Definable Subsets Research Articles

Vaught’s Two-Cardinal Theorem and Notions of Minimality in Continuous Logic

Much of the work in this thesis was motivated by an effort to prove a continuous analogue of the Baldwin-Lachlan characterization of uncountable categoricity: a theory T in a countable language is uncountably categorical (has one model up to isomorphism of size kappa for some uncountable kappa) if and only if T has no Vaughtian pairs and T is omega-stable. As in the classical setting, we approach the forward direction by proving a continuous version of Vaught's Two-Cardinal theorem. A continuous theory T has a (kappa, lambda)-model if there is M, a model of T, with density character kappa which has a definable subset with density character lambda. We show that if T has a (kappa, lambda)-model for infinite cardinals kappa>lambda, then T has an (aleph_1, aleph_0)-model. We also show that with the additional assumption that T is omega-stable, if T has an (aleph_1, aleph_0)-model, then for any uncountable kappa, T has a (kappa, aleph_0)-model. This provides us with the tools necessary to prove the forward direction of the Baldwin-Lachlan characterization of uncountable categoricity for continuous logic. Towards the reverse direction, we introduce a continuous notion of strong minimality, saying that a set is minimal if every subset which is the zero set of a definable predicate is totally bounded, or its approximate complements are all totally bounded. We show that this characterization is equivalent to the classical definition of strong minimality, and that it allows us to use algebraic closure to define a notion of dimension which determines models up to isomorphism. However, the only known examples of strongly minimal theories in the continuous setting are classical theories viewed as continuous with the discrete metric, and we see that this notion lacks the machinery necessary to prove the reverse direction of the Baldwin-Lachlan characterization of uncountable categoricity. In an effort to better understand minimality in continuous logic, we also introduce a continuous notion of dp-minimality, and provide a few equivalent characterizations. We use these to show that the theory of Infinite Dimensional Hilbert spaces is a natural example of a dp-minimal continuous theory.

Definable one-dimensional topologies in O-minimal structures

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right) $$ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

What Becomes of the Broken-Hearted? Unconscionable Conduct, Emotional Dependence, and the ‘Clouded Judgment’ Cases

The High Court’s decision in Louth v Diprose that emotional dependence significantly contributed to special disadvantage was a significant development within the doctrine of unconscionable conduct. The decision in Louth established a template of sorts that found useful application in the later cases of Williams v Maalouf, Xu v Lin and Mackintosh v Johnson. Though they are few, these cases form definable subset within the broader doctrine of unconscionable conduct that might broadly be termed ‘clouded judgment’ cases. These cases quite arguably blur the lines between the doctrines of unconscionable conduct and undue influence. There is a discernible pattern to these matters. In these cases, the donor has formed an attachment to the object of his or her affection. To put matters gently, the affection is misplaced. Nonetheless, the donor makes a gift to the object of his or her affection. Subsequent developments lead the donor to realise that the gift was both improvident and bestowed upon an undeserving party. This article argues that Louth v Diprose is a troublesome precedent. First, the primacy of deception, which was a key issue in Louth, is unduly reductive. It obscures the overall context of the defendant’s conduct. Secondly, the High Court in Louth overlooked facts that might have undermined the finding that the plaintiff was at a special disadvantage. Thirdly, the case reflects a concept, known as the ‘presumption of competency’ that unhelpfully tilted the balance in favour of the plaintiff. This presumption appears to have been somewhat reversed in Mackintosh.

INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

AbstractWe give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

The canonical topology on dp-minimal fields

We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

Definable sets containing productsets in expansions of groups

Abstract We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x ⋅ y ∈ A {x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.

Existentially generated subfields of large fields

We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated.Let L be a large field of characteristic exponent p, and let E⊆L be an infinite existentially generated subfield. We show that E contains L(pn), the pn-th powers in L, for some n<ω. This generalises a result of Fehm from [4], which shows E=L, under the assumption that L is perfect. Our method is to first study existentially generated subfields of henselian fields. Since L is existentially closed in the henselian field L((t)), our result follows.

Ordinal definable subsets of singular cardinals

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ⊆ κ.

Characterization of Mouse γδ T Cell Subsets in the Setting of Type-2 Immunity.

Accumulating evidence indicates that γδ T cells are a critical component of type-2 immunity. However, the role of these cells in type-2 immune responses seems to be divergent. γδ T cells are heterogeneous lymphocytes that can be further divided into TCR-Vγ/δ definable subsets. Different subsets have distinct and sometimes opposite function during immune responses. In this chapter, we describe the detailed protocol for characterization of γδ T cell subsets in a mouse model of ovalbumin (OVA)/alum-induced type-2 immunity. Our protocol includes identifying γδ T cell subsets by flow cytometry, functionally inactivating individual subsets in vivo, purifying γδ T cell subsets, and using adoptive cell transfer to explore the role of individual subsets in OVA/alum-induced IgE responses.

