A diagrammatic view of differential equations in physics

Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.Comment: 69 page

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of C-sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All of the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.Comment: In Proceedings ACT 2021, arXiv:2211.0110

Novelty seeking is linked to openness and extraversion, and can lead to greater creative performance

Item does not contain fulltextObjective: Novelty seeking (the tendency to explore things novel and unfamiliar) has been extensively researched in the clinical and health domains, but its effects on creative performance are largely unknown. We examined whether creativity-related personality traits (openness to experience and extraversion) are associated with novelty seeking, and whether novelty seeking is linked to, and facilitates, creativity. Method: In Study 1a (Nn=n230; Magen=n20; 64% females) and Study 1b (Nn=n421; Magen=n19; 65% females), we measured extraversion, openness to experience, novelty seeking, and divergent thinking. To provide causal evidence for the relation between novelty seeking and creativity, in Study 2 (Nn=n147; Magen=n27; 75% females), we manipulated people's motivation to seek novelty and then measured subsequent divergent thinking. Results: In Studies 1a and 1b, we demonstrated that trait novelty seeking is associated with openness and extraversion, on the one hand, and divergent thinking on the other. In Study 2, the novelty seeking manipulation led to greater divergent thinking. Conclusions: We conclude that novelty seeking is linked to openness to experience and extraversion, and that it can lead to greater divergent thinking.15 p

Predicting Virality on Networks Using Local Graphlet Frequency Distribution

The task of predicting virality has far-reaching consequences, from the world of advertising to more recent attempts to reduce the spread of fake news. Previous work has shown that graphlet distribution is an effective feature for predicting virality. Here, we investigate the use of aggregated edge-centric local graphlets around source nodes as features for virality prediction. These prediction features are used to predict expected virality for both a time-independent Hawkes model and an independent cascade model of virality. In the Hawkes model, we use linear regression to predict the number of Hawkes events and node ranking, while in the independent cascade model we use logistic regression to predict whether a k-size cascade will multiply by a factor X in size. Our study indicates that local graphlet frequency distribution can effectively capture the variances of the viral processes simulated by Hawkes process and independent-cascade process. Furthermore, we identify a group of local graphlets which might be significant in the viral processes. We compare the effectiveness of our methods with eigenvector centrality-based node choice

Advancing Transportation Equity: Research and Practice

Yingling Fan is the corresponding author: [email protected] contributes to many societal outcomes, including employment, health, and wealth. However, disparities and inequities in transportation systems, services, and decision-making processes disproportionately impact underserved and underrepresented communities. This study seeks to create a better understanding of current research and practice and recommend future research and practice that can advance transportation equity in Minnesota. To that end, the research team conducted a literature review that summarizes recent developments in the field of transportation equity, reviewed existing equity-focused programs within and beyond the transportation sector, and engaged multiple stakeholder groups, including a project advisory group with experts in addressing disparities and inequities, a group of transportation users and equity stakeholders, and community members. The study presents a working definition of transportation equity, recommends action steps for MnDOT and its partners to consider in advancing transportation equity, and identifies directions for future research and practice that can advance transportation equity in the state of Minnesota

Curvature Invariants for the Alcubierre and Nat\'ario Warp Drives

A process for using curvature invariants is applied to evaluate the metrics for the Alcubierre and the Natario warp drives at a constant velocity.Curvature invariants are independent of coordinate bases, so plotting these invariants will be free of coordinate mapping distortions. As a consequence, they provide a novel perspective into complex spacetimes such as warp drives. Warp drives are the theoretical solutions to Einstein's field equations that allow the possibility for faster-than-light (FTL) travel. While their mathematics is well established, the visualisation of such spacetimes is unexplored. This paper uses the methods of computing and plotting the warp drive curvature invariants to reveal these spacetimes. The warp drive parameters of velocity, skin depth and radius are varied individually and then plotted to see each parameter's unique effect on the surrounding curvature. For each warp drive, this research shows a safe harbor and how the shape function forms the warp bubble. The curvature plots for the constant velocity Natario warp drive do not contain a wake or a constant curvature indicating that these are unique features of the accelerating Natario warp drive.Comment: 41 Pages, 15 figure