A direct proof of the Feferman-Vaught theorem and other preservation theorems in products

1991 ◽  
Vol 56 (2) ◽  
pp. 632-636 ◽  
Author(s):  
Yiannis Vourtsanis
Keyword(s):  
Direct Proof ◽  
The Other ◽  
Direct Products ◽  
First Order ◽  
New Methods ◽  
Preservation Theorems ◽  
Order Language ◽  
Reduced Products ◽  
The One ◽  
By Products

Here we give short and direct proofs of the Feferman-Vaught theorem and other preservation theorems in products of structures. In 1952, Mostowski [5] first showed the preservation of ≡ωω by direct powers of structures. Subsequently, in 1959, Feferman and Vaught [2] proved the preservation of ≡ωω by arbitrary direct products and also by reduced products with respect to cofinite filters. In 1962, Frayne, Morel and Scott [3] noticed that the results extend to arbitrary reduced products. In 1970, Barwise and Eklof [1] showed the preservation of ≡∞λ by products and in 1971 Malitz [4] showed the preservation of ≡κλ with κ strongly inaccessible (or ∞) by products. Below, we give short proofs of the above results. The ideas used here have initiated, in [7], [8], [9], [10], [11], the introduction of several new methods in the theory of products, which on the one hand give new, direct proofs of the known results in the area, including generalizations or strengthenings of some of those, and, on the other hand, give several new results as well in the theory of products and related areas.Below, L denotes a (first order) language, and by a structure we mean an L-structure. 0 and 1 denote the logically valid and false sentence, respectively. We may write ā ∈ A for ā ∈ An for some n. Also, the values 1 and 2 of a parameter l in the definitions below express a duality corresponding to disjunctive and conjunctive forms.

scholarly journals L(Q)-preservation theorems

1975 ◽  
Vol 40 (3) ◽  
pp. 410-418
Author(s):  
Jörg Flum
Keyword(s):  
Direct Products ◽  
First Order ◽  
Product Operation ◽  
Preservation Theorems ◽  
Order Language

Much effort has been spent to prove that the reduced product operation preserves, and sometimes even strengthens the L-equivalence of structures, where L is some infinitary language. A similar result is suggested by the following well-known fact:Assume D is a nonprincipal ultrafilter on ω and, for n Є ω, Cn is a set. If the ultra-product ΠωCn/D is infinite, it has a cardinality ≥ Hence, by Łos' theorem, (i) if and φ(x) is a first-order formula, then iff where Q is the unary quantifier “there are many.”We shall prove some generalizations of (i). In particular, we show(ii) if D is a nonprincipal ultrafilter over I = ω, and then where L(Q) is the language obtained from the first-order language by adding the quantifier Q.(ii) remains true, if D is an ω-regular or an atomless filter over a set I.Lipner [7] proved that if is regular, then the L(Q)-equivalence is preserved under direct products. We show that the assumption “ is regular” is necessary.

Table II : Quantitative determination of carbonyl compounds at different odour sources (concentrations in ppb) Rendering plant Gelatine plant neighbourhood neighbourhood Formaldehyde 40 16 Acetaldehyde 39 24 Acetone 36 73 Prcpanal 10 -Isobutyraldehyde 10 30 Pentanal 15 19 Hexanal 3.52 Heptanal 12.5 Octanal 10.5 Nonanal 1 2 acids (figure 7). However extractions always involve a serious decrease in sensitivity, while evaporation of the extract produces a solution in 0.1-0.5 ml of solvent, and only 1 pi of it can be brought in the gas chromatograph. Therefore work is in progress to enhance sensitivity by converting acids in­ to halogenated derivatives, which can be GC-analysed with the more sensitive electron-capture detector. For thiols a similar procedure is investigated as with aldehydes. One possibility is absorption of thiols in an alkaline solution and reaction with 2,4-dinitrochlorobenzene, yielding 2,4-dinitrofenylsulfides, which are analysed by HPLC (9). Sane improvements on removal of reagents at the one hand and on separation of sane by-products on the other hand have to be achieved in order to in­ crease the sensitivity with another factor of ten. 5. CONCLUSION The actual scope and limitations of chemical analysis of odour show that all problems can be tackled as far as emission is concerned. For iititiission measurements seme progress is necessary, but there is no essential reason why chemical analysis would be unable to attain the desired sensitivity for all types of odorants. There is no doubt that in a few years the last dif­ ficulties will be solved. In order to achieve real control of odour nui­ sance, automatic measurement is necessary on a long time basis. There again seme technical development is to be expected. Does this mean that machines are going to decide if an odour is pre­ sent or not? By no means, while the population will always be the reference, and psychophysical measurements will be necessary to make chemical analysis possible.

