The evolution of urban areas plays a major role in crafting public policy and spurring investment. We propose a discrete model to study the existence and dynamics of spatial wealth concentration, centered on a two-dimensional lattice with wealth and an inherent amenities interacting via a feedback process. Various parameter regimes beget distinct dynamics, including that of spatial wealth concentration—as consistent with empirical observation. To enable a more rigorous analysis we derive a continuum model of partial differential equations (PDEs) from the discrete model and analyze the instability regime of the continuum system; the resultant regime agrees with observations of simulation of the discrete model. We also perform a sensitivity analysis on the continuum model to determine how small changes in parameters affect solutions.

This is a case study of teaching 3D design and 3D printing in a project-based computing course for undergraduate math majors. This article discusses content organization, implementation, project grading, and includes a personal reflection. There is an emphasis on lessons learned and how to encourage student creativity and artistic expression. An appendix details 3D design techniques in Mathematica.

We show existence, uniqueness, and directedness properties for infinite geodesics in the FPP model. After giving the fundamental definitions, we describe results by Newman and collaborators giving existence and uniqueness of directed geodesics under an unproven curvature assumption. We then give two proofs of the existence of at least two infinite geodesics under no unproven assumptions. In the final two sections, we give proofs of directedness statements for infinite geodesics using more recent methods which give information even under no unproven assumptions and prove a generalized uniqueness statement for infinite geodesics.

The circular law theorem states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. In these expository notes, we present a proof in a Gaussian case, due to Mehta and Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. We also discuss some related models and open problems.

This article surveys the Euler calculus - an integral calculus based on Euler characteristic - and its applications to data, sensing, networks, and imaging.

In a previous paper, we showed that many important quantum information-theoretic phenomena, including the no-cloning and no-broadcasting theorems, are in fact generic in all non-classical probabilistic theories. An exception is teleportation, which most such theories do not support. In this paper, we investigate which probabilistic theories, and more particularly, which composite systems, {\em do} support a teleportation protocol. We isolate a natural class of composite systems that we term {\em regular}, and establish necessary and sufficient conditions for a regular tripartite system to support a conclusive, or post-selected, teleportation protocol. We also establish a sufficient condition for deterministic teleportation that yields a large supply of theories, neither classical nor quantum, that support such a protocol.