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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.

  • Many forcing axioms for all regular uncountable cardinals
    arXiv: Logic, 2013
    Co-Authors: Saharon Shelah
    Abstract:

    Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the Set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many Stationary Sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some Stationary Set, does hold for some Stationary Set. More specifically we try to get a universe satisfying GCH such that e.g. for regular kappa < lambda there are pairs (S,B), S \subSeteq S^\lambda_\kappa Stationary, B \subSeteq H (lambda), which satisfies some pregiven forcing axiom related to (S,B), (so (lambda\ S)-complete, i.e. "trivial outside S) but no more, i.e. slightly stronger versions fail. So Set theoretically we try to get a universe satisfying G.C.H. but still satisfies "many", even for a maximal family in some sense, of forcing axioms of the form "for some Stationary" while preserving GCH. As completion of the work lagged for a while, here we deal only with the Set theory.

  • The Stationary Set Splitting Game
    arXiv: Logic, 2010
    Co-Authors: Paul B. Larson, Saharon Shelah
    Abstract:

    The \emph{Stationary Set splitting game} is a game of perfect information of length $\omega_{1}$ between two players, \unspls and \spl, in which \unspls chooses stationarily many countable ordinals and \spls 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 $\Sigma^{2}_{2}$ maximality with a predicate for the nonStationary ideal on $\omega_{1}$, and an example of a consistently undetermined game of length $\omega_{1}$ with payoff definable 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.

  • Reflection implies the SCH
    arXiv: Logic, 2004
    Co-Authors: Saharon Shelah
    Abstract:

    We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every Stationary family of countable subSets of mu^+ reflect in some subSet of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad Stationary Set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four Stationary Sets.

  • Ladder gaps over Stationary Sets
    arXiv: Logic, 2004
    Co-Authors: Uri Abraham, Saharon Shelah
    Abstract:

    For a Stationary Set S subSeteq omega_1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some Stationary Set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over omega_1 Setminus S there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c poSets is again a polarized c.c.c poSet.

  • Ladder gaps over Stationary Sets
    Journal of Symbolic Logic, 2004
    Co-Authors: Uri Abraham, Saharon Shelah
    Abstract:

    For a Stationary Set S ⊆ ω 1 , and a ladder system C over S , a new type of gaps called C -Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some Stationary Set S , for every ladder C over S , every gap contains a subgap that is C -Hausdorff. But for every ladder E over ω 1 ∖ S there exists a gap with no subgap that is E -Hausdorff. A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c. poSets is again a polarized c.c.c. poSet.

Teturo Kamae - One of the best experts on this subject based on the ideXlab platform.

  • Uniform Sets and super-Stationary Sets over general alphabets
    Ergodic Theory and Dynamical Systems, 2011
    Co-Authors: Teturo Kamae
    Abstract:

    Uniform Sets and super-Stationary Sets over the binary alphabet have been extensively studied. In this paper, they are generalized to general alphabets. We generalize the fact that any uniform Set contains a super-Stationary Set so that any uniform complexity is realized by a super-Stationary Set. This gives a formula to calculate the uniform complexity functions. We also give characterizations of the class of super-Stationary Sets in general Settings in two somewhat different ways than in the binary case. Super-Stationary Sets are considered as phenomena which are independent of the time scale, but sensitive only to the direction of time, or dependent just on the order of events in time series. Hence, characterizations of super-Stationary Sets give insights into what is time, what looks like a history without a description of time duration, or what remains meaningful after we lose the quantitative sense of time.

  • super Stationary Set subword problem and the complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae, Hui Rao, Bo Tan, Yumei Xue
    Abstract:

