Sulla completezza e sulla categoricità della teoria delleW-algebre semplici

Riassunto SeW è un'algebra funzionalmente completa si dimostra che le algebre semplici e non degeneri di Var (W) sono tutti e soli i modelli di Th(W); inoltre si dimostra che se card(W=ℵ₀ Th (W) è categorica in ℵ₀, mentre esistono modelli non isomorfi di cardinalità . Se card(W)>ℵ₀ si danno alcuni risultati particolari sulla non categoricità in alcuni cardinali maggiori o uguali a card (W).

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Summary LetW be a functionally complete algebra. We proof that simple and non trivial algebras of variety generated byW are all and only the models of Th (W). Let card(W)=ℵ₀; the we proof Th (W) is ℵ₀-categorical, while there exist not isomorphic models with cardinality . Let card(W)>ℵ₀; then we have some particular results which prove that Th (W) is not α-categorical for some α≥card (W).

     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >ℵ₁ to ℵ₀. So, using the right iteractions, we can iterate such forcing without collapsing ℵ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for everyS ⊑ ℵ₂,S or ℵ₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filterD on (3) for every is regular inL (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper. The author thanks the United States-Israel Binational Science Foundation for supporting the research by grant 1110.     

Two theorems about projective sets

In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ₁ which are in the codes are constructible; thus it extends the familiar theorem of Shoenfield that subsets of ω are constructible. The second is concerned with largest countable sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals. Y. N. Moschovakis is a Sloan Foundation Fellow. During the preparation of this paper, both authors were partially supported by NSF Grant GP-27964.     

More on powers of singular cardinals

We give bounds for where cfδ=ℵ₁, (∀a<δ) , in cases which previously remained opened, including the first such cardinal: theω ₁-th cardinal inC _ω=∩n<ω C _n whereC ₀ is the cardinal andC _n+1 the set of fixed points ofC _n. No knowledge of earlier results is required. A subsequent work generalizing this was applied to many more cardinals ([Sh 7]). The author would like to thank the Canadian NSERC for supporting this research by Grant A3040 and the Israel Academy of Science for supporting it.     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Categoricity in ℵ1 of sentences inL_{\omega _1 ,\omega } (Q)

We investigate the categoricity and number of non-isomorphic models in ℵ₁ of sentences in . AssumingV=L we prove that no sentence in has exactly one uncountable model. Thus partially answering problem 24 of a problem list by Friedman.     

On the <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript> limits of subsets of the real line

We point out that it is consistent with ZFC that 2 ^ω > ℵ₁ and every subset of ℝ is the ω ₁ limit of a sequence of G _δ sets in ℝ. We prove also that assuming cov ( ) > ℵ₁, not every set in ℝ is the ω ₁ limit of a sequence of measurable sets. This solves two problems of T. Natkaniec and J. Wesołowska.      

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Autostability of hyperarithmetical models

Let be a Δ ₁ ¹ -constructivizable model. If its Scott rank is strictly less than ω ₁ ^CK , then it is proved autostable. But if , then there exists an ordinal α<ω ₁ ^CK for which is not autostable in any degree O^(γ+1) for all γ>α. We also consider some problems concerning Δ ₁ ¹ -autostability of Δ ₁ ¹ -constructivizable Boolean algebras. Supported through the FP “Integration” and the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 198–205, March–April, 2000.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Adjoint subspaces in Banach spaces,with applications to ordinary differential subspaces

Summary Given two subspaces A₀ ⊂ A₁ ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A₀ ⊂ A ⊂ A₁, A ₁ ^* ⊂ A∗ ⊂ A ₀ ^* ⊂ W⁺=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary differential subspaces in W=L^q(ι) ⊕ L^r(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C( ) ⊕ C( ), where is a compact interval. Entrata in Redazione il 21 febbraio 1977. The work of EarlA. Coddington was supported in part by the National Science Foundation under NSF Grant No. MCS-76-05855.     

Martin’s axiom and the transitivity ofP c-points

It is shown to be consistent with Martin’s Axiom and >ℵ₁ that every twoP _c-points inβℕ\ℕ have the same topological type. This research was partially supported by NSERC. The author would also like to thank Saharon Shelah for making some enlightening remarks, on an earlier version of this paper, which resulted in Definition 2.3.     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

S-forcing IIa: Adding diamonds and more applications: Coding sets,Arhangel’skii’s problem and ℒ[Q_1^{< \omega } ,Q_2^1 ]

We continue our earlier paper [20] by proving the equivalence, for regularκ>ω, of the existence of (κ, 1) morasses with built-in ♦ sequences and a strengthening, S_K◊ , of the forcing principle, S_K◊ of [20]. We obtain various applications of S_K◊, to wit: the existence of a stationary subset of [K₊]_<K with sup as coding function, the existence of a counterexample to Arhangel’skii’s conjecture (κ=ℵ₁) and compactness, axiomatizability and transfer properties for the Magidor-Malitz language ℒ (κ=ℵ₁). Research partially supported by NSF Grant MCS 8301042.