HAPPY AND MAD FAMILIES INL(ℝ)

2018 ◽  
Vol 83 (2) ◽  
pp. 572-597 ◽  
Author(s):  
ITAY NEEMAN ◽  
ZACH NORWOOD
Keyword(s):  
Generic Absoluteness ◽  
Mad Families ◽  
Recent Theorem ◽  
Image Position ◽  
Solovay Model ◽  
Happy Family

AbstractWe prove that, in the choiceless Solovay model, every set of reals isH-Ramsey for every happy familyHthat also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under$A{D^ + }$.

RESTRICTED MAD FAMILIES

2019 ◽  
Vol 85 (1) ◽  
pp. 149-165
Author(s):  
OSVALDO GUZMÁN ◽  
MICHAEL HRUŠÁK ◽  
OSVALDO TÉLLEZ
Keyword(s):  
Mad Families ◽  
Image Position

AbstractLet ${\cal I}$ be an ideal on ω. By cov${}_{}^{\rm{*}}({\cal I})$ we denote the least size of a family ${\cal B} \subseteq {\cal I}$ such that for every infinite $X \in {\cal I}$ there is $B \in {\cal B}$ for which $B\mathop \cap \nolimits X$ is infinite. We say that an AD family ${\cal A} \subseteq {\cal I}$ is a MAD family restricted to${\cal I}$ if for every infinite $X \in {\cal I}$ there is $A \in {\cal A}$ such that $|X\mathop \cap \nolimits A| = \omega$. Let a$\left( {\cal I} \right)$ be the least size of an infinite MAD family restricted to ${\cal I}$. We prove that If $max${a,cov${}_{}^{\rm{*}}({\cal I})\}$ then a$\left( {\cal I} \right) = {\omega _1}$, and consequently, if ${\cal I}$ is tall and $\le {\omega _2}$ then a$\left( {\cal I} \right) = max$ {a,cov${}_{}^{\rm{*}}({\cal I})\}$. We use these results to prove that if c$\le {\omega _2}$ then o$= \overline o$ and that as$= max${a,non$({\cal M})\}$. We also analyze the problem whether it is consistent with the negation of CH that every AD family of size ω1 can be extended to a MAD family of size ω1.

Solovay models and forcing extensions

2004 ◽  
Vol 69 (3) ◽  
pp. 742-766 ◽  
Author(s):  
Joan Bagaria ◽  
Roger Bosch
Keyword(s):  
Generic Absoluteness ◽  
Solovay Model ◽  
Mahlo Cardinals

Abstract.We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly- absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.

scholarly journals A short proof of a recent theorem of G. Szekeres

1976 ◽  
Vol 21 (3) ◽  
pp. 277-277
Author(s):  
Charles H. C. Little
Keyword(s):  
Directed Graph ◽  
Short Proof ◽  
Symmetric Difference ◽  
Minus Sign ◽  
Recent Theorem ◽  
Quick Proof ◽  
Image Position

The purpose of this note is to apply the work of Kasteleyn (1967) on the enumeration of I-factors of a graph to derive a quick proof of a theorem of Szekeres (1973). In the following, G is understood to be a finite, directed graph. If u, v are adjacent vertices of G, we denote by (u, v) an edge of G directed from u to v. Let{f1, f2,…, fm} be a set of 1-factors of G, and for all i write where n is half the number of vertices of G. (Here we regard a 1-factor as a set of edges). Then Kasteleyn associates with fi a plus sign if ui1vi1ui2vi2…uinvin is an even permutation of u11v11u12v12…u1nv1n, and a minus sign otherwise. The symmetric difference of two 1-factors is a collection of circuits, called alternating circuits. An alternating circuit of G is said to be clockwise even if the number of its edges that are directed in agreement with the clockwise sense is even; otherwise it is clockwise odd. Since the length of any alternating circuit is even, these definitions are not dependent on the sense designated as clockwise. It follows from the work of Kasteleyn that two given 1-factors in a directed graph agree in sign if and only if the number of clockwise even alternating circuits in their symmetric difference is even.

scholarly journals SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS

2018 ◽  
Vol 83 (3) ◽  
pp. 1282-1305 ◽  
Author(s):  
GUNTER FUCHS ◽  
KAETHE MINDEN
Keyword(s):  
Forcing Axioms ◽  
First Order ◽  
Generic Absoluteness ◽  
Subcomplete Forcing ◽  
Image Position ◽  
Aronszajn Trees ◽  
Order Structures ◽  
Forcing Axiom

AbstractWe investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.

scholarly journals Orthogonal families of real sequences

1998 ◽  
Vol 63 (1) ◽  
pp. 29-49
Author(s):  
Arnold W. Miller ◽  
Juris Steprans
Keyword(s):  
Hilbert Space ◽  
Inner Product ◽  
Partial Functions ◽  
Perfect Set ◽  
Mad Families ◽  
Real Model ◽  
Cohen Real ◽  
Image Position ◽  
Orthogonal Set ◽  
Orthogonal Sequences

For x, y ϵ ℝω define the inner productwhich may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0.Define lp to be the set of all x ϵ ℝω such thatFor Hilbert space, l2, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4].Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2.It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence.It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel.Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω.Abian raised the question of what are the possible cardinalities of maximal orthogonal families.Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ.By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ.

UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS

2017 ◽  
Vol 82 (4) ◽  
pp. 1229-1251
Author(s):  
TREVOR M. WILSON
Keyword(s):  
Measurable Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Generic Absoluteness ◽  
Relative Consistency ◽  
Consistency Results ◽  
High Consistency ◽  
Image Position ◽  
General Method

AbstractWe prove several equivalences and relative consistency results regarding generic absoluteness beyond Woodin’s ${\left( {{\bf{\Sigma }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ generic absoluteness result for a limit of Woodin cardinals λ. In particular, we prove that two-step $\exists ^&#x211D; \left( {{\rm{\Pi }}_1^2 } \right)^{{\rm{uB}}_\lambda } $ generic absoluteness below a measurable limit of Woodin cardinals has high consistency strength and is equivalent, modulo small forcing, to the existence of trees for ${\left( {{\bf{\Pi }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many “failures of covering” for the models $L\left( {T,{V_\alpha }} \right)$ for α below a measurable cardinal.

AXIOM I0 AND HIGHER DEGREE THEORY

2015 ◽  
Vol 80 (3) ◽  
pp. 970-1021 ◽  
Author(s):  
XIANGHUI SHI
Keyword(s):  
Critical Point ◽  
Degree Theory ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Generic Absoluteness ◽  
Perfect Set ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Measurable Cardinals

AbstractIn this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

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