1986 ◽  
pp. 171-171
Keyword(s):  
Chemical Analysis ◽  
Quantitative Determination ◽  
Alkaline Solution ◽  
Automatic Measurement ◽  
The Other ◽  
Long Time ◽  
Time Basis ◽  
The One ◽  
By Products

On Axiomatizability of Non-Commutative Lp-Spaces

2007 ◽  
Vol 50 (4) ◽  
pp. 519-534
Author(s):  
C. Ward Henson ◽  
Yves Raynaud ◽  
Andrew Rizzo
Keyword(s):  
Hilbert Spaces ◽  
Normed Space ◽  
The Other ◽  
Space Structures ◽  
Operator Spaces ◽  
Infinite Dimensions ◽  
Schatten Classes ◽  
Lp Spaces ◽  
First Order ◽  
Order Language

AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

scholarly journals Role-Value Maps and General Concept Inclusions in the Minimal Description Logic with Value Restrictions or Revisiting Old Skeletons in the DL Cupboard

2020 ◽  
Vol 34 (3) ◽  
pp. 291-301
Author(s):  
Franz Baader ◽  
Clément Théron
Keyword(s):  
Description Logic ◽  
Direct Proof ◽  
General Concept ◽  
The Other ◽  
Other Hand ◽  
The One ◽  
Minimal Description ◽  
The Impact

Abstract We investigate the impact that general concept inclusions and role-value maps have on the complexity and decidability of reasoning in the description logic $$\mathcal{FL}_0$$ FL 0 . On the one hand, we give a more direct proof for ExpTime-hardness of subsumption w.r.t. general concept inclusions in $$\mathcal{FL}_0$$ FL 0 . On the other hand, we determine restrictions on role-value maps that ensure decidability of subsumption, but we also show undecidability for the cases where these restrictions are not satisfied.

scholarly journals Zinc(II) and copper(II) complexes with pheophytin and mesoporphyrin and their stability to UV-B irradiation: Vis spectroscopy studies

2012 ◽  
Vol 77 (2) ◽  
pp. 187-199 ◽  
Author(s):  
Jelena Zvezdanovic ◽  
Dejan Markovic ◽  
Sanja Milenkovic
Keyword(s):  
Heavy Metal ◽  
The Other ◽  
First Order ◽  
Absorbance Spectroscopy ◽  
First Order Kinetics ◽  
Porphyrin Derivatives ◽  
Vis Spectroscopy ◽  
Photochemical Decomposition ◽  
Spectroscopy Studies ◽  
The One

Stability of Zn(II) and Cu(II) complexes of porphyrin derivatives (pheophytin and mesoporphyrin) to UV-B -irradiation has been studied by absorbance spectroscopy in 95% ethanol. The chosen porphyrins as well as their heavy metal complexes undergo photochemical decomposition obeying first-order kinetics. In general, pheophytin is more stable than mesoporphyrin to UV-B irradiation. On the other hand, stability of Zn(II)-complex is smaller than Cu(II)-complex both for pheophytin and mesoporphyrin; however while Cu(II)-complex with pheophytin is more stable than the one with mesoporphyrin, with Zn(II)-complex the situation is vice versa.

FIRST ORDER PHASE TRANSITIONS, WHEN AND FROM WHERE DO THEY EMERGE? —THE CASE OF GELS

1995 ◽  
Vol 09 (07) ◽  
pp. 737-749 ◽  
Author(s):  
KEN SEKIMOTO
Keyword(s):  
Phase Transition ◽  
Transition Temperature ◽  
Transition Point ◽  
The Other ◽  
Basic Fact ◽  
Theoretical Understanding ◽  
First Order ◽  
First Order Phase Transition ◽  
The One ◽  
First Order Phase

