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1.
Yasunari Higuchi 《Probability Theory and Related Fields》1993,97(4):489-514
We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature >0, there exists a critical valueh
c
() of external magnetic fieldh such that the following two statements hold.
We also shows that the percolation probability is continuous in (,h) except on the half line {(, 0);
c
}. 相似文献
(i) | Ifh>h c (), then the percolation probability (i.e., the probability that the origin is in the infinite cluster of + spins) with respect to the Gibbs state ,h for the parameter (,h) is positive. |
(ii) | Ifhh c (), then the connectivity function ,h + (0,x) (the probability that the origin is connected by + spins tox with respect to ,h ) decays exponentially as |x|. |
2.
Given a pointx in a convex figureM, let(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.
相似文献
(i) | (x)2 for every pointx intM. |
(ii) | either(x)3 or(x) on intM. Furthermore, the setB={x intM:(x) is either odd or infinite } is dense inM. |
3.
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:
相似文献
(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. |
4.
Ulrich Kohlenbach 《Archive for Mathematical Logic》1992,31(4):227-241
A pointwise version of the Howard-Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel'sT as well as Kleene/Feferman's) is applied to systems of intuitionistic and classical arithmetic (H andH
c) in all finite types with full induction as well as to the corresponding systems with restricted induction andc.
相似文献
1) | H and are closed under a generalized fan-rule. For a restricted class of formulae this also holds forH c andc. |
2) | We give a new and very perspicuous proof that for each one can construct a functional such that is a modulus of uniform continuity for on {1n(nn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for . |
3) | The type structure of all pointwise majorizable set-theoretical functionals of finite type is used to give a short proof that quantifier-free choice with uniqueness (AC!)1,0-qf. is not provable within classical arithmetic in all finite types plus comprehension [given by the schema (C):y 0x (yx=0A(x)) for arbitraryA], dependent -choice and bounded choice. Furthermore separates several -operators. |
5.
B. Le Gac 《Analysis Mathematica》1992,18(2):103-109
(X
k
),k=1,2,... —
k
2
>1; (X
k
) , E(X
k
X
t
)=0 p
k<>(p+1)
(p,k,l=1, 2, ...) , , ,
相似文献
6.
A 2-periodic continuous real functionf is said to beperiodically monotone if it has the following property: there exist numbert
1t
2t
3t
1+2 such thatf is nonincreasing fort
1t
2 and nondecreasing int
2tt
3. For any 2-periodic, integrable real functiong with
0
2
|g(t|dt) we define
|