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)     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

When countably paracompact,locally compact,screenable spaces are paracompact

Z. Balogh has shown that all normal, locally compact, screenable spaces are paracompact; it is known that the countably paracompact analogue of this theorem holds assuming V = L; in this paper we show that it also holds assuming Axiom R + MA_ω1, both of which follow from Martin's Maximum or PFA⁺(1), strengthenings of the Proper Forcing Axiom.     

Semi‐proper forcing,remarkable cardinals,and Bounded Martin's Maximum

We show that L(?) absoluteness for semi‐proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L(?) absoluteness for proper forcings. By [7], L(?) absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi‐Proper Forcing Axiom (BSPFA) is equiconsistent with the Bounded Proper Forcing Axiom (BPFA), which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum (BMM) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V , X^# exists. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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)     

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).     

A Relationship Among Gentzen's Proof-Reduction,Kirby-Paris' Hydra Game and Buchholz's Hydra Game

We first note that Gentzen's proof-reduction for his consistency proof of PA can be directly interpreted as moves of Kirby-Paris' Hydra Game, which implies a direct independence proof of the game (Section 1 and Appendix). Buchholz's Hydra Game for labeled hydras is known to be much stronger than PA. However, we show that the one-dimensional version of Buchholz's Game can be exactly identified to Kirby-Paris' Game (which is two-dimensional but without labels), by a simple and natural interpretation (Section 2). Jervell proposed another type of a combinatorial game, by abstracting Gentzen's proof-reductions and showed that his game is independent of PA. We show (Section 3) that this Jervell's game is actually much stronger than PA, by showing that the critical ordinal of Jervell's game is φ_ω (0) (while that of PA or of Kirby-Paris' Game is φ₁ (0) = ?₀) in the Veblen hierarchy of ordinals.     

A splitting theorem for the Medvedev and Muchnik lattices

This is a contribution to the study of the Muchnik and Medvedev lattices of non‐empty Π⁰₁ subsets of 2^ω. In both these lattices, any non‐minimum element can be split, i. e. it is the non‐trivial join of two other elements. In fact, in the Medvedev case, ifP > _M Q, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ?‐theories.     

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.     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On κ‐hereditary Sets and Consequences of the Axiom of Choice

We will prove that some so‐called union theorems (see [2]) are equivalent in ZF⁰ to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ‐hereditary sets yields equivalents to statements about the transitive closure of κ‐narrow relations. The instance κ = ω₁ (i. e., hereditarily countable sets) yields an equivalent to Howard‐Rubin's Form 172 (the transitive closure Tc(x) of every hereditarily countable set x is countable). In particular, the countable union theorem (Howard‐Rubin's Form 31) and, a fortiori, the axiom of countable choice imply Form 172.     

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.     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

The Computational Power of ℳ︁ω

We prove that the Kleene schemes for primitive recursion relative to the μ‐operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene‐Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳ︁^ω we prove Nigg's conjecture that ℳ︁^ω is strictly weaker than Plotkin's PCF + PA.     

Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H¹‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the L^λ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρ_S separated from zero, we establish the uniform‐in‐time H¹‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρ_S separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.     

Atomic decomposition for the vorticity of a viscous flow in the whole space

We show that the vorticity of a viscous flow in ?³ admits an atomic decomposition of the form ω(x, t) = ω_k(x – x_k, t), with localized and oscillating building blocks ω_k, if such a property is satisfied at the beginning of the evolution. We also study the long time behavior of an isolated coherent structure and the special behavior of flows with highly oscillating vorticities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pf ≠ NPf for almost all f

We discuss the question of Ralf‐Dieter Schindler whether for infinite time Turing machines P^f = NP^f can be true for any function f from the reals into ω₁. We show that “almost everywhere” the answer is negative.     

Martin's Axiom and the Dual Distributivity Number

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ₁. This answers a question of Halbeisen.