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The ADE correspondences are ubiquitous in mathematics. We begin with the regular polyhedra (known to the ancient Greeks) and invite the reader on a journey of discovery.
For the benefit of students, we provide an introduction to areas of mathematics we need: vector spaces, polytopes, groups (discrete and continuous), conjugacy representations, etc.
We treat some more advanced topics: monstrous (and other) moonshine, Monster and E_8, Niemeier lattices, the triangle property, generalized line graphs, quiver representations, cluster algebras, von Neumann algebras, catastrophes, Calabi–Yau, elliptic fibrations.
We discuss some areas where the ADE classification arises: polytopes, tessellations, root systems, Coxeter groups, spectra of graphs, binary polyhedral groups, reflections, Clifford algebras, Lie groups and algebras.
The ADE diagrams, shown on the cover, constitute one of the most universal and mysterious patterns in all of mathematics. John McKay's remarkable insights unveiled a connection between the 'double covers' of the groups of regular polyhedra, known since ancient Greek times, and the exceptional Lie algebras, recognised since the late nineteenth century. The correspondence involves the ADE diagrams being interpreted in different ways: as quivers associated with the groups and as Dynkin diagrams of root systems of Lie algebras. The ADE diagrams arise in many areas of mathematics, including topics in algebraic geometry, string theory, spectral theory of graphs and cluster algebras. Accessible to students, this book explains these connections with exercises and examples throughout. An excellent introduction for students and researchers wishing to learn more about this unifying principle of mathematics, it also presents standard undergraduate material from a novel perspective.
Background: Deep brain stimulation (DBS) in Parkinson’s disease (PD) requires extensive trial-and-error programming, often taking over a year to optimize. An objective, rapid biomarker of stimulation success is needed. Our team developed a functional magnetic resonance imaging (fMRI)-based algorithm to identify optimal DBS settings. This study prospectively compared fMRI-guided programming with standard-of-care (SoC) clinical programming in a double-blind, crossover, non-inferiority trial. Methods: Twenty-two PD-DBS patients were prospectively enrolled for fMRI using a 30-sec DBS-ON/OFF cycling paradigm. Optimal settings were identified using our published classification algorithm. Subjects then underwent >1 year of SoC programming. Clinical improvement was assessed under SoC and fMRI-determined stimulation conditions. Results: fMRI optimization significantly reduced the time required to determine optimal settings (1.6 vs. 5.6 months, p<0.001). Unified Parkinson’s Disease Rating Scale (UPDRSIII) improved comparably with both approaches (23.8 vs. 23.6, p=0.9). Non-inferiority was demonstrated within a predefined margin of 5 points (p=0.0018). SoC led to greater tremor improvement (p=0.019), while fMRI showed greater bradykinesia improvement (p=0.040). Conclusions: This is the first prospective evaluation of an algorithm able to suggest stimulation parameters solely from the fMRI response to stimulation. It suggests that fMRI-based programming may achieve equivalent outcomes in less time than SoC, reducing patient burden while potentially enhancing bradykinesia response.
Background: Patients with severe traumatic brain injury (TBI) are at uniquely high risk of venous thromboembolism (VTE), but the benefits of VTE prophylaxis must be weighed against the risk of intracranial hemorrhage expansion. Current guidelines are heterogenous in their recommendations for chemical VTE prophylaxis (cVTEp) in this high-risk cohort. We conducted a systematic review to identify the optimal timing of cVTEp in severe TBI patients. Methods: We executed a systematic search of the literature to identify adult severe TBI patients treated with cVTEp. Results were pooled, analyzed using random-effects models, and presented as Forest plots and odds ratios. Results: We included 21 studies representing 322,735 patients. The odds of VTE were 0.47 (95% CI: 0.37,0.60) when using the authors’ own criteria for early initiation, and the odds of VTE remained significantly decreased in subgroup analysis (<24h, <48 and <72h). Early VTEp both as defined by authors and in subgroup analysis did not significantly impact the odds of hemorrhage progression or mortality; except for initiation <48h which showed a positive impact on mortality (OR: 0.74, 95% CI: 0.63-0.87). Conclusions: This study supports early initiation of cVTEp in reducing the odds of VTE events without significantly increasing the risk of adverse events.
