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Curving Towards Bézout: An Examination Of Plane Curves And Their Intersection, Camron Alexander Robey Cohen
Curving Towards Bézout: An Examination Of Plane Curves And Their Intersection, Camron Alexander Robey Cohen
Honors Papers
One area of interest in studying plane curves is intersection. Namely, given two plane curves, we are interested in understanding how they intersect. In this paper, we will build the machinery necessary to describe this intersection. Our discussion will include developing algebraic tools, describing how two curves intersect at a given point, and accounting for points at infinity by way of projective space. With all these tools, we will prove Bézout’s theorem, a robust description of the intersection between two curves relating the degrees of the defining polynomials to the number of points in the intersection.
Chaos And Learning In Discrete-Time Neural Networks, Jess M. Banks
Chaos And Learning In Discrete-Time Neural Networks, Jess M. Banks
Honors Papers
We study a family of discrete-time recurrent neural network models in which the synaptic connectivity changes slowly with respect to the neuronal dynamics. The fast (neuronal) dynamics of these models display a wealth of behaviors ranging from simple convergence and oscillation to chaos, and the addition of slow (synaptic) dynamics which mimic the biological mechanisms of learning and memory induces complex multiscale dynamics which render rigorous analysis quite difficult. Nevertheless, we prove a general result on the interplay of these two dynamical timescales, demarcating a regime of parameter space within which a gradual dampening of chaotic neuronal behavior is induced …