TY - JOUR AB - Animals exhibit a variety of behavioural defences against socially transmitted parasites. These defences evolved to increase host fitness by avoiding, resisting or tolerating infection. Because they can occur in both infected individuals and their uninfected social partners, these defences often have important consequences for the social group. Here, we discuss the evolution and ecology of anti-parasite behavioural defences across a taxonomically wide social spectrum, considering colonial groups, stable groups, transitional groups and solitary animals. We discuss avoidance, resistance and tolerance behaviours across these social group structures, identifying how social complexity, group composition and interdependent social relationships may contribute to the expression and evolution of behavioural strategies. Finally, we outline avenues for further investigation such as approaches to quantify group-level responses, and the connection of the physiological and behavioural response to parasites in different social contexts. AU - Stockmaier, Sebastian AU - Ulrich, Yuko AU - Albery, Gregory F. AU - Cremer, Sylvia AU - Lopes, Patricia C. ID - 12765 IS - 4 JF - Functional Ecology SN - 0269-8463 TI - Behavioural defences against parasites across host social structures VL - 37 ER - TY - JOUR AB - The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines. AU - Ivanov, Grigory AU - Köse, Seyda ID - 12680 IS - 6 JF - Discrete Mathematics SN - 0012-365X TI - Erdős-Ko-Rado and Hilton-Milner theorems for two-forms VL - 346 ER - TY - JOUR AB - In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12792 JF - Communications in Mathematical Physics SN - 0010-3616 TI - On the spectral form factor for random matrices VL - 401 ER - TY - JOUR AB - Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. AU - Corbet, René AU - Kerber, Michael AU - Lesnick, Michael AU - Osang, Georg F ID - 12709 JF - Discrete and Computational Geometry SN - 0179-5376 TI - Computing the multicover bifiltration VL - 70 ER - TY - JOUR AB - Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality. AU - Boissonnat, Jean Daniel AU - Wintraecken, Mathijs ID - 12763 JF - Journal of Applied and Computational Topology SN - 2367-1726 TI - The reach of subsets of manifolds VL - 7 ER -