A note on parameter free Π1‐induction and restricted exponentiation

We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})$ (= Boolean combinations of Σ₁) theorems of IΠ^?₁ in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ^?_{n + 1} is conservative over IΣ^?_n with respect to $\mathcal {B}(\Sigma _{n+1})$ sentences cannot be extended to Π_{n + 2} sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Arbres distingués,bi-arbres et théorèmes de relèvement

We introduce the notion of distinguished tree relation and give applications. We prove in ZFC an old conjecture of A.V. Ostrovsky about the image of a Borel space under a compact covering mapping. We also prove that if $\text{?}{}_{\mathrm{\xi +1}}{}^{\mathrm{L}}$ is countable then any compact covering mapping from a Δ¹₁ space onto a Σ⁰_1+ξ+2 or Π⁰_1+ξ+2 space is inductively perfect. The converse of last statement is shown. To cite this article: G. Debs, J. Saint Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Estimation of the algorithmic complexity of classes of computable models

We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ ₂ ⁰ -complete), computable models with ω-categorical theories (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), prime models (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), models with ω ₁-categorical theories (Δ _ω ⁰ -complex Σ _ω+1 ⁰ -set. We obtain a universal lower bound for the model-theoretic properties preserved by Marker’s extensions (Δ _ω ⁰ .     

On universality of countable powers of absolute retracts

We construct an absolute retract X of arbitrarily high Borel complexity such that the countable power X ^ω is not universal for the Borelian class A ₁ of sigma-compact spaces, and the product X ^ω x ∑, where ∑ is the radial interior of the Hilbert cube, is not universal for the Borelian class A ₂ of absolute G _δσ-spaces.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Consistency statements and iterations of computable functions in IΣ1 and PRA

In this paper we will state and prove some comparative theorems concerning PRA and IΣ₁. We shall provide a characterization of IΣ₁ in terms of PRA and iterations of a class of functions. In particular, we prove that for this class of functions the difference between IΣ₁ and PRA is exactly that, where PRA is closed under iterations of these functions, IΣ₁ is moreover provably closed under iteration. We will formulate a sufficient condition for a model of PRA to be a model of IΣ₁. This condition is used to give a model-theoretic proof of Parsons’ theorem, that is, IΣ₁ is Π₂-conservative over PRA. We shall also give a purely syntactical proof of Parsons’ theorem. Finally, we show that IΣ₁ proves the consistency of PRA on a definable IΣ₁-cut. This implies that proofs in IΣ₁ can have non-elementary speed up over proofs in PRA.     

Sampling distribution of Gini's index of diversity

In the case of a multinomial distribution Π₁(1-Π₁)+Π₂(1-Π₂)+…+ Π_k(1-Π_k) is at times referred to as Gini's index of diversity. In this paper, we present the distributional properties of the statistic

, based on samples of size n for a homogeneous multinomial distribution. For 2≤n≤ 12 and 2≤k≤12, we give a short table of the pdf, cdf, and the moments of D. For large values of n, we mention some results for the asymptotic distribution of D for the general multinomial distribution.     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type $$\frac{{|f^{(n)} (z)|}}{{n!}} \leqslant M_n (z,\Omega ,\Pi )\frac{{(\lambda _\Omega (z))^n }}{{\lambda _\Pi (f(z))}},z \in \Omega$$ are considered where M _n (z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that $$M_n (z,\Delta ,\Pi ) = (1 + |z|)^{n - 1}$$ if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh. Furthermore, we show that $$C_n (\Omega ,\Pi ) = sup\left\{ {M_n (z,\Omega ,\Pi )|z \in \Omega } \right\} \leqslant 4^{n - 1}$$ holds for arbitrary simply connected domains whereas the inequality 2 ^n-1 ≤ C _n (Ω,Π) is proved only under some technical restrictions upon Ω and Π .     