Depression is a complex mental health disorder with highly heterogeneous symptoms that vary significantly across individuals, influenced by various factors, including sex and regional contexts. Network analysis is an analytical method that provides a robust framework for evaluating the heterogeneity of depressive symptoms and identifying their potential clinical implications.
Objective:
To investigate sex-specific differences in the network structures of depressive symptoms in Asian patients diagnosed with depressive disorders, using data from the Research on Asian Psychotropic Prescription Patterns for Antidepressants, Phase 3, which was conducted in 2023.
Methods:
A network analysis of 10 depressive symptoms defined according to the National Institute for Health and Care Excellence guidelines was performed. The sex-specific differences in the network structures of the depressive symptoms were examined using the Network Comparison Test. Subgroup analysis of the sex-specific differences in the network structures was performed according to geographical region classifications, including East Asia, Southeast Asia, and South or West Asia.
Results:
A total of 998 men and 1,915 women with depression were analysed in this study. The analyses showed that all 10 depressive symptoms were grouped into a single cluster. Low self-confidence and loss of interest emerged as the most central nodes for men and women, respectively. In addition, a significant difference in global strength invariance was observed between the networks. In the regional subgroup analysis, only East Asian men showed two distinct clustering patterns. In addition, significant differences in global strength and network structure were observed only between East Asian men and women.
Conclusion:
The study highlights the sex-specific differences in depressive symptom networks across Asian countries. The results revealed that low self-confidence and loss of interest are the main symptoms of depression in Asian men and women, respectively. The network connections were more localised in men, whereas women showed a more diverse network. Among the Asian subgroups analysed, only East Asians exhibited significant differences in network structure. The considerable effects of neurovegetative symptoms in men may indicate potential neurobiological underpinnings of depression in the East Asian population.
It remains unclear which individuals with subthreshold depression benefit most from psychological intervention, and what long-term effects this has on symptom deterioration, response and remission.
Aims
To synthesise psychological intervention benefits in adults with subthreshold depression up to 2 years, and explore participant-level effect-modifiers.
Method
Randomised trials comparing psychological intervention with inactive control were identified via systematic search. Authors were contacted to obtain individual participant data (IPD), analysed using Bayesian one-stage meta-analysis. Treatment–covariate interactions were added to examine moderators. Hierarchical-additive models were used to explore treatment benefits conditional on baseline Patient Health Questionnaire 9 (PHQ-9) values.
Results
IPD of 10 671 individuals (50 studies) could be included. We found significant effects on depressive symptom severity up to 12 months (standardised mean-difference [s.m.d.] = −0.48 to −0.27). Effects could not be ascertained up to 24 months (s.m.d. = −0.18). Similar findings emerged for 50% symptom reduction (relative risk = 1.27–2.79), reliable improvement (relative risk = 1.38–3.17), deterioration (relative risk = 0.67–0.54) and close-to-symptom-free status (relative risk = 1.41–2.80). Among participant-level moderators, only initial depression and anxiety severity were highly credible (P > 0.99). Predicted treatment benefits decreased with lower symptom severity but remained minimally important even for very mild symptoms (s.m.d. = −0.33 for PHQ-9 = 5).
Conclusions
Psychological intervention reduces the symptom burden in individuals with subthreshold depression up to 1 year, and protects against symptom deterioration. Benefits up to 2 years are less certain. We find strong support for intervention in subthreshold depression, particularly with PHQ-9 scores ≥ 10. For very mild symptoms, scalable treatments could be an attractive option.
Spontaneous flow reversals in buoyancy-driven flows are ubiquitous in many fields of science and engineering, often characterized by violent, intermittent occurrences. In this study, we present a complex-network-based reduced-order model to analyse intermittent events in turbulent flows, using temporal and spatial snapshot data. This framework combines elements of dynamical system theory with network science. We demonstrate its utility by applying it to data sequences from intermittent flow reversal events in two-dimensional thermal convection. This approach has proven robust in detecting and quantifying structures and predicting reversals. Additionally, it provides a perspective on the physical mechanisms underlying flow reversals through cluster evolution. This purely data-driven methodology shows the potential to enhance our understanding, prediction and control of turbulent flows and complex systems.