This research investigates generalized pseudo-inverses in matrix algebras, focusing on how inverse conditions enable matrix decompositions. We study inner inverses, outer inverses, with reflexive inverses satisfying both, and relate these to von Neumann regular elements and regular matrices. Stronger notions such as unit regularity and strong regularity are analyzed via the existence of inverses involving units, commuting conditions, and idempotent “spectral” factors. We also examine Drazin inverses, and Moore-Penrose inverses in the natural *involution context. Our project highlights when these properties force structured decompositions of matrices into sums/products of units, idempotents, and nilpotents, supported by examples and counterexamples.
This project investigates patterns in the decimal expansions of fractions of the form 𝑛/(3^𝑥), with particular focus on n/81. The central research question asks: which digits fail to appear in these repeating decimal representations, and can their absence be predicted using modular arithmetic? While repeating decimals are a familiar concept, the structural constraints governing their digit composition are less commonly examined.To explore this question, I analyzed decimal expansions in base 10 through the lens of modular arithmetic, examining how powers of 3 interact with powers of 10. By studying residue classes and cyclic behavior, I identified patterns that restrict which digits can occur in specific expansions. Preliminary results show that the structure of the multiplicative group modulo 3^𝑥 imposes predictable limitations on digit appearance. These findings provide a systematic method for forecasting digit absence in fractions with denominator 3^x, revealing deeper connections between modular arithmetic and decimal representation.
Knowing what types of enzymes a molecule will interact with can aid drug development by minimizing side effects due to unwanted interactions. In this project, we built and interpreted models for classifying enzyme substrates, aiming to address an information gap in this understanding: the distinguishing properties of the substrates of each major enzyme class. We utilized the machine learning technique XGBoost in Python to build a predictive model for each enzyme class using molecular data as well as top linear combinations of the data obtained using Principal Components Analysis. We will discuss the algorithm we developed to automatically tune the parameters of XGBoost to optimize the model. We will also present examples of how to interpret these models using graphs to visualize the impact of variables in each model and identifying common factors in the top contributing variables of significant principal components to characterize each enzyme class. For example, we found that the probability of a molecule interacting with oxidoreductase enzymes is positively associated with the number of nonpolar regions. A particular descriptor is NumHeteroAtoms, the number of non-carbon atoms in the molecule, which was negatively associated with the probability of interacting with oxidoreductases.
Graph-based codes allow us to visualize error-correcting codes and construct systems of low-complexity decoding. However, certain roadblocks- called stopping sets- can prevent complete error correction. This raises a question: how can we design encoding strategies that avoid such roadblocks? We investigate a setting where we must look at partitions of variable nodes with the goal of avoiding stopping sets in at least one part. Specifically, we examine an example with six variable nodes in a Tanner graph and its corresponding 4X6 parity-check matrix. We present a proof for the partial error correction for two out of three parts in the partition. Looking forward, we aim to determine the probability of encountering stopping sets in a topological lifting of the graph.
A C-complex is a union of Seifert surfaces for the components of a link which intersect each other in clasps. The clasp number of a link is the minimal number of clasps amongst all C-complexes bounded. It gives a measure of complexity and can be used to provide bounds on other useful characteristics of a link. This paper provides a new lower bound for the number of clasps of all C-complexes bounded by a given 3-component link improving results of Amundsen-Anderson-D.-Guyer. Furthermore, we construct links that achieve these bounds. In order to do so, we express the triple linking numbers as the area bounded by three curves, called word curves, and then perform the geometry and discrete optimization needed to minimize the length of these curves.
