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Abstract. A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes.We consider the class P of 3-dimensional combinatorial simple polytopes P , different from a tetrahedron, whose facets do not form 3-and 4-belts. This class includes mathematical fullerenes, i. e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a right-angled realisation in Lobachevsky 3-space, which is unique up to isometry.Our families of smooth manifolds are associated with polytopes from the class P. The first family consists of 3-dimensional small covers of polytopes from P, or hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes from P. Our main result is that both families are cohomologically rigid, i. e. two manifolds M and M from either of the families are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that if M and M are diffeomorphic, then their corresponding polytopes P and P are combinatorially equivalent. These results are intertwined with the classical subjects of geometry and topology, such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.

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Sericinus montela, a globally threatened butterfly species, feeds exclusively on Aristolochia contorta (Northern pipevine). Field surveys and glasshouse experiments were conducted to obtain a better understanding of the relationship between the two species. Interviews with the persons concerned with A. contorta were conducted to collect information about the site management measures. We found that management practices to control invasive species and manage the riverine areas might reduce the coverage of A. contorta and the number of eggs and larvae of S. montela. Our results indicated that the degraded quality of A. contorta may result in a decrease in S. montela populations by diminishing their food source and spawning sites. This study implies that ecological management in the riverine area should be set up to protect rare species and biodiversity.

Purpose: To quantify and predict the magnitude of multi‐leaf collimator (MLC) positional errors in volumetric modulated arc therapy (VMAT) plans using statistical learning techniques to allow more accurate representation of the dose distribution expected to be delivered. Methods: A total of 74 VMAT plans used for patient treatments from three separate institutions were acquired. All plans were delivered using a Varian Millennium 120 MLC. The plans were split into training (N=3), validation (N=6) and testing (N=65) sets. From these, numerical features such as individual leaf position and velocity, and categorical features such as whether the leaf was moving towards or away from the isocenter, the bank the leaf was a part of, and the control point (CP) at which the error occurred were extracted. The differences between planned and delivered leaf positions in the training data were used as a target response for the development of a linear regression model, a decision tree model, and a random forest model. Optimized model parameters were found using cross‐validation on the validation set. Performance of each model in predicting the positional errors was assessed using mean absolute error (MAE) and root mean square error (RMSE) on the held‐out test set. Results: The MAE between planned and delivered positions for moving MLCs was 1.27 mm (RMSE = 1.60 mm). The decision tree model had the best performance, the predictions of which had MAE for moving MLCs of 0.27 mm (RMSE = 0.39 mm). Leaf velocity significantly predicted position errors, (β = 0.128, p<0.0001), and explained a significant amount of the variance (r2=0.90, p<0.0001). Conclusion: The decision tree model accurately predicted actual MLC leaf positions during delivery. Incorporating predicted errors into the planned MLC positions leads to a more realistic representation of the leaf locations which can be expected during treatment delivery to the patient.