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1.
The slow growing hierarchy is commonly defined as follows: G0( x) = 0, Gx−1( x) := Gx( x) + 1 and Gλ( x) := Gλ[x]( x) where λ< 0 is a limit and ·[·]: 0∩ Lim × ω → 0 is a given assignment of fundamental sequences for the limits below 0. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy ( Gx) x<0 is slow growing, i.e. each Gx is majorized by a Kalmar elementary recursive function. It is shown in this paper that the slow growing hierarchy (Gx)x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by Gx for some <0. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences. The paper is essentially self-contained. Only little knowledge about ordinals less than 0 — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed. 相似文献
2.
The following results are proved: (a) In a model obtained by adding 2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof([μ]0,)=μ+ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of
is equivalent to the weak Freese-Nation property of any of
or
for uncountable κ. (d) Modulo the consistency of (ω+1,ω)(1,0), it is consistent with GCH that
does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ω Cohen reals destroys the weak Freese-Nation property of
. These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke. 相似文献
3.
Let R[ f] be the remainder of some approximation method, having estimates of the form f; R[ f] f; ρ i ; f(i) for i = 0,…, r. In many cases, ρ 0 and ρ r are known, but not the intermediate error constants ρ 1,…,ρ r−1. For periodic functions, Ligun (1973) has obtained an estimate for these intermediate error constants by ρ 0 and ρ r. In this paper, we show that this holds in the nonperiodic case, too. For instance, the estimates obtained can be applied to the error of polynomial or spline approximation and interpolation, or to numerical integration and differentiation. 相似文献
4.
Though the sample mean
is a natural estimator for the mean μ of an Inverse Gaussian (IG) distribution having another parameter λ, when a guess μ 0 for μ seems plausible, an alternative adaptive estimator which shrinks towards μ 0 when a preliminary test for H 0:μ = μ 0 is tenable or else towards
, is considered a suitable competitor. Certain numerical illustrations are presented showing higher efficiency of such a testimator over
in several situations when the sample size is small and λ is either known or unknown. 相似文献
5.
Let G be a metrizable topological group. Denote by itb( G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ -bounded if itb( G) 0. The group G is called o-bounded if for every sequence ( Un) nω of neighborhoods of the identity in G there exists a sequence ( Fn) nω of finite subsets in G such that G= nωFn· Un; G is called strictly o-bounded (respectively OF- determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood Un of the identity in G and a finite subset Fn of G, respectively. The player II wins if G= nωFn· Un. For a second countable group G the following results are proven.
. If G is strictly o-bounded, then itb(G)1 and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies
. G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii)
; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup. 相似文献
6.
It was proved by Dow and Simon that there are 2 ω1 (as many as possible) pairwise nonhomeomorphic compact, T2, scattered spaces of height ω 1 and width ω. In this paper, we prove that if is an ordinal withω 1 < ω 2 and θ = κ ξ: ξ < is a sequence of cardinals such that either κ ξ = ω or κ ξ = ω 1 for every ξ < , then there are 2 ω1 pairwise nonhomeomorphic compact, T2, scattered spaces whose cardinal sequence is θ. 相似文献
7.
Let π i : Ei→ M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections si : M→ Ei. Suppose that U is an open neighborhood of s1( M) in E1 and F : U→ E2 a smooth embedding so that π 2Fs1 : M→ M is homotopic to a diffeomorphism f. We show that if k>[( n+1)/2]+1 then E1 and the induced bundle f*E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [( n+1)/2]+1< kn we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f*E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[( n+1)/2]+1 and M is a homotopy n-sphere. 相似文献
8.
A construction is given for a ( p2a( p+1), p2, p2a+1( p+1), p2a+1, p2a( p+1)) ( p a prime) divisible difference set in the group H× Z2pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ 1≠0, and those are fairly rare. We also give a construction for a ( pa−1+ pa−2+…+ p+2, pa+2, pa( pa+ pa−1+…+ p+1), pa( pa−1+…+ p+1), pa−1( pa+…+ p2+2)) divisible difference set in the group H× Zp2× Zap. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ 1=λ 2, and this corresponds to the parameters for the ordinary Menon difference sets. 相似文献
9.
A bisequence of complex numbers {μ n} −∞∞ determines a strong moment functional
satisfying L[ xn] = μ n. If
is positive-definite on a bounded interval ( a, b) R{0}, then
has an integral representation
, n=0, ±1, ±2,…, and quadrature rules { wni, xni} exist such that μ k = ∑ i=innsnikwni. This paper is concerned with establishing certain extremal properties of the weights wni and using these properties to obtain maximal mass results satisfied by distributions ψ( x) representing
when only a finite bisequence of moments {μ k} k=−nn−1 is given. 相似文献
10.
