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We introduce gradient flow aggregation, a random growth model. Given existing particles $\{x_1,\ldots,x_n\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at $\infty$ and flows in direction of the vector field $\nabla E$ where $ E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}$, $0 < \alpha < \infty$. The case $\alpha = 0$ refers to the logarithmic energy ${-}\sum\log\|x-x_i\|$. Particles stop once they are at distance 1 from one of the existing particles, at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten’s method, can be used to deduce sub-ballistic growth for $0 \leq \alpha < 1$, $\text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}$. This is optimal when $\alpha = 0$. The case $\alpha = 0$ leads to a ‘round’ full-dimensional tree. The larger the value of $\alpha$, the sparser the tree. Some instances of the higher-dimensional setting are also discussed.

Summary

The prologue sets the stage for the book that follows. It describes where the book will go and how it will get there. It defines “enablers as followers who allow or even encourage their leaders to engage in, and then to persist in, behaviors that are destructive.” The prologue explores the all-important concept of followership, provides a brief history of relations between leaders and followers, and introduces the concept of the “leadership system,” that is, the equal importance of 1) leaders; 2) followers; 3) contexts. The concluding section of the prologue introduces distinctions among President Donald Trump’s enablers by dividing them into Trump’s Tribe and Trump’s Team. It similarly introduces the context of the book, which is the coronavirus crisis in the United States, especially during the first six months, January through June 2020.

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Random growth via gradient flow aggregation

Prologue – Enabler

Summary

Vision for Coalition Health Service Support, 2020 Time Frame

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3 results

Random growth via gradient flow aggregation

Prologue – Enabler

Summary

Vision for Coalition Health Service Support, 2020 Time Frame