Review: Swimming in Hong Kong

A review of Swimming in Hong Kong (2016), a short story collection by Stephanie Han

Kuroda's formula and arithmetic statistics

Kuroda's formula relates the class number of a multi-quadratic number field $K$ to the class numbers of its quadratic subfields $k_i$. A key component in this formula is the unit group index $Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]$. We study how $Q(K)$ behaves on average in certain natural families of totally real biquadratic fields $K$ parametrized by prime numbers

Pipelining Saturated Accumulation

Aggressive pipelining and spatial parallelism allow integrated circuits (e.g., custom VLSI, ASICs, and FPGAs) to achieve high throughput on many Digital Signal Processing applications. However, cyclic data dependencies in the computation can limit parallelism and reduce the efficiency and speed of an implementation. Saturated accumulation is an important example where such a cycle limits the throughput of signal processing applications. We show how to reformulate saturated addition as an associative operation so that we can use a parallel-prefix calculation to perform saturated accumulation at any data rate supported by the device. This allows us, for example, to design a 16-bit saturated accumulator which can operate at 280 MHz on a Xilinx Spartan-3(XC3S-5000-4) FPGA, the maximum frequency supported by the component's DCM

The 33-isogeny Selmer groups of the elliptic curves y2=x3+n2y^2=x^3+n^2

Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\phi$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hat{\phi}:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_{\phi}(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hat{\phi}}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.Comment: 23 page

Integral points on the congruent number curve

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal{E}_D(\mathbb{Q})$ in $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6)^{\omega(abcd)}$ integer solutions.Comment: 23 page

Coping with Acculturative Stress: MDMA Usage among Asian American Young Adults in the Electronic Dance Music Scene

The intersection of Asian American identity and illicit substance use is greatly understudied in psychological literature, especially with matters of mental health and drug use being stigmatized by Asian cultural norms. However, with an increasingly alarming number of fatal drug overdoses by Asian Americans at electronic dance music (EDM) events, attention must be drawn to the needs of this unique population. The present study characterizes this community by drawing from data of 1,290 Asian American young adults who participate in the EDM scene. This study also hypothesizes the impact of acculturative stress and feelings of social belonging on MDMA usage patterns. Analysis reveals a population of largely East and Southeast Asian, 2nd generation, college-educated young adults with strikingly high usage rates of MDMA, an illicit drug linked to the EDM scene. Multiple regression models were created that could predict MDMA use through various measures related to acculturative stress and social belonging. Findings revealed the significant impact of acculturation, acculturative stress, mental health, peer relationships, and desires for social belonging on this population’s MDMA usage patterns, providing an important platform from which future research may launch much-needed additional studies of Asian American young adults and illicit drug use