Let U be the unitary group over a finite field K where [ K]≠3 and char K≠2. Every transformation π in U with detπ = ±1 is a product of reflections. We determine the length t of every such transformation π i.e. for each π we find reflections σ i in U such that π=σ 1 σ t and so that no factorization of π into fewer than t reflections exists. 相似文献
11.
For a 1-dependent stationary sequence { Xn} we first show that if u satisfies p1= p1( u)= P( X1> u)0.025 and n>3 is such that 88 np131, then P{max(X1,…,Xn)u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3, where ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1−p2+p3−p4+2p12+3p22−5p1p2)−1 with pk=pk(u)=P{min(X1,…,Xk)>u}, k1 and From this result we deduce, for a stationary T-dependent process with a.s. continuous path { Ys}, a similar, in terms of P{max 0skTYs< u}, k=1,2 formula for P{max 0stYsu}, t>3 T and apply this formula to the process Ys= W( s+1)− W( s), s0, where { W( s)} is the Wiener process. We then obtain numerical estimations of the above probabilities. 相似文献
12.
It is shown that there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1,ω 2 and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [κ +] 2, when κ is an infinite cardinal. 相似文献
13.
Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ 1. We consider the power method, i.e. that of choosing a vector v0 and setting vk = Akv0; then the Rayleigh quotients Rk = ( Avk, vk)/( vk, vk) usually converge to λ 1 as k → ∞ (here ( u, v) denotes their inner product). In this paper we give two methods for determining how close Rk is to λ 1. They are both based on a bound on λ 1 − Rk involving the difference of two consecutive Rayleigh quotients and a quantity ω k. While we do not know how to directly calculate ω k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ 1 − Rk which is proportional to (λ 2/λ 1) 2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ). 相似文献
14.
This paper considers a class of nonlinear difference equations Δ3yn + ƒ(n, yn, yn−r) = 0, n N (n0) . A necessary and sufficient condition for the existence of a bounded nonoscillatory solution is given. 相似文献
15.
Let { pk} k≥3 be a sequence of nonnegative integers which satisfies 8 + Σ k≥3 ( k-4) pk = 0 and p4 ≥ p3. Then there is a convex 4-valent polytope P in E3 such that P has exactly pk k-gons as faces. The inequality p4 ≥ p3 is the best possible in the sense that for c < 1 there exist sequences that are not 4-realizable that satisfy both 8 + Σ k ≥3 ( k - 4) pk = 0 and p4 > cp3. When Σ k ≥ 5 pk ≠ 1, one can make the stronger statement that the sequence { pk} is 4-reliazable if it satisfies 8 + Σ k ≥ 3 ( k - 4) pk = 0 and p4 ≥ 2Σ k ≥ 5 pk + max{ k ¦ pk ≠ 0}. 相似文献
16.
If a˜cardinal κ 1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ 0 and its new cofinality, ρ, is less than κ 0, then, under some additional assumptions, each cardinal λ>κ 1 less than cc( P(κ 1)/[κ 1] <κ1) is collapsed to κ 0 as well. If in addition N= M[ f], where f : ρ→κ 1 is an unbounded mapping, then N is a˜|λ|=κ 0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba. 相似文献
17.
A directed graph D with vertex set V is called cyclically h-partite ( h2) provided one can partition V= V0+ V1++ Vh−1 so that if ( u, υ) is an arc of D then uε Vi, and υε Vi+1 (notation mod h). In this communication we obtain a characterization of cyclically h-partite strongly connected digraphs. As a consequence we obtain a sufficient condition for a digraph to have a h-kernel. 相似文献
18.
In Ramm, Phys. Lett. 99A, (1983), 258-260, it is proved that a compactly supported inhomogeneity in the velocity profile is uniquely determined by the values of the acoustic pressure collected for all positions of the source and receiver on the surface of the earth ( on the whole plane P) at low frequencies. Here it is proved that the data collected on ω 1 x ω 2 suffice for the uniqueness theorem to hold, where ω 1 and ω 2 are arbitrary open sets on the plane P. This result holds also for the data collected on ω 1 × ω 2 at a fixed frequency. 相似文献
19.
There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω 1, friction-free contact in domain Ω 2, there are no tangential contact stresses, and “film” contact without normal force in domain Ω 3 (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω 1 Ω 2 Ω 3. The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions. A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system. These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them. 相似文献
20.
Consider two transient Markov processes ( Xvt) tεR·, ( Xμt) tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process. We show that there exist (randomized) stopping times S for (Xvt), T for (Xμt) with common final distribution, L(XvS|S < ∞) = L(XμT|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown where
denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed. 相似文献
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