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  1. Significance

    Understanding how development and evolution shape functional morphology is a basic question in biology. A paradigm of this is the finch’s beak that has adapted to different diets and behaviors over millions of years. We take a mathematical and physical perspective to quantify the nature of beak shape variations, how they emerge from changes to the development program of the birds, and their functional significance as a mechanical tool.

     
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  2. Free, publicly-accessible full text available September 1, 2024
  3. Recent advances in high-resolution imaging techniques and particle-based simulation methods have enabled the precise microscopic characterization of collective dynamics in various biological and engineered active matter systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. By contrast, learning macroscopic hydrodynamic equations for active matter directly from experiments or particle simulations remains a major challenge, especially when continuum models are not known a priori or analytic coarse graining fails, as often is the case for nondilute and heterogeneous systems. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through a range of applications, from a chiral active particle model mimicking nonidentical swimming cells to recent microroller experiments and schooling fish. In all these cases, our scheme learns hydrodynamic equations that reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data.

     
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  4. The persistence of imperfect mimicry in nature presents a challenge to mimicry theory. Some hypotheses for the existence of imperfect mimicry make differing predictions depending on how mimetic fidelity is measured. Here, we measure mimetic fidelity in a brood parasite–host system using both trait-based and response-based measures of mimetic fidelity. Cuckoo finchesAnomalospiza imberbislay imperfectly mimetic eggs that lack the fine scribbling characteristic of eggs of the tawny-flanked priniaPrinia subflava, a common host species. A trait-based discriminant analysis based on Minkowski functionals—that use geometric and topological morphometric methods related to egg pattern shape and coverage—reflects this consistent difference between host and parasite eggs. These methods could be applied to quantify other phenotypes including stripes and waved patterns. Furthermore, by painting scribbles onto cuckoo finch eggs and testing their rate of rejection compared to control eggs (i.e. a response-based approach to quantify mimetic fidelity), we show that prinias do not discriminate between eggs based on the absence of scribbles. Overall, our results support relaxed selection on cuckoo finches to mimic scribbles, since prinias do not respond differently to eggs with and without scribbles, despite the existence of this consistent trait difference.

     
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  5. Geometric graph models of systems as diverse as proteins, DNA assemblies, architected materials and robot swarms are useful abstract representations of these objects that also unify ways to study their properties and control them in space and time. While much work has been done in the context of characterizing the behaviour of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, underconstrained networks that are far more common in nature and technology. Here, we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.

     
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  6. Image registration has been widely studied over the past several decades, with numerous applications in science, engineering and medicine. Most of the conventional mathematical models for large deformation image registration rely on prescribed landmarks, which usually require tedious manual labeling. In recent years, there has been a surge of interest in the use of machine learning for image registration. In this paper, we develop a novel method for large deformation image registration by a fusion of quasiconformal theory and convolutional neural network (CNN). More specifically, we propose a quasiconformal energy model with a novel fidelity term that incorporates the features extracted using a pre-trained CNN, thereby allowing us to obtain meaningful registration results without any guidance of prescribed landmarks. Moreover, unlike many prior image registration methods, the bijectivity of our method is guaranteed by quasiconformal theory. Experimental results are presented to demonstrate the effectiveness of the proposed method. More broadly, our work sheds light on how rigorous mathematical theories and practical machine learning approaches can be integrated for developing computational methods with improved performance.

     
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