An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive, a definition introduced in a previous work by Johnson and Paparella (2016). For a given matrix S, the spectracone is a polyhedral cone formed by the vectors that produce nonnegative matrices when multiplied on the left by S and the right by S-inverse. The spectratope is a set defined similarly, with the added condition that the included vectors have an infinity norm of one. This work identifies previously unknown relationships between Perron similarities and their Kronecker products. First, if two matrices are Perron similarities, their Kronecker product is also a Perron similarity. Second, the Kronecker product of spectracones (spectratopes) is a subset of the spectracone (spectratope) of the Kronecker product. In addition, if two matrices are Perron similarities with dimensions greater than 1, then the Kronecker product of the spectracones (spectratopes) is properly contained in the spectracone (spectratope) of the Kronecker product. Another significant result is if S is a Perron similarity, then the ray through the all-ones vector is properly contained in the spectracone of S, but the converse is not true. One reason this work is relevant is that Perron similarities were previously defined to have a necessary and sufficient condition that the ray through the all-ones vector is properly contained in the spectracone. With the publication of this article, the condition was found to only be necessary, and the definition was corrected. Perron similarities are important when examining the nonnegative inverse eigenvalue problem, or NIEP, which is concerned with the spectrum of nonnegative matrices. This research expands on previous work related to the NIEP by examining relationships between Perron similarities and their Kronecker products.

Asthma, a chronic disease of the lungs, affects approximately 262 million people and caused 461,000 deaths worldwide, just in 2019. Asthma can result in frequent emergency department visits, hospitalizations, and premature deaths. To effectively manage their condition, asthma patients need to accurately track their medication usage. But it is a manual task, prone to human error. To improve the lives of asthma patients, we built a smart asthma-inhaler system to automatically record patients' use of medication. Our system is made of a Bluetooth-enabled inhaler, a smartphone app, and a Bluetooth-enabled wearable sensor. The inhaler records when the patients took their medications and sends the data to the smartphone app. The app retrieves weather information including temperature, humidity, and Air quality index from the web. The wearable sensor measures the particulate matter in the air surrounding the patient when the inhaler was used. This system should allow patients to accurately record when and why they used their inhalers. Physicians can then use this information to better diagnose and treat asthma. We hope to marketize this system to mitigate the damages of asthma.