arXiv subjects

Bjorn Sandstede

Publications and source records attributed to Bjorn Sandstede.

At least 19 recordsLinked to original sources

arXiv2025

Quantitative metrics for trait and identity distributions

Understanding the role of demographic diversity in group settings requires effective quantitative metrics. Intersectional feminist theory has highlighted that demographic identities can intersect in complex ways, but most metrics used to study these traits are one-dimensional. In their paper "Diversity, identity, and data" (2025), Topaz et al. introduced two novel metrics that capture multiple aspects of demographic identities among group members: "intersecting diversity" and "shared identity". We present a mathematical framework to provide probabilistic interpretations for both metrics. Using these interpretations, we prove that these two measures are anti-correlated and establish tight bounds on their possible combined values, demonstrating that there is no clear "optimal" point that maximizes both metrics. We apply these metrics in three case studies on Hollywood movies, the television show "Survivor", and a random sample of North American companies in which we explore their bounds and anti-correlation as well as their relationship to group performance in these settings. By formalizing the mathematical structure for these metrics and demonstrating their empirical relevance, we provide a foundation for researchers across the social sciences, mathematics, and related fields to more precisely quantify distributions of intersecting traits within groups and better understand their implications for group dynamics and performance.

math.GM↗

arXiv2025

Homoclinic snaking of contact defects in reaction-diffusion equations

We apply spatial dynamical-systems techniques to prove that certain spatiotemporal patterns in reversible reaction-diffusion equations undergo snaking bifurcations. That is, in a narrow region of parameter space, countably many branches of patterned states coexist that connect at towers of saddle-node bifurcations. Our patterns of interest are contact defects, which are 1-dimensional time-periodic patterns with a spatially oscillating core region that at large distances from the origin in space resemble pure temporally oscillatory states and arise as natural analogues of spiral and target waves in one spatial dimension. We show that these solutions lie on snaking branches that have a more complex structure than has been seen in other contexts. In particular, we predict the existence of families of asymmetric travelling defect solutions with arbitrary background phase offsets, in addition to symmetric standing target and spiral patterns. We prove the presence of these additional patterns by reconciling results in classic ODE studies with results from the spatial-dynamics study of patterns in PDEs and using geometrical information contained in the stable and unstable manifolds of the background wavetrains and their natural equivariance structure.

math.DS↗

arXiv2025

Branches of localized patterned states

Motivated by theoretical analyses of spatially localized structures with arbitrarily long periodic plateaus, we provide a framework of assumptions that simplifies their analysis and leads to a topological criterion for when localized patterned structures lie on a discrete stack of loops or on a single unbounded branch. The framework proposed here also connects closely with continuation algorithms that are often used to verify the hypotheses that guarantee the emergence of localized patterned states.

math.DS↗

arXiv2025

Localized synchronous patterns in weakly coupled bistable oscillators

Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with nonlinearities of Lambda-Omega type and establish the existence of localized synchrony patterns in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Depending on the coupling, localized synchrony patterns lie on a discrete stack of isola branches or on a single connected snaking branch.

math.DS↗

arXiv2024

Most probable escape paths in perturbed gradient systems

Stochastic systems are used to model a variety of phenomena in which noise plays an essential role. In these models, one potential goal is to determine if noise can induce transitions between states, and if so, to calculate the most probable escape path from an attractor. In the small noise limit, the Freidlin-Wentzell theory of large deviations provides a variational framework to calculate these paths. This work focuses on using large deviation theory to calculate such paths for stochastic gradient systems with non-gradient perturbations. While for gradient systems the most probable escape paths consist of time-reversed heteroclinic orbits, for general systems it can be a challenging calculation. By applying Melnikov theory to the resulting Euler-Lagrange equations recast in Hamiltonian form, we determine a condition for when the optimal escape path is the heteroclinic orbit for the perturbed system. We provide a numerical example to illustrate how the computed most probable escape path compares with the theoretical result.

