Auswahl der wissenschaftlichen Literatur zum Thema „Stationary set preserving“

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

AbstractA classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that ω2L is countable: {X ∈ L ∣ X ⊆ ω1L and X has a CUB subset in a cardinal-preserving extension of L} is constructive, as it equals the set of constructible subsets of ω1L which in L are stationary. Is there a similar such result for subsets of ω2L? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are “solvable” in the above sense, as well as denning a notion of reduction between them.

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

Todorčević showed that Rado's Conjecture (RC) implies CC*, a strengthening of Chang's Conjecture. We generalize this by showing that also CC**, a global version of CC*, follows from RC. As a corollary we obtain that RC implies Semistationary Reflection and (†), i.e. the statement that all forcings that preserve the stationarity of subsets of ω1 are semiproper.

Modal parameters can reflect the dynamic characteristics of the structure and can be used to control vibration. To identify the operational modal parameters of linear slow-time-varying structures only from non-stationary vibration response signals, a method based on moving window locality preserving projections (MWLPP) algorithm is proposed. Based on the theory of “time freeze”, the method selects a fixed length window and takes the displacement response signal in each window as a stationary random sequence. The locality preserving projections algorithm is used to identify the transient modal frequency and modal shape of the structure at this window. The low-dimensional embedding of the displacement response data set calculated by locality preserving projections (LPP) corresponds to the modal coordinate response matrix, and the transformation matrix corresponds to the modal shape matrix. The simulation results of the mass slow-time-varying three degree of freedom (DOF) and the density slow-time-varying cantilever beam show that the new method can effectively identify the modal shape and modal natural frequency of the linear slow-time-varying only from the non-stationary vibration response signal, and the performance is better than the moving window principal component analysis (MWPCA).

AbstractIn this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings, we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving linear-elliptic operators for the extension and the effect of additional nonlinear advection on the set of reachable shapes. It is moreover discussed how the computational complexity of the proposed algorithm can be reduced. The benefit of the nonlinearity in the extension operator is substantiated by several numerical test cases of stationary, incompressible Navier–Stokes flows in 2d and 3d.

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

Electronic tongue mimics human gustatory sensation and is used to characterize and discriminate beverages and foods. Feature extraction plays a key role in improving the classification accuracy by preserving the distinct characteristics while reducing high dimensionality of data generated from electronic tongue. This paper presents a new feature extraction method based on stationary wavelet singular entropy for a developed electronic tongue system to classify pasteurized cow milk. The electronic tongue consists of an array of five working electrodes along with a reference and a counter electrode to characterize milk sample. The feature extraction of acquired data is done by computing stationary wavelet transform to obtain detail and approximate coefficients at different level of decomposition. These coefficients are processed using singular value decomposition followed by calculation of entropy to obtain stationary wavelet singular entropy values. These values form the feature set and feed to two classifiers, k-nearest neighbor and back propagation artificial neural network, and their classification accuracy is evaluated with variation in their model parameters. The proposed method is compared with other wavelet transform-entropy methods in terms of classification accuracy, which indicates that the proposed method is more effective in discriminating milk samples.