We briefly review the recent theoretical understanding of the first order phase transition undergone by gels with an emphasis on physical concepts, deliberately excluding details of modeling and analytic methods. The density of a gel changes discontinuously at the transition point. A variety of features of the transition result from the basic fact that the inhomogeneity of the density of the gel inevitably causes shear deformation. This deformation, on the one hand, reflects the geometry of the sample and, on the other hand, may alter the transition temperature.

scholarly journals PROGNOSTICATION IN CHINESE BUDDHIST HISTORICAL TEXTS THE GĀOSĒNG ZHUÀN AND THE XÙ GĀOSĒNG ZHUÀN

2020 ◽  
Vol 73 (1) ◽  
pp. 1-45
Author(s):  
Gang Yang ◽  
Christoph Anderl
Keyword(s):  
The Other ◽  
Buddhist Literature ◽  
Historical Texts ◽  
Historical Literature ◽  
Short Survey ◽  
The Future ◽  
Future Events ◽  
The One ◽  
By Products

This paper explores topics and techniques of prognostication as recorded in medieval Buddhist historical literature, with an emphasis on the Gāosēng zhuàn 高僧傳 (GSZ) and Xù gāosēng zhuàn 續高僧傳 (XGSZ). The paper first provides a short survey of how prognostication is treated in Chinese Buddhist translated texts. In these ‘canonical’ sources there is clear ambiguity over the use of supernatural powers: on the one hand, such practices are criticised as non-Buddhist or even heterodox; on the other, narratives on Śākyamuni’s former and present lives as well as accounts of other buddhas, bodhisattvas, and the Buddha’s disciples abound with descriptions of their special abilities, including knowledge of the future. In contrast, the GSZ and XGSZ display a clear standpoint concerning mantic practices and include them as integral aspects of monastics’ lives. The two texts articulate that the ability to predict the future and other supernatural powers are natural by-products of spiritual progress in the Buddhist context. This paper discusses the incorporation of various aspects of the Indian and Chinese traditions in monastics’ biographies, and investigates the inclusion of revelations of future events (for example, in dreams) and mantic techniques in these texts. In addition, it traces parallels to developments in non-Buddhist literature and outlines some significant differences between the GSZ and the XGSZ.

Arctic Nomadic Design (The Nenets Case)

2020 ◽  
Vol 24 (1) ◽  
pp. 111-142
Author(s):  
Andrei V. Golovnev
Keyword(s):  
Extreme Environment ◽  
Space Time ◽  
3D Modelling ◽  
The Other ◽  
The Arctic ◽  
Yamal Peninsula ◽  
Gps Tracking ◽  
New Methods ◽  
Movement Effect ◽  
The One

The nomadic technologies of reindeer herders from Yamal Peninsula, in their multidimensional complexity – from the space-time continuity of mobile camps on the open tundra to multi-functionality of material things – enable and facilitate mobility in the extreme environment of the Arctic. The nomadic tradition contains a whole array of concepts (or principles), which, on the one hand, are ultimately practical and, on the other, deserve a theoretical projection. These include: nomadic transformer, mobile module, movement effect, techno-animation, material austerity, space-time continuity, arctic aesthetics. A study of nomadic design implies the usage of new methods of movement recording – MTA (mapping–tracking–acting), including visual data such as UAV mapping, GPS-tracking and 3D-modelling.

The power of locality domains in phonology

Phonology ◽  
2017 ◽  
Vol 34 (2) ◽  
pp. 385-405 ◽  
Author(s):  
Thomas Graf
Keyword(s):  
Computational Complexity ◽  
The Other ◽  
Adequate Model ◽  
First Order ◽  
Other Hand ◽  
Integral Role ◽  
Domain Restrictions ◽  
The One

Domains play an integral role in linguistic theories. This paper combines locality domains with current models of the computational complexity of phonology. The first result is that if a specific formalism – strictly piecewise grammars – is supplemented with a mechanism to enforce first-order definable domain restrictions, its power increases so much that it subsumes almost the full hierarchy of subregular languages. However, if domain restrictions are based on linguistically natural intervals, we instead obtain an empirically more adequate model. On the one hand, this model subsumes only those subregular classes that have been argued to be relevant for phonotactic generalisations. On the other hand, it excludes unnatural generalisations that involve counting or elaborate conditionals. It is also shown that strictly piecewise grammars with interval-based domains are theoretically learnable, unlike those with arbitrary, first-order domains.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

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