real-valuedg: M R issubadditive ifg(x + y) g(x) + g(y) for allx, y M. The following hold:
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(1) | IfM andM * are modules inA andM M * A + then a subadditiveg:M R can always be extended to a subadditive functionF:M * R when card(M) = 0 and card(M * ) 1, or wheneverM * possesses a countable dense subset. |
(2) | IfZ A is a subgroup (whereZ denotes the integers) andg:Z + R is subadditive with g(n)/n = – theng cannot be subadditively extended toA + whenA does not contain an unbounded subset of cardinality . |
(3) | Assuming the Continuum Hypothesis, there is an ordered abelian groupA of cardinality 1 with a moduleM and elementA + /M for whichA + = M(), and a subadditiveg:M R which does not extend toA +. This even happens withg 0. |
(4) | Letg:A + R be subadditive on the positive halfA + ofA. Then the necessary and sufficient condition forg to admit a subadditive extension to the whole groupA is: sup{g(x + y) – g(x)x –y} < +="> for eachy <> inA. |
(5) | IfM is a subgroup of any abelian groupA andg:M K is subadditive, whereK is an ordered abelian group, theng admits a subadditive extensionF:A K. |
(6) | IfA is any abelian group andg:A R is subadditive, theng = + where:A R is additive and 0 is a non-negative subadditive function:A R. IfA is aQ-vector space may be takenQ-linear. |
(7) | Ifg:V R is a continuous subadditive function on the real topological linear spaceV then there exists a continuous linear functional:V R and a continuous subadditive:V R such thatg = + and 0. ifV = R n this holds for measurable subadditiveg with a continuous and measurable. |
2.
The Komlós-Révész theorem states: For r.v.s.X
n
with X
n
1M there exists a subsequenceX
k
n
and a r.v.X with X1M such that
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3.
Let C be a simply connected domain, 0, and let n,nN, be the set of all polynomials of degree at mostn. By n() we denote the subset of polynomials p n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|<1}, and=" by=">1},>we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate n (or some important aspects of it) to the imagesf
s
(D), wheref
s
(z):=f[(1–s)z], 0s<1. for=" instance=" we=" prove=" the=" existence=" of=" a=" universal=">1.>c
0 such that, forn2c
0, 相似文献
4.
Bruck nets,codes, and characters of loops 总被引:1,自引:1,他引:0
G. Eric Moorhouse 《Designs, Codes and Cryptography》1991,1(1):7-29
Numerous computational examples suggest that if
k-1
k
are (k- 1)- and k-nets of order n, then rank
p
k
- rank
p
k-1 n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if p
e n, then rank
p
3 3n - 2 - e, where equality holds if and only if the loop G coordinatizing 3 has a normal subloop K such that G/K is an elementary abelian group of order p
e
. Furthermore if n is squarefree, then rank
p
= 3n - 3 for every prime p ¦ n, if and only if 3 is cyclic (i.e., 3 is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n 2 mod 4, is in fact desarguesian of prime order. 相似文献
5.
LetV
n
={1, 2, ...,n} ande
1,e
2, ...,e
N
,N=
be a random permutation ofV
n
(2). LetE
t={e
1,e
2, ...,e
t} andG
t=(V
n
,E
t
). If is a monotone graph property then the hitting time() for is defined by=()=min {t:G
t
}. Suppose now thatG
starts to deteriorate i.e. loses edges in order ofage, e
1,e
2, .... We introduce the idea of thesurvival time =() defined by t = max {u:(V
n, {e
u,e
u+1, ...,e
T
}) }. We study in particular the case where isk-connectivity. We show that
|