American Mathematical Society

Adinkras are highly structured graphs developed to study 1-dimensional supersymmetry algebras. A cyclic ordering of the edge colors of an Adinkra, or rainbow, determines a Riemann surface and a height function on the vertices of the Adinkra determines a divisor on this surface. We study the induced map from height functions to divisors on the Jacobian of the Riemann surface. In the first nontrivial case, a 5-dimensional hypercube corresponding to a Jacobian given by a product of 5 elliptic curves each with j -invariant 2048, we develop and characterize a purely combinatorial algorithm to compute height function images. We show that when restricted to a single elliptic curve, every height function is a multiple of a specified generating divisor, and raising and lowering vertices corresponds to adding or subtracting this generator. We also give strict bounds on the coefficients of this generator that appear in the collection of all divisors of height functions.

In this paper I present a study of the Collatz problem and its conjecture using graph theory and the properties of the digital root of natural numbers.

A hallmark of science is the open exchange of knowledge. At this time of crisis, it is more important than ever for scientists around the world to openly share their knowledge, expertise, tools, and technology. Scientific models are critical tools for anticipating, predicting, and responding to complex biological, social, and environmental crises, including pandemics. They are essential for guiding regional and national governments in designing health, social, and economic policies to manage the spread of disease and lessen its impacts. However, presenting modeling results alone is not enough. Scientists must also openly share their model code so that the results can be replicated and evaluated.

Model predictive control (MPC) is of increasing interest in applications for constrained control of multivariate systems. However, one of the major obstacles to its broader use is the computation time and effort required to solve a possibly nonconvex optimal control problem (OCP) online. This article introduces a sensitivity-based warmstarting strategy for systems with nonlinear dynamics and polyhedral constraints with the goal of reducing the computational footprint of MPC controllers. It predicts changes in the solution of the parameterized OCP as the parameter varies by calculating the semiderivative of the solution mapping. The main novelty of this article is that the polyhedrality of the constraints allows us to avoid imposing any constraint qualification conditions or strict complementarity assumptions. A numerical study featuring MPC applied to unmanned aerial vehicles illustrates the proposed approach.

We consider an optimal control problem involving a nonlinear ODE with control, an integral cost functional, and a control constraint. Our main assumptions include a coercivity condition and the condition that the optimal control is an isolated solution of the variational inequality appearing in the first-order optimality condition. We show that the optimal open-loop control is Lipschitz continuous in time; moreover, we identify the dependence of the Lipschitz constant of the optimal control on the data of the problem. Then, we establish the existence of a Lipschitz continuous optimal feedback control. As an application, we study regularity properties of the optimal value function. A main tool for obtaining these results is the property of uniform strong metric regularity.

We use data from papers posted to the Mathematics section of the arXiv to explore the representation of women in mathematics research. We show that women are under-represented as authors of mathematics papers on the arXiv, even in comparison to the proportion of women who hold full-time positions in mathematics departments. However, some subfields have much greater participation than others.