Displaying similar documents to “Splitting stationary sets in κ λ for λ with small cofinality”

Some properties of stationary sets

C. A. Di Prisco, W. Marek

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CONTENTSIntroduction..................................................................51. Derivative of a stationary set...................................72. Stationary degrees ...............................................133. Subsets of P ϰ ( λ ) ..............................................194. Stationary subsets of P ϰ ( λ ) .............................255. Superstationary substes of P ϰ ( λ ) ....................326. End-stationary subsets of P ϰ ( λ ) ......................34References................................................................37 ...

Partition ideals below ℵ ω

P. Dodos, J. Lopez-Abad, S. Todorcevic (2012)

Fundamenta Mathematicae

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Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form [ ℵ k ] ω and use this to refine the well-known high-dimensional polarized partition relation for ℵ ω of Shelah.

Transitive Properties of Ideals on Generalized Cantor Spaces

Jan Kraszewski (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we show that there exists a null set A ⊆ 2 ω ₁ such that for every null set B ⊆ 2 ω ₁ we can find x ∈ 2 ω ₁ such that A ∪ (A+x) cannot be covered by any translation of B.

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function φ reflects an infinite cardinal κ , if given a topological space X with φ ( X ) ≥ κ , there exists Y ∈ [ X ] ≤ κ with φ ( Y ) ≥ κ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

The nonexistence of robust codes for subsets of ω₁

David Asperó (2005)

Fundamenta Mathematicae

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Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence ⟨ S α : α < ω ₁ ⟩ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that S ∈ L [ r , ⟨ S * α : α < ω ₁ ⟩ ] whenever each S * α is equal to S α modulo nonstationary changes, or may have the weaker meaning that S ∈ L [ r , ⟨ S α ∩ C : α < ω ₁ ⟩ ] for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of P κ ( X ) , where κ ⊆ X. Suppose μ < κ, μ < μ = μ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in [ H ( λ ) ] μ ⁺ which are internally club but not internally approachable.

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We analyze a jump processes Z with a jump measure determined by a “memory” process S . The state space of ( Z , S ) is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of ( Z , S ) is the product of the uniform probability measure and a Gaussian distribution.

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on κ λ is semiproper if the corresponding poset ℙ I is semiproper. In this paper we investigate properties of semiproper ideals on κ λ .

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put λ ( α ) = s ∈ ( α ) : s ( 0 ) = λ = m i n [ s ( β ) : β < α ] . We show that f ∈ (α) iff for some natural number n there are infinite cardinals λ ₀ i > λ ₁ > . . . > λ n - 1 and ordinals α ₀ , . . . , α n - 1 such that α = α ₀ + ⋯ + α n - 1 and f = f ₀ ⏜ f ₁ ⏜ . . . ⏜ f n - 1 where each f i ∈ λ i ( α i ) . Under GCH we prove that if α < ω₂ then (i) ω ( α ) = s ∈ α ω , ω ₁ : s ( 0 ) = ω ; (ii) if λ > cf(λ) = ω, λ ( α ) = s ∈ α λ , λ ⁺ : s ( 0 ) = λ , s - 1 λ i s ω ₁ - c l o s e d i n α ; (iii) if cf(λ) = ω₁, λ ( α ) = s ∈ α λ , λ ⁺ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α ; (iv) if cf(λ) > ω₁, λ ( α ) = α λ . This yields a complete characterization of the classes (α) for all...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class 𝒮 of morphisms in a locally presentable category 𝒞 of structures, the orthogonal class of objects...

Separating by G δ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint G δ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of L λ ⁺ ω , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with ⟨ D ℳ , ≺ ℳ ⟩ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where ⟨ D ℳ ' , ≺ ℳ ' ⟩ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L λ ⁺ ω is exactly ℶ δ ( λ ) . It was proved in [BK71] that if ℵ₀ < λ < κ a r e r e g u l a r c a r d i n a l n u m b e r s , t h e n t h e r e i s a f o r c i n g e x t e n s i o n , p r e s e r v i n g c o f i n a l i t i e s , s u c h t h a t i n t h e e x t e n s i o n 2λ = κ a n d δ ( λ ) < λ ⁺ ⁺ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e ℵ ₀ < λ < θ ≤ κ a r e c a r d i n a l n u m b e r s s u c h t h a t ∙ λ < λ = λ ; ∙ cf(θ) ≥ λ⁺ and μ λ < θ whenever μ < θ; ∙ κ λ = κ . Then there...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α ≤ 2 , a topological group G such that G γ is countably compact for all cardinals γ < α, but G α is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has...