    Let Ω⊂{0,1}NΩ⊂{0,1}N be a nonempty closed Set with N={0,1,2,…}N={0,1,2,…}. For N={N0Stationary Set if Ω[N]=ΩΩ[N]=Ω holds for any infinite subSet NN of NN. Denoting Ω′Ω′ the derived Set (i.e. the Set of accumulating points) of ΩΩ and degΩ=inf{d;Ω(d+1)=0} with Ω(1)=Ω′,Ω(2)=(Ω′)′,…, it is known [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subSet ΩΩ of {0,1}N{0,1}N such that there exists an infinite subSet NN of NN with degΩ[N]<∞degΩ[N]<∞, there exists an infinite subSet MM such that Ω[M]Ω[M] is a super-Stationary Set. Moreover, if degΩ[N]=∞degΩ[N]=∞ for any infinite subSet NN of NN, then the maximal pattern complexity [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] pΩ∗(k) is 2k(k=1,2,…). Thus, the uniform complexity functions are realized by the super-Stationary Sets [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call ξ∈{0,1}∗ξ∈{0,1}∗ a super-subword of ω∈{0,1}Nω∈{0,1}N if there exists S={s1Set of ω∈{0,1}Nω∈{0,1}N having no super-subword ξξ. Denote Q(Ξ)=∪ξ∈ΞP(ξ)andP(Ξ)=∩ξ∈ΞP(ξ), where Ξ⊂{0,1}∗Ξ⊂{0,1}∗. In this paper, we prove that the class of super-Stationary Sets other than {0,1}N{0,1}N coincides with the class of Q(Ξ)Q(Ξ) for nonempty finite Sets Ξ⊂{0,1}+Ξ⊂{0,1}+. Moreover, it also coincides with the class of P(L(Ξ)) for nonempty finite Sets Ξ⊂{0,1}+Ξ⊂{0,1}+, where L(Ξ) is the Set of minimal covers of ΞΞ. Using these expressions, we can calculate the complexity of super-Stationary Sets and prove that the complexity function of a super-Stationary Set in kk is either 2k2k or a polynomial function of kk for large kk. We also discuss the word problems related to the super-subwords.

  • Uniform Sets and complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae
    Abstract:

    For a nonempty closed subSet @W of {0,1}^@S, where @S is a countably infinite Set, let p"@W(S)@?#@p"S@W be the complexity function depending on the nonempty finite Sets S@?@S, where # denotes the number of elements in a Set and @p"S:{0,1}^@S->{0,1}^S is the projection. Define the maximal pattern complexity function p"@W^*(k)@?sup"S";"#"S"="kp"@W(S) as a function of k=1,2,.... We call @W a uniform Set if p"@W(S) depends only on #S=k, and the complexity function p"@W(k)@?p"@W(S) as a function of k=1,2,... is called the uniform complexity function of @W. Of course, we have p"@W(k)=p"@W^*(k) in this case. Such uniform Sets appear, for example, as the partitions generated by congruent Sets in a space with optimal positionings, or they appear as the restrictions of a symbolic system to optimal windows. Let @W^' be the derived Set (i.e. the Set of accumulating points) of @W and deg@W@?inf{d;@W^(^d^+^1^)=0@?} with @W^(^1^)=@W^',@W^(^2^)=(@W^')^',.... We prove that for any nonempty closed subSet @W of {0,1}^N, where N={0,1,2,...}, such that deg(@W@?@r) N, there exists an increasing injection @f:N->N such that @W@?@f@?@j=@W@?@f for any increasing injection @j:N->N. Such a Set @W@?@f is called a super-Stationary Set. Moreover, if deg(@W@?@r)=~ for any injection @r:N->N, then p"@W^*(k)=2^k(k=1,2,...) holds. A uniform Set @W@?{0,1}^@S is said to have a primitive factor [@W@?@f] if there exists an injection @f:N->@S such that @W@?@f is a super-Stationary Set, where [@W@?@f] is the isomorphic class containing @W@?@f. Then, any uniform Set has at least one primitive factor, and hence, any uniform complexity function is realized by the uniform complexity function of a super-Stationary Set. It follows that the uniform complexity function p"@W(k) is either 2^k for any k or a polynomial function of k for large k.

  • Super-Stationary Set, subword problem and the complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae, Hui Rao, Bo Tan, Yumei Xue
    Abstract:

    Let Ω⊂{0,1}NΩ⊂{0,1}N be a nonempty closed Set with N={0,1,2,…}N={0,1,2,…}. For N={N0

César Rosales - One of the best experts on this subject based on the ideXlab platform.

  • Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ^ n
    The Journal of Geometric Analysis, 2006
    Co-Authors: Manuel Ritoré, César Rosales
    Abstract:

    In this article we study Sets in the (2 n + 1)- dimensional Heisenberg group ℍ^ n which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ^ n . We define a notion of mean curvature for hypersurfaces and we show that the boundary of a Stationary Set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones . Our main result describes which are the CMC hypersurfaces of revolution in ℍ^ n . The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric Sets in ℍ^ n .

  • Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍn
    Journal of Geometric Analysis, 2006
    Co-Authors: Manuel Ritoré, César Rosales
    Abstract:

    In this article we study Sets in the (2n + 1)-dimensional Heisenberg group ℍnwhich are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a Stationary Set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones.

  • Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n
    arXiv: Differential Geometry, 2005
    Co-Authors: Manuel Ritoré, César Rosales
    Abstract:

    In this paper we study Sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a Stationary Set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in $\hhn$. The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence we classify the rotationally invariant isoperimetric Sets in $\hhn$.

Yumei Xue - One of the best experts on this subject based on the ideXlab platform.

