0690 An Evaluation Of Genioglossus Strengthening On Obstructive Sleep Apnea Treatment Outcomes

Introduction Obstructive sleep apnea (OSA) is characterized by repetitive episodes of pharyngeal collapse. The genioglossus is a major upper airway dilator muscle thought to be important in OSA pathogenesis. Upper airway (UA) muscle training has reported benefits in some OSA patients. Our goal was to assess the effect of upper airway muscle training on OSA outcomes. Methods Sixty five patients with OSA (AHI>10/h) were divided in three subgroups: 1) Treated with auto-CPAP (n=21), 2) Previously failed or refused CPAP therapy (no treatment), (n=24), 3) Currently treated with an oral appliance who still have residual OSA (AHI>10/h), (n=20). All subjects were given a custom-made tongue strengthening device. Within each group we conducted a prospective, randomized, controlled study examining the effect of upper airway muscle training. In each subgroup, subjects were randomized to UA muscle training (volitional protrusion against resistance) or sham group (negligible resistance), with 1:1 ratio over 6 weeks of treatment (twice daily for 20 min/session). In the baseline and the final visit, subjects completed home sleep testing, questionnaires (ESS, PSQI), acoustic pharynogometry, Iowa Oral Performance Instrument (IOPI), and Psychomotor Vigilance Test (PVT). Results Results remain blinded; 33 patients received treatment Y and 32 patients received treatment Z. To date, we have not observed a main effect of treatment group on several measures of OSA severity. Some changes in subjective measures over time were observed but difficult to interpret until unblinding occurs. Conclusion Treatment of OSA using upper airway muscle training exercises requires further study. Whether muscle training is a viable approach for a definable subset of OSA patients remains unclear. Support R01HL085188-05A1 (U.S. NIH Grant/Contract)

Definable sets containing productsets in expansions of groups

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\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.

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

We 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 main theorem

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.

Γ‎-internal spaces

This chapter describes the topological structure of Γ‎-internal spaces. Let V be an algebraic variety over a valued field. An iso-definable subset X of unit vector V is said to be Γ‎-internal if it is in pro-definable bijection with a definable set which is Γ‎-internal. A number of delicate issues arise here. A pro-definable subset X of unit vector V is Γ‎-parameterized if there exists a definable subset Y of Γ‎ⁿ, for some n, and a pro-definable map g : Y → unit vector V with image X. The chapter presents an example showing that there exists Γ‎-parameterized subsets of unit vector V which are not iso-definable, whence not Γ‎-internal. It also presents the main results about the topological structure of Γ‎-internal spaces.

Strongly stably dominated points

This chapter focuses on the properties of strongly stably dominated types over valued fields bases. In this setting, strong stability corresponds to a strong form of the Abhyankar property for valuations: the transcendence degrees of the extension coincide with those of the residue field extension. The chapter proves a Bertini type result and shows that the strongly stable points form a strict ind-definable subset Vsuperscript Number Sign of unit vector V. It then proves a rigidity statement for iso-definable Γ‎-internal subsets of maximal o-minimal dimension of unit vector V, namely that they cannot be deformed by any homotopy leaving appropriate functions invariant. The chapter also describes the closure of iso-definable Γ‎-internal sets in Vsuperscript Number Sign and proves that Vsuperscript Number Sign is exactly the union of all skeleta.

CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

We 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.

VARIATIONS SUR UN THÈME DE ALDAMA ET SHELAH

We consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

EXISTENTIAL-IMPORT MATHEMATICS

First-order logic has limited existential import: the universalized conditional ∀x [S(x) → P(x)] implies its corresponding existentialized conjunction ∃x [S(x) & P(x)] in some but not all cases. We prove the Existential-Import Equivalence:∀x [S(x) → P(x)] implies ∃x [S(x) & P(x)] iff ∃x S(x) is logically true.The antecedent S(x) of the universalized conditional alone determines whether the universalized conditional has existential import: implies its corresponding existentialized conjunction.A predicate is a formula having only x free. An existential-import predicate Q(x) is one whose existentialization, ∃x Q(x), is logically true; otherwise, Q(x) is existential-import-free or simply import-free. Existential-import predicates are also said to be import-carrying.How widespread is existential import? How widespread are import-carrying predicates in themselves or in comparison to import-free predicates? To answer, let L be any first-order language with any interpretation INT in any [sc. nonempty] universe U. A subset S of U is definable in L under INT iff for some predicate Q(x) in L, S is the truth-set of Q(x) under INT. S is import-carrying definable iff S is the truth-set of an import-carrying predicate. S is import-free definable iff S is the truth-set of an import-free predicate.Existential-Importance Theorem: Let L, INT, and U be arbitrary. Every nonempty definable subset of U is both import-carrying definable and import-free definable.Import-carrying predicates are quite abundant, and no less so than import-free predicates. Existential-import implications hold as widely as they fail.A particular conclusion cannot be validly drawn from a universal premise, or from any number of universal premises.—Lewis-Langford, 1932, p. 62.