math.DS↗

arXiv2024

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners

The stability of nonlinear waves on spatially extended domains is commonly probed by computing the spectrum of the linearization of the underlying PDE about the wave profile. It is known that convective transport, whether driven by the nonlinear pattern itself or an underlying fluid flow, can cause exponential growth of the resolvent of the linearization as a function of the domain length. In particular, sparse eigenvalue algorithms may result in inaccurate and spurious spectra in the convective regime. In this work, we focus on spiral waves, which arise in many natural processes and which exhibit convective transport. We prove that exponential weights can serve as effective, inexpensive preconditioners that result in resolvents that are uniformly bounded in the domain size and that stabilize numerical spectral computations. We also show that the optimal exponential rates can be computed reliably from a simpler asymptotic problem posed in one space dimension.

math.NA↗

arXiv2024

Parameter identifiability in PDE models of fluorescence recovery after photobleaching

Identifying unique parameters for mathematical models describing biological data can be challenging and often impossible. Parameter identifiability for partial differential equations models in cell biology is especially difficult given that many established \textit{in vivo} measurements of protein dynamics average out the spatial dimensions. Here, we are motivated by recent experiments on the binding dynamics of the RNA-binding protein PTBP3 in RNP granules of frog oocytes based on fluorescence recovery after photobleaching (FRAP) measurements. FRAP is a widely-used experimental technique for probing protein dynamics in living cells, and is often modeled using simple reaction-diffusion models of the protein dynamics. We show that current methods of structural and practical parameter identifiability provide limited insights into identifiability of kinetic parameters for these PDE models and spatially-averaged FRAP data. We thus propose a pipeline for assessing parameter identifiability and for learning parameter combinations based on re-parametrization and profile likelihoods analysis. We show that this method is able to recover parameter combinations for synthetic FRAP datasets and investigate its application to real experimental data.

q-bio.QM↗

arXiv2023

Truncation of contact defects in reaction-diffusion systems

Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with an embedded defect in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.

math.DS↗

arXiv2022

Quantifying different modeling frameworks using topological data analysis: a case study with zebrafish patterns

Mathematical models come in many forms across biological applications. In the case of complex, spatial dynamics and pattern formation, stochastic models also face two main challenges: pattern data is largely qualitative, and model realizations may vary significantly. Together these issues make it difficult to relate models and empirical data -- or even models and models -- limiting how different approaches can be combined to offer new insights into biology. These challenges also raise mathematical questions about how models are related, since alternative approaches to the same problem -- e.g., cellular Potts models; off-lattice, agent-based models; on-lattice, cellular automaton models; and continuum approaches -- treat uncertainty and implement cell behavior in different ways. To help open the door to future work on questions like these, here we adapt methods from topological data analysis and computational geometry to quantitatively relate two different models of the same biological process in a fair, comparable way. To center our work and illustrate concrete challenges, we focus on the example of zebrafish-skin pattern formation, and we relate patterns that arise from agent-based and cellular automaton models.

q-bio.CB↗

arXiv2022

The Drift of #MyBodyMyChoice Discourse on Twitter

#MyBodyMyChoice is a well-known hashtag originally created to advocate for women's rights, often used in discourse about abortion and bodily autonomy. The Covid-19 outbreak prompted governments to take containment measures such as vaccination campaigns and mask mandates. Population groups opposed to such measures started to use the slogan "My Body My Choice" to claim their bodily autonomy. In this paper, we investigate whether the discourse around the hashtag #MyBodyMyChoice on Twitter changed its usage after the Covid-19 outbreak. We observe that the conversation around the hashtag changed in two ways. First, semantically, the hashtag #MyBodyMyChoice drifted towards conversations around Covid-19, especially in messages opposed to containment measures. Second, while before the pandemic users used to share content produced by experts and authorities, after Covid-19 the users' attention has shifted towards individuals.

