Nigel Boston
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Heuristics for 2-class Towers of Cyclic Cubic Fields Open
We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-…
The Distribution of the Number of Real Solutions to the Power Flow Equations Open
In this paper we study the distributions of the number of real solutions to the power flow equations over varying electrical parameters. We introduce a new monodromy and parameter homotopy continuation method for quickly finding all soluti…
Exploiting symmetry in the power flow equations using monodromy Open
We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in…
Heuristics for p-class towers of real quadratic fields Open
Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian gene…
The weight distribution of quasi-quadratic residue codes Open
We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a stron…
The $2$-Class Tower of $\mathbb{Q}(\sqrt{-5460})$ Open
The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case…
Exploiting Algebraic Structure in Global Optimization and the Belgian Chocolate Problem Open
The Belgian chocolate problem involves maximizing a parameter δ over a non-convex region of polynomials. In this paper we detail a global optimization method for this problem that outperforms previous such methods by exploiting underlying …
Exploiting Algebraic Structure in Global Optimization and the Belgian\n Chocolate Problem Open
The Belgian chocolate problem involves maximizing a parameter {\\delta} over a\nnon-convex region of polynomials. In this paper we detail a global optimization\nmethod for this problem that outperforms previous such methods by exploiting\n…
The Weight Distribution of Quasi-quadratic Residue Codes Open
In this paper, we begin by reviewing some of the known properties of QQR codes and proved that $PSL_2(p)$ acts on the extended QQR code when $p \equiv 3 \pmod 4$. Using this discovery, we then showed their weight polynomials satisfy a stro…
A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion Open
Low-rank matrix completion (LRMC) problems arise in a wide variety of\napplications. Previous theory mainly provides conditions for completion under\nmissing-at-random samplings. This paper studies deterministic conditions for\ncompletion.…