scholarly journals Aleph–zero categorical Stone algebras

1978 ◽  
Vol 26 (3) ◽  
pp. 337-347 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Boolean Algebra ◽  
Characterization Problem ◽  
A Chain ◽  
Dense Set

AbstractThis paper is a contribution to the problem of characterizing the ℵ0-categorical Stone algebras. If the dense set is either finite or a chain, the problem is solved by reducing it to the ℵ0-categoricity of the skeleton and the dense set, solutions for these being known. If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras. For generalized Post algebras the characterization problem is solved completely.

scholarly journals A class of projective Stone algebras

1981 ◽  
Vol 24 (1) ◽  
pp. 133-147 ◽  
Author(s):  
Ivo Düntsch
Keyword(s):  
Boolean Algebra ◽  
Stone Algebra ◽  
Double Stone Algebra ◽  
Dense Set

We prove that a regular double Stone algebra is protective in the category of Stone algebras if and only if its centre is a projective Boolean algebra and its dense set is countably generated as a filter. It follows that every countable regular double Stone algebra is projective as a Stone algebra.

Chains and antichains in interval algebras

1994 ◽  
Vol 59 (3) ◽  
pp. 860-867
Author(s):  
M. Bekkali
Keyword(s):  
Boolean Algebra ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Interval Algebra ◽  
A Chain

AbstractLet κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2ℵ0 = ℵω1 that there is a boolean algebra B of size ℵω1 such that length(B) = ℵω1 is not attained and the incomparability of B is less than ℵω1. Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].

scholarly journals Strictly positive measures on Boolean algebras

2008 ◽  
Vol 73 (4) ◽  
pp. 1416-1432 ◽  
Author(s):  
Mirna Džamonja ◽  
Grzegorz Plebanek
Keyword(s):  
Boolean Algebra ◽  
Positive Measure ◽  
Finite Measure ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Radon Measures ◽  
Finitely Additive Measures ◽  
Nonatomic Measure ◽  
A Chain ◽  
Additive Measures

AbstractWe investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.

One more aspect of forcing and omitting types

1976 ◽  
Vol 41 (1) ◽  
pp. 73-80
Author(s):  
Zofia Adamowicz
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Canonical Models ◽  
Defining Relations ◽  
Original Sense ◽  
Tarski Algebra ◽  
Cohen Forcing ◽  
Omitting Types ◽  
True Relation ◽  
Dense Set

As has been shown by the investigations of some mathematicians, there are numerous analogies between forcing and omitting types. Evidence of that can also be found in the application of forcing to model theory. Both methods use the Rasiowa-Sikorski lemma on the existence in a Boolean algebra of an ultrafilter intersecting every dense set from a given denumerable family.In this paper we will not use the terms Cohen forcing and omitting types in their original sense. We shall deal mainly with the Scott-Boolean kind of forcing as a transformation of Cohen's idea and with methods used in logic that consist in finding a suitable ultrafilter in the Lindenbaum-Tarski algebra for a given theory and defining canonical models—under the name of omitting types.The two methods will be confronted to show that Cohen's forcing stems from logical methods. Attention will also be drawn to some differences between forcing in set theory and the general methods of logic.In logic we construct a model by defining relations in a set of constants. In particular when defining a model for set theory we define a certain relation E in a set of constants. It is usually immaterial what E is, in particular whether it is the true relation of membership up to isomorphism.

Observation of Atom Rows in Thorium Pyromellitate Using a Field Emission STEM

1977 ◽  
Vol 35 ◽  
pp. 80-81
Author(s):  
H. Todokoro ◽  
S. Nomura ◽  
T. Komoda
Keyword(s):  
Field Emission ◽  
Amorphous Carbon ◽  
Carbon Film ◽  
Dark Field Image ◽  
Dark Field ◽  
Amorphous Carbon Film ◽  
Atomic Size ◽  
Wide Range ◽  
A Chain

It is interesting to observe polymers at atomic size resolution. Some works have been reported for thorium pyromellitate by using a STEM (1), or a CTEM (2,3). The results showed that this polymer forms a chain in which thorium atoms are arranged. However, the distance between adjacent thorium atoms varies over a wide range (0.4-1.3nm) according to the different authors.The present authors have also observed thorium pyromellitate specimens by means of a field emission STEM, described in reference 4. The specimen was prepared by placing a drop of thorium pyromellitate in 10-3 CH3OH solution onto an amorphous carbon film about 2nm thick. The dark field image is shown in Fig. 1A. Thorium atoms are clearly observed as regular atom rows having a spacing of 0.85nm. This lattice gradually deteriorated by successive observations. The image changed to granular structures, as shown in Fig. 1B, which was taken after four scanning frames.