cs.CY↗

arXiv2022

Spatially Localized Structures in Lattice Dynamical Systems

We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the profile is close to the second bistable state outside this plateau. We show that the existence branches of such patterns generically form either an infinite stack of closed loops (isolas) or intertwined s-shaped curves (snaking). We then use bifurcation theory near the anti-continuum limit, where the coupling between edges in the lattice vanishes, to prove existence of isolas and snaking in a bistable discrete real Ginzburg--Landau equation. We also provide numerical evidence for the existence of snaking diagrams for planar localized patches on square and hexagonal lattices and outline a strategy to analyse them rigorously.

math.DS↗

arXiv2022

Localized patterns in planar bistable weakly coupled lattice systems

Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov--Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold.

math.DS↗

arXiv2022

Localized radial roll patterns in higher space dimensions

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves ("isolas"), or the length increases to infinity so that branches are unbounded in function space ("snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyse the structure of branches of localized radial roll solutions in dimension 1+$\varepsilon$, with $0<\varepsilon\ll1$, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.

math.DS↗

arXiv2021

Enabling equation-free modeling via diffusion maps

Equation-free modeling aims at extracting low-dimensional macroscopic dynamics from complex high-dimensional systems that govern the evolution of microscopic states. This algorithm relies on lifting and restriction operators that map macroscopic states to microscopic states and vice versa. Combined with simulations of the microscopic state, this algorithm can be used to apply Newton solvers to the implicitly defined low-dimensional macroscopic system or solve it more efficiently using direct numerical simulations. The key challenge is the construction of the lifting and restrictions operators that usually require a priori insight into the underlying application. In this paper, we design an application-independent algorithm that uses diffusion maps to construct these operators from simulation data. Code is available at https://doi.org/10.5281/zenodo.5793299.

math.DS↗

arXiv2021

A Novel Corpus of Discourse Structure in Humans and Computers

We present a novel corpus of 445 human- and computer-generated documents, comprising about 27,000 clauses, annotated for semantic clause types and coherence relations that allow for nuanced comparison of artificial and natural discourse modes. The corpus covers both formal and informal discourse, and contains documents generated using fine-tuned GPT-2 (Zellers et al., 2019) and GPT-3(Brown et al., 2020). We showcase the usefulness of this corpus for detailed discourse analysis of text generation by providing preliminary evidence that less numerous, shorter and more often incoherent clause relations are associated with lower perceived quality of computer-generated narratives and arguments.

cs.CL↗

arXiv2021

Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix

Spiral waves emerge in numerous pattern forming systems and are commonly modeled with reaction-diffusion systems. Some systems used to model biological processes, such as ion-channel models, fall under the reaction-diffusion category and often have one or more non-diffusing species which results in a rank-deficient diffusion matrix. Previous theoretical research focused on spiral spectra for strictly positive diffusion matrices. In this paper, we use a general two-variable reaction-diffusion system to compare the essential and absolute spectra of spiral waves for strictly positive and rank-deficient diffusion matrices. We show that the essential spectrum is not continuous in the limit of vanishing diffusion in one component. Moreover, we predict locations for the absolute spectrum in the case of a non-diffusing slow variable. Predictions are confirmed numerically for the Barkley and Karma models.

math.DS↗

arXiv2021

Snaking bifurcations of localized patterns on ring lattices

We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, whilst all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.

math.DS↗

arXiv2019

Modeling stripe formation on growing zebrafish tailfins

As zebrafish develop, black and gold stripes form across their skin due to the interactions of brightly colored pigment cells. These characteristic patterns emerge on the growing fish body, as well as on the anal and caudal fins. While wild-type stripes form parallel to a horizontal marker on the body, patterns on the tailfin gradually extend distally outward. Interestingly, several mutations lead to altered body patterns without affecting fin stripes. Through an exploratory modeling approach, our goal is to help better understand these differences between body and fin patterns. By adapting a prior agent-based model of cell interactions on the fish body, we present an in silico study of stripe development on tailfins. Our main result is a demonstration that two cell types can produce stripes on the caudal fin. We highlight several ways that bone rays, growth, and the body-fin interface may be involved in patterning, and we raise questions for future work related to pattern robustness.

q-bio.CB↗