A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ ⁺). Then with further forcing we show that it is consistent that GCH fails at ? _ω , ? _ω strong limit, while there is a lightface definable wellorder of H(? _ω+1) (“definable failure” of the singular cardinal hypothesis at ? _ω ). The large cardinal hypothesis used is the existence of a κ ⁺⁺-strong cardinal, where κ is κ ⁺⁺-strong if there is an embedding j: V → M with critical point κ such that H(κ ⁺⁺) ? M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ⁺⁺-strong cardinal is used throughout.     

Pressing Down Lemma for λ-trees and its applications

For any ordinal λ of uncountable cofinality, a λ-tree is a tree T of height λ such that |T _α| < cf(λ) for each α < λ, where T _α = {x ∈ T: ht(x) = α}. In this note we get a Pressing Down Lemma for λ-trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that |T _α | ? ω for each α < η, then the tree T is collectionwise Hausdorff if and only if for each antichain C ? T and for each limit ordinal α ? η with cf(α) > ω, {ht(c): c ∈ C} ∩ α is not stationary in α. In the last part of this note, we investigate some properties of κ-trees, κ-Suslin trees and almost κ-Suslin trees, where κ is an uncountable regular cardinal.     

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

Parabolic Itô equations with monotone nonlinearities

In this paper the equation u_t = Lu ? F(u) + α(t, ω) is studied, where u(t) ?B₀ a Banach space. L is an unbounded self-adjoint negative definite operator. F is a monotone nonlinear potential operator. α(t, ω) is a white noise process on B₀. With suitable further restrictions on L and F it is proved that the equation has a unique solution. As t → ∞ the distribution of u(t, ω) approaches a stationary distribution which is calculated explicitly.     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Partition relations for uncountable ordinals

Partition relations of the form α→(α,m)², where α is an ordinal andm is a positive integer, are considered. Let κ be a cardinal. The following are proved: If κ is singular and 2^K=K ⁺ then (K⁺)²?((K⁺)²,3)². If κ is a strong limit cardinal, then², iff ((cfκ)²→((cfκ)²,m)². If κ is regular and K²→(K²,3)², then the κ-Souslin hypothesis holds. If K^ω<α+ and cfα=cfκ>ω, then α?(α,3)².     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

Various types of Whitney maps on n-dimensional compact connected polyhedra (n ⩾ 2)

In [6], we defined an adequate index n(P) for 1-dimensional compact connected polyhedron P and we showed that Fd ω⁻¹(t) ⩽ n(P) − 1 for any Whitney map ω for C(P) and any t ∈ [0, ω(P)]. In this paper, we show that the similar result is not always true if P is any n-dimensional compact connected polyhedron (n ⩾ 2). In fact, we show the following: Let P be any n-dimensional compact connected polyhedron (n ⩾ 2). Let m be any natural number such that m ⩾ 2. Then there exists a Whitney map ω for C(P) such that the m-sphere S^m is homotopically dominated by ω⁻¹(t) for some t ∈ (0, ω(P)). In particular, Fd ω⁻¹(t) ⩾ m.     

On uncountable cardinal sequences for superatomic Boolean algebras

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results:

1. Ifθ = 〈κ _α :α <ω ₁〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.
2. Ifκ is an uncountable cardinal such thatκ ^<κ =κ andθ = 〈κ _α :α <κ ⁺〉 is a cardinal sequence such thatκ _α ≥κ for everyα <κ ⁺ andκ _α =κ for everyα <κ ⁺ such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.

     

Countably compact hyperspaces and Frolík sums

Let H⁰(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω^∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T₂-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T₂ and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T₃ space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T₃, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product _∏α<κH⁰(Xα) embeds into .     

Modules with absolute endomorphism rings

Eklof and Shelah [8] call an abelian group absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. More generally, we say that an R-module is absolutely rigid if its endomorphism ring is just the ring of scalar multiplications by elements of R in every generic extension of the universe. In [8] it is proved that there do not exist absolutely rigid abelian groups of size ≥ κ(ω), where κ(ω) is the first ω-Erd?s cardinal (for its definition see the introduction). A similar result holds for rigid systems of abelian groups. On the other hand, recently Göbel and Shelah [15] proved that for modules of size < κ(ω) this phenomenon disappears. Their result on R _ω -modules (i.e. on R-modules with countably many distinguished submodules) that establishes the existence of ‘well-behaving’ fully rigid systems of abelian groups of large sizes < κ(ω) will be extended here to a large class of R-modules by proving the existence of modules of any sizes < κ(ω) with endomorphism rings which are absolute. In order to cover rings as general as possible, we utilize a method developed by Brenner, Butler and Corner (see [2, 3, 5]) to reduce the number of distinguished submodules required in the construction from ?₀ to five.We give several applications of our results. They include modules over domains with four pairwise comaximal prime elements, and modules over quasi-local rings whose completions contain at least five algebraically independent elements.     

The critical Fujita exponent for a diffusion equation with a potential term

We study the initial-boundary-value problem of the diffusion equation u _t = Δu ^m ? V (x)u ^m + u ^p in a conelike domain D = [1,∞) × Ω, where V (x) ~ ω ₂ |x| ^?2 with ω ₂ > 0. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω, and let l denote the positive root of l ² + (n ? 2)l = ω ₁ + ω ₂. We prove that if m < p ≤ m + 2/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + 2/(n + l), then the problem has global solutions for some \( {u_0}\gneq 0 \) .     

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α ]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Resolvability vs. almost resolvability

A space X is 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 of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

High order stiffly stable composite multistep methods for numerical integration of stiff differential equations

Let \(\dot y\) =f(y,t) withy(t ₀)=y ₀ possess a solutiony(t) fort≧t ₀. Sett _n=t ₀+nh, n=1, 2,.... Lety ₀ denote the approximate solution ofy(t _n) defined by the composite multistep method with \(\dot y_n \) =f(y _n ,t _n ) andN=1, 2,.... It is conjectured that the method is stiffly stable with orderp=l for alll≧1 and shown to be so forl=1,..., 25. The method is intrinsically efficient in thatl future approximate solution values are established simultaneously in an iterative solution process with only one function evaluation per iteration for each of thel future time points. Step and order control are easily implemented, in that the approximate solution at only one past point appears in each component multistep formula of the method and in that the local truncation error for the first component multistep formula of the method is easily evaluated as $$T^{[l]} = \frac{h}{{t_{Nl} - t_{(N - 1)l - 1} }}\{ y_{Nl}^{PRED} - y_{Nl} \} ,$$ wherey _Nl ^PRED denotes the value att _Nl of the Lagrange interpolating polynomial passing through the pointsy _(N?1)l+j att _(N?1)l+j withj=?1, 0,...,l ? 1.     

The covering number and the transitive covering number may be totally different

We construct a translation invariant σ-ideal T(κ) (where κ is an infinite cardinal number) such that cov_t (T(κ)) = 2^κ while cov (T(κ)) = cof (T(κ)) = ω₁. The constructions can be carried out in R as well.     

Noetherian types of homogeneous compacta and dyadic compacta

The Noetherian type of a space is the least κ such that it has a base that is κ-like with respect to reverse inclusion. Just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type ω. Assuming GCH, every point in a homogeneous compactum X has a local base that is c(X)-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local π-base.