Fragments of Martin's Maximum in generic extensions

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω₁ + 1)‐game‐closed forcings. PFA can be destroyed by a strongly (ω₁ + 1)‐game‐closed forcing but not by an ω₂‐closed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Local Density of Kleene Degrees

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K ?^ωω, there are N₁ sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these theorems by showing that if V = L, then for all K ? ^ωω, Kleene semirecursive in K Kleene degrees are dense above K.     

The canonical function game

The canonical function game is a game of length ω ₁ introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ² ₂ absoluteness, cardinality spectra and Π₂ maximality for H(ω ₂) relative to the Continuum Hypothesis.     

The Strengths of Some Violations of Covering

We consider two models V₁, V₂ of ZFC such that V₁ ⊆ V₂, the cofinality functions of V₁ and of V₂ coincide, V₁ and V₂ have that same hereditarily countable sets, and there is some uncountable set in V₂ that is not covered by any set in V₁ of the same cardinality. We show that under these assumptions there is an inner model of V₂ with a measurable cardinal κ of Mitchell order κ⁺⁺. This technical result allows us to show that changing cardinal characteristics without changing cofinalities or ω‐sequences (which was done for some characteristics in [13]) has consistency strength at least Mitchell order κ⁺⁺. From this we get that the changing of cardinal characteristics without changing cardinals or ω‐sequences has consistency strength Mitchell order ω₁, even in the case of characteristics that do not stem from a transitive relation. Hence the known forcing constructions for such a change have lowest possible consistency strength. We consider some stronger violations of covering which have appeared as intermediate steps in forcing constructions.     

On the Degree of Caustics by Reflection

Given a point S ∈ ?²: = ?²(?) and an irreducible algebraic curve 𝒞 of ?² (with any type of singularities), we consider the lines ? _m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The caustic by reflection Σ _S (𝒞) is classically defined as the Zariski closure of the envelope of the reflected lines ? _m . We identify this caustic with the Zariski closure of Φ(𝒞), where Φ is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve 𝒞 (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of Φ-polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of 𝒞 with its polar at any point and to the intersection numbers of 𝒞 with its Hessian curve. These computations are linked with generalized Plücker formulas for the class and for the number of inflection points of 𝒞.     

Sacks forcing does not always produce a minimal upper bound

Theorem. There is a countable admissible set, $\text{Ol}$, with ordinal ω^CK₁ such that if S is Sacks generic over $\text{Ol}$ then ω₁^S > ω^CK₁ and S is a nonminimal upper bound for the hyperdegrees in $\text{Ol}$. (The same holds over $\text{Ol}$ for any upper bound produced by any forcing which can be construed so that the forcing relation for Σ₁ formulas is Σ₁.) A notion of forcing, the “delayed collapse” of ω^CK₁, is defined. The construction hinges upon the symmetries inherent in how this forcing interacts with Σ₁ formulas. It also uses Steel trees to make a certain part of the generic object Σ₁ over the final inner model, $\text{Ol}$, and, indeed, over many generic extensions of $\text{Ol}$.     

The algebraic structure of the isomorphic types of tally,polynomial time computable sets

 We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P _T into six parts: D ⁻, D^ ⁻ , C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets. 1. is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in can be distinguished by first order formulas. 3. There exist infinitely many nontrivial automorphisms for . 4. is not distributive, but any interval in is a countable distributive lattice. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID="ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID="ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies. Received: 23 July 1999 / Published online: 27 March 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=" ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID=" ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.     

Some Boolean Algebras with Finitely Many Distinguished Ideals I

We consider the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th_prin and Th_sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th_prin) ? intalg(ω⁴), Sentalg(Th_max) ? intalg(ω³) ? Sentalg(Th_ac), and Sentalg(Th_sa) ? intalg(ω² + ω²). For Th_max and Th_ac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic.     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On potential isomorphism and non-structure

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game. The research of the first and second author was partially supported by Academy of Finland grant 40734 Mathematics Subject Classification (2000): 03C55, 03C45     

Generic compactness reformulated

We point out a connection between reflection principles and generic large cardinals. One principle of pure reflection is introduced that is as strong as generic supercompactness of ₂ by -closed forcing. This new concept implies CH and extends the reflection principles for stationary sets in a canonical way.Mathematics Subject Classification (2000): 03E50, 03E55     

H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π

We consider a biharmonic problem Δ²u_ω = f_ω with Navier type boundary conditions u_ω = Δu_ω = 0, on a family of truncated sectors Ω_ω in ?² of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (f_ω)_{ω∈(2π/3, π]} is assumed to depend smoothly on ω in L²(Ω_ω). The main result is that u_ω converges to u_π when ω → π with respect to the H²-norm. We can also show that the H²-topology is optimal for such a convergence result.

     

The modal logic of the countable random frame

 We study the modal logic M L ^r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L ^r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. Received: 1 May 2000 / Revised version: 29 July 2001 / Published online: 2 September 2002 Mathematics Subject Classification (2000): 03B45, 03B70, 03C99 Key words or phrases: Modal logic – Random frames – Almost sure frame validity – Countable random frame – Axiomatization – Completeness     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

Polynomial Representations via Spectral Decompositions

We show using functional analysis methods, that under some conditions, such as the positivity of a homogeneous polynomial, one can give a representation of polynomials which are positive on semi-algebraic sets, allowing irrational fractions with denominators of the form (1+t² ₁) _1/2 ...(1+t ² _n) _n/2, _iZ₊, i = 1, ..., n     

Forking and Incomplete Types

Let Δ be a set of formulas. In this paper we study the following question: under what assumptions on Δ, the concept “a complete Δ-type p over B does not fork over A ? B” behaves well. We apply the results to the structure theory of ω₁-saturated models. Mathematics Subject Classification: 03C45.     

On End-Extensions of Models of ¬exp

Every model of IΔ₀ is the tally part of a model of the stringlanguage theory Th-FO (a main feature of which consists in having induction on notation restricted to certain AC⁰. sets). We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ₀ + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models. Mathematics Subject Classification: 03F30, 03C62, 68Q15.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).