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Abstract We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schrödinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents. 

Abstract The Arctic marginal ice zone (MIZ) is the transitional region between dense pack ice and open ocean. As an increasingly important component of the polar marine environment, recent investigations have focused on changes in MIZ size and location as the climate has warmed. Fractal geometry offers a universal measure of complexity, shape, and self-similarity across scales, and a powerful tool for characterizing MIZ evolution. Here we analyze the fractal dimension of the Arctic MIZ boundary and find a pronounced seasonal cycle that is repeated almost exactly each year, with a sharp maximum in late summer. The long-term trend is slight, with a decrease of less than 2% over the satellite era, while MIZ width has increased over the same period by almost 40%. Our results have important implications for climatic and ecological processes which depend critically on MIZ geometry. We demonstrate thermodynamic feedbacks through statistical analysis and provide context for future applications. 

Abstract Autonomous recording units (ARUs) are recognized for their use in detecting vocalizing bird species to assess presence, occupancy, and density, but their potential to monitor reproductive status of individuals and reproductive rates is not well known. We investigated whether song rates derived from ARU data, when combined with the known date, can be used to predict the proportion of male songbirds in 3 breeding status classes (single, paired, and feeding young). We monitored breeding status with weekly field visits and collected daily ARU recordings at 46 olive‐sided flycatcher (Contopus cooperi) breeding territories in northwestern Canada in 2016–2017. We tested 4 variations of a hierarchical multinomial regression model that used time of day, day of year, and song rate derived from 2‐minute recordings to predict breeding status, and evaluated models using a novel, likelihood‐based approach. We found the top model correctly estimated 79% of the observed proportions of birds in each breeding status across the length of the breeding season. Although date was the primary predictor of breeding status, singing rate reduced some of the uncertainty and provided more accurate estimates for a given time. A major challenge to prediction accuracy and data interpretation was accounting for bird movement and the associated impact on detection, which we partly addressed by limiting our study to individuals who were detected on at least 30% of ARU sampling days. We demonstrate that ARUs can be used to assess breeding status in a cryptic, low‐density species at risk such as the olive‐sided flycatcher, suggesting this method could be applied to a wider range of species to better understand demographics and population dynamics, and inform management decisions, for bird species of concern. 

Abstract We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal homogenization theory in Lipschitz domains of Kenig et al. A key idea, to deal with the hard constraint on the volume, is a combination of a large scale almost dilation invariance with a selection principle argument. 

Abstract Ultrasound‐directed self‐assembly (DSA) utilizes the acoustic radiation force associated with a standing ultrasound wave field to organize particles dispersed in a fluid medium into specific patterns. State‐of‐the‐art ultrasound DSA methods use single‐frequency ultrasound wave fields, which only allow organizing particles into simple, periodic patterns, or require a large number of ultrasound transducers to assemble complex patterns. In contrast, this work introduces multi‐frequency ultrasound wave fields to organize particles into complex patterns. A method is theoretically derived to determine the operating parameters (frequency, amplitude, phase) of any arrangement of ultrasound transducers, required to assemble spherical particles dispersed in a fluid medium into specific patterns, and experimentally validated for a system with two frequencies. The results show that multi‐frequency compared to single‐frequency ultrasound DSA enables the assembly of complex patterns of particles with substantially fewer ultrasound transducers. Additionally, the method does not incur a penalty in terms of accuracy, and it does not require custom hardware for each different pattern, thus offering reconfigurability, which contrasts, e.g., acoustic holography. Multi‐frequency ultrasound DSA can spur progress in a myriad of engineering applications, including the manufacturing of multi‐functional polymer matrix composite materials that derive their structural, electric, acoustic, or thermal properties from the spatial organization of particles in the matrix. 

Abstract Ultrasound‐directed self‐assembly (DSA) uses ultrasound waves to organize and orient particles dispersed in a fluid medium into specific patterns. Combining ultrasound DSA with vat photopolymerization (VP) enables manufacturing materials layer‐by‐layer, wherein each layer the organization and orientation of particles in the photopolymer is controlled, which enables tailoring the properties of the resulting composite materials. However, the particle packing density changes with time and location as particles organize into specific patterns. Hence, relating the ultrasound DSA process parameters to the transient local particle packing density is important to tailor the properties of the composite material, and to determine the maximum speed of the layer‐by‐layer VP process. This paper theoretically derives and experimentally validates a 3D ultrasound DSA model and evaluates the local particle packing density at locations where particles assemble as a function of time and ultrasound DSA process parameters. The particle packing density increases with increasing particle volume fraction, decreasing particle size, and decreasing fluid medium viscosity is determined. Increasing the particle size and decreasing the fluid medium viscosity decreases the time to reach steady‐state. This work contributes to using ultrasound DSA and VP as a materials manufacturing process. 

Abstract Classical multidimensional scaling is a widely used dimension reduction technique. Yet few theoretical results characterizing its statistical performance exist. This paper provides a theoretical framework for analyzing the quality of embedded samples produced by classical multidimensional scaling. This lays a foundation for various downstream statistical analyses, and we focus on clustering noisy data. Our results provide scaling conditions on the signal-to-noise ratio under which classical multidimensional scaling followed by a distance-based clustering algorithm can recover the cluster labels of all samples. Simulation studies confirm these scaling conditions are sharp. Applications to the cancer gene-expression data, the single-cell RNA sequencing data and the natural language data lend strong support to the methodology and theory. 

Abstract There is often considerable uncertainty in parameters in ecological models. This uncertainty can be incorporated into models by treating parameters as random variables with distributions, rather than fixed quantities. Recent advances in uncertainty quantification methods, such as polynomial chaos approaches, allow for the analysis of models with random parameters. We introduce these methods with a motivating case study of sea ice algal blooms in heterogeneous environments. We compare Monte Carlo methods with polynomial chaos techniques to help understand the dynamics of an algal bloom model with random parameters. Modelling key parameters in the algal bloom model as random variables changes the timing, intensity and overall productivity of the modelled bloom. The computational efficiency of polynomial chaos methods provides a promising avenue for the broader inclusion of parametric uncertainty in ecological models, leading to improved model predictions and synthesis between models and data. 

Abstract Archetypal analysis (AA) is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of AA in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we introduce two preprocessing techniques to reduce the dimension and representation cardinality of the data, respectively. We prove that provided data are approximately embedded in a low-dimensional linear subspace and the convex hull of the corresponding representations is well approximated by a polytope with a few vertices, our method can effectively reduce the scaling of AA. Moreover, the solution of the reduced problem is near-optimal in terms of prediction errors. Our approach can be combined with other acceleration techniques to further mitigate the intrinsic complexity of AA. We demonstrate the usefulness of our results by applying our method to summarize several moderately large-scale datasets.