scholarly journals MEASURING CLUB-SEQUENCES TOGETHER WITH THE CONTINUUM LARGE

2017 ◽  
Vol 82 (3) ◽  
pp. 1066-1079
Author(s):  
DAVID ASPERÓ ◽  
MIGUEL ANGEL MOTA
Keyword(s):  
Closed Subset ◽  
Finite Support ◽  
Symmetry Constraints ◽  
Image Position ◽  
Construction Works ◽  
The Continuum

AbstractMeasuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta < {\omega _1}}}$ with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$, a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$. We answer a question of Justin Moore by building a forcing extension satisfying measuring together with ${2^{{\aleph _0}}} > {\aleph _2}$. The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Set Theory ◽  
Closed Subset ◽  
Choice Function ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Finite Support ◽  
Compact Spaces ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

scholarly journals On pseudo-distributive near-rings

1985 ◽  
Vol 28 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Gordon Mason
Keyword(s):  
Group Ring ◽  
Finite Support ◽  
First Case ◽  
Image Position

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfiesfor all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

An alternative to Wiener-Hopf methods for the study of bounded processes

1964 ◽  
Vol 1 (01) ◽  
pp. 85-120 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Waiting Time ◽  
Finite Interval ◽  
Time Process ◽  
First Passage ◽  
Discrete Parameter ◽  
Waiting Time Process ◽  
Image Position ◽  
Wald's Identity ◽  
Independent Identically Distributed ◽  
The Continuum

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξ n are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

Comment on the paper ‘On the influence of the Hall effect on the spectrum of the ideal magnetohydrodynamic cylindrical pinch’ by U. Schaper

1984 ◽  
Vol 31 (1) ◽  
pp. 173-175 ◽  
Author(s):  
A. Lifshitz ◽  
E. Fedorov ◽  
U. Schaper
Keyword(s):  
Eigenvalue Problem ◽  
Hall Effect ◽  
Distributional Sense ◽  
General Properties ◽  
First Case ◽  
Ideal Magnetohydrodynamic ◽  
Image Position ◽  
The Continuum ◽  
The Ideal

General properties of the eigenvalues of Schaper (1983), concerning the continuum of eigenvalues on p. 7, needs correction. The solutions in the distributional sense of the eigenvalue problem (5.3), will be given for two cases. The first case has been solved by A. Lifshitz and E. Fedorov and concerns continuous eigenvalues. In the second case, the solution for the points of accumulation of discrete eigenvalues is discussed.

An alternative to Wiener-Hopf methods for the study of bounded processes

1964 ◽  
Vol 1 (1) ◽  
pp. 85-120 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Waiting Time ◽  
Finite Interval ◽  
Time Process ◽  
First Passage ◽  
Discrete Parameter ◽  
Waiting Time Process ◽  
Image Position ◽  
Wald's Identity ◽  
Independent Identically Distributed ◽  
The Continuum

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξn are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

scholarly journals COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS

2018 ◽  
Vol 83 (1) ◽  
pp. 208-236 ◽  
Author(s):  
VERA FISCHER ◽  
SY D. FRIEDMAN ◽  
DIEGO A. MEJÍA ◽  
DIANA C. MONTOYA
Keyword(s):  
Three Dimensional ◽  
Finite Support ◽  
Coherent Systems ◽  
Cichoń’S Diagram ◽  
Image Position ◽  
Generic Extensions

AbstractWe introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń’s diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ${\rm{\Delta }}_3^1$ well-order of the reals.

scholarly journals Determination of the Interstellar Extinction Toward WN Stars from Observed UBV Color Indices

1991 ◽  
Vol 143 ◽  
pp. 640-640
Author(s):  
William D. Vacca ◽  
Werner Schmutz
Keyword(s):  
Power Law ◽  
Theoretical Models ◽  
Interstellar Extinction ◽  
High Quality ◽  
Central Wavelength ◽  
Color Indices ◽  
Image Position ◽  
The Continuum

Recent observations of Wolf-Rayet stars by Massey (1984) and Torres-Dodgen and Massey (1988) have yielded high quality, absolutely calibrated spectra of nearly all known Wolf- Rayet objects. These observations also indicate that discontinuities, or “jumps”, are present in the continuum spectra of some Wolf-Rayet stars. Such continuum jumps are predicted by the current theoretical models of Wolf-Rayet atmospheres. In general, these models provide good fits to the observed spectra of Wolf-Rayet stars. The models also indicate that, between jumps, the intrinsic continuum can be closely approximated by a power law in wavelength. In this case, we have the following relation between the intrinsic colors: where and Z3645 is the strength of the He II (η = 4) jump at 3645 Å in mags. This relation holds because the central wavelength of the u filter (λu = 3650 Å) is nearly coincident with that of the He II jump. In addition, models of WN stars with helium-dominated atmospheres predict a correlation between D3645 and (6 - v)o:

An unbounded spectral mapping theorem

1968 ◽  
Vol 8 (1) ◽  
pp. 119-127 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Complex Numbers ◽  
Spectral Mapping Theorem ◽  
Complex Polynomials ◽  
Spectral Mapping ◽  
Image Position ◽  
Densely Defined

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

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