On hyper‐torre isols

In this paper we present a contribution to a classical result of E. Ellentuck in the theory of regressive isols. E. Ellentuck introduced the concept of a hyper‐torre isol, established their existence for regressive isols, and then proved that associated with these isols a special kind of semi‐ring of isols is a model of the true universal‐recursive statements of arithmetic. This result took on an added significance when it was later shown that for regressive isols, the property of being hyper‐torre is equivalent to being hereditarily odd‐even. In this paper we present a simplification to the original proof for establishing that equivalence. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existentially Incomplete Tame Models and a Conjecture of Ellentuck

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II₂ arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some do not.     

A Fine Structure in the Theory of Isols

In this paper we introduce a collection of isols having some interesting properties. Imagine a collection W of regressive isols with the following features: (1) u, v ? W implies that u ? v or v ? u, (2) u ? v and v ? W imply u ? W, (3) W contains ? = {0,1,2,…} and some infinite isols, and (4) u e? W, u infinite, and u + v regressive imply u + v ? W. That such a collection W exists is proved in our paper. It has many nice features. It also satisfies (5) u, v ? W, u ? v and u infinite imply v ? g(u) for some recursive combinatorial function g, and (6) each u ? W is hereditarily odd-even and is hereditarily recursively strongly torre. The collection W that we obtain may be characterized in terms of a semiring of isols D(c) introduced by J. C. E. Dekker in [5]. We will show that W = D(c), where c is an infinite regressive isol that is called completely torre.     

Tychonoff products of compact spaces in ZF and closed ultrafilters

Let {(X_i, T_i): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??_R(X) denotes the family of all closed subsets H = V × ΠX_i of X, where V is a non‐trivial closed subset of ΠX_i and Q_H is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(X_i, T_i): i ∈ I } of compact spaces, every filterbase ?? ? ??_R(Y), Y = Π_{i ∈I}Y_i, extends to a ??_R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(X_i, T_i): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??_R(X), X = Π_{i ∈ω}X_i, extends to a ??_R(X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(X_i, T_i): i ∈ ω } of compact topological spaces, every countable family ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of lattices associated with n-cubes

A Lattice L(X) is defined starting from a cubical lattice L and an increasing diagonally closed subset X of L (Section 1). The lattice L(X) are proved to be—up to isomorphism—precisely those of signed simplexes of a simplical complex (Section 2); furthermore, an algebraic combinatorial characterization of the lattices L(X) is given (Section 3).     

On Banach spaces X for which Π2 (X,L 1) = N1(X,L 1)

We prove that every 2-summing operator from a Banach space X into an L ₁-space is nuclear if and only if X is isomorphic to a Hilbert space. Then we study the class of Banach spaces X for which Π₂(l ₂, X) = N ₁(l ₂, X).     

The SL(<Emphasis Type="Italic">V</Emphasis>)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by L _ω the ample line bundle corresponding to the polarization on the product X = Π _i=1 ^m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L _ω. We give a sufficient and necessary condition on ω such that X ^ss (L _ω) ≠ \(\not 0\) (resp., X ^s (L _ω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.     

Forcing axioms and the continuum hypothesis

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π₂-sentences over the structure (H(ω ₂), ∈, NS _ω1), in the sense that its (H(ω ₂), ∈, NS _ω1) satisfies every Π₂-sentence σ for which (H(ω ₂), ∈, NS _ω1) ? σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π₂-sentences over the structure (H(ω ₂), ∈, ω ₁) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $ . In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.     

Existentially Complete Nerode Semirings

Let Λ denote the semiring of isols. We characterize existential completeness for Nerode subsemirings of Λ, by means of a purely isol-theoretic “Σ₁ separation property”. (A “concrete” characterization that is not Λ-theoretic is well known: the existentially complete Nerode semirings are the ones that are isomorphic to Σ₁ ultrapowers.) Our characterization is purely isol-theoretic in that it is formulated entirely in terms of the extensions to Λ of the Σ₁ subsets of the natural numbers. Advantage is taken of a special kind of isol first conjectured to exist by Ellentuck and first proven to exist by Barback (unpublished). In addition, we strengthen the negative part of [13] by showing that existential completeness is not secured, for a given Nerode semiring, by either (i) a certain “functional closure” property for the extensions of partial recursive functions or (ii) the property of “pulling in” some portion of each partial recursive fiber; these latter results are perhaps a little surprising.     

