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1.
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.
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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. |
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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. |
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Charles M. Newman 《Constructive Approximation》1991,7(1):389-399
LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4
–
exp(ixt–V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga. 相似文献
6.
Let be a domain in C, 0, and let
n
0
() be the set of polynomials of degreen such thatP(0)=0 andP(D), whereD denotes the unit disk. The maximal range
n
is then defined to be the union of all setsP(D),P
n
0
(). We derive necessary and, in the case of ft convex, sufficient conditions for extremal polynomials, namely those boundaries whose ranges meet
n
. As an application we solve explicitly the cases where is a half-plane or a strip-domain. This also implies a number of new inequalities, for instance, for polynomials with positive real part inD. All essential extremal polynomials found so far in the convex cases are univalent inD. This leads to the formulation of a problem. It should be mentioned that the general theory developed in this paper also works for other than polynomial spaces.Communicated by J. Milne Anderson. 相似文献
7.
We study quadrilateralsQ which are given by two intervals on {:Im = 0} and {:Im = 1}, and two Jordan arcs
1,
2, in {:0 Im 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is long, i.e., if
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А. О. кАРАпЕтьН 《Analysis Mathematica》1994,20(3):185-203
Let B n be a domain and (y), y B and arbitrary positive, continuous function. If p, s (0, +), denote byH
s,
p
(T
B
) the class of the functionsf(z)f(x+iy), holomorphic in the tube domain
|