|The Overlays of Riordan Arrays
An array (formally, a function 𝐴: ℤ2 → 𝔽) can be thought of as a matrix that extends infinitely in all directions. We can discuss operators 𝑋 and 𝑌 which shift an array right by one column or up by one row, respectively. A polynomial in 𝑋 and 𝑌 that annihilates 𝐴 is called a “template” whereas an “overlay” is an array with finite support that contains the coefficients of a template. Our research looks at new ways in which machinery such as templates and overlays can be applied to well-studied arrays, specifically focusing on Riordan arrays.
Riordan arrays are lower triangular arrays that extend infinitely only to the right and downwards. They were originally motivated by a generalization of Pascal’s triangle and can be described in one of two ways: Either column-wise by their generating functions or row-wise by their associated patterned sequence. This pattern sequence often corresponds to a natural representation as an overlay. Our research focuses on exploring that representation and the requirements for such a representation to exist.
Furkan CakmakPolitics, Philosophy, and Public Affairs
|When Losing Matters: Emotions, Media Choice and Affective Polarization
Affective political polarization has become a popular topic of study in American politics in recent years. With hostility among partisan groups increasing, many scholars are researching its causes. One of the possible causes is argued to be media. However, reasons behind why media influence some viewers and not others are unclear. Thus, the mechanism between media choice and affective polarization is not well understood. My research aims to tackle this challenge by incorporating emotions into the mechanism between media choice and affective polarization. My central research question is “Is there a relationship between emotional state and media choice and how does the ability to choose one’s media influence affective polarization?”. My argument is that emotions will be the defining factor in terms of whether partisan viewers will expose themselves to media to be polarized. Their willingness to polarize will increase as they experience more negative emotions.
To test my claims, I employ a mixed-method approach. I first analyze panel data to investigate voters' media habits before and after significant political events. I also look into relationship between level of affective polarization and news media viewership. Next, I carry out a survey experiment to expose respondents to different political events with varying newsworthiness. This allows me to analyze how partisans react to political events emotionally and how they utilize media to regulate their emotional reactions. Finally, I interview partisans to discover discrete emotions common in politics and reasons behind choosing partisan outlets or avoiding news media completely.
My initial survey experiment showed that when exposed to an event with winners and losers, partisans on the losing side will be more likely to experience negative emotions and avoid news media. Instead, they choose entertainment media to improve their moods. As a result, emotions are shown to be a defining factor on media habits of partisans. This research has the potential to better understand why polarization is on the rise in our society today. More importantly, it can provide real-world solutions on how to decrease hostility among partisans and increase deliberation in politics.
|Biomineralization for Effective Remediation of Radionuclides in the Environment
Nuclear power is an efficient source of energy but comes with many negative stigma due to the mishandling of radioactive materials and dispersion during natural disasters in the past. Reducing current environmental contamination and preventing future accumulation is a major concern moving forward with nuclear energy. A few species of bacteria that are commonly found in groundwater ecosystems have been shown to form insoluble minerals in water contaminated with heavy metals, including actinides. Integrating natural biomineralization processes of microorganisms into the framework for actinide remediation requires thorough investigation of the mineral product and environmental behavior. In this work a model system of Shewanella putrefaciens will be used to cultivate a uranium phosphate mineral. Recovered product will then be analyzed through thermodynamics, solubility, and soil mobility experiments to determine the reactivity, longevity, and transport within environmental conditions.
|A rational approach to create proto-replicators and self-replicating polymers
The synthetic effort to create self-replicating polymer machines that would allow for complete selectivity in chain size and monomeric pattern. Tectons are the molecules which self-assemble through intermolecular hydrogen bonding. The synthetic effort focuses on generating two types of tectons hydrogen Donor-Acceptor-Donor (DAD) and a hydrogen Acceptor-Donor-Acceptor (ADA),which will bind to each other through a specific hydrogen bonding network. Hypothetically, these tectons are either natively fluorescent or will become fluorescent on pairing. Careful consideration of the physical properties of each tecton will be assayed using single molecule experiments.
Tectons are made up of DAD/ADA heads attached to group 14 dye. The DAD head is a pyridinederivative and ADA head is a uracil derivative and are synthesized in house. After completion of the synthesis, these tectons will be subjected to the single molecule experiments like Fluorescence Cross Correlation Spectroscopy (FCCS), Fluorescence Lifetime Correlation Spectroscopy (FLCS), and Total Internal Reflection Fluorescence (TIRF) to find various thermodynamic properties, on instruments which are designed by Brozik group. Computational investigations to further understand the nature of molecular orbitals of different readers and how is the electrons are being excited.
Through this study, we were able to synthesis the building blocks of tectons. Using single-molecule analysis, thermodynamic parameters can be determined, and from this, we can conclude that these are indeed natively fluorescent after interacting through hydrogen bonding. This is the foundational work that will lead to future studies of proto-replicator and self-replicating polymer machines under different environmental conditions.
|Studying the responses of polymeric materials containing light-activated shape-changing molecules
Smart shape-changing materials are the materials that are able to change their shape upon exposure to stimuli such as heat, stress, and light and relax back to their original state when the stimulus is removed. Disperse Red 1 (DR1) dye is a molecule that changes its shape in response to light and adding this molecule to the polymer can make a smart shape-changing material that has a broad range of applications in areas such as optical devices, robotics, aerospace, medical, etc.
This research seeks to understand the origin of the response in the shape-changing materials made with poly (methyl methacrylate) (PMMA) polymer doped with DR1(DR1-doped PMMA). The main responses in these materials are due to heating the polymer and shape changing of DR1 molecules. The importance of the research is to understand the contribution of each response and to figure out which response is dominant in the light-activated shape-changing polymers.
The approach to answering this question is to couple the experiments and theories. The temperature-dependent and light-dependent experiments are conducted to record the responses in PMMA fiber doped with DR1. Temperature-dependent experiments isolate thermal processes while light-dependent experiments separate contributions from thermal and molecular shape changes. Developing a theory as a function of multiple measured parameters can solve the puzzle. The results of the experiments and the developed theories determined that the dominant response inDR1-doped PMMA fiber is due to heating and the effect of light-induced molecular shape change of DR1 is negligible.
|Third-degree Analogs to Continued Fractions
Continued fractions have been studied for many years, as they can be used as representations of real numbers with many useful applications. Of particular interest is when patterns occur in these representations and their ability to provide amazingly accurate approximations of real numbers. We can encode these continued fractions as 2×2 matrices. Generalizing this algorithm to larger matrices helps us construct analogs of continued fractions in higher dimensions. Our goal is to look at analogs of this algorithm when applied to 3×3 and 4×4 companion matrices to analyze when periodic patterns arise and how these analogs relate to other mathematical constructs, such as Riordan arrays.
Saptarshi Rajan SarkarPhysics
|Turbulence: Through the lens of Quantum Fluids
Some of the most interesting phenomena in the universe are observed in dynamical systems with flows. The natural cycles, formation of waves, tracks of pathogens in air are some of the most relevant turbulent systems around us. Although we can exactly simulate the air molecules to an extremely high degree of precision, we know that even the best weather models can only forecast up to a couple weeks in advance. Simulating very large number of microscopic particles can be difficult in itself, while the emergent turbulence at large scales makes it rather intractable. Turbulent systems, however, possess characteristic rotating structures called eddies. Once we can count the number of eddies in a turbulent system, we can develop a thermodynamic model. We can then use statistical mechanics to average out over properties of the system to get an effective macroscopic description of the system, which can be both experimentally measured and numerically modeled.