We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or geometric structure, including distance-regular graphs and graphs arising from finite geometries, among others. Our proofs use techniques from spectral graph theory, linear optimization, finite geometry, and probability, yielding new machinery for analysing edge-expansion phenomena in highly structured graphs.
Accurate production forecasts are essential for the integration of renewable energy sources into the power grid. This paper illustrates how to obtain probabilistic forecasts of wind power generation using gradient boosting trees and an ensemble of weather forecasts. To this end, we perform a comparative analysis across three state-of-the-art probabilistic prediction methods-conformalized quantile regression, natural gradient boosting and conditional diffusion models-all of which can be combined with tree-based machine learning. The methods are validated using four years of data for all Belgian offshore wind farms. We benchmark the models against the power curve and a calibrated wake model as well as a probabilistic method using stochastic variational Gaussian process regression. The tree-based models significantly reduce the mean absolute error in comparison to the deterministic baselines. Additionally, all three methods outperform the Gaussian process baseline in probabilistic skill, while two out of the three also improve point forecast accuracy. The conditional diffusion model attains the best performance, with improvements of 5% in mean absolute error and 12% in continuous rank probability score compared to the probabilistic baseline. Last, the results indicate an average improvement in point forecast accuracy of 17% by using an ensemble of weather forecasts instead of a single provider.
Kirkland and Šmigoc constructed a family of stochastic matrices realising the Type III boundary polynomials of the Karpelevič region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range 0<α≤1, for the reduced Ito polynomialsfα(x)=xy(xq−(1−α))d−αd,n=qd+y, associated with Type III Farey-neighbour arcs of order n. For 0<α<1, the boundary theorem of Dmitriev and Dynkin restricts the possible nonzero entries of a realising matrix. After relabelling the vertices, its directed graph consists of the edges i→i+n1 and a selection of edges i→i+n(n+1−q), with indices taken modulo n. Coates' coefficient formula and a weighted form of Turán's theorem then show that the directed cycles of length q fall into d classes. Any two cycles in the same class share a vertex, while cycles in different classes have no vertices in common. Each class has the same sum of cycle weights, where the weight of a cycle is the product of its edge weights. The positions of these cycles along the n-cycle then yield the product condition required by Kirkland and Šmigoc. The case α=1 is treated separately, and an explanation is provided for the failure when α=0.
Background Armed conflict may disrupt immunization services and increase barriers to childhood vaccination. This study examined determinants of full basic childhood vaccination in Mali and assessed whether they differed between conflict-affected and non-conflict regions. Methods We conducted a secondary analysis of the 2023–2024 Mali Demographic and Health Survey, including 2859 children aged 12–23 months. Full basic vaccination was defined as receipt of Bacille Calmette–Guérin vaccine, three doses of pentavalent vaccine, three doses of oral polio vaccine (excluding the birth dose), and measles vaccine. Survey-weighted logistic regression models were fitted nationally and separately for conflict-affected and non-conflict regions. Interactions between conflict status and selected explanatory variables were tested. Results Full basic vaccination coverage was 51.87%. Coverage did not differ significantly between conflict-affected and non-conflict regions (50.08% vs 52.64%; p = 0.467), and conflict status was not independently associated with vaccination after adjustment. Antenatal care attendance and facility-based delivery were consistently associated with higher odds of full vaccination. In conflict-affected regions, media exposure and shorter perceived distance to health facilities were also associated with higher vaccination uptake. A significant interaction was observed between conflict status and distance to health facility (p = 0.01). Conclusions Childhood vaccination coverage in Mali remains suboptimal. Although overall coverage was similar across conflict and non-conflict regions, determinants differed by context, highlighting the need for tailored strategies to improve vaccine uptake.
The microscopic mechanisms by which Ti substitution modifies the structural, electronic and magnetic properties of M-type barium hexaferrite remain incompletely understood. In this work, first-principles density functional theory (DFT) calculations combined with structural and magnetic measurements were employed to establish the relation between Ti site occupancy, electronic structure, exchange interactions and magnetic behavior in BaFe12-xTixO19. The formation-energy calculations reveal that Ti4+ preferentially occupies the 12k sites at low concentrations and progressively extends to the 2b sites with increasing Ti content. Occupation matrix analysis indicates that charge compensation occurs through localized Fe3+-> Fe2+ reduction exclusively at the 12k sublattice. This electronic redistribution modifies the Fe-O hybridization, producing a half-metallic electronic structure for 0.5 <= x <= 1.5, selectively weakens the dominant Fe-O-Fe superexchange interactions, and progressively suppresses the magnetocrystalline anisotropy. The X-ray diffraction and magnetic measurements performed on co-precipitated BaFe12-xTixO19 nanopowders confirm the predicted increase in the cell parameters, the systematic reduction of the saturation magnetization, magnetocrystalline anisotropy constant, anisotropy field, remanent magnetization and coercive field with increasing Ti content. The excellent agreement between theory and experiment establishes a unified microscopic mechanism linking Ti site occupancy, charge compensation, electronic structure, exchange interactions and magnetic softening in M-type barium hexaferrite.
