A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $\lambda_s$ (the algebraic convergence) and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $\lambda_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $\lambda_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement "The convergence $\lambda_{ls}$ on the collapsing algebra B=\ro ((\omega_2)^{<\omega}) is weakly topological" is independent of ZFC

A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Weight Gain Is Associated with Medial Contact Site of Subthalamic Stimulation in Parkinson's Disease

The aim of our study was to assess changes in body-weight in relation to active electrode contact position in the subthalamic nucleus. Regular body weight measurements were done in 20 patients with advanced Parkinson's disease within a period of 18 months after implantation. T1-weighted (1.5T) magnetic resonance images were used to determine electrode position in the subthalamic nucleus and the Unified Parkinson's disease rating scale (UPDRS-III) was used for motor assessment. The distance of the contacts from the wall of the third ventricle in the mediolateral direction inversely correlated with weight gain (r = −0.55, p<0.01) and with neurostimulation-related motor condition expressed as the contralateral hemi-body UPDRS-III (r = −0.42, p<0.01). Patients with at least one contact within 9.3 mm of the wall experienced significantly greater weight gain (9.4±(SD)4.4 kg, N = 11) than those with both contacts located laterally (3.9±2.7 kg, N = 9) (p<0.001). The position of the active contact is critical not only for motor outcome but is also associated with weight gain, suggesting a regional effect of subthalamic stimulation on adjacent structures involved in the central regulation of energy balance, food intake or reward

Dystonia in neurodegeneration with brain iron accumulation: outcome of bilateral pallidal stimulation

Neurodegeneration with brain iron accumulation encompasses a heterogeneous group of rare neurodegenerative disorders that are characterized by iron accumulation in the brain. Severe generalized dystonia is frequently a prominent symptom and can be very disabling, causing gait impairment, difficulty with speech and swallowing, pain and respiratory distress. Several case reports and one case series have been published concerning therapeutic outcome of pallidal deep brain stimulation in dystonia caused by neurodegeneration with brain iron degeneration, reporting mostly favourable outcomes. However, with case studies, there may be a reporting bias towards favourable outcome. Thus, we undertook this multi-centre retrospective study to gather worldwide experiences with bilateral pallidal deep brain stimulation in patients with neurodegeneration with brain iron accumulation. A total of 16 centres contributed 23 patients with confirmed neurodegeneration with brain iron accumulation and bilateral pallidal deep brain stimulation. Patient details including gender, age at onset, age at operation, genetic status, magnetic resonance imaging status, history and clinical findings were requested. Data on severity of dystonia (Burke Fahn Marsden Dystonia Rating Scale—Motor Scale, Barry Albright Dystonia Scale), disability (Burke Fahn Marsden Dystonia Rating Scale—Disability Scale), quality of life (subjective global rating from 1 to 10 obtained retrospectively from patient and caregiver) as well as data on supportive therapy, concurrent pharmacotherapy, stimulation settings, adverse events and side effects were collected. Data were collected once preoperatively and at 2–6 and 9–15 months postoperatively. The primary outcome measure was change in severity of dystonia. The mean improvement in severity of dystonia was 28.5% at 2–6 months and 25.7% at 9–15 months. At 9–15 months postoperatively, 66.7% of patients showed an improvement of 20% or more in severity of dystonia, and 31.3% showed an improvement of 20% or more in disability. Global quality of life ratings showed a median improvement of 83.3% at 9–15 months. Severity of dystonia preoperatively and disease duration predicted improvement in severity of dystonia at 2–6 months; this failed to reach significance at 9–15 months. The study confirms that dystonia in neurodegeneration with brain iron accumulation improves with bilateral pallidal deep brain stimulation, although this improvement is not as great as the benefit reported in patients with primary generalized dystonias or some other secondary dystonias. The patients with more severe dystonia seem to benefit more. A well-controlled, multi-centre prospective study is necessary to enable evidence-based therapeutic decisions and better predict therapeutic outcomes

Loss-of-Function Variants in HOPS Complex Genes VPS16 and VPS41 Cause Early Onset Dystonia Associated with Lysosomal Abnormalities.

OBJECTIVES: The majority of people with suspected genetic dystonia remain undiagnosed after maximal investigation, implying that a number of causative genes have not yet been recognized. We aimed to investigate this paucity of diagnoses. METHODS: We undertook weighted burden analysis of whole-exome sequencing (WES) data from 138 individuals with unresolved generalized dystonia of suspected genetic etiology, followed by additional case-finding from international databases, first for the gene implicated by the burden analysis (VPS16), and then for other functionally related genes. Electron microscopy was performed on patient-derived cells. RESULTS: Analysis revealed a significant burden for VPS16 (Fisher's exact test p value, 6.9 × 109 ). VPS16 encodes a subunit of the homotypic fusion and vacuole protein sorting (HOPS) complex, which plays a key role in autophagosome-lysosome fusion. A total of 18 individuals harboring heterozygous loss-of-function VPS16 variants, and one with a microdeletion, were identified. These individuals experienced early onset progressive dystonia with predominant cervical, bulbar, orofacial, and upper limb involvement. Some patients had a more complex phenotype with additional neuropsychiatric and/or developmental comorbidities. We also identified biallelic loss-of-function variants in VPS41, another HOPS-complex encoding gene, in an individual with infantile-onset generalized dystonia. Electron microscopy of patient-derived lymphocytes and fibroblasts from both patients with VPS16 and VPS41 showed vacuolar abnormalities suggestive of impaired lysosomal function. INTERPRETATION: Our study strongly supports a role for HOPS complex dysfunction in the pathogenesis of dystonia, although variants in different subunits display different phenotypic and inheritance characteristics. ANN NEUROL 2020;88:867-877