he emergence of vascular networks is a long-standing problem which has been the subject of intense research in the past decades. One of the main reasons being the widespread applications that it has in tissue regeneration, wound healing, cancer treatment, etc. The mechanisms involved in the formation of vascular networks are complex and despite the vast amount of research devoted to it, there are still…
Polytopes (also known in dimensions zero through three as "points", "line segments", “polygons", and “polyhedra") have been objects of interest to mathematicians throughout the recorded history of mathematics. Most notably, the five Platonic solids were probably known at least a thousand years before Plato. Regular polytopes are "as symmetric as possible" in a sense that…
Let be a free associative algebra over generated by for in some indexing set and let be a poset. For and we define an action of on (the complex vector space with basis ) in a way such that either annihilates or sends it to where covers and we extend multiplicatively and linearly. Let be the two-sided ideal which annihilates all elements of . We characterize when is Young's Lattice and we discuss the…
Let Q be an orientation of a type A Dynkin diagram. An eta map corresponding to Q is a surjection from the weak order on permutations to a Cambrian lattice (of triangulations of a polygon). We give a new geometric way to construct the Auslander-Reiten quiver of the quiver representations rep(Q). We use it to naturally define…
The Penrose triangle, also known as the impossible tribar is an icon for cohomology. It is literally the icon for Cech cohomology on Wikipedia. The idea goes back to a paper by Roger Penrose in 1992, but was first reported by Penrose several years earlier. There, he shows how the impossibility of the figure depends…
Much of the literature on the design and analysis of supersaturated designs (SSDs), in which the number of factors exceeds the number of runs, rests on design principles assuming a least-squares analysis. More recently, researchers have discovered the potential of analyzing SSDs with penalized regression methods like the LASSO and Dantzig selector estimators. There exists much theoretical work for these methods…
Machine learning has become widely popular in fields like computer vision, natural language processing, and speech recognition, often performing tasks better than humans. A fundamental building block of many of these algorithms is a neural network known as a multilayer perceptron. In this tutorial we will discuss how to construct these networks and how to train them using back propagation…
Register now for a weekend anniversary celebration marking 130 years of mathematics education and research here at NC State. The history of mathematics at the university dates to 1889, when math courses were among the first taught as NC State ushered in its inaugural class of students. Harrelson Hall was the former home of the…
We present a general formula describing the joint distribution of two permutation statistics—the peak number and the descent number—over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formula involves a certain kind of plethystic substitution on quasisymmetric generating functions. We apply this result to cyclic permutations, involutions, and derangements, and…
We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational…
When running any iterative algorithm it is useful to know when to stop. Here we review LSQR and LSLQ, two iterative methods for solving \min_x \|Ax-b\|_2 based on the Golub-Kahan bidiagonalization process, as well as estimates for the 2-norm error \|x-x_*\|_2, where x_* is the minimum norm solution. We also review the closely related Craig's…
The ill-posedness and nonlinearity are two factors causing the phenomenon of multiple local minima and ravines of conventional least squares cost functionals for Coefficient Inverse Problems. Since any minimization method can stop at any point of a local minimum, then the problem of numerical solution of any Coefficient Inverse Problems becomes inherently unstable and so…
Mike Thompson, the Managing Director and CEO of First Analytics, will be giving a seminar on how machine learning and mathematics are used to handle large data analytics programs at First Analytics.
Chaos refers to seemingly random and unpredictable dynamics of a system that evolves in time. Certain chaotic systems exhibit an additional level of complexity: intermittent extreme events that are noticeably distinct from the usual chaotic dynamics. These extreme events include ocean rogue waves, extreme weather patterns, and epileptic seizure. I will discuss several examples of these…