Les principaux résultats de cette thèse sont liés au forcing, mais notre présentation bénéficie de sa mise en relation avec un autre domaine de la logique: la théorie des modèles des logiques infinitaires. Une idée clé de notre travail, qui était plus ou moins implicite dans les recherches de nombreux auteurs, est que le forcing joue un rôle en logique infinitaire similaire à celui joué par le théorème de compacité en logique du premier ordre. Plus précisément, de la même manière que le théorème de compacité est l'outil clé pour produire des modèles de théories du premier ordre, le forcing peut être l'outil clé pour produire les modèles des théories infinitaires. La première partie de cette thèse explore la relation entre les logiques infinitaires et les modèles à valeurs booléennes. Une propriété de consistance est une famille d'ensembles de formules non contradictoires, fermée sous certaines opérations logiques naturelles. Les propriétés de consistance reproduisent dans le contexte des logiques infinitaires la technique donnée par la méthode de résolution pour produire des modèles d'une formule du premier ordre; elles sont l'outil standard pour produire des modèles de formules infinitaires non contradictoires. Le premier résultat majeur que nous établissons dans cette thèse est le Théorème d'Existence des Modèles Booléens, affirmant que toute formule dans un ensemble qui est dans une propriété de consistance possède un modèle à valeurs booléennes avec la propriété de "mixing", et renforce le résultat original de Mansfield. Le Théorème d'Existence des Modèles Booléens nous permet de prouver trois résultats supplémentaires dans la théorie des modèles des logiques infinitaires munis de la sémantique des modèles à valeurs booléennes avec la propriété de ``mixing": un théorème de complétude par rapport à un calcul de type Gentzen, un théorème d'interpolation et un théorème d'omission des types. Cependant, nous croyons que le résultat central de cette partie de la thèse est le Théorème de Compacité Conservative. Dans la poursuite d'une généralisation de la compacité du premier ordre pour les logiques infinitaires, nous introduisons le concept de "renforcement conservatif" et de "conservativité finie". Nous soutenons que la généralisation appropriée de la consistance finie (relative à la sémantique de Tarski pour la logique du premier ordre) est la conservativité finie (relative à la sémantique donnée par les modèles à valeurs booléennes). À notre avis, ces résultats nous permettent de soutenir que: Les modèles à valeurs booléennes avec la propriété de "mixing" fournissent une sémantique naturelle pour les logiques infinies. Dans la seconde partie de la thèse, nous nous appuyons sur les résultats de la première partie pour aborder la question suivante: pour quelle famille de formules infinitaires peut-on forcer l'existence d'un modèle de Tarski sans détruire les sous-ensembles stationnaires? Kasum et Velickovic ont introduit une caractérisation des formules pour lesquelles un modèle de Tarski peut être forcé par un forcing préservant les ensembles stationnaires (AS-goodness). Leur travail s'appuie sur le résultat révolutionnaire d'Asperò et Schindler. Nous définissons la propriété ASK - une variante de l'AS-goodness - que nous utilisons également de la même manière que Kasum et Velickovic. Il est démontré que pour toute formule ayant la propriété ASK, on peut forcer l'existence d'un modèle de Tarski d'une manière qui préserve les ensembles stationnaires. La preuve de ce résultat s'appuie sur la perspective de la théorie des modèles de forcing présentée dans la première partie de la thèse, tout en introduisant une nouvelle notion de forcing itéré. Cette présentation du forcing itéré est étroitement liée au Théorème de Compacité Conservateur, soulignant à nouveau l'analogie entre les paires (forcing, logiques infinitaires) et (compacité, logique du premier ordre)
The main results of this thesis are related to forcing, but our presentation benefits from relating them to another domain of logic: the model theory of infinitary logics. In the 1950s, after the basic framework of first-order model theory had been established, Carol Karp, followed by Makkai, Keisler and Mansfield among others, developed the area of logic known as "infinitary logics". One key idea from our work, which was more or less implicit in the research of many, is that forcing plays a role in infinitary logic similar to the role compactness plays in first-order logic. Specifically, much alike compactness is the key tool to produce models of first-order theories, forcing can be the key tool to produce the interesting models of infinitary theories. The first part of this thesis explores the relationship between infinitary logics and Boolean valued models. Leveraging on the translation of forcing in the Boolean valued models terminology, this part lays the foundations connecting infinitary logics to forcing. A consistency property is a family of sets of non-contradictory sentences closed under certain natural logical operations. Consistency properties are the standard tools to produce models of non-contradictory infinitary sentences. The first major result we establish in the thesis is the Boolean Model Existence Theorem, asserting that any sentence which belongs to some set which is in some consistency property has a Boolean valued model with the mixing property, and strengthens Mansfield's original result. The Boolean Model Existence Theorem allows us to prove three additional results in the model theory of Boolean valued models for the semantics induced by Boolean valued models with the mixing property: a completeness theorem, an interpolation theorem, and an omitting types theorem. These can be shown to be generalizations of the corresponding results for first order logic in view of the fact that a first order sentence has a Tarski model if and only if it has a Boolean valued model. However we believe that the central result of this part of the thesis is the Conservative Compactness Theorem. In pursuit of a generalization of first-order compactness for infinitary logics, we introduce the concepts of conservative strengthening and of finite conservativity. We argue that the appropriate generalization of finite consistency (relative to Tarski semantics for first order logic) is finite conservativity (relative to the semantics given by Boolean valued models). The Conservative Compactness Theorem states that any finitely conservative family of sentences admits a Boolean valued model with the mixing property. In our opinion these results support the claim: Boolean-valued models with the mixing property provide a natural semantics for infinitary logics. In the second part of the thesis we leverage on the results of the first part to address the following question: For what family of infinitary formulae can we force the existence of a Tarski model for them without destroying stationary sets? Kasum and Velickovic introduced a characterization of which sentences can be forced by a stationary set preserving forcing (AS-goodness). Their work builds on the groundbreaking result of Asperò and Schindler. We define the ASK property -a variant of AS-goodness- which we also employ to the same effect of Kasum and Velickovic. It is shown that for any formula with the ASK-property, one can force the existence of a Tarski model in a stationary set preserving way. The proof of this result builds on the model theoretic perspective of forcing presented in the first part of the thesis, and does so introducing a new notion of iterated forcing. This presentation of iterated forcing is strictly intertwined with the Conservative Compactness Theorem, thereby emphasizing again the analogy between the pairs (forcing, infinitary logics) and (compactness, first-order logic)

The use of solid catalysts has a number of advantages compared to catalysis in solution. The most important advantage is their discrete state, usually stationary, which enables easy separation of the product from the catalyst. Although catalysis by solids in organic technology was largely restricted till about the mid-1970s to bulk chemicals produced by continuous processes, it has since been extended to organic intermediates and fine chemicals (which are usually medium to small-volume production in batch processes). We devote this chapter to a brief review of the major types of solid catalysts used in the production of intermediates and fine chemicals. Though these reactions can be carried out in both the vapor and liquid phases, the substrates used in organic synthesis are often relatively complex liquid molecules which tend to decompose under harsh conditions. Hence it is usually desirable to operate under softer conditions, thus preserving the liquid state of the substrate and preventing any likely decomposition to unwanted products. Because catalysis by solids will almost certainly play a major role in organic syntheses of the future, surface science studies involving complex organic molecules are being increasingly undertaken (see, e.g., Rylander, 1979, 1985; Molnar, 1985; Kim and Barteau, 1989; Joyner, 1990; Idriss et al., 1992; Schulz and Cox, 1992, 1993; Pierce and Barteau, 1995; and the recent review by Smith, 1996). However, this book will not be concerned with such mechanistic considerations. There are a few classes of catalysts that have acquired a degree of prominence during the last decade in the synthesis of organic intermediates and fine chemicals that marks them as uniquely relevant in the context of industry’s irreversible shift to green technology. In addition to the homogeneous catalysts considered subsequently in Chapter 9, they include a wide variety of solid catalysts. These catalysts can be classified in two ways: (1) as distinct classes of catalysts that cut across different types of reactions, including dissolved catalysts supported on solids; and (2) as catalysts specific to different types of reactions. It is also possible to control catalytic action by using appropriate solvents/additives.