Time-Resolved Cryo-Electron Microscopy of Oscillating Microtubules

1990 ◽  
Vol 48 (1) ◽  
pp. 502-503
Author(s):  
Eva-Maria Mandelkow ◽  
Ron Milligan
Keyword(s):  
Electron Microscopy ◽  
Dynamic Instability ◽  
Entire Solution ◽  
Reaction Scheme ◽  
Time Resolved ◽  
X Ray ◽  
X Ray Scattering ◽  
A Chain ◽  
Ray Scattering

Microtubules form part of the cytoskeleton of eukaryotic cells. They are hollow libers of about 25 nm diameter made up of 13 protofilaments, each of which consists of a chain of heterodimers of α-and β-tubulin. Microtubules can be assembled in vitro at 37°C in the presence of GTP which is hydrolyzed during the reaction, and they are disassembled at 4°C. In contrast to most other polymers microtubules show the behavior of “dynamic instability”, i.e. they can switch between phases of growth and phases of shrinkage, even at an overall steady state [1]. In certain conditions an entire solution can be synchronized, leading to autonomous oscillations in the degree of assembly which can be observed by X-ray scattering (Fig. 1), light scattering, or electron microscopy [2-5]. In addition such solutions are capable of generating spontaneous spatial patterns [6].In an earlier study we have analyzed the structure of microtubules and their cold-induced disassembly by cryo-EM [7]. One result was that disassembly takes place by loss of protofilament fragments (tubulin oligomers) which fray apart at the microtubule ends. We also looked at microtubule oscillations by time-resolved X-ray scattering and proposed a reaction scheme [4] which involves a cyclic interconversion of tubulin, microtubules, and oligomers (Fig. 2). The present study was undertaken to answer two questions: (a) What is the nature of the oscillations as seen by time-resolved cryo-EM? (b) Do microtubules disassemble by fraying protofilament fragments during oscillations at 37°C?

Do Event-Related Brain Potentials Reflect Mental (Cognitive) Operations?

2002 ◽  
Vol 16 (3) ◽  
pp. 129-149 ◽  
Author(s):  
Boris Kotchoubey
Keyword(s):  
Cognitive Processes ◽  
Point Of View ◽  
Internal Variables ◽  
Brain Potentials ◽  
Cognitive Operations ◽  
External Variables ◽  
Series Of Experiments ◽  
Mental Operations ◽  
A Chain ◽  
Processing Mechanisms

Abstract Most cognitive psychophysiological studies assume (1) that there is a chain of (partially overlapping) cognitive processes (processing stages, mechanisms, operators) leading from stimulus to response, and (2) that components of event-related brain potentials (ERPs) may be regarded as manifestations of these processing stages. What is usually discussed is which particular processing mechanisms are related to some particular component, but not whether such a relationship exists at all. Alternatively, from the point of view of noncognitive (e. g., “naturalistic”) theories of perception ERP components might be conceived of as correlates of extraction of the information from the experimental environment. In a series of experiments, the author attempted to separate these two accounts, i. e., internal variables like mental operations or cognitive parameters versus external variables like information content of stimulation. Whenever this separation could be performed, the latter factor proved to significantly affect ERP amplitudes, whereas the former did not. These data indicate that ERPs cannot be unequivocally linked to processing mechanisms postulated by cognitive models of perception. Therefore, they cannot be regarded as support for these models.

Right-Hemisphere Specialization for Contour Grouping

2014 ◽  
Vol 61 (5) ◽  
pp. 331-339 ◽  
Author(s):  
Gregor Volberg
Keyword(s):  
Right Hemisphere ◽  
Contour Detection ◽  
Global Processing ◽  
Detection Accuracy ◽  
Frequency Spectra ◽  
Visual Hemifield ◽  
Spatial Frequencies ◽  
A Chain ◽  
The Right ◽  
Hemisphere Specialization

Previous studies often revealed a right-hemisphere specialization for processing the global level of compound visual stimuli. Here we explore whether a similar specialization exists for the detection of intersected contours defined by a chain of local elements. Subjects were presented with arrays of randomly oriented Gabor patches that could contain a global path of collinearly arranged elements in the left or in the right visual hemifield. As expected, the detection accuracy was higher for contours presented to the left visual field/right hemisphere. This difference was absent in two control conditions where the smoothness of the contour was decreased. The results demonstrate that the contour detection, often considered to be driven by lateral coactivation in primary visual cortex, relies on higher-level visual representations that differ between the hemispheres. Furthermore, because contour and non-contour stimuli had the same spatial frequency spectra, the results challenge the view that the right-hemisphere advantage in global processing depends on a specialization for processing low spatial frequencies.

A chain model for chromosomes

1961 ◽  
Vol 58 ◽  
pp. 1072-1077 ◽  
Author(s):  
Frank Stahl
Keyword(s):  
Chain Model ◽  
A Chain
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