Locally Σ₁-definable well-orders of H(κ⁺)

Peter Holy, Philipp Lücke (2014)

Fundamenta Mathematicae

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Given an uncountable cardinal κ with κ = κ < κ and 2 κ regular, we show that there is a forcing that preserves cofinalities less than or equal to 2 κ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein...

On reduction of two-parameter prediction problems

J. Friedrich, L. Klotz, M. Riedel (1995)

Studia Mathematica

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We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group G 2 (whose dual is a Polish space) with known values of the processes on a separable subset S 2 ⊆ G 2 to results for weakly stationary processes on G 1 × G 2 with observed values on G 1 × S 2 . In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

On families of Lindelöf and related subspaces of 2 ω ₁

Lúcia Junqueira, Piotr Koszmider (2001)

Fundamenta Mathematicae

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We consider the families of all subspaces of size ω₁ of 2 ω ₁ (or of a compact zero-dimensional space X of weight ω₁ in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω₁-sequences. Various relations among these families modulo the club filter in [ X ] ω ₁ are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩ M for an elementary submodel M of size ω₁. Various results with this flavor are obtained. Another...

On the bounding, splitting, and distributivity numbers

Alan S. Dow, Saharon Shelah (2023)

Commentationes Mathematicae Universitatis Carolinae

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The cardinal invariants 𝔥 , 𝔟 , 𝔰 of 𝒫 ( ω ) are known to satisfy that ω 1 ≤ 𝔥 ≤ min { 𝔟 , 𝔰 } . We prove that all inequalities can be strict. We also introduce a new upper bound for 𝔥 and show that it can be less than 𝔰 . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets

Athanassios Tzouvaras (2004)

Fundamenta Mathematicae

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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures ( 2 κ , [ 2 κ ] > κ , < ) , ( 2 λ , [ 2 λ ] > λ , < ) are...

( δ , 2 ) -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with nonzero identity, let ℐ ( ℛ ) be the set of all ideals of R and δ : ℐ ( ℛ ) → ℐ ( ℛ ) an expansion of ideals of R defined by I ↦ δ ( I ) . We introduce the concept of ( δ , 2 ) -primary ideals in commutative rings. A proper ideal I of R is called a ( δ , 2 ) -primary ideal if whenever a , b ∈ R and a b ∈ I , then a 2 ∈ I or b 2 ∈ δ ( I ) . Our purpose is to extend the concept of 2 -ideals to ( δ , 2 ) -primary ideals of commutative rings. Then we investigate the basic properties of ( δ , 2 ) -primary ideals and also discuss the relations among ( δ , 2 ) -primary, δ -primary...

Estimating the conditional expectations for continuous time stationary processes

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

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One of the basic estimation problems for continuous time stationary processes X t , is that of estimating E { X t + β | X s : s ∈ [ 0 , t ] } based on the observation of the single block { X s : s ∈ [ 0 , t ] } when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If A is a domain with the ascending chain condition on (integral) invertible ideals, then the group I ⁢ ( A ) of its invertible ideals is generated by the set I m ⁢ ( A ) of maximal invertible ideals. In this note we study some properties of I m ⁢ ( A ) and we prove that, if I ⁢ ( A ) is a free group on I m ⁢ ( A ) , then A is a locally factorial Krull domain.

On the strong Brillinger-mixing property of α -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

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First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C ( x , y ) defining an α -determinantal point process (DPP). Assuming absolute integrability of the function C 0 ( x ) = C ( o , x ) , we show that a stationary α -DPP with kernel function C 0 ( x ) is “strongly” Brillinger-mixing, implying, among others, that its tail- σ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...