The limits of determinacy in second order arithmetic: consistency and complexity strength

We prove that determinacy for all Boolean combinations of \({F_{\sigma \delta }}\) (Π ₃ ⁰ ) sets implies the consistency of second-order arithmetic and more. Indeed, it is equivalent to the statement saying that for every set X and every number n, there exists a β-model of Π _n ¹ -comprehension containing X. We prove this result by providing a careful level-by-level analysis of determinacy at the finite level of the difference hierarchy on \({F_{\sigma \delta }}\) (Π ₃ ⁰ ) sets in terms of both reverse mathematics, complexity and consistency strength. We show that, for n ≥ 1, determinacy for sets at the nth level in this difference hierarchy lies strictly between (in the reverse mathematical sense of logical implication) the existence of β-models of Π _n+2 ¹ -comprehension containing any given set X, and the existence of β-models of Δ _n+2 ¹ -comprehension containing any given set X. Thus the nth of these determinacy axioms lies strictly between Π _n+2 ¹ -comprehension and Δ _n+2 ¹ -comprehension in terms of consistency strength. The major new technical result on which these proof theoretic ones are based is a complexity theoretic one. The nth determinacy axiom implies closure under the operation taking a set X to the least Σ _n+1 admissible containing X (for n = 1; this is due to Welch [9]).     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

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.     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

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 $\text{Ol}$.     

Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ₁ induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

1. P ^? + BΣ₁ + Exp ? There is a proper d-r.e. degree.
2. P ^? +BΣ₁+ Exp ? IΣ₁ ? There is a proper d-r.e. degree below 0′.
3. P ^? + BΣ₁ + Exp ? There is a non-r.e. degree below 0′.

     

Maps of Borel sets

Questions concerning the structure of Borel sets were raised in special cases by Luzin, Aleksandrov, and Uryson as the problems of distinguishing the sets with certain homogeneous properties in Borel classes and determining the number of such pairwise nonhomemorphic sets. The universal homogeneity, i.e., the property to contain an everywhere closed copy of any Borel set of the same or smaller class, was considered by L.V. Keldysh. She called the sets of classes Π _α ⁰ , α > 2, of first category in themselves that possess this homogeneity property canonical and proved their uniqueness. Thus she revealed the central role of the universality property when describing homeomorphic Borel sets. These investigations led her to the problem of universality of Borel sets and to the problem of finding conditions under which there exists an open map between Borel sets. In this paper, such conditions are presented and similar questions are considered for closed, compact-covering, harmonious, and other stable maps.     

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.     

Some Structural Aspects of the Katětov Order on Borel Ideals

We prove that the Katětov order on Borel ideals (1) contains a copy of \(\mathcal {P}(\omega )/\mathbf {Fin}\), consequently it has increasing and decreasing chains of lenght ??; (2) the sequence F i n ^α (α < ω ₁) is a strictly increasing chain; and (3) in the Cohen model, Katětov order does not contain any increasing nor decreasing chain of length ??, answering a question of Hru?ák (2011).     

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C^k-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ¹=Σ¹(X) of X by the following property: x∈Σ¹ if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ¹ of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σ_c¹=Σ¹∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim_cX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σ_c¹=Σ¹∩{?=c} are compact spaces foliated by holomorphic curves.     

A dual form of Ramsey's Theorem

Let k?ω, where ? is the set of all natural numbers. Ramsey's Theorem deals with colorings of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions of ω. Let (ω)^k (respectively (ω)^ω) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X? (ω)^ω let (X)^k be the set of all Y? (ω)^k such that Y is coarser than X. Dual Ramsey Theorem. If (ω)^k = C₀ ∪ … ∪ C_t?1 where each C_i is Borel then there exists X? (ω)^ω such that (X)^k ? C_i for some i <l. Dual Galvin-Prikry Theorem. Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [Trans. Amer. Math. Soc.159 (1971), 257–292] and a new proof of the Halpern-Läuchli Theorem [Trans. Amer. Math. Soc.124 (1966), 360–367].