Operator Ranges in Banach Spaces,I

Let X be a Banach space. A subspace L of X is called an operator range if there exists a continuous linear operator T defined on some Banach space Y and such that TY = L. If Y = X then L is called an endomorphism range. The paper investigates operator ranges under the following perspectives: (1) Existence (Section 3), (2) Inclusion (Section 4), and (3) Decomposition (Section 5). Section 3 considers the existence in X of operator ranges satisfying certain conditions. The main result is the following: if X and Fare separable Banach spaces and T : Y → X is a continuous operator with nonclosed range, then there exists a nuclear operator R:Y→X such that T + R is injective and has nonclosed dense range (Theorem 3.2). Section 4 seeks to determine conditions under which every nonclosed operator range in a Banach space is contained in the range of some injective endomorphism with nonclosed dense range. Theorem 4.3 contains a sufficient condition for this. Examples of spaces satisfying this condition are c₀, l_p (1 < p < ∞), L_q (1 < q < 2) and their quotients. In particular, this answers a question posed by W. E. Longstaff and P. Rosenthal (Integral Equations and Operator Theory 9 , (1986), 820-830. Section 5 discusses the possibility of representing a given dense nonclosed operator range as the sum of a pair L_1, L₂ of operator ranges with zero intersection in the cases where (a) L₁ and L₂ are dense, (b) L₁ and L₂ are closed. The results generalize corresponding results, for endomorphisms in Hilbert space, of J. Dixmier (Bull. Soc. Math. France 77 (1949), 11-101 and P. A. Fillmore and J. P. Williams (Adv. Math. 7 (1971), 254-281. A final section is devoted to open problems.     

Spaces which contain a copy of Sω or S2 and their applications

Let S_ω and S₂ denote the sequential fan and Arens' space, respectively. In this paper, we shall prove the following main results. (1) If Π^∞_i=1 X_i contains a copy of S_ω (S₂), then some Πⁿ_i=1 X_i contains a copy of S_ω (S_ω or S₂, respectively). (2) Let f: X → Y be a closed map such that any f^-1(y) contains no closed copy of S_ω (S₂). If X contains a closed copy of S_ω (S₂), then Y contains a closed copy of S_ω (S_ω or S₂, respectively).As applications of (1) and (2), we shall consider the Fréchet or strongly Fréchetness, or sequentiality of products of finitely or countably many spaces.     

Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝^N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫_Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space L_ω(w). For a Young function Ω ∉ Δ₂(∞) we present a necessary and sufficient conditions in order that L_ω(w) = L_ω(X_Ω) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from L_ω into itself when ω Δ₂(∞) and obtain necessary and sufficient condition in order that the composition operator maps L_ω. continuously onto L_ω.     

ℙmax variations related to slaloms

We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π₂-statements in the structure 〈H (ω ₂), ∈, NS_{ω 1}, R 〉 for some set R of reals in L (ℝ), for the following statements: (1) The cofinality of the null ideal is ℵ₁. (2) There exists a good basis of the strong measure zero ideal. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical magnetic susceptibility in the spin-fermion model for cuprate superconductors

Using the method of diagram techniques for the spin and Fermi operators in the framework of the SU(2)-invariant spin-fermion model of the electron structure of the CuO₂plane of copper oxides, we obtain an exact representation of the Matsubara Green’s function D_⊥(k, iω _m ) of the subsystem of localized spins. This representation includes the Larkin mass operator Σ_L(k, iω _m ) and the strength and polarization operators P(k, iω _m ) and Π(k, iω _m ). The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green’s function D_⊥(k, iω _m ) explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator Π(k, iω _m ).     

A local normal form theorem for infinitary logic with unary quantifiers

We prove a local normal form theorem of the Gaifman type for the infinitary logic L_∞ω( Q _u)^ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht‐Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L_∞ω( Q _u)^ω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (?^≥iy)ψ(y), where ψ(y) has counting quantifiers restricted to the (2^n–1 – 1)‐neighborhood of y. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pluriadjoint bundles of polarized surfaces

LetX be a complex connected projective smooth algebraic surface and letL be an ample line bundle onX. The maps associated with the pluriadjoint bundles (K _X L) ¹,t2, are studied by combining an ampleness result forK _X L with a very recent result by Reider. It turns out that apart from some exceptions and up to reductions, 1) (K _X L)³ is very ample; 2) (K _X L) ² is ample and spanned by global sections and is very ample unless eitherg (L)=2 (arithmetic genus ofL) orX contains an elliptic curveE withE ²=0,E·L=1;3) when (K _X L) ² is not very ample, the associated map has degree 4, equality implying thatg (L)=2 and .     

Successor levels of the Jensen hierarchy

I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X ′ be the closure of X ∪ {X } under rudimentary functions. Given a Σ₀‐formula φ (x) and a code c for a rudimentary function f, T (φ, c, ) is a Σ_ω ‐formula such that for any ∈ X, X ′ ? φ [f ( )] iff X ? T (φ, c, )[ ]. I make this precise and show relativized versions of this. As an application, I prove that under certain conditions, if Y is the Σ_ω extender ultrapower of X with respect to some extender F that also is an extender on X ′, then the closure of Y ∪ {Y } under rudimentary functions is the Σ₀ extender ultrapower of X′ with respect to F, and the ultrapower embeddings agree on X. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)