  • super Stationary Set subword problem and the complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae, Hui Rao, Bo Tan, Yumei Xue
    Abstract:

    Let Ω⊂{0,1}NΩ⊂{0,1}N be a nonempty closed Set with N={0,1,2,…}N={0,1,2,…}. For N={N0Stationary Set if Ω[N]=ΩΩ[N]=Ω holds for any infinite subSet NN of NN. Denoting Ω′Ω′ the derived Set (i.e. the Set of accumulating points) of ΩΩ and degΩ=inf{d;Ω(d+1)=0} with Ω(1)=Ω′,Ω(2)=(Ω′)′,…, it is known [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subSet ΩΩ of {0,1}N{0,1}N such that there exists an infinite subSet NN of NN with degΩ[N]<∞degΩ[N]<∞, there exists an infinite subSet MM such that Ω[M]Ω[M] is a super-Stationary Set. Moreover, if degΩ[N]=∞degΩ[N]=∞ for any infinite subSet NN of NN, then the maximal pattern complexity [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] pΩ∗(k) is 2k(k=1,2,…). Thus, the uniform complexity functions are realized by the super-Stationary Sets [T. Kamae, Uniform Set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call ξ∈{0,1}∗ξ∈{0,1}∗ a super-subword of ω∈{0,1}Nω∈{0,1}N if there exists S={s1Set of ω∈{0,1}Nω∈{0,1}N having no super-subword ξξ. Denote Q(Ξ)=∪ξ∈ΞP(ξ)andP(Ξ)=∩ξ∈ΞP(ξ), where Ξ⊂{0,1}∗Ξ⊂{0,1}∗. In this paper, we prove that the class of super-Stationary Sets other than {0,1}N{0,1}N coincides with the class of Q(Ξ)Q(Ξ) for nonempty finite Sets Ξ⊂{0,1}+Ξ⊂{0,1}+. Moreover, it also coincides with the class of P(L(Ξ)) for nonempty finite Sets Ξ⊂{0,1}+Ξ⊂{0,1}+, where L(Ξ) is the Set of minimal covers of ΞΞ. Using these expressions, we can calculate the complexity of super-Stationary Sets and prove that the complexity function of a super-Stationary Set in kk is either 2k2k or a polynomial function of kk for large kk. We also discuss the word problems related to the super-subwords.

  • Super-Stationary Set, subword problem and the complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae, Hui Rao, Bo Tan, Yumei Xue
    Abstract:

    Let Ω⊂{0,1}NΩ⊂{0,1}N be a nonempty closed Set with N={0,1,2,…}N={0,1,2,…}. For N={N0

Thomas Jech - One of the best experts on this subject based on the ideXlab platform.

  • Projective Stationary Sets and Strong Reflection Principle
    arXiv: Logic, 1994
    Co-Authors: Qi Feng, Thomas Jech
    Abstract:

    We study projective Stationary Sets. The Projective Stationary Reflection principle is the statement that every projective Stationary Set contains an increasing continuous $\in$--chain of length $\omega_1$. We show that if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also it is equivalent to the Strong Reflection Principle. We show that the saturation of the nonStationary ideal on $\omega_1$ is equivalent to a certain kind of reflection.

  • Full reflection at a measurable cardinal
    Journal of Symbolic Logic, 1994
    Co-Authors: Thomas Jech, Jiří Witzany
    Abstract:

    AbstractA Stationary subSet S of a regular uncountable cardinal κreflects fully at regular cardinals if for every Stationary Set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a Stationary subSet of α. Full Reflection states that every Stationary Set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.

  • Full Reflection at a Measurable Cardinal
    arXiv: Logic, 1993
    Co-Authors: Thomas Jech, Jiří Witzany
    Abstract:

    A Stationary subSet $S$ of a regular uncountable cardinal $\kappa$ {\it reflects fully} at regular cardinals if for every Stationary Set $T \subSeteq \kappa$ of higher order consisting of regular cardinals there exists an $\alpha \in T$ such that $S \cap \alpha$ is a Stationary subSet of $\alpha$. {\it Full Reflection} states that every Stationary Set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $\kappa$ having Mitchell order $\kappa^{++}$ it is consistent that Full Reflection holds at every $\lambda \leq \kappa$ and $\kappa$ is measurable.

  • Full reflection of Stationary Sets at regular cardinals
    arXiv: Logic, 1992
    Co-Authors: Thomas Jech, Saharon Shelah
    Abstract:

    A Stationary subSet S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every Stationary Set T subSeteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a Stationary subSet of alpha. We prove that the Axiom of Full Reflection which states that every Stationary Set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.