Semilinear stars are contractible

Let $ {\mathcal {R}}$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture of Edmundo et al. (201

Universally and existentially definable subsets of global fields

Universally and existentially definable subsets of global fields

Topological structures of a type of granule based covering rough sets

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

General Extensional Mereology is Finitely Axiomatizable

Mereology is the theory of the relation “being a part of”. The first exact formulation of mereology is due to the Polish logician Stanislaw Leśniewski. But Leśniewski’s mereology is not first-order axiomatizable, for it requires every subset of the domain to have a fusion. In recent literature, a first-order theory named General Extensional Mereology (GEM) can be thought of as a first-order approximation of Leśniewski’s theory, in the sense that GEM guarantees that every definable subset of the domain has a fusion, and this has been achieved by positing an axiom schema which in effect defines infinitely many axioms. Intuitively, in order to range over every definable subset, such an axiom schema seems unavoidable. But this paper will show that GEM is finitely axiomatizable and that all we need are just finitely many instances of the said axiom schema.

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

AbstractWe extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ&lt;λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from elements ofλOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal aboveλ. We further obtain some related results about definable functions onλλand consequences of resurrection axioms for definable subsets ofλλ.

Local Ramsey theory: an abstract approach

AbstractGiven a topological Ramsey space , we extend the notion of semiselective coideal to sets and study conditions for that will enable us to make the structure a Ramsey space (not necessarily topological) and also study forcing notions related to which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space (cf. [][]). This extends results from [][] to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space , under suitable large cardinal hypotheses every semiselective ultrafilter is generic over ; and that given a semiselective coideal , every definable subset of is ‐Ramsey. This generalizes the corresponding results for the case when is equal to Ellentuck's space (cf. [][]).

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

The role of pertinently diversified and balanced training as well as testing data sets in achieving the true performance of classifiers in predicting the antifreeze proteins

Antifreeze proteins (AFPs) are those proteins, which inhibit the ice nucleation process and thereby enabling certain organisms to survive under sub-zero temperature habitats. AFPs are supposed to be evolved from different types of protein families to perform the unique function of antifreeze activity and turn out to be the classical example of convergent evolution. The common sequence similarity search methods have failed to predict putative AFPs due to poor sequence and structural similarity that exists among the different sub-types of AFP. The machine learning techniques are the viable alternative approaches to predict putative AFPs. In this paper, we have discussed about the criteria (like apposite feature selection, balanced data sets and complete learning) that are needed to be taken into account for successful application of machine learning methods and implemented these criteria by using a clustering procedure in order to achieve the true performance of the learning algorithms. Diversified and representative training and testing data sets are very crucial for perfect learning as well as true testing of machine learning based prediction methods for two reasons: first is that a training dataset that lacks definable subset of input patterns makes prediction of patterns belonging to this subset either difficult or unfeasible (thus resulting in incomplete learning) and secondly a testing data set that lacks definable subset of input patterns does not tell about whether this subset of patterns can be correctly predicted by the classifier or not (thus resulting in incomplete testing). Moreover, balanced training and testing data sets are equally important for achieving the true (robust) performance of classifiers because a well-balanced training set eliminates bias of the classifier toward particular class/sub-class due to over-representation or under-representation of input patterns belonging to those classes/sub-classes. We have used K-means clustering algorithm for creating the diversified and balanced training as well as testing data sets, to overcome the shortcoming of random splitting, which cannot guarantee representative training and testing sets. The current clustering based optimal splitting criteria proved to be better than random splitting for creating training and testing set in terms of superior generalization and robust evaluation.

Definable Groups for Dependent and 2-Dependent Theories

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\bar\kappa$ type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is $\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.

Identifying socio-demographic and clinical characteristics associated with medication beliefs about aromatase inhibitors among postmenopausal women with breast cancer.

Non-adherence/persistence to adjuvant endocrine therapy can negatively impact survival. Beliefs about medicines are known to affect adherence. This study aims to identify socio-demographic and clinical characteristics associated with medication beliefs among women taking aromatase inhibitors (AIs). Women completed an online survey on beliefs about AI therapy [Beliefs about Medicines Questionnaire (BMQ)], beliefs about breast cancer [Assessment of Survivor Concerns scale (ASC)], and depression [Personal Health Questionnaire depression scale (PHQ-8)]. Socio-demographic and clinical characteristics were collected. Bivariate analyses and linear regression models were performed to investigate relationships between variables. A total of 224 women reported currently taking AI therapy and were included in the analysis. Significantly higher concern beliefs were found among women who had at least mild depression, experienced side effects from AIs, and previously stopped therapy with another AI. Significant correlations were found between concern and necessity beliefs and cancer and health worry. Women age 70 and older displayed less fear of cancer recurrence and health worry, and a trend towards lower necessity and concern beliefs. No differences were found for other variables. In the regression model, greater necessity beliefs were found with increases in the number of current prescription medications (B=1.06, 95% CI 0.31-1.81, p=0.006) and shorter duration of current AI therapy (B=-0.65, 95% CI -1.23 to -0.07, p=0.029), whereas greater concern beliefs were associated with higher depression scores (B=1.19, 95% CI 0.35-2.03, p=0.006). Medication necessity and concern beliefs were associated with a definable subset of patients who may be at higher risk for non-persistence.

CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